Analytic Continuation of Liouville Theory
SSU-ITP-11/42
Analytic Continuation of Liouville Theory
Daniel Harlow, a Jonathan Maltz, a Edward Witten a,ba
Stanford Institute for Theoretical Physics, Department of Physics, Stanford University,Stanford CA 94305. b School of Natural Sciences, Institute for Advanced Study, Princeton NJ 08540.
E-mail: [email protected] , [email protected] , [email protected] Abstract:
Correlation functions in Liouville theory are meromorphic functions of theLiouville momenta, as is shown explicitly by the DOZZ formula for the three-pointfunction on S . In a certain physical region, where a real classical solution exists, thesemiclassical limit of the DOZZ formula is known to agree with what one would expectfrom the action of the classical solution. In this paper, we ask what happens outsideof this physical region. Perhaps surprisingly we find that, while in some range of theLiouville momenta the semiclassical limit is associated to complex saddle points, ingeneral Liouville’s equations do not have enough complex-valued solutions to accountfor the semiclassical behavior. For a full picture, we either must include “solutions”of Liouville’s equations in which the Liouville field is multivalued (as well as beingcomplex-valued), or else we can reformulate Liouville theory as a Chern-Simons theoryin three dimensions, in which the requisite solutions exist in a more conventional sense.We also study the case of “timelike” Liouville theory, where we show that a proposalof Al. B. Zamolodchikov for the exact three-point function on S can be computed bythe original Liouville path integral evaluated on a new integration cycle. a r X i v : . [ h e p - t h ] O c t ontents B.1 Hypergeometric Series 90B.2 Hypergeometric Differential Equation 91B.3 Riemann’s Differential Equation 91B.4 Particular Solutions of Riemann’s Equation 92B.5 Integral Representations of Hypergeometric Functions 96
C Gamma Functions and Stokes Phenomena 99
C.1 Generalities 99C.2 Analysis Of The Gamma Function 102C.3 The Inverse Gamma Function 105
D Semiclassical Conformal Blocks 108E An Integral Expression for log(Gamma(z)) 113 – 1 –
Integral over the SL(2,C) Moduli of the Saddle Point for Three LightOperators 114
Quantum Liouville theory has been studied extensively since it was first introducedby Polyakov several decades ago in the context of non-critical string theory [1]. Sincethen it has been invoked as a model for higher-dimensional Euclidean gravity, as a non-compact conformal field theory, and as a dilaton background in string theory. Amongmore recent developments, Liouville theory has been found [2] to have a connection tofour-dimensional gauge theories with extended supersymmetry and has emerged as animportant component of speculative holographic duals of de Sitter space and the multi-verse [3–5]. In many of these applications the Liouville objects of interest are evaluatedat complex values of their parameters. The goal of this paper is to understand to whatextent these analytically continued objects are computed by appropriately continuedversions of the Liouville path integral.Liouville theory has been studied from many points of view, but the essential pointfor studying the question of analytic continuation is that several nontrivial quantitiesare, remarkably, exactly computable. A basic case is the expectation value on a two-sphere S of a product of three primary fields of Liouville momentum α i , i = 1 , , (cid:42) (cid:89) i =1 e α i φ ( z i ) (cid:43) . (1.1)For this correlation function, there is an exact formula – known as the DOZZ formula[6, 7]. The existence of such exact formulas makes it possible to probe questions thatmight otherwise be out of reach. We will exploit this in studying the semiclassical limitof Liouville theory in the present paper.Liouville theory is conveniently parametrized in terms of a coupling constant b ,with the central charge being c = 1 + 6 Q , where Q = b + b − . For a semiclassicallimit, we take b →
0, giving two interesting choices for the Liouville momenta. Wecan consider “heavy” operators, α i = η i /b , with η i fixed as b →
0. The insertion ofa heavy operator changes the saddle points which dominate the functional integral.Alternatively, we can consider “light” operators, α i = bσ i , where now we keep σ i fixedas b →
0. Light operators do not affect a saddle point; they just give us functions thathave to be evaluated at a particular saddle point. We will consider both cases in thepresent paper. – 2 – real saddle point in the Liouville path integral is simply a real solution of theclassical equations of motion − ∂ a ∂ a φ + Q (cid:101) R + 8 πbµe bφ = 0 . (1.2)Such a solution is a critical point of the classical action S . Path integrals are mostsimple if they are dominated by a real saddle point. For the Liouville correlationfunction of three heavy fields on S , there is a real saddle point that dominates thesemiclassical limit of the path integral if and only if the η i are real, less than 1 /
2, andobey (cid:80) i η i >
1. These inequalities, which define what we will call the physical region,were described in [8] and the explicit solution was described and its action computedin [7]. The action evaluated at the classical solution is of the form S = G ( η , η , η ) /b where the function G can be found in (4.26). In [7], it was shown that, in the physicalregion, the weak coupling limit of the three-point function of heavy fields is (cid:42) (cid:89) i =1 exp(2 η i φ/b )( z i ) (cid:43) ∼ exp( − S ) = exp (cid:0) − G ( η , η , η ) /b + O (1) (cid:1) , (1.3)as one would expect. What happens when we leave the physical region? There is no problem in continuingthe left hand side of (1.3) beyond the physical region. Indeed, the DOZZ formula showsthat the left hand side of the three-point function (1.1) is, for fixed b , a meromorphicfunction of the variables α i , and in particular one can analytically continue the η i toarbitrary complex values. Similarly, the DOZZ formula is analytic in b for b > G extends to a multivalued function of complex variables η i . This multivaluedness takes a very simple form. There are branch points at integervalues of the η i or of simple sums and differences such as η + η − η . The monodromyaround these branch points takes the form G → G + 2 πi (cid:32) n + (cid:88) i =1 m i η i (cid:33) , (1.4)where n and the m i are integers and the m i are either all even or all odd.There actually is one specific region of complex η i – the case that Re η i = 1 / η i >
0, so that the external states are normalizable states in the sense of [8] – in When heavy operators are present, they add delta function terms in (1.2) and also make contri-butions to the action S . In this introduction, we will omit such details. – 3 –hich a semiclassical interpretation of the DOZZ formula is available [9, 10] in termsof real singular solutions of Liouville’s equations that have a natural interpretationin terms of hyperbolic geometry. The action of these singular solutions is given by aparticular branch of the multivalued function G , for the values of the η i in question.(The solutions themselves are given by an analytic continuation of those constructed in[7] and thus are a special case of the solutions we discuss later.) In the present paper,we aim to interpret the DOZZ formula semiclassically for arbitrary complex η i .Our investigation started with the question of how to interpret the multivaluedness(1.4). The most obvious interpretation is that the branches of G might correspond tothe actions of complex solutions of the Liouville equation. Outside the physical region,the correlation function of a product of heavy fields does not have a real saddle point,but one might hope that it would have one or more complex saddle points.The most obvious notion of a complex saddle point is simply a complex-valuedsolution of the classical Liouville equation (1.2). Such a solution is a critical point ofthe Liouville action S , interpreted now as a holomorphic function of a complex Liouvillefield φ . Optimistically, one would think that Liouville theory for the case of three heavyoperators on S has complex-valued solutions parametrized by the integers n and m i that appeared in eqn. (1.4). Then one would hope that for any given values of the η i ,the path integral could be expressed as a sum of contributions from the complex saddlepoints. Which saddle points must be included (and with what coefficients) would ingeneral depend on the η i , as Stokes phenomena may intrude.To appreciate the analytic continuation of path integrals, one needs to know thatto a given saddle point one can associate, in principle, much more than a perturbativeexpansion. The basic machinery of complex saddle points and Stokes phenomena saysthe following. Let S be the set of complex saddle points; these are also known ascritical points of the complexified action. To each ρ ∈ S , one associates an integrationcycle C ρ in the complexified path integral. Roughly speaking, C ρ is defined by steepestdescent, starting at the critical point ρ and descending by gradient flow with respectto the “Morse function” h = − Re S . The C ρ are well-defined for generic values of theparameters; in our case, the parameters are b and the η i . The definition of C ρ fails iftwo critical points ρ and (cid:101) ρ have the same value of Im S and unequal values of Re S .In this case, the difference S ρ − S (cid:101) ρ is either positive or negative (we write S ρ for thevalue of S at ρ and similarly h ρ for the value of h ); if for example it is positive, then See for example section 2 of [11] for an elementary explanation, much more detailed and precisethan we can offer here. Some familiarity with this material is necessary for a full appreciation of thepresent paper. An integration “cycle” is simply the multi-dimensional analog of an integration “contour.” Forsimplicity, we assume that the critical points are isolated and nondegenerate. – 4 –he jumping of integration cycles takes the form C ρ → C ρ + n C (cid:101) ρ , (1.5)for some integer n .Any integration cycle C on which the path integral converges can always be ex-pressed in terms of the C ρ : C = (cid:88) ρ ∈S a ρ C ρ , (1.6)with some coefficients a ρ . In particular – assuming that the machinery of critical pointsand Stokes walls applies to Liouville theory, which is the hypothesis that we set outto test in the present paper – the integration cycle for the Liouville path integral musthave such an expansion. The subtlety is that the coefficients in this expansion are noteasy to understand, since one expects them to jump in crossing Stokes walls. However,there is one place where the expansion (1.6) takes a simple form. In the physical region,one expects Liouville theory to be defined by an integral over the ordinary space of real φ fields. On the other hand, in the physical region, there is a unique critical point ρ corresponding to a real solution. Starting at a real value of φ and performing gradientflow with respect to h , φ remains real. (When φ is real, h is just the ordinary realLiouville action.) So C ρ is just the space of real φ fields as long as the η i are in thephysical region. In the physical region, the expansion (1.6) collapses therefore to C = C ρ . (1.7)In principle – if the machinery we are describing does apply to Liouville theory – theexpansion (1.6) can be understood for any values of the η i and b by starting in thephysical region and then varying the parameters at will, taking Stokes phenomena intoaccount.If one knows the coefficients in the expansion (1.6) for some given values of theparameters, then to determine the small b asymptotics of the path integral Z = (cid:90) C Dφ exp( − S ) (1.8)is straightforward. One has Z = (cid:80) ρ a ρ Z ρ , with Z ρ = (cid:82) C ρ Dφ exp( − S ). On the otherhand, the cycle C ρ was defined so that along this cycle, h = − Re S is maximal at thecritical point ρ . So for small b , Z ρ ∼ exp( − S ρ ) . (1.9)The asymptotic behavior of Z is thus given by the contributions of those critical pointsthat maximize h ρ = Re( − S ρ ), subject to the condition that a ρ (cid:54) = 0.– 5 –t this point, we can actually understand more explicitly why (1.7) must hold inthe physical region. A look back to (1.4) shows that as long as b and the η i are real,all critical points have the same value of Re S . So all critical points with a ρ (cid:54) = 0 areequally important for small b in the physical region. Thus, the computation of [7]showing that in the physical region the Liouville three-point function is dominated bythe contribution of the real critical point also shows that in this region, all other criticalpoints have a ρ = 0. The Gamma function gives a practice case for some of these ideas. (For a previousanalysis along similar lines to what we will explain, see [12]. For previous mathematicalwork, see [13, 14].) The most familiar integral representation of the Gamma functionis Γ( z ) = (cid:90) ∞ dt t z − exp( − t ) , Re z > . (1.10)A change of variables t = e φ converts this toΓ( z ) = (cid:90) ∞−∞ dφ exp( − S ) , (1.11)where the “action” is S = − zφ + e φ . (1.12)This integral is sometimes called the minisuperspace approximation [8, 15, 16] to Li-ouville theory, as it is the result of a truncation of the Liouville path integral to theconstant mode of φ (and a rescaling of φ to replace e bφ by e φ ).If z is real and positive, the action S has a unique real critical point at φ = log z ,and this is actually the absolute minimum of S (on the real φ axis). We call this criticalpoint ρ . Gradient flow from ρ keeps φ real, so the corresponding integration cycle C ρ is simply the real φ axis. If z is not real but Re z >
0, then C ρ , defined by gradient flowfrom ρ , is not simply the real φ axis, but is equivalent to it modulo Cauchy’s Residuetheorem. The original integral (1.10) or (1.11) can be approximated for z → ∞ in thehalf-space Re z > ρ , at which the value ofthe action is S = − z log z + z . The contribution of this critical point leads to Stirling’sformula Γ( z ) ∼ exp( z log z − z + O (log z )) , Re z > z , defined in the whole complex plane with poles at non-positive integers. This is analogous to the fact that the exact Liouville three-pointfunction (1.1) is a meromorphic function of the α i , even when we vary them to a regionin which the path integral over real φ does not converge. The analytic continuation of– 6 –he Gamma function to negative Re z can be exhibited by deforming the integrationcontour in (1.11) into the complex φ plane as z varies. To understand the behaviorof the integral for Re z ≤
0, it helps to express this integral in terms of contributionsof critical points. The complex critical points of S are easily determined; they are thepoints ρ n with φ n = log z + 2 πin, n ∈ Z . (1.13)For Re z >
0, the integration contour defining the Gamma function is simply C ρ , butfor negative Re z , the integration contour has a more elaborate expansion (cid:80) n ∈T C ρ n ,where T is determined in Appendix C. Once one determines T , the analog of Stirling’sformula for Re z ≤ z varies, but its expression as a sum ofcritical point contours C ρ n jumps in crossing the Stokes walls at Re z = 0. The presentpaper is an attempt to understand to what extent the machinery just sketched actuallyapplies to the full Liouville theory, not just the minisuperspace approximation. The result of our investigation has not been as simple as we originally hoped. Theclassical Liouville equations do not have enough complex critical points to account forthe multivaluedness (1.4), at least not in a straightforward sense.As soon as one allows the Liouville field φ to be complex, one meets the fact thatthe classical Liouville equations are invariant under φ → φ + ikπ/b, k ∈ Z . (1.14)This assertion, which extends what we just described in the minisuperspace approxi-mation, actually accounts for part of the multivaluedness (1.4). The shift (1.14) gives G → G + 2 πik (1 − (cid:80) i η i ).This is all we get by considering complex solutions of the Liouville equations in asimple way. For example, in the physical region, even if we allow φ to be complex, themost general solution of Liouville’s equations is the one described in [7], modulo a shift(1.14). We prove this in section 3 by adapting standard arguments about Liouville’sequations in a simple way.Outside of the physical region, the solutions of the complex Liouville equations areno more numerous. One can try to find some complex solutions by directly generalizingthe formulas of [7] to complex parameters. For a certain (difficult to characterize) rangeof the parameters η i and b , this procedure works and gives, again, the unique solutionof the complex Liouville equations, modulo a transformation (1.14). In other ranges of– 7 –he parameters, the formulas of [7] do not generalize and one can then argue that thecomplex Liouville equations have no solutions at all.The way that the formulas of [7] fail to generalize is instructive. In general, whenone extends these formulas to complex values of η i and b , branch points appear in thesolution and φ is not singlevalued. (The quantities such as exp(2 bφ ) that appear inthe classical Liouville equations remain singlevalued.) Singlevaluedness of φ places aserious constraint on the range of parameters for which a complex critical point exists.We will show that, after taking into account the symmetry (1.14), ordinary, single-valued complex solutions of Liouville’s equations suffice for understanding the semiclas-sical asymptotics of the Liouville two-point function, and also for understanding thesemiclassical asymptotics of the three-point function in a somewhat larger region thanconsidered in [7]. In particular we will see that the old “fixed-area” prescription forcomputing correlators outside the physical region can be replaced by the machinery ofcomplex saddlepoints, which makes the previously-subtle question of locality manifest.But for general values of the η i , there are not enough singlevalued complex solutionsto account for the asymptotics of the three-point function.What then are we to make of the semiclassical limit outside of the region wheresolutions exist? Rather surprisingly, we have found that allowing ourselves to use themultivalued “solutions” just mentioned in the semiclassical expansion enables us toaccount for the asymptotics of the DOZZ formula throughout the full analytic contin-uation in the η i . There is a simple prescription for how to evaluate the action of these“solutions”, and which has them as stationary points. This prescription agrees withthe conventional Liouville action on singlevalued solutions and produces its analyticcontinuation when evaluated on the multivalued “solutions”. In particular once φ ismultivalued, to evaluate the action one must pick a branch of φ at each insertion pointof a heavy vertex operator exp(2 η i φ/b ); allowing all possible choices, one does indeedrecover the multivaluedness expected in (1.4). We do not have a clear rationale for whythis is allowed. For one thing if we do not insist on expanding on cycles attached tocritical points as in (1.6) then it seems clear that for any value of η i we can always findan integration cycle that passes only through single-valued field configurations simplyby arbitrarily deforming the original cycle in the physical region in a manner that pre-serves the convergence as we continue in η i . It is only when we try to deform this cyclein such a way that the semiclassical expansion is transparent that we apparently need This was anticipated by A. B. Zamolodchikov, whom we thank for discussions. As explained in section 5.1 the prescription is essentially to re-express the Liouville field in termsof the “physical metric” g ij = e bφ (cid:101) g ij , which is always single valued. The branch points of φ becomeisolated divergences of the metric that have a very specific form, and which turn out to be integrableif a “principal value” regularization is used. – 8 –o consider these exotic integration cycles attached to multivalued “solutions”.We also attempt to probe the classical solutions that contribute to the three-pointfunction of heavy primary fields by inserting a fourth light primary field. This is notexpected to significantly modify the critical points contributing to the path integral,but should enable one to “measure” or observe those critical points. If the light primaryfield is “degenerate” in the sense of [17], then one can obtain a very concrete formulafor the four-point function, and this formula supports the idea that the three-pointfunction is dominated by the multivalued classical solution. When the light operatoris non-degenerate the situation is more subtle, a naive use of the multivalued solutionsuggests an unusual singularity in the Liouville four-point function which we are ableto prove does not exist. We speculate as to the source of this discrepancy, but we havebeen unable to give a clear picture of how it is resolved. Since it somewhat strains the credulity to believe that the Liouville path integralshould be expanded around a multivalued classical solution, we have also looked foranother interpretation. Virasoro conformal blocks in two dimensions have a relation toChern-Simons theory in three-dimensions with gauge group SL (2 , C ) that was identi-fied long ago [18]. An aspect of this relation is that quantization of Teichmuller space[19], which is an ingredient in SL (2 , C ) Chern-Simons theory, can be used to describeVirasoro conformal blocks [20]. Since Liouville theory can be constructed by gluingtogether Virasoro conformal blocks for left- and right-movers, it should also have anexpression in terms of Chern-Simons theory; one hopes to express Liouville theory ona Riemann surface Σ in terms of Chern-Simons on Σ × I , where I is a unit interval.The boundary conditions required at the ends of I are those of [18]. These bound-ary conditions have recently been reconsidered and the relation between Liouville andChern-Simons theory developed in more detail [21].Given these facts, instead of looking for complex solutions of Liouville theory on Σ,we can look for complex solutions of SL (2 , C ) Chern-Simons theory on Σ × I with theappropriate boundary conditions. Here we find a simpler story than was summarized insection 1.2. Solutions are precisely parametrized by the integers n and m i of eqn. (1.4)and the action depends on those parameters in precisely the expected fashion. So apossible interpretation of the results of the present paper is that if one wishes to applythe machinery of complex saddle points and integration cycles to Liouville theory in aconventional way, one should use the Chern-Simons description. Possibly this reflectsthe fact that the gradient flow equations of complex Chern-Simons theory are elliptic(as analyzed in [11]); this is not so for complexified Liouville theory.– 9 – .4 Timelike Liouville Theory As an application of these ideas, we will consider the case of what we will calltimelike Liouville theory, or Liouville theory with large negative central charge. Thisis the case that b is small and imaginary, so that b <
0. If b is imaginary, then theexponential term exp(2 bφ ) in the Liouville action is of course no longer real. One cancompensate for this by taking φ → iφ , but then the kinetic energy of the Liouville fieldbecomes negative, and the Liouville field becomes timelike in the sense of string theory.From that point of view, ordinary Liouville theory, in which the kinetic energy of φ hasthe usual sign, might be called spacelike Liouville theory. We will use that terminologyoccasionally. Timelike Liouville theory has possible applications in quantum cosmology[3–5], and also as the worldsheet description of closed string tachyon condensation [22].It was shown in [23] that the DOZZ formula, when analytically continued in b ,has a natural boundary of holomorphy on the imaginary axis. On the other hand, itwas also shown that the Ward identities that lead to the DOZZ formula have a secondsolution – which we will call the timelike DOZZ formula – that is well-behaved on theimaginary b axis, but runs into trouble if analytically continued to real b . If b is neitherreal nor imaginary, the two formulas are both well-behaved but different. The timelikeDOZZ formula has also been independently derived as a possible “matter” theory to becoupled to spacelike Liouville in [24] and further studied in [25, 26]. Its first appearanceseems to be as equation (4.5) in [27], where it appeared as an intermediate step in aproposal for the c = 1 limit of Liouville. From the perspective of the present paper, with all fields and parameters poten-tially continued to complex values, timelike Liouville theory and ordinary or spacelikeLiouville theory are the same theory, possibly with different integration cycles. We will In [27] it was argued that the timelike DOZZ formula should be multiplied by various nonanalyticfactors in order for it to describe timelike Liouville. This proposal seems to work only when b = ip/q with p, q ∈ Z , and does not allow continuation to generic α . These modifications are allowed becausefor these special values of b the uniqueness argument for the timelike DOZZ formula breaks down.Some interesting applications of timelike Liouville seem to require generic values of b and α , for examplein coupling to a “matter” CFT with generic c >
25, so we are interested in describing a theory thatworks for generic imaginary b . From the point of view of this paper analyticity in b is also morenatural to consider; since the integrand of the path integral is an analytic function of its parametersthe integral should be analytic as well. We will see below that evaluating the Liouville path integralin the timelike regime does not produce any nonanalytic factors, so they would have to be put in byhand. Schomerus’s justification of the extra factors involves wanting the three point function to reduceto the two point function, which may be an appropriate requirement in a theory that works preciselyat b = ip/q (and may be related to Virasoro minimal models). The timelike DOZZ formula does nothave this property, but we suggest an alternative interpretation in section 7.4 that does not requirethe new factors. – 10 –nvestigate this question and show that the timelike DOZZ formula can indeed comefrom the same path integral that gives the ordinary or spacelike DOZZ formula, withan extra factor that represents a change in the integration cycle.It was shown in [22, 23, 27] that timelike Liouville theory does not at first seemto have all of the usual properties of a conformal field theory; this issue was discussedfurther in [28] but not resolved. The simplest problem in interpreting the timelike DOZZformula in terms of conformal field theory is that naively it appears that the two-pointfunction is not diagonal in the conformal dimensions. Our path integral interpretationof the timelike Liouville correlators sheds some light on this question; we will arguethat the two-point function is indeed diagonal and conjecture that the problems whichhave been identified have to do with the existence of new degenerate fields that donot decouple in the conventional way. This is possible because of the intrinsicallynonunitary nature of timelike Liouville. We have not been able to answer the moresubtle question of which states to factorize correlators on. For a minisuperspace analysisof this problem, see [29]. An outline of this paper is as follows. In section 2, we review Liouville theory. Insection 3, we study complex solutions of Liouville’s equations on the sphere with heavyoperators. In section 4, show that the analytic continuation of the DOZZ formula in arestricted region can be interpreted in terms of complex classical solutions. In section5, we study the full analytic continuation and confront the issue of the nonexistence ofnonsingular solutions. We then use a fourth light primary field to probe the classicalconfigurations contributing to the three-point function of heavy primaries, confirmingour explanation of the DOZZ analytic continuation in terms of singular “solutions”. Insection 6, we reinterpret the question of complex classical solutions in terms of Chern-Simons theory. In section 7, we consider timelike Liouville theory. In section 8 we givea brief summary of our results and suggest directions for future work. Finally, in aseries of appendices, we describe a variety of useful technical results.The length of the paper is partly the result of an attempt to keep it self-contained.We have written out fairly detailed accounts of a variety of results that are known butare relatively hard to extract from the literature. This is especially so in section 2 andin some of the appendices. Casual readers are welcome to skip to the conclusion, whichcontains the highlights in compact form.– 11 –
Review Of Liouville Theory
We begin with an overview of Liouville theory. The goal is to present and motivateall the existing results that we will need in following sections; there are no new resultshere. Some relatively modern reviews on Liouville theory are [30, 31]; a much olderone is [8]. Our conventions are mostly those of [7].
The Liouville action, obtained for example by gauge fixing a generic conformal fieldtheory coupled to two-dimensional gravity [32], is S L = 14 π (cid:90) d ξ (cid:112)(cid:101) g (cid:104) ∂ a φ∂ b φ (cid:101) g ab + Q (cid:101) R φ + 4 πµe bφ (cid:105) . (2.1)Here Q = b + b , and the exponential operator is defined in a renormalization schemeusing (cid:101) g to measure distances. The metric (cid:101) g is referred to as the “reference” metric ( (cid:101) R is its scalar curvature), while the quantity g ab = e Q φ (cid:101) g ab is referred to as the “physical”metric. Since we are viewing Liouville theory as a complete theory in and of itself, the“physical” metric is no more physical than the reference one, but it is extremely usefulfor semiclassical intuition so we will often discuss it in what follows. This theory isinvariant, except for a c -number anomaly, under conformal transformations: z (cid:48) = w ( z ) φ (cid:48) ( z (cid:48) , z (cid:48) ) = φ ( z, z ) − Q (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂z (cid:12)(cid:12)(cid:12)(cid:12) . (2.2)Here we use a complex coordinate z = ξ + iξ , and w ( z ) is any locally holomorphicfunction. Under these transformations the renormalized exponential operators haveconformal weights ∆( e αφ ) = ∆( e αφ ) = α ( Q − α ) , (2.3)as we explain in section 2.2. The stress tensor is T ( z ) = − ( ∂φ ) + Q∂ φ, (2.4)and the central charge of the conformal algebra is c L = 1 + 6 Q = 1 + 6( b + b − ) . (2.5) In the terminology that we adopt, the scaling dimension of an operator is ∆ + ∆, which is twicethe weight for a scalar operator. – 12 –e will study this theory on a two-sphere. It is convenient to take the referencemetric to be the flat metric ds = dz dz , with φ = − Q log r + O (1) as r → ∞ , r = | z | , (2.6)which ensures that the physical metric is a smooth metric on S . This ensures that φ is nonsingular at infinity with respect to (2.2). The intuition for the condition (2.6) isthat there is an operator insertion at infinity representing the curvature of S , whichhas been suppressed in taking the reference metric to be flat.Though the use of a flat reference metric is convenient, with this choice there issome subtlety in computing the action; one must regulate the region of integration andintroduce boundary terms. Following [7], we let D be a disk of radius R , and definethe action as the large R limit of S L = 14 π (cid:90) D d ξ (cid:2) ∂ a φ∂ a φ + 4 πµe bφ (cid:3) + Qπ (cid:73) ∂D φdθ + 2 Q log R. (2.7)The last two terms ensure finiteness of the action and also invariance under (2.2). The semiclassical limit b → φ c = 2 bφ ,in terms of which the action becomes b S L = 116 π (cid:90) d ξ (cid:2) ∂ a φ c ∂ a φ c + 16 πµb e φ c (cid:3) + 12 π (cid:73) ∂D φ c dθ + 2 log R + O ( b ) , (2.10)and the boundary condition becomes φ c ( z, z ) = − zz ) + O (1) as | z | → ∞ . (2.11)The equation of motion following from this action is ∂∂φ c = 2 πµb e φ c . (2.12)If we now define λ ≡ πµb to be fixed as b →
0, then φ c will have a fixed limit for b → Since g ab = e φ c δ ab , the physical metric has a good limit as well. The equation One way to interpret them is to note that if we begin with the original Liouville action (2.1) withround reference metric ds = 4(1 + r ) (cid:0) dr + r dθ (cid:1) , (2.8)then the field redefinition φ → φ − Q log (cid:18)
21 + r (cid:19) (2.9)produces exactly the action (2.7) up to a finite field-independent constant. Rather than trying to keeptrack of this, we will just take the action (2.7) as our starting point. Intuitively this choice of scaling ensures that the radius of curvature λ − / of the physical metricis large in units of the “microscopic” scale µ − . – 13 –f motion is equivalent to the condition of constant negative curvature of g ab , and thisis the source of the classical relationship between Liouville’s equation (2.12) and theuniformization of Riemann surfaces. Because of the unusual nature of the transformation (2.2), we can guess that it willbe exponentials of φ that transform with definite conformal weights. Classically we seethat e αφ (cid:48) ( z (cid:48) ,z (cid:48) ) = (cid:18) ∂w∂z (cid:19) − αQ (cid:18) ∂w∂z (cid:19) − αQ e αφ ( z,z ) , (2.13)so that classically V α ≡ e αφ is a primary conformal operator with conformal weights∆ = ∆ = αQ [17]. α is called the Liouville momentum. Quantum mechanically, theconformal weights of these operators are modified. In free field theory, normal orderingcontributes a well-known additional term − α to each weight. In Liouville theory,the quantum correction is exactly the same, since we can compute the weight of theoperator V α by considering correlations in a state of our choice. We simply considercorrelations in a state in which φ <<
0, thus turning off the Liouville interactionsand reducing the computation of operator scaling to the free field case. So V α hasconformal weight α ( Q − α ), as in (2.3).In this subsection we will discuss the properties of these operators and their correla-tors in more detail at the semiclassical level, in particular seeing how this factor emergesin the formula for ∆. In the following subsections we will review the exact constructionof Liouville theory that confirms this expression for ∆ beyond the semiclassical regime.We will now consider correlation functions of primary fields, (cid:28) V α ( z , z ) · · · V α n ( z n , z n ) (cid:29) ≡ (cid:90) D φ c e − S L n (cid:89) i =1 exp (cid:18) α i φ c ( z i , z i ) b (cid:19) . (2.14)We would like to approximate this path integral using the method of steepest descentfor small b , but to do so we must decide how the α i ’s scale with b . The action (2.10)scales like b − , so for an operator to have a nontrivial effect on the saddle points wemust choose its Liouville momentum α to scale like b − . Thus if we want an operatorto affect the saddle point, we take α = η/b and keep η fixed for b →
0. This giveswhat is conventionally called a “heavy” Liouville primary field. Asymptotically sucha field has ∆ = η (1 − η ) /b for b →
0. One can also define “light” operators with α = bσ , where σ is kept fixed for b →
0. Light operators have fixed dimensions in the We will see below that this argument requires Re α < Q/
2, since otherwise the backreaction ofthe operator will prevent φ << – 14 –emiclassical limit. Insertion of such an operator has no effect on the saddle point φ c ,and to lowest order in b can be approximated by a b -independent factor of e σ i φ c ( z,z ) .Semiclassically the insertion of a heavy operator has the effect of adding an addi-tional delta function term to the action, leading to a new equation of motion: ∂∂φ c = 2 πµb e φ c − π (cid:88) i η i δ ( ξ − ξ i ) (2.15)Let us assume that in the vicinity of one of the operator insertions we may ignore theexponential term. This equation then becomes Poisson’s equation: ∇ φ c = − πη i δ ( ξ − ξ i ) . (2.16)This has the solution φ c ( z, z ) = C − η i log | z − z i | , (2.17)so we find that in a neighborhood of a heavy operator we have φ c ( z, z ) = − η i log | z − z i | + O (1) as z → z i . (2.18)We also find that that the physical metric in this region has the form: ds = 1 r η i ( dr + r dθ ) (2.19)We can insert this solution back into the equation of motion to check whether theexponential is indeed subleading. We find that this is the case if and only ifRe( η i ) < . (2.20)If this inequality is not satisfied, then the interactions affect the behaviour of the fieldarbitarily close to the operator. In [8], this was interpreted as the non-existence oflocal operators with Re( η ) > , and the condition that “good” Liouville operators haveRe( η ) < is referred to as the Seiberg bound. The modern interpretation of this result,as we will see in the following section, is that both α and Q − α correspond to the same quantum operator, with a nontrivial rescaling: V Q − α = R ( α ) V α . (2.21) R ( α ) is referred to as the reflection coefficient, for reasons explained in [8, 16]. Either α or Q − α will always obey the Seiberg bound, and we can always choose that one when Note the the convention that 4 ∂∂ = ∇ . – 15 –tudying the semiclassical limit. We will thus focus only on semiclassical solutions forwhich all operators have Re( η i ) < .We will in general be interested in complex values of η i , so the metric (2.19) willbe complex and thus not admit a simple geometric interpretation. For the next fewparagraphs, however, we will assume that η i is real to enable us to develop some usefulintuition. We first observe that since η i < , we can do a simple change of variables tofind a metric ds = dr (cid:48) + r (cid:48) dθ (cid:48) , (2.22)where the coordinate ranges are r (cid:48) ∈ [0 , ∞ ) and θ (cid:48) ∈ [0 , (1 − η i )2 π ). Thus we caninterpret the effect of the operator as producing a conical singularity in the physicalmetric, with a conical deficit for 0 < η i < and a conical surplus for η i <
0. Findingreal saddle points in the presence of heavy operators with real η ’s is thus equivalentto finding metrics of constant negative curvature on the sphere punctured by conicalsingularities of various strength.An interesting additional constraint comes from the Gauss-Bonnet theorem. Theintegrated curvature on a sphere must be positive to produce a positive Euler character,so for a metric of constant negative curvature to exist on a punctured sphere thepunctures must introduce sufficient positive curvature to cancel the negative curvaturefrom the rest of the sphere. By integrating equation (2.15) and using the boundarycondition (2.11) we find a real solution φ c can exist only if (cid:88) i η i > S , theinequalities together imply 0 < η i < . Unless we satisfy these inequalities, there is noreal saddle point for the Liouville path integral, even if the η i are all real. The Gauss-Bonnet constraint (2.23) also implies that there is no real saddle point for a product oflight fields on S ; this case amounts to setting all η i to zero. In particular, there is noreal saddle point for the Liouville partition function on S . This has traditionally beendealt with by fixing the area (calculated in the physical metric) and then attemptingto integrate over the area; the fixed area path integral has a real saddle point. We willdevelop an alternative based on complex saddle points.More generally, if the η ’s are complex, then as we mentioned above a saddle point φ c will in general be complex and there is no reason to impose (2.23).So far we have not encountered the renormalization issues mentioned at the begin-ning of the section. But if we try to evaluate the action (2.10) on a solution obeying(2.18), then we find that both the kinetic term and the source term contributed by– 16 –he heavy operator are divergent. To handle this, again following [7], we performthe action integral only over the part of the disk D that excludes a disk d i of radius (cid:15) about each of the heavy operators. We then introduce “semiclassically renormalized”operators V ηib ( z i , z i ) ≈ (cid:15) η ib exp (cid:18) η i π (cid:73) ∂d i φ c dθ (cid:19) . (2.24)It is easy to check that this operator multiplied by the usual integrand of the pathintegral (the exponential of minus the action) has a finite limit as (cid:15) → (cid:15) η ib in (2.24) contributes a term − η i /b to the scaling dimension of the operator V η i /b ; this is a contribution of − η i /b to both∆ and ∆, consistent with the quantum shift − α i of the operator weights. We can thusincorporate the effects of all the heavy operators by introducing a modified action: b (cid:101) S L = 116 π (cid:90) D −∪ i d i d ξ (cid:0) ∂ a φ c ∂ a φ c + 16 λe φ c (cid:1) + 12 π (cid:73) ∂D φ c dθ + 2 log R − (cid:88) i (cid:18) η i π (cid:73) ∂d i φ c dθ i + 2 η i log (cid:15) (cid:19) (2.25)The equations of motion for this action automatically include both Liouville’s equation(2.12) and the boundary conditions (2.11) and (2.18). The final semiclassical expressionfor the expectation value of a product of heavy and light primary fields is (cid:68) V η b ( z , z ) · · · V ηnb ( z n , z n ) V bσ ( x , x ) · · · V bσ m ( x m , x m ) (cid:69) ≈ e − (cid:101) S L [ φ η ] m (cid:89) i =1 e σ i φ η ( x i ,x i ) . (2.26)Here there are n heavy operators and m light operators, and φ η is the solution of (2.15)obeying the correct boundary conditions. In this formula effects that are O ( b ) in theexponent have been kept only if they depend on the positions or conformal weights ofthe light operators. We will do light operator computations in sections (4.3, 5.4, 5.5,7.2.3), and we will be more careful about these corrections there. If there is more thanone solution, and we will find that in general there will be, then the right hand side of(2.26) will include a sum (or integral) over these saddlepoints. In two-dimensional conformal field theory, the expectation value of a product ofthree primary operators on S is determined up to a constant by conformal symmetry The exponential term is finite since we are assuming (2.20). – 17 –17]. We saw in the previous section that the operators V α are primaries of weight∆ = α ( Q − α ), so their three-point function must be of the form (cid:104) V α ( z , z ) V α ( z , z ) V α ( z , z ) (cid:105) = C ( α , α , α ) | z | +∆ − ∆ ) | z | +∆ − ∆ ) | z | +∆ − ∆ ) . (2.27)Here z ij = z i − z j . The function C ( α , α , α ) is the main dynamical data of anyCFT. In a CFT with only finitely many primaries, matrix elements of C are oftencalled structure constants, but this terminology does not seem entirely felicitous when C depends on continuous variables. The DOZZ formula is an analytic expression for C in Liouville theory [6, 7]. This proposal satisfies all the expected conditions in Liouvilletheory, and is the unique function that does so; in particular, it is the unique solutionof recursion relations that were derived in [30, 33]. Knowing C ( α , α , α ), along withrules for a sewing construction of higher order amplitudes can be viewed as an exactconstruction of the quantum Liouville theory.The DOZZ formula reads: C ( α , α , α ) = (cid:104) πµγ ( b ) b − b (cid:105) ( Q − (cid:80) α i ) /b × Υ Υ b (2 α )Υ b (2 α )Υ b (2 α )Υ b ( α + α + α − Q )Υ b ( α + α − α )Υ b ( α + α − α )Υ b ( α + α − α ) . (2.28)Here Υ b ( x ) is an entire function of x defined (for real and positive b ) bylog Υ b ( x ) = (cid:90) ∞ dtt (cid:34) ( Q/ − x ) e − t − sinh (( Q/ − x ) t )sinh tb sinh t b (cid:35) < Re( x ) < Q. (2.29)Though this integral representation is limited to the strip 0 < Re( x ) < Q , Υ b ( x ) has ananalytic continuation to an entire function of x . This follows from recursion relationsthat are explained in Appendix A, along with other properties of Υ b . Υ is definedas d Υ b ( x ) dx | x =0 , and γ ( x ) ≡ Γ( x )Γ(1 − x ) . In the following section we will discuss some of themotivation for this formula, but for the moment we will just make three observations:(1) This expression obeys C ( Q − α , α , α ) = R ( α ) C ( α , α , α ) with R ( α ) = (cid:104) πµγ ( b ) b − b (cid:105) (2 α − Q ) /b Υ b (2 α − Q )Υ(2 α )= (cid:2) πµγ ( b ) (cid:3) (2 α − Q ) /b b γ (2 α/b − − /b ) γ (2 bα − b ) ; (2.30)this justifies the reflection formula (2.21). To derive this, one uses the reflectionsymmetry Υ b ( Q − x ) = Υ b ( x ) and also the recursion relations for Υ b .– 18 –2) The entire expression (2.28) is almost invariant under b → b , and it becomes soif we also send µ → (cid:101) µ , with π (cid:101) µγ (1 /b ) = (cid:2) πµγ ( b ) (cid:3) b (2.31)This is a weak-strong duality, in the sense that if µ scales like b − to produce goodsemiclassical saddle points with finite curvature as b →
0, then (cid:101) µ (cid:101) b = (cid:101) µ/b will beextremely singular in the same limit so the dual picture will not be semiclassical.(3) C ( α , α , α ) as defined in (2.28) is a meromorphic function of the α i , with theonly poles coming from the zeros of the Υ b ’s in the denominator. In particular itis completely well-behaved in regions where the inequalities (2.23) and (2.20) areviolated. That said, the integral representation of Υ b is only valid in the strip0 < Re( x ) < Q , and in the semiclassical limit, for four of the Υ b ’s in (2.28), theboundary of the strip is precisely where the inequality (2.23) or (2.20) breaksdown. This can lead to a change in the nature of the semiclassical limit. Inparticular when all three α ’s are real and obey the Seiberg and Gauss-Bonnetinequalities, all seven Υ b ’s can be evaluated by the integral (2.29). This is not anaccident; in particular, we will argue below that analytically continuing past theline Re( η + η + η ) = 1 corresponds to crossing a Stokes line in the Liouvillepath integral; the number of contributing saddle points increases as we do so. We will for the most part be studying the semiclassical limit of the DOZZ formula,but we will find it extremely helpful to also consider certain four-point functions. Intwo-dimensional CFT, the four-point function on S is the first correlation functionwhose position dependence is not completely determined by conformal symmetry. Itis strongly constrained, but unfortunately there is much freedom in how to apply theconstraint and there do not seem to be standard conventions in the literature. We willdefine: (cid:28) V α ( z , z ) V α ( z , z ) V α ( z , z ) V α ( z , z ) (cid:29) = | z | − ∆ − ∆ − ∆ ) | z | +∆ − ∆ − ∆ ) | z | − | z | +∆ − ∆ − ∆ ) G ( y, y ) , (2.32) The material discussed here is mostly not required until the final two parts of section 5, so thereader who is unfamiliar with the CFT techniques of [17] may wish to stop after equation (2.38) andpostpone the rest. – 19 –ith the harmonic ratio y defined as: y = z z z z . (2.33)This parametrization is chosen so that the limit z → ∞ , z → z → y , and z → z →∞ | z | (cid:10) V α (0 , V α ( y, y ) V α (1 , V α ( z , z ) (cid:11) = G ( y, y ) (2.34)Using radial quantization as in [17], we can write this as G ( y, y ) = (cid:104) α | V α (1 , V α ( y, y ) | α (cid:105) . (2.35)We can also write C as C ( α , α , α ) = (cid:104) α | V α (1 , | α (cid:105) . (2.36)In a conventional two-dimensional CFT, these two equations are the starting point forthe conformal bootstrap program [17]. In this program, one expresses the four-pointfunction (2.35) in terms of products of three points functions in two different ways,either by inserting a complete set of states between the fields V α (1 ,
1) and V α ( y, y )in (2.35), or by using the operator product expansion to replace the product of thosetwo fields with a single field. In Liouville, the situation is more subtle since α is acontinuous label with complex values and it is not immediately clear what is meantby a complete set of states. Similarly, in making the operator product expansion,one expands the product of two fields in terms of a complete set of fields, and it isagain not clear how to formulate this. This problem was solved by Seiberg [8], whoargued using minisuperspace that the states with α = Q + iP are indeed delta-functionnormalizable for real and positive P , and moreover that these states along with theirVirasoro descendants are a complete basis of normalizable states. One can check thefirst of these assertions directly from the DOZZ formula by demonstrating that lim (cid:15) → C ( Q/ iP , (cid:15), Q/ iP ) = 2 πδ ( P − P ) G ( Q/ iP ) , (2.37)with the two-point normalization G ( α ) given by G ( α ) = 1 R ( α ) = 1 b (cid:2) πµγ ( b ) (cid:3) ( Q − α ) /b γ (2 α/b − − /b ) γ (2 bα − b ) . (2.38) In showing this, one uses the fact that the numerator of the DOZZ formula has a simple zero for (cid:15) →
0, while the denominator has a double zero for (cid:15) → P − P →
0. One encounters therelation lim (cid:15) → (cid:15)/ (( P − P ) + (cid:15) ) = πδ ( P − P ). – 20 –eiberg also argued semiclassically that the state V α ( y, y ) | α (cid:105) with both α ’s real andless than Q/ α + α > Q . This follows from the Gauss-Bonnett constraint. If we assume that α and α are in this range, then we can expandthis state in terms of the normalizable states | Q/ iP, k, k (cid:105) . Here | Q/ iP, k, k (cid:105) isa shorthand notation for V Q/ iP (0 , | vac (cid:105) and its Virasoro descendants. Similarly if α + α > Q the state (cid:104) α | V α (1 ,
1) is also normalizable, and we can evaluate (2.32)by inserting a complete set of normalizable states. Using (2.21), (2.36), and (2.37) thisleads to G ( y, y ) = 12 (cid:90) ∞−∞ dP π C ( α , α , Q/ iP ) C ( α , α , Q/ − iP ) × F (∆ i , ∆ P , y ) F (∆ i , ∆ P , y ) . (2.39)Here ∆ P = P + Q /
4, and the function F is the familiar Virasoro conformal block[17], expressible as F (∆ i , ∆ P , y ) = y ∆ P − ∆ − ∆ ∞ (cid:88) k =0 β P,k (cid:104) α | V α (1 , | Q/ iP, k, (cid:105) C ( α , α , Q/ iP ) y k . (2.40)The sum over k is heuristic; it really represents a sum over the full conformal familydescended from V Q/ iP . The power of y for a given term is given by the level of thedescendant being considered, so for example L − L − | Q/ iP (cid:105) contributes at order y . β P,k is defined in [17], it appears here in the expansion of V α | α (cid:105) via V α ( y, y ) | α (cid:105) = (cid:90) ∞ dP π C ( α , α , Q/ iP ) R ( Q/ iP ) | y | P − ∆ − ∆ ) × ∞ (cid:88) k,k =0 β P,k β P,k y k y k | Q/ iP, k, k (cid:105) . (2.41)Both β P,k and the conformal block itself are universal building blocks for two-dimensionalCFT’s, and conformal invariance completely determines how they depend on the con-formal weights and central charge.We can then define the general four-point function away from the specified region of α , α by analytic continuation of (2.39). As observed in [7], this analytic continuationchanges the form of the sum over states. The reason is that as we continue in the α ’s,the various poles of the C ’s can cross the contour of integration and begin to contributediscrete terms in addition to the integral in (2.39).One final tool that will be useful for us is the computation of correlators thatinclude degenerate fields. A degenerate field in 2D CFT is a primary operator whose– 21 –escendants form a short representation of the Virasoro algebra, and this implies thatcorrelation functions involving the degenerate field obey a certain differential equation[17]. Such short representations of the Virasoro algebra can arise only for certain valuesof the conformal dimension. In Liouville theory the degenerate fields have α = − n b − mb , (2.42)where n and m are nonnegative integers [33]. In particular we see that there areboth light and heavy degenerate fields. We will be especially interested in the lightdegenerate field V − b/ , so we observe here that the differential equation its correlatorobeys is (cid:32) ∂ ∂z − n (cid:88) i =1 ∆ i ( z − z i ) − n (cid:88) i =1 z − z i ∂∂z i (cid:33) × (cid:28) V − b/ ( z, z ) V α ( z , z ) · · · V α n ( z n , z n ) (cid:29) = 0 . (2.43)Here ∆ is the conformal weight of the field V − b/ . An identical equation holds forcorrelators involving V − b , with ∆ now being the weight of V − b . For example, byapplying this equation to the three-point function (cid:104) V − b/ V α V α (cid:105) and using also the factthat it must take the form (2.27), one may show that this three-point function vanishesunless α = α ± b/
2. (This relation is known as the degenerate fusion rule.) Wecan check that the DOZZ formula indeed vanishes if we set α = − b/ α , α , but there is in important subtlety in that if we simultaneously set α = α ± b/ α = − b/ C ( α , α , α ) is indeterminate. (The numerator anddenominator both vanish.) The lesson is that correlators with degenerate fields cannotalways be simply obtained by specializing generic correlators to particular values.One can actually obtain a good limit for the four-point function with a degenerateoperator from the integral expression (2.39) [30]. The evaluation is subtle in that thereare poles of C ( α , α , Q/ iP ) that cross the contour as we continue α → − b/
2, andin particular there are two separate pairs of poles that merge as α → − b/ α ( P ) = α ± b/
2. If we are careful toperform the integral with α = − b/ (cid:15) and then take (cid:15) →
0, we find that the formulafor the four-point function involving the light degenerate field V − b/ simplifies into adiscrete formula of the usual type [33]: G ( y, y ) = (cid:88) ± C ± C ± F (∆ i , ∆ ± , y ) F (∆ i , ∆ ± , y ) . (2.44)– 22 –ere we have taken α = − b/
2, and ± corresponds to the operator V α ± b/ . The raisedindex ± is defined using the two-point function (2.38), so: C ± = C ( α ± b/ , α , − b/ R ( α ± b/
2) = C ( α , − b/ , Q − α ∓ b/ . (2.45)As just discussed the value of the structure constant on the right cannot be determinedunambiguously from the DOZZ formula, but the contour manipulation of the four-pointfunction gives C ( α , − b/ , Q − α ∓ b/ ≡ lim δ → (cid:104) lim (cid:15) → (cid:15) C ( α , − b/ δ, Q − α ∓ b/ (cid:15) − δ ) (cid:105) . (2.46)This definition agrees with a Coulomb gas computation in free field theory [7]. Ex-plicitly, from the DOZZ formula we find C + 12 = − πµγ ( − b ) γ (2 α b ) γ (2 + b − bα ) C − = 1 (2.47)As shown in [17], by applying the differential equation (2.43) to (2.44) we canactually determine F in terms of a hypergeometric function. This involves using SL (2 , C ) invariance to transform the partial differential equation (2.43) into an ordinarydifferential equation, which turns out to be hypergeometric. The analysis is standardand somewhat lengthy, so we will only present the result: F (∆ i , ∆ ± , y ) = y α ∓ (1 − y ) β F ( A ∓ , B ∓ , C ∓ , y ) , (2.48)with: ∆ ± = ( α ± b/ Q − α ∓ b/ −
12 + 3 b α ∓ = ∆ ± − ∆ − ∆ β = ∆ − − ∆ − ∆ A ∓ = ∓ b ( α − Q/
2) + b ( α + α − b ) − / B ∓ = ∓ b ( α − Q/
2) + b ( α − α ) + 1 / C ∓ = 1 ∓ b (2 α − Q ) . That computation is based on the observation that for the α i ’s occuring in this structure constant,the power of µ appearing in the correlator is either zero or one. This suggests computing the correlatorby treating the Liouville potential as a perturbation of free field theory and then computing theappropriate perturbative contribution to produce the desired power of µ . Hypergeometric functions will appear repeatedly in our analysis, so in Appendix B we present aself-contained introduction. – 23 –sing this expression and formula (B.15) from the Appendix, Teschner showed that(2.44) will be singlevalued only if the structure constant obeys a recursion relation [33]: C ( α , α , α + b/ C ( α , α , α − b/
2) = − γ ( − b ) πµ × γ (2 α b ) γ (2 bα − b ) γ ( b ( α + α − α ) − b / γ ( b ( α + α − α ) − b / γ ( b ( α + α − α ) − b / γ ( b ( α + α + α ) − − b / V − b , heshowed that the DOZZ formula is the unique structure constant that allows both four-point functions to be singlevalued. In this version of the logic, C ± is determined bythe Coulomb gas computation rather than the limit (2.46) of the DOZZ formula. Thisat last justifies the DOZZ formula (2.28). In this section we will describe the most general complex-valued solutions of Liouville’sequation on S with two or three heavy operators present. The solutions we will presentare simple extensions of the real solutions given for real η ’s in [7]. We will emphasize thenew features that emerge once complex η ’s are allowed and also establish the uniquenessof the solutions. One interesting issue that will appear for the three-point function isthat for many regions of the parameters η , η , η , there are no nonsingular solutionsof Liouville’s equation with the desired properties, not even complex-valued ones. Wewill determine the analytic forms of the singularities that appear and comment on theirgenericity. We will first determine the local form of a solution Liouville’s equation with flatreference metric: ∂∂φ c = 2 λe φ c . (3.1)We have defined λ = πµb , which we hold fixed for b → φ c in terms of e φ c ( z,z ) = 1 λ f ( z, z ) , (3.2)which gives equation of motion ∂∂f = 1 f ( ∂f ∂f − . (3.3)– 24 –here is a classic device [34] that allows the transformation of this partial differentialequation into two ordinary differential equations, using the fact that the stress tensor(2.4) is holomorphic. In particular, the holomorphic and antiholomorphic componentsof the stress tensor are proportional to W = − ∂ f /f and (cid:102) W = − ∂ f /f respectively.We thus have: ∂ f + W ( z ) f = 0 (3.4) ∂ f + (cid:102) W ( z ) f = 0 (3.5)with W and (cid:102) W holomorphic. In these equations, we may treat z and z independently, sowe must be able to write f locally as a sum of the two linearly independent holomorphicsolutions of the W ( z ) equation with coefficients depending only on z : f = u ( z ) (cid:101) u ( z ) − v ( z ) (cid:101) v ( z ) (3.6)Plugging this ansatz into the (cid:102) W equation, we see that (cid:101) u and (cid:101) v are anti-holomorphicsolutions of that equation. Going back to the original Liouville equation, we find:( u∂v − v∂u )( (cid:101) u∂ (cid:101) v − (cid:101) v∂ (cid:101) u ) = 1 . (3.7)The first factor is a constant since it is the Wronskian evaluated on two solutions of the W ( z ) equation, and similarly the second factor is constant. Both Wronskian factorsmust be nonzero to satisfy this equation, so u and v are indeed linearly independent,and similarly (cid:101) u and (cid:101) v . So each pair gives a basis of the two linearly independentholomorphic or antiholomorphic solutions of the appropriate equation. We thus arriveat a general form for any complex solution of Liouville’s equation, valid locally as longas the reference metric is ds = dz ⊗ dz : e φ c = 1 λ u ( z ) (cid:101) u ( z ) − v ( z ) (cid:101) v ( z )) , (3.8)with u and v obeying ∂ g + W ( z ) g = 0 (3.9)and (cid:101) u and (cid:101) v obeying ∂ (cid:101) g + (cid:102) W ( z ) (cid:101) g = 0 . (3.10)The representation in (3.8) is not quite unique; one can make an arbitrary invertiblelinear transformation of the pair (cid:18) uv (cid:19) , with a compensating linear transformation of (cid:0)(cid:101) u (cid:101) v (cid:1) . – 25 –o specify a particular solution, we need to choose W and (cid:102) W and also a basis forthe solutions of (3.9), (3.10). These choices are constrained by the boundary conditions,in particular (2.11) and (2.18). If this problem is undetermined then there are modulito be integrated over, while if it is overdetermined there is no solution.In the following subsections, we we will show what happens explicitly in the specialcases of two and three heavy operators on the sphere. But we first make some generalcomments valid for any number of such operators. The presence of heavy operatorsrequires the solution φ c to be singular at specific points z i . In terms of f , we need f ( z, z ) ∼ | z − z i | η i as z → z i . (3.11)Looking at the form (3.8), there are two possible sources of these singularities. Thefirst is that at least one of u , v , (cid:101) u , (cid:101) v is singular. The second is that all four functions arenonzero but u (cid:101) u − v (cid:101) v = 0, because of a cancellation between the two terms. Assumingthat this cancellation happens at a place where none of u , v , (cid:101) u , (cid:101) v are singular, we canexpand f ∼ A ( z − z ) + B ( z − z ) + O ( | z − z | ) as z → z . (3.12)Inserting this into (3.3), we find that AB = 1 and thus A and B are both nonzero. Itthus cannot produce the desired behavior (3.11).We will have more to say about this type of singularity later, but for now we willfocus on singularities that occur because some of the functions are singular. In orderto produce the behavior (3.11) from singularities of the individual functions, u and v must behave as linear combinations of ( z − z i ) η i and ( z − z i ) − η i for z → z i , with similarbehavior for (cid:101) u , (cid:101) v . To get this behavior, W and (cid:102) W must have double poles at z = z i ,with suitably adjusted coefficients. A double pole of W or (cid:102) W in a differential equationof the form (3.4) is called a regular singular point. A double pole is the expectedbehavior of the stress tensor at a point with insertion of a primary field.Moreover, for the solution to be regular at the point at infinity on S , we need(2.11), which translates into f ( z, z ) ∼ | z | as | z | → ∞ . (3.13)To achieve this, the two holomorphic solutions of the differential equation ( ∂ z + W ) g = 0should behave as 1 and z , respectively, near z = ∞ . Asking for this equation to have asolution of the form a z + a + a − z − + . . . with arbitrary a − and a and no logarithmsin the expansion implies that W vanishes for z → ∞ at least as fast as 1 /z . This isalso the expected behavior of the stress tensor in the presence of finitely many operatorinsertions on R . Given this behavior, the differential equation again has a regularsingular point at z = ∞ . – 26 –e do not want additional singularities in W or (cid:102) W as they would lack a physicalinterpretation. To be more precise, a pole in W leads to a delta function or derivativeof a delta function in ∂W . Liouville’s equation implies that ∂W = 0, and a deltafunction correction to that equation implies the existence of a delta function sourceterm in Liouville’s equation – that is, an operator insertion of some kind.Thus for a finite number of operator insertions, W and (cid:102) W have only finitely manypoles, all of at most second order. In particular, W and (cid:102) W are rational functions. Theparameters of these rational functions must be adjusted to achieve the desired behaviornear operator insertions and at infinity. We now study this problem in the special casesof two or three heavy operators. Specializing to the case of two operators, W should have two double poles andshould vanish as 1 /z for z → ∞ ; (cid:102) W should be similar. Up to constant multiples, thesefunctions are determined by the positions of the poles: W ( z ) = w (1 − w ) z ( z − z ) ( z − z ) (cid:102) W ( z ) = (cid:101) w (1 − (cid:101) w ) z ( z − z ) ( z − z ) . (3.14)We have picked a convenient parametrization of the constants. In this case, the ODE’scan be solved in terms of elementary functions. A particular basis of solutions is g ( z ) = ( z − z ) w ( z − z ) − w g ( z ) = ( z − z ) − w ( z − z ) w (cid:101) g ( z ) = ( z − z ) (cid:101) w ( z − z ) − (cid:101) w (cid:101) g ( z ) = ( z − z ) − (cid:101) w ( z − z ) (cid:101) w . (3.15)It remains to determine w and (cid:101) w in terms of η and η and to write the u ’s and v ’s interms of this basis. In doing this we need to make sure that (3.11) is satisfied, and alsothat the product of the Wronskians obeys (3.7). Up to trivial redefinitions, the resultis that we must have η = η = w = (cid:101) w ≡ η , also having u ( z ) = g ( z ) (3.16) v ( z ) = g ( z ) (cid:101) u ( z ) = κ (cid:101) g ( z ) (cid:101) v ( z ) = (cid:101) g ( z ) κ (1 − η ) | z | – 27 –his leads to the solution e φ c = 1 λ (cid:16) κ | z − z | η | z − z | − η − κ − η ) | z | | z − z | − η | z − z | η (cid:17) . (3.17)The criterion η = η is expected, since in conformal field theory, the two-point functionfor operators of distinct conformal weights always vanishes. κ is an arbitrary complexnumber, but it is slightly constrained if we impose as a final condition that f benonvanishing away from the operator insertions. The denominator in (3.17) can vanishonly if κ lies on a certain real curve (cid:96) in the complex plane ( (cid:96) is simply the real axis if η is real). Omitting the curve (cid:96) from the complex κ plane, and taking into account thefact that the sign of κ is irrelevant, we get a moduli space of solutions that has complexdimension one and that as a complex manifold is a copy of the upper half-plane H .Returning now to the general form (3.17), we will make two comments:(i) Suppose that η is real. To avoid singularities, we cannot have κ be real, but wecan instead choose it to be purely imaginary. e φ c will then be real but negativedefinite, and φ c will be complex. Nonetheless this situation still has a simplegeometric interpretation: we can define a new metric − e φ c δ ab , which is indeeda genuine metric on the sphere, and because of the sign change it has positive curvature! It has two conical singularities, and for positive η (cid:48) s it describes theintrinsic geometry of an American football. This observation is a special case of ageneral bijection between saddle points of spacelike and timelike Liouville, whichwill be explored later.(ii) Eqn. (3.17) gives the most general form of e φ c , but this leaves the possibility ofadding to φ c itself an integer multiple of 2 πi , as in eqn. (1.14). Thus the modulispace of solutions has many components and is isomorphic to H × Z . For the case of three heavy operators, the potentials W and (cid:102) W must now be rationalfunctions with three double poles. Their behaviour at infinity determines them up toquadratic polynomials in the numerator, which we can further restrict by demandingthe correct singularities of u , v , (cid:101) u , and (cid:101) v at the operator insertions. There will be anew challenge, however; while we can easily choose a basis of solutions of (3.9) and(3.10) with the desired behavior near any one singular point, it is nontrivial to arrangeto get the right behavior at all three singular points.– 28 –nsisting that the residues of the poles in W and (cid:102) W have the correct forms toproduce (3.11) leads to unique expressions for W and (cid:102) W : W ( z ) = (cid:20) η (1 − η ) z z z − z + η (1 − η ) z z z − z + η (1 − η ) z z z − z (cid:21) z − z )( z − z )( z − z ) (cid:102) W ( z ) = (cid:20) η (1 − η ) z z z − z + η (1 − η ) z z z − z + η (1 − η ) z z z − z (cid:21) z − z )( z − z )( z − z )(3.18)With these potentials, the differential equation of interest becomes essentially the hy-pergeometric equation, modulo an elementary normalization. So the solutions can beexpressed in terms of hypergeometric functions, or equivalently, but slightly more ele-gantly, in terms of Riemann’s P functions. P functions are solutions of a differentialequation with three regular singularities at specified points, and with no singularity atinfinity. The equations (3.9) and (3.10) are not quite of this form since they do havea regular singular point at infinity, but we can recast them into Riemann’s form bydefining g ( z ) = ( z − z ) h ( z ) and (cid:101) g ( z ) = ( z − z ) (cid:101) h ( z ). One can check that the equationsobeyed by h and (cid:101) h are special cases of Riemann’s equation B.5, with the parameters α = η α (cid:48) = 1 − η (3.19) β = − η β (cid:48) = η − γ = η γ (cid:48) = 1 − η . We now observe that the boundary conditions (3.11) ensure that without loss of gener-ality we can choose u , v , (cid:101) u , and (cid:101) v to diagonalize the monodromy about any particularsingular point, say z . Also picking a convenient normalization of these functions, wehave u ( z ) = ( z − z ) P η ( x ) (3.20) v ( z ) = ( z − z ) P − η ( x ) (cid:101) u ( z ) = a ( z − z ) P η ( x ) (cid:101) v ( z ) = a ( z − z ) P − η ( x ) In Appendix B, we give a self-contained development of the minimum facts we need concerninghypergeometric and P -functions. The reader unfamiliar with these functions is encouraged to readthis appendix now. The unpleasant asymmetry of the second line follows from the definition of h , but a symmetricdefinition introduces significant complication in the formulas that follow so we will stay with thischoice. – 29 –ere a , a are complex numbers to be determined, x = z ( z − z ) /z ( z − z ), and the P functions explicitly are related to hypergeometric functions by P η ( x ) = x η (1 − x ) η F ( η + η − η , η + η + η − , η , x ) P − η ( x ) = x − η (1 − x ) − η F (1 − η + η − η , − η − η − η , − η , x ) . (3.21)We can determine the product a a by imposing (3.7); by construction we know that u∂v − v∂u and (cid:101) u∂ (cid:101) v − (cid:101) v∂ (cid:101) u are both constant, so to make sure their product is 1 itis enough to demand it in the vicinity of z = z . This is easy to do using the seriesexpansion for the hypergeometric function near x = 0, leading to a a = | z | | z | | z | (1 − η ) . (3.22)It is clear from the above formulas that f = u (cid:101) u − v (cid:101) v is singlevalued about z = z . Forthis to also be true near z , z is a non-trivial constraint, which can be evaluated usingthe connection formulas (B.11). For example, f | z − z | − = a P η ( x ) P η ( x ) − a P − η ( x ) P − η ( x )will be singlevalued near z = z , which corresponds to x = 1, only if a a η ,η a η , − η = a a − η ,η a − η , − η . (3.23)The connection coefficients a ij are given by (B.12), so combining this with (3.22) wefind ( a ) = | z | | z | | z | γ ( η + η − η ) γ ( η + η − η ) γ ( η + η + η − γ (2 η ) γ ( η + η − η ) (3.24)Thus both a and a are determined (up to an irrelevant overall sign) , so the solutionis completely determined. The reader can check that with the ratio given by (3.23)the solution is also singlevalued near z . This is a nontrivial computation using (B.13),but it has to work, since the absence of monodromy around z , z , and ∞ implies thatthere must also be none around z .The final form of the solution is thus e φ c = 1 λ | z − z | − [ a P η ( x ) P η ( x ) − a P − η ( x ) P − η ( x )] . (3.25)In the end, this is simply the analytic continuation in η i of the real solution presentedin [7], but our argument has established its uniqueness.– 30 –here is still a potential problem with the solution. The coefficients a , a werecompletely determined without any reference to avoiding cancellations between theterms in the denominator, and it is not at all clear that the denominator has no zeroesfor generic η ’s. It is difficult to study the existence of such cancellations analyticallyfor arbitrary η ’s, but we have shown numerically that they indeed happen for genericcomplex η ’s. If we assume that such a singularity is present at z = z , then we sawabove that its analytic form is given by (3.12).For real η ’s, we can say more. When the η ’s are real, the right hand side of (3.24)is real so a is either real or imaginary. If it is imaginary, then (3.22) shows that a will also be imaginary and with opposite sign for its imaginary part. Moreover forreal η ’s, P η ( x ) P η ( x ) and P − η ( x ) P − η ( x ) are strictly positive. Thus when a ispurely imaginary, both terms in the denominator have the same phase and there canbe no singularities arising from cancellation. The metric e φ c δ ab will however be negativedefinite, so this will be a complex saddle point for φ c . If we start with such η ’s andallow them to have small imaginary parts then cancellations do not appear at once,but we find numerically that if we allow the imaginary parts to become large enoughthen cancellations in the denominator do occur.We can also consider the case that the η ’s are real and a is also real. a will thenbe real and with the same sign as a , so cancellations are now possible. We learned insection 2.2 that real solutions can only occur if certain inequalities (2.20) and (2.23) aresatisfied. So if the η ’s are real but violate the inequalities, the denominator in (3.25)definitely vanishes somewhere. On the other hand, if the η ’s are real and satisfy theinequalities, then a real metric of constant negative curvature corresponding to a realsolution of Liouville’s equations does exist. It can be constructed by gluing together twohyperbolic triangles, or in any number of other ways. So in this case, the denominatorin (3.25) is positive definite away from the operator insertions. This is the regionstudied in [7].We conclude with two remarks about the nature of these singularities near a zero ofthe denominator in the formula for e φ c . We first observe that the singularities naturallycome in pairs since the denominator of (3.25) is symmetric under exchanging x and x , sofor example if we choose the z i to be real then the solution is symmetric under reflectionacross the real z -axis. We secondly comment on the stability of these singularities: thegeneral local expansion (3.12) near a zero involves two complex coefficients A and B .When these are of unequal magnitude, the existence of a zero of f is stable under smallperturbations. This is because one can associate to an isolated zero of the complexfunction f an integer-valued invariant, the winding number. To define it, set f = se iψ We show this explicitly below in Appendix B.5. – 31 –here s is a positive function and ψ is real. Supposing that f has an isolated zeroat z = z , consider e iψ as a function defined on the circle z = z + (cid:15)e iθ , for somesmall positive (cid:15) and real θ . The winding number is defined as π (cid:72) π dθ dψ/dθ , and isinvariant under small changes in f . (If f is varied so that several zeroes meet, thenonly the sum of the winding numbers is invariant, in general.) In the context of (3.12),the winding number is 1 for | A | > | B | , and − | A | < | B | , and depends on higherterms in the expansion if | A | = | B | . In the case of a zero of the denominator in (3.25),one generically has | A | (cid:54) = | B | if the η ’s are complex, so isolated singularities arisingby this mechanism are stable against small perturbations. When the η ’s are real, thebehavior near singularities of this type requires more examination. In this section, we use the complex classical solutions constructed in the previous sectionto interpret the analytic continuation first of the two-point function (2.38) and thenof the three-point function as given by the DOZZ formula (2.28). We will find thatfor the two-point function there is a satisfactory picture in terms of complex saddlepoints, which agrees with and we believe improves on the old fixed-area results in thesemiclassical approximation. For the three-point function we will find that the situationis more subtle; we will be able to “improve” on the fixed-area result here as well, but tounderstand the full analytic continuation we will need to confront the singularities atwhich the denominator of the solution vanishes. For ease of presentation we postponeour discussion of those singularities until section 5, and we here focus only on the partof the analytic continuation that avoids them. We also include the case of three lightoperators as check at the end of the section.
We saw in section 2.4 that the DOZZ formula implies that the Liouville two-pointfunction takes the form (cid:104) V α ( z , z ) V α ( z , z ) (cid:105) = | z | − α ( Q − α ) πb (cid:2) πµγ ( b ) (cid:3) ( Q − α ) /b γ (2 α/b − − /b ) γ (2 bα − b ) δ (0) (4.1)The factor of δ (0) is a shorthand which reflects the continuum normalization of theoperators with α = Q + iP and the fact that we have taken the two fields in (4.1)to have the same Liouville momentum. It may seem unphysical to study the analytic– 32 –ontinuation of a divergent quantity, but as we will review, the divergence has a simplesemiclassical origin that is independent of α . This “exact” result for the two-point function does not come from a real Liouvillepath integral, even if α is real. One can easily show that, for the two-point function on S , the path integral over real Liouville fields does not converge. Consider a smooth realfield configuration that obeys the boundary conditions (2.11) and (2.18). The modifiedaction (2.25) will be finite. Now consider adding a large negative real number ∆ φ c to φ c . The kinetic term will be unaffected and the exponential term will become smaller inabsolute value, but the boundary terms will add an extra term ∆ φ c (1 − η ). Recallingthat we always choose the Seiberg bound to be satisfied, we see that by taking ∆ φ c tobe large and negative we can thus make the action as negative as we wish. The pathintegral therefore cannot converge as an integral over real φ c ’s [8]. The original approach to resolve this divergence, proposed in [8], was to restrict thepath integral only to field configurations obeying (cid:82) d ξe φ c = A . This clearly avoids thedivergence. However, if one tries to integrate over A , one would get back the originaldivergence, while on the other hand if one simply keeps A fixed, one would not expectto get a local quantum field theory. As an alternative proposal, we claim that (4.1) iscomputed by a local path integral over a complex integration cycle. This is analogous tothe suggestion [35] of dealing with a somewhat similar divergence in the path integral ofEinstein gravity by Wick rotating the conformal factor of the metric to complex values.To motivate our proposal, we will show that the semiclassical limit of (4.1), with α scaling as η/b , is reproduced by a sum over the complex saddle points with two heavyoperators that we constructed in section 3.2. We interpret this as suggesting that thepath integral is evaluated over a cycle that is a sum of cycles attached to complex saddlepoints, as sketched in section 1.1. The requisite sum is an infinite sum, somewhat likewhat one finds for the Gamma function for Re z <
0, as described in Appendix C.We will also find that the set of contributing saddle points jumps discontinuously as η crosses the real axis. This again parallels a result for the Gamma function, and weinterpret it as a Stokes phenomenon. Indeed if we were to use the Liouville theory as part of a gravity theory where conformal symmetryis gauged, then to compute a two-point function of integrated vertex operators we would partially fixthe gauge by fixing the positions of the two operators and then divide by the volume of the remainingconformal symmetries. This would remove this divergent factor. This divergence should not be confused with the factor of δ (0), which we will see has to do withan integral over a noncompact subgroup of SL (2 , C ). In particular we can make the same argumentfor the three-point function with three real α ’s and find the same divergence if (cid:80) i α i < Q , and sincethe DOZZ formula does not have any δ (0) it is clear that this is a different issue [8]. – 33 – .1.1 Evaluation of the Action for Two-Point Solutions In computing the action of the two-point solution (3.17), we first need to deal withtaking the logarithm to get φ c . The branch cut in the logarithm makes this a nontrivialoperation. To make the following manipulations simpler, we will here relabel κ = i (cid:101) κ ,so the solution becomes e φ c = − λ (cid:101) κ (cid:16) | z − z | η | z − z | − η + (cid:101) κ (1 − η ) | z | | z − z | − η | z − z | η (cid:17) . (4.2)We choose (cid:101) κ to ensure that the denominator has no zeroes. Since we are imposing theSeiberg bound, we have Re(1 − η ) >
0. There is a sign choice in defining (cid:101) κ , so wewill choose it to have positive real part. In particular note that if η is real then we canhave (cid:101) κ be real and positive. Our prescription for taking the logarithm will then be φ c,N ( z, z ) = iπ + 2 πiN − log λ − (cid:101) κ − (cid:18) | z − z | η | z − z | − η + 1 (cid:101) κ (1 − η ) | z | | z − z | − η | z − z | η (cid:19) (4.3)The choice of branch for the final logarithm is inessential, in the sense that making adifferent choice would be equivalent to shifting the integer N in (4.3). We will choosethe branch such that the final logarithm behaves like − η log | z − z | + (4 η −
4) log | z | near z . Its value away from z is defined by continuity; there is no problem in extendingthis logarithm throughout the z -plane (punctured at z and z ). We will have no suchluck for the three-point function; in that case, zeroes of the logarithm are essential.We will see momentarily that to compute the action, we need to know the leadingbehaviour near z and z , so we observe that φ c,N ( z, z ) → − η log | z − z i | + C i as z → z i , (4.4)with C = 2 πi (cid:18) N + 12 (cid:19) − log λ − (cid:101) κ + (4 η −
4) log | z | C = 2 πi (cid:18) N + 12 (cid:19) − log λ + 2 log (cid:101) κ + 4 η log | z | + 4 log(1 − η ) . (4.5) Because of the boundary conditions (3.11), there cannot be monodromy of this logarithm about z , z even though its argument vanishes there. – 34 –o verify that the same integer N appears in both C and C , we note that this isclear for real η and (cid:101) κ , since then the final logarithm in (4.3) has no imaginary part; ingeneral it then follows by continuity.Now to compute the modified action (2.25), we use a very helpful trick from [7].This is to compute d (cid:101) S L /dη when (cid:101) S L is evaluated on a saddle point. A priori , therewould be η dependence both implicitly through the functional form of the saddle pointand explicitly through the boundary terms in (cid:101) S L , but the variation of (2.25) withrespect to φ c is zero when evaluated on a solution and only the explicit η -dependencematters. We thus have the remarkably simple equation: b d (cid:101) S L dη = − C − C = − πi (2 N + 1) + 2 log λ + (4 − η ) log | z | − − η ) (4.6)We can thus determine (cid:101) S L [ φ c,N ] up to a constant by integrating this simple function,and we can determine the constant by comparing to an explicit evaluation of the actionwhen η = 0. When η is zero, the saddle point (4.3) becomes an SL (2 , C ) transformationof a metric which is just minus the usual round sphere φ c = iπ + 2 πiN − log λ − zz ) . (4.7)For this solution we can evaluate the action (2.25) explicitly, finding b (cid:101) S = 2 πi ( N + ) − log λ −
2. Now doing the integral of (4.6) our final result for the action (2.25) withnonzero η is thus b (cid:101) S L =2 πi ( N + 1 / − η ) + (2 η − λ + 4( η − η ) log | z | + 2 [(1 − η ) log (1 − η ) − (1 − η )] . (4.8)We can observe immediately that the z dependence is consistent with the two-pointfunction of a scalar operator of weight ( η − η ) /b . This action is independent of (cid:101) κ , sowhen we integrate over it this will produce a divergent factor, which we interpret asthe factor δ (0) in (4.1).Before moving on to the exact expression, we we will observe here that this actionis multivalued as a function of η , with a branch point emanating from η = 1 /
2, wherethe original solution (4.2) is not well-defined. Under monodromy around this point, N shifts by 2, so all even and likewise all odd values of N are linked by this monodromy.Of course, to see the monodromy, we have to consider paths in the η plane that violatethe Seiberg bound Re( η ) < . We thank X. Dong for a discussion of this point and for suggesting the following line of argument. – 35 – .1.2 Comparison with Limit of Exact Two-Point Function
We now compute the semiclassical asymptotics of (4.1). We can easily find that (cid:104) V α ( z , z ) V α ( z , z ) (cid:105) ∼ δ (0) | z | − η (1 − η ) /b λ (1 − η ) /b (cid:20) γ ( b ) b (cid:21) (1 − η ) /b γ (cid:18) (2 η − b (cid:19) (4.9)The first three factors obviously match on to the result (4.8) that we found in the previ-ous section, but the last two have more subtle semiclassical limits. It is not hard to seethat the factor involving γ ( b ) is asymptotic for small positive b to exp (cid:110) − − η ) log bb (cid:111) ,but to understand the final factor, we need to understand the asymptotics of the Γfunction at large complex values of its argument. For real positive arguments, thisis the well-known Stirling approximation, but for complex arguments, the situation ismore subtle: Γ( x ) = (cid:40) e x log x − x + O (log x ) Re( x ) > e iπx − e − iπx e x log( − x ) − x + O (log( − x )) Re( x ) < . (4.10)This result can be obtained in a variety of ways; because of the fact (see [8, 15, 16] andsection 1.1.1) that the integral representation of the Gamma function is a minisuper-space approximation to Liouville theory, we present in Appendix C a derivation usingthe machinery of critical points and Stokes lines. Using (4.10), we find γ (cid:18) (2 η − b (cid:19) ∼ e iπ (2 η − /b − e − iπ (2 η − /b exp (cid:20) (4 η − b (log(1 − η ) − b − (cid:21) . (4.11)So we can write the semiclassical limit as (cid:104) V α ( z , z ) V α ( z , z ) (cid:105) ∼ δ (0) | z | − η (1 − η ) /b λ (1 − η ) /b × e − b [(1 − η ) log(1 − η ) − (1 − η )] e iπ (2 η − /b − e − iπ (2 η − /b . (4.12)All factors now clearly match (4.8) except for the last. To complete the argument,setting y = e iπ (2 η − /b , we need to know that the function 1 / ( y − y − ) can be expandedin two ways: 1 y − y − = ∞ (cid:88) k =0 y − (2 k +1) = − ∞ (cid:88) k =0 y k +1 . (4.13)One expansion is valid for | y | > | y | <
1. So either way, there is a set T of integers with 1 e iπ (2 η − /b − e − iπ (2 η − /b = ± (cid:88) N ∈ T e πi ( N ∓ / η − /b . (4.14)– 36 – consists of nonnegative integers if Im ((2 η − /b > η − /b <
0. We have to interpret the line Im ((2 η − /b ) = 0 as a Stokes linealong which the representation of the integration cycle as a sum of cycles associatedto critical points changes discontinuously. If b is real, the criterion simplifies and onlydepends on the sign of Im η . The sign in (4.14) has an analog for the Gamma functionand can be interpreted in terms of the orientations of critical point cycles. We will now briefly discuss how to relate this point of view to the more traditionalfixed-area technique [8]. For this section we restrict to real α ’s. We begin by definingthe fixed-area expectation value for a generic Liouville correlator as (cid:104) V α · · · V α n (cid:105) A ≡ ( µA ) ( (cid:80) i α i − Q ) /b
1Γ (( (cid:80) i α i − Q ) /b ) (cid:104) V α · · · V α n (cid:105) . (4.15)Assuming that Re( (cid:80) i α i − Q ) >
0, an equivalent formula is (cid:104) V α · · · V α n (cid:105) = (cid:90) ∞ dAA e − µA (cid:104) V α · · · V α n (cid:105) A . (4.16)With the A dependence of (cid:104) V α · · · V α n (cid:105) A being the simple power of A given on theright hand side of (4.15), the A integral in (4.16) can be performed explicitly, leadingback to (4.15). So far this is just a definition, but comparison of (4.16) to the originalLiouville path integral suggests an alternate proposal for how to compute the fixed-area expectation value: evaluate the Liouville path integral dropping the cosmologicalconstant term and explicitly fixing the physical area (cid:82) d ξe φ A = A . Semiclassically wecan do this using a Lagrange multiplier, which modifies the equation of motion: ∂∂φ A = 2 πA ( (cid:88) i η i − e φ A − π (cid:88) η i δ ( ξ − ξ i ) . (4.17)The point to notice here is that when (cid:80) i η i <
1, if we define φ c,N = iπ + 2 πiN + φ A and λ = ( (cid:80) i η i − /A , the solutions of this equation are mapped exactly into the complexsaddle points we have been discussing. One can check explicitly for the semiclassicaltwo-point function we just computed that the various factors on the right hand side of(4.15) conspire to remove the evidence of the complex saddle points and produce theusual fixed-area result [7]: (cid:104) V η/b (1 , V η/b (0 , (cid:105) A ≡ πδ (0) G A ( η/b ) ≈ πδ (0) e − b (1 − η )(log Aπ +log(1 − η ) − . (4.18) In eqn. (4.17), we set the Lagrange multiplier to the value at which the equation has a solution.To find this value, one integrates over the z -plane, evaluating the integral of the left hand side withthe help of (2.6). – 37 –istorically the proposal was to use (4.15) in the other direction, as a way to definethe Liouville correlator when (cid:80) i η i <
1, but it was unclear that this would be validbeyond the semiclassical approximation. We see now how it emerges naturally fromthe analytic continuation of the Liouville path integral.
We now move on to the three-point function. We will initially focus on two partic-ular regions of the parameter space of the variables η i , i = 1 , . . . ,
3. In what we willcall Region I, we require that (cid:80) i Re( η i ) >
1, and that the imaginary parts Im( η i ) aresmall enough that the solution (3.25) does not have singularities coming from zeroes ofthe denominator. The inequality (cid:80) i Re( η i ) > φ c from diverging at large negative φ c , as discussed above in the context of thetwo-point function. When the η i are actually real and less than 1 /
2, we get the physicalregion studied in [7], which is the only range of η i in which Liouville’s equation has realnonsingular solutions. In this sense the three-point function is a simpler case than thetwo-point function, since in that case no choice of η allowed a real integration cycle forthe path integral.We will also be interested in the region defined by0 < Re( η i ) < (cid:88) i Re( η i ) < < Re( η i + η j − η k ) ( i (cid:54) = j (cid:54) = k ) , where again the imaginary parts are taken to be small enough that there are no singu-larities from zeroes of the denominator. We will refer to this as Region II. Note that ifthe imaginary parts are all zero, we can see from (3.22) and (3.24) that a and a willbe purely imaginary in this region and there will be no singularities. The third line of(4.19) has not appeared before in our discussion; we call it the triangle inequality. Itsmeaning is not immediately clear. It is automatically satisfied when (cid:80) i Re( η i ) > η i ) < , but when (cid:80) i Re( η i ) < η ’s, a , a are imaginary. The metric − e φ c δ ab is thus well defined and has constant positivecurvature. Since we have taken the η i to be positive, the metric has three conicaldeficits. Such metrics have been studied in both the physics and math literature [36, 37],and they can be constructed geometrically in the following way. Suppose that we canconstruct a geodesic triangle on S whose angles are θ i = (1 − η i ) π . We can gluetogether two copies of this triangle by sewing the edges together, and since the edges– 38 –re geodesics they have zero extrinsic curvature and the metric will be smooth accrossthe junction. The angular distance around the singular points will be 2 θ i = (1 − η i )2 π ,and as explained in the discussion of (2.22), this is the expected behavior for a classicalsolution with the insertion of primary fields of Liouville momenta η i /b . So this givesa metric of constant positive curvature with the desired three singularities. For thisconstruction to work, we need only make sure that a triangle exists with the specifiedangles. First note that because of the positive curvature of S , we must have (cid:80) i θ i > π ,which gives (cid:80) i η i <
1. We can choose one of vertices of the triangle, say the one labeledby η , to be the north pole, and then the two legs connected to it must lie in greatcircles passing through both the north and the south pole. If we extend these legs allthe way down to the south pole, then the area between them is a “diangle,” as shownin Figure 1. The third leg of the triangle then splits the diangle into two triangles,the original one and its complement, labelled A and B respectively in the figure. Theinequality (cid:80) i θ i > π applied to the complementary triangle then gives η + η − η > B becomes smaller and smaller and the originaltriangle A degenerates into a diangle. Once the inequality is violated, no metric withonly the three desired singularities exists. AB
12 3
Figure 1 . Spherical triangles.
We will now compare the semiclassical actions of the complex saddle points (3.25) inthese two regions with the semiclassical limit of the DOZZ formula. We will see that inRegion I only the real saddle point contributes (this is expected for reasons explained insection 1.1) while in Region II, similarly to what we found for the two-point function,infinitely many contribute. We interpret this change as a Stokes phenomenon; thecondition Re( η + η + η ) = 1 separating the two regions evidently defines a Stokes– 39 –all. In Region II, we will initially assume that all three operators are heavy, but in afinal subsection we will treat the case that they are light and again find evidence for apath integral interpretation of the DOZZ formula. To evaluate the action for a saddle point contributing to the three-point function, wecan again use the trick of differentiating the action with respect to η i . So we need todetermine the asymptotic behaviour of (3.25) near z i . We denote as φ c,N the solutioncorresponding to (3.25), where the subscript N labels the possibility of shifting φ c by2 πiN . We will again have φ c,N ( z, z ) → − η log | z − z i | + C i as z → z i , (4.20)and to determine C i we again need to confront the problem of defining the logarithmof f . We will first treat Region I, where we define φ c,N ( z, z ) = 2 πiN − log λ − | z − z |− (cid:0) a P η ( x ) P η ( x ) − a P − η ( x ) P − η ( x ) (cid:1) . (4.21)The branch in the logarithm is chosen so that using (3.24) and the series expansion of P η , we find C = 2 πiN − log λ − (1 − η ) log | z | | z | | z | − log γ ( η + η − η ) γ ( η + η − η ) γ ( η + η + η − γ (2 η ) γ ( η + η − η ) . (4.22)The function φ c,N ( z, z ) that we get by continuation away from z will be singlevaluedby the same argument as for the two-point function. To find C and C , we can use theconnection coefficients B.13, B.12, but it is easier to just permute the indices to find C = 2 πiN − log λ − (1 − η ) log | z | | z | | z | − log γ ( η + η − η ) γ ( η + η − η ) γ ( η + η + η − γ (2 η ) γ ( η + η − η ) , (4.23) C = 2 πiN − log λ − (1 − η ) log | z | | z | | z | − log γ ( η + η − η ) γ ( η + η − η ) γ ( η + η + η − γ (2 η ) γ ( η + η − η ) . (4.24)– 40 –s with the two-point function, we can justify the equality of N in the vicinity ofdifferent points by observing that we may begin with real η ’s obeying (cid:80) i η i >
1, forwhich the argument of the logarithm is real and positive. We then continue to thedesired value of η on a path that remains in Region I. As before, by continuity N cannot change. As a check of this claim, we observe that paths within Region I cannotactivate the branch cuts of the logarithms in these expressions for C i . Indeed, for anyset of η i ’s which is in Region I, all of the arguments of γ ( · ) have real part betweenzero and one. γ ( · ) has no zeros or poles in this strip, so any loop in Region I can becontracted to a point without changing the monodromy of the logarithm. Thus thereis no monodromy.To compute the action, we need to integrate b ∂ (cid:101) S L ∂η i = − C i , (4.25)which gives b (cid:101) S L = (cid:32)(cid:88) i η i − (cid:33) log λ + ( δ + δ − δ ) log | z | + ( δ + δ − δ ) log | z | + ( δ + δ − δ ) log | z | + F ( η + η − η ) + F ( η + η − η ) + F ( η + η − η )+ F ( η + η + η − − F (2 η ) − F (2 η ) − F (2 η ) − F (0)+ 2 πiN (1 − (cid:88) i η i ) . (4.26)Here we have F ( η ) ≡ (cid:90) η log γ ( x ) dx, (4.27)with the contour staying in the strip 0 < Re( x ) <
1, and also δ i ≡ η i (1 − η i ) . The η i -independent constant was determined in [7] by explicitly evaluating the actionin the case (cid:80) i η i = 1, with the result b (cid:101) S L = (cid:80) i 1) = − x − γ ( x ) to make sure that when we take the η ’s to bereal (and in Region II), the only imaginary parts comes from the first term. We canagain permute to find: C =2 πi (cid:18) N + 12 (cid:19) − log λ − (1 − η ) log | z | | z | | z | + 2 log (cid:32) − (cid:88) i η i (cid:33) − log γ ( η + η − η ) γ ( η + η − η ) γ ( η + η + η ) γ (2 η ) γ ( η + η − η ) , (4.29) C =2 πi (cid:18) N + 12 (cid:19) − log λ − (1 − η ) log | z | | z | | z | + 2 log (cid:32) − (cid:88) i η i (cid:33) − log γ ( η + η − η ) γ ( η + η − η ) γ ( η + η + η ) γ (2 η ) γ ( η + η − η ) . (4.30)Finally we can again integrate this to find b (cid:101) S L = (cid:32)(cid:88) i η i − (cid:33) log λ + ( δ + δ − δ ) log | z | + ( δ + δ − δ ) log | z | + ( δ + δ − δ ) log | z | + F ( η + η − η ) + F ( η + η − η ) + F ( η + η − η )+ F ( η + η + η ) − F (2 η ) − F (2 η ) − F (2 η ) − F (0)+ 2 (cid:34) (1 − (cid:88) i η i ) log(1 − (cid:88) i η i ) − (1 − (cid:88) i η i ) (cid:35) + 2 πi ( N + 1 / − (cid:88) i η i ) . (4.31)Here we determined the constant by matching to η i = 0, which as we found before givesan action 2 πi ( N + 1 / − log λ − F ( η ). It is clear from thedefinition (4.27) that F ( η ) has branch points at each integer η . The form of the branchpoints for η = − n with n = 0 , , , . . . is − ( η + n ) log( η + n ), while for η = m with– 42 – = 1 , , . . . it is ( η − m ) log( m − η ). We thus find that the monodromy of F ( η ) aroundany loop in the η -plane is F ( η ) → F ( η ) + ∞ (cid:88) m =1 ( η − m )2 πiN m − ∞ (cid:88) n =0 ( η + n )2 πiN n , (4.32)where N n and N m count the number of times the loop circles the branch points ina counterclockwise direction. Now applying this to (4.26), we see that continuationin the η i can produce far more branches than can be accounted for by nonsingularcomplex solutions. In particular, the various nonsingular solutions can only accountfor multivaluedness of the form 2 πiN (1 − (cid:80) i η i ), while continuation around a loop inthe general η i parameter space can easily produce shifts of the action by terms suchas 2 πiN ( η + η − η ). There thus seems to be a mismatch between the branches ofthe action (4.26) and the available saddle points. One might be tempted to interpretthis multivaluedness as indicating the existence of additional solutions, but we showedin section 3.3 that there are no more solutions. We will suggest a mechanism forexplaining this additional multivaluedness in section 5, as part of our discussion of thesingularities that appear in the case of general η i , and another possible interpretationin section 6. The situation however is simpler for continuations that stay in Region Iand/or Region II. Such a continuation will only activate the branch cuts in F ( (cid:80) i η i − (cid:80) i η i = 1. We now compute the semiclassical limit of the DOZZ formula (2.28) with three heavyoperators in Regions I and II. The semiclassical behavior of the prefactor of the DOZZ formula is clear: (cid:104) λγ ( b ) b − b (cid:105) ( Q − (cid:80) i α i ) /b → exp (cid:34) − b (cid:40)(cid:32)(cid:88) i η i − (cid:33) log λ − (cid:32)(cid:88) i η i − (cid:33) log b (cid:41)(cid:35) . (4.33)To study the remaining terms, we need the b → b ( η/b ). In AppendixA, we show thatΥ b ( η/b ) = e b [ F ( η ) − ( η − / log b + O ( b log b ) ] 0 < Re( η ) < . (4.34) This computation was done in [7] in Region I with real η i , and our computations here are simpleextensions of that. – 43 –n Region I, all of the Υ b ’s have their arguments in the region of validity for this formula,so we find that they asymptote to:exp (cid:104) b (cid:110) F (2 η ) + F (2 η ) + F (2 η ) + F (0) − F ( (cid:88) i η i − − F ( η + η − η ) − F ( η + η − η ) − F ( η + η − η ) − (cid:32)(cid:88) i η i − (cid:33) log b (cid:111)(cid:105) . (4.35)Combining these two contributions, we find complete agreement with (4.31) with N =0. Thus in Region I, only one saddle point contributes and we can interpret the pathintegral as being evaluated on a single integration cycle passing through it.In Region II, the only new feature is that Υ b ( (cid:80) i α i − Q ) is no longer in the regionwhere we can apply (4.34). To deal with this, we can use the recursion relation (A.4)to move the argument back to the region where we can use (4.34):Υ b (cid:32)(cid:32)(cid:88) i η i − (cid:33) /b (cid:33) = γ (cid:32)(cid:32)(cid:88) i η i − (cid:33) /b (cid:33) − b − (cid:80) i η i − /b Υ b (cid:32)(cid:88) i η i /b (cid:33) . (4.36)Using also (4.10) for the asymptotics of the Gamma function (and hence of γ ( x ) =Γ( x ) / Γ(1 − x )), we finally arrive at C ( η i /b ) ∼ exp (cid:34) − b (cid:40) (cid:32)(cid:88) i η i − (cid:33) log λ − F (2 η ) − F (2 η ) − F (2 η ) − F (0)+ F (cid:0) (cid:88) i η i (cid:1) + F ( η + η − η ) + F ( η + η − η ) + F ( η + η − η )+ 2 (cid:104) (cid:32) − (cid:88) i η i (cid:33) log (cid:32) − (cid:88) i η i (cid:33) − (cid:32) − (cid:88) i η i (cid:33) (cid:105)(cid:41)(cid:35) × e iπ ( (cid:80) i η i − /b − e − iπ ( (cid:80) i η i − /b . (4.37)This is in complete agreement with (4.31), provided that as with the two-point functionwe interpret the final factor as coming from a sum over infinitely many complex saddlepoints. Rather as before, the saddle points that contribute are N = {− , − , . . . } when Im (( (cid:80) i η − /b ) < N = { , , , . . . } when Im (( (cid:80) i η i − /b ) > 0. Thecondition Im (( (cid:80) i η i − /b ) = 0 defines a Stokes wall.– 44 – .3 Three-Point Function with Light Operators So far we have considered only correlators where all operators are heavy. As a finalcheck we will compute the semiclassical limit of the DOZZ formula (2.28) with threethree light operators of Liiouville momenta α i = bσ i , with σ i fixed for b → 0, andcompare it with a semiclassical computation based on equation (2.26). This computionis essentially a repackaging of a fixed-area computation outlined in [7]; we include it asan additional illustration of the machinery of complex saddle points and also becausemany of the details were omitted in [7]. In section 7, we will also use the same tools todo a new check in the context of timelike Liouville, so it is convenient to first presentthem in a more familiar context.We begin by computing the asymptotics of the DOZZ formula with three lightoperators; in order to capture the nontrivial effects of the operators we need to computeto higher order in b than before. To order b in the exponent the prefactor not involvingΥ b ’s becomes (cid:104) λγ ( b ) b − b (cid:105) ( Q − (cid:80) i α i ) /b = b − /b +2 (cid:80) i σ i − λ /b +1 − (cid:80) i σ i e − γ E + O ( b log b ) (4.38)Here γ E is the Euler-Mascheroni constant γ E ≡ lim n →∞ (cid:0)(cid:80) nk =1 1 k − log n (cid:1) . To take thelimits of the Υ b functions, we need the the asymptotics of Υ b ( σb ) as b → 0. This isgiven by equation (A.10):Υ b ( bσ ) = Cb / − σ Γ( σ ) exp (cid:20) − b log b + F (0) b + O ( b log b ) (cid:21) . (4.39)Here C is a constant that will cancel in the final result. This along with (A.11) issufficient to determine the asymptotics of all parts of the DOZZ formula except for theΥ b involving (cid:80) i σ i . For this one we can use the recursion relation:Υ b (cid:104)(cid:0) (cid:88) i σ i − (cid:1) b − /b (cid:105) = γ (cid:16) (cid:88) i σ i − − /b (cid:17) − b /b − (cid:80) i σ i Υ b (cid:104)(cid:0) (cid:88) i σ i − (cid:1) b (cid:105) (4.40)To evaluate the semiclassical limit of this we need the corrections to (4.10). We canget these by using the machinery of Appendix C, but we can simplify the discussionusing Euler’s reflection formula Γ( x )Γ(1 − x ) = π sin( πx ) : γ ( x − /b ) = π Γ(1 − x + 1 /b ) sin (cid:0) π ( x − /b ) (cid:1) (4.41)The Γ function appearing on the right hand side of this equation always has positivereal part as b → 0, so we can simply include the first subleading terms in Stirling’sformula to find Γ(1 − x + 1 /b ) = √ πb − /b +2 x − e − /b (1 + O ( b )) . (4.42)– 45 –his then gives γ ( (cid:88) i σ i − − /b ) = ie iπ ( (cid:80) i σ i − − /b ) − e − iπ ( (cid:80) i σ i − − /b ) b /b − (cid:80) i σ i +6 e /b (1 + O ( b )) . (4.43)Combining all these results together we can write: C ( σ b, σ b, σ b ) = ib − λ /b +1 − (cid:80) i σ i e /b − γ E + O ( b log b ) e iπ ( (cid:80) i σ i − − /b ) − e − iπ ( (cid:80) i σ i − − /b ) × Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ + σ − σ )Γ(2 σ )Γ(2 σ ) . (4.44)We now compare this result to an appropriate refinement of (2.26). There areseveral subtleties to consider. With all operators light the appropriate saddle pointis the sphere (4.7). As with the saddle point (3.17) for the two-point function withheavy operators there will be a moduli space of such solutions, in this case givenby the quotient SL (2 , C ) /SU (2), since the subgroup of SL (2 , C ) that leaves fixed aparticular round sphere metric is a copy of SU (2). The light operator insertions willdepend explicitly on these moduli, so we need the general SL (2 , C ) transformation ofthe saddlepoint (4.7). From (2.2) this is given by φ c,N ( z, z ) = 2 πi ( N + 1 / − log λ − (cid:0) | αz + β | + | γz + δ | (cid:1) , (4.45)with α, β, γ, δ ∈ C and obeying αδ − βγ = 1. In using (2.26) we will need to integratethe right hand side over all such saddlepoints.An additional subtlety is that in (2.26) all effects of the operator insertions are O ( b ) in the exponent. To precisely include all effects of this order, we would need tocarefully compute the renormalized fluctuation determinant about each saddle point,and also include the O ( b ) corrections to the action (2.10). Moreover we would needthe Jacobian in transforming the integral over φ c into an integral over the parameters α, β, γ, δ . We will include the subleading terms in the action explicitly, but to simplifythings we will represent the fluctuation determinant and Jacobian as a b -dependentprefactor A ( b ) which is at most O (log b ) in the exponent. Note that neither of thesethings should be affected by shifting the saddlepoint by 2 πi so we expect A ( b ) to beindependent of N . It is also independent of σ i since neither effect has anything to do We do NOT need to include O ( b ) corrections to the saddlepoint (4.45) even though they arepresent. The reason is that the leading order saddlepoints are stationary points of the leading orderaction, so perturbing the solution at O ( b ) does not affect the action until O ( b ), which is beyond ourinterest. – 46 –ith the operator insertions. With this convention, we can now write a more preciseversion of (2.26) that is appropriate for comparison with (4.44): (cid:104) V bσ ( z , z ) V bσ ( z , z ) V bσ ( z , z ) (cid:105) ≈ A ( b ) (cid:88) N ∈ T e − S L [ φ c,N ] (cid:90) dµ ( α, β, γ, δ ) (cid:89) i =1 e σ i φ c,N ( z i ,z i ) . (4.46)Here T is some set of integers and dµ ( α, β, γ, δ ) = 4 δ ( αδ − βγ − d α d β d γ d δ is the invariant measure on SL (2 , C ) [7]. The integrals over over the full α, β, . . . planes.The O ( b ) correction to the action (2.10) is given by π (cid:72) ∂D φ c dθ + 4 log R . For thesaddle point (4.45) the leading part was computed above (4.8), and now including thesubleading term we find S L [ φ c,N ] = 1 b (cid:104) πi ( N + 1 / − log λ − (cid:105) + 2 πi ( N + 1 / − log λ + O ( b ) . (4.47)The integral over the moduli is quite difficult, we will simplify it some here and thenrelegate the final computation to an appendix. Our technique is identical to that in[7]. We first note that (cid:90) dµ ( α, β, γ, δ ) (cid:89) i =1 e σ i φ c,N ( z i ,z i ) = λ − (cid:80) i σ i e πi ( N +1 / (cid:80) i σ i × (cid:90) dµ ( α, β, γ, δ ) (cid:16) | αz + β | + | γz + δ | (cid:17) σ (cid:16) | αz + β | + | γz + δ | (cid:17) σ (cid:16) | αz + β | + | γz + δ | (cid:17) σ . (4.48)The position dependence of this integral can be extracted by using its SL (2 , C ) trans-formation properties; changing variables by the transformation which sends z → z → 1, and z → ∞ we find the usual three-point function behaviour (cid:90) dµ (cid:89) i =1 e σ i φ c,N = λ − (cid:80) i σ i e πi ( N +1 / (cid:80) i σ i | z | σ − σ − σ ) | z | σ − σ − σ ) | z | σ − σ − σ ) I ( σ , σ , σ )(4.49)with I ( σ , σ , σ ) ≡ (cid:90) dµ ( α, β, γ, δ ) (cid:16) | β | + | δ | (cid:17) σ (cid:16) | α + β | + | γ + δ | (cid:17) σ (cid:16) | α | + | γ | (cid:17) σ . (4.50)– 47 –he result of this integral was quoted in [7], but many steps were omitted and the fullevaluation is quite sophisticated. For completeness we have included a full derivationin Appendix F. The result is I ( σ , σ , σ ) = π Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ + σ − σ )Γ(2 σ )Γ(2 σ ) . (4.51)Using this along with (4.47) and (4.49), we find that (4.46) gives C ( σ i b ) ≈ π A ( b ) λ /b +1 − (cid:80) i σ i e /b (cid:88) N ∈ T e πi ( N +1 / (cid:80) i σ i − − /b ) × Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ + σ − σ )Γ(2 σ )Γ(2 σ ) (4.52)Comparing this with the DOZZ asymptotics (4.44) we find complete agreement, withthe saddle points included depending on the sign of Im( (cid:80) i σ i − /b ). We also seethat apparently A ( b ) = iπ − b − e − γ E , which would be interesting to check by explictlytreating the measure. That it is imaginary is unsurpising given the complex integrationcycle. This concludes our argument that the analytic continuation of the DOZZ formulain Regions I and II is described by the Liouville path integral evaluated on a complexintegration cycle that changes as we cross Stokes lines. The behaviour is completelyanalogous to that of the Gamma function as described in Appendix C. This has aqualitative explanation that was explained in section 1.1.1. The integral representationof the Gamma function is the zero mode part of the Liouville path integral, and thecomplex saddle points that we studied for Regions I and II differed only by shifting thezero mode. What we learned in this section is that in Regions I and II there are noadditional subtleties in the analytic continuation in η i beyond those that are alreadyapparent in the zero mode. We now confront the issue first raised in section 3.3: for most complex values of the η i , there are no nonsingular solutions of Liouville’s equation with the desired boundary– 48 –onditions. The candidate solution (3.25) fails to be a solution because of zeroes of thedenominator function f ( z, z ) = A ( z − z ) + B ( z − z ) + . . . . (5.1)At such a zero, φ c = − f − log λ is singular, and perhaps more seriously, it is alsogenerically multivalued. Around a zero of f with winding number k , φ c changes by − πik .This seems to raise a serious challenge to any attempt to interpret the full analyticcontinuation of the DOZZ formula in terms of conventional path integrals. In thissection we will study this further. We will make three arguments that even when φ c ismultivalued, the expression (3.25) still makes some sense and controls the asymptoticbehaviour of the DOZZ formula. We will first show that there is a minor redefinition ofthe action which agrees with the formula (2.25) when there are no singularities but isfinite even in the presence of zeroes of the denominator. Moreover it correctly producesthe analytic continuation of (2.25). We will then show that the presence of singularitiesactually allows the full multivaluedness of the action (4.31) to be realized by analyticcontinuation of the “solutions.” Finally we will probe the saddle points that dominatethe three-point function by including a fourth light operator. For the case that we areable to implement this test – the case that the light operator is degenerate – we willfind agreement with the (3.25) for all values of the η i . We will close by commenting onthe implications for general four-point functions. We begin by observing that in Region I defined at the beginning of subsection4.2, we included a restriction on the imaginary parts of the η i ’s to ensure that thedenominator in (3.25) did not vanish away from the operator insertions. However, theformulas that followed seemed to know nothing about this additional restriction; themultivaluedness in the expressions for C i and (cid:102) S L cannot be activated without violatingthe conditions Re ( (cid:80) i η i ) < η i < , regardless of the imaginary parts of the η i ’s. Moreover the expression (4.31) for the action can easily be continued to valuesof η i where the denominator vanishes, and its value is perfectly finite there. This isperhaps unexpected because near a zero of the denominator, one has φ c ( z, z ) ≈ − A ( z − z ) + B ( z − z )] , (5.2)which has a logarithmic singularity as well as a branch cut discontinuity. With suchdiscontinuous behavior, the kinetic term in the Liouville action (cid:82) d ξ∂ a φ c ∂ a φ c certainly We consider the case that | A | (cid:54) = | B | , which is generically true for complex η ’s. When the η i ’s arereal and a is also real then we can have | A | = | B | , we will comment on this below. – 49 –iverges. The finite analytic continuation of the action therefore cannot be computedby naive application of (2.25).We begin by observing that for solutions with no additional singularities we canrewrite (2.25) as b (cid:101) S L = 1 π (cid:90) D −∪ d i d ξ (cid:2) ∂f ∂f /f + 1 /f (cid:3) + boundary terms . (5.3)As before, the d i are small discs centered around z i . We propose that even in thepresence of zeroes of the denominator of e φ c , this is still the correct form of the action,with the integral defined by removing a small disc of radius (cid:15) centered around eachzero and then taking (cid:15) → 0. The divergence from the discontinuity in φ c is avoidedsince f is continuous, but we still need to show that there is no divergence as (cid:15) → 0. Inparticular near a zero at z = z , we have the expansion (5.1), so we can approximatethe contribution to the integral from the vicinity of z as1 π (cid:90) (cid:15) drr (cid:90) π dθ AB + 1( Ae iθ + Be − iθ ) . (5.4)The radial integral is logarithmically divergent, but as long as | A | (cid:54) = | B | the angularintegral is zero! The higher order corrections to f will produce manifestly finite correc-tions to the action, and in fact one can show that this definition of the action is invariantunder coordinate transformations of the form z → z + O ( z ). This is thus analogousto the principal value prescription for computing the integral of 1 /x across x = 0. Weclaim that the action computed this way agrees with what one gets by analytic contin-uation in η i . To justify this, we need to show that we can continue to use the trick ofdifferentiating with respect to η i to calculate the action. This requires a demonstrationthat a multivalued “solution” is a stationary point of the improved action. To showthis we can compute the variation of the improved action under f → f + δf with δf continuous; most terms are clearly zero when evaluated on a multivalued “solution,”but a potentially nontrivial boundary term is generated by the integration by parts:∆ (cid:101) S L = − (cid:15) πb (cid:90) π dθ ∂ r ff δf (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( z − z )= (cid:15)e iθ . (5.5)For intuition, we observe that this boundary term is also present near each of theoperator insertions. Near the operator at z i , we have f ∼ r η i , and the boundary termproduces a nontrivial variation − η i (cid:15) η δf . This variation is cancelled by the variation − η i π (cid:82) π dθφ c of the regulated operator. The point however is that for f obeying (5.1),this boundary term is automatically zero by itself since the angular integral vanishes.So in this sense, a multivalued “solution” is a stationary point of the action.– 50 –or orientation, perhaps we should mention that a singlevalued φ c with singular-ities away from the operator insertions can never be such a stationary point. Indeed,generalities about elliptic differential equations ensure that a solution of the complexLiouville equations is smooth away from operator insertions. In the singular case, it isonly because φ c is multivalued that it may be, in some sense, a stationary point of theaction.With this explanation of what the action means in the presence of singularities,we may drop the conditions on the imaginary parts of η i from both Regions I and IIand the story of the previous section goes through unchanged. This argument does failin the special cases where | A | = | B | , for which higher order terms near the singularityare important and the singularity may be non-isolated. We will view this just as adegenerate limit of the more general situation. In particular we can continue fromRegion I to anywhere else in the η i -plane without passing through a configuration witha singularity with | A | = | B | , so this subtlety should not affect our picture of the analyticcontinuation of (2.28).Before we move on, we observe that there are two different kinds of multivaluednessbeing discussed in this section. One is with respect to η i , and the other is with respectto z, z . For convenience we summarize the multivaluedness of various quantities in thefollowing table: z, z behaviour at fixed η i η i behaviour at fixed z, ze φ c singlevalued singlevalued b (cid:102) S L trivial defined up to addition of2 πi ( (cid:80) i η i m i + n ) with m i alleven or all odd C i trivial defined up to addition of 2 πia trivial defined up to multiplicationby a sign a /a trivial singlevalued f singlevalued defined up to multiplicationby a z, z -independent sign φ c possibly singlevalued, possi-bly monodromy of additionof 4 πi about points where f = 0 defined up to addition of 2 πi – 51 – .2 Multivaluedness of the Action We saw in subsection 4.2 that the action (4.26) is highly multivalued as a functionof the η i , with the multivaluedness arising from the function F ( η ) defined in (4.27).We can now interpret this multivaluedness of the action as a consequence of themultivaluedness in z, z that φ c acquires in the presence of zeroes of f . This multi-valuedness does not affect the kinetic and potential terms of the action as defined insection 5.1, since they depend only on f , which is singlevalued as a function of z, z .But the terms − (cid:80) i η i π (cid:82) π dθφ c that come from the regulated operator insertions aresensitive to this multivaluedness. Their contribution to the action is∆ (cid:101) S L = − b (cid:88) i η i C i (5.6)where C i is the constant term in φ c near the operator insertion. Using the formulas(4.22)-(4.24) for C i , we see that continuing along a closed path in the parameter spaceof the η i can shift C i by an integer multiples of 2 πi , hence shifting the action by aninteger linear combination of the quantities 2 πiη i . We can see the same effect in theformula (4.26) for the action; the same processes that cause a shift in the C i cause anequivalent shift in the function F in this formula, leading to the same multivaluedness.For example, on a path on which η + η − η circles around an integer value, shifting C and C by 2 πi and C by − πi , there is a corresponding shift in the action from F ( η + η − η ).It is important to note that it is only because φ c can be multivalued as a functionof z, z that we can realize the full multivaluedness of the action in η . We argued belowequation (4.24) that any continuation in η i that passes only through continuous φ c ’scannot produce monodromy for the difference of any two C i ’s because of continuity.But once we allow paths in η i that pass through multivalued (and thus discontinuous) φ c ’s, these differences can have the nontrivial monodromy necessary to produce thefull set of branches of the action. Thus the multivaluedness of the action in η i has anatural interpretation once we allow solutions of the complex Liouville equations thatare multivalued in z, z . We are finally ready to consider in general the semiclassical asymptotics of theDOZZ formula (2.28). The DOZZ formula is constructed from the function Υ b ( η/b ),where η is a linear combination of the η i . In all, seven Υ b functions appear in thenumerator or denominator of the DOZZ formula. To evaluate the small b asymptoticsof this formula, one needs the small b asymptotics of the Υ b functions. This is given in– 52 –A.7) for η in a certain strip in the complex plane; it can be determined in general byusing the recursion relations (A.3) to map η into the desired strip. In the process, therecursion relation generates a function that can be expanded as a sum of exponentials,as in (4.14); we interpret this as a sum over different complex critical points.For generic η i , when evaluating the asymptotics of the DOZZ formula using theasymptotic formula (A.7), we will need to apply the recursion relations to all of theΥ b ’s. There is just one crucial difference from the derivation of eqn. (4.37). The finalfactor in that formula has an expansion in positive or negative powers of exp(2 πi (cid:80) i η i ),where (cid:80) i η i entered because in that derivation, we had to apply the recursion relationonly to one of the Υ b functions, namely Υ b ( (cid:80) i η i /b ). In general, we have to allow forthe possibility that the argument of any one of the seven Υ b functions in the DOZZformula may leave the favored strip. So (cid:80) i η i may be replaced by the equivalentexpression appearing in any one of the other Υ b functions, namely 2 η , η + η − η , orany permutation thereof.In the process, it is not quite true that the action can be shifted by 2 πi (cid:80) i m i η i forarbitrary integers m i . Rather, the m i are either all even or all odd. This holds becausesimilarly the Υ b functions in the DOZZ formulas are all functions of (cid:80) i c i η i /b , wherethe c i are all even (the factors in the numerator of the DOZZ formula) or all odd (thefactors in the denominator). One interesting point about this is that for some values of the η i , singlevalued complexsolutions of Liouville’s equations do exist. But even in such regions, we may need to usethe recursion relations to compute the asymptotics of the DOZZ formula, and hencewe seem to need the full multivaluedness of the action, even though from the presentpoint of view this multivaluedness seems natural only when the classical solutions arethemselves multivalued. The reason that this happens is that in continuing in η i fromRegion I to these regions we necessarily pass through regions where φ c is multivalued in z . When we arrive at the region of interest it is then possible that although a continuoussingle-valued solution exist we have actually landed on a discontinuous one. The For a simple example of this phenomenon, consider the function h ( x, x, η ) = log (cid:18) | x | + η | x − | (cid:19) . For η real and positive we can define the branch of the logarithm so that h is a continuous functionwith an ambiguity of an overall additive factor of 2 πiN . But if we choose such a branch and then ateach point x continue in η around a circle containing η = 0, this will produce a shift of 2 πi near x = 1but not near x = 0; the resulting function will thus be discontinuous even though a continuous choiceof branch exists. – 53 –ocations and strengths of these discontinuities will depend on the path in η . Thisallows the full multivaluedness of the action to be realized, since the discontinuities willnot affect the kinetic term when written in terms of f but they will allow independentshifts of φ c by 2 πiN near the operator insertions and infinity.These discontinuities are admittedly unsettling so we note here that in section 6,we explain a different point of view in which the full multivaluedness of the action isequally natural for any values of the η i . The previous two arguments for the role of multivalued “solutions” in the Liouvillepath integral were rather indirect. We give here a more direct argument. In section2.4, we reviewed Teschner’s formula (2.44) for the exact four-point function of a lightdegenerate field V − b/ with three generic operators V α i . This expression is meromorphicin α i , and choosing all three α i ’s to scale like 1 /b we can study its semiclassical limitfor any values of the η i . Moreover we can compare this to (2.26), which says that inthe semiclassical limit this correlator can be evaluated by replacing the operator V − b/ by the function exp( − bφ ) = exp( − φ c / φ c is the saddle point determined bythe three heavy operators. If there are several relevant saddle points φ c,N , N ∈ T , withaction (cid:101) S L,N , then (2.26) gives (cid:10) V η /b ( z , z ) V η /b ( z , z ) V − b/ ( z , z ) V η /b ( z , z ) (cid:11) ≈ (cid:88) N e − φ c,N ( z ,z ) / e − (cid:101) S L,N . Using the definitions (2.27) and (2.32) and also (3.2), this implies G ( x, x ) ≈ √ λ | z || z || z || z | (cid:88) N f N ( z , z ) e − (cid:101) S ,N . (5.7) (cid:101) S ,N is a branch of (4.26) without its position-dependent terms, with the branchlabelled by N , and with the replacement η → η . Explicitly: b (cid:101) S ,N = ( η + η + η − 1) log λ + F ( η + η − η ) + F ( η + η − η ) + F ( η + η − η )+ F ( η + η + η − − F (2 η ) − F (2 η ) − F (2 η ) − F (0)+ 2 πi ( n + m η + m η + m η ) . (5.8) As discussed below (2.26), we have omitted z -independent factors that are O ( b ) in the exponent.These come from the functional determinant and corrections to the action (2.25). These factors willcancel between the two sides in (5.14) below. – 54 –ere n, m i are integers determined by the branch N . We saw in section 3.3 that e φ c,N = 1 /f N is uniquely determined (independent of N ), so f N is uniquely determinedup to sign. (The sign comes from the choice of square root in defining a .) By comparingthis with the semiclassical limit of Teschner’s formula (2.44), we can thus explicitlycheck the position dependence of the saddle point f ( z, z )!Rewriting Teschner’s proposal (2.44) with a condensed notation, we have G ( x, x ) = C − C − (cid:20) F − ( x ) F − ( x ) + C + 12 C C − C − F + ( x ) F + ( x ) (cid:21) . (5.9)Teschner’s recursion relation can be rewritten as C C + 12 C − C − = − − b (2 α − b )) × γ (2 b (2 α − γ ( b ( α + α − α − b/ γ ( b ( α + α − α )) γ ( b ( α + α − α − b/ γ ( b ( α + α + α − Q − b/ , (5.10)and using (3.22) and (3.23) and taking the semiclassical limit this becomes C C + 12 C − C − → − a a . (5.11)Here a and a are the constants in the semiclassical solution (3.25), with the replace-ment η → η . In the same limit, we can see from (3.21) that F + ( x ) → P − η ( x ) F − ( x ) → P η ( x ) . (5.12)In checking this, it is useful to recall that we can send η → − η in the definitionof P − η ( x ) without changing the function since this is one of Kummar’s permutationsfrom Appendix B. We thus find that in the semiclassical limit we have G ( x, x ) = C − (cid:20) P η ( x ) P η ( x ) − a a P − η ( x ) P − η ( x ) + O ( b ) (cid:21) . (5.13)With the help of (3.25), we find that this will agree with (5.7) if e − (cid:101) S − ,N = a ,N √ λ | z || z || z | e − (cid:101) S ,N + O ( b ) . (5.14) In deriving this formula, we neglected O ( b ) terms in the exponent of C − . These are the sameterms that we previously neglected on the right-hand side of (5.7), since the difference between η and η − b/ O ( b )in the exponent. – 55 –eginning with this equation we explictly include the branch dependence of a for therest of the section. Semiclassically the structure constants C and C − are in thesame region of the η i plane since their η i ’s differ by something that is O ( b ), so wecan assume they are both a sum over the same set of branches N . This justifies ourequating the sums term by term in (5.14). Using (5.8), we see that: (cid:101) S ,N − (cid:101) S − ,N = 12 log λ + iπm + 12 (cid:104) log γ ( η + η − η ) + log γ ( η + η − η )+ log γ ( η + η + η − − log γ ( η + η − η ) − γ (2 η ) (cid:105) + O ( b ) . (5.15)Comparing with (3.24), we see that (5.14) is clearly satisfied up to an overall branch-dependent sign.To see that this sign works out, we need to give a more careful argument. First wecan define a ,N = | z || z || z | exp (cid:104) log γ ( η + η − η ) + log γ ( η + η − η ) + log γ ( η + η + η − − log γ ( η + η − η ) − γ (2 η ) + iπ (cid:101) m (cid:105) . (5.16)The logarithms are defined by continuation from real η ’s in Region I along a specificpath, which gives an unambiguous meaning to (cid:101) m . The signs will match in (5.14) if m = (cid:101) m . To demonstrate this, recall that near z we may write φ c,N = − η log | z − z | + C ,N , (5.17)with C ,N = − πim − log λ − (1 − η ) log | z | | z | | z | − log γ ( η + η − η ) − log γ ( η + η − η ) − log γ ( η + η + η − γ ( η + η − η ) + 2 log γ (2 η ) . (5.18)Here the logarithms are defined by analytic continuation along the same path as indefining a ,N . Since ∂ (cid:101) S L,N ∂η = − C ,N , we are justified using m in this formula. Finallynear z = z we have e − φ c,N / ≡ √ λf N = | z − z | − η e − C ,N [1 + O ( | z − z | )] , (5.19)so in (5.7) we should choose the branch of f N , and thus of a ,N , with (cid:101) m = m .This completes our demonstration of (5.14). We consider this to be very strongevidence that at least for the case of the degenerate four-point function, the Liouvillepath integral is controlled by singular “solutions” throughout the full η i three-plane. It does not matter what the path is, but we need to choose one. – 56 – .5 Four-Point Function with a General Light Operator The discussion of the previous section showed that a certain type of four-point func-tion is semiclassically described by singular “solutions” of Liouville’s equation. Morespecifically, the nontrivial position dependence of the correlator (5.7) was captured bythe function f N ( z , z ). The effect of the singularities is rather benign, however; thecorrelator simply has nontrivial zeros as a function of the position of the light operator.As argued at the end of section 3, the zeros of f N are generically stable under quantumcorrections and thus are actually zeros of the exact four-point function (2.44). There isnothing inherently wrong with such zeros, but this observation is troubling nonetheless.The reason is that these zeros are smooth only because the light operator is exactlydegenerate. If instead of the operator e − φ c / we had considered a more general lightoperator e σφ c , then a semiclassical computation based on equation (2.26) (the otherthree operators are still heavy) would have given G ( x, x ) ≈ G λ − σ | z | σ | z | σ | z | σ | z | σ (cid:88) N f N ( z , z ) − σ e − (cid:101) S ,N . (5.20)Here G is a O ( b ) factor from the fluctuation determinant and the corrections tothe action, both of which we expect to be independent of z , and (cid:101) S ,N is given by(5.8). The problem however is that in the vicinity of a point z where f N ( z , z ) ≈ A ( z − z ) + B ( z − z ), this correlator is generically singular and discontinuous! Wecan quantify the nature of these singularities by using the winding number introducedat the end of section 3, and we find that the semiclassical correlator has winding number − σ around z if | A | > | B | and winding number 2 σ if | A | < | B | . The winding number isnot an integer because the function is discontinuous. It cannot be changed significantlyby small corrections, and since it is generically nonzero we are tempted to concludethat the exact four-point function must also be discontinuous as a function of the lightoperator position at finite but sufficiently small b ! This situation would not be entirely without precedent; in the SL (2 , R ) WZNWmodel appropriate for studying strings in AdS [38–41] it was shown in [42, 43] that One might hope that the discontinuity could cancel in the sum over the different branches N , butthis will not work because for any given generic values of η , η , η , σ there will be a single dominantsaddlepoint that is parametrically larger as b → In general it is of course possible for a smooth function to have a semiclassical approximationwhich is discontinuous, a simple example is x/λ ) , which has a line of zeros turn into a branch cut as λ → 0. A more sophisticated example that we have been studying extensively in this paper is Υ b ( x/b ),which exhibits the same phenomenon. That this does not happen for the four-point function underconsideration is a special consequence of the semiclassical formula (5.20) for the correlator, where thenontrivial z -dependence is all in a factor that is finite as b → e − /b is independent of z . – 57 –he exact 4-point function of certain operators has singularities when all four operatorsare at distinct positions. This could be seen semiclassically from stringy instantonsgoing “on-shell” and was reproduced exactly using the machinery of the Knizhnik-Zamolodchikov equation [44]. In that situation however the singularities were localizedto isolated points and the correlator was continuous away from those points. In theremainder of this section we will give an argument that in Liouville there are in fact nosingularities, isolated or otherwise, in the exact four-point function when the operatorpositions do not coincide. We will then close the section with some speculation aboutwhere our semiclassical argument goes wrong. We caution however that we will usesome plausible pieces of lore that have not strictly been proven, so our argument isslightly heuristic.We take as a starting point the exact formula (2.39) for the Liouville 4-point func-tion, which we reproduce here for convenience G ( x, x ) = 12 (cid:90) ∞−∞ dP π C ( α , α , Q/ iP ) C ( α , α , Q/ − iP ) × F (∆ i , ∆ P , x ) F (∆ i , ∆ P , x ) . (5.21)This formula is strictly true only when Re( α + α ) > Q/ α + α ) > Q/ x, x . The first is singularities of the Virasoro conformal blocks F as afunction of x , and the second is possible divergence of the integral over P for particularvalues of x . We will address each of these issues, beginning with possible singularitiesof the conformal blocks.The conformal blocks are expected to have branch points at x = 0 , , ∞ , whichcorrespond to the UV singularities of the correlator when the operator at z approachesthe operators at z , z , or z . The singularity at x = 0 is manifest from the definition(2.40), and the singularity at x = 1 arises from the nonconvergence of the series in(2.40) when | x | = 1. When all operator weights are real and positive the fact thatthe radius of convergence of this series is indeed one follows from the convergence ofinserting a complete set of states in unitary quantum mechanics. The convergencefor generic complex operator weights has actually never been proven in the literature,although it was conjectured to be true in [45] and discussed more recently in [30, 46].In [46] it was proven that if the radius of convergence is indeed one, then there are noother singularities with | x | > c → ∞ limit of the conformal block, which turns out to meanincluding only descendants of the form ( L − ) n | Q/ iP (cid:105) in the sum in (2.40), can beevaluated explictly from the definition and gives [45, 47]lim c →∞ F (∆ i , ∆ P , x ) = x ∆ P − ∆ − ∆ F (∆ P + ∆ − ∆ , ∆ P + ∆ − ∆ , P , x ) . (5.22)As discussed in Appendix B, this hypergeometric function is singular only at x =0 , , ∞ . So any additional singularities of F would have to disappear in the c → ∞ limit, which seems unnatural. When c is finite but one of the external legs is degeneratewe can again compute the conformal block, with result (2.48). Again it only hassingularities in the expected places. In 27 years of studying these functions as far as weknow no evidence has emerged for singularities at any other points in x , so from nowon we assume that they do not exist.The other possible source of singularities in the four point function (5.21) is diver-gence of the integral over P . To study this further, we need large P expressions bothfor the structure constants and the conformal blocks. The appropriate asymptotics forΥ b are quoted (with some minor typos we correct here) as equation 14 in [30]:log Υ b ( x ) = x log x − x ∓ iπ x + O ( x log x ) Im x → ±∞ . (5.23)We will not derive this, but it isn’t hard to get these terms from our expression (4.34)for the semiclassical limit of Υ b with x scaling like 1 /b . Using this in (2.28), we findthat at large real P we have C ( α , α , Q/ ± iP ) = 16 − P + O ( P log P ) . (5.24)The structure of the conformal blocks at large P was studied by Al. B. Zamolodchikovin a series of papers [48, 49], ; he obtained the following remarkable result: F (∆ i , ∆ , x ) =(16 q ) ∆ − c − x c − − ∆ − ∆ (1 − x ) c − − ∆ − ∆ × θ ( q ) c − − +∆ +∆ +∆ ) (1 + O (1 / ∆)) . (5.25) To do this, we observe that for large η we can use Stirling’s formula to approximate log γ ( x ) insidethe integral expression (4.27) for F ( η ). This is not quite the same as a full finite- b derivation since inprinciple there could be subleading terms in b that become important for sufficiently large η , but wehave checked this formula numerically at finite b with excellent agreement so apparently this does nothappen. The formula (4.34) was valid only for η in a certain region, but using the recursion relationsto get to other regions will not affect things to the order we are working in (5.23) so (5.23) is valid forarbitrary Re( x ). English translations are availiable online but hard to find. This also especially the case for reference[45], which gives a beautiful exposition of the general formalism of [17] that is more complete thananything else in the literature. The most accessible place to find the formula quoted here seems to bein section 7 of [7], but beware of a notational difference in that our conventions are related to theirsby 1 ↔ – 59 –ere θ ( q ) = (cid:80) ∞ n = −∞ q n and q = exp [ − πK (1 − x ) /K ( x )], with K ( x ) = 12 (cid:90) dt (cid:112) t (1 − t )(1 − xt ) . (5.26)This q can be interpreted as exp( iπτ ), where τ is the usual modular parameter of theelliptic curve y = t (1 − t )(1 − xt ). So in particular Im τ is always positive and onealways has | q | < x (cid:54) = 0 , , ∞ ). Forfun we note that, like most things in this paper, K ( x ) is actually a hypergeometricfunction: equation (B.18) gives K ( x ) = π F (1 / , / , , x ).The derivation of (5.25) uses certain reasonable assumptions about the semiclassicallimits of correlation functions; we will be explicit about them in Appendix D, wherefor convenience we review the origin of the leading behaviour F (∆ i , ∆ P , x ) ∼ (16 q ) P . (5.27)This will be sufficient for our study of the integral in (5.21); combining it with (5.24)we see that the integral will converge, for x (cid:54) = 0 , , ∞ , given the fact that | q | < 1. Thusthe integral cannot generate any new singularities. This completes our argument thatthe Liouville four-point function (5.21) cannot have any new singularities in x .So what is wrong with our semiclassical argument for such singularities in thebeginning of the section? To really understand this we would have to compute thesemiclassical limit of (5.21) and compare it to our formula (5.20). For the moment, thisis beyond our ability. We may guess however that the problem lies in our assumptionthat the factor G is independent of z . This was true for the degenerate computationin the previous section, but the singularity we discovered here perhaps suggests thatmore sophisticated renormalization of the nondegenerate light operator is required inthe vicinity of any singular points of the “solution”. It is at first unsettling that therenormalization of the operator should depend on the positions and strengths of theother heavy operators, but we already saw in section 5.1 that even the “principal value”prescription for evaluating the action depended on these things at distances arbitrarilyclose to the singular point. Thus we expect that once an appropriate renormalizationis performed, the semiclassical singularity in (5.20) will be smoothed out. It would begood to be more explicit about what this renormalization is, but we will not try to doso here.A different perspective on this four-point function is provided by the Chern-Simonsformulation of Liouville theory, which we will introduce momentarily. In this formu-lation it seems clear that there are conventional nonsingular solutions that exist forany η i and which can be used to study the semiclassical limit of this correlator; in thisversion of things it seems apparent that no singularity can emerge.– 60 – Interpretation In Chern-Simons Theory In section 3.1, to a solution of Liouville’s equations we associated a holomorphicdifferential equation (cid:18) ∂ ∂z + W ( z ) (cid:19) f = 0 (6.1)and also an antiholomorphic differential equation (cid:18) ∂ ∂z + (cid:102) W ( z ) (cid:19) f = 0 . (6.2)Locally, (6.1) has a two-dimensional space of holomorphic solutions, and (6.2) hasa two-dimensional space of antiholomorphic solutions. We constructed a solution ofLiouville’s equation from a basis (cid:18) uv (cid:19) of holomorphic solutions of (6.1) along witha basis (cid:18)(cid:101) u (cid:101) v (cid:19) of antiholomorphic solutions of (6.2). This construction applies on anyRiemann surface Σ, though we have considered only S in the present paper.Globally, in passing around a noncontractible loop in Σ, or around a point at whichthere is a singularity due to insertion of a heavy operator, the pair (cid:18) uv (cid:19) has in generalnon-trivial monodromy. The monodromy maps this pair to another basis of the sametwo-dimensional space of solutions, so it takes the form (cid:18) uv (cid:19) → (cid:18)(cid:98) u (cid:98) v (cid:19) = M (cid:18) uv (cid:19) , (6.3)where M is a constant 2 × M is 1, so M takes values in SL (2 , C ). One way to prove this is to use the fact that the Wronskian u∂v − v∂u is independent of z , so it must have the same value whether computed in thebasis u, v or the basis (cid:98) u, (cid:98) v . This condition leads to det M = 1. Alternatively, we mayobserve that the differential equation (6.1) may be expressed in terms of an SL (2 , C )flat connection. We introduce the complex gauge field A defined by A z = (cid:18) − W ( z ) 0 (cid:19) , A z = 0 . (6.4)Since these 2 × A as a connection with gaugegroup SL (2 , C ). On the other hand, a short calculation shows that the condition for Our convention for non-abelian gauge theory is that D µ = ∂ µ + A µ , so in particular F µν =[ D µ , D ν ] = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] and the gauge transformation is D µ → gD µ g − , with g ∈ G . – 61 – pair (cid:18) fg (cid:19) to be covariantly constant with respect to this connection is equivalent torequiring that f is a holomorphic solution of the equation (6.1) while g = ∂f /∂z . Thusparallel transport of this doublet around a loop, which we accomplish by multiplyingby U = P e − (cid:72) A z dz , is the same as analytic continuation around the same loop. Inparticular if we define the matrix S = (cid:18) u v∂u ∂v (cid:19) , then we have U S = SM T . Takingthe determinant of this equation, we find that M ∈ SL (2 , C ).Similarly, the antiholomorphic differential equation (6.2) has monodromies valuedin SL (2 , C ). This may be proved either by considering the Wronskian or by introducingthe corresponding flat connnection (cid:101) A , defined by (cid:101) A z = 0 , (cid:101) A z = (cid:32) (cid:102) W ( z ) − (cid:33) . (6.5)(It is sometimes convenient to take the transpose in exchanging z and z , and we havedone so, though this will not be important in the present paper.)The connections A and (cid:101) A have singularities near points with heavy operator in-sertions. The monodromy around these singularities can be inferred from the local be-havior of the solutions of the differential equation. For example, the solutions of (6.1)behave as z η and z − η near an operator insertion at z = 0 with Liouville momentum α = η/b . The monodromies of these functions under a circuit in the counterclockwisedirection around z = 0 are exp( ± πiη ). An invariant way of describing this, withoutpicking a particular basis of solutions, is to say thatTr M = 2 cos(2 πη ) . (6.6)Similarly, the behavior of the local solutions near z = 0 implies that the monodromy ofthe antiholomorphic equation (6.2) around z = 0 has the same eigenvalues, and henceagain obeys (6.6).More generally, the two flat connections A and (cid:101) A are actually gauge-equivalent and have conjugate monodromies around all cycles, including noncontractible cycleson Σ (if its genus is positive) as well as cycles of the sort just considered. This isguaranteed by the fact that f = u (cid:101) u − v (cid:101) v has no monodromy, since the Liouville field φ c is e φ c = 1 /λf .So a solution of Liouville’s equations – real or complex – gives us a flat SL (2 , C )connection over Σ that can be put in the gauge (6.4) and can also be put in the The explicit gauge transformation between them is g = (cid:18) ∂u∂ (cid:101) u + ∂v∂ (cid:101) v − u∂ (cid:101) u − v∂ (cid:101) v − (cid:101) u∂u − (cid:101) v∂v u (cid:101) u + v (cid:101) v (cid:19) . – 62 –auge (6.5). The basic idea of the present section is that, by a complex solution ofLiouville’s equations, we should mean in general a flat SL (2 , C ) connection, up togauge transformation, which can be gauge-transformed to either of those two forms.We do not worry about what sort of expression it has in terms of a Liouville field.The attentive reader may notice that we have cut some corners in this explanation,because in section 3 the reference metric was chosen to be flat in deriving the holomor-phic differential equations. This is not possible globally if Σ has genus greater than1, and even for Σ = S , it involves introducing an unnatural singularity at infinity. Amore precise description is to say that A is a flat connection that locally, after picking alocal coordinate z , can be put in the form (6.5), in such a way that in the intersection ofcoordinate patches, the gauge transformation required to compare the two descriptionsis lower triangular g = (cid:18) ∗ ∗ ∗ (cid:19) . (6.7)A flat connection with this property is known as an oper. This notion is explainedin section 3 of [21], but we will not need that degree of detail here. The global char-acterization of (cid:101) A has the same form (with upper triangular matrices replacing lowertriangular ones, given the choice we made in (6.5)). Our proposal then is that a classi-cal solution of Liouville theory is a flat connection whose holomorphic structure is thatof an oper, while its antiholomorphic structure is also that of an oper. A few elementary observations may give us some practice with these ideas. Let usfirst consider the main example of this paper, namely S with insertions of three heavyoperators of Liouville momenta α i = η i /b , at positions z = z i . The monodromies M i around the three points will have to obeyTr M i = 2 cos(2 πη i ) , det M i = 1 (6.8)In addition, the product of the three monodromies must equal 1: M M M = 1 . (6.9)Equivalently M M = M − , (6.10)from which it follows thatTr M M = Tr M − = 2 cos(2 πη ) . (6.11)– 63 –nd of course we are only interested in a flat bundle up to conjugacy M i → gM i g − , g ∈ SL (2 , C ) . (6.12)To start with, let us just ignore the oper condition and ask how many choices ofthe M i there are, up to conjugacy, that obey the conditions in the last paragraph. Wecan partially fix the gauge invariance by setting M = (cid:18) e πiη e − πiη (cid:19) . (6.13)The remaining freedom consists of diagonal gauge transformations g = (cid:18) λ λ − (cid:19) , (6.14)where we only care about λ up to sign, since a gauge transformation by g = − M = (cid:18) p qr s (cid:19) (6.15)If we look for a solution with q = 0, we soon find that, for generic values of the η i , oncewe adjust p and r to get the right values of Tr M and det M , we cannot also satisfy(6.11). So we take q (cid:54) = 0, in which case λ in (6.14) can be chosen uniquely, up to sign,to set q = 1. Then, imposing det M = 1, we get M = (cid:18) p pq − q (cid:19) . (6.16)Now the conditions Tr M = 2 cos(2 πη ), Tr M M = 2 cos(2 πη ) give two linear equa-tions for p and q which generically have a unique solution. So M and therefore M = M − M − are uniquely determined.The conclusion is that a flat bundle on the three-punctured sphere with prescribedconjugacy classes of the monodromies M i is unique, up to gauge equivalence, even ifwe do not require the oper conditions. The unique SL (2 , C ) flat bundle with thesemonodromies can be realized by a holomorphic differential equation and also by an This remains true for non-generic values of the η i where the derivation in the last paragraph doesnot quite apply, unless the η i equal (0 , , 0) or a permutation of (0 , / , / i , Tr M i = ± 2, so that the eigenvalues of M i are equal, then one shouldnot assume that M i can be diagonalized; its Jordan canonical form may be ± (cid:18) (cid:19) . – 64 –ntiholomorphic one. The proof of this statement is simply that functions W and (cid:102) W with the right singularities do exist, as in (3.18).Since it can be realized by both a holomorphic differential equation and an anti-holomorphic one, the unique SL (2 , C ) flat bundle on the three-punctured sphere withmonodromies in the conjugacy classes determined by the η i is a complex solution ofLiouville’s equations in the sense considered in the present section. What one wouldmean by its action and why this action is multivalued will be explained in section 6.3.It is instructive to consider a more generic case with s > η i , inserted at points z i ∈ S . Now there are s monodromies M i , i = 1 , . . . , s .They are 2 × M i = 2 cos(2 πη i ) , det M i = 1 . (6.17)We also require M M . . . M s = 1 , (6.18)and we are only interested in the M i up to conjugacy M i → gM i g − . (6.19)A simple parameter count shows that the moduli space M of flat bundles over the s -punctured sphere that obey these conditions has complex dimension 2( s − V ⊂ M consisting of flat bundles that can be realizedby a holomorphic differential equation (6.2). We already know that the potential W that appears in the holomorphic differential equation is unique for s = 3. When oneincreases s by 1, adding a new singularity at z = z i for some i , one adds anotherdouble pole to W , giving a new contribution ∆ W = c/ ( z − z i ) + c (cid:48) / ( z − z i ). But c isdetermined to get the right monodromy near z i , so only c (cid:48) is a new parameter (usuallycalled the accessory parameter). Hence the dimension of V is s − V is a middle-dimensional subspace of M . (For a more complete account of this standard result, seesection 8 of [21].) Similarly, the subspace (cid:101) V of flat bundles that can be realized by anantiholomorphic differential equation is middle-dimensional.A complex solution of Liouville theory in the sense that we consider in the presentsection corresponds to an intersection point of V and (cid:101) V . As V and (cid:101) V are both middle-dimensional, it is plausible that their intersection generically consists of finitely manypoints, or possibly even that it always consists of just one point. Unfortunately we donot know if this is the case. All we really know is that for any s , if the η i are real andobey the Seiberg and Gauss-Bonnet bounds, then there is a real solution of Liouville’sequations, and this corresponds to an intersection point of V and (cid:101) V .– 65 – .3 Interpretation In Chern-Simons Theory To explain what one would mean in this language by the action of a classical solu-tion, and why it is multivalued, the main idea is to relate Liouville theory on a Riemannsurface Σ to Chern-Simons theory on Σ × I . The basic reason that there is such a rela-tion is that Virasoro conformal blocks (which can be understood as building blocks ofLiouville theory) can be viewed as physical states in three-dimensional Chern-Simonstheory. This was first argued in [18] and has been reconsidered much more recently[21].We start with an SL (2 , C ) connection A with Chern-Simons action S CS = 14 πib (cid:90) M Tr (cid:18) A ∧ d A + 23 A ∧ A ∧ A (cid:19) (6.20)on a three-manifold M . S CS is invariant under gauge transformations that are con-tinuously connected to the identity, but not under homotopically non-trivial gaugetransformations. For M = S , the homotopically non-trivial gauge transformations areparametrized by π ( SL (2 , C )) = Z . The integer invariant of a gauge transformation isoften called winding number. In defining S CS , we have picked a convenient normaliza-tion, such that under a homotopically nontrivial gauge transformation on S of windingnumber n , S CS transforms by S CS → S CS + 2 πinb . (6.21)With b understood as the Liouville coupling parameter, this matches the multival-uedness of Liouville theory that comes from the trivial symmetry φ c → φ c + 2 πi .Conventionally, the Chern-Simons action is normalized to make S CS singlevalued mod2 πi Z and the homotopically nontrivial gauge transformations are regarded as symme-tries [50]. For our purposes, this would be far too restrictive (since we do not wantto assume that b is the inverse of an integer). Rather, in the path integral, we con-sider integration cycles that are not invariant under homotopically non-trivial gaugetransformations, and we do not view homotopically nontrivial gauge transformationsas symmetries of the theory. In other words, we adopt the perspective of [11]. The inte-gration cycles are middle-dimensional in the space of SL (2 , C )-valued flat connections.A basis of the possible integration cycles is given by the cycles that arise by steepestdescent from a critical point of the action.The Yang-Mills field strength is defined, as usual, by F = d A + A∧A . The classicalequations of motion of Chern-Simons theory are simply F = 0 . (6.22)– 66 –e take the three-manifold M to be simply M = Σ × I , where I is a unit interval and Σis the Riemann surface on which we want to do Liouville theory. The fundamental groupof M is therefore the same as that of Σ, and so a solution of the classical equation ofmotion (6.22) is just an SL (2 , C ) flat connection on Σ. This being so, one may wonderwhat we have gained by introducing a third dimension.The answer to this question is that to do Liouville theory, we need more than an SL (2 , C ) flat connection on Σ. It must obey two conditions: (i) it can be describedby a holomorphic differential equation, and (ii) it can also be described by an anti-holomorphic differential equation. It is possible to pick boundary conditions at the twoends of I so that condition (i) is imposed at one end and condition (ii) at the otherend. Such boundary conditions were introduced in Chern-Simons theory in [18] andused to relate that theory to Virasoro conformal blocks; a variant related to Nahm’sequations and other topics in mathematical physics has been described in [21].We may call these oper or Nahm pole boundary conditions. For the very schematicpurposes of the present paper, almost all that we really need to know about them isthat they completely break the SL (2 , C ) gauge symmetry down to the center ± × I .The gauge transformation is described by a map g from Σ × I to SL (2 , C ). At theleft end of Σ × I , g must equal 1 or − 1. For the present paper, an overall gaugetransformation by the center of SL (2 , C ) will be of no interest, since it acts triviallyon all gauge fields, so we can assume that at the left end, g = 1. But then there aretwo choices at the right end, namely g = 1 and g = − 1. After we make this choice,the remaining freedom in describing g topologically is given by π ( SL (2 , C )) = Z .The homotopy classification of gauge transformations is by Z × Z , where the Z factorclassifies the relative value of g at the two ends and Z classifies the twist by π ( SL (2 , C )).This last statement is not completely trivial; we must verify that the homotopyclassification is by a simple product Z × Z and not by a nontrivial extension 0 → Z → Γ → Z → 0. Concretely, the question is whether a gauge transformation with g = − g = − g to be a function on Σ × I that only depends on the second factor;such a map can be constructed from a path from 1 to − SL (2 , C ). (Theconclusion just stated remains valid when monodromy defects are included, as we domomentarily. For this, it suffices to note that by continuity the question is independentof the values of the η i , while if one of the η i vanishes, we can forget the correspondingdefect.) – 67 – igure 2 . Monodromy defects, depicted as horizontal dotted lines, in Σ × I . In the exampleshown, Σ = S and the number of monodromy defects is 3. Some basic points about how to interpret various Liouville fields in the Chern-Simonsdescription have been explained in [21]. The main point of concern to us is how toincorporate a primary field of Liouville momentum α = η/b at a point p ∈ Σ. Theanswer is simply that in the Chern-Simons description, the gauge field A should havean appropriate monodromy around the codimension two locus p × I ⊂ Σ × I . This canbe achieved by requiring A to have a suitable singularity along Σ. If z = re iθ is a localcoordinate that vanishes at p , then we require that the singular behavior of A shouldbe A = i dθ (cid:18) η − η (cid:19) + . . . , (6.23)where the ellipsis refers to less singular terms. This singular behavior has been chosenso that the eigenvalues of the monodromy are exp( ± πiη ). We call a singularity ofthis kind in the gauge field a monodromy defect. (What happens when a monodromydefect meets the Nahm pole singularity at the ends of I has been analyzed in section3.6 of [51]. The details are not important here.) If in the Liouville description, thereare several primary fields, say at points p i given by z = z i , then in the Chern-Simonsdescription we include monodromy defects on p i × I for each i (Fig. 2).Now we want to work out the topological classification of gauge transformations inthe presence of monodromy defects. The ansatz (6.23) is only invariant under diagonalgauge transformations along p . So now a gauge transformation g is constrained asfollows: at the ends of Σ × I , it equals ± SL (2 , C ).Let us look at what happens along a particular monodromy defect. A diagonal– 68 –auge transformation can be written g = (cid:18) ρe iϑ ρ − e − iϑ (cid:19) , (6.24)where ρ is positive and ϑ is real. Let us parametrize the interval I by a parameter y that equals, say, 0 at the left end of I and 1 at the right end. We constrain ϑ to vanishat y = 0 (where g = 1), and to equal πm at y = 1, where m is even if g = 1 on theright end of Σ × I , and m is odd if g = − m the winding number alongthe defect. It is sometimes useful to write it as m = 1 π (cid:90) dy dϑdy . (6.25)We have normalized m so that wrapping once around the maximal torus of SO (3) corre-sponds to m = 1, while wrapping once around the maximal torus of SU (2) correspondsto m = 2.So in total, with s monodromy defects, the topological classification of gauge trans-formations is by integers ( n, m , m , . . . , m s ), where n is the bulk winding number and m i , i = 1 , . . . , s are winding numbers defined just along the monodromy defects. The m i are either all even or all odd, since their oddness or evenness is determined by thebehavior of g .In the presence of a monodromy defect, it is necessary to add one more term tothe action. The reason is that in the presence of a monodromy defect, we would like aflat bundle on the complement of the monodromy defect that has the singularity (6.23)along the defect to be a classical solution. Such a flat bundle obeys F = 2 πi (cid:18) η − η (cid:19) δ K . (6.26)Here we write K for the support of the monodromy defect, and δ K is a two-form deltafunction supported on K . But in the presence of the Chern-Simons action (6.20) only,the equation of motion is simply F = 0 rather than (6.26). To get the equation wewant, we must add to the action a term S K = − b (cid:90) K Tr A (cid:18) η − η (cid:19) . (6.27)Finally, we can determine how the action transforms under a gauge transformationin the presence of a monodromy defect. (See section 4.2.6 of [11] for an alternativeexplanation.) We have already discussed the behavior of the Chern-Simons term under– 69 –auge transformation, so what remains is to understand what happens to the newinteraction S K . For K = p × I , only A y , the component of A in the y direction,appears in (6.26). Under a diagonal gauge transformation (6.24), the diagonal matrixelements of A y are shifted by ∓ d log( ρe iϑ ) /dy . Taking the trace and integrating over y , we find that S K is shifted by2 iηb (cid:90) dy dϑdy = 2 πiηmb . (6.28)(There is no contribution involving d log ρ , since (cid:82) dy ( d log ρ/dy ) = 0, as ρ = 1 at bothendpoints.)More generally, let us go back to the case of s heavy operator insertions, withLiouville parameters η i , i = 1 , . . . , s , inserted at points p i ∈ Σ. In Chern-Simons theory,they correspond to monodromy defects, supported on K i = p i × I . In classifying gaugetransformations, we introduce a winding number m i associated to each monodromydefect. There is also a bulk winding number n . The shift in the total action S = S CS + (cid:80) i S K i under a gauge transformation is S → S + 2 πib (cid:32) n + s (cid:88) i =1 m i η i (cid:33) . (6.29) The moral of the story is that in the Chern-Simons description, critical points areflat connections on Σ × I , with prescribed behavior near the ends and near monodromydefects, modulo topologically trivial gauge transformations. Topologically nontrivialgauge transformations cannot be regarded as symmetries because they do not leave theaction invariant. Instead, they generate new critical points from old ones.For the main example of this paper – three heavy operators on S – all criticalpoints are related to each other by topologically nontrivial gauge transformations. Thismeans that there is a simple way to compare the path integrals over cycles associatedto different critical points.In fact, let A be a connection that represents a critical point ρ . Suppose a gaugetransformation g with winding numbers n and m , . . . , m s acts on A to produce a newcritical point A (cid:48) . Let Z ρ and Z ρ (cid:48) be the path integrals over integration cycles associatedto A and to A (cid:48) , respectively. Z ρ (cid:48) and Z ρ are not equal, since the gauge transformation g does not preserve the action. But since g transforms the action by a simple additive c -number (6.29), there is a simple exact formula that expresses the relation between Z ρ (cid:48) and Z ρ : Z ρ (cid:48) = Z ρ exp (cid:32) − (2 πi/b )( n + (cid:88) i m i η i ) (cid:33) . (6.30)– 70 –e will discuss the interpretation of this formula in Liouville theory in section 7.3below. So far in this paper we have analytically continued in α but not in b . From the pointof view advocated in the introduction, this is somewhat artificial; we should allow our-selves to consider the path integral with arbitrary complex values of all parameters andthen study which integration cycles to use to reproduce the analytic continuation fromthe physical region. For complex b ’s with positive real part, we can simply continuethe DOZZ formula and the machinery of the preceding sections is essentially unmodi-fied. Indeed numerical results for complex b were given in [7], confirming the crossingsymmetry of the four-point function based on the DOZZ formula. As mentioned inthe introduction, in various cosmological settings it is desireable to define a version ofLiouville that has real central charge that is large and negative. The most obvious wayto try to do this is to continue the DOZZ formula all the way to purely imaginary b ,since the formula (2.5) will then be in the desired range [22]. This has been shown tofail rather dramatically [23], as we will discuss in the following subsection. However,we first introduce some conventional redefinitions to simplify future formulas when b isimaginary. We begin with b = − i (cid:98) b (7.1) φ = i (cid:98) φ (7.2) Q = i (cid:18) (cid:98) b − (cid:98) b (cid:19) ≡ i (cid:98) Q, (7.3)after which the action (2.1) becomes S L = 14 π (cid:90) d ξ (cid:112)(cid:101) g (cid:104) − ∂ a (cid:98) φ∂ b (cid:98) φ (cid:101) g ab − (cid:98) Q (cid:101) R (cid:98) φ + 4 πµe (cid:98) b (cid:98) φ (cid:105) . (7.4)The theory with this action is conventionally referred to as “timelike” Liouville theory,since the kinetic term has the wrong sign. In this equation, superscripts are procreatingat an alarming rate, so we pause to remind the reader that (cid:101) g is the reference metricand does not undergo analytic continuation. We will use “hat”, as in (cid:98) b , exclusively torefer to the timelike analogues of standard Liouville quantities. The central charge isnow c L = 1 − (cid:98) Q , (7.5)– 71 –hich for small real (cid:98) b accomplishes our goal of large negative central charge. Thephysical metric becomes g ab = e (cid:98) Q (cid:98) φ (cid:101) g ab , so the boundary condition on (cid:98) φ at infinity is (cid:98) φ ( z, z ) = − (cid:98) Q log | z | + O (1) . (7.6)To talk about exponential operators, it is convenient to make one final definition α = i (cid:98) α, (7.7)which gives conformal weights∆ (cid:16) e − (cid:98) α (cid:98) φ (cid:17) = ∆ (cid:16) e − (cid:98) α (cid:98) φ (cid:17) = (cid:98) α ( (cid:98) α − (cid:98) Q ) . (7.8)In the presence of heavy operators (cid:98) α i = η i / (cid:98) b , the generalized action (2.25) for therescaled field φ c = 2 bφ = 2 (cid:98) b (cid:98) φ with a flat reference metric is (cid:101) S L = − π (cid:98) b (cid:90) D −∪ i d i d ξ (cid:16) ∂ a φ c ∂ a φ c − (cid:98) λe φ c (cid:17) − (cid:98) b (cid:18) π (cid:73) ∂D φ c dθ + 2 log R (cid:19) + 1 (cid:98) b (cid:88) i (cid:18) η i π (cid:73) ∂d i φ c dθ i + 2 η i log (cid:15) (cid:19) . (7.9)Here (cid:98) λ = πµ (cid:98) b = − λ , and in fact other than an overall sign change this is the onlydifference from the expression of this action in terms of the “unhatted” variables. Notethat φ c and η i do not need to be “hatted” since they are the same before and after theanalytic continuation. The equation of motion is now ∂∂φ c = − (cid:98) λe φ c − π (cid:88) i η i δ ( ξ − ξ i ) , (7.10)which for positive µ is just the equation of motion for constant positive curvature withconical deficits at the heavy operators. When (cid:98) b and η i are real and η i is in Region II,described by (4.19), this equation has a real solution. As discussed below (4.19), thissolution can be constructed from spherical triangles. In the FRW/CFT application oftimelike Liouville, this real saddle point is identified with the asymptotic metric in aColeman-de Luccia bubble [3, 4]. The redefinitions of the previous section make clear that at the classical level therelationship between spacelike and timelike Liouville is straightforward. Much lessclear is the question of the appropriate integration cycle for the path integral when b – 72 –s imaginary. One way to attempt to specify a cycle is to try to continue the DOZZformula from real b . As just mentioned, this does not work. We can see why byconsidering more carefully the analytic properties of Υ b ( x ) in b [23]. From (A.1) we seethat the defining integral for Υ b ( x ) does not converge for any x when b is imaginary,which is already a sign of trouble, but this could possibly be avoided by deforming thecontour. A more sophisticated argument from [23] is as follows: consider the function H b ( x ) = Υ b ( x )Υ ib ( − ix + ib ) , (7.11)where for the moment we take b to have positive real part and negative imaginary partto ensure that both Υ’s can be defined by the integral (A.1). This function is entireand has simple zeros everywhere on the lattice generated by b and 1 /b , as illustratedin Figure 3. Using the recursion relations (A.3) we can show that H b obeys: H b ( x + b ) = e iπ (2 bx − H b ( x ) H b ( x + 1 /b ) = e iπ (1 − x/b ) H b ( x ) (7.12)It is convenient to here introduce a Jacobi θ -function b x Figure 3 . Zeros of H b ( x ). The solid circles come from the zeros of Υ b ( x ) while the emptycircles come from zeros of Υ ib ( − ix + ib ). – 73 – ( z, τ ) = i ∞ (cid:88) n = −∞ ( − n e iπτ ( n − / +2 πiz ( n − / Im τ > , (7.13)which is entire in z for any Im τ > θ ( z + 1 , τ ) = e − iπ θ ( z, τ ) θ ( z + τ, τ ) = e iπ (1 − τ − z ) θ ( z, τ ) . (7.14)By cancelling the terms n = 1 , , ... with n = 0 , − , ... we see that it has a zero at z = 0,and by applying these recursion relations we see that it has zeros for all z = m + nτ with m, n ∈ Z . In fact these zeros are simple and are the only zeros, which follows fromthe standard product representation of the theta function. This function is useful forus because we can now observe that e iπ ( x + x/b − xb ) θ ( x/b, /b ) (7.15)obeys the same recursion relations and has the same zeros as H b ( x ). Their ratio isdoubly periodic and entire in x , and must therefore be a function only of b . We candetermine this function by setting x = b + b and recalling that Υ b ( Q/ 2) = 1. Theresult is that H b ( x ) = e iπ (cid:16) x + xb − xb + b − b − (cid:17) θ ( x/b, /b ) θ ( + b , /b ) . (7.16)We can now use this formula to study the behaviour of Υ b near imaginary b ; sinceΥ ib ( − ix + ib ) = H b ( x )Υ b ( x ) , if we move b up towards the postive real axis then Υ ib will ap-proach the region of interest. But (7.16) reveals that doing this continuation requires θ to approach the real τ -axis. This is actually a natural boundary of analytic contin-uation for θ , with a nonlocal and extremely violent singularity running all along thereal τ -axis. The detailed form of the approach to the singularity depends strongly on z ,so there is no possibility of cancellation between the two θ ’s in H b except for at specialvalues of z . This shows that for generic values of x , Υ b simply cannot be continuedto generic imaginary b [23]. This is the origin of the failure of [22] to make sense oftimelike Liouville theory in this way.What then are we to do? One possibility is to restrict to special values of b and α where the continuation can still be nonsingular; this is explored in [27]. We areinterested however in generic complex values of the parameters so this will not workfor us. A very interesting proposal was made by Al. B. Zamolodchikov [23], and also Conventions for Jacobi θ -functions are rather inconsistent, so we note that this definition is im-plemented in Mathematica as EllipticTheta (cid:2) , πz, e iπτ (cid:3) . This formula was also derived in [27]. – 74 –ndependently by Kostov and Petkova [24–26]. A key observation is that althoughwe cannot continue the DOZZ formula to imaginary b , we can continue Teschner’srecursion relations. For real b the essentially unique solution of these recursion relationsis the DOZZ formula, but for generic complex b the solution is not unique since wecan engineer (cid:98) α -dependent combinations of θ -functions by which we can multiply anysolution of the recursion relations to produce a new solution. But when we get toimaginary b , it turns out that there is again an (almost) unique solution, which isnot given by analytic continuation of the DOZZ formula. This solution is not quiteunique because one can multiply it by an (cid:98) α -independent arbitrary function of b withoutaffecting the recursion relations. Fixing this normalization in a way that we explainmomentarily, and which slightly differs from the choice in [23], the solution is: (cid:98) C ( (cid:98) α , (cid:98) α , (cid:98) α ) = 2 π (cid:98) b (cid:104) − πµγ ( − (cid:98) b ) (cid:98) b (cid:98) b (cid:105) ( (cid:80) i (cid:98) α i − (cid:98) Q ) / (cid:98) b e − iπ ( (cid:80) i (cid:98) α i − (cid:98) Q ) / (cid:98) b Υ (cid:98) b ( (cid:98) α + (cid:98) α − (cid:98) α + (cid:98) b )Υ (cid:98) b ( (cid:98) α + (cid:98) α − (cid:98) α + (cid:98) b )Υ (cid:98) b ( (cid:98) α + (cid:98) α − (cid:98) α + (cid:98) b )Υ (cid:98) b ( (cid:98) α + (cid:98) α + (cid:98) α − (cid:98) Q + (cid:98) b )Υ (cid:98) b ( (cid:98) b )Υ (cid:98) b (2 (cid:98) α + (cid:98) b )Υ (cid:98) b (2 (cid:98) α + (cid:98) b )Υ (cid:98) b (2 (cid:98) α + (cid:98) b ) . (7.17)We will refer to this as the timelike DOZZ formula. The power of πµγ ( − (cid:98) b ) differsslightly from C.10 in [23], but the choice we have made here is given by a scaling ar-gument in the path integral and is required for our interpretation of timelike Liouvilleas being a different integration cycle of ordinary Liouville. We have also divided C.10from [23] by a factor of (cid:98) b π γ (1 − (cid:98) b ) γ (2 − / (cid:98) b ); as just mentioned these choices do notaffect the recursion relations and can be interpreted as an ambiguity in the normaliza-tion of the operators, but we will see below in section 7.2.3 that the choice we havemade here is supported by semiclassical computation. Moreover in section 7.3 we willsee exactly that it is the natural choice for our interpretation of the timelike Liouvillepath integral.We can also write down an exact two-point function. Since unlike the three-pointfunction the two-point function (2.38) does have a good analytic continuation to imag-inary b , it is natural to choose the timelike 2-point function to agree with this analytic The reason that purely real and purely imaginary b have essentially unique solutions of the recur-sion relations is that the lattice generated by b and 1 /b becomes degenerate in these two cases andfunctions with two real periodicities are highly constrained. The freedom involving multiplying by θ -functions of the (cid:98) α ’s goes away since in these cases some of the θ functions are always evaluated at τ = 0. The original DOZZ formula (2.28) and the formula (7.17) below are related at complex b bysuch a factor, which is why they can not be continued into each other. For more details see [23, 33]. – 75 –ontinuation. This gives (cid:98) G ( (cid:98) α ) = − (cid:98) b π (cid:98) C (0 , (cid:98) α, (cid:98) α ) = − (cid:98) b (cid:104) − πµγ ( − (cid:98) b ) (cid:105) (2 (cid:98) α − (cid:98) Q ) / (cid:98) b e − iπ (2 (cid:98) α − (cid:98) Q ) / (cid:98) b γ (2 (cid:98) α (cid:98) b + (cid:98) b ) γ (cid:18) (cid:98) b − (cid:98) α (cid:98) b − (cid:19) . (7.18)Note that for real (cid:98) α , this expression is not positive-definite, as expected from thewrong-sign kinetic term. Its relation to the three-point function is somewhat arbitrary,unlike in spacelike Liouville where there was a clear rationale for the formula (2.37).In particular setting one of the (cid:98) α ’s to zero in the timelike DOZZ formula does NOTproduce a δ -function. Indeed the timelike DOZZ formula has a finite and nonzerolimit even when (cid:98) α → , (cid:98) α (cid:54) = (cid:98) α . In [23], this was observed as part of a largerissue whereby the degenerate fusion rules mentioned below equation (2.43) are notautomatically satisfied by the timelike DOZZ formula. In [28], this was interpretedas the two-point function being genuinely non-diagonal in the operator weights. Wewill not be able to explain this in a completely satisfactory way, but we will suggest apossible resolution below in section 7.4. In this section we will show in three different cases, analogous to the three casesstudied above for the spacelike DOZZ formula, that the semiclassical limits of (7.17) and(7.18) are consistent with our claim that they are produced by the usual Liouville pathintegral on a different integration cycle. This task is greatly simplified by observingthat we can trivially reuse all of our old solutions (or “solutions”) and expressionsfor the action. Say that we have a solution φ c,N ( η i , λ, b, z, z ) of the original Liouvilleequation of motion (2.15). It is easy to check that (cid:98) φ c,N ( η i , (cid:98) λ, (cid:98) b, z, z ) ≡ φ c,N ( η i , (cid:98) λ, (cid:98) b, z, z ) − iπ (7.19)obeys (7.10). We can also compute the action by noting that if we define the originalmodified action (2.25) as (cid:101) S L [ η i , λ, b, z i , z i ; φ c,N ( η i , λ )], then we have (cid:101) S L (cid:104) η i , − (cid:98) λ, − i (cid:98) b, z i , z i ; (cid:98) φ c,N ( η i , (cid:98) λ ) (cid:105) ≡ (cid:98)(cid:101) S L (cid:104) η i , (cid:98) λ, (cid:98) b, z i , z i ; (cid:98) φ c,N ( η i , λ ) (cid:105) = − (cid:101) S L (cid:104) η i , (cid:98) λ, (cid:98) b, z i , z i ; φ c,N ( η i , (cid:98) λ ) (cid:105) + iπ (cid:98) b (cid:32) − (cid:88) i η i (cid:33) . (7.20)The left hand side of this is just (7.9), so we can thus compute the action for timelikeLiouville theory by simple modification of our previous results.– 76 – .2.1 Two-point Function Using (7.20) and (4.8), we find that the timelike version of the saddlepoint (4.3) hastimelike action (cid:98)(cid:101) S L = − (cid:98) b (cid:34) πiN (1 − η ) + (2 η − (cid:98) λ + 2 (cid:16) (1 − η ) log(1 − η ) − (1 − η ) (cid:17) + 2( η − η ) log | z | (cid:35) . (7.21)The semiclassical limit of (7.18) with α heavy is: (cid:98) G ( η ) → (cid:16) e πi (1 − η ) / (cid:98) b − (cid:17) exp (cid:26) (cid:98) b (cid:104) − (1 − η ) log (cid:98) λ + 2 (cid:0) (1 − η ) log(1 − η ) − (1 − η ) (cid:1)(cid:105)(cid:27) , (7.22)which is matched by a sum over the two saddle points N = 0 and N = 1 with actionsgiven by (7.21). Note that the integral over the moduli would again produce a diver-gence, but that unlike in the DOZZ case this divergence did not seem to be producedby the limit α → , α = α . Note also that there is now no Stokes line in the η plane, there are always only two saddle points that contribute. This is analogous tothe integral representation of 1 / Γ( z ) as discussed in appendix C. Similarly for three heavy operators in Region I, the timelike version of (4.21) hastimelike action (cid:98)(cid:101) S L = − (cid:98) b (cid:34) − (cid:32) − (cid:88) i η i (cid:33) log (cid:98) λ − ( (cid:98) δ + (cid:98) δ − (cid:98) δ ) log | z | − ( (cid:98) δ + (cid:98) δ − (cid:98) δ ) log | z | − ( (cid:98) δ + (cid:98) δ − (cid:98) δ ) log | z | + F ( η + η − η ) + F ( η + η − η ) + F ( η + η − η )+ F ( η + η + η − − F (2 η ) − F (2 η ) − F (2 η ) − F (0)+ 2 πi ( N − / − (cid:88) i η i ) (cid:35) , (7.23)where we have defined (cid:98) δ i ≡ η i ( η i − (cid:98)(cid:101) S L = − (cid:98) b (cid:34) − (cid:32) − (cid:88) i η i (cid:33) log (cid:98) λ − ( (cid:98) δ + (cid:98) δ − (cid:98) δ ) log | z | − ( (cid:98) δ + (cid:98) δ − (cid:98) δ ) log | z | − ( (cid:98) δ + (cid:98) δ − (cid:98) δ ) log | z | + F ( η + η − η ) + F ( η + η − η ) + F ( η + η − η )+ F ( η + η + η ) − F (2 η ) − F (2 η ) − F (2 η ) − F (0)2 (cid:40) (1 − (cid:88) i η i ) log(1 − (cid:88) i η i ) − (1 − (cid:88) i η i ) (cid:41) + 2 πiN (1 − (cid:88) i η i ) (cid:35) . (7.24)To compare these with the timelike DOZZ formula, we can again make use of theasymptotic formula (4.34). The terms in (7.17) that don’t involve Υ (cid:98) b approach e − (cid:80) i ηi (cid:98) b ( iπ +2 log (cid:98) b − log (cid:98) λ )+ O (1 / (cid:98) b ) , (7.25)and using (4.34) we find that in Region I the Υ (cid:98) b ’s combine with this to give (cid:98) C ( η i / (cid:98) b ) ∼ exp (cid:40) (cid:98) b (cid:34) − (cid:32) − (cid:88) i η i (cid:33) log (cid:98) λ + F ( η + η − η ) + F ( η + η − η )+ F ( η + η − η ) + F ( η + η + η − − F (2 η ) − F (2 η ) − F (2 η ) − F (0) + iπ (1 − (cid:88) i η i ) (cid:35)(cid:41) . (7.26)Comparing with (7.23), we see that only the saddle point with N = 0 contributes. InRegion II as before (4.37) we need to use (A.4) to shift one of the Υ (cid:98) b ’s before using theasymptotic formula (4.34), giving:Υ (cid:98) b (cid:16) (cid:80) i η i − (cid:98) b + 2 (cid:98) b (cid:17) ∼ γ (cid:16) ( (cid:88) i η i − / (cid:98) b (cid:17) − (cid:98) b (cid:98) b ( − (cid:80) i η i ) − ( (cid:80) i η i − / ) e (cid:98) b F ( (cid:80) i η i ) . (7.27)The result in Region II is (cid:98) C ( η i / (cid:98) b ) ∼ exp (cid:40) (cid:98) b (cid:34) − (cid:32) − (cid:88) i η i (cid:33) log (cid:98) λ + F ( η + η − η ) + F ( η + η − η )+ F ( η + η − η ) + F ( η + η + η − − F (2 η ) − F (2 η ) − F (2 η ) − F (0) + 2(1 − (cid:88) i η i ) log(1 − (cid:88) i η i ) − − (cid:88) i η i )+ iπ (1 − (cid:88) i η i ) (cid:35)(cid:41)(cid:16) e iπ (1 − (cid:80) i η i ) / (cid:98) b − e − iπ (1 − (cid:80) i η i ) / (cid:98) b (cid:17) . (7.28)– 78 –omparing this with (7.24), we see it matches a sum over two saddle points with N = 0and N = 1. Unlike the spacelike DOZZ formula there are no Stokes walls in Region II,in complete analogy with the situation for 1 / Γ( z ) explained in appendix C. As a final semiclassical check of the timelike DOZZ formula (7.17), we will calculateits b → σ i = α i b = − (cid:98) α i (cid:98) b , whichgives ∆ → σ as (cid:98) b → 0. Manipulations similar to those leading up to (4.44) now give (cid:98) C ( − σ (cid:98) b, − σ (cid:98) b, − σ (cid:98) b ) = − πi (cid:98) b − (cid:98) λ − (cid:80) i σ i − / (cid:98) b e − / (cid:98) b − γ E + O ( (cid:98) b log (cid:98) b ) (cid:16) e πi ( (cid:80) i σ i − / (cid:98) b ) − (cid:17) × Γ(1 − σ )Γ(1 − σ )Γ(1 − σ )Γ(1 + σ − σ − σ )Γ(1 + σ − σ − σ )Γ(1 + σ − σ − σ )Γ(2 − (cid:80) i σ i ) . (7.29)From the structure of this formula, it appears that we will be able to interpret as asum over two complex saddle points as with Region II in the previous section. Thereis a subtlety however in that to produce the Γ-functions that will emerge from themodular integral in our imminent semiclassical computation, we need to apply theEuler reflection formula Γ( x )Γ(1 − x ) = π/ sin πx to each of them. Anticipating thisresult, we write: (cid:98) C ( − σ (cid:98) b, − σ (cid:98) b, − σ (cid:98) b ) = (cid:98) b − (cid:98) λ − (cid:80) i σ i − / (cid:98) b e − / (cid:98) b − γ E + O ( (cid:98) b log (cid:98) b ) (cid:16) e πi ( (cid:80) i σ i − / (cid:98) b ) − (cid:17) × Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( (cid:80) i σ i − σ )Γ(2 σ )Γ(2 σ ) × (cid:0) e πi ( σ + σ − σ ) − (cid:1) (cid:0) e πi ( σ + σ − σ ) − (cid:1) (cid:0) e πi ( σ + σ − σ ) − (cid:1) (cid:0) e πi ( σ + σ + σ ) − (cid:1) ( e πiσ − 1) ( e πiσ − 1) ( e πiσ − . (7.30)The structure of the terms in the third line show that a much more complicated set ofsaddlepoints are needed to explain this result than in the spacelike case (4.44). At theend of this section we will explain why this happens.The semiclassical formula analogous to (4.46) for this correlation function is (cid:104) e σ φ c ( z ,z ) e σ φ c ( z ,z ) e σ φ c ( z ,z ) (cid:105) ≈ A ( − i (cid:98) b ) (cid:88) N ∈ T e − S L [ (cid:98) φ c,N ] (cid:90) dµ ( α, β, γ, δ ) (cid:89) i =1 e σ i (cid:98) φ c,N ( z i ,z i ) . (7.31)Here we have assumed that the fluctuation determinant and Jacobian parametrized by A ( − i (cid:98) b ) are just the analytic continuations of their spacelike counterparts. This should– 79 –e true if our path integral interpretation is correct, and we will see momentarily thatthis works out. (cid:98) φ c,N is the timelike “solution” with branch choice N , related to theusual spacelike “solution” by (7.19). Explicitly (cid:98) φ c ( z, z ) = 2 πiN ( z, z ) − log (cid:98) λ − | αz + β | + | γz + δ | ) . (7.32)Based on (7.30), we have allowed N to vary with position to allow the different branchesof the action to be realized. This is one of the situations discussed in section 5.3.1 wherediscontinuous “solutions” must be included even though single-valued solutions exist.Computing the action (7.4) and simplifying the modular integral as in section 4.3 wefind (cid:98) C ( − σ (cid:98) b, − σ (cid:98) b, − σ (cid:98) b ) ≈ A ( − i (cid:98) b ) (cid:98) λ − (cid:80) i σ i − / (cid:98) b e − / (cid:98) b I ( σ , σ , σ ) (cid:88) N ∈ T e − πi ( (cid:80) i m i σ i + n/ (cid:98) b ) (7.33)Here − n is the value of N at ∞ and − m i is its value near the various insertions.Using (4.51) and comparing this with (7.30), we find complete agreement providedthat A ( − i (cid:98) b ) = (cid:98) b − π − e − γ E . Recalling that at the end of section 4.3 we found A ( b ) = ib − π − e − γ E , this indeed works out as expected.The set T of included branches is now rather complex; it can be read off from(7.30) but we will not try to characterize it more precisely. We observe however thatmany branches that correspond to discontinuous “solutions” are now definitely needed.This is different than what we found for spacelike Liouville in section 4.3, where thecontributing saddle points were single-valued and continuous and just the same as inRegion II for heavy operators. The reason for this distinction is that, as explainedin appendix F, the modular integral over SL (2 , C ) converges only when the σ ’s obeycertain inequalities (F.20). In spacelike Liouville with the σ ’s in Region II the integralis convergent, so we can evaluate it without any contour deformation. In timelikeLiouville, when the σ ’s are in Region II many of the inequalities are violated andthe integral must be defined by analytic continuation. This continuation results inadditional Stokes phenomena, which changes the contributing saddle-points. The checks of the previous section were semiclassical, but we will now give an exactargument that the timelike DOZZ formula (7.17) is produced by evaluating the usualLiouville path integral on a new integration cycle. We will show that the ratio of thespacelike and timelike DOZZ formulae must have a specific form and then demonstratethat it does. – 80 –e begin by defining: Z ρ ( α i , z i , z i ) = (cid:90) C ρ D φ c V α ( z , z ) ...V α n ( z n , z n ) e − S L , (7.34)where here ρ is a critical point of the action with heavy operators as sources, and thepath integral is evaluated on the steepest descent cycle C ρ that passes through ρ . Asdiscussed in the introduction, this quantity is not in general equal to the Liouvillecorrelator; we need to sum over such cycles with integer coefficients a ρ as in (1.6). Wewill now argue however that the ρ -dependence of Z ρ is quite simple. First recall theexact version of the action (2.10) S L = 116 πb (cid:90) D d ξ (cid:2) ( ∂φ c ) + 16 λe φ c (cid:3) + 12 πb (1 + b ) (cid:73) ∂D φ c dθ + 2 b (1 + 2 b + b ) log R. (7.35)We note that under the transformation φ c → φ c +2 πiN , we have S L → S L + πiNb (1+ b ).Semiclassically the operator V α defined by (2.24) transforms as V α → V α e πiα/b underthe same transformation. Since the Seiberg bound ensures that the renormalizationneeded to define this operator precisely is the same as in free field theory, this isactually the exact transformation of V α . Moreover the path-integral measure D φ c isinvariant under the shift. This means that if two ρ ’s differ only by adding 2 πiN , thenwith a slight abuse of notation we have the simple relation Z ρ +2 πiN = e πiN ( (cid:80) i α i /b − /b ) Z ρ . (7.36)This result is exact, and more generally it shows that the result of integrating over asum of integration cycles of this type can be factored out from the correlator: Z = ∞ (cid:88) N = −∞ a ρ +2 πiN Z ρ +2 πiN = Z ρ ∞ (cid:88) N = −∞ a ρ +2 πiN e πiN ( (cid:80) i α i /b − /b ) . (7.37)Thus in general the ratio of Z ’s which are computed on different cycles, both of theform (cid:80) ∞ N = −∞ a ρ +2 πiN C ρ +2 πiN , will be expressible as a ratio of Laurent expansions in e πi ( (cid:80) i α i /b − /b ) with integer coefficients. This is a rather nontrivial constraint; forexample it implies that the ratio is invariant under shifting any particular α i by α i → α i + b . There is also a more subtle invariance of the form b → b √ b and α i → α i √ b .Unfortunately as discussed in section 5, to understand the DOZZ formula in thefull range of α i ’s it is not sufficient to only consider integration cycles that differ by When we discuss discontinuous “solutions” momentarily, the kinetic term should really be under-stood to be expressed in terms of f in equation (5.3). – 81 – global addition of 2 πiN . We found semiclassically in (5.2) that to fully explain theDOZZ formula it was necessary to consider discontinuous “solutions” that differ bydifferent multiples of 2 πi at the different operator insertions. To proceed further weneed to assume that we can apply the machinery of the previous paragraph to these“solutions” and their associated “integration cycles of steepest descent.” The ideais that the action (7.35), with the kinetic term expressed in terms of f as in (5.3),changes only by an overall c -number if we shift the field configuration by 2 πiN witha position-dependent N ∈ Z . The change in the action depends on the value of N atinfinity, and the contributions of operator insertions also shift in a way that dependson the value of N in their vicinity. For the reader who is uncomfortable with this,we note that in the Chern-Simons interpetation espoused in section 6, these additional“solutions” were just as valid and conventional as the usual ones. So one could inprinciple rephrase what follows in Chern-Simons language, which would perhaps makeit sound more plausible. We will henceforth assume that the relationship between Z ρ and Z ρ (cid:48) is given by a formula analogous to (6.30) in the Chern-Simons version: Z ρ (cid:48) = Z ρ e − πib (cid:0) n + (cid:80) i m i α i b (cid:1) (7.38)Here n and m i are the differences in N at infinity and near the various operator inser-tions, and m i are either all even or all odd.We will now compute the ratio of the spacelike and timelike DOZZ formulas fora region of b where both make sense, with the goal being to check that their ratio isconsistent with this result. Using (7.17) expressed in terms of the “unhatted” variablesas well as (2.28), (7.16), and (7.11), we find: (cid:98) C ( − iα , − iα , − iα ) C ( α , α , α ) = − πib lim (cid:15) → Υ b ( (cid:15) )Υ H b ( (cid:15) ) e iπ (1 /b − b )( Q − (cid:80) i α i ) × H b ( (cid:80) i α i − Q ) H b ( α + α − α ) H b ( α + α − α ) H b ( α + α − α ) H b (2 α ) H b (2 α ) H b (2 α )= − πie − πib ( (cid:80) i α i b − (1+ b ) / ) × θ ( (cid:80) i α i − Qb , b ) θ ( α + α − α b , b ) θ ( α + α − α b , b ) θ ( α + α − α b , b ) θ (cid:48) (0 , b ) θ (2 α /b, b ) θ (2 α /b, b ) θ (2 α /b, b ) . (7.39)Here θ (cid:48) ( z, τ ) ≡ ∂θ ∂z ( z, τ ). We can simplify this a bit by using (7.14) to shift theargument of one of the θ -functions: (cid:98) C ( − iα , − iα , − iα ) C ( α , α , α ) = 2 πi θ ( (cid:80) i α i b , b ) θ ( α + α − α b , b ) θ ( α + α − α b , b ) θ ( α + α − α b , b ) θ (cid:48) (0 , b ) θ (2 α /b, b ) θ (2 α /b, b ) θ (2 α /b, b ) . (7.40)– 82 –e’d now like to express this as a ratio of sums of terms of the form e − πib (cid:0) n + (cid:80) i m i α i b (cid:1) with integer coefficients. To facilitate this, we define (cid:101) θ ( z, τ ) ≡ − ie − iπτ/ iπz θ ( z, τ ) = ∞ (cid:88) n = −∞ ( − n e iπτn ( n − πizn (cid:101) θ ( τ ) ≡ − π e − iπτ/ θ (cid:48) (0 , τ ) = ∞ (cid:88) n = −∞ ( − n ne iπτn ( n − , (7.41)in terms of which we have: (cid:98) C ( − iα , − iα , − iα ) C ( α , α , α ) = (cid:101) θ ( (cid:80) i α i b , b ) (cid:101) θ ( α + α − α b , b ) (cid:101) θ ( α + α − α b , b ) (cid:101) θ ( α + α − α b , b ) (cid:101) θ ( b ) (cid:101) θ (2 α /b, b ) (cid:101) θ (2 α /b, b ) (cid:101) θ (2 α /b, b ) . (7.42)From (7.41), we see that the right hand side of this equation now explicitly is a ratio ofthe desired form. This completes our demonstration that the timelike DOZZ formulais given by the ordinary Liouville path integral evaluated on a different integrationcycle. Note in particular that the ratio of the two is bad both for purely real andpurely imaginary b , which illustrates the failure to directly continue between the two.This argument also confirms our choice of prefactor in the timelike DOZZ formula,since other choices, including the one made in C.10 from [23], would have polluted thisresult. Unlike most sections which are titled by questions, in this case our answer will be anoptimistic “maybe”. We have established that the timelike DOZZ formula is computedby evaluating the Liouville path integral on a particular choice of cycle, which meansthat its correlation functions will necessarily obey the usual conformal Ward identities.So in the sense that any local path integral which computes correlators that obey theconformal Ward identities is a conformal field theory, it is clear that timelike Liouvilletheory fits the bill. For example as a consequence of this our semiclassical computationsconfirmed the usual position dependence of the two- and three-point functions. But thereal meat of this question is understanding to what extent timelike Liouville theory fitsinto the standard conformal field theory framework of [17]. At least one thing that seemsto work is that the derivation of the timelike DOZZ formula from the recursion relationsconfirms that the four-point function with a single degenerate operator constructed inthe standard way is crossing symmetric. There has however been justifiable concern inthe literature [22, 23, 28] about the fact that the timelike DOZZ formula does not obeythe degenerate fusion rules when its arguments are specialized to degenerate values.– 83 –he simplest manifestation of this is the nonvanishing of (cid:98) C (0 , (cid:98) α , (cid:98) α ) when (cid:98) α (cid:54) = (cid:98) α ,as discussed below (7.18).The reason that this is troubling is that semiclassically it seems obvious thatlim α → e αφ = 1. If this were really true as an operator equation, it would imply thatthe timelike Liouville two-point function is nondiagonal in the operator dimensions.Since the diagonal nature of this function is a consequence only of the Ward identities,and we know the Ward identities are satisfied just from the path integral, somethinghas to give. What the Timelike DOZZ formula seems to tell us is that sending α → not produce the identity operator, but instead producesanother operator of weight zero that does not obey the degenerate fusion rule. Theexistence of such an operator is usually forbidden by unitarity, but timelike Liouvilleis necessarily nonunitary so this does not contradict anything sacred. We believe thatthis is the correct interpretation. As evidence for this proposal, we consider the differential equation obeyed by u, v for the semiclassical three-point function with three heavy operators: ∂ u + W ( z ) u = 0with W ( z ) = (cid:20) η (1 − η ) z z z − z + η (1 − η ) z z z − z + η (1 − η ) z z z − z (cid:21) z − z )( z − z )( z − z ) . We observe that if η → 0, there is still a regular singular point at z = z that onlycancels if we also have η = η . When η (cid:54) = η , the solution will generically have alogarithmic singularity at z = z . In this limit the standard semiclassical solution (3.25)that we reviewed previously breaks down, and a new solution needs to be constructed.We interpret this as the three-point function of a new nontrivial operator of weight zerowith two conventional Liouville operators. In the spacelike case this also could havehappened, but since for real b spacelike Liouville is unitary such an operator cannot Recall that even in spacelike Liouville there was a subtlety with computing degenerate correlatorsby specializing the general correlators to degenerate values, as discussed below (2.43). We thank V. Petkova for useful correspondence on this point. He points out that this nondecou-pling happens in the c < We are here assuming the Seiberg bound Re( η i ) < / The monodromy matrix M of the differential equation about z in this limit has in some basis theform (cid:18) λ (cid:19) , with λ some function of η and η . This matrix has one eigenvector with eigenvalue 1, butis not diagonalizeable. In the Chern-Simons interpretation this means that there is still a monodromydefect in the gauge field even after we send η → – 84 –xist and the O ( b ) corrections to the saddlepoint conspire to set the correlator to zero.In timelike Liouville there is no reason for this conspiracy to happen, and indeed fromthe timelike DOZZ formula we see that it does not. We take the fact that this extrasingularity disappears only when η = η as evidence that, contrary to the worriesexpressed in [22, 23, 28], the real two-point function of timelike Liouville theory isindeed diagonal.Perhaps a natural framework to discuss a CFT that includes an extra operator ofdimension 0 that does not decouple is “logarithmic” CFT. Something which remainsmysterious about this interpretation however is that there does not seem to be anycandidate primary operator we can express in terms of the Liouville field to play thisrole. We leave this unresolved for future work, but we note that in the Chern-Simonsformulation it is straightforward to describe the nondegenerate primary of dimension0; it corresponds to a monodromy defect with unipotent monodromy as explained inthe footnote on the previous page.The main open question that would allow a more systematic understanding oftimelike Liouville as a CFT is to identify the set of states on which we should factor-ize correlation functions. In spacelike Liouville theory this question was answered bySeiberg [8], and is formalized by the expression (2.39) for the four-point function. Inthat case the key insight came from study of minisuperspace and the analogy to scat-tering off of an exponential potential. A similar analysis for timelike Liouville theorywas initiated in [29] and studied further in [28], but the Hamiltonian is non-hermitianand subtle functional analysis seems to be called for. We have not tried to extend theminisuperspace analysis of [28, 29] to the full timelike Liouville theory, but it seemsthat this would be the key missing step in establishing the appropriate basis of statesto factorize on. This would then allow construction of four-point functions as in (2.39)for spacelike Liouville, and one could check numerically if they are crossing symmetric.Since in the end of the day we know that the path integral does produce crossing-symmetric four-point functions that obey the Ward identities, it seems certain thatsuch a construction is possible; it would be good to understand it explicitly. In this conclusion we summarize our main results and suggest a few directions for futurework. We began by trying to assign a path integral interpretation to the full analyticcontinuation of the DOZZ formula (2.28) for the three-point function of Liouville pri-mary operators. Our technique was to compare the semiclassical limit of the DOZZformula and various other correlators to the classical actions of complex solutions ofLiouville’s equation. We found that for certain regions of the parameters the analytic– 85 –ontinuation is well described by the machinery of Stokes walls and complex saddlepoints, and in particular we showed that the old transition to the “fixed-area” region[8] can be reinterpreted in this manifestly-local language. The main surprise was thatin order to properly account for the full analytic continuation it was necessary to in-clude multivalued/discontinuous “solutions”, whose actions were defined according to asimple prescription in section 5.1. In section 5.5,we saw that these singularities naivelysuggested singularities in the four-point function, but argued that they were in factresolved by quantum corrections. One is tempted to declare this an example of thequantum resolution of two-dimensional gravitational singularities.Two situations come to mind where these ideas may be relevant. In [52] a statisticalmodel of bubble collisions in three-dimensional de Sitter space was constructed in whichthe 4-point function had singularities when the operators were not coincident, similar tothe naive result in section 5.5. This theory has a yet to be well-understood relationshipto dS/CFT in three dimensions, which is expected to have a Liouville sector coupled toa nonunitary CFT [5]. Perhaps in a more refined version of this theory the singularitycould be resolved as in section 5.5. Secondly, in three-dimensional Euclidean quantumgravity with negative cosmological constant, it was found in [53] that including only realsmooth solutions in the path integral produces a partition function that does not havethe correct form to come from a CFT computation. Perhaps other complex “solutions”need to be included? More generally the use of singular “solutions” seems to be anew phenomenon in field theory and we wonder where else it could appear.We then discussed how the question of analytic continuation could be reformulatedin the Chern-Simons description of Liouville theory, where we found that the pictureof analytic continuation in terms of Stokes phenomenon is more conventional and allrelevant solutions seem to be nonsingular. It would be interesting to get a more precisepicture of these solutions; since explicit formulas exist on the Liouville side it seemsplausible that they may also be achievable on the Chern-Simons side. This would allowa more concrete realization of the ideas suggested in section 6.Finally we used the tools developed in the previous sections to discuss an expression(7.17), proposed in [23], for an exact three-point function in timelike Liouville theory.We found that we could interpret this formula as being the result of performing the usualLiouville path integral on a different integration cycle, which we demonstrated bothsemiclassically and exactly. We also discussed the extent to which timelike Liouvilletheory can be understood as a conformal field theory, arguing that it probably can butthat the spectrum of states to factorize on needs to be understood before the question For quantum gravity in higher dimensions it is unlikely that the path integral makes sense beyondthe semiclassical expansion about any particular background, so in particular it is probably not well-defined enough for us to ask about Stokes phenomenon. – 86 –an be decisively settled. Even before this question is addressed however, we alreadyconsider our results sufficient motivation to begin using the formula of [23] to studythe various proposed applications of timelike Liouville theory to closed string tachyoncondensation [22] and FRW/CFT duality [3]. Acknowledgments We thank A. B. Zamolodchikov for discussions, advice, and sometechnical assistance. We also thank L. Susskind for raising the question, S. Shenkerand S. Hellerman for many enlightening discussions, and X. Dong, R. Mazzeo, Y.Nakayama, E. Shaghoulian, T. Takayanagi, and J. Teschner for useful conversations.Finally we thank L. Hadasz, Z. Jaskolski, V Petkova, R. Schiappa, and V. Schomerus foruseful correspondence on a previous version of this work. Research of EW is supportedin part by NSF Grant PHY-0969448. JM and DH are both supported in part byNSF grant PHY-0756174. DH was also partially supported by a Stanford GraduateFellowship and the Howrey Term Endowment Fund in Memory of Dr. Ronald Kantor.JM was also partially supported by the Mellam Family foundation and the StanfordSchool of Humanities and Sciences Fellowship. DH thanks the Perimeter Institute forTheoretical Physics for hospitality during part of the completion of this work. A Properties of the Υ b Function The function Υ b has now become standard in the literature on Liouville theory, butfor convenience we here sketch derivations of its key properties. The function can bedefined bylog Υ b ( x ) = (cid:90) ∞ dtt (cid:34) ( Q/ − x ) e − t − sinh (( Q/ − x ) t )sinh tb sinh t b (cid:35) < Re( x ) < Re( Q ) . (A.1)Here Q = b + b . The definition reveals that Υ b ( Q − x ) = Υ b ( x ). When x = 0 thesecond term in the integral diverges logarithmically at large t , and at small but finite x it behaves like log x . Υ b therefore has a simple zero at x = 0 as well as x = Q .To extend the function over the whole x -plane, we can use the identity log Γ( x ) = (cid:90) ∞ dtt (cid:20) ( x − e − t − e − t − e − xt − e − t (cid:21) Re( x ) > This identity is derived in Appendix E. – 87 –o show that in its range of definition Υ b obeysΥ b ( x + b ) = γ ( bx ) b − bx Υ b ( x )Υ b ( x + 1 /b ) = γ ( x/b ) b xb − Υ b ( x ) . (A.3)Where: γ ( x ) ≡ Γ( x )Γ(1 − x )These recursion relations are the crucial property of Υ b from the point of view ofLiouville theory, among other things they are what allow a solution of Teschner’s re-cursion relations to be expressed in terms of Υ b . The recursion relations also showthat the simple zeros at x = 0 , Q induce more simple zeros at x = − mb − n/b and x = ( m (cid:48) + 1) b + ( n (cid:48) + 1) /b , with m, m (cid:48) and n, n (cid:48) all non-negative integers. It it is alsouseful to record the inverse recursion relations:Υ b ( x − b ) = γ ( bx − b ) − b bx − − b Υ b ( x )Υ b ( x − /b ) = γ ( x/b − /b ) − b b − xb Υ b ( x ) . (A.4)We will also need various semiclassical limits of Υ b . Rescaling t by b and usingthe identity log x = (cid:90) ∞ dtt (cid:2) e − t − e − xt (cid:3) Re( x ) > , we see that b logΥ b ( η/b + b/ 2) = − (cid:18) η − (cid:19) log b + (cid:90) ∞ dtt (cid:34)(cid:18) η − (cid:19) e − t − t (cid:18) − t b 24 + . . . (cid:19) sinh (cid:2) ( η − ) t/ (cid:3) sinh t (cid:35) . (A.5)When 0 < Re( η ) < 1, the subleading terms in the series 1 + t b + . . . can be integratedterm by term, with only the 1 contributing to nonvanishing order in b . From theidentity (A.2), we can find F ( η ) ≡ (cid:90) η / log γ ( x ) dx = (cid:90) ∞ dtt (cid:34) ( η − / e − t − t sinh (( η − / t )sinh( t ) (cid:35) < Re( η ) < , so using this we find the asymptotic formula:Υ b ( η/b + b/ 2) = e b [ − ( η − / log b + F ( η )+ O ( b ) ] 0 < Re( η ) < . (A.6) We thank A. Zamolodchikov for suggesting the use of (A.2) in the first of these derivations. – 88 –n particular if we choose η to be constant as b → b ( η/b ) = e b [ F ( η ) − ( η − / log b + O ( b log b ) ] 0 < Re( η ) < , (A.7)which is useful for our heavy operator calculations in section 4.For light operator calculations we will also be interested in the situation where theargument of Υ b scales like b . Looking at the b → b (( σ + 1) b ) ≈ σb Υ b ( σb ) . (A.8)One solution to this relation is Υ b ( σb ) ≈ b − σ Γ( σ ) h ( b ) , (A.9)where h ( b ) is independent of σ . Unfortunately this solution is not unique since we canmultiply it by any periodic function of σ with period one and still obey the recursionrelation. We see however that it already has all of the correct zeros at σ = 0 , − , − , ... to match the Υ b function, so we might expect that this periodic function is a constant.This periodic function in any case is nonvanishing and has no poles, so it must be theexponential of an entire function. If the entire function is nonconstant then it mustgrow as σ → ∞ , which seems to be inconsistent with the nice analytic properties of Υ b .In particular (A.6) shows no sign of such singularities in η as η → 0. We can derive h ( b ) analytically, up to a b -independent constant which we determine numerically. Themanipulations are sketched momentarily in a footnote, the result isΥ b ( σb ) = Cb / − σ Γ( σ ) exp (cid:20) − b log b + F (0) /b + O ( b log b ) (cid:21) . (A.10)The numerical agreement of this formula with the asymptotics of the integral (A.1) isexcellent;in particular we find C = 2 . The constant C will cancel out of all To do this numerical comparison, it is very convenient to first note that for Re( (cid:101) σb ) > 0, we havelog Υ b (cid:0) ( (cid:101) σ + ) b (cid:1) = − ( b − (cid:101) σb ) log b + b F ( (cid:101) σb ) + I ( (cid:101) σ, b ), with I ( (cid:101) σ, b ) = (cid:90) ∞ dtt (cid:18) t − t/ (cid:19) sinh (cid:2) ( b − (cid:101) σ ) t/ (cid:3) sinh t b . This integral approaches a finite limit as b → 0, which makes it easy to extract the leading termsin (A.10) and also to do the numerical comparison with (A.1). Although this derivation requiredrestrictions on σ , the final result does not since it can be continued throughout the σ plane using therecursion relation. Of course the asymptotic series is only useful when σ is O ( b ). – 89 –f our computations since we are always computing ratios of equal numbers of Υ b ’s.This precise numerical agreement also confirms our somewhat vague argument for theabsence of an additional periodic function in σ . As an application of this formula wecan find the asymptotics of Υ from the DOZZ formula:Υ = Cb / exp (cid:20) − b log b + F (0) /b + O ( b log b ) (cid:21) . (A.11) B Theory of Hypergeometric Functions This appendix will derive the results we need about hypergeometric and P -functions[54]. No prior exposure to either is assumed. Our initial approach is rather pedestrian;it is aimed at producing concrete formulas (B.10-B.13) which illustrate the monodromyproperties of various solutions of Riemann’s hypergeometric differential equation andthe “connection coefficients” relating them. This “toolbox” approach is convenient forpractical computations, but the disadvantage is that it involves complicated expressionsthat obscure some of the underlying symmetry. In section B.5 we give a more elegantgeneral formulation in terms of the integral representation, which illustrates the basiclogic of the previous sections in a simpler way but is less explicit about the details. Italso allows us to recast the three-point solutions of section 3.3 in an interesting way. B.1 Hypergeometric Series We begin by studying the series F ( a, b, c, z ) = ∞ (cid:88) n =0 ( a ) n ( b ) n ( c ) n n ! z n . (B.1)Here ( x ) n ≡ x ( x + 1) · · · ( x + n − 1) = Γ( x + n )Γ( x ) . It is easy to see using the ratio test that if c is not a negative integer, then for any complex a and b the series converges absolutelyfor | z | < 1, diverges for | z | > 1, and is conditional for | z | = 1. It is also symmetric in a and b , and we can observe that if either a or b is a nonpositive integer then the seriesterminates at some finite n . One special case which is easy to evaluate is F (1 , , , z ) = − log(1 − z ) z , which shows that the analytic continuation outside of the unit disk is not necessarilysinglevalued.We will also need the value of the series at z = 1. We will derive this from theintegral representation in section B.5: F ( a, b, c, 1) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) Re( c − a − b ) > . (B.2)– 90 – .2 Hypergeometric Differential Equation By direct substitution one can check that the function F ( a, b, c, z ) obeys the follow-ing differential equation: z (1 − z ) f (cid:48)(cid:48) + ( c − ( a + b + 1) z ) f (cid:48) − abf = 0 . (B.3)This second-order equation has three regular singular points, at 0,1, and ∞ . Since F ( a, b, c, z ) is manifestly nonsingular at z = 0, its analytic continuation has potentialsingularities only at 1 and ∞ . There is the possibility of a branch cut running between1 and ∞ . We saw this in the special case we evaluated above, and it is standardto choose this branch cut to lie on the real axis. We can determine the monodromystructure of a general solution of this equation by studying its asymptotic behaviour inthe vicinity of the singular points. By using a power-law ansatz it is easy to see thatany solution generically takes the form f ( z ) ∼ A ( z ) + z − c B ( z ) as z → z − a A ∞ (1 /z ) + z − b B ∞ (1 /z ) as z → ∞ A (1 − z ) + (1 − z ) c − a − b B (1 − z ) as z → , (B.4)with A i ( · ) , B i ( · ) being holomorphic functions in a neighborhood of their argument beingzero. The solution of (B.3) defined by the series (B.1) is a case of (B.4), with A ( z ) = F ( a, b, c, z ) and B = 0. We will determine the A i and B i at the other two singularpoints later. These expressions confirm that a solution of (B.3) will generically havebranch points at 0, 1 and ∞ . B.3 Riemann’s Differential Equation It will be very convenient for our work on Liouville to make use of Riemann’shypergeometric equation, of which (B.3) is a special case. This more general differentialequation is: f (cid:48)(cid:48) + (cid:26) − α − α (cid:48) z − z + 1 − β − β (cid:48) z − z + 1 − γ − γ (cid:48) z − z (cid:27) f (cid:48) + (cid:26) αα (cid:48) z z z − z + ββ (cid:48) z z z − z + γγ (cid:48) z z z − z (cid:27) f ( z − z )( z − z )( z − z ) = 0 (B.5)along with a constraint: α + α (cid:48) + β + β (cid:48) + γ + γ (cid:48) = 1 . (B.6) When the two exponents given here become equal at one or more of the singular points, there areadditional asymptotic solutions involving logs. We will not treat this special case, although it appearsfor the particular choice a = b = 1 / c = 1 in appendix D. – 91 –ere z ij ≡ z i − z j , the parameters α , β , γ , α (cid:48) , β (cid:48) , γ (cid:48) are complex numbers, and theconstraint is imposed to make the equation nonsingular at infinity. The points z i areregular singular points. This is in fact the most general second-order linear differentialequation with three regular singular points and no singularity at infinity. To see thatthis reduces to the hypergeometric equation (B.3), one can set z = 0, z = ∞ , z = 1, α = γ = 0, β = a , β (cid:48) = b , and α (cid:48) = 1 − c .Solutions to Riemann’s equation can always be written in terms of solutions of thehypergeometric equation; this is accomplished by first doing an SL (2 , C ) transformationto send the three singular points to 0, 1, and ∞ , followed by a nontrivial rescaling. Tosee this explicitly, say that g ( a, b, c, z ) is a solution of the differential equation (B.3),not necessarily the solution given by (B.1). Then a somewhat tedious calculation showsthat f = (cid:18) z − z z − z (cid:19) α (cid:18) z − z z − z (cid:19) γ g (cid:18) α + β + γ, α + β (cid:48) + γ, α − α (cid:48) , z ( z − z ) z ( z − z ) (cid:19) (B.7)is a solution of the differential equation (B.5).Near the singular points any solution behaves as f ( z ) ∼ A ( z − z ) α + B ( z − z ) α (cid:48) as z → z A ( z − z ) β + B ( z − z ) β (cid:48) as z → z A ( z − z ) γ + B ( z − z ) γ (cid:48) as z → z , (B.8)so the monodromies are simply expressed in terms of α, α (cid:48) , β, . . . B.4 Particular Solutions of Riemann’s Equation We now construct explicit solutions of Riemann’s equation that have simple mon-odromy at the three singular points in terms of the hypergeometric function (B.1).Given equation (B.7), the most obvious solution we can write down is f ( α ) ( z ) ≡ (cid:18) z − z z − z (cid:19) α (cid:18) z − z z − z (cid:19) γ F (cid:18) α + β + γ, α + β (cid:48) + γ, α − α (cid:48) , z ( z − z ) z ( z − z ) (cid:19) . We denote it f ( α ) because the holomorphy of the series (B.1) at 0 ensures that anynontrivial monodromy near z comes only from the explicit factor ( z − z ) α . Thedifferential equation is invariant under interchanging α ↔ α (cid:48) , so we can easily write As before, there is a caveat that when the two exponents are equal at one or more of the singularpoints, there are additional solutions involving logs. – 92 –own another solution that is linearly independent with the first (assuming that α (cid:54) = α (cid:48) ): f ( α (cid:48) ) ( z ) ≡ (cid:18) z − z z − z (cid:19) α (cid:48) (cid:18) z − z z − z (cid:19) γ F (cid:18) α (cid:48) + β + γ, α (cid:48) + β (cid:48) + γ, α (cid:48) − α, z ( z − z ) z ( z − z ) (cid:19) . This solution has the alternate monodromy around z = 0.The differential equation is also invariant under β ↔ β (cid:48) and γ ↔ γ (cid:48) : the formerleaves the solutions f ( α ) , f ( α (cid:48) ) invariant and can be ignored, but the latter apparentlygenerates two additional solutions. We can find even more solutions by simultaneouslypermuting { z , α, α (cid:48) } ↔ { z , β, β (cid:48) } ↔ { z , γ, γ (cid:48) } , so combining these permutations wefind a total of 4x6=24 solutions, known as “Kummer’s Solutions”. Since these are allsolutions of the same 2nd-order linear differential equation, any three of them must belinearly dependent.To pin down this redundancy, it is convenient to define a particular set of sixsolutions, each of which has simple monodromy about one of the singular points. Thedefinition is somewhat arbitrary as one can change the normalization at will as well asmove around the various branch cuts. We will choose expressions that are simple whenall z -dependence is folded into the harmonic ratio x ≡ z ( z − z ) z ( z − z ) . (B.9) This derivation of Kummer’s Solutions using the symmetric equation (B.5) is quite straightforward,but if we had used the less symmetric equation (B.3) then they would seem quite mysterious. – 93 –ur explicit definitions are the following: P α ( x ) = x α (1 − x ) γ F ( α + β + γ, α + β (cid:48) + γ, α − α (cid:48) , x )= x α (1 − x ) − α − β F (cid:18) α + β + γ, α + β + γ (cid:48) , α − α (cid:48) , xx − (cid:19) P α (cid:48) ( x ) = x α (cid:48) (1 − x ) γ (cid:48) F ( α (cid:48) + β + γ (cid:48) , α (cid:48) + β (cid:48) + γ (cid:48) , α (cid:48) − α, x )= x α (cid:48) (1 − x ) − α (cid:48) − β F (cid:18) α (cid:48) + β + γ, α (cid:48) + β + γ (cid:48) , α (cid:48) − α, xx − (cid:19) P γ ( x ) = x α (1 − x ) γ F ( α + β + γ, α + β (cid:48) + γ, γ − γ (cid:48) , − x )= x α (cid:48) (1 − x ) γ F ( α (cid:48) + β + γ, α (cid:48) + β (cid:48) + γ, γ − γ (cid:48) , − x ) P γ (cid:48) ( x ) = x α (1 − x ) γ (cid:48) F ( α + β + γ (cid:48) , α + β (cid:48) + γ (cid:48) , γ (cid:48) − γ, − x )= x α (cid:48) (1 − x ) γ (cid:48) F ( α (cid:48) + β + γ (cid:48) , α (cid:48) + β (cid:48) + γ (cid:48) , γ (cid:48) − γ, − x ) P β ( x ) = x α (1 − x ) − α − β F (cid:18) α + β + γ, α + β + γ (cid:48) , β − β (cid:48) , x − (cid:19) = x α (cid:48) (1 − x ) − α (cid:48) − β F (cid:18) α (cid:48) + β + γ, α (cid:48) + β + γ (cid:48) , β − β (cid:48) , x − (cid:19) P β (cid:48) ( x ) = x α (1 − x ) − α − β (cid:48) F (cid:18) α + β (cid:48) + γ, α + β (cid:48) + γ (cid:48) , β (cid:48) − β, x − (cid:19) = x α (cid:48) (1 − x ) − α (cid:48) − β (cid:48) F (cid:18) α (cid:48) + β (cid:48) + γ, α (cid:48) + β (cid:48) + γ (cid:48) , β (cid:48) − β, x − (cid:19) . (B.10)These formulas are somewhat intimidating, but they follow from the simple permuta-tions just described. For convenience in the following derivation we give two equivalentforms of each. More symmetric integral expressions for them will be described in sectionB.5.Since only two of these can be linearly independent, there must exist coefficients a ij such that P α ( x ) = a αγ P γ ( x ) + a αγ (cid:48) P γ (cid:48) ( x ) P α (cid:48) ( x ) = a α (cid:48) γ P γ ( x ) + a α (cid:48) γ (cid:48) P γ (cid:48) ( x ) . (B.11)These coefficients are called connection coefficients. To determine them we can evaluate– 94 –hese two equations at x = 0 and x = 1, which gives a αγ = Γ(1 + α − α (cid:48) )Γ( γ (cid:48) − γ )Γ( α + β + γ (cid:48) )Γ( α + β (cid:48) + γ (cid:48) ) a αγ (cid:48) = Γ(1 + α − α (cid:48) )Γ( γ − γ (cid:48) )Γ( α + β + γ )Γ( α + β (cid:48) + γ ) a α (cid:48) γ = Γ(1 + α (cid:48) − α )Γ( γ (cid:48) − γ )Γ( α (cid:48) + β + γ (cid:48) )Γ( α (cid:48) + β (cid:48) + γ (cid:48) ) (B.12) a α (cid:48) γ (cid:48) = Γ(1 + α (cid:48) − α )Γ( γ − γ (cid:48) )Γ( α (cid:48) + β + γ )Γ( α (cid:48) + β (cid:48) + γ ) . In solving these equations one uses (B.2). Similarly one can find: a αβ = Γ(1 + α − α (cid:48) )Γ( β (cid:48) − β )Γ( α + β (cid:48) + γ )Γ( α + β (cid:48) + γ (cid:48) ) a αβ (cid:48) = Γ(1 + α − α (cid:48) )Γ( β − β (cid:48) )Γ( α + β + γ )Γ( α + β + γ (cid:48) ) a α (cid:48) β = Γ(1 + α (cid:48) − α )Γ( β (cid:48) − β )Γ( α (cid:48) + β (cid:48) + γ )Γ( α (cid:48) + β (cid:48) + γ (cid:48) ) (B.13) a α (cid:48) β (cid:48) = Γ(1 + α (cid:48) − α )Γ( β − β (cid:48) )Γ( α (cid:48) + β + γ )Γ( α (cid:48) + β + γ (cid:48) ) . Finally we note that our expressions for the connection coefficients allow us toderive some beautiful facts about the original hypergeometric function F ( a, b, c, z ).First making the replacements mentioned below (B.5), we see that (B.11) gives: F ( a, b, c, z ) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) F ( a, b, a + b − c, − z )+ Γ( c )Γ( a + b − c )Γ( a )Γ( b ) (1 − z ) c − a − b F ( c − a, c − b, c − a − b, − z ) . (B.14)This gives explicit expressions for A (1 − z ) and B (1 − z ) for F ( a, b, c, z ), aspromised above. We can also set α = 0, α (cid:48) = 1 − c , β = 0, β (cid:48) = c − a − b , γ = a , γ (cid:48) = b , z = 0, z = 1, and z = ∞ , in which cases B.11 gives: F ( a, b, c, z ) = Γ( c )Γ( b − a )Γ( c − a )Γ( b ) ( − z ) − a F ( a, − c + a, − b + a, z − )+ Γ( c )Γ( a − b )Γ( c − b )Γ( a ) ( − z ) − b F ( b, − c + b, b − a, z − ) . (B.15) Two identities which are useful are sin( x ) sin( y ) = sin( x + y − z ) sin( z ) + sin( z − x ) sin( z − y ) andΓ( x )Γ(1 − x ) = π sin( πx ) . – 95 –his expression gives A ∞ (1 /z ) and B ∞ (1 /z ) for F ( a, b, c, z ), and in fact it gives thefull analytic continuation of the series (B.1) in the region | z | > 1, since the hypergeo-metric series on the right hand side converge in this region. We can thus observe thatindeed the only singular behaviour of the function F ( a, b, c, z ) is a branch cut runningfrom one to infinity. B.5 Integral Representations of Hypergeometric Functions We now consider the integral representations of the hypergeometric and P functions.We begin by defining I C ( a, b, c, z ) = (cid:90) C ds s a − c ( s − c − b − ( s − z ) − a , (B.16)where C is some contour to be specified in the s -plane. If we insert this integral intothe hypergeometric differential equation (B.3) we get (cid:90) C ds dds (cid:2) s a − c +1 ( s − c − b ( s − z ) − a − (cid:3) = 0 , (B.17)so this function will be a solution of the equation as long as C has the same and initialand final values for the quantity in square brackets. As an example we can choose C torun from from one to infinity, which is allowed when Re c > Re b > 0. The monodromyof the integrand as z circles zero is trivial everywhere on the contour, so we expect thissolution to be proportional to the original hypergeometric function (B.1). Indeed wehave F ( a, b, c, z ) = Γ( c )Γ( b )Γ( c − b ) (cid:90) ∞ ds s a − c ( s − c − b − ( s − z ) − a Re c > Re b > . (B.18)To establish this we can use the binomial expansion(1 − z/s ) − a = ∞ (cid:88) n =0 Γ( a + n )Γ( a )Γ( n + 1) (cid:16) zs (cid:17) n (B.19)to expand the integrand. We then change variables to t = 1 /s and use Euler’s integralfor the Beta function β ( x, y ) ≡ Γ( x )Γ( y )Γ( x + y ) = (cid:90) dt t x − (1 − t ) y − Re x > , Re y > . (B.20)This representation allows an easy determination of the value of the hypergeometricfunction at z = 1. Changing variables s = 1 /t and using the β function integral we– 96 – igure 4 . The Pochhammer contour. find: F ( a, b, c, 1) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) Re ( c − a − b ) > . (B.21)The main point however is that by integrating on other contours it is possible toget other solutions of the hypergeometric differential equation in a straightforward way.For example if we integrate from zero to z , which requires 2 > b > Re c > w = z/s it is easy to see that we get z − c F (1 + b − c, a − c, − c, z ) = Γ(2 − c )Γ(1 + b − c )Γ(1 − b ) (cid:90) z ds s b − c (1 − s ) c − a − ( z − s ) − b , (B.22)which is the other linearly independent solution of the hypergeometric differential equa-tion with simple monodromy at z = 0. More generally there are four singular pointsof the integrand, and placing the contour between any two gives six different solutionswhich correspond to the six solutions that have simple monodromy at 0 , , ∞ .Unfortunately these simple contours require strange inequalities on a, b, c to besatisfied which we certainly do not expect to hold for the general solutions we areconsidering in Liouville theory. To find contours that produce solutions for arbitrary a, b, c is more subtle. The trick is to use a closed contour that winds around bothpoints of interest twice, but in such a way that all branch cuts are crossed a net zeronumber of times. This is called a Pochhammer contour, it is illustrated in Figure 4. Ifthe inequalities we’ve been assuming are satisfied then we can neglect the parts of thecontour that circle the endpoints and it collapses to the one that runs between the twopoints times a simple factor that depends on the choice of branch of the integrand. Ingeneral, if the inequalities are not satisfied, we can just use the Pochhammer contour.For example we can write One might guess that this formula should also require Re c > Re b > 0, but these conditions area relic of our simple choice of contour. The Pochhammer contour we discuss below will remove theseextra conditions, but it cannot remove the condition Re ( c − a − b ) > z = 1 if this is violated. – 97 – ( a, b, c, z ) = Γ( c )Γ( b )Γ( c − b ) 1(1 − e − πib )(1 − e πi ( b − c ) ) (cid:90) C ds s a − c ( s − c − b − ( s − z ) − a , (B.23)where C is a Pochhammer contour involving 1 and ∞ and the extra factor − e − πib )(1 − e πi ( b − c ) ) cancels the sum over four traversals out to infinity and back. This expression givesthe full analytic continuation of the hypergeometric function in all of its parameters.We can also construct an integral representation of an arbitrary solution of Rie-mann’s more general differential equation. We begin by defining P C =( z − z ) α ( z − z ) β ( z − z ) γ × (cid:90) C ds ( s − z ) α (cid:48) + β + γ − ( s − z ) α + β (cid:48) + γ − ( s − z ) α + β + γ (cid:48) − ( s − z ) − α − β − γ . (B.24)By explicit substition into Riemann’s equation and rewriting as a total derivative as in(B.17), we see that see that this integral will be a solution as long as the quantity V = ( s − z ) α (cid:48) + β + γ ( s − z ) α + β (cid:48) + γ ( s − z ) α + β + γ (cid:48) ( s − z ) − α − β − γ − (B.25)has the same value at both ends of the contour. For a closed contour this meansthe contour must cross each branch cut of the integrand a net zero number of times.We may thus choose Pochhammer type contours involving any two of the four branchpoints z , z , z , z . There are six such choices, and the integral evaluated on these sixdifferent choices is proportional to our P α , P α (cid:48) , . . . defined by (B.10). The detailedproportionality depends on the branch choices and we will not work it out here.So far what we have gained in elegance over our previous formulation in terms ofthe series ( B. 1) we have arguably lost in the sophistication of the contours and thebranch choices. But for η ’s in the physical region this presentation allow a very nicerepackaging of the Liouville solution (3.25). The quantity f ( z, z ) = u (cid:101) u − v (cid:101) v definedby (3.2) has an enlightening expression in terms of the types of integrals we have beenconsidering so far. Our claim is that f = f | z − z | | z − z | α | z − z | β | z − z | γ × (cid:90) d s | s − z | α (cid:48) + β + γ − | s − z | α + β (cid:48) + γ − | s − z | α + β + γ (cid:48) − | s − z | − α + β + γ ) , (B.26)with the parameters α, α (cid:48) , . . . given by equation (3.19) and the integral being taken overthe full s -plane. As long as this integral converges it is clearly monodromy invariant, The precise form of this factor depends on a choice of branch for the integrand. A different choicewould multiply the integral by a z -independent constant. – 98 –nd it solves the holomorphic and antiholomorphic differential equations (3.4,3.5) bythe same argument as just given for the integral (B.24). We saw in section 3.3 that thesetwo properties were sufficient to uniquely determine f up to an overall normalization, sothis establishes our claim. The conditions for the convergence of this integral, expressedin terms of the η i ’s, are Re( η + η − η ) < η + η − η ) < η + η − η ) < η + η + η ) > . These are certainly obeyed in the “physical” region in Liouville. Are they equivalent toit, or more precisely to Region I from section 4.2? Actually, they imply 0 < Re η i < i , while in Region I we would have had 0 < Re η i < / 2. But for operatorsobeying the Seiberg bound, the integral converges only in Region I. For more general η i , such an expression would require a more sophisticated type of integral. For real η ’sin Region I, this expression is manifestly positive and it shows that there cannot be anyzeros of f , something that was not clear from our old expression (3.25). We wonder ifLiouville solutions for correlators with more than three heavy operators can be writtenin terms of generalizations of this integral. C Gamma Functions and Stokes Phenomena C.1 Generalities In this appendix, we review the Stokes phenomena that occur for the Gammafunction and its reciprocal, as they are closely related to the zero mode integrals ofspacelike and timelike Liouville theory. Γ( z ) has the following integral representation[56] for Re z > 0: Γ( z ) = (cid:90) ∞ t z − e − t dt = z z (cid:90) ∞−∞ e − z ( e φ − φ ) dφ, (C.1)(With a slightly different change of coordinates by t = e φ rather then t = ze φ , we couldhave put this in the form of the Liouville zero mode integral, as in eqn. (1.11).)The exponent I = − z ( e φ − φ ) in (C.1) and (C.14) has critical points at φ n = 2 πin, n ∈ Z . (C.2) For a much more detailed introduction to Stokes phenomena, see section 2 of [55]. For a treat-ment of the Gamma function along lines similar to what follows, see [12]; see [13, 14] for previousmathematical work. – 99 –o each such critical point φ n , one attaches an integration contour C n . This is a contourthat passes through the critical point φ n and along which the exponent I has stationaryphase while Re I has a local maximum. More briefly, we call this a stationary phasecontour. Alternatively, the contour C n can be defined as a contour of steepest descentfor h = Re ( − z ( e φ − φ )). For a steepest descent contour C n , it is straightforward todetermine the large z behavior of the integral (cid:90) C n dφ exp( I ) . (C.3)The maximum of Re I along the cycle C n is, by the steepest descent condition, at φ n .For large z , the integral can be approximated by the contribution of a neighborhoodof the critical point. In our case, the value of I at a critical point is − z (1 − πin ), soasymptotically (cid:90) C n dφ exp( − z ( e φ − φ )) ∼ exp( − z (1 − πin )) (C.4)(times a subleading factor that comes from approximating the integral near the criticalpoint).Now let us consider the integral (C.1, initially assuming that z is real and positive.The Gamma function is then defined by the integral (C.1), with the integration cycle C being the real φ axis. On the real axis, there is a unique critical point at φ = 0.Moreover, for real z , I is real on the real axis, and the contour of steepest descent from φ = 0 is simply the real axis. Thus, if z is real and positive, the integration cycle C in the definition of the Gamma function is the same as steepest descent cycle C , onwhich the asymptotics are given by (C.4). So we get the asymptotic behavior of theGamma function on the real axis: Γ( z ) ∼ z z e − z . (C.5)This is essentially Stirling’s formula (the factor 1 / √ πz in Stirling’s formula comesfrom a Gaussian approximation to the integral (C.4) near its critical point).Now let us vary z away from the positive z axis. The Gamma function is still definedby the integral (C.1), taken along the real φ axis, as long as Re z > 0. As soon as z isnot real, it is no longer true that the steepest descent contour C coincides with the realaxis. However, as long as Re z > In complex dimension 1, the stationary phase contour through a critical point coincides with thesteepest descent contour, but in higher dimension, the stationary phase condition is not enough todetermine C n and one must use the steepest descent condition. For much more on such matters, seesection 2 of [55]. Notice that, in our case, because the function I has opposite sign in (C.14) relativeto (C.1), the cycle C n is different in the two cases, though we do not indicate this in the notation. – 100 –escent contour C is equivalent to the real axis, modulo a contour deformation that isallowed by Cauchy’s theorem. Hence, Stirling’s formula remains valid throughout thehalf-plane Re z > z , in generalwe will have to vary the integration contour C away from the real axis. To analyticallycontinue beyond the region Re z > 0, we can let the integration contour C move awayfrom the real φ axis, so that the integral still converges and varies analytically with z .In the case of the Gamma function, there is some restriction on the ability to do this,since the Gamma function actually has poles at z = 0 and along the negative z axis.Now we come to the essential subtlety that leads to Stokes phenomena. As onevaries the parameters in an integral such as (C.1), the steepest descent contours C n generically vary smoothly, but along certain “Stokes lines” (or Stokes walls in a problemwith more variables), they jump. In our case, the only relevant parameter is z , so theStokes lines will be defined in the z -plane. For generic values of z , the C n are copies of R (topologically) with both ends at infinity in the complex φ plane. For example, for z real and positive, C n is defined by Im z = 2 πn and actually is an ordinary straight linein the φ plane. However, for special values of z , steepest descent from one critical point p leads (in one direction) to another critical point p (cid:48) . Whether this occurs dependsonly on the argument of z , so it occurs on rays through the origin in the z plane; theserays are the Stokes lines. As one varies z across a Stokes line (cid:96) , the steepest descentcontour from p will jump (on one side of (cid:96) , it passes by p (cid:48) on one side; on the otherside of (cid:96) , it passes by p (cid:48) on the other side and then heads off in a different direction).For the Gamma function, we can easily find the Stokes lines. Since the steepestdescent cycles have stationary phase, they can connect one critical point p to anothercritical point p (cid:48) only if the phase of I is the same at p and at p (cid:48) . For the critical pointat φ = 2 πin , the value of Im I is c n = Im ( − z (1 − πin )). So c n = c n (cid:48) for n (cid:54) = n (cid:48) if andonly if Re z = 0. We really should remove from this discussion the point z = 0 whereour integral is ill-defined for any noncompact contour (and the Gamma function has apole), so there are two Stokes lines in this problem, namely the positive and negativeimaginary z axis.There is one more basic fact about this subject. Away from Stokes lines, thesteepest descent contour C n are a basis for the possible integration cycles (on whichthe integral of interest converges) modulo the sort of contour deformations that arepermitted by Cauchy’s theorem. So any integration contour C – such as the one for theanalytically continued Gamma function – always has an expansion C = (cid:88) n a n C n (C.6)– 101 –here the a n are integers, and the relation holds modulo contour deformations thatare allowed by Cauchy’s theorem. Since the integral over any of the C n always has thesimple asymptotics (C.4), the asymptotics of the integral over C are known if one knowsthe coefficients a n . As one varies z in the complex plane, C will vary continuously, butthe C n jump upon crossings Stokes lines. So the asymptotic behavior of the integralfor large z will jump in crossing a Stokes line. The well-behaved problem of large z asymptotics is therefore to fix an angular sector in the complex z -plane between twoStokes lines and consider the behavior as z → ∞ in the given angular sector. Actually,this would be a full picture if there were only finitely many critical points. In the caseof the Gamma function, it will turn out that the sum (C.6) is an infinite sum if Re z < z is real and negative. To get a simpleproblem of large z asymptotics, one must keep away from the negative z axis, wherethe Gamma function has its poles, as well as from the Stokes lines. C.2 Analysis Of The Gamma Function Now let us make all this concrete. The integral (C.1) converges along contoursthat begin and end in regions where Re( − z ( e φ − φ )) → −∞ . These regions have beenshaded in Fig, 5 for the case that z is real and positive. The C n are the horizontallines Im φ = 2 πn . One can see by hand in this example that any integration cyclethat begins and ends in the shaded regions is a linear combination of the C n , as in eqn.(C.6). The real φ axis – which is the integration contour C in (C.1) – coincides with C , and is indicated in the figure as the Relevant Contour.In Fig, 6, we have sketched how the steepest descent contours C n are deformedwhen z is no longer real but still has positive real part. In passing from Fig, 5 to Fig, 6,the C n evolve continuously and it remains true that the integration contour C definingthe Gamma function is just C , modulo a deformation allowed by Cauchy’s theorem.However, the C n jump upon crossing the Stokes lines at Re z = 0. This is shownin Fig. C.2 for the case Im z > 0. While in Fig, 6, the steepest descent contours forthe Gamma function have one end in the upper left and one end to the right, in Fig.C.2, they end to the right in both directions. As a result, although nothing happensto the contour C that defines the Gamma function in going from Fig, 6 to Fig. C.2, toexpress C as a linear combination of the C n ’s, we must in Fig, C.2 take an infinite sum C = (cid:88) n ≥ C n . (C.7)– 102 – igure 5 . For the Gamma function integral (C.1) to converge, the integration contour mustbegin and end in regions of the φ plane with Re( − z ( e φ − φ )) → −∞ . These regions areshaded here for the case that z is real and positive. In addition, we show the critical pointsat φ n = 2 πin (represented in the figure by dots) and the steepest descent cycles C n , whichare the horizontal lines Im φ = 2 πn . The dependence on n of the Gamma function integral over C n is very simple: (cid:90) C n dφ exp( − z ( e φ − φ )) = exp(2 πinz ) (cid:90) C dφ exp( − z ( e φ − φ )) . (C.8)This is because a shift φ → φ + 2 πin maps C to C n and shifts I by I → I + 2 πinz .This formula is an analog for the Gamma function of eqn. (6.30) for Liouville theory.Using (C.7) and (C.8), we find that in the quadrant Re z < 0, Im z > 0, the Gammafunction is Γ( z ) = z z (cid:90) C dφ exp( − z ( e φ − φ )) = z z (cid:88) n ≥ (cid:90) C n dφ exp( − z ( e φ − φ )) (C.9)= z z ∞ (cid:88) n =0 exp(2 πinz ) (cid:90) C dφ exp( − z ( e φ − φ )) . (C.10)From (C.9), we can read off the asymptotic behavior of the Gamma function in thequadrant in question. Approximating the integral over C by the value at the maximum,and performing the sum over n , we getΓ( z ) ∼ z z e − z − exp(2 πiz ) . (C.11)– 103 – igure 6 . This figure is equivalent to Fig. 5, except that now z is complex but still withRe z > 0. (In drawing the figure, we have taken the case Im z > z is real, but the steepest descent contours are changed. Thephrase Relevant Contour labels the contour C that controls the asymptotics of the Gammafunction in this region. (Again, a prefactor analogous to the 1 / √ πz in Stirling’s formula can be found byevaluating the integral over C more accurately.) From this point of view, the polesof the Gamma function at negative integers arise not because of a problem with theintegral over C but because of a divergence of the geometric series. The factor 1 / (1 − exp(2 πiz )) in this formula is important only near the negative real axis.For use in the main text of the paper we note that a similar analysis of the casewhere Re z < 0, Im z > z ) ∼ z z e − z − exp( − πiz ) , (C.12)and that these formula can all be combined to giveΓ( z ) = (cid:40) e z log z − z + O (log z ) Re( z ) > e iπz − e − iπz e z log( − z ) − z + O (log( − z )) Re( z ) < , (C.13)where the logarithms are always evaluated on the principal branch.– 104 – igure 7 . This is the analog of Fig, 6, but now for the case that Re z < 0, Im z > 0. (ForRe z < 0, Im z < 0, just turn the figure upside down.) A Stokes phenomenon has occurred,relative to Fig, 6. In Fig, 6, each steepest descent curve for the Gamma function connectsthe shaded region in the upper left to one of the shaded regions on the right. This is also thebehavior of the contour C that defines the Gamma function. However, for Re z. Im z < 0, thesteepest descent contours connect two adjacent shaded regions on the right. To construct thecontour C that controls the Gamma function, one must take an infinite sum (cid:80) n ≥ C n . C.3 The Inverse Gamma Function We can play the same game for the inverse of the Gamma function, starting withthe integral representation1Γ( z ) = 12 πi (cid:73) C t t − z e t dt = − πi z − z +1 (cid:73) C e z ( e − φ + φ ) e − φ dφ. (C.14)In (C.14), C t starts at real −∞ − i(cid:15) , encircles the branch cut along the negative real t axis, and ends up at −∞ + i(cid:15) .To arrive at the right hand side of (C.14), we have made the coordinate change t = ze − φ , which differs slightly from the transformation t = ze φ used in deriving(C.1). The critical points are still at φ n = 2 πin . Once again the shaded regions inFig, C.3 are the ones in which the integral is convergent, as Re I → −∞ , where now I = z ( e − φ + φ ). The steepest descent contours are shown in Fig, C.3 and connectadjacent shaded regions on the left of the figure. For z real and positive, the image– 105 – igure 8 . For the inverse Gamma function integral, with z real and positive, the criticalpoints and steepest descent contours are as shown here. Regions in which the integral isconvergent are again shaded. The function 1 / Γ( z ) is defined by an integral over a contour C that coincides with the steepest descent contour C associated to the critical point at z = 0.This contour connects two adjacent shaded regions on the left of the figure. in the φ plane of the contour C t in the t plane is the steepest descent contour C thatpasses through the critical point at φ = 0. Since C is a steepest descent cycle, theasymptotic behavior of the integral in this region is just e I , with I evaluated at thecritical point. So in this region, 1Γ( z ) ∼ z − z +1 e z . (C.15)(As always, subleading factors, including powers of z , can be determined by evaluatingthe integral more accurately near the critical point.)When we vary z , the regions in the φ plane in which the integral converges moveup and down slightly. The integration contour C defining the inverse Gamma functionvaries smoothly and continues to connect two adjacent shaded regions. However, whenRe z < 0, the qualitative behavior of the steepest descent contours C n changes. As– 106 – igure 9 . For Re z < z > 0, the steepest descent contours C n connect a region onthe left to the upper right, as shown here. On the other hand, after varying z , the integrationcontour C that defines the inverse Gamma function still connects two adjacent regions on theleft. (It is not drawn here.) It is therefore no longer true that C equals one of the steepestdescent contours; rather, C is a difference C − C . sketched in Fig, 9, for such values of z , each C n connects a shaded region in the left ofthe φ plane to the upper right. The formula expressing C in terms of steepest descentcontours is now C = C − C . (Here C − C starts in a shaded region on the left, headsto the upper right, and then doubles back to an adjacent shaded region on the left –thus reproducing C .) Accordingly, (C.15) is modified to1Γ( z ) ∼ (1 − e πiz ) z − z +1 e z . (C.16)Of course, this result for the asymptotic behavior of 1 / Γ( z ) is equivalent to theresult that we had earlier for the asymptotic behavior of Γ( z ). However, seeing thisbehavior directly from the Stokes phenomena that affect an integral (rather than theinverse of an integral) is useful background for the body of this paper.Just as in the relation between spacelike and timelike Liouville theory as studiedin this paper, the Gamma function and the inverse Gamma function are essentiallygiven by the same integral, evaluated on different integration contours. This fact is– 107 –lso related to the functional equation obeyed by the Gamma function. We have2 πi Γ(1 − z ) = (cid:90) C t t z − e t dt. (C.17)The integrand on the right hand side can be converted to the integrand of the Gammafunction integral (C.1) if we substitute t → − t . This maps C t to another integrationcontour C (cid:48) t and gives 2 πi Γ(1 − z ) = exp( iπ ( z − (cid:90) C (cid:48) t t z − e − t dt. (C.18)So the inverse Gamma function, apart from some elementary factors and a substitution z → − z , is given by the same integral as the Gamma function, but with a differentintegration contour. D Semiclassical Conformal Blocks In this appendix we give a derivation, based on [48], of an asymptotic formula for theVirasoro conformal block at large intermediate operator weight. The original argumentwas somewhat terse and implicitly involved certain assumptions about the semiclassicallimit of correlators, so in our view its validity has not been established completelyrigorously from the definition (2.40). It has however survived stringent numerical tests[7], and in our discussion in section 5.5 we were comfortable assuming it to be true.We will expand out the argument here and try to be clear about what the assumptionsare. The asymptotic formula is F (∆ i , ∆ , x ) ∼ (16 q ) ∆ , (D.1)with q = exp [ − πK (1 − x ) /K ( x )] (D.2)and K ( x ) = 12 (cid:90) dt (cid:112) t (1 − t )(1 − xt ) . (D.3)The idea is to study a five-point function of a light degenerate primary operatorwith four primary scalar fields of generic operator weight in the semiclassical limit c (cid:29) 1. The external conformal weights ∆ i are taken to be of order c , and the internalweight ∆ p is initially also taken to be of this order; it will eventually be taken much– 108 –arger than c . If we parameterize the central charge as c = 1 + 6( b + 1 /b ) , thenthe conformal weight of the light degenerate operator is − − b . We will write thecorrelator as (cid:104)O ( z , z ) O ( z , z )Ψ( z, z ) O ( z , z ) O ( z , z ) (cid:105) , (D.4)where Ψ is the degenerate operator and we choose | z | > | z | > | z | > | z | > | z | . Thiscorrelator obeys equation (2.43), which here gives (cid:34) b ∂ z + (cid:88) i =1 (cid:18) ∆ i ( z − z i ) + 1 z − z i ∂ i (cid:19)(cid:35) (cid:104)O O Ψ O O (cid:105) = 0 . (D.5)We can also expand the correlator using the operator product expansion [17] O ( z , z ) O ( z , z ) = (cid:88) p C p | z | p − ∆ − ∆ ) (cid:88) k, (cid:101) k ( z ) k ( z ) (cid:101) k β pk β p, (cid:101) k O { k, (cid:101) k } p ( z , z ) , (D.6)which allows us to extract all dependence on z : (cid:104)O O Ψ O O (cid:105) = (cid:88) p C p | z | p − ∆ − ∆ ) × (cid:88) k, (cid:101) k ( z ) k ( z ) (cid:101) k β pk β p, (cid:101) k (cid:104)O O Ψ O { k, (cid:101) k } p (cid:105) . (D.7)In these formulae, as discussed below equation (2.40) the sum over k is only heuristicand more precisely includes a sum over all descendants at a given level k . The operators O { k, (cid:101) k } p are the Virasoro descendants of the primary O p . Now say that we defineΨ p ( z, z ; z i , z i ) ≡ (cid:104)O O Ψ O p (cid:105)(cid:104)O O O p (cid:105) . (D.8)In the semiclassical limit we can think of the function Ψ p as the classical expectationvalue of the degenerate operator in the presence of the other operators, and since thelight operator has weight of order c we expect it to have a finite limit at c → ∞ . InLiouville theory, this is the statement that light operators just produce O ( b ) factorsin the correlation function as in equation (2.26). This argument could be applied to any CFT with a c that can be large and primary operators withthe desired weights. ”Light” in this context just means that the operator weight of the degeneratefield scales like c . Liouville is a theory that fits the bill, but we will use general CFT language toavoid the subtleties of Liouville factorization. We assume here that as in Liouville, the only primaries that appear in the operator productexpansion are scalars. We could drop this assumption at the cost of slightly more complicated formulas,the result (D.1) would be the same. – 109 –o far we have written only exact formulas, but we now come to the first ap-proximation: in the semiclassical limit we claim that the same formula holds also fordescendants of O p with the same function Ψ p : (cid:104)O O Ψ O { k, (cid:101) k } p (cid:105) ≈ Ψ p ( z, z ; z i , z i ) (cid:104)O O O { k, (cid:101) k } p (cid:105) . (D.9)The justification for this is that the correlator (cid:104)O O Ψ O { k, (cid:101) k } p (cid:105) can be written in termsof a series of differential operators acting on (cid:104)O O Ψ O p (cid:105) . The differential operatorsare of the form [17] L − m = (cid:88) i =3 , ,z (cid:20) ( m − i ( z i − z ) m − z i − z ) m − ∂ i (cid:21) . (D.10)Here ∆ z ≡ − − b . Similarly the correlator (cid:104)O O O { k, (cid:101) k } p (cid:105) can be written in terms of asimilar series of differential operators acting on (cid:104)O O O p (cid:105) , but with the sum in (D.10)being only over 3,4. The point however is that in the semiclassical limit we expect (cid:104)O O O p (cid:105) ∼ e − c S cl (D.11)for some S cl , and since we have taken ∆ , ∼ c while Ψ p and ∆ z are both O ( c ), we seethat the i = z term in (D.10) becomes unimportant, and also that in the i = 3 , p . This establishes (D.9). In the semiclassicalapproximation we thus have (cid:104)O O Ψ O O (cid:105) ≈ (cid:88) p C p | z | p − ∆ − ∆ ) Ψ p ( z, z ; z , z , z , z , z , z ) × (cid:88) k, (cid:101) k ( z ) k ( z ) (cid:101) k β p,k β p, (cid:101) k (cid:104)O O O { k, (cid:101) k } p (cid:105) . (D.12)We can make this formula more elegant by defining F (∆ i , ∆ p , z i ) ≡ ( z ) ∆ p − ∆ − ∆ (cid:88) k ( z ) k β p,k (cid:104)O O O { k, } p (cid:105)(cid:104)O O O p (cid:105) , (D.13)after which we get (cid:104)O O Ψ O O (cid:105) ≈ (cid:88) p Ψ p C p (cid:104)O O O p (cid:105) F (∆ i , ∆ p , z i ) F (∆ i , ∆ p , z i ) . (D.14)We note for future convenience that F becomes the Virasoro conformal block F after sending z → ∞ , z → , z → x, z → 0. Based on its definition, we can guessthat F has a semiclassical limit [48] of the form F ∼ e − c f cl , (D.15)– 110 –here f cl is called the “semiclassical conformal block”. We will now study the implications of the expressions (D.14, D.15) for the differen-tial equation (D.5). It will be convenient to view z and z as independent. We observethat for generically different ∆ p ’s, the various terms in the sum over p in (D.14) havedifferent monodromy as z circles z . For each p the different terms in the differentialequation (D.5) have the same monodromy, so in order for the equation to be solved by(D.14) it seems reasonable to expect that it must actually be solved separately for each p . In the semiclassical limit the action of the derivatives with respect to z i on Ψ p issuppressed as in our discussion below (D.10), so we find that the differential equationcan be converted into an ordinary differential equation just involving Ψ p : (cid:34) ∂ z + (cid:88) i =1 (cid:18) δ i ( z − z i ) − C i z − z i (cid:19)(cid:35) Ψ p = 0 , (D.16)with C i = ∂ i ( S cl + f cl ) . (D.17)Here δ i = b ∆ i . In this type of differential equation the parameters C i are referred toas “accessory parameters”; clearly if we can learn something about them then we arelearning about the semiclassical conformal block. In [48] it was shown that for ∆ (cid:29) c ,a combination of symmetry and the WKB approximation allows a determination of all C i , and thus of the semiclassical conformal block in that limit.This argument begins with the observation that there are three linear relations onthe accessory parameters which come from demanding that the term in round bracketsin (D.16), which is related to the semiclassical limit of the stress tensor by the Wardidentity (cid:104) T ( z ) O O O O (cid:105) = (cid:80) i =1 (cid:16) ∆ i ( z − z i ) + z − z i ∂ i (cid:17) (cid:104)O O O O (cid:105) , vanishes like z − atinfinity: (cid:88) i C i = 0 (cid:88) i ( C i z i − δ i ) = 0 (cid:88) i (cid:0) C i z i − δ i z i (cid:1) = 0 . (D.18) In fact this exponentiation has never actually been proven directly from the definition (2.40),although it has been checked to high order numerically. We thank A. B. Zamolodchikov for a discussionof this point, and for a summary of his unpublished numerical work. – 111 –here is thus only one independent accessory parameter, which we take to be C . Ifwe then take the limit z → ∞ , z → , z → x, z → 0, equation (D.16) becomes (cid:20) ∂ z + δ z + δ ( z − x ) + δ (1 − z ) + δ + δ + δ − δ z (1 − z ) − C x (1 − x ) z ( z − x )(1 − z ) (cid:21) Ψ p = 0 . (D.19)In the same limit (D.17) simplifies to C = ∂ x f cl . (D.20)One way to parametrize the effect of C is to study the monodromy of Ψ p as z circles both x and z . We will work out this relationship below, but we first notethat this will give us what we want because we can also determine this monodromyfrom the definition of Ψ p . The reason is that as discussed in section 2.4, the four-point function (cid:104)O O Ψ O p (cid:105) receives contributions only from two intermediate conformalweights. If we parametrize conformal weights as ∆ = α ( b + 1 /b − α ) then these are∆ ± = α ± ( b + 1 /b − α ± ), with α ± = α p ± b/ 2. These contributions behave near z = 0like z ∆ ± − ∆ p − ∆ z , which semiclassically becomes z (cid:16) ± √ − b ∆ p (cid:17) , so their monodromymatrix in this basis is M = e iπ (cid:16) √ − b ∆ p (cid:17) e iπ (cid:16) − √ − b ∆ p (cid:17) . (D.21)So far everything we have said is valid for ∆ p ∼ c , but we now observe that if we take∆ p (cid:29) c then we can solve (D.19) in the WKB approximation, where we include onlythe first and last terms. This gives approximate solution:Ψ p ∼ exp (cid:34) ± (cid:112) x (1 − x ) C (cid:90) zz dz (cid:48) (cid:112) z (cid:48) (1 − z (cid:48) )( z (cid:48) − x ) (cid:35) . (D.22)Comparison with the mondromy matrix (D.21) in the same limit gives C ≈ − π b ∆ p x (1 − x ) K ( x ) . (D.23)Finally we can integrate this by observing that K ( x ) ∂ x K (1 − x ) − K (1 − x ) ∂ x K ( x ) = − π x (1 − x ) , which follows from the Wronskian of the hypergeometric differential equationobeyed by K ( x ) = π F ( , , , x ), with the normalization determined by expanding near x = 0. This gives f cl ≈ πb ∆ p K (1 − x ) K ( x ) + constant. (D.24)– 112 –e can determine the constant to be − b ∆ log 16 by matching the series expansionnear x = 0 to the normalization of the leading term in the conformal block F (∆ i , ∆ p , x ) = x ∆ p − ∆ − ∆ (1 + O ( x )) , (D.25)which at last gives F ∼ (16 q ) ∆ . (D.26) E An Integral Expression for log Γ( z ) In this appendix we derive the identitylog Γ( z ) = (cid:90) ∞ dtt (cid:104) ( z − e − t − e − t − e − zt − e − t (cid:105) Re[ z ] > . (E.1)We by differentiating the definition (C.1) of the Gamma function with respect to z : Γ (cid:48) ( z ) = d (cid:82) ∞ t z − e − t dtdz = (cid:90) ∞ t z − log | t | e − t dt (E.2)= (cid:90) ∞ dtt z − e − t (cid:90) ∞ dss [ e − s − e − st ] = lim (cid:15) → (cid:90) ∞ dtt z − e − t (cid:90) ∞ (cid:15) dss [ e − s − e − st ](E.3)= lim (cid:15) → (cid:90) ∞ dtt z − e − t (cid:90) ∞ (cid:15) dss e − s − lim (cid:15) → (cid:90) ∞ dtt z − (cid:90) ∞ (cid:15) dss e − t e − st (E.4)Here we have used integral identity log b = (cid:82) ∞ dxx ( e − x − e − bx ). While the integralcurrently is convergent, we have to reexpress the lower limit in terms of (cid:15) in order tobreak the integral up into divergent pieces. We do this to allow us to enact coordinatechanges on the second term. The divergences will cancel each other out. We make thecoordinate change ρ = t (1 + s ) on the second term. The change is linear in t . Weexchange the ρ and s integrals and evaluate the t and ρ integrals givinglim (cid:15) → (cid:90) ∞ dtt z − e − t (cid:90) ∞ (cid:15) dss e − s − lim (cid:15) → (cid:90) ∞ (cid:15) dss (cid:90) ∞ dtt z − e − t e − st (E.5)= Γ( z ) lim (cid:15) → (cid:90) ∞ (cid:15) dss e − s − lim (cid:15) → (cid:90) ∞ (cid:15) dss (cid:90) ∞ dρ s (cid:16) ρs + 1 (cid:17) z − e − ρ (E.6)= Γ( z ) (cid:104) lim (cid:15) → (cid:90) ∞ (cid:15) dss e − s − lim (cid:15) → (cid:90) ∞ (cid:15) dss (1 + s ) (cid:16) s + 1 (cid:17) z − (cid:105) . (E.7)We see that the Gamma function factors out of the integral representation of Γ (cid:48) ( z ).To continue, we need to make one more coordinate change before putting the integrals– 113 –ack together 1 + s = e (cid:101) s which yields ds = e (cid:101) s d (cid:101) s . Since (cid:15) → 0, when s ∼ (cid:15) then1 + s = e (cid:101) s = 1 + (cid:101) s + O ( (cid:101) s ) ∼ (cid:15) and (cid:101) s ∼ (cid:15) . The lower limit remains unchanged inthe second integral. (E.7) then becomesΓ (cid:48) ( z ) = Γ( z ) (cid:104) lim (cid:15) → (cid:90) ∞ (cid:15) dss e − s − lim (cid:15) → (cid:90) ∞ (cid:15) dss (1 + s ) (cid:16) s + 1 (cid:17) z − (cid:105) (E.8)= Γ( z ) (cid:104) lim (cid:15) → (cid:90) ∞ (cid:15) dss e − s − lim (cid:15) → (cid:90) ∞ (cid:15) d (cid:101) se (cid:101) s ( e (cid:101) s − e (cid:101) s ( e − (cid:101) s ) z − (cid:105) (E.9)= Γ( z ) (cid:104) lim (cid:15) → (cid:90) ∞ (cid:15) dss e − s − lim (cid:15) → (cid:90) ∞ (cid:15) d (cid:101) s (1 − e − (cid:101) s ) e − z (cid:101) s (cid:105) (E.10)Now we can put the integrals back together and take the limit (cid:15) → 0, resulting inΓ (cid:48) ( z )Γ( z ) = (cid:90) ∞ ds (cid:104) s e − s − e − zs (1 − e − s ) (cid:105) . (E.11)Integrating (E.11) with respect to z yields (E.1)log Γ( z ) = (cid:90) z d (cid:101) z Γ (cid:48) ( (cid:101) z )Γ( (cid:101) z ) = (cid:90) ∞ dss (cid:104) ( z − e − s − e − s − e − zs (1 − e − (cid:101) s ) (cid:105) (E.12) F Integral over the SL(2, C ) Moduli of the Saddle Point forThree Light Operators In section 4.3, we encountered the following integral: I ( σ , σ , σ ) ≡ (cid:90) dµ ( α, β, γ, δ ) (cid:16) | β | + | δ | (cid:17) σ (cid:16) | α + β | + | γ + δ | (cid:17) σ (cid:16) | α | + | γ | (cid:17) σ . (F.1)In the text we claimed this integral is given by I ( σ , σ , σ ) = π Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ + σ − σ )Γ(2 σ )Γ(2 σ ) , (F.2)and in this appendix we will show it. We will see along the way that the integral isdivergent unless certain inequalities involving the σ i ’s are satisfied, so we will assumethem as we come to them and then in the end define the integral away from thoseregions by analytic continuation.Following [7], we begin by performing the coordinate change ξ = βδ , ξ = α + βγ + δ , and ξ = αγ . The measure becomes dµ ( α, β, γ, δ ) = d ξ d ξ d ξ | ( ξ − ξ )( ξ − ξ )( ξ − ξ ) | , and the integral– 114 –ecomes I ( σ i ) = (cid:90) d ξ d ξ d ξ | ξ | − − ν | ξ | − − ν | ξ | − − ν × (1 + | ξ | ) − σ (1 + | ξ | ) − σ (1 + | ξ | ) − σ (F.3)Here ν = σ − σ − σ , ν = σ − σ − σ , ν = σ − σ − σ . This expression is invariantunder the SU (2) subgroup of SL (2 , C ) given by ξ i → fξ i + g − gξ i + f , with | f | + | g | = 1. Wecan use this to send ξ → ∞ : I ( σ , σ , σ ) = π (cid:90) d ξ d ξ | ξ − ξ | − − ν (1 + | ξ | ) − σ (1 + | ξ | ) − σ . (F.4)From here the result is quoted in [7] without further explanation, we will fill in thesteps. The evaluation will involve repeated use of the defining integrals of the Γ and β functions, which we reproduce for convenience:Γ( x ) = (cid:90) ∞ t x − e − t dt Re[ x ] > β ( x, y ) = Γ( x )Γ( y )Γ( x + y ) = (cid:90) λ x − (1 − λ ) y − dλ = (cid:90) ∞ dt t x − (1 + t ) x + y (F.6)Re[ x ] > 0, Re[ y ] > . We will also need a lesser-known version of the Feynman parameters which is used inclosed string theory. | z | A = 1Γ( A/ (cid:90) ∞ dtt A/ − e − t | z | with Re A > 0. (F.7)Now to business. Starting with integral (F.4), we convert it into four real integralsdefined by the coordinate change ξ = x + iy ξ = u + iv. (F.8)Noting that d ξ = − i ( dξ ∧ dξ ) = − i ( dx + idy ) ∧ ( dx − idy ) = dx ∧ dy with a similaridentity for d ξ , we find equation (F.4) becomes I = π (cid:90) ∞−∞ dx dy du dv (( x − u ) + ( y − v ) ) − − ( σ − σ − σ ) (1 + x + y ) − σ (1 + u + v ) − σ . (F.9) This identity can be easily derived by changing variables to (cid:101) t = t | z | . – 115 –e now use the identity (F.7) three times with A = 4 σ , σ , ν , which requiresRe σ > , Re σ > , Re ( σ + σ − σ ) < 1, to get: I = π (cid:90) ∞−∞ dx dy du dv (cid:90) ∞ dψ dχ dκ ψ σ − σ − σ χ σ − κ σ − Γ(1 + σ − σ − σ )Γ(2 σ )Γ(2 σ ) (F.10) × exp [ −{ ψ [( x − u ) + ( y − v ) ] + χ (1 + x + y ) + κ (1 + u + v ) } ] . Collecting powers of x, y, u, v we have I = π (cid:90) ∞−∞ dx dy du dv (cid:90) ∞ dψ dχ dκ ψ σ − σ − σ χ σ − κ σ − Γ(1 + σ − σ − σ )Γ(2 σ )Γ(2 σ ) (F.11) × exp [ −{ ( x + y )( ψ + χ ) + ( u + v )( ψ + κ ) − ψ ( ux + yv ) + χ + κ } ] . Completing the square for x and y and factoring out ( ψ + χ ) gives I = π (cid:90) ∞−∞ dx dy du dv (cid:90) ∞ dψ dχ dκ ψ σ − σ − σ χ σ − κ σ − Γ(1 + σ − σ − σ )Γ(2 σ )Γ(2 σ ) (F.12) × exp [ − ( ψ + χ ) (cid:110)(cid:16) x − uψψ + χ (cid:17) + (cid:16) y − vψψ + χ (cid:17) (cid:111) ] × exp − ( ψ + χ ) (cid:110) ( u + v ) ψ + κψ + χ − ( u + v ) (cid:16) ψ ( ψ + χ ) (cid:17) + χ + κψ + χ (cid:111) ] . The x, y integral is now a straightforward Gaussian integral. Changing variables by thelinear shift s = x − uψψ + χ , t = y − vψψ + χ and performing the integral over s and t we find I = π (cid:90) ∞−∞ du dv (cid:90) ∞ dψ dχ dκ ψ σ − σ − σ χ σ − κ σ − ( ψ + χ )Γ(1 + σ − σ − σ )Γ(2 σ )Γ(2 σ ) (F.13) × exp [ − (cid:110) ( u + v ) (cid:104) ψ + κ − ψ ψ + χ (cid:105) + χ + κ (cid:111) ] . We can also do the u and v integrals, which give I = π Γ(1 + σ − σ − σ )Γ(2 σ )Γ(2 σ ) (cid:90) ∞ dψ dχ dκ ψ σ − σ − σ χ σ − κ σ − e − χ − κ [( ψ + κ )( ψ + χ ) − ψ ] . (F.14)So far the required manipulations have been fairly obvious, but to proceed furtherwe need to perform a rather nontrivial coordinate change on ψ , χ , and κ . We will For the u , v integrals in (F.13) to converge the prefactor of the u and v exponetial terms in (F.13): (cid:2) ψ + κ − ψ ψ + χ (cid:3) ≥ 0. It can be shown that this is so, by putting everthing over a common denominator,and noting that the term becomes a sum of positive quantities. – 116 –otivate it by answer analysis of (F.2). Summing the exponents of the ψ , χ , and κ terms in the numerator of (F.14), we get one minus the argument of the first Gammafunction in (F.2). This implies that in the new coordinates, ψ , χ , and κ must have somecommon factor ρ in order to produce this first Gamma function. This also means ρ mustequal χ + κ due to definition (F.5). Now χ and κ are linearly independent, therefore wepropose the coordinate change: χ = ρ cos θ and κ = ρ sin θ . To determine ψ we seethat if we take the arguments of last two Gamma functions in the numerator of (F.2),they add up to 2 σ . That means the factor Γ( σ + σ − σ )Γ( σ + σ − σ σ ) = β ( σ + σ − σ , σ + σ − σ ) must be a factor in the integral. This Beta function will involve factors the ofsin θ and cos θ from χ and κ . A quick glance at the factors of χ and κ in (F.14) showsthat they are not correct. However, if ψ instead included a factor of cos ( θ ) sin ( θ ),we would get the proper factors for the Beta function in terms of the integral over θ . ψ must then have one factor of ρ and one factor of cos ( θ ) sin ( θ ). Lastly ψ must belinearly independent from χ and κ since we want a one to one mapping, so ψ musthave a third factor which we will call ζ . The coordinate change is then ψ = ρζ cos ( θ ) sin ( θ ) χ = ρ cos ( θ ) κ = ρ sin ( θ ) . (F.15)The Jacobian for this is dψdχdκ = 2 ρ cos ( θ ) sin ( θ ) dρdθdζ. (F.16)After the performing the transformation and collecting common terms, (F.14) be-comes I = 2 π Γ(1 + σ − σ − σ )Γ(2 σ )Γ(2 σ ) (cid:90) ∞ (cid:90) ∞ (cid:90) π/ dζ dρ dθ ρ σ + σ + σ − − ζ σ − σ − σ − (1 + ζ ) × (cos θ ) σ +2 σ − σ − (sin θ ) σ +2 σ − σ − exp [ − ρ ] . (F.17)We now change the θ coordinate to λ = sin θ which gives dθ = dλ √ λ (1 − λ ) and thus I = π Γ(1 + σ − σ − σ )Γ(2 σ )Γ(2 σ ) (cid:90) ∞ (cid:90) ∞ (cid:90) dζ dρ dλ ρ σ + σ + σ − − ζ σ − σ − σ − (1 + ζ ) × (1 − λ ) σ + σ − σ − λ σ + σ − σ − exp [ − ρ ] . (F.18)These transformations have completely factorized the integral, which we can at lastevaluate using (F.5) and (F.6). The result is: I = π Γ( σ + σ + σ − σ + σ − σ )Γ( σ + σ − σ )Γ( σ + σ − σ )Γ(2 σ )Γ(2 σ )Γ(2 σ ) . (F.19)– 117 –n addition to the inequalities mentioned below (F.9), in doing these final integralswe had to assume that Re( σ + σ − σ ) > σ + σ − σ ) > σ + σ − σ ) > σ + σ + σ ) > . These inequalities are easy to understand from equation (F.4); they come from conver-gence of the integral when ξ → ξ and when either ξ or ξ or both go to infinity. Theinequalities Re( σ ) > 0, Re( σ ) > σ + σ − σ ) < 1, does not. This finalinequality is somewhat mysterious becuse it breaks the symmetry between σ , σ , σ and does not follow in any obvious way from the convergence of (F.4). In fact it isa relic of our method of evaluating the integral; in deriving (F.10) we used (F.7) tointroduce a factor of σ − σ − σ ) which then cancelled out of the integral in the end.We could have avoided this inequality by deforming the contour in (F.7). For anotherway to see that the inequality is spurious, we observe that if we had used the symmetryto send ξ or ξ to infinity instead of ξ in deriving (F.4) then a different inequalityrelated to this one by symmetry would have appeared that we did not need to use inour evaluation. 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