PPrepared for submission to JHEP
Summing over Geometries in String Theory
Lorenz Eberhardt
School of Natural Sciences, Institute for Advanced Study,Einstein Drive 1, Princeton, NJ 08540, USA
E-mail: [email protected]
Abstract:
We examine the question how string theory achieves a sum over bulkgeometries with fixed asymptotic boundary conditions. We discuss this problemwith the help of the tensionless string on M × S × T (with one unit of NS-NSflux) that was recently understood to be dual to the symmetric orbifold Sym N ( T ).We strengthen the analysis of [1] and show that the perturbative string partitionfunction around a fixed bulk background already includes a sum over semi-classicalgeometries and large stringy corrections can be interpreted as various semi-classicalgeometries. We argue in particular that the string partition function on a Euclideanwormhole geometry factorizes completely into factors associated to the two bound-aries of spacetime. Central to this is the remarkable property of the moduli spaceintegral of string theory to localize on covering spaces of the conformal boundary of M . We also emphasize the fact that string perturbation theory computes the grandcanonical partition function of the family of theories (cid:76) N Sym N ( T ). The boundarypartition function is naturally expressed as a sum over winding worldsheets, each ofwhich we interpret as a ‘stringy geometry’. We argue that the semi-classical bulkgeometry can be understood as a condensate of such stringy geometries. We alsobriefly discuss the effect of ensemble averaging over the Narain moduli space of T and of deforming away from the orbifold by the marginal deformation. a r X i v : . [ h e p - t h ] F e b ontents backgrounds 15 Hyperbolic 3-manifolds 63
A.1 Hyperbolic 3-space 63A.2 General properties 64A.3 (Co)homology of 3-hyperbolic manifolds 64A.4 Examples 65
B Some facts about Riemann surfaces 67
B.1 Differentials 67B.2 Divisors 67B.3 Classification of line bundles 68B.4 Spin structures 69B.5 Riemann-Roch theorem 69
C Subgroups of the fundamental group and covering spaces 70
C.1 Regular covering spaces 70C.2 Relation to homomorphisms to S N D Uniformization 73
D.1 Teichm¨uller space and the mapping class group 73D.2 Fuchsian uniformization 74D.3 Schottky uniformization 76D.4 Simultaneous uniformization 77
E (Branched) complex projective structures 78
E.1 Complex projective structures 78E.2 Branched complex projective structures 81
F Topologically twisted partition function 83
F.1 Topological twist 83F.2 Change of variables 84
The AdS/CFT correspondence has provided us with a unique glimpse into the prop-erties of quantum gravity and consistency of the correspondence is still surprisingfrom a variety of angles.There are essentially two classes of proposals that seem to have qualitativelydifferent properties. On the one hand, there are ‘top down’ constructions derived– 1 – (S (cid:116) S ) = + + . . . Figure 1 . The gravitational computation of the partition function with two boundarycircles in a two-dimensional theory of gravity. The presence of the wormhole geometrygenerically destroys factorization of the answer and leads to an ensemble dual on the blueboundary. from string theory, such as the duality between type IIB string theory on AdS × S and N = 4 SYM or the duality between string theory on AdS × S × T and adeformation of the symmetric orbifold CFT Sym N ( T ) [2, 3]. On the other hand,there are ‘bottom up’ constructions of dual pairs, which start from a semiclassicalgravity theory. Since it is not known how to quantize gravity in higher dimensions,these examples are all low-dimensional. The prime example is JT-gravity [4, 5],whose boundary dynamics is given by the Schwarzian theory [6–8]. The Schwarziantheory in turn describes the infrared dynamics of the SYK model [9, 10] that is alsodescribed by a double-scaled matrix model [11, 12]. The main difference is thatthe duality in these cases is between a gravity theory and an ensemble of CFTs.There has been recently also a proposal for an exotic U(1)-gravity theory in threedimensions, that is described holographically by free bosons averaged over the modulispace of Narain lattices [13, 14].The averaged examples behave more intuitively from a gravity point of view.They naturally involve a sum over all bulk geometries, including also Euclideanwormhole geometries that are responsible for much of the recent progress on theinformation paradox [15, 16]. Inclusion of wormhole geometries spoils factorizationof the boundary partition function on disconnected boundaries and leads to theensemble interpretation [17, 18]. See Figure 1. The typical member of the ensemblein those dualities also generically seems to exhibit chaos. The dual gravitationaltheory captures only the averaged signal and it is an important open problem toexplain where the random noise comes from in the gravitational description. See[19–26] for recent progress on this.It has been a subject of debate how to relate these two classes of proposals.The bulk properties of the stringy examples of the correspondence away from theirsupergravity regime are much more alien to us, which is mainly due to our lack of un-derstanding and computational power of string theory (or M-theory). In particular,for such stringy examples of holography, there cannot be a non-zero wormhole correc-– 2 –ion to the partition function, since it would be inconsistent with a single boundarytheory. In the supergravity approximation, the stringy examples AdS × S andAdS × S × T admit Euclidean wormhole solutions [27–29], in tension with a singlelocal boundary CFT. This already indicates that string theory modifies the ‘sumover geometries’ prescription in a non-trivial way.In this paper, we revisit the question directly within string theory. Our exampleis the symmetric orbifold CFT Sym N ( T ). This precise theory is conjectured to bedual to ‘tensionless’ string theory on AdS × S × T [30–32], see also [33, 34] forearlier work. The string background is supported only by one unit of NS-NS flux(and no R-R flux). This duality has been subjected to very stringent tests: not onlyhas the full spectrum been matched with the symmetric product orbifold [30], butalso some correlators [31, 32, 35] (including higher genus correlators [36, 37]) havebeen compared.While this model is under very good control, it also has some downsides. Thefirst of these is the non-geometric and non-local nature of the theory. A small ten-sion means that strings are very floppy and generically can wind around asymptoticregions or cycles of the geometry with little cost of energy. The generic state of thetheory has lots of winding strings. This is in particular true for the graviton andthus there does not seem to be a local notion of geometry. However, it is somewhatpremature to disregard classical geometry entirely. When treating the string per-turbatively, we start with a sigma model on a fixed background and of course thestring should ‘feel’ the background geometry. Thus, we can still ask questions aboutsums over geometries. While the concept of a background geometry is not very well-defined in the regime in which we are working, the concept of a worldsheet theoryis. For a large background, these are equivalent – any classical background gives riseto a worldsheet theory. Thus, we will essentially replace the notion of summing overgeometries by sums over different worldsheet theories.While questions such as ‘which manifolds should we sum over to obtain theboundary partition function?’ have a clear answer in this framework, the result issomewhat difficult to interpret from a semiclassical gravity point of view. Indeed, theanswer to this question that we shall advocate in this paper is that the string partitionfunction is independent of the background bulk manifold, large stringy correctionsaround the given background ensure that all the other semi-classical bulk geometriesare automatically taken into account. This even extends to wormhole geometries.So instead of Figure 1, the correct answer in the tensionless string is to take eitherof the two geometries and consider all the stringy excitations on it – the resultwill be the same! In this sense, the two contributions in Figure 1 should not beincluded separately, since we would count the same state multiple times. We already By a wormhole solution we mean here and in the following a connected bulk manifold whoseconformal boundary is disconnected. – 3 –onjectured this to be the case in [1].This answer was already anticipated long ago from a purely boundary perspectivein free N = 4 SYM [38]. There it was seen that the thermal partition function offree N = 4 SYM exhibits a phase transition due to an exponential number of lightstrings. This is the Hawking-Page transition emulated by a large number of stringycorrections. However, a bulk description was missing at the time.This proposal seems counterintuitive at first glance. The action of a classicalbackground is of order O ( G − ), whereas quantum corrections around it should beof order 1 in G N . This seems to make it impossible for the above statement to betrue. This naive argument is circumvented as follows. It is true that the, say, toruspartition function of a single string is of order one, but only as long is that stringis ‘short’. The contribution can be enhanced by a factor of N by taking a stringthat either winds N times around a cycle or asymptotic region of the geometry or N strings that each wind once around it. In the example at hand there can be avery large quantum correction to the classical result due to strings that effectivelywind G − times such cycles. This is essentially the Hagedorn transition, since thevery large number of light strings can lead to a macroscopic contribution. These areheavy enough to backreact on the geometry and change it effectively into a differentgeometry. This still does not explain intuitively why the wormhole partition function fac-torizes. The answer that we find to this question is perhaps a bit disappointing.Naively, one could have expected that there are string configurations like the firstpicture of Figure 2 that stretch between the two boundaries and cause a correlationbetween the two boundary theories, thus leading to non-factorization. However, suchstrings actually do not exist in the model. It turns out that all the strings of themodel stay close to the boundary of the space as in the second picture of Figure 2.Since the geometry is asymptotically AdS, they actually stay asymptotically far outand do not explore the bulk. This picture makes it intuitively clear how factorizationis achieved. One could even say that there is no bulk, since an observer in such astringy universe would have no way of detecting it.We should mention that the actual mechanism that achieves this ‘localization’of the string worldsheets to the boundary is quite surprising and works thanks to anactual localization in the moduli space of Riemann surfaces. The string path integrallocalizes to such Riemann surfaces that cover the boundary holomorphically, whichreduces the integral over the moduli space to a sum. This localization principle wasdiscussed and proved in [31, 32, 36, 37] for correlation functions in global AdS . Weextend the proof to arbitrary higher genus worldsheets and all (possibly singular)hyperbolic three manifolds, which are the manifolds that can serve as background In [39, 40], the question was examined in vector and matrix models and a similar result wasfound. – 4 – igure 2 . The wormholes with perturbative string excitations on top of them. Thesepictures are cartoons, since the actual model that we will consider is three-dimensionaland instead of the green worldline curves, we will have worldsheets wrapping the geometry.It turns out that the model realizes only the possibility in the right picture: there are noperturbative string excitations that connect the two asymptotic regions, but only separatestring excitations that stay close to the two boundaries. geometry for the string (modulo some technical issues that will be explained). Eventhough the main focus of this work lies on partition functions, we explain that ourproof also goes through for correlation functions, in which case the worldsheet local-izes on certain ramified covers of the boundary.We will not be able to compute the actual value of the string partition function,but explain the general mechanisms behind the independence of the string partitionfunction on the bulk geometry in this model. Even after the localization propertyhas been demonstrated, this is non-trivial because the sphere partition function ofthe worldsheet theory does not follow the localization principle and is not confinedto the boundary of the bulk. Thus, the typical string configuration actually looksroughly like in Figure 3. It is therefore not true that the quantum corrections aroundthe background know nothing of the background. We argue that their dependenceon the bulk geometry is such that they cancel the sphere partition function and thecombined string partition function is independent of the chosen bulk manifold. Itis also interesting that the sum over all possible covering maps of the boundary canarise in very different manners for different bulk manifolds.We have just explained that the tensionless string of this model does not ‘see’ thebulk. It is therefore surprising that actually all classical bulk manifolds make an ap-pearance in the boundary theory. For this, it is crucial to work in a grand canonicalensemble, where the number of strings is not fixed, since this is the natural ensem-ble of string perturbation theory. The dual ‘CFT’ is hence actually (cid:76) N Sym N ( T )with an appropriate chemical potential conjugate to N held fixed. When tuningthis chemical potential to special values, the grand canonical partition function is– 5 – igure 3 . A typical string configuration for string theory on the wormhole geometry. Asin Figure 6, this picture is drawn in one dimension lower. The small strings in the middleare spheres, whereas the winding strings close to the boundary are genus g ≥ dominated by very large N and classical geometry seems to emerge as a conden-sate of the winding strings. We confirmed this explicitly only for geometries with asingle torus boundary such as thermal AdS, since the simplest wormhole geometryhas genus 2 boundaries, which makes explicit computations difficult. The emergenceof classical geometries as condensates of ‘stringy’ geometries (i.e. winding strings)becomes physically clearer once we average the boundary CFT over a suitable setof parameters. As an example, we discuss this for the Narain moduli space of T .Using the results of [13, 14], the averaged string partition function is then expressedas a sum over ‘micro-geometries’, i.e. geometries that fill in worldsheets (or connectthem by ‘micro-wormholes’). Such geometries have a very large number of sheetsthat meet asymptotically at the boundary of the bulk. When these geometries alignproperly, they can form an emergent macroscopic geometry in the classical sense. Outline.
The following is an outline of the paper. In the remaining part of theintroduction, we first explain our philosophy on how we (approximately) computestring partitions with fixed boundary conditions using string perturbation theory.We then discuss in Section 1.2 some of the ideas of this paper with the help of a verysimple toy model in two bulk dimensions, where instead of worldsheets, the boundaryis covered by worldlines. This model exhibits already some (but not all!) featuresof the tensionless string and is technically much simpler to treat. The remainingpart of the paper is roughly divided into two parts which can be read more or lessindependently.Sections 2–5 are more technical and their main goal is to establish the indepen-dence of the string partition function on the bulk geometry. Of those, Section 2reviews the computation of the partition function of the symmetric orbifold andSection 3 reviews and develops the formulation of the tensionless string on different– 6 –ulk manifolds. We use this formalism to demonstrate that the string path inte-gral does indeed localize in the moduli space of Riemann surface. We explore theconsequences of this property in Section 5 and explain how to introduce the grandcanonical potential from the bulk point of view. These sections make use of sometechnology from the theory of Riemann surfaces and hyperbolic manifolds. For thebenefit of the reader, we collected the relevant material in Appendices B, C, D, E andA. Appendix F contains a discussion of the topologically twisted partition functionof the sigma model on T that is a part of the worldsheet theory.Section 6 is more physical in nature and discusses the physical interpretationof classical geometries emerging as condensates from ‘stringy geometries’. We alsodiscuss the effect of introducing an ensemble average in the symmetric orbifold andthe ‘micro-geometries’ that we mentioned above.We summarize our main findings in Section 7 and discuss open problems andfuture directions. Suggested reading.
To simplify the reading process, we have depicted the depen-dencies of the various sections in Figure 4. We suggest that the reader can jumpdirectly after the introduction to section 6 and check back on the other sections asneeded. Appendices contain background information and are included to make thepaper self-contained. They are mostly not necessary to understand the main text.
Here, we make some comments about computing the full non-perturbative string pathintegral with fixed boundary conditions from the bulk point of view. We discuss thisusing string perturbation theory. The question should properly be addressed withinstring field theory. but to make computation feasible we use string perturbationtheory as an approximation. We will treat spacetime as Euclidean.The problem in string perturbation theory is that we treat the background geom-etry as fixed and consider stringy excitations around the background. Let us fix someasymptotic boundary conditions of the bulk spacetime manifolds (such as asymptot-ically AdS). Then to compute the string partition function with these boundaryconditions, we should in principle sum over all bulk manifolds with the appropriateboundaries and include stringy corrections around those bulk geometries: Z string ( N ) = (cid:88) bulk saddle geometries M with ∂ M = N Z string ( M ) . (1.1)Here, Z string ( M ) is the perturbative string partition function around the background M and Z string ( N ) is the string partition function with fixed boundary conditions.This is essentially how we would compute compute a semiclassical gravity partitionfunction. There are several problems with this:– 7 –. Introduction2. Partitionfunction ofthe symmetricorbifold 3. Tensionlessstring theoryon locally AdS backgrounds4. Localization5. Backgroundindependence6. Stringyand classicalgeometry Appendix AAppendix BAppendix CAppendix DAppendix EAppendix FAppendix AAppendix B Figure 4 . The basic organization of the paper. Arrows indicate strong logical dependen-cies and we recommend to read the relevant sections first. Dashed lines indicate weakerdependencies.
1. String theory also includes non-perturbative objects (in g string ) and they shouldin principle also be included in the string partition function on a fixed manifold M .2. In general, we expect some backreaction of the string on the geometry. Thus,when including very heavy string excitations in the partition function, they canchange the background geometry. Hence we should only include ‘light’ stringexcitations around a fixed background.3. String theory contains also other massless fields besides the metric. Thus, thesum over geometries should rather be a sum over supergravity backgrounds.4. The background geometries M should be saddles, i.e. satisfy the supergravityequation of motion. These equations get α (cid:48) corrected and in principle therecould be also be also background values for all the massive string modes. Themore correct statement would be to sum instead of over bulk manifolds over– 8 –ifferent worldsheet sigma-models with the correct asymptotic boundary con-dition.5. In the gravitational path integral, we should sum over all bulk manifolds,whether they are saddles or not. Of course, saddle geometries lead to a dom-inant contribution. In string perturbation theory, we do not know even inprinciple how to include non-saddle geometries, since these do not correspondto consistent worldsheet theories.These issues make it clear that (1.1) can at best be an approximation (or at leastthe definition of Z string ( M ) is not straightforward).In the example that we consider in this paper, we will argue that (1.1) fails muchmore profoundly. For the tensionless string on a manifold M × S × T the sumover manifolds is superfluous. It is already fully contained Z string ( M ), as long as weinclude also heavy string excitations. We are lucky that in this instance the answerwe compute turns out to be exact and no effect of backreaction has to be taken intoaccount. So instead of the sum in (1.1), we have Z string ( N ) = Z string ( M ) for any bulkmanifold M with ∂ M = N . Indications for this in other models were also found in[39–41]. Some of the physical intuition for the setup in this paper can understood from avery simple toy model. Let us consider a quantum mechanical model with a one-dimensional Hilbert space. The unique state in this model is taken to have energy E . We now consider N copies of the model and gauge the obvious S N -symmetrythat permutes the copies. This gauging implements physically the intuition that thedifferent copies are indistinguishable.Let us compute the partition function of this theory. The N -fold product ofthe original theory has still a one-dimensional Hilbert space and the unique state isinvariant under the S N symmetry. Thus, the partition function is simply Z N = tr (cid:0) e − βH (cid:1) = e − βNE . (1.2)Let us see how this simple result arises from a path integral point of view. TheEuclidean quantum mechanics is considered on a thermal circle of length β . Gaugingof S N -symmetry instructs us to sum over all S N -bundles over the thermal circle. The N copies of the model can be viewed as N separate thermal circles. Gauging of S N sums over all possible joinings of these circles. We displayed the possibilities for N = 2 and N = 3 in Figure 5. In general such a bundle is determined by a partitionof N that labels the lengths of the connected components of the bundle. These S N -bundles have nontrivial symmetry factors that one needs to take into account. For– 9 – igure 5 . The S N -bundles in the toy model for N = 2 and N = 3. .a partition N = (cid:80) ∞ m =1 N m m , the symmetry factor is (cid:89) m N m ! m N m , (1.3)which accounts for the fact that we can perform a cyclic relabelling of the coveringwithin each connected components and permute identical components. We then have Z N = e − βNE (cid:88) partitions N = (cid:80) m N m m (cid:89) m N m ! m N m = e − βNE . (1.4)The last identity follows from the fact that partitions label conjugacy classes of S N and the size of these conjugacy classes is N ! times the symmetry factor (1.3). Grand canonical ensemble.
We now want to interpret this system holographi-cally. For this, the path integral point of view is useful. We view the covering spacesof the boundary circle as particles that propagate near the boundary of the Euclideanbulk spacetime. This is depicted in Figure 6. In this sense, our very simple toy modelis dual to free particles in the bulk. Hence the spacetime theory would be a QFTon AdS . In a QFT, the number of particles in the background is usually not fixed.Thus, we would like to consider both sides of the ‘holographic correspondence’ inthe grand canonical ensemble, where N is not fixed. Instead, we fix a correspondingfugacity variable p . The boundary grand canonical partition partition function reads Z = ∞ (cid:88) N =0 p N Z N = ∞ (cid:88) N =0 p N e − βNE = 11 − p e − βE . (1.5) Worldline action.
We now argue that this model can indeed be taken seriouslyby constructing a worldline action for the particle. Let us choose polar coordinatesto describe the bulk. For definiteness, let us take the bulk to have a flat metricd s = d r + r d φ , (1.6)where 0 ≤ r ≤ β π and 0 ≤ φ ≤ π . Then we take the length of the worldline as itsaction: S worldline = E (cid:90) d τ (cid:112) G µν ∂ τ X µ ∂ τ X ν . (1.7)– 10 – bulkworldline Figure 6 . We fill the boundary thermal circle with a bulk spacetime. On this bulkspacetime, we let free particles propagate. In the picture, the worldline represents thesecond covering space of Figure 5. where X ( τ ) = ( r ( τ ) , φ ( τ )) are the embedding coordinates in spacetime and G µν is the spacetime metric. Of course, the equations of motion of this theory allowonly for straight worldlines, but one could cure this by adding a very steep Mexicanhat-type potential that confines the worldline to the boundary region of the bulk.Then possible worldlines are labelled by their topological winding number near theboundary of spacetime. This can be viewed as an (approximate) type of localization:only worldlines that cover the boundary isometrically are allowed to contribute tothe partition function. Hence the length of the worldsheet is dβ for some integer d ≥
1. The connected partition function of this model is simply Z connbulk = ∞ (cid:88) d =1 d e − βdE = − log (cid:0) − e − βE (cid:1) . (1.8)We can further refine this partition function by counting the winding number. Wewill do this here in an-hoc manner and simply posit that Z connbulk = ∞ (cid:88) d =1 p d d e − βdE = − log (cid:0) − p e − βE (cid:1) = ⇒ Z bulk = 11 − p e − βE , (1.9)in agreement with what we found in the boundary theory. Bulk independence.
We see that the bulk result didn’t depend a lot on the detailsof the bulk. We could have equally well chosen a different bulk manifold, such as We are suppressing some details here that are not important to understand the physics. Thereare one-loop determinants around the classical solutions that should be taken into account for acomplete description. – 11 –
Figure 7 . The indistinguishability of different bulk manifolds. Since the red worldlinestays close to the boundary, different topologies are not distinguishable for it. a genus 1 surface with a circle removed. We only cared about the properties of thebulk manifold near its boundary (thanks to the Mexican hat potential). This isessentially because of our choice of potential that pushes the worldline out towardsthe boundary of the bulk. In pictures, this is Figure 7.
Condensation.
One might say that the ‘bulk’ theory essentially knows nothingabout the bulk, since all the degrees of freedom seem to be localized near the bound-ary. This is not entirely true. These worldline geometries do reflect some parts of theclassical geometry. For this, we notice that the grand canonical partition functionhas a pole at p = e βE . We shall argue that the meaning of this pole is the following.When tuning p close to e βE , worldline geometries with large N dominate the defi-nition of the grand canonical partition function (1.5). This means that the typicalbulk picture features an extremely large number of worldlines. On the pole, theseworldlines essentially reconstruct the classical bulk geometry, or in a precise sensethat we discuss in Section 6.2 the bulk geometry is a condensate of the worldlines.In our toy example, we do not know anything about the effective spacetime theory,so we cannot say which classical bulk geometry this should be. In the string theoryexample that we examine in this paper, we see all bulk geometries emerging thatsatisfy the classical equations of motion. The location of the pole in p is related tothe on-shell spacetime action of the geometry. The residue is essentially the one-loopdeterminant around that geometry (that is trivial in our toy example). This allowsone to reconstruct the grand canonical partition function also alternatively from allthe classical partition functions. Euclidean wormholes.
One can consider our toy model also on wormhole ge-ometries. It is clear that what we said for the bulk independence continues to holdfor wormhole geometries. If several boundaries are present, we can introduce differ-ent chemical potentials for the various boundaries and the grand canonical partitionfunction simply factorizes into its constituents. In particular, the multi-boundary– 12 –rand canonical partition function in out model is simply n (cid:89) i =1 − p i e − β i E . (1.10)The poles in this partition function should again correspond to different bulk ‘ge-ometries’. However, they clearly only account for disconnected geometries. For thestring the situation is much more complicated, since (saddle) wormholes are onlyexpected for genus ≥ Averaging.
In this final paragraph, we modify our theory a bit. We want tointroduce an average of theories, but with our simple theory with a single state, thisis not possible. We replace our simple quantum mechanical theory with anotherquantum mechanical theory X . We will assume that X itself has some parametersover which we can average (for example the SYK model). We then take N copies of X and gauge the permutation symmetry. For a single realization, we can again expressthe boundary partition function as a sum over all possible covering maps of theboundary circle. But we can now interpret the partition functions on the coveringmaps themselves using holography. Once we average over the parameters of thetheory, they are computed holographically by filling in the covering geometry in allpossible ways, thus leading to a sort of ‘micro-geometry’ consisting of N sheets. Thisincludes wormholes that connect different disconnected components of the coveringspace. The model that is constructed in this way is in some sense stringy, sinceit has exponentially more geometries (in N ) then X itself. Moreover, these ‘micro-geometries’ can align themselves to lie on top of each other and form one macroscopicgeometry. This leads to a more concrete picture of the condensation process. Let us review the partition function of the symmetric orbifold Sym N ( X ) for somebase theory X , following [42, 43]. We consider a Riemann surface Σ g of genus g . Because of the conformal anomaly,the partition function does not only depend on the moduli of the surface, but alsoon an explicit metric. Let us fix the hyperbolic metric, i.e. the metric with constantnegative curvature − g (or a flat metric in the case g = 1) and writeΣ g = H / Γ F g , (2.1) Averaging over E would essentially average over the size of the bulk, which is not what wewant. – 13 –or some discrete (Fuchsian) subgroup of Γ F g ⊂ PSL(2 , R ) and H the upper halfplane. Γ F g is determined up to overall conjugation. The upper half plane with thePoincar´e metric induces the hyperbolic metric on Σ g . It is convenient to compute partition functions in a grand canonical ensemble. Holo-graphically, the grand canonical ensemble corresponds to the situation where thenumber of strings in the background is not kept fixed. We will discuss this in Sec-tion 5 and 6 in detail. From a CFT point of view, we are computing the generatingfunction of the partition function Z Sym( X ) = ∞ (cid:88) N =0 p N Z Sym N ( X ) , (2.2)where we suppressed the dependence on the moduli of the surface. Here p = e πiσ isthe corresponding fugacity. We set by convention Z Sym ( X ) = 1.In general, the genus g partition function of a permutation orbifold with group S N on a genus g surface can be written as follows: Z Sym( X ) (cid:0) H / Γ F g (cid:1) = exp (cid:88) subgroups H of Γ F g up to conjugation p [Γ F g :H] [Γ F g : H] Z X (cid:0) H / H (cid:1) , (2.3)where Z X is the partition function of the seed theory X . Here, we assumed that thetheory X is bosonic, see the discussion below for the case with fermions. Relation to covering maps.
More geometrically, the sum over subgroups of π (Σ g ) up to conjugation can be viewed as a sum over all possible unbranchedconnected covering surfaces of the original theory. [Γ F g : H] is the degree of thecorresponding covering map: Z Sym( X ) (Σ g ) = exp ∞ (cid:88) N c =1 p N c N c (cid:88) connected covering surfaces (cid:101) Σ N c( g − of degree N c Z X (cid:16)(cid:101) Σ N c ( g − (cid:17) . (2.4)It does not matter whether the sum also extends over infinite coverings, becausethe denominator ensures that they do not contribute. The exponential generatesalso disconnected covering surfaces and inserts the correct combinatorial symmetryfactors. Here and in the following we use N c to refer to the degree of connected The discussion also holds for g = 1, in which case the universal covering space is C and therelavant group Γ = Z × Z acts by translations. – 14 –overing maps and N to refer to the degree of disconnected covering maps. In thisform, the formula even holds true without specifying a constant curvature metric,because there is a natural metric on the covering surface that is the pull back ofthe metric on the base space along the covering map. Enumerating subgroups ofa Fuchsian group Γ F g systematically is quite difficult, but counting the number ofterms is possible. We have collected some relevant facts in Appendix C. There wealso explained the role of the symmetric group in this construction. Fermions.
Finally, let us consider the case where X contains fermions and we wantto compute the partition function of the symmetric orbifold on Σ g with a fixed spinstructure. Such a spin structure on the base surface corresponds to a spin bundle S , whose pullback along the covering map induces a natural spin structure on thecovering surface. This is the spin structure wich enters the right hand side of eq. (2.4). backgrounds In this section, we set up the general framework to describe the tensionless stringon Euclidean backgrounds of the form M × S × T , where M is a hyperbolic 3-manifold, i.e. a space that is locally Euclidean AdS . This section is partially a reviewand is based on [30, 32, 37]. We employ the hybrid formalism [44] that continuesto be well-defined in the tensionless limit. We start with global AdS , in which casethe space has PSL(2 , R ) × PSL(2 , R ) symmetry (or in the Euclidean case PSL(2 , C )symmetry). The hybrid formalism starts with the following worldsheet theory:PSU(1 , | k ⊕ top. twisted T ⊕ ghosts . (3.1)Here, PSU(1 , | k is the WZW model on the supergroup PSU(1 , | T sigmamodel has N = 4 supersymmetry and is topologically twisted and hence contributes c = 0 towards the central charge. Finally, the ghost sector consists of the usual bc ghosts of the bosonic sector and an additional ρ -ghost that replaces the βγ -ghost.It is a timelike free boson with screening charge and contributes c = 28 towards thecentral charge.The parameter k corresponds to the amount of NS-NS flux in the backgroundand the tensionless string is obtained for k = 1. We will make this choice in theremainder of the paper. BRST cohomology.
The BRST operator is relatively complicated to write downin these variables, but we shall not have need of its explicit form. The hybrid string– 15 –s formulated as an N = 4 topological string. Thus, there is a topologically twisted N = 4 algebra on the worldsheet whose supercharges we denote byconformal weight 1: G + , ˜ G + , (3.2)conformal weight 2: G − , ˜ G − , (3.3)since as usual due to the topological twist, the conformal weights of the superchargesare shifted. This theory actually has two BRST operators given by G +0 and ˜ G +0 . Thephysical state subspace of the Hilbert space is identified with the double cohomology(similarly for the right-movers): G +0 | Φ (cid:105) = 0 , ˜ G +0 | Φ (cid:105) = 0 , | Φ (cid:105) ∼ | Φ (cid:105) + G +0 ˜ G +0 | Ψ (cid:105) . (3.4)Physical states are moreover the top components of spin multiplets of the R-symmetry of the algebra. This is necessary for the integrated vertex operator (cid:90) G − ¯ G − Φ (3.5)to be an su (2) singlet. The second cohomology achieves the restriction of physicalstates to the small Hilbert space, which in the RNS formalism is often imposed byhand. Correlation functions.
Correlation functions are defined as in the N = 4 topo-logical string [44, 45] (cid:90) M g (cid:42) g − (cid:89) i =1 | G − ( µ i ) | (cid:20)(cid:90) ˜ G + ¯˜ G + (cid:21) g − (cid:90) J ¯ J n (cid:89) i =1 (cid:90) G − ¯ G − Φ i (cid:43) . (3.6)Here J is the Cartan-element of the R-symmetry SU(2) (that has conformal weight1). G − ( µ ) is the usual pairing between Beltrami differentials and conformal fieldsof weight 2. For the N = 4 algebra, there are many inequivalent ways of doing thetopological twist. They are related by rotating the supercharges. Effectively, onejust replaces some of the G − by ˜ G − or equivalently, some of the ˜ G + by G + . It wasexplained in [32] that (for k = 1) there is a unique choice up to equivalence that isnon-vanishing. The amplitudes are defined by (cid:90) M g (cid:42) g − (cid:89) i =1 | G − ( µ i ) | g − (cid:89) i = g | ˜ G − ( µ i ) | (cid:20)(cid:90) ˜ G + ¯˜ G + (cid:21) g − (cid:90) J ¯ J n (cid:89) i =1 (cid:90) ˜ G − ¯˜ G − Φ i (cid:43) . (3.7)Compared to the previous formula, some supercharges G − have been replaced with˜ G − (and similarly for the right-movers). The result is independent on the precisechoice of these switches (up to normalization).– 16 – .2 Free field realization The formalism simplifies considerably for k = 1, since there is a free field realizationof PSU(1 , | in terms of symplectic bosons and free fermions [30, 46]. We follow theconventions of [32]. The free fields have the following defining (anti-)commutationrelations: [ ξ αr , η βs ] = ε αβ δ r + s, , { ψ αr , χ βs } = ε αβ δ r + s, . (3.8)Here, ξ α and η β are spin- symplectic bosons and ψ α and χ β are usual spin- fermions. α , β take values in { + , −} and are SU(2) spinor indices. These freefields generate the superalgebra u (1 , | . The generators of this superalgebra areidentified with the bilinears J m = − ( η + ξ − ) m − ( η − ξ + ) m , K m = − ( χ + ψ − ) m − ( χ − ψ + ) m , (3.9a) J ± m = ( η ± ξ ± ) m , K ± m = ± ( χ ± ψ ± ) m , (3.9b) S αβ + m = ( χ β ξ α ) m , S αβ − m = − ( η α ψ β ) m , (3.9c) U m = − ( η + ξ − ) m + ( η − ξ + ) m , V m = − ( χ + ψ − ) m + ( χ − ψ + ) m . (3.9d)It is also convenient to define the combinations Z = U + V and Y = U − V . Thesuperalgebra u (1 , | contains the algebra su (1 , | as a subalgebra that consistsof all the currents except for Y . su (1 , | in turn is obtained as a central extensionof psu (1 , | , the central generator being Z . Thus, we essentially need to set Z = 0in order to recover psu (1 , | from u (1 , | . This is done by introducing thefollowing BRST operator: Q = cZ , (3.10)where we introduce a new bc ghost system with h ( b ) = 1 and h ( c ) = 0. Note that[ Q , Y ( z )] = ∂c ( z ). Thus Q has the effect of removing both Y and Z from the freefield realization. Fortunately the Z – Z OPE does not have a central term and hencewe have indeed Q = 0. Representations.
Representations of psu (1 , | are straightforward to describein the free field representation. We will be brief here, for more details see [30, 32].Unflowed representations are are identified with the R-sector representation of thefree fields. Positive modes of the free fields annihilate the primary state | m , m (cid:105) .Zero modes of the symplectic bosons act as ξ +0 | m , m (cid:105) = (cid:12)(cid:12) m , m + (cid:11) , η +0 | m , m (cid:105) = 2 m (cid:12)(cid:12) m + , m (cid:11) , (3.11) ξ − | m , m (cid:105) = − (cid:12)(cid:12) m − , m (cid:11) , η − | m , m (cid:105) = − m (cid:12)(cid:12) m , m − (cid:11) . (3.12) The spin structures of the symplectic bosons and the fermions are coupled since the supercharges S αβγ have to be single-valued. – 17 –he sl (2 , R ) spin of the representation is given by j = m − m . On top of this, thefermions span a 2 = 4-dimensional Clifford module. We define χ +0 | m , m (cid:105) = ψ +0 | m , m (cid:105) = 0 (3.13)so that the four states are obtained by action of χ − and ψ − . The four states consistof two R-symmetry su (2) singlets and an su (2) doublet. Evaluating the zero mode Z on the states leads to Z = (cid:40) m − m − , su (2) doublet ,m − m − ± , su (2) singlets . (3.14)Thus, the only possible BRST invariant representation of psu (1 , | has the bosonicsubrepresentations ( j = 0 , ) ⊕ ( j = 1 , ) ⊕ ( j = , ) [30]. Spectral flow.
More representations are obtained from acting with spectral flowon the symplectic bosons. There are two independent spectral flow symmetries thatwe denote by σ (+) and σ ( − ) . σ ( ± ) ( ξ ∓ r ) = ξ ∓ r ± , σ ( ± ) ( η ± r ) = η ± r ∓ , (3.15a) σ ( ± ) ( ψ ∓ r ) = ψ ∓ r ∓ , σ ( ± ) ( χ ± r ) = χ ± r ± . (3.15b)Unspecified actions are trivial. The composition σ = σ (+) ◦ σ ( − ) is the usual spectralflow action on psu (1 , | ; it leaves the generators Y and Z invariant. There is alsothe opposite composition ˆ σ = σ (+) ◦ ( σ ( − ) ) − . The existence of these spectral flowautomorphism allows one to define spectrally flowed representations that are obtainedby composing the unflowed representations with the spectral flow automorphism.Focussing on the the spectral flow automorphism σ , this allows one to define thecorresponding vertex operators V wj,h ( x, z ) . (3.16)Here, w ∈ Z ≥ is the amount of spectral flow. We traded the labels m and m forthe more physical labels of sl (2 , R ) spin j and spacetime conformal weight h . Theadditional label x corresponds to the position of the vertex operator in spacetime.We take x to be complex (and the corresponding variable for the right-movers is the We suppress here again the right-movers, the vertex operator would be properly denoted by V wj,h, ¯ h ( x, ¯ x, z, ¯ z ). These vertex operators are affine primary (in a spectrally flowed sense) with respect to the sl (2 , R ) subalgebra. It will not be important for us what the precise properties of the vertexoperators are with respect to the su (2) subalgebra and we may take them to be affine primary (inthe unflowed sense). We can restrict to w ≥
1, since only these fields correspond to fields in spacetime. – 18 –omplex conjugate), which specifies the reality of the theory and implies that we areworking in Euclidean AdS .Due to the existence of the second spectral flow operator ˆ σ , the same psu (1 , | representation actually appears multiple times in the free field representation. In-deed, by spectrally flowing with respect to ˆ σ , one replicates the same representationof psu (1 , | . In particular, we can look at the spectrally flowed images of the vac-uum representation. This leads to copies of the vacuum representation of psu (1 , | ,but they are not in the vacuum w.r.t. the free fields. In particular, we shall makeuse of the field W ( u ) that is obtained by twice flowing the vacuum representationw.r.t. ˆ σ [32]. Focussing on the bosons, it has defining OPEs ξ α ( z ) W ( u ) ∼ O ( z − u ) , η α ( z ) W ( u ) ∼ O (( z − u ) − ) . (3.17) String correlators.
Let us now discuss how the prescription for correlators in thehybrid formalism (3.7) combines with the free field realization. It was discovered in[32], that the replacement G − → ˜ G − effectively shifts the sl (2 , R ) spin j of the vertexoperators down by 1 unit. It does not matter how we distribute these shifts on thevertex operators. This potentially violates the conservation of the Y -current of thefree field realization. In order to get a non-zero result for the correlator one has toinsert n + 2 g − W -fields. Thus, one is lead to study the correlators (cid:42) g − (cid:89) α =1 W ( u α ) n (cid:89) i =1 V w i j i ,h i ( x i , z i ) (cid:43) (3.18)in the PSU(1 , | WZW model. Conservation of the U(1) current Y imposes thecondition (cid:88) i j i = 2 − g − n . (3.19)These correlators are the main object of study in the hybrid formalism. Even thoughwe are mainly interested in studying the partition functions, we see that we have toinsert at least one vertex operator satisfy this constraint (except for g = 1). Thus,we will keep most parts of this paper general and they also apply to correlationfunctions.Until now, we have not properly implemented the gauging of the U(1)-current.Besides introducing a pair of bc -ghosts with BRST operator (3.10), we also have tointegrate over the moduli space of flat U(1)-bundles over the Riemann surface, asindicated by the presence of g zero modes of b . The moduli space of line bundles onthe Riemann surface is the Jacobian, defined byJac(Σ g ) = C g / ( Z g ⊕ Ω Z g ) , (3.20)where Ω is the period matrix. We have collected some background on Riemannsurfaces in Appendix B. The isomorphism between the moduli space of line bundlesand the Jacobian is given by the Abel Jacobi map.– 19 –he prescription of the correlation functions for the hybrid string (3.7) alreadyincorporates g additional integrals besides the moduli space integrals. We can in-terpret these integrals as an integral over the Jacobian. Of these g integrals, g − T sigma-model is topologically twisted, itscharge conservation is anomalous. Thus to get a non-trivial partition function wehave to insert charged operators in the correlator. This is precisely achieved by thedescription (3.7). We now want to reduce this theory from global Euclidean AdS to a quotient spaceAdS / Γ for a discrete group Γ ⊂ PSL(2 , C ). We do not necessarily require Γ toact properly discontinuously, which means that the quotient space can have orbifoldsingularities. Consistency of the theory seems to require the deficit angles to havethe form 2 π (1 − M − ) for M ∈ Z ≥ . Groups Γ that act properly discontinuously areknown as Kleinian groups and we collected some background material in Appendix A. Spin structure.
There is a small subtlety here since we are considering a super-symmetric theory. We also want to specify a spin structure on the target manifold.This is essentially achieved by lifting Γ ⊂ PSL(2 , C ) to a subgroup of SL(2 , C ). Thedifferent liftings correspond to the different spin structures. In the following, we con-sider the spin structure part of the spacetime geometry and hence always work withthe lift Γ ⊂ SL(2 , C ). This lift is discussed further in the context of uniformizationsin Appendix D. Action on worldsheet fields.
Being a subgroup of SL(2 , C ), the action of Γ onthe various fields in the worldsheet theory is described by the action of the generators J a . Thus, we are essentially taking a usual orbifold in CFT. In particular, Γ does notact on the fields W ( u α ). Γ does however act on the vertex operators V w i j i ,h i ( x i , z i ) andcorrespondingly, one can define twisted vertex operators. They are not needed todiscuss partition functions. We will only consider untwisted vertex operators in thispaper. The partition function is obtained in the special case when we chose w i = 1and h i = 0 for all vertex operators. In this case the vertex operators correpond tothe vacuum of the dual CFT and are hence actually independent of x . Note that itis not possible to choose n = 0, since it would be impossible to satisfy the constraint(3.19) (except for g = 1). Twisted sectors.
When computing the orbifold correlators, we are effectively per-forming a gauging by a discrete subgroup Γ ⊂ SL(2 , C ). This gauging is achieved bysumming over all principal Γ-bundles over the Riemann surface Σ. We can write (cid:104)· · · (cid:105) = 1 | Γ | (cid:88) Γ-bundles ρ Γ (cid:104)· · · (cid:105) ρ Γ , (3.21)– 20 –here in (cid:104)· · · (cid:105) ρ Γ all fields have twisted boundary conditions. Γ-bundles are specifiedby a homomorphism ρ Γ : π (Σ g ) −→ Γ , (3.22)that tells us how the fields are twisted when we move around the cycles of Σ g andhence we simply identify ρ Γ with the bundle. π (Σ g ) depends of course on a choice ofbase point. Changing the base point results in an overall conjugation of the homomor-phism and specifies the same bundle. Thus Γ-bundles are in 1-to-1 correspondencewith the set Hom( π (Σ g , z ) , Γ) / Γ, which we will just write by Hom( π (Σ g ) , Γ). Inthe following all homomorphisms from π (Σ g ) are understood up to overall conjuga-tion. This is the analogue of the more well-known situation at genus 1, where onehas to sum over all twisted boundary conditions. Since π ( T ) = Z , we haveHom( π ( T ) , Γ) ∼ = (cid:104) a, b ∈ Γ | ab = ba , ( a, b ) ∼ ( gag − , gbg − ) (cid:105) , (3.23)where the group elements a and b are the images under the homomorphism ρ Γ of the α - and β -cycle of the torus.Recall that we also perform a U(1)-gauging via the BRST operator Q (3.10)to reduce the free theory to PSU(1 , | and correspondingly integrate over all flatU(1)-bundles that can also be specified by a homomorphism ρ U(1) : π (Σ g ) −→ U(1). Actually, we do not need any reality and consider the homomorphism into thecomplexified group ρ C × : π (Σ g ) −→ C × consisting of all complex numbers excludingzero. It is convenient to combine these two homomorphisms as follows ρ ≡ ρ Γ ⊗ ρ C × : π (Σ g ) −→ Γ × C × ⊂ SL(2 , C ) × C × . (3.24) Normalization.
We should also mention that the orbifold has another effect – itmultiplies the partition function by | Γ | − . Since the orbifold group is infinite, thisfactor is naively zero. However, we expect that upon regularization, we can get anon-zero value. Since we will in any case not be able to determine the precise valueof the partition function (or correlation functions), we will not discuss this factorfurther. For simple geometries it can be calculated explicitly [1]. Let us summarize the most important features of the formalism. The form of thestring partition function/correlation function (3.7) is basically dictated by demandingcharge conservation. The g additional integrals can be interpreted as integrals overthe Jacobian and implement precisely the gauging that reduces the free fields to psu (1 , | . There is more freedom in the abelian case and several homomorphisms can correspond to thesame bundle. See Appendix B.3. – 21 –he natural prescription for the string correlation function in the free-field real-ization takes the following schematic form: (cid:88)
Γ-bundles ρ Γ (cid:90) M g,n (cid:90) Jac(Σ g ) (cid:42) g − n (cid:89) α =1 W ( u α ) ∂H ( z ) g − (cid:89) β =1 e − iH ( v β ) n (cid:89) i =1 V w i j i ,h i ( x , z i ) (cid:43) ρ Γ ⊗ ρ C × . (3.25)Here, we have suppressed insertions of the ghosts σ and ρ (but they are neededto obtain a non-vanishing correlator). We have also suppressed right-movers. Theinsertion of the fields W ( u α ) is necessary in the free-field realization. ∂H is theR-symmetry U(1) current of the internal CFT on T . Since the internal CFT istopologically twisted, the U(1) charge conservation is anomalous which necessitatesthe inclusion of these terms. We also have inserted the ∂H -current, which comes fromthe J -current in (3.7) and is needed for a non-vanishing correlator, see Appendix F.We have not been very precise about these terms. Since we will not be able tocompute them fully, we only need the qualitative structure. The summation overΓ-bundles reduces the theory from global AdS to AdS / Γ. The integrand doesdepend on the locations of u α and v β , but does so in a trivial free way. The spins j i satisfy the following constraint: (cid:88) i j i = 2 − g − n . (3.26)In the following, we demonstrate that the integrand localizes in the total space M g,n × Jac(Σ g ). Thus, the correlation functions reduce to a discrete sum insteadof integrals.
We now show that the worldsheet partition function localizes on covering surfaces ofthe boundary. Our argument generalizes the argument of [31, 32, 36, 37] and proceedsin several steps. Readers only interested in the result may skip to Section 4.3. ξ ± . The strategy is to consider the expressions λ ± ( z ) = (cid:42) ξ ± ( z ) g − n (cid:89) α =1 W ( u α ) n (cid:89) i =1 V w i j i ,h i ( x i , z i ) (cid:43) ρ . (4.1)The subscript ρ means that we compute the correlation function with twisted bound-ary conditions that are specified by the homomorphism ρ = ρ Γ ⊗ ρ C × : π (Σ g ) −→ Γ × C × ⊂ SL(2 , C ) × C × , (4.2) Of course, this space is not a direct product, but we continue to denote it like this for notationalsimplicity. – 22 –s described in Section 3.3. We stress that SL(2 , C ) × C × (and not GL(2 , C )) is thecorrect group once we remember that there are also the free fermions ψ ± (and χ ± )that have twisted boundary conditions in the path integral, see the free field realiza-tion (3.9). Those twisted boundary conditions are specified by the homomorphism ρ C × .It is sometimes useful to combine λ = ( − λ − , λ + ), which takes values in the two-dimensional holomorphic vectorbundle S ⊗ E ρ , where S is a fixed spin structure and E ρ is the flat bundle determined by ρ . For a fixed spin structure S on the worldsheet,any other spin structure can be obtained by tensoring with a Z -bundle. Viewing Z ⊂ C × as a subgroup, we can combine the necessary sum over spin structures withthe sum (or integral) over non-trivial U(1)-bundles. We will see below that there isa natural spin structure S , but for now we keep it arbitrary. To summarize, let uscollect the properties of λ ± ( z ):1. λ ( z ) is a section of the holomorphic vectorbundle S ⊗ E ρ .2. Both components of λ ( z ) have single zeros at z = u α .3. λ ( z ) has only poles near z i . More precisely, the behaviour is λ + ( z ) = O (cid:16) ( z − z i ) − wi +12 (cid:17) , (4.3a)( λ − ( z ) + x i λ + ( z )) = O (cid:16) ( z − z i ) wi +12 (cid:17) . (4.3b)This follows from translating the representations described in Section 3.2 intoOPEs. This is done explicitly in [32].The existence of such a holomorphic section is extremely constraining. Of course,we could simply have identically λ ≡
0, but this also implies the vanishing of the fullpartition function. This follows by considering the OPE limit z → z i . The leadingterm in the OPE is ξ + ( z ) V w i j i ,h i ( x , z i ) = ( z − z i ) − wi +12 V w i j i − ,h i + ( x , z i ) (4.4)and thus the leading term in the singularity of λ + ( z ) captures the correlation func-tion of the primaries itself. Vanishing of λ would hence imply vanishing of the fullcorrelator. So let us assume that λ (cid:54)≡ Construction of a meromorphic 1-form.
Given this data, we can construct ameromorphic 1-form ω ( z ) with twisted U(1) boundary conditions as follows. Moreprecisely, ω is a meromorphic section of K ⊗ L ρ C × , where K is the canonical bundleand L ρ C × the flat line bundle determined by the restriction of ρ to the C × subgroup.The poles of ω ( z ) are precisely given by the insertion points z = z i (and are singlepoles) and ω ( z ) has single zeros at all the z = u α (and no other zeros). We set ω ( z ) = (cid:112) λ − ( z ) ∂λ + ( z ) − λ + ( z ) ∂λ − ( z ) . (4.5)– 23 –o show that this is indeed a well-defined 1-form, we check the following properties:1. λ − ( z ) ∂λ + ( z ) − λ + ( z ) ∂λ − ( z ) is a meromorphic section of the line bundle K ⊗ L ρ C × . For this, one simply has to check that the SL(2 , C )-part of the homo-morphism ρ cancels out in this combination. Moreover, even though we havenot used a covariant derivative, this expression transforms covariantly.2. λ − ( z ) ∂λ + ( z ) − λ + ( z ) ∂λ − ( z ) has (at most) double poles at z = z i and no otherpoles.3. λ − ( z ) ∂λ + ( z ) − λ + ( z ) ∂λ − ( z ) has (at least) double zeros at z = u α .4. λ − ( z ) ∂λ + ( z ) − λ + ( z ) ∂λ − ( z ) has precisely these zeros and poles with all doublemultiplicity. The line bundle K ⊗ L ρ C × has degree 2 deg( K ) = 4 g − L ρ C × isflat and hence does not contribute to the degree). The number of zeros minusthe number of poles of any meromophic section of this bundle is hence 4 g − λ − ( z ) ∂λ + ( z ) − λ + ( z ) ∂λ − ( z ) and this argument shows that thislist is complete.5. λ − ( z ) ∂λ + ( z ) − λ + ( z ) ∂λ − ( z ) possesses a well-defined square root. This followsfrom the fact that all its zeros and poles are second order and thus taking thesquare root is well-defined up to an overall sign.This shows all the desired properties. There is a small caveat: the square root isonly guaranteed to be a section of K ⊗ L ρ C × ⊗ L Z , where L Z is a Z -bundle thatsquares to the trivial line bundle. We will resolve this problem below and show that L Z is absent when choosing a suitable spin structure. ω ( z ) is a simple quantity, since we can apply the technology of line bundles anddivisors to it. Note first that the existence of such a meromorphic section ω ( z ) isvery constraining. It is a meromorphic differential with n poles, but 2 g − n prescribed zeros. By the Riemann-Roch theorem, this is generically impossible andthus non-vanishing of ω ( z ) imposes a non-trivial constraint. We can quantify thisvery precisely. Fix any meromorphic 1-form ˜ ω ∈ K ⊗ L ρ C × ( ⊗ L Z ). We suppress the Z -factor in the following. Then the ratio ω ( z ) / ˜ ω ( z ) is a meromorphic function onthe Riemann surface. Correspondingly, it’s divisor is principal. The divisor of ω ( z )is D = − n (cid:88) i =1 z i + g − n (cid:88) α =1 u α . (4.6)Thus, we need that the divisor D − K − D ( L ρ C × ) to be principal. By the Abel-Jacobitheorem, this is equivalent to the statement that the image under the Abel-Jacobimap vanishes. This specifies the line bundle L ρ C × uniquely and shows that there is exactly one line bundle for which ω ( z ) can be non-vanishing. The Abel-Jacobi mapcan be evaluated more explicitly for L ρ C × , which is explained in Appendix B.3.– 24 – onstructing γ ( z ). As a next step, we construct a map γ ( z ) that will turn outto be a branched covering map from the worldsheet to the boundary. We define it as γ ( z ) ≡ − λ − ( z ) λ + ( z ) . (4.7) γ ( z ) has again a number of properties that are straightforward to check:1. γ ( z ) is a (multi-valued) function on the Riemann surface Σ g . π (Σ g ) acts on itby M¨obius transformations.2. ∂γ ( z ) (cid:54) = 0 and ∂ ( γ ( z ) − ) (cid:54) = 0 for all z (cid:54) = z i . This follows from ∂γ ( z ) = λ + ( z ) ∂λ − ( z ) − λ − ( z ) ∂λ + ( z ) λ + ( z ) = ω ( z ) λ + ( z ) . (4.8)Zeros of ∂γ ( z ) originate either from zeros of ω ( z ) or from poles of λ + ( z ). Inboth cases, they cancel out by our previous analysis. The argument for γ ( z ) − is analogous.3. γ ( z ) = x i + O (( z − z i ) w i ) for i = 1 , . . . , n .The second property means that γ ( z ) maps into CP and is branched over the z i . Tostate the first point more clearly, let us uniformize the worldsheet Riemann surfaceusing a Fuchsian uniformization Σ g = H / Γ F g .
12 13
We have then π (Σ g ) ∼ = Γ F g . γ ( z )can be viewed as a single-valued map from the upper half plane H to CP . It is anequivariant map in the following sense: γ ( g ( z )) = ρ Γ ( g )( γ ( z )) (4.9)for z ∈ H and g ∈ π (Σ g ). Here ρ Γ ( g ) ∈ SL(2 , C ) acts on γ ( z ) via M¨obius trans-formations. Such a map is known as a (branched) developing map on the Riemannsurface. It defines a branched complex projective structure on the surface Σ g . Wehave collected some facts about (branched) complex projective structures in Ap-pendix E. Reconstructing λ ± ( z ). Using ( λ + ( z ) , λ − ( z )) we have constructed the two quan-tities ω ( z ) and γ ( z ). The two are actually equivalent to the original data, since wecan recover λ ( z ) = ω ( z ) (cid:112) ∂γ ( z ) (cid:18) γ ( z )1 (cid:19) . (4.10) For worldsheet genus 1, the uniformization is C / ( Z × Z ), but all the following arguments areunchanged. In order to be consistent with our notation, we denote by Γ F g ⊂ PSL(2 , R ) the genus g Fuchsianuniformization group and by Γ ⊂ SL(2 , C ) the orbifold group. Other names are deformation or geometric realization . – 25 –he square root of ∂γ ( z ) is well-defined, because ∂γ ( z ) has no zeros and all poles aredouble poles away from z = z i . The quantities ω ( z ) and γ ( z ) are far more convenient,since they separate the C × -part and the SL(2 , C )-part of the problem. Thus, we willcontinue to work with them without losing any information. Spin structure.
We have been slightly cavalier with the square root, they could inprinciple introduce signs in both ω ( z ) and (cid:112) ∂γ ( z ) around the cycles of the Riemannsurface. This does not happen, provided that we choose the correct spin structureon the worldsheet. Let us see this in more detail. ∂γ ( z ) satisfies ∂g ( z )( ∂γ )( g ( z )) = ( ∂ρ ( g ))( γ ( z )) ∂γ ( z ) , (4.11)for g ∈ Γ, where we view group elements sometimes as M¨obius transformations. Weare trying to define a square root of this transformation behaviour. We know how totake the square root of ( ∂ρ ( g ))( γ ( z )) – this is dictated by the lift of Γ from PSL(2 , C )to SL(2 , C ). For ρ ( g )( z ) = az + bcz + d , ( ∂ρ ( g ))( z ) = 1( cz + d ) , (4.12)we can define the square root to be (cid:112) ( ∂ρ ( g ))( z ) ≡ cz + d . (4.13)To make sense of (cid:112) ∂γ ( z ), we also need to define a square root (cid:112) ∂g ( z ). This isanalogous to the above situation: (cid:112) ∂g ( z ) is not well-defined for Γ F g ⊂ PSL(2 , R ),but only once we lift it to (cid:101) Γ F g ⊂ SL(2 , R ), which defines a spin structure on theworldsheet. This is explained also in Appendix D.2. We thus conclude that everysuch map γ ( z ) naturally induces a spin structure on the worldsheet. With thesedefinitions, 1 (cid:112) ∂γ ( z ) (cid:18) γ ( z )1 (cid:19) (4.14)is by construction a section of S − ⊗ E ρ and hence ω is a section of K ⊗ L ρ C × , thuseleminating the possibility of an additional Z bundle that could appear in the squareroot. In the beginning of this Section, we fixed a spin structure S . We now see thatthis is not arbitrary and we should identify S with the induced spin structure so thatthe formula (4.10) becomes correct. Localization in moduli space of Riemann surfaces.
We again note that theexistence of such a map γ ( z ) is extremely constraining. Viewing γ ( z ) as a map onthe upper half-plane, we can look at its Schwarzian S ( γ )( z ) = ∂ γ ( z ) ∂γ ( z ) − ∂ γ ( z )) ∂γ ( z )) , (4.15)– 26 –hich defines a meromorphic quadratic differential on the Riemann surface. It isholomorphic away from z = z i because ∂γ ( z ) (cid:54) = 0. Near z = z i , it has a double pole S ( γ )( z ) = − w i − z − z i ) + O (( z − z i ) − ) . (4.16)It is also periodic around the cycles of the Riemann surface because the Schwarzianderivative is invariant under M¨obius transformations. Thus S ( γ )( z ) is indeed a mero-morphic section of K .From here, we already see that there will be a further localization in the param-eters of the problem. Let us first discuss the case w i = 1 for all vertex operators,where the Schwarzian is a holomorphic quadratic differential. The homomorphism ρ Γ depends on 6 g − g ≥ ρ ( α ) , . . . , ρ ( α g ), ρ ( β ) , . . . , ρ ( β g ). The − g (cid:89) I =1 [ ρ ( α I ) , ρ ( β I )] = 1 . (4.17)Another 3 parameters are redundant, because they correspond to overall conjugation.Thus, the representation variety Hom( π (Σ g ) , SL(2 , C )) (up to overall conjugation)has complex dimension 6 g −
6. Most ρ Γ ’s will not be associated to a map γ ( z ).Such maps are in 1-1 correspondence with quadratic differentials S (Γ)( z ) (again upto overall composition with a M¨obius transformation). The (complex) dimension ofthe space of quadratic differentials is only 3 g −
3. Thus, we conclude that there are3 g − ρ Γ is fixed, there are only discrete points in the moduli space ofRiemann surfaces for which λ ± ( z ) can be non-zero. If we also want to integrate overthe positions of the vertex operators, then there are additional constraints becausewe also want to require γ ( z i ) = x i . This leads to n further constraints and thus thestring integrand localizes in M g,n .In the case with poles, one has to be careful, because the correspondence betweenmeromorphic quadratic differentials that satisfy (4.16) and developing maps γ is nolonger 1–1. Instead, the quadratic differentials have to satisfy an extra condition,that is called integrability [47]. This condition ensures that γ has trivial monodromyaround the insertion points. This imposes n conditions on the quadratic differential.Another n conditions are imposed by requiring that the solution γ ( z ) satisfies γ ( z i ) = x i . Thus the space of quadratic differentials with these properties is 3 g − − n . If we would compute correlation functions with twisted vertex operators, then we would specifynon-trivial monodromy around the insertion points. Of course these are actually n − x i areinvisible for the Schwarzian. We already took M¨obius transformations into account and in order tohave a uniform presentation we count them as n constraints. – 27 –omparing again with the the dimension of Hom( π (Σ g ) , SL(2 , C )), we see that 3 g − n conditions have to be obeyed, which shows that generically λ ± ( z ) can onlyexist on isolated points in M g,n . Due to the non-abelian nature of the problem, thislocus is much harder to quantify than for the case of line bundles. Relation to covering maps.
Ideally, we would like to conclude that the map γ ( z )is a covering map to the boundary surface. This is almost true, but unfortunatelyour input is not enough to decide this question.The missing part in establishing this is to show that γ ( z ) only maps to the regionof discontinuity Ω of the boundary. See Appendix A for an explanation of the regionof discontinuity. For global AdS , this problem is non-existent, since Ω = CP . If thiswere the case, then we could compose with the canonical covering map π : Ω −→ Σ G in order to construct a covering map (here Σ G is the boundary Riemann surface ofgenus G ). In the unbranched case (i.e. when w i = 1 for every i ), it is known that thefollowing conditions are equivalent [48]:1. γ ( z ) (cid:54) = CP γ is a covering map on its image.3. ρ Γ ( π (Σ g )) acts discontinuously on the image (i.e. γ ( z ) maps into Ω).Unfortunately, there are situations where neither of these conditions is satisfied. See[47] for an explicit counterexample.Thus, while this argument shows that a non-trivial λ ( z ) can only exist when boththe U(1)-bundle localizes and the complex structure of the Riemann surface localizes,we cannot exactly predict the localization locus. However, when the worldsheet isa covering surface of the boundary, we can explicitly construct such a section λ ( z ),see e.q. (4.10) (this holds for arbitrary correlators).Physically, there is reason to believe that only the covering maps should appearin the localization locus. If a map does not satisfy these conditions, then there is apoint z ∗ on the worldsheet such that γ ( z ∗ ) (cid:54)∈ Ω. In a small neighborhood of z ∗ , we canstill view Γ( z ) as a map from the worldsheet to the boundary surface, but the mapis undefined at z = z ∗ . In fact, the behaviour of γ ( z ) is very similar to an essentialsingularity: in an arbitrarily small neighborhood of z ∗ , the map takes every possiblevalue in CP infinitely many times (with the possible exception of up to two pointsaccording to Picard’s theorem). A good analogy is the “covering map” CP −→ CP given by γ ( z ) = e z . While γ ( z ) satisfies ∂γ ( z ) (cid:54) = 0 (and ∂ ( γ ( z ) − ) (cid:54) = 0) everywhere,it is undefined at z = 0 and has an essential singularity there. Of course, we knowthat CFT correlators on the sphere cannot have essential singularities which is whythis issue does not arise. For non-trivial boundaries, this is not automatic, but westill find it reasonable that CFT correlators are free from essential singularities inthis sense. We could view such a map γ ( z ) also as a covering map of infinite degree.– 28 –overing spaces are suppressed by a factor of the degree in the partition functionof the symmetric orbifold, see eq. (2.4). Thus, also from this point of view, it isnatural that these non-covering maps do not contribute. We shall assume this in thefollowing. The discussion simplifies considerably when either the worldsheet or the boundarysurface has genus 1. We discuss these cases here separately. We restrict to theunbranched case.
Boundary genus 1.
In this case, we can conjugate ρ Γ and let it map into theaffine group. Thus, γ ( z ) defines in this case a complex affine structure (following theterminology of [48].) It is a known result that such an affine structure only existswhen also the worldsheet has genus 1. This implies that in spaces with a boundarytorus, all higher genus corrections have to vanish. This was already conjectured tobe the case in [1].This result is simple to prove, so let us repeat the proof here. If ρ Γ maps in thegroup of affine transformations, ∂γ ( z ) only transforms muliplicatively around thecycles of the Riemann surface. Hence ∂γ ( z ) is an element of a line bundle K ⊗ L .The line bundle L captures the multiplicative factors by which ∂γ ( z ) transforms.Since L is again specified by a homomorphism (the projection of ρ Γ to the rotationalsubgroup of the affine group), L is a flat line bundle. Thus, deg( K ⊗ L ) = 2 g − ∂γ ( z ) has 2 g − ∂γ ( z ) hasno zeros and hence g ≤ C / ( Z ⊕ Z τ ). There are three different cases:1. The boundary torus is obtained by Schottky uniformization: T = ( CP \{ , ∞} ) / Z . The group action in the boundary identifies the boundary coordi-nate x ∈ CP as x ∼ e πit x , (4.18)where t is the boundary modular parameter. Thus, we search for a map γ ( z )(viewed as a meromorphic map on C ) that satisfies ∂γ ( z ) (cid:54) = 0, ∂ ( γ ( z ) − ) (cid:54) = 0and γ ( z + 1) = e πict γ ( z ) , γ ( z + τ ) = e − πidt γ ( z ) , (4.19)for two integers c and d . Consider f ( z ) = γ ( z ) ∂γ ( z ) . f ( z ) is clearly periodic and isthus an elliptic function. f ( z ) has no poles, since the only possible poles arelocated at the poles of γ ( z ) and cancel out. This implies that f ( z )is boundedand hence constant. Thus γ ( z ) = B exp (2 πiAz ). We can now solve for the– 29 –eriodicity conditions. The first implies that A = ct − a for some integer a .The second periodicity condition implies the relation( cτ + d ) t = aτ + b (4.20)for integers a, b, c, d . Notice that the exponential map automatically maps inΩ = CP \{ , ∞} and thus we can construct a well-defined map from the world-sheet to the boundary torus. Thus the localization locus indeed correspondsto all covering surfaces of the boundary torus.2. The boundary torus is obtained by the following quotient: x ∼ e πitmM + πinM x (4.21)for m ∈ Z and n ∈ Z M . The corresponding bulk geometries are conical defects.The maps with these periodicity conditions are again exponential maps andone finds the same localization locus as before.3. The boundary torus is obtained from the standard uniformization x ∼ x + 1 ∼ x + t . (4.22)Thus, we search now for a map satisfying ∂γ ( z ) (cid:54) = 0, ∂ ( γ ( z ) − ) (cid:54) = 0 and γ ( z + 1) = ct − a + γ ( z ) , γ ( z + τ ) = − dt + b + γ ( z ) (4.23)for four integers a, b, c, d . ( ∂γ ( z )) − is an elliptic function without poles andhence constant. Thus γ ( z ) = Az for some A . The first periodicity conditionimplies A = ct − a and the second leads to the same condition( cτ + d ) t = aτ + b (4.24)as above. Worldsheet genus 1.
This case is also similar to the previous one. Let us assumethat the boundary has genus G ≥
2, since we already analyzed the genus 1 case.Since π (Σ g ) is abelian, also the image ρ Γ ( π (Σ g )) is abelian. We may hence usean overall conjugation and conjugate the image ρ Γ ( π (Σ g )) into the affine group.Abelian (Kleinian) subgroups of SL(2 , C ) are precisely given by the three cases thatwe discussed above. We thus learn again that γ ( z ) takes one of the simple formsthat we described above. In any case, the image of γ ( z ) is CP with either one ortwo points removed. However, for a boundary with genus G ≥
2, the limit set hasnecessarily infinitely points. Thus, we conclude that the image of γ does not lie in theregion of discontinuity Ω of the boundary. So while the worldsheet partition functionindeed localizes, the localizing surface here does not correspond to a covering map.This exemplifies the problem that we encountered in the last subsection. There are also a couple of other possibilities, that are excluded by inspection such as the order2 abelian subgroup generated by inversion x (cid:55)→ − x . – 30 – orldsheet genus 0. Finally, let us also comment on the genus 0 case. In thiscase, our argument completely trivializes. Since π (Σ ) is the trivial group, thereare no non-trivial Γ-bundles over which we could sum and no non-trivial moduli inwhich the partition function could localize. In fact, one can see that the existence of λ ± does not impose any constraints. Thus, we expect the sphere partition functionto be generically non-vanishing.
We can now say something about the full worldsheet correlation function (with theadditional operator insertions that appear in (3.25)). It should take the followingqualitative form: (cid:42) g − n (cid:89) α =1 W ( u α ) ∂H ( z ) g − (cid:89) β =1 e − iH ( v β ) n (cid:89) i =1 V w i j i ,h i ( x , z i ) (cid:43) (Σ g,n ,L g ) = (cid:88) (Σ (cid:48) ,L (cid:48) ) δ (6 g − n ) (Σ g,n − Σ (cid:48) ) δ (2 g ) ( L g − L (cid:48) ) × (cid:42) g − n (cid:89) α =1 W ( u α ) ∂H ( z ) g − (cid:89) β =1 e − iH ( v β ) n (cid:89) i =1 V w i j i ,h i ( x , z i ) (cid:43) (cid:48) (Σ (cid:48) ,L (cid:48) ) . (4.25)Here, the notation emphasizes the dependence of the correlator on the Riemannsurface Σ g (or the punctured Riemann surface Σ g,n ) and the line bundle L g thatspecifies the periodicity properties of the free fields. The prime on the correlator onthe RHS indicates that the delta function is factored out. Let us be summarize whatwe have exactly shown:1. We have shown that the correlation function vanishes, except when (Σ g,n , L g )coincides with special (punctured) Riemann surfaces with corresponding linebundles.2. From counting dimensions, we have shown that these special Riemann surfacesare isolated points in the M g,n × Jac(Σ g ). Thus, the sum appearing on theRHS is indeed a discrete sum.3. Whenever Σ (cid:48) is a ramified cover of the boundary surface with ramificationindices specified by spectral flow, there exists a pair (Σ (cid:48) , L (cid:48) ) that satisfies allthe constraints and that should appear in the sum.4. We have not shown that all the surfaces that satisfy the constraints we haveanalyzed are ramified covers and in general this is not true. But we have argued We consider a two-point function with w = w = 1, since this the lowest number of fieldswhere the formalism makes sense. – 31 –hat nonetheless only such surfaces should appear in the partition function andassume this in the following to be the case.5. We have only shown that the correlation function vanishes generically. As suchit is a distribution with point-like support in Σ g,n × Jac(Σ g,n ). Such distributionsare a finite sum of δ -functions and derivatives [49, Theorem 6.25]. It is naturalto assume that no derivatives appear. This is the case for genus 0 correlatorsfor global AdS, which under much better analytic control [31, 32], but we donot know an argument for this in our more general setting. We shall assumethis in the following.Let us in the following normalize the delta function such that (cid:90) M g,n δ (6 g − n ) (Σ g,n − Σ (cid:48) ) f (Σ g,n ) = f (Σ (cid:48) ) , (4.26) (cid:90) Jac(Σ g ) δ (2 g ) ( L − L (cid:48) ) f ( L ) = f ( L (cid:48) ) . (4.27)for any functions f . We view the delta-functions as top-forms on the respectivespaces so that no measure is necessary. The primed correlator appearing in (4.25)should in particular be understood as a function on moduli space. Remaining correlator.
Let us now discuss the remaining correlator. As we al-ready stressed, we do not know how to compute it, but we know a few qualitativeproperties that it should satisfy. The first is a simple counting argument for the de-grees of freedom. It was already explained in [30, 33] that the tensionless string hasonly four transverse oscillators and these correspond roughly to the oscillators of T .In other words, the remaining correlation function should essentially only capture thetopologically twisted partition function of T . This topologically twisted partitionfunction has a direct relation with the untwisted partition function (evaluated on aspecific metric), provided that the insertion points v β are chosen judiciously. This isexplained in Appendix F. Spin structure.
The question remains with which spin structure the T partitionfunction should be evaluated. The natural guess (that is confirmed in the specificexample of a torus in [1]) is to take the induced spin structure on the worldsheetthat we discussed in Section 4.1, since there is no other distinguished spin structureon the worldsheet. This spin bundle in fact coincides with the pull-back bundle γ ∗ S of the spin bundle on the boundaries that is defined by lifting Γ ⊂ PSL(2 , C ) toSL(2 , C ) as discussed in Section 3.3. Full integrand.
The upshot of this is that the worldsheet integrand should takethe following form: I = (cid:88) ( γ,L (cid:48) ) δ (6 g − n ) (Σ g,n − Σ γ ) δ (2 g ) ( L g − L (cid:48) ) Z classical (Σ γ ) Z T (Σ γ , γ ∗ S ) , (4.28)– 32 –here S is the spin structure of the boundary component to which γ maps and Σ γ the covering surface. Classical part.
We have included a factor Z classical (Σ γ ) in the ansatz for the inte-grand (4.29). Such a factor necessarily is present because of the conformal anomaly.The central charge of the (not topologically twisted) sigma model on T is c = 6 andconsequently Z T (Σ γ , γ ∗ S ) depends on the metric on the worldsheet. This metricdependence is cancelled by Z classical (Σ γ ). However, this property does not determine Z classical (Σ γ ) completely. Comparing thermal AdS with the BTZ black hole and theconical defect, it was seen in [1] explicitly that Z classical does depend on the precisebackground. However, we should expect that Z classical does actually not depend onthe moduli of the worldsheet and we have I = Z classical (cid:88) ( γ,L (cid:48) ) δ (6 g − n ) (Σ g,n − Σ γ ) δ (2 g ) ( L g − L (cid:48) ) Z T (Σ γ , γ ∗ S ) . (4.29)One argument for this is modular invariance. This expression is modular invariant.If the classical part would also depend on the moduli, then it would need to do so in away that preserves modular invariance, which is typically impossible for a ‘classical’partition function. This independence is also observed in simple examples [1]. After having established that the worldsheet partition function of the tensionlessstring localizes in the moduli space of Riemann surfaces, we want to go a step furtherand argue that the string partition function is actually independent of the precisegeometry in which we place the string, but depends only on the boundary geometry.
Before we start, let us review some concepts of the fundamental group. Let us fixa geometry AdS / Γ an orbifold group Γ. Let us denote the boundary surface byΣ G , which we assume for now to be connected. The boundary is obtained as Ω / Γ,where Ω ⊂ CP is the region of discontinuity of the action of Γ on CP . Ω is anintermediate covering space of Σ Γ . Thus, we have the hierarchy of coverings (cid:101) Σ G −→ Ω −→ Σ G , (5.1)where (cid:101) Σ G = H for G ≥
2. The covering map Ω −→ Σ G is given by identifyingpoints in the orbits of the group action. Such a covering space is called regular (or These cases are a bit special because the worldsheet torus can cover the boundary torus withan arbitrarily high degree and the classical part depends on the degree of the covering map. Thisissue does not arise for the higher genus situations. – 33 – ormal ). Γ is the group of deck transformations of this covering map, i.e. the groupof automorphisms of Ω which leave the covering map unchanged. Regularity impliesthat there is a short exact sequence of groups [50, Proposition 1.40] −→ π (Ω) −→ π (Σ G ) −→ Γ −→ . (5.2)In particular, there is a canonical projection map p : π (Σ G ) −→ Γ.For disconnected boundaries, the relevant groups are strictly speaking no longergroups because we can define a fundamental group for every connected componenton the boundary. In this case there is a projection p : π (Σ ( i ) G i ) −→ Γ for everycomponent Σ ( i ) G i of the boundary. In this subsection, we explain how string theory on different backgrounds can beequivalent. Twisted sectors of the orbifold partition function with orbifold group Γare labelled by (not necessarily injective) homomorphisms ρ : π (Σ g ) −→ Γ , (5.3)up to overall conjugation. We used to denote this homomorphism by ρ Γ in the lastsection, but since ρ C × will not appear again, we simply denote it by ρ .We also argued that the worldsheet partition function localizes on covering sur-faces. Covering surfaces in a given twisted sector are specified by injective homomor-phisms ι : π (Σ g ) −→ π (Σ G ) such that p ◦ ι = ρ . In case the boundary is discon-nected, ι could map into the fundamental group of any component. To understandthis, suppose that we have given an injective homomorphism ι : π (Σ g ) −→ π (Σ G )with the property p ◦ ι = ρ . The image of ι defines a subgroup of π (Σ G ) and hence acovering space. However, such a homomorphism actually specifies a marked coveringspace, i.e. a covering space together with canonical generators for the fundamentalgroup. These canonical generators are given by the images under ι of the generatorsof π (Σ g ). Thus, we will actually count identical covering spaces, but with differentmarkings several times. This is appropriate if we want the partition function to bea function on Teichm¨uller space and not on moduli space. Let us also recall that weunderstand all homomorphisms ι : π (Σ g ) −→ π (Σ G ) up to inner automorphisms,i.e. overall conjugation.In a commutative diagram, we can summarize the situation as follows: π (Σ g )1 π (Ω) π (Σ G ) Γ 1 pι ρ , (5.4) We are not precise about the base points here. All statements should be understood with afixed base point. – 34 –here the horizontal maps are exact. Thus, the worldsheet partition function natu-rally is of the following form for a connected boundary Z worldsheet = (cid:88) ρ : π (Σ g ) −→ Γ (cid:88) ι : π (Σ g ) −→ π (Σ G )injective , p ◦ ι = ρ Z ι . (5.5)For a disconnected boundary, the change is very minimal. Since the worldsheet cancover either boundary, we get an additional sum over boundary components: Z worldsheet = (cid:88) i (cid:88) ρ : π (Σ g ) −→ Γ (cid:88) ι : π (Σ g ) −→ π (Σ ( i ) G )injective , p ◦ ι = ρ Z ι . (5.6) Let us exemplify this structure with some examples.
Cusp geometry.
This is a particularly simple case that actually was not discussedin [1]. In this case, the boundary of the space is a single torus and the orbifold groupis Z ⊕ Z , acting on the boundary by identification of x ∼ x + 1 ∼ x + t . In thiscase Ω = C and hence π (Ω) = { } is the trivial group. So, p is simply the identitymap, which implies that ρ = ι . Thus, out of the two sums that are present in (5.5),actually only the sum over twisted sectors is present. In every twisted sector, thepartition function localizes on exactly one configuration. This configuration alreadyappeared in our discussion in Section 4.2. In this case, ρ = ι is a homomorphismfrom Z to Z and hence naturally identified with an integer 2 × Thermal AdS . In this case Ω = CP \ { , ∞} and the short exact sequence offundamental groups reads1 −→ π (Ω) ∼ = (cid:104) α (cid:105) −→ π (Σ G ) ∼ = (cid:104) α (cid:105) ⊕ (cid:104) β (cid:105) −→ Z ∼ = (cid:104) β (cid:105) −→ . (5.7)Here, we wrote (cid:104) α (cid:105) for the abelian group { nα | n ∈ Z } . The map from π (Ω) to π (Σ G ) is then the obvious inclusion map and the second map is the projection mapthat sends α (cid:55)→ β (cid:55)→ β . Homomorphisms ι are still naturally identified with2 × Z ∼ = (cid:104) α (cid:105) ⊕ (cid:104) β (cid:105) both in the boundaryand the bulk, we can write ι = (cid:18) a bc d (cid:19) , p ◦ ι = (cid:0) c d (cid:1) . (5.8)Thus, twisted sectors are naturally labelled by the two integers ( c, d ) and the local-ization in each twisted sector is labelled by two integers ( a, b ). This is precisely thestructure that was observed in [1]. For the Euclidean BTZ black hole black hole,the situation is analogous, except that the role of α and β is interchanged and hence p ◦ ι = (cid:0) a b (cid:1) . – 35 – onical defect in thermal AdS . This case is very similar. The short exact offundamental groups is now1 −→ π (Ω) ∼ = (cid:104) α (cid:105) −→ π (Σ G ) ∼ = (cid:104) α (cid:105) ⊕ (cid:104) β (cid:105) −→ Z ⊕ Z M ∼ = (cid:104) β (cid:105) ⊕ (cid:104) θ (cid:105) −→ . (5.9)The first map is now the embedding α (cid:55)→ ( M α, θ is the additional generator oforder M . Under the same identification of ι with a 2 × p ◦ ι = (cid:18) a mod M b mod
Mc d (cid:19) . (5.10) Handlebodies.
Moving on to a single higher-genus boundary, our description be-comes less explicit. The orbifold group is a Schottky group Γ S G of genus G and assuch is a free group with generators B , . . . , B G . The fundamental group of theboundary surface is generated by α , . . . , α G , β , . . . , β G with the usual relation (cid:81) I [ α I , β I ] = 1. According to the short exact sequence (5.2), the fundamental groupof the domain Ω is the smallest normal subgroup of π (Σ G ) that contains α , . . . , α G . The surjection π (Σ G ) −→ Γ S G simply sends α I (cid:55)−→ β I (cid:55)−→ B I . Thisdefines the handlebody that belongs to this marking. For any different marking ofthe boundary surface, we obtain a different handlebody. Unfortunately, describinghomomorphisms ρ : π (Σ g ) −→ Γ S G is quite complicated and we will not be moreexplicit than this. Until now, we have not discussed how the chemical potential of the grand canonicalensemble enters the worldsheet theory. In the boundary theory, we have discussedit in Section 2.2. In general, the boundary of the hyperbolic 3-manifold in questionmight have several components. In this case, from a boundary point of view, wecan actually introduce several chemical potentials. From a worldsheet point of view,these chemical potentials enter in a slightly assymmetric way. We first discuss the‘diagonal’ chemical potential (by this we mean the average of all boundary chemicalpotentials) and postpone the others to Section 5.5. There are two ways of introducingit, which seem to be equivalent. Basically, the chemical potential can be introducedby adding the spacetime identity vertex operator on the worldsheet. This correspondsto the zero mode of the dilaton in spacetime and hence the chemical potential can alsobe identified with the string coupling constant. We begin with the latter formulation.
Identification with string coupling.
The fugacity p = e πiσ is closely related tothe string coupling constant. Consider adding the topological term λ π (cid:90) √ γ R (5.11)to the worldsheet action. Here γ is the worldsheet metric and R the correspondingRicci scalar. This can be evaluated using the Gauss-Bonnet theorem and naively– 36 –quals λ (2 − g ). However, the Riemann surface is punctured and one has to be a bitcareful about the insertions of vertex operators, where the metric becomes singular.We want to exclude the insertion points from the integration. Near these insertions,the metric is determined by the condition that the worldsheet is a holomorphic coverof the boundary theory.To analyze what happens close to the insertion, let us choose local coordinatessuch that a vertex operator is inserted close to z i = 0 and the corresponding insertionin the boundary is also x i = 0. The covering map behaves locally like γ ( z ) = a i z w i + O ( z w i +1 ) . (5.12)Thus, the metric behaves asd s ∝ | ∂γ ( z ) | d z d¯ z = A | z | w i − d z d¯ z + O ( | z | w i − ) . (5.13)For a metric of the form d s = e φ ( z, ¯ z ) d z d¯ z , we have √ γ R = − ∆ φ ( z, ¯ z ) . (5.14)In our case, φ ( z, ¯ z ) = ( w i −
1) log | z | + regular terms, thus leading to √ γ R = − π ( w i − δ (2) ( z ) + finite . (5.15)Integrating over the full Riemann surface leads by the Gauss-Bonnet theorem tothe Euler characteristic. Thus, after excising the singular points (or alternativelyintegrating only over the regular part of the curvature), we obtain λ π (cid:90) Σ g \{ z ,...,z n } √ γ R = λ (cid:32) − g + n (cid:88) i =1 ( w i − (cid:33) . (5.16)The Riemann-Hurwitz formula connects the degree N c = deg( γ ), the worldsheet andboundary genus and the ramification indices as follows:2 − g = N c (2 − G ) − n (cid:88) i =1 ( w i − . (5.17)Thus, we can express the parentheses in terms of the degree N c of the relevantcovering map and the boundary genus G as follows: λ π (cid:90) Σ g \{ z ,...,z n } √ γ R = λN c (2 − G ) . (5.18)This is exactly what we want: this term in the action weighs different coveringmaps according to their degree, which is analogous to the situation in the symmetricorbifold (2.4). For this to work literally, we have to identify λ (2 − G ) = 2 πiσ . (5.19)– 37 –n terms of the string coupling constant, we have p = g G − , (5.20)where p = e πiσ and g string = e − λ . Here we assume that there is only one bound-ary surface or that all boundary surfaces have the same genus. If this is not thecase, the situation can be corrected by including off-diagonal chemical potentials,see Section 5.5. The spacetime identity operator.
Alternatively, the grand canonical partitionfunction is obtained by adding the following term to the worldsheet action [51]: δS = 2 πiσ (cid:90) Σ I ( z, ¯ z ) . (5.21)Here, I ( z, ¯ z ) is the worldsheet operator that corresponds to the spacetime identity.Note that δS is a an SL(2 , C ) singlet and as such is a marginal operator that wecan add to the worldsheet theory. I ( z, ¯ z ) is actually quite easy to construct, itsunintegrated version corresponds to the vertex operator I (0) ( z, ¯ z ) = V w =1 j,h =0 ( x, ¯ x, z, ¯ z ) (5.22)Since h = 0, the RHS does not depend on x and gives hence a well-defined operator I (0 ( z, ¯ z ). The worldsheet conformal weight also vanishes. Thus, I (0) ( z, ¯ z ) correspondsto an unintegrated vertex operator on the worldsheet. To obtain the integratedversion, we apply the descent formalism. For the N = 4 topological string, oneapplies the twisted supercharges ˜ G −− ¯˜ G −− to I (0) ( z, ¯ z ). One needs to pick ˜ G −− andnot G −− to preserve conservation of the various ghost currents of the full worldsheettheory. Since we want the vertex operator to have also vanishing Z -charge (seeSection 3.2), we need to pick also j = . Applying ˜ G −− lowers the value of the sl (2 , C ) spin by one unit and hence the integrated vertex operator has j = − . Thisleaves the constraint (3.19) unchanged.The integrated vertex operator I = (cid:82) ˜ G −− ¯˜ G −− I (0) ( z, ¯ z ) plays then the role ofthe spacetime identity. Up to normalization, it survives the orbifold, since it is an sl (2 , C ) singlet. As such, it is a central term of the operator algebra. Since all it’sOPEs are trivial, it simply produces an eigenvalue when inserted in a correlator, (cid:42) I g − n (cid:89) α =1 W ( u α ) n (cid:89) i =1 V w i j i ,h i ( x i , z i ) (cid:43) = λ (cid:42) g − n (cid:89) α =1 W ( u α ) n (cid:89) i =1 V w i j i ,h i ( x i , z i ) (cid:43) . (5.23)To determine this eigenvalue, one can invoke various consistency conditions. First ofall, the eigenvalue λ cannot depend on either j i , h i , x i or z i , since these quantitiescan be changed by acting with an element in the chiral operator algebra on thevertex operators and I should be central in this operator algebra. Similarly, we– 38 –ould consider any descendant vertex operator that is obtained by the action of thechiral operator algebra.From general properties of OPEs, it follows that the eigenvalue λ should be addi-tive in the involved quantum numbers. Adding another unintegrated identity vertexoperator should not change the result (since we can pull out the vertex operators oneafter another). This has the effect of adding both one more W and a V . When theboundary theory is a sphere, every vertex operator V w i j i ,h i ( x i , z i ) contributes w i tothe eigenvalue [51, 52]. Since I is a SL(2 , C ) singlet under the orbifold, this shouldnot change in the topologically non-trivial situation. These arguments together showthat V w i j i ,h i ( x i , z i ) contributes w i to the eigenvalue of I and each W contributes − ,thus giving in total λ = A (cid:32) n (cid:88) i =1 w i − (2 g − n ) (cid:33) , (5.24)where we allowed for an arbitrary normalization A of I . We set A = (1 − G ) − ,where G is the boundary genus so that λ = 12 − G (cid:32) − g + n (cid:88) i =1 ( w i − (cid:33) . (5.25)This fits with the sphere result of [51]. Of course, this constant only makes sensewhen the corresponding correlator is non-vanishing, otherwise the statement is void.For this a covering map from the worldsheet to the boundary has to exist.From the Riemann Hurwitz formula (5.17), we conclude that λ = N c = deg( γ ),i.e. the eigenvalue of the spacetime identity operator is again the degree of the asso-ciated covering map. Thus we see that adding the identity operator is equivalent tochanging the string coupling. This makes sense because both these operators can beidentified with the dilaton zero mode. Since the string coupling is easier to introduce,we will continue to work with the string coupling.
Sphere partition function.
We saw that the chemical potential p of the grandcanonical ensemble is essentially mapped to the string coupling constant on theworldsheet according to p = g G − . It is important to keep in mind that the wholediscussion only applies for worldsheet genera g ≥
1. There is also a non-vanishingsphere partition function that we need to treat separately. We do not know how thesphere partition function is computed in general, because the volume of the residualM¨obius symmetry is difficult to regularize. However, we expect it to capture the‘classical part’ of the string partition function. We thus expect that Z ∝ I bulk ( H / Γ) ∝ vol( H / Γ) (5.26) We assume that G (cid:54) = 1. For G = 1, both the parenthesis and the prefactor vanish so that theratio can still be non-trivial, but is more subtle to define. This statement makes again sense for a boundary torus. – 39 –s proportional to the volume of spacetime. Here I bulk ( M ) is the spacetime on-shellaction evaluated on hyperbolic manifold M . This volume is anomalous and has tobe regularized. The regularization depends on the boundary metric (and not only onits conformal class), which gives rise to the conformal anomaly. This computationis standard, see [53, 54]. We normalize H by setting the Ricci scalar to −
6. Theconstant of proportionality is then essentially given by the central charge of theboundary theory Z = c π vol( H / Γ) ≡ − c I bulk ( H / Γ) , (5.27)so that e Z captures the conformal anomaly of a central charge c CFT. We normalize I bulk ( H / Γ) such that it precisely accounts for the conformal anomaly of a c = 1CFT. As we shall see below, we should take c = 6, the central charge of a single T ,since the effect of the grand canonical potential is already incorporated in the stringcoupling g − that multiplies the sphere partition function. We constructed the ‘diagonal’ chemical potential above. In general, we can howeverhave different chemical potentials at the boundaries of the bulk. From a worldsheetperspective, these additional chemical potentials can be introduced by consideringnon-trivial discrete torsion. We first recall the concept of discrete torsion and thenapply it to our situation.
Discrete torsion.
For an orbifold CFT, the genus g partition function takes ingeneral the following form 1 | Γ | (cid:88) ρ : π (Σ g ) → Γ ε ( ρ ) Z ρ , (5.28)where Γ is again the orbifold group and homomorphisms ρ up to conjugation specifyΓ-bundles. ε ( ρ ) is a phase depending on the Γ-bundle over which we sum. It isconstrained through various consistency conditions and in general allowed valuesare classified by the Schur multiplier of the orbifold group – the cohomology groupH (Γ , U(1)). Given a 2-cocycle φ ∈ H (Γ , U(1)), one can define [55–57] ε ( ρ ) = g (cid:89) I =1 φ ( ρ ( α I ) , ρ ( β I )) φ ( ρ ( β I ) , ρ ( α I )) , (5.29)where α , . . . , α g , β , . . . , β g are a canonical homology basis of the worldsheet, see(B.1). Such a local spacetime action does not exist in this regime. When we talk about the spacetimeeffective action, we mean the supergravity action which applies in the regime k (cid:29) k = 1. This seems to give the correct results, but is not a very satisfying procedure. – 40 – chur multiplier for Kleinian groups. Let us discuss the Schur multiplier forKleinian groups, i.e. the orbifold groups Γ that reduce H to H / Γ. Let us assumethat
M ∼ = H / Γ is a non-singular manifold. The group cohomology of Γ can becomputed by noticing that H / Γ is an Eilenberg-MacLane space K (Γ , π ( H / Γ) = Γ and π n ( H / Γ) = 0 for n >
1. It is known that the group coho-mology coincides with the usual singular cohomology of its corresponding Eilenberg-MacLane space. Hence the Schur multiplier can be interpreted more geometricallyas the singular cohomology group H ( H / Γ , U(1)).From a worldsheet action point of view, an element B ∈ H ( H / Γ , U(1)) is a two-form with d B = 0 (up to exact forms). Hence we can add the following topologicalterm to the worldsheet action: 2 πi (cid:90) Σ g ∗ B , (5.30)where g : Σ −→ H is the embedding coordinate of the string in H / Γ. If B ∈ H ( H / Γ , Z ), then this term is in 2 πi Z and hence has no effect in the path integral.Thus, inequivalent topological terms are indeed classified by the cohomology groupH ( H / Γ , U(1)). We explain in Appendix A that H ( H / Γ , U(1)) ∼ = U(1) R is torsionfree. It is naturally related to the torsion free homology group H ( H / Γ , Z ) ∼ = Z R . R can be computed in terms of Γ and the boundary genera of the manifold M , seeeq. (A.7).In the following, we will discuss some very natural generators of H ( H / Γ , Z ),namely the boundary components of H / Γ. Let ∂ H / Γ = Σ (1) (cid:116) · · · (cid:116) Σ ( n ) be theboundary components of the space. Then each Σ ( i ) defines a cocycle in [Σ ( i ) ] ∈ H ( H / Γ , Z ). Of course, (cid:80) ni =1 [Σ ( i ) ] is the boundary of H / Γ and is thus null-homologous. This is the only relation in homology and thus we obtain n − ( H / Γ , Z ). Identification with off-diagonal chemical potentials.
We are now arguingthat additional chemical potentials that can be introduced for multiple boundariescan indeed be identified with the discrete torsion parameters. Let us pick as genera-tors of H ( H / Γ , Z ) the fundamental classes [Σ (1) ], . . . , [Σ ( n − ]. We are omitting thelast boundary to keep the generators independent (we have [Σ ( n ) ] = − [Σ (1) ] − · · · − [Σ ( n − ]). As we mentioned, there could be more generators that do not originatefrom the boundaries which we shall ignore. Let now B (1) , . . . , B ( n − the correspond-ing generators of H ( H / Γ , Z ) such that (cid:82) Σ ( i ) B ( j ) = δ ij for i = 1 , . . . , n −
1. Then we Otherwise we have to employ orbifold cohomology in the following discussion. We do not understand the meaning of additional generators of H ( H / Γ , Z ) in general. However,for the conical defect, H (Γ , U(1)) has an additional torsion generator and its effect can be absorbedin the definition of the boundary chemical potential [1]. Thus, it has no physical effect in this case. – 41 –an consider the topological terms S top = 2 πi n − (cid:88) i =1 σ i (cid:90) Σ g ∗ B ( i ) . (5.31)We can evaluate this term on the configurations that appear in the path integral.Let us assume that the worldsheet Σ covers the boundary Σ ( i ) holomorphically (for i = 1 , . . . , n − S top = 2 πi deg( γ ) n − (cid:88) j =1 σ j (cid:90) Σ ( i ) B ( j ) = 2 πi deg( γ ) σ i . (5.32)If it covers instead Σ ( n ) , we obtain S top = 2 πi deg( γ ) n − (cid:88) i =1 σ i (cid:90) Σ ( n ) B ( i ) = − πi deg( γ ) n − (cid:88) i =1 σ i . (5.33)This confirms that these terms indeed correspond to the off-diagonal chemical po-tentials. The quasi-Fuchsian wormhole.
For the two-sided wormhole that is obtained as H / Γ QF G for Γ QF G a genus G quasi-Fuchsian group (see Appendix A.4 for the relevantdefinitions), we can be quite explicit. There is a single generator of H ( H / Γ QF G , Z ),which we can take to be the fundamental class of either boundary component. Thus,the topological term weighs contributions that cover the left boundary with oppositephases than those that cover the right boundary. Eq. (5.6) takes the form Z worldsheet = (cid:16) p p − (cid:17) g − G − (cid:88) ρ : π (Σ g ) −→ Γ QF G (cid:88) ι : π (Σ g ) −→ π (Σ (1) G ) , injective , p ◦ ι = ρ Z ι + (cid:16) p − p (cid:17) g − G − (cid:88) ρ : π (Σ g ) −→ Γ QF G (cid:88) ι : π (Σ g ) −→ π (Σ (2) G ) , injective , p ◦ ι = ρ Z ι . (5.34)Upon setting the string coupling constant to g G − = p p , we can arrange it thatcovering maps mapping to the left (right) boundary are weighted by the fugacities p ( p ). We shall now restrict the discussion to partition functions and tie various observa-tions together. Our ultimate goal is to compute the string partition function in thebackground AdS / Γ × S × T in the grand canonical ensemble. Let us first restrict to– 42 –ackgrounds with a single boundary so that there is no discrete torsion. The grandcanonical string partition function takes the general form˜ Z = exp (cid:32) ∞ (cid:88) g =1 g g − Z g (cid:33) = exp (cid:32) ∞ (cid:88) g =1 p g − G − Z g (cid:33) , (5.35)where Z g is the genus g worldsheet partition function. The localization propertyimplies that that the worldsheet genus takes the form g = N c ( G −
1) + 1 (withthe exception of genus g = 0). To take into account the sphere partition functionwith g = 0, we proceed as follows. The sphere partition function in the canonicalensemble should provide a contribution − γ ) I bulk , where the factor 6 comesfrom the fact that the seed theory of the boundary has central charge c = 6, seeeq. (5.27). Thus, in the grand canonical ensemble, the effect of the sphere partitionfunction is a renormalization p (cid:55)−→ p e − I bulk (5.36)or σ (cid:55)−→ σ + iπ I bulk .Thus, the string partition function becomes Z = exp (cid:32) ∞ (cid:88) N c =1 (cid:0) p e − I bulk (cid:1) n Z N c ( G − (cid:33) . (5.37)The partition function Z N c ( G − is in turn obtained by integrating (4.29). Werefine (4.29) by splitting the sum over covering maps as discussed in Section 5.2, seeeq. (5.5): Z g = Z classical , g (cid:90) M g (cid:88) ρ : π (Σ g ) −→ G (cid:88) ι : π (Σ g ) −→ π (Σ G ) , injective , p ◦ ι = ρ δ (6 g − (Σ g , Σ ι ) Z T (Σ ι , S ι ) . (5.38)Here, we changed the notation slightly and denote by Σ ι the marked covering surfacethat is determined by the homomorphism ι . We integrate over Σ g in the expression.Finally, S ι is the induced spin structure on Σ ι . Markings and integration.
We should note that the surface Σ ι comes naturallywith a marking, i.e. a set of generators α , . . . , α g , β , . . . , β g of π (Σ g ) up to inner automorphisms. The ambiguity up to an inner automorphism comes from the factthat we haven’t chosen a basepoint for π (Σ g ) and neither for the boundary. Thus,the same surface appears several times in the sum of the integrand, since we are alsosumming over markings. Recall that the moduli space of Riemann surfaces is relatedto Teichm¨uller space as follows: M g = T g / MCG(Σ g ) , (5.39)– 43 –here MCG(Σ g ) = Out( π (Σ g )) is the mapping class group. We have collected a fewrelevant facts in Appendix D.1. T g is the moduli space of marked Riemann surfaces.Hence, we can denote the result of the integral schematically as Z g = Z classical , g (cid:88) ρ : π (Σ g ) −→ Γ (cid:88) ι : π (Σ g ) −→ π (Σ G ) , injective , p ◦ ι = ρ Z T (Σ ι , S ι ) (cid:30) Out( π (Σ g )) , (5.40)which means that we pick only one arbitrary marking for each covering surface. Ifwe combine the two sums into a single sum of arbitrary homomorphisms of π (Σ g )to π (Σ G ), then this ‘gauging’ by Out( π (Σ g )) is easy to implement. An injectivehomomorphism ι : π (Σ g ) −→ π (Σ G ) up to conjugation and outer automorphismis fully characterized by its image ι ( π (Σ g )) ⊂ π (Σ G ) up to conjugation. Thus thesum becomes a sum over subgroups of π (Σ G ) up to conjugation of finite index d .Putting the pieces together, we obtain for the full string partition function Z = exp ∞ (cid:88) N c =1 (cid:0) p e − I bulk (cid:1) N c Z classical , N c ( G − (cid:88) H ⊂ π (Σ G ) up to conjugation , [ π (Σ G ):H]= N c Z T (Σ H , S H ) . (5.41)Here we changed notation again slightly to account for the fact that we are labelling(unmarked) covering surfaces and their spin structures by subgroups of π (Σ G ) (upto conjugation). The classical part.
We see that this formula is very close to the partition functionof the symmetric orbifold (2.4). It becomes equal provided that Z classical , N c ( G − = e N c I bulk N c . (5.42)This was observed to be true in cases with a torus boundary in [1], but we do notknow a general argument for this formula. Background independence.
In particular, once we make this identification, thebackground dependence of the string partition function completely cancels out. I bulk depends on the precise orbifold group that we use to engineer the bulk manifold,whereas the sum over subgroups only depends on the boundary surface, but noton the bulk three-manifold. While we are not able to give a general proof of thisphenomenon, we hopefully elucidated the mechanism behind it. Disconnected boundary.
Let us now consider the generalization to a discon-nected boundary. We consider for simplicity the case of a wormhole obtained by aquasi-Fuchsian group, with boundary components two genus G surfaces, that wedenote by Σ (1) G and Σ (2) G . The discussion is entirely analogous once we identify– 44 – G − = p p with the diagonal chemical potential, see eq. (5.34). The spherepartition function again renormalizes the chemical potentials. Since in the canon-ical ensemble, the sphere partition function should again lead to the contribution − γ ) I bulk , it does not distinguish covering maps that cover the left- or the rightboundary. This means that only the diagonal chemical potential is renormalized asfollows: p p (cid:55)−→ p p e I bulk , (5.43)where p and p are the fugacities of the left and right boundaries. In order toreproduce the symmetry orbifold partition function, both terms in (5.6) have theclassical contribution Z ( i )classical ,d ( G − = d − e dI bulk , as in eq. (5.42). Thus, the stringpartition function on the wormhole manifestly factorizes, Z wormhole ( p , p ) = Z Sym ( p ) Z Sym ( p ) . (5.44) In this section, we would like interpret our results more geometrically. In the previoussections, we provided evidence that the worldsheet partition function is independentof the bulk geometry and actually only depends on the boundary geometry.
We have seen that the symmetric orbifold partition function is naturally expressedin terms of (possibly disconnected) covering spaces and these are interpreted holo-graphically as the worldsheet. Thus, there is apparently no such concept as bulkgeometry, there is only the geometry of the covering space, i.e. the worldsheet.Even though our results defy the intuition of semiclassical gravity, they are stillvery geometric – as long as one replaces the concept of a bulk manifold by thecollection of worldsheets, which is a notion of stringy geometry in this concept. Theworldsheets themselves can in good approximation be treated semiclassically.
Sometimes, a collection of these stringy geometries can be interpreted as classical ge-ometries. For this to be meaningful, we want to talk about large N in the symmetricorbifold. Since we have argued that string theory describes the grand canonical en-semble, N gets replaced by the fugacity p = e πiσ . Initially, the grand canonicalensemble is only well-defined if we take Im σ big enough, since otherwise the defini-tion of the grand canonical partition function (2.2) does not converge. We define thefunction (Im σ ) min as the minimal chemical potential for which the grand canonicalpotential is still well-defined. This is an interesting function on the moduli space ofthe boundary surface(s). For Im σ close to (Im σ ) min , the grand canonical partition– 45 –unction is dominated by contributions from large N and we can expect semiclassicalbulk geometry to emerge.Let us consider a bulk background geometry with a torus boundary such asthermal AdS or the Euclidean BTZ black hole. In this case, we can be very explicitand (Im σ ) min is only a function of the boundary modular parameter τ bdry . Thebehaviour of the symmetric orbifold partition function in the canonical ensemble isknown explicitly in a large N limit [58]:log Z NSSym N ( T ) ( τ bdry ) ∼ N π max ( c,d )=1 , c + d odd Im τ bdry | cτ bdry + d | . (6.1)The maximum is taken over all coprime pairs of integers whose sum is odd. Weconsidered the NS spin structure. The expression is manifestly invariant under therelevant modular groupΓ NS = (cid:26) (cid:18) a bc d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) a + d even , b + c even , c + d odd (cid:27) . (6.2)This translates into (Im σ ) min = 12 max ( c,d )=1 , c + d odd Im τ bdry | cτ bdry + d | . (6.3)This function has phase transitions. Conventionally, these terms are interpreted asclassical bulk geometries that dominate the partition function in different regimes.For purely imaginary τ bdry , there is a phase transition at τ = i corresponding tothe transition between thermal AdS (( c, d ) = (0 , c, d ) = (1 , Z NSSym( T ) = exp ∞ (cid:88) m,w =1 (cid:88) r ∈ Z / Z w p mw mw Z T (cid:20) r + m w (cid:21) (cid:18) mτ bdry + r (cid:19) . (6.4)This is a way of writing a sum over all covering tori, compare to eq. (2.4) for thegeneral case. The degree of the relevant covering map takes the form N c = mw . Theparenthesis with entries r + m ∈ Z / Z and w ∈ Z / Z indicates the spin structure ofthe T partition function. Intuitively, he parameter w is the winding number of theworldsheet around the spatial cycle of the boundary torus and the parameter m is thewinding number around the temporal cycle. Close to (Im σ ) max , some covering mapsare dominating the partition function. In the example of a boundary torus, theseare simple to describe. For the term ( c, d ) = (0 , τ covering = N conn τ bdry , where N conn isthe degree of the (connected) covering map. This corresponds to the terms w = 1,– 46 – = 1 and m = N c in the eq. (6.4). Taking the exponential into account, we geta collection of (possibly disconnected) dominating worldsheets for every partition N = N + N + . . . of N . This is simply the untwisted sector of the symmetricorbifold, which describes N fundamental strings that wind each once around thermalAdS , appropriately symmetrized to account for their statistics. For the other terms,different covers dominate. For instance, in the case of ( c, d ) = (1 , τ covering = τ bdry N conn , which is dominant in thelimit τ bdry →
0. In general, for every choice of coprime ( c, d ) with c + d odd, there isexactly one dominating connected covering surface of every degree. We remark that for each degree there are only finitely many covering surfaces,and thus for every finite degree (i.e. any finite N ), there are some classical geometriesthat the stringy geometries fail to distinguish. In the grand canonical ensemble, allthe different geometries are resolved eventually. This qualitative feature should alsohold for more complicated bulk/boundary geometries. Poles from bulk geometries.
What we just observed experimentally in the sym-metric orbifold is expected to hold true generally. Assume that the canonical bound-ary partition function is dominated by a semiclassical single bulk geometry. Fromthe general ideas of holography, we expected that Z Sym N ( T ) ∼ e − NI bulk , (6.5)up to an order 1 quantum correction. Here, we normalized I bulk as in Section 5.4. Theexplicit factor of N in the exponent comes from the fact that the Newton-constantsatisfies G N ∝ N − as follows from the Brown-Hennaux formula [59]. For the grandcanonical partition function, this implies that Z Sym( T ) ∼ − p e − I bulk . (6.6)Thus, we expect the appearance of a pole at p = e I bulk (or at σ = − iπ I bulk ). Symmetric orbifold of the monster.
This behaviour was also observed in atoy model – the symmetric product orbifold of the (holomorphic) Monster CFT[60]. Holomorphicity gives much stronger control in this case and in fact the grandcanonical partition function is determined by Borcherds formula: Z Sym( M ) = e − πiσ j ( σ ) − j ( τ bdry ) , (6.7)where j is the Klein j -invariant. In this case, condensation to thermal AdS happensnear σ = τ bdry . In fact, it was argued in [60] that this condensation is a Bose-Einstein The relation between ( c, d ) and ( m, w, r ) is in general cumbersome to describe and we omit it.But there is a very clear geometrical interpretation of the relation which is given by the rules 1–3below. – 47 –ondensation – near σ = τ bdry , a finite fraction of CFTs is in the ground state. Theformula has also poles whenever σ is related to τ bdry by a modular transformation,which can be interpreted as condensation of black hole geometries. This qualitativebehaviour carries over to the non-holomorphic symmetric orbifold of T . Analytic continuation and pole structure.
The toy model makes it clear thatit is useful to consider the grand canonical partition function as a meromorphicfunction in σ . While the definition of the symmetric orbifold partition function (2.4)only converges for Im σ > (Im σ ) min , it is easy to analytically continue in σ . This ismost conveniently done using the product formula of the symmetric orbifold partitionfunction [61, 62]. Through analytic continuation, one can actually see that the grandcanonical partition function has additional poles whenever σ = i Im τ bdry M , (6.8)where M is an odd positive integer. In fact, the grand canonical partition functionhas (at least) the following poles: σ = i M Im τ bdry | cτ bdry + d | (6.9)for c + d is odd and M odd. Interpretation as bulk geometries.
We can interpret all these poles as bulkgeometries. Integers
M > π (1 − M − ). The constraint on c + d and M odd comes from the spin structure,since only in this case there is a bulk spin structure compatible with the givenboundary spin structure [63]. Thus, we argued that whenever the chemical potentialapproaches a pole, the tensionless strings form a condensate and the classical bulkgeometry emerges.We should note that there can be even more poles for σ in the upper half plane. There is a pole in the grand canonical partition function for every scalar state in theCFT T that satisfies ∆ < . Such states may or may not be present depending onthe precise shape of the torus. We will see a similar problem appearing for highergenus partition functions. For these values of the chemical potential, there is asingularity appearing from a large number of excited tensionless strings. We do notknow how to interpret these singularities from a semiclassical bulk perspective.The phase diagram of the torus partition function for purely imaginary τ bdry takes the form as depicted in Figure 8. Near the edge, stringy geometry condenses These additional poles did not show up in the Monster toy example because the orbifold pro-jection acts assymmetrically and projects them out. Formally, there are even more poles in the lower half-plane. However, one can see quite easilythat the poles accumulate on the rationals (this is obvious in the Monster example eq. (6.7)). Thus,it is not clear to us whether analytic continuation to the lower half-plane is sensible. – 48 – o n d e n s a t i o n (Im σ ) min grand canonical partitionfunction converges Im τ bdry Im σ Figure 8 . The behaviour of the grand canonical torus partition function for purely imag-inary τ bdry . Blue lines correspond to poles with M = 1 and red lines to poles with M = 3. into classical geometry. One can analytically continue below the solid blue line thatrepresents (Im σ ) min to observe also the other condensing geometries. Residue.
The residue of the grand canonical potential can be interpreted as theinfinite N stringy one-loop determinant around the respective background. Let usfocus on the thermal AdS pole at σ = i Im τ bdry . Writing Z T ( τ bdry ) = e − π Im τ bdry Z T qu ( τ bdry ) , (6.10)we separate out the classical part of the T partition function. The residue takes theform Res σ = i Im τ bdry Z NSSym( T ) = exp ∞ (cid:88) m, w =1 (cid:48) (cid:88) r ∈ Z / Z w e − πm ( w − w ) Im τ bdry mw Z T qu (cid:20) r + m w (cid:21) (cid:18) mτ bdry + rw (cid:19) . (6.11)The prime signifies that the summation over m for the divergent term is omitted – thisis the term with w = 1 and the leading term in Z T qu coming from the vacuum. Thisis the one-loop determinant of the symmetric orbifold Sym ∞ ( T ). It was computedin this form from string theory in [30]. Aside from possible convergence issues thatwe have not analyzed, we can reconstruct the full grand canonical partition functionfrom the infinite N one-loop determinants: Z NSSym( T ) = (cid:88) bulk geometries M Z qu ( M ) σ + iπ I bulk ( M ) + poles from light scalars in T , (6.12)– 49 –here Z qu ( M ) is the suitable one-loop determinant for the respective bulk geometry.While this is under good control for a torus boundary, we know much less about theanalytic behaviour for higher genus boundaries. There could be branch cuts, higherorder poles etc. By taking suitable contour integrals, one may also reconstruct thefinite N canonical partition function. We should note that this ‘Farey tail’ relieson holomorphicity in σ and not on holomorphicity in the modular parameter andthus does not need to assume holomorphicity of the partition function or the ellipticgenus [64, 65]. Rules for the bulk geometry.
Given that the grand canonical ensemble toruspartition function can be interpreted as a sum over bulk geometry, we may try tocome up with a general rule on how to reconstruct the bulk geometry/topology fromthe respective family of covering maps. One can analyze the geometry of the coveringmaps that leads to the poles (6.9). The relation between condensing worldsheets andbulk geometries is in this case the following:1. If every condensing worldsheet wraps a boundary cycle only once, then thisboundary cycle is contractible in the emerging classical geometry.2. If every condensing worldsheet wraps a boundary cycle exactly M times, thenthis boundary cycle is bounded by a disk in the bulk with a conical defect withdeficit angle 2 π (1 − M − ).3. For a non-contractible cycle in the emerging bulk geometry, there is no boundin the condensate for how many times the cycle is wrapped.The second rule can be motivated from the first, since a string that winds M timesaround a conical deficit with deficit angle 2 π (1 − M − ) can be viewed by unwrappingthe geometry as a string that winds once around the same geometry without deficitangle.These rules make it geometrically clear that there is exactly one connected cover-ing map of every degree contributing for the non-singular geometries and one coveringmap whose degree is a multiple of M for the conical defect geometries. There is alast geometry that we have not discussed: the cusp geometry which we mentionedin Section 5.3. It has a somewhat singular role, since it can be understood as the M → ∞ limit of the conical defects. This means that the potential pole it causesin the grand canonical partition function is σ = 0, which is the accumulation pointof poles of the conical defects. Thus, it is a delicate issue whether it should be in-cluded in the sum over bulk geometries. Since it is obtained as the limit M → ∞ ofthe conical defects, our rules seem to suggest that is not associated to any stringygeometry of covering maps. For the symmetric orbifold of the Monster CFT, this was already discussed in [60]. – 50 – igher genus boundary.
We expect the behaviour of the higher genus partitionfunctions to be similar, but there are some important qualitative differences. In thehigher genus case, the number of connected covering spaces is (super)exponentiallylarge in the degree N c , compared to roughly O ( N c ) connected covering spaces inthe case of a single torus boundary. It is explained in Appendix C that there are O (2 N c ( N c !) g − ) connected covering surfaces of degree N c if g ≥ N c (cid:29)
1. Thisenormous qualitative difference is also reflected in the classical geometries: for genus g ≥
2, there are many more possible bulk manifolds – not only handlebodies, butalso non-handlebody geometries.Correspondingly, the expected pole structure of the grand canonical partitionfunction and the function (Im σ ) min is much more complicated. Besides the conformalstructure, it also depends on the metric on the boundary surface due to the conformalanomaly. Under a Weyl transformation, (Im σ ) min transforms as(Im σ ) min (cid:55)−→ (Im σ ) min − πS [ φ ] , (6.13)where S [ φ ] is the Liouville action capturing the conformal anomaly of the Weyltransformation. This ambiguity does not affect the behaviour of the poles and fordefiniteness, we might hence fix the hyperbolic metric on the boundary Riemannsurface.The natural guess for the function (Im σ ) min is now(Im σ ) min = 3 iπ max (singular) hyperbolic 3-manifolds compatiblewith boundary spin structure M I bulk ( M ) , (6.14)where the maximum is taken over all possibly singular hyperbolic three-manifoldswith the correct conformal boundary. It was claimed that this is not true for highergenus partition functions if there is a light enough scalar in the spectrum, whichhappens e.g. for a very large or small T [66]. Such a condensing bulk geometrywould no longer have the form AdS / Γ × S × T , but would also involve the internalfactors non-trivially. This is similar to what we observed for the pole structure of thetorus partition function, except that the ‘excited geometries’ originating from the T excitations can conceivably even be dominant in the higher genus case.It is plausible that all the handlebodies themselves can be realized as particularstringy geometries. In fact, there seems to be a natural class of covering maps as-sociated to every handlebody. Recall that a handlebody is specified by a choice ofLagrangian sublattice of H (Σ G , Z ) in the boundary, corresponding to the homologycycles that become contractible in the bulk. Let us consider the Lagrangian sublat-tice generated by the standard generators (cid:104) β , . . . , β G (cid:105) , which would be the highergenus analogue of thermal AdS . In this case, there is a natural class of covering The number of connected covering spaces of degree N c is actually σ ( N c ), the divisor function,for a torus boundary. – 51 –urfaces that are candidates to form a bulk condensate. In general, degree N cover-ing surfaces are specified by homomorphisms φ : π (Σ G ) −→ S N , see the discussionin Appendix C.2. We propose to consider homomorphisms for which φ ( α I ) = , theidentity permutation. The resulting covering maps hence have only their β I -cyclesunwrapped. This is motivated by the rules (1)–(3) above that we observed for atorus boundary. We do not know whether φ ( β I ) should satisfy further constraints inorder to contribute to the semiclassical geometry. This assignment is also motivatedby the fact that the β I cycles are the analogue of the thermal cycle in the torus case.If we instead want to create a conical defect for the α cycle, say, then we wouldrequire that φ ( α ) is some fixed cyclic permutation of length M .It is then plausible that also other non-handlebody solutions geometries lead topoles in the grand canonical partition function. There are however several issuesthat are unclear to us:1. We do not know whether there exists a well-defined analytic continuation.There could be more exotic phenomena for higher genus boundaries such asbranch cuts etc.2. In the case of a genus G (cid:54) = 1 boundary, we can add a counter term π (cid:82) R√ g =2 − G to the boundary action, where g is the boundary metric. This has theeffect of renormalizing the partition function. At least at the face of it, we canchoose these counterterms independently for all N in the canonical ensemble.Changing these counterterms can change all the locations of the poles in thegrand canonical partition function simultaneously and it can also change theirorder.3. We saw in the case of the torus partition function that the cusp geometry ismore singular than conical defects, since it would be located at accumulationpoints of the chemical potential. Such accumulation points should also exist forthe higher genus boundaries. At least some of these correspond to analogues ofcusp geometries (that we define as geometries with codimension 3 singularities).4. We do not know the fate of codimension 1 singular bulk geometries. A par-ticularly important codimension 1 singular bulk geometry is obtained by orb-ifolding the Fuchsian wormhole by the reflection symmetry. The geometry isdescribed in more detail in Appendix A.4. If such a geometry were to emergefrom stringy covering maps, it would need to do so in a way that treats allcycles symmetrically.
5. Some geometries only exist for specific choices of the boundary moduli. Forexample, if the boundary surface has an involution without fixed points, then This particular bulk geometry is non-orientable in the sense that the reflection symmetry thatwe use in the orbifold reverses orientation. Thus, the geometry might not appear for this reason. – 52 –e can consider the quotient of the Fuchsian wormhole by the combined reflec-tion and involution. If the correspondence between bulk geometries and polesin the grand canonical partition function holds, then it seems that there mustbe new ‘exceptional’ poles for these special values of the moduli.
We have seen that at least in some cases, one can convincingly identify geometrieswith a single boundary as a family of covering maps. It is an important problemto extend this to geometries with disconnected boundaries. In this case, there aretwo chemical potentials σ and σ . Of course, the purpose of our discussion ofthe tensionless string was to establish that the grand canonical partition functionfactorizes, both from a bulk and a boundary perspective and the disconnected grandcanonical partition function simply takes the form Z (1) ( σ ) Z (2) ( σ ), where Z ( i ) arethe grand canonical partition functions of the two boundaries. In view of the abovediscussion the poles of this product grand canonical potential in ( σ , σ ) are obviouslyjust a reflection of all the disconnected geometries.We now speculate that even though the grand canonical partition function fac-torizes, it might still contain the information about all the classical wormholes. Tosee wormholes, it is much more natural to change the basis of the chemical potentialsand define σ = σ θ , σ = σ − θ , (6.15)so that σ is the ‘diagonal’ chemical potential and θ is the ‘off-diagonal’ chemicalpotential. We now want to ‘entangle’ the left and right boundaries, since this shouldcreate a correlation between the two boundaries. We view this as a version of astringy ER=EPR [67]. In the present instance, this is naturally done by integratingover θ . From the definition of the grand canonical ensemble, this has the followingeffect: (cid:101) Z Sym( T ) ( σ ) = (cid:90) d θ Z (1)Sym( T ) (cid:16) σ θ (cid:17) Z (2)Sym( T ) (cid:16) σ − θ (cid:17) (6.16)= ∞ (cid:88) N =0 , N =0 (cid:90) d θ Z (1)Sym N ( T ) Z (2)Sym N ( T ) e πiσ ( N + N ) e πiθ ( N − N ) (6.17)= ∞ (cid:88) N =0 Z (1)Sym N ( T ) Z (2)Sym N ( T ) e πiσN . (6.18)For this to make sense, we want thatIm σ > σ ) (1)min , σ ) (2)min , (6.19)so that the series converges uniformly and we are allowed to interchange the sumand the integral. Here, the superscripts (1) and (2) refer as usual to the left and right– 53 –oundary. The resulting expression still has poles for every disconnected geometrythat lead to a pole earlier. The location of the pole in σ is σ (1)p + σ (2)p , where σ (1)p and σ (2)p are two poles of Z (1)Sym( T ) and Z (2)Sym( T ) , respectively. Thus, these geometries arenaturally still interpreted as the disconnected geometries with bulk action I bulk ( M (1)3 (cid:116) M (2)3 ) = I bulk ( M (1)3 ) + I bulk ( M (2)3 ) = σ (1)p + σ (2)p . (6.20)However, it could be possible that (cid:101) Z Sym( T ) ( σ ) has poles that are not associatedto any of the disconnected geometries. For example, if Z (1 , T ) has branch cuts,then the integral over θ leads to an integral around these branch cuts, which is notassociated with any pole and hence no disconnected geometry. This does not seem tohappen for the torus partition function, because it only has simple poles. But this isexpected, since there is no connected bulk geometry with two torus boundaries. Itseems a very difficult problem to explicitly compute Z (1)Sym( T ) and Z (2)Sym( T ) for genus 2boundaries, do the analytic continuation and understand the pole structure. Hencewe have not been able to determine whether wormhole geometries can actually appearin the grand canonical partition function. One could introduce an ensemble average in the symmetric orbifold. The presum-ably easiest way would be by averaging over the Narain moduli space of T , as wasdone in [13, 14] for a bosonic sigma-model on T . This is problematic to do forthe partition function of the symmetric orbifold itself, because the average does notconverge. One can get a convergent result for the logarithm of the torus partitionfunction. This is essentially the quenched free energy [68] (in contrast to the annealed free energy, where one first averages and then takes a logarithm). For a single torusboundary, this quantity takes the formlog Z Sym( T ) = ∞ (cid:88) m, w =1 (cid:88) r ∈ Z / Z w p mw mw Z T (cid:20) r + m w (cid:21) (cid:18) mτ bdry + rw (cid:19) , (6.21)where overline denotes the ensemble average. The averaged T partition functioncan be interpreted as a sum over bulk geometries. These bulk geometries are ge-ometries that fill in the covering space and not the original boundary of AdS . Wehave learned that the covering space is to be interpreted as the worldsheet. Thus,averaged connected partition function can be interpreted as a sum over all fillings More precisely, there is no hyperbolic 3-manifold that has two tori as conformal boundaries.This is quite simple to see from a realization of the hyperbolic manifold from a Kleinian group, seee.g. [1] for an explanation. I thank Alex Belin for discussion about this. This is because averages (cid:104) Z G T · · · Z G n T (cid:105) for a single T sigma model only converge for (cid:80) i G i ≤ G i denotes as usual the genus of the surface. – 54 –f all connected worldsheets. Disregarding convergence issues, the same is true for Z Sym( T ) itself: it can be interpreted as a sum over all possible geometries (in thesense of U(1)-gravity [14]) that fills in the possibly disconnected worldsheet.Such geometries are topological spaces, where one is allowed to make additionalidentifications on the boundary to obtain the correct boundary manifold. For exam-ple, the totally disconnected contribution in thermal AdS would be the space N (cid:71) i =1 thermal AdS ( i )3 (cid:46) ∼ , (6.22)where ∼ identifies all the boundaries of the thermal AdS spaces. The number ofcopies is fixed to N in the canonical ensemble, but is again arbitrary in the grandcanonical ensemble. We emphasize however that there are a lot of possible geometriesof this sort. We could replace some of the thermal (6.22) by black holes or couldreplace pairs (or higher tuples) of AdS spaces by wormholes, etc. We could alsoreplace a pair of thermal AdS ’s by one thermal AdS whose length of the thermalcycle is twice as large, but which has an additional identification on its boundarytorus.In this averaged form, condensation of such averaged stringy geometries intoclassical geometries becomes much clearer. For example, (6.22) is one geometry thatfor large N resembles more and more thermal AdS itself, because the different copiesof thermal AdS start to form a continuum. For example, the two condensing stringygeometries for a boundary torus and N = 2 are depicted in Figure 9. It might alsohelp if we picture only the t = 0 slices of the geometries in Figure 9. For this we cutthe torus open and look only at the cut. This yields the first picture in Figure 10.The difference between the two geometries is invisible on the level of the t = 0 slice,their difference lies in whether the blue and red sheets are interchanged when movingaround the thermal circle or not. In the ensemble average picture, there is also awormhole geometry that we did not picture in Figure 9, but whose t = 0 surface is thesecond picture in Figure 10. This by far does not exhaust the list of possible stringygeometries with two sheets. The conical defect geometry is pictured in Figure 11.There are also other geometries that do not exhibit a clear Lorentzian interpretation.For example, thermal AdS (cid:116) Euclidean BTZ with identified boundaries is such a case.This geometry can be pictured geometrically as two interlocking solid tori, which isin fact a description of the three-sphere as a Heegaard splitting. We discuss somefurther speculations on these geometries in the Discussion 7.2. The reader might feel uneasy about this identification. We perform this identification mainlyfor the physical interpretation’s sake, but it doesn’t change any physical quantity that we arecomputing. – 55 – igure 9 . The two averaged stringy geometries that condense to classical thermal AdS for a degree two covering surface. In this figure, the boundaries of the inner and outertorus are identified. Figure 10 . The t = 0 surfaces of some string geometries with two sheets. The boundaryis the spatial boundary circle of global AdS . In the previous subsection, we explored ensemble averaging as a way to let thesemiclassical bulk geometry emerge semiclassically. The averaging process turns onan ‘interaction’ between the different covering spaces, since it allows for wormholesconnecting them. However, there is a more traditional way of turning on such aninteraction, namely by deforming the symmetric orbifold away from its orbifold point.We shall see that the two procedures are in some ways similar. A large deformationin the moduli space of the symmetric orbifold is expected to connect the tensionlessstring to the supergravity regime, where gravity can be treated semiclassically [69].This means in particular that the deformation will need to connect the ‘stringy’gravity picture that we discussed smoothly with a semiclassical picture.– 56 – igure 11 . The t = 0 surface of the stringy conical deficit geometry with M = 2. Thisgeometry is obtained from the disc { z ∈ C | | z | ≤ } by the identification z ∼ − z on theboundary. Marginal operator.
There is a marginal operator in the theory that turns on thedeformation. Its construction is as follows. The twist field of a transposition hasconformal weight h =
624 2 − + · = . The first contribution is the standardground state energy of the twist 2 sector. The second term is the ground stateenergy of the Ramond sector of four free fermions, since the relevant spin structurein evenly twisted sectors is the Ramond sector. The ground states transform inthe spinor representation of so (4) that decomposes under the R-symmetry subgroup su (2) as ⊕ · . In particular, the highest weight state of the doublet is a BPSstate of the N = 4 algebra. Thus it gives rise to a marginal operator (actuallyfour marginal operators) by taking the level superconformal descendants. Let Ψ α ˙ α be the BPS field (where α and ˙ α are spinor indices of the left- and right-movingR-symmetry), then the marginal operator is given byΦ A ˙ A = ε αβ ε ˙ α ˙ β G βA − ¯ G ˙ β ˙ A − Ψ α ˙ α (6.23)Here, capital indices are spinor indices of the outer automorphism group of the N = 4superconformal algebra. Conformal perturbation theory.
We now imagine to compute the symmetricorbifold partition function perturbatively in the deformation parameter. The per-turbed partition function takes the form Z ( λ )Sym( T ) = (cid:68) e − λ (cid:82) Σ Φ (cid:69) = ∞ (cid:88) n =0 ( − λ ) n n ! (cid:28)(cid:18)(cid:90) Σ Φ (cid:19) n (cid:29) , (6.24)where we take for definiteness Φ ≡ ε A ˙ A Φ A ˙ A the singlet with respect to the diagonalautomorphism group. Thus, in perturbation theory, we need to compute correlation We evade the question on how to correctly normalize Φ in the grand canonical ensemble. The following discussion is independent of this choice. – 57 – igure 12 . Degree 2 branched covering maps appearing in conformal perturbation awayfrom the symmetric product orbifold. The left figure shows the surface that appears atsecond order in conformal perturbation theory. The right surface appears at higher orderin perturbation theory. functions of twist 2 operators on the torus. They are as usual given by summing overall covering surfaces of the torus, but now these covering surfaces are branched over n points at n -th order in perturbation theory. The branch points are simple, i.e. oforder 2. Odd orders in the perturbation theory actually vanish. This can be seengeometrically from the Riemann-Hurwitz formula (5.17), since for a branched coverthe sum (cid:80) i ( w i −
1) over ramification indices has to be even.
Interactions between covering surfaces.
Inclusion of twist fields in correlatorseffectly turns on an interaction of the different covering spaces. Let us explain thisat the simplest example of a symmetric orbifold with two copies. In the undeformedtheory, there are four covering spaces in total of which one is disconnected. Atsecond order in conformal perturbation theory, covering surfaces cover the torusholomorphically with two ramification points of order 2. Such a covering surfaceis of course connected. Schematically, we can write such a surface as in Figure 12.Here, the dashed lines correspond to the branch points of the covering and in betweenthem are branch cuts. Emergence of semi-classical bulk geometry.
We now offer a speculative pic-ture on how the semi-classical bulk geometry emerges at very high order in conformalperturbation theory. As we just saw, the different sheets of the stringy geometry areindependent in the symmetric orbifold point, but become more and more correlated It follows from the Burnside formula for covering maps that there are always four coveringmaps at every even order in perturbation theory. The surfaces depicted in Figure 12 are thoseoriginating from the disconnected covering surface in the undeformed symmetric orbifold. Thereare also surfaces originating from the connected covering surfaces. – 58 –s we deform the theory. Because of the independence of the covering surface atthe tensionless point in moduli space, there are also independent bulk geometries forall the different covering surfaces. These are the multi-sheeted geometries that weconstructed in the previous subsection. As we turn on the interaction between thecovering surfaces, we should also turn on an interaction between the different sheetsof the geometry. It is plausible that configurations where the sheets of the micro-geometries align have a much lower action once this interaction is turned on. Oncewe make the interaction strong enough, only those bulk geometries with aligningsheets survive and form semi-classical bulk geometries.
Let us summarize the most important points of the paper. We put the fine print infootnotes.1. The worldsheet theory of the tensionless string in a background M × S × T ,where M is a hyperbolic 3-manifold, localizes on covering surfaces of theboundary Riemann surface(s) Σ (1) (cid:116) · · · (cid:116) Σ ( n ) .
2. We have argued that the string partition function on different hyperbolic 3-manifolds with the same boundary Riemann surface(s) Σ (1) (cid:116) · · · (cid:116) Σ ( n ) agreeto all orders in string perturbation theory.
3. The natural ensemble for string perturbation theory in AdS / CFT is the grandcanonical ensemble. Instead of fixing the number of fundamental strings in thebackground, an associated chemical potential is fixed, which on the string sideis related to the string coupling constant.4. For special values of the chemical potential, the grand canonical partition func-tion can have poles. These poles can be associated to (possibly singular) on-shell classical bulk geometries. We argued that the classical bulk geometryshould be understood as a condensate of covering maps. We proposed a set of Such an interaction is necessarily non-local from the point of view of the ‘micro-geometries’,but can be local from the point of view of the emergent semi-classical geometry. We have established that the worldsheet theory localizes on a discrete subset of the modulispace of Riemann surfaces. Covering surfaces are contained in this subset, but we have not beenable to show equality of the two sets. In establishing these facts, we emphasized the role of acomplex projective structure that emerges naturally from the worldsheet theory (branched complexprojective structure in the case of correlation functions). We realized M = H / Γ as a quotient of global hyperbolic space. Different bulk manifoldscorrespond to different orbifold groups Γ and different choices of Γ reorganize the sum over coveringmaps. – 59 –incomplete) rules how the classical geometry and the stringy geometries arerelated. The full grand canonical partition function can be recovered by a fareytail sum over classical geometries with a 1-loop determinant, see eq. (6.12).
5. The grand canonical partition function can depend on several chemical poten-tials – one for each boundary in the geometry. The remaining chemical poten-tials can be introduced on the worldsheet via discrete torsion. While wormholegeometries cannot lead to poles in the full grand canonical partition functionbecause it factorizes, we speculated that wormholes could become visible assingularities in the chemical potential once we integrate out the off-diagonalchemical potentials.6. Averaging over the Narain moduli space of T makes the relation betweenstringy geometries and classical geometries more manifest. Averaged stringygeometries can be interpreted as all possible fillings of the worldsheet, whichare geometries with multiple sheets that meet at the boundary of AdS . Whentuning the chemical potential to the critical value, these multiple sheets cancondensate to a single classical geometry. Deforming away from the symmetricorbifold has also the effect of introducing an interaction between the coveringsurfaces. We argued that also for strong interaction, the interaction favors thosemicro-geometries that align properly and leads to an emerging semi-classicalbulk geometry. Let us discuss some open issues. The first points are technical and the later onesmore conceptual.
The hybrid formalism in the free-field variables.
Our main technical goal inthis paper was to compute the string partition function of the tensionless string onvarious hyperbolic manifolds. Even though the tensionless string enjoys a free fieldrepresentation in the hybrid formalism, its BRST structure is quite complicated andwe have not attempted to fully define the string partition function in these free fieldvariables. In order to put this and similar computations on a firmer footing, it isindispensable to develop the hybrid formalism in these variables more rigorously.
Localization to covering maps.
We have not succeeded to demonstrate fullythat the string partition function localizes only on covering maps. We gave a simplecounterexample in Section 4.2: the worldsheet torus partition function seems ingeneral to be non-zero also in geometries with higher genus boundaries. However, We did this analysis very explicitly for a torus boundary, where the appearing geometries areSL(2 , Z ) family of black holes and conical defects. For more complicated geometries, our assertionis much more speculative. – 60 –ince the orbifold is infinite, the inclusion of the order of the orbifold group is ratherdelicate and can suppress these contributions. We do not know whether the ‘fake’covering surfaces that appear have a deeper meaning or are just a complication ofthe formalism. Worldsheet theory from the boundary CFT.
The complications of perturba-tive string theory are striking, when compared to the simplicity of the result from adual CFT point of view. String theory has to deal with lots of additional structuresthat are not visible in the final result, such as BRST cohomologies, integrals over M g,n × Jac(Σ g ) etc. One could hope to construct a version of string perturbationtheory that does not have all this additional structure. The idea of reconstructingthe worldsheet from the boundary CFT is an old one, see [70–73] and was recentlymade concrete for symmetric orbifolds in [74]. Backreaction.
One might question the validity of string perturbation theory inour work. We have trusted string perturbation theory far beyond the regime whereit is usually applicable, since we have considered arbitrary numbers of highly wind-ing strings. We would expect such heavy string configurations to backreact on thegeometry and change it. This is in fact precisely what we saw, since we can interpretcondensates of string configurations as new classical geometries. Nonetheless, thestring perturbative description seems to work well in that it succeeds to describethe full boundary theory correctly. One potential reason why this happens is per-haps the fact that the worldsheets (except for the sphere) seem to stay close to theboundary of the bulk. Thus even though the interior of the bulk geometry changes,perturbative strings do not get affected by this change.
Non-saddle geometries.
Recently, the importance of including non-saddle ge-ometries in the gravitational path integral has been emphasized [11, 21, 75]. If theseappear in the grand canonical partition function, they are hard to isolate. Let usdiscuss this for the boundary torus partition function. The reason is that they areonly expected to have an order 1 action and thus would lead to poles in the grandcanonical partition function at p = 1, i.e. at σ = 0. However, σ = 0 is already anaccumulation point of poles of the partition function. Thus, non-saddle geometriesdo not seem to naturally appear in the grand canonical farey tail sum (6.12). Black hole-string and wormhole-string transition.
It has been proposed thatblack holes of string size transition into a very long string winding around the horizon[76–78]. Our results of this paper that this is true very generally in our setting. Whileall computations have been done in Euclidean signature, the qualititative picture ofwinding strings should be preserved under Wick rotation. In some sense or analysisthus leads to more general wormhole-string transitions etc.– 61 – ulk emergence.
While the condensation of covering maps to classical geometriesseems to work quantiatively, it does not explain how the bulk emerges from the two-dimensional worldsheets. Ultimately, we should not only see the bulk action emerge,but also more local notions like bulk reconstruction from entanglement wedges. Itremains to be seen whether this toy model can be used to obtain more fine-grainedinformation.
Ensemble average and chaos.
The tensionless string lacks any chaotic behaviour.Even after ensemble averaging any member of the ensemble is a solvable CFT. Inparticular, indicators such as the spectral form factor and out of time order correla-tors diagnose a clearly non-chaotic behaviour of the symmetric orbifold [79, 80]. Thisis in sharp contrast with the situation in theories of pure gravity, such as JT-gravity[11]. Thus our analysis corroborates that the ensemble is a feature of low-energyeffective descriptions of the theory.
Baby universes and third quantization.
We have seen in Section 6.4 that onecan introduce an ensemble average in the symmetric orbifold that seems to make thenature of the stringy geometries somewhat more manifest. It is interesting that theaverage of a single partition function already contains many geometries that featuredisconnected wormholes, even though the asymptotic boundaries of the wormholeare in fact identified, see Figure 10. Perhaps these ‘micro geometries’ can be viewedas baby universes [24, 81–83]. Deformation away from symmetric orbifold.
We have started to analyze thebehaviour of the symmetric orbifold partition function when deforming away fromthe symmetric orbifold and shown that it turns on an interaction of the coveringsurfaces. It would be very interesting to make our speculations about the emergenceof the bulk geometry more concrete.
Effective spacetime theory.
The description of the tensionless string is formu-lated on the worldsheet. It would be far more enlightening to understand the corre-sponding spacetime theory. At the tensionless point, such a theory would be highlynon-local. Constructing the spacetime theory explicitly would presumably shed fur-ther light on the emergence of a local bulk dual when deforming away from thesymmetric product. For vector-models, the corresponding higher spin spacetimetheory was recently constructed in [84].
AdS × S . There is the obvious question how this generalizes to higher dimen-sional examples of the AdS/CFT correspondence, most notably AdS × S . The dual N = 4 SYM theory possesses a free point that corresponds to tensionless strings in In the examples of Figures 10 and 11, the Cauchy slice is such that the spatial surface has nocompact disconnected region. However, one can see that one can arrange the Cauchy slice in morecomplicated examples such that all the sheets are disconnected universes. – 62 –he bulk. While there is not yet a complete formulation of the corresponding world-sheet theory (see however [85, 86] for attempts and progress), the question can bestudied from the boundary. This has a long history [38, 87]. We hope that theinsights of this paper can be useful also in higher dimensions.
Acknowledgements
I would like to thank Alexandre Belin, Andrea Dei, Bob Knighton, Shota Komatsu,Juan Maldacena, Tom´aˇs Proch´azka, Bo Sundborg and Edward Witten for discus-sions. I would also like thank Andrea Dei, Bob Knighton, Shota Komatsu and espe-cially Bo Sundborg for very helpful comments on the manuscript. I thank Alessan-dro Sfondrini for organizing the workshop “Correlation functions in low-dimensionalAdS/CFT” in Castasegna, where this work was started. This work is supported bythe IBM Einstein Fellowship at the Institute for Advanced Study.
A Hyperbolic 3-manifolds
In this Appendix, we collect some facts about hyperbolic 3-manifolds. These arelocally Euclidean AdS spaces and serve as bulk manifolds in our investigation. Everyhyperbolic 3-manifold can be written as a quotient H / Γ for a discrete subgroupΓ ⊂ PSL(2 , C ), where H is hyperbolic 3-space. Thus the study of hyperbolic 3-manifolds is equivalent to studying discrete subgroups Γ ⊂ PSL(2 , C ) – so-called Kleinian groups . A.1 Hyperbolic 3-space
We work with the Poincar´e ball model, where H is identified with the unit ballinside R , equipped with the metric4 (cid:80) i =1 d x i (1 − | x | ) . (A.1)The isometry group is the conformal group PSL(2 , C ). The boundary of H is theRiemann sphere CP and PSL(2 , C ) acts by M¨obius transformations on it. H ⊂ H can then be thought of as the equatorial plane x = 0, which ispreserved by a subgroup PSL(2 , R ) ⊂ PSL(2 , C ). The boundary of H inside H canbe identified with the equatorial circle R ∪ {∞} of CP , which is indeed preserved byPSL(2 , R ). This equatorial plane corresponds to the t = 0 spacelike surface in global Lorentzian
AdS . – 63 – .2 General properties While a Kleinian group Γ acts properly discontinuously on H , it typically does noton the boundary CP . Let Ω be the maximal open set in CP on which it doesact properly discontinuously. We shall in the following assume that Ω (cid:54) = ∅ . Thecomplement Λ = CP \ Ω is called the domain of discontinuity or the limit set. ByAhlfohr’s measure theorem, Λ has measure zero. The set Λ is however typically verydiscontinuous and fractal.Let us furthermore assume that Γ is finitely generated (which is the case in allexamples of interest to us). Then Ahlfohr’s finiteness theorem states that Ω / Γ is afinite union of punctured Riemann surfaces. Ω itself can have either 1, 2 or infinitelymany connected components.Hyperbolic 3-manifolds are rigid, which means that for each boundary geometryand topological type, there is at most one hyperbolic structure. As a consequence, thesum over geometries in 3d gravity is indeed a sum and thus not contain continuouspieces.
A.3 (Co)homology of 3-hyperbolic manifolds
In the main text, we need some basic aspects of the topology of hyperbolic manifolds.In this discussion, we assume the manifold M = H / Γ to be smooth and do not allowsingularities. We determine the integer (co)homology groups in terms of the Kleiniangroup Γ and basic geometric invariants of the boundary. We have of course the basicgroups H ( M , Z ) ∼ = H ( M , Z ) ∼ = Z (A.2)and H ( M , Z ) ∼ = H ( M , Z ) ∼ = 0, since the manifold is non-compact (we will assumein the following that it has at least one boundary). Higher (co)homology groups alsovanish. By the Hurewicz theorem, we haveH ( M , Z ) ∼ = π ( M ) ab ∼ = Z r ⊕ Γ tor . (A.3)Here, Γ tor is the torsion subgroup of the abelianization of π ( M ) ∼ = Γ (that is notnecessarily the torsion subgroup of Γ). r is the rank of the abelianization of Γ. Bythe universal coefficients theorem, we haveH ( M , Z ) ∼ = Z r , H ( M , Z ) ∼ = Z R ⊕ Γ tor , H ( M , Z ) ∼ = Z R . (A.4)We will determine the rank R below. H ( M , Z ) has no torsion, since otherwise by theuniversal coefficients theorem, the torsion would also appear in H ( M , Z ). Next, wedetermine the rank R of H ( M , R ). We have the long exact sequence of (relative) It is more convenient to use real coefficients here, since we already determined all the torsionparts. – 64 –omology groups0 −→ H ( M , ∂ M ; R ) ∼ = R −→ H ( ∂ M , R ) ∼ = R n −→ H ( M , R ) ∼ = R R −→ H ( M , ∂ M ; R ) ∼ = R r −→ H ( ∂ M , R ) ∼ = R (cid:80) i G i −→ H ( M , R ) ∼ = R r −→ H ( M , ∂ M , R ) ∼ = R R −→ H ( ∂ M , R ) ∼ = R n −→ H ( M , R ) ∼ = R −→ . (A.5)Here, n is the number of boundaries. We used Poincar´e-Lefschetz duality, which isthe generalization of Poincar´e duality to orientable manifolds with boundary,H k ( M , R ) ∼ = H − k ( M , ∂ M , R ) , H k ( M , R ) ∼ = H − k ( M , ∂ M , R ) (A.6)to express the relative homology groups in term of known quantities. Because thealternating sum of the ranks of the group has to vanish in any exact sequence, wecan calculate R from this and obtain R = n + r − (cid:88) i G i − r − − (cid:88) i ( G i − . (A.7)Thus, everything is expressible in terms of the group Γ and the Euler characteristicsof the boundary surfaces.For our discussion, we also need H ( M , U(1)). From the universal coefficientstheorem, it follows that H ( M , U(1)) ∼ = U(1) R . (A.8) A.4 Examples
Here we discuss examples of Kleinian groups. These are essentially the bulk manifoldsfor the uniformizations of the boundary surfaces that we reviewed in Appendix D.
Schottky groups.
Since the boundary of H is CP , it is natural to consider thequotient space H / Γ S G , where Γ S G is a genus G Schottky group. The resulting 3-manifold is a handlebody and can be thought of as the interior of a Riemann surfacewhen embedded in 3-space.The characteristic feature of handlebodies is that some homology cycles of Rie-mann surface become contractible in the bulk. The α I cycles of the boundary can beidentified with the circles C I bounding the fundamental domain. They can be seen tobe contractible in the bulk. Topologically, this yields a classification of handlebodies:for every Lagrangian sublattice of H (Σ G , Z ) (i.e. a G -dimensional sublattice withtrivial induced intersection form), there is a corresponding handlebody where thesecycles can be contracted. However for genus G ≥ Lorentzian wormholes. This analytic continuation can be performed in different waysleading to different physical interpretations [53].– 65 – uchsian groups.
We can also act with a Fuchsian group on H . Since the Fuch-sian group Γ F G fixes the extended real line R ∪{∞} ⊂ CP , the domain of discontinuityΩ equals the union of the upper and lower half-plane. Consequently, the boundaryΩ / Γ F G of the resulting hyperbolic 3-manifold coincides with two copies of the Rie-mann surface Σ G = H / Γ F G . Thus the resulting geometry is a Euclidean wormholeconnecting the two copies of the surface. The metric can be written down veryexplicitly [27, 88] d s H / Γ F G = d χ + cosh χ d s G , (A.9)where d s G is the hyperbolic metric on the Riemann surface. χ ∈ R parametrizes thelocation along the throat of the wormhole. The thinnest part of the throat is locatedat χ = 0. The solution is reflection symmetric χ (cid:55)→ − χ . We could further quotientby this isometry to obtain a geometry with a single boundary. It is singular, becausethe { χ = 0 } surface is fixed, but we can get a non-singular geometry by assumingthat the Riemann surface Σ G has a non-trivial involution (automorphism of order 2)without fixed points and by composing it with the reflection χ (cid:55)→ − χ . The resultinggeometry has only a single boundary, but is not a handlebody. In fact, this geometryis very simple from a topological point of view. The wormhole is homotopy equivalentto the Riemann surface Σ G and thus the geometry obtained by further dividing outthe Z action is homotopy equivalent to Σ G / Z . Depending on whether the involutionis orientation preserving or reversing, Σ G / Z is either a Riemann surface of genus G +12 (for this G has to be odd), or it is the non-orientable Klein bottle of genus G .In the two cases, the non-trivial homology groups are orientation preserving involution: H ( M , Z ) ∼ = Z G +1 , H ( M , Z ) ∼ = Z , (A.10a)orientation reversing involution: H ( M , Z ) ∼ = Z G ⊕ Z , H ( M , Z ) ∼ = 0 , (A.10b)respectively. This shows that the resulting manifold is not equivalent to a handle-body. The reader can easily verify that the formula (A.7) holds in both cases. More-over, the first case gives an example with H ( M , Z ) (cid:54) = 0, which means that anadditional chemical potential can be introduced in the theory using discrete torsion,see Section 5.5. However, in this case M is also non-orientable and it is not clearto us whether one can formulate the worldsheet theory on a non-orientable targetspace. Quasi-Fuchsian groups.
We can similarly consider the manifold H / Γ QF G for Γ QF G a quasi-Fuchsian group, see also Appendix D.4. In this case the resulting hyperbolicmanifold can be thought of as a Euclidean wormhole connecting two genus G surfaceswith possibly different moduli. Since χ (cid:55)→ − χ also reverses the orientation, the whole manifold is orientable when the involutionis orientation-reversing and vice-versa. – 66 – Some facts about Riemann surfaces
In this Appendix, we will briefly recall some facts about Riemann surfaces. ThisAppendix is mostly meant for reference and to fix conventions. We shall denote byΣ g a Riemann surface (with no restrictions on the genus). B.1 Differentials
We fix canonical generators of π (Σ g ), α , . . . , α g , β , . . . , β g , satisfying (cid:81) gI =1 [ α I , β I ] =1. We often view these generators as elements of H (Σ g , Z ), where they have canon-ical intersection products: α I ∩ α J = β I ∩ β J = 0 , α I ∩ β J = δ IJ , (B.1)for I = 1 , . . . , g . Let ω , . . . , ω g be the corresponding dual basis of H (1 , (Σ g , C ): (cid:90) α I ω J = δ IJ , (cid:90) β I ω J = Ω IJ , (B.2)where Ω IJ is the period matrix. It satisfies Ω = Ω T Im Ω > . (B.3)A change of generators α , . . . , α g , β , . . . , β g is an element of Aut( π (Σ g )) and inducesan automorphism of H (Σ g , Z ). Such an automorphism preserves the intersectionproduct and can thus be identified with an element of Sp(2 g, Z ). It acts on theperiod matrix by fractional linear transformations: Ω (cid:55)→ ( A Ω + B )( C Ω + D ) − , (cid:18) A BC D (cid:19) ∈ Sp(2 g, Z ) . (B.4) B.2 Divisors
A divisor is a formal finite sum of points on the surface: D = (cid:88) i n i z i , n i ∈ Z , z i ∈ Σ . (B.5)The group of divisors is the free abelian group on the points of the surface. For ameromorphic function f on Σ, define the principal divisor as ( f ) = (cid:80) z ord z ( f ) z ,where ord z ( f ) is the order of vanishing of f at z (negative when f has a pole at z ).Since ( f g ) = ( f ) + ( g ), principal divisors form a subgroup of all divisors and we canform equivalence classes D ∼ D + ( f ), which are the divisor classes. The degree ofa divisor is | D | = (cid:88) i n i ∈ Z , (B.6)– 67 –hich is well-defined also on divisor classes since | ( f ) | = 0. One can define divisorsalso for 1-forms etc., since the order of vanishing is always a well-defined concept forany tensorfield.A particularly important divisor is the canonical divisor K , given by the divisorof any meromophic one-form. The equivalence class is well-defined, since the ratio ω /ω of any two one-forms is a meromorphic function. The degree of the canonicaldivisor is | K | = 2 g − D on a Riemann surface determines a line bundle O ( D ), whose sectionsare the space of meromorphic functions on Σ which vanish at least as fast at z i asprescribed by the divisor D . B.3 Classification of line bundles
Here, we sketch the classification of line bundle on Riemann surfaces, since this playsa role in Section 4. Line bundles form a group with respect to the tensor product.Topologically, line bundles are just classified by their first Chern class (which is agroup homomorphism from the line bundles to the H (Σ g , Z ) ∼ = Z ), which coincideswith the degree of the corresponding divisor. We are however interested in theanalytic classification. We can restrict ourselves to line bundles with vanishing firstChern class, since any line bundle of degree d can be obtained by tensoring the flatline bundle with a fixed line bundle of degree d . Line bundles with vanishing firstChern class carry a connection with vanishing curvature and are hence flat. Theycan be characterized by a homomorphism ρ : π (Σ g ) −→ C × , (B.7)that describes the holonomy of the bundle. Since the target group is abelian, sucha homomorphism can be also understood as a homomorphism from H (Σ g , Z ) into C × . We can in turn identify H (Σ g , Z ) with Z g by employing the canonical basis α , . . . , α g , β , . . . , β g . Thus, we can set ρ ( α I ) = e πis I and ρ ( β I ) = e πit I . Mostof these parameters are actually redundant. Consider the following quasiperiodicfunction in z : exp (cid:32) − πi (cid:88) I s I (cid:90) zz ω I (cid:33) . (B.8)This function can be viewed as a holomorphic transformation between line bundleswith different multipliers around the cycles. By definition, its holonomies around the α -cycles remove the phases given by s I completely. Thus, we can set s I = 0. Thereare also some identifications on the parameters t I . We have of course t I ∼ t I + 1because of the exponential map. We can also use the same quasiperiodic functionwith s I ∈ Z (so that its holonomies around the α I -cycles are still trivial). This hasthe effect of changing t I (cid:55)→ t I − Ω IJ s J . In summary, the parameters t I form the spaceJac(Σ g ) = C g / ( Z g ⊕ Ω Z g ) . (B.9)– 68 –his is a g -dimensional complex torus – the so-called Jacobian. This is the modulispace of flat line bundles on Σ g . It carries a natural complex metric and has volumevol(Jac(Σ g )) = det Im Ω . (B.10)The same can also be seen by using the Abel-Jacobi map, that maps injectively(for g ≥
1) Σ g −→ Jac(Σ g ): u : Σ g −→ Jac(Σ g ) , (B.11) z (cid:55)−→ (cid:90) zz ω . (B.12)Note that since the integration path is unspecified, the result is only well-definedas an element of the Jacobian. z is an arbitrary reference point on Σ g . This mapextends naturally to divisors by linearity. For a flat line bundle, the associated divisorhas degree 0 and is hence independent of the choice of z . The Abel-Jacobi map nowyields a isomorphism between line bundles (or their divisors) and the Jacobian. Adivisor D is hence principal if and only if | D | = 0 and u ( D ) = 0 . (B.13) B.4 Spin structures
In this work, we consider fields with half-integer spin on the Riemann surface Σ g .They are sections of a spin bundle S . S satisfies S = K . The degree of any spinstructure is | S | = g −
1. Consequently, 2 u ( S ) = u ( K ). Since the Jacobian is areal 2 g -dimensional torus, there are 2 g ways to choose u ( S ) ∈ Jac(Σ g ) such that2 u ( S ) = u ( K ), corresponding to the 2 g spin structures on the Riemann surface.Notice that for two spin bundles S and S , S ⊗ S − is a flat line bundle on Σ g and satisfies 2 u ( S ⊗ S − ) = 0 ∈ Jac(Σ g ). Thus given a fixed spin structure, weobtain any other spin structure by tensoring with such a special flat line bundle withstructure group Z ⊂ C × .Spin structures can be further divided into even and odd spin structures ac-cording to whether they have an even or odd number of holomorphic sections (oralternatively via theta-characteristics). There are 2 g − (2 g + 1) even spin structuresand 2 g − (2 g −
1) odd spin structures.
B.5 Riemann-Roch theorem
The Riemann-Roch theorem is a statement about the number of holomorphic sectionsof a line bundle L . Denoting the space of sections by H (Σ g , L ), the theorem statesthat dim H (Σ g , L ) − dim H (Σ g , K ⊗ L − ) = | L | + 1 − g . (B.14)– 69 –or example, we frequently make use of the following well-known statements (validfor g ≥ (Σ g , K ) = g , (B.15a)dim H (Σ g , K ) = 3 g − . (B.15b)For a spin bundle S , the Riemann-Roch theorem makes no predictions. The dimen-sion of sections of S is generically 1 for an odd spin structure and 0 for an even spinstructure, but can jump on subloci of M g [89]. C Subgroups of the fundamental group and covering spaces
This Appendix is more algebraic in nature and describes how to efficiently listthe subgroups of the fundamental group of Riemann surfaces. The fundamentalgroup is generated by α , . . . , α g , β , . . . , β g with (cid:81) I [ α I , β I ] = 1, as described inAppendix B.1. C.1 Regular covering spaces
Connected covering spaces of degree N c of the genus g Riemann surface are in one-to-one correspondence to subgroups of π (Σ g ) up to conjugacy. (We are not interestedin covering spaces of punctured spaces and hence consider conjugate subgroups tobe equivalent).Regular (or normal) covering spaces are those for which the corresponding sub-group H of π (Σ g ) is normal. In this case G = π (Σ g ) / H is a finite group of order N c .The covering space Σ N c ( g − enjoys a group action of G and Σ N c ( g − / G = Σ g .Starting from degree 3, not all covering spaces are regular. C.2 Relation to homomorphisms to S N A different useful perspective to think about covering spaces is to label them bygroup homomorphisms φ : π (Σ g ) −→ S N , (C.1)whose image acts transitively on { , . . . , N } . If we drop the transitivity conditionthan we also get disconnected covering spaces. This is because a covering space can beunderstood as a fibration with fibre { , . . . , N } and structure group S N that permutesthe different sheets. A covering space that is obtained by relabelling { , . . . , N } isequivalent and thus we are again interested in group homomorphisms up to conjugacy.Such group homomorphisms are much more managable since we only have tospecify them on the generators. We only have to ensure that the relation g (cid:89) I =1 [ φ ( α I ) , φ ( β I )] = 1 (C.2)– 70 –s satisfied. The relation to subgroups of π (Σ g ) is obtained by settingH φ = Stab( { } ) , (C.3)which by the orbit-stabilizer theorem is an index N subgroup (which is not necessarilynormal). C.3 Number of subgroups
Let us next discuss the number of subgroups of a fixed degree. This question isanswered by Hurwitz theory. The number of disconnected covering surfaces of degree N of a genus N surfaces is given by Burnside’s formula: H disc ( N, g ) = (cid:88) R (cid:18) dim RN ! (cid:19) − g , (C.4)where the sum extends over all irreducible representations of the symmetric group S N . Here, H stands for Hurwitz. In this formula, covering surfaces with non-trivialautomorphism group are weighted by the inverse order of their automorphism group.To find the number of connected coverings one can form the Hurwitz potential: H disc g ( q ) = 1 + ∞ (cid:88) d =1 H disc ( N, g ) q d . (C.5)Then the connected covering surfaces can be obtained by taking a logarithm, whichpasses to the connected Hurwitz potential: H conn g ( q ) = log H disc g ( q ) = ∞ (cid:88) N c =1 H conn ( N c , g ) q N c . (C.6)The actual number of connected covering spaces is then given by N c H conn (c , g ). Thefactor of the automorphism group also appears in the symmetric orbifold partitionfunction (2.4). For low values of N c , these values are listed in Table 1. The asymp-totics d → ∞ is easy to describe (for genus g ≥ N disc ( d, g ) ∼ N conn ( d, g ) ∼ d !) g − . (C.7)Passing to the connected Hurwitz numbers does not change the asymptotics. This is just the genus g partition function of two-dimensional gauge theory with gauge group S N . This gauge theory counts covering spaces, because every covering space can be viewed as afibre bundle where the fiber consists of N points, as discussed in C.2. Since gauge theory countsthe number of S N -bundles, it counts the number of covering spaces. – 71 –ndex number of subgroups up to conjugacy2 4 · x −
13 6 · x + 3 · x − · x + 14 8 · x + 4 · x + 8 · x − · x − · x − · x + 8 · x − Table 1 . The number of subgroups of the fundamental group of a Riemann surface of agiven index. Here, x = 2 g −
2. The structure (cid:80) a | d ! n a a x for n a integers persists also athigher index orders. C.4 Enumerating subgroups
Actually enumerating subgroups is difficult. We do so for illustration for low indices.
Index 2.
For index 2, the situation is simple, because every subgroup is normal.Consequently, every index 2 subgroup is obtained as the kernel of a surjective ho-momorphism φ : π (Σ g ) −→ Z . Such a homomorphism is entirely determined byspecifying it on its generators. For every generator, there are 2 choices and in totalthere are hence 2 g − Index 3.
For index 3 subgroups, the situation is much more difficult, because mostsubgroups are actually not normal. We proceed by constructing all homomorphisms φ : π (Σ g ) −→ S . Let us first disregards the transitivity condition. If the imageof the homomorphism lies actually in Z , then the covering space is regular. Let usnow look at a general homomorphism into S and discuss the constraint (C.2). Forany choice of φ on the generators, the product in (C.2) is an even permutation. Ifthe image of the homomorphism lies not in Z , then there is at least one generatorthat is mapped to a transposition. By changing the transposition, one changes theresulting permutation in the product of (C.2). Thus one sees that when the imageof the homomorphism lies not in Z , exactly of the choices satisfy the constraint(C.2). If the image lies in Z , the constraint is trivially satisfied. Hence the numberof homomorphisms φ : π (Σ g ) −→ S is given by13 (6 g − g ) + 3 g . (C.8)Relabeling would divide this number further by 6. The second term corresponds tothe regular covering spaces. This way of counting quickly becomes complicated. Index 4.
Let us only mention here that there are different regular covering spacesfor degree 4 coverings, corresponding to the cases where the image of φ lies in Z × Z or in Z . – 72 – Uniformization
In this appendix, we survey the existing (simultaneous) uniformizations of Riemannsurfaces. In the main text, we use these sometimes for the worldsheet and sometimesfor the boundary surfaces. In the main text, we use g as the worldsheet genus and G as the boundary genus. To keep notation in this appendix uniform, we use g inthe following. D.1 Teichm¨uller space and the mapping class group
We start by recalling some facts about the structure of the moduli space of Rie-mann surfaces. The dimension of the moduli space of Riemann surfaces M g isdim( M g ) = 3 g − g ≥ g = 1). Deformations of the complexstructure are parametrized by Beltrami-differentials µ z ¯ z , which form the tangentspace T Σ g M g . The cotangent space is in turn naturally identified with the space ofquadratic differentials (i.e. holomorphic sections of K ). The structure of the modulispace M g itself is quite complicated:1. M g is not compact, but can be compactified in a canonical way by includingnodal Riemann surfaces (the Deligne-Mumford compactification M g ). In thiswork, the integrands of string path integrals are supported only in the interiorof the moduli space and all our statements are independent of the compactifi-cation.2. M g has the structure of an orbifold that keeps track of the automorphismgroups of the Riemann surfaces. The moduli spaces M and M are unstablebecause they have a continuous automorphism group. All Riemann surfaceswith g ≥ g − M g .3. M g naturally carries a K¨ahler metric, the so-called Weil-Petersson metric. Itcan be described as follows. For two tangent-vectors (Beltrami differentials) atΣ g ∈ M g , define an inner product as follows: (cid:104) µ z ¯ z | ν z ¯ z (cid:105) = (cid:90) Σ g d z √ γ µ z ¯ z ν z ¯ z . (D.1)This inner product depends on the choice of the metric on the surface γ ab . TheWeil-Petersson metric is defined by choosing the unique metric with constantnegative curvature on Σ g (see uniformization theorems below).4. M g is not simply connected. Its universal covering space is Teichm¨uller space T g . On T g , the mapping class group MCG(Σ g ) acts and T g / MCG(Σ g ) ∼ = M g .– 73 –he mapping class group consists of all the orientation-preserving homeomor-phisms of Σ g modulo those that are continuously connected to the identity andis isomorphic to the outer automorphism group of the fundamental group of agenus g surface (by the Dehn-Nielsen-Baer theorem):MCG(Σ g ) ∼ = Out( π (Σ g )) . (D.2)The mapping class group for genera g ≥ π (Σ g ) induces an outer automorphism on its abelianizationH (Σ g , Z ). Generators of the mapping class group preserve the intersectionproduct on H (Σ g , Z ) and one obtains a surjectionMCG(Σ g ) −→ Sp(2 g, Z ) . (D.3)The kernel of this morphism is non-trivial for genera g ≥ g ). It is sometimes useful to consider the Torelli space U g = T g / Tor(Σ g ), that can be described as the space of Riemann surfacestogether with a choice of canonical homology cycles.For g = 1, most of these assertions become trivial. We have T = H , the upperhalf-plane. The Weil-Petersson metric on T coincides with the Poincar´e upper halfplane metric. The mapping class group for genera g = 1 is the well-known modulargroup SL(2 , Z ) ∼ = Sp(2 , Z ) and since the fundamental group is abelian, the Torellisubgroup is trivial. Consequently, T and U coincide.For genus 2 and 3, the situation is similar. While the Torelli subgroup is non-trivial, the Torelli space U g is still simple to describe. The period mapping Σ (cid:55)→ Ω Σ embeds the Torelli space in the Siegel upper half plane H g = { Ω ∈ C g × g | Ω T = Ω , Im Ω > } . (D.4)For g = 2 and 3, this is actually an isomorphism and thus U g ∼ = H g . The period mapceases to be surjective for g = 4 and the image becomes much harder to describe.The moduli space of Riemann surfaces is then given by M g ∼ = H g / Sp(2 g, Z ) in thesegenera. D.2 Fuchsian uniformization
For genera g ≥
2, the universal covering space Σ g is the upper half plane H . Hencewe can write Σ g = H / Γ F g for some discrete subgroup Γ F g ⊂ PSL(2 , R ) that actsproperly discontinuously on the upper half-plane. Such a group is called a Fuchsiangroup. Since Γ F g ∼ = π (Σ g ), Γ F g is represented by 2 g matrices satisfying (cid:40) A , . . . , A g , B , . . . , B g ∈ PSL(2 , R ) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:89) I =1 [ A I , B I ] = (cid:41) . (D.5)– 74 –uch a collection of matrices depends on 6 g − real parameters. Since collectionsof matrices that are related by an overall conjugation lead to the same Riemannsurface, a Fuchsian group depends on 6 g − g − g -gon that is obtain by cutting the Riemannsurface Σ g along the cycles α I and β I . The Fuchsian uniformization does not makethe complex structure of moduli space manifest. However, it naturally realizes theunique metric with constant negative curvature on Σ g . Since the upper half-planemetric on H is preserved by the action of PSL(2 , R ), it descends to a well-definedmetric on Σ g with constant negative curvature. Group cohomology.
For our application, the cohomology of Fuchsian groupsplays some role, since we are computing an orbifold with Fuchsian group. Σ g isan Eilenberg MacLane space K (Γ F g , Since the group cohomology can be com-puted in terms of the (singular) cohomology of the corresponding Eilenberg MacLanespace, we have H n (Γ F g , M ) = H n (Σ g , M ) = M , n = 0 , ,M g , n = 1 , , n > . (D.6)for any abelian group M . In particular, H (Γ F g , U(1)) = U(1) and thus, there shouldbe a phase that we can freely choose when orbifolding by a Fuchsian group. Thecorresponding cocycle can be written down very explicitly. Let [ g ] A I and [ g ] B I denotethe number of A I ’s (or B I ’s) in g when g is written as a word in the generators. Sincethe constraint in (D.5) satisfies [ · · · ] A I = [ · · · ] B I = 0, this remains also well-definedin the Fuchsian group. The homomorphisms [ · ] A I and [ · ] B I can be taken as thegenerators of H (Γ F g , Z ). The generator of H (Γ F g , Z ) can be chosen to take the form ϕ ( g, h ) = g (cid:88) I =1 ([ g ] A I [ h ] B I − [ g ] B I [ h ] A I ) . (D.7)An element for the U(1) cohomology is then given by exp (2 πiθϕ ( g, h )) for θ ∈ [0 , Spin structure.
We can describe spin structures very naturally using Fuchsianuniformization. The idea is to identify spin structures with lifts (cid:101) Γ F g of Γ F g to SL(2 , R ).There are 2 g such lifts, since we may choose the sign for every generator freely. Anysuch lift is compatible with the relation (cid:81) I [ A I , B I ] = 0. Group elements γ acts by This means that the only non-trivial homotopy group π n occurs for n = 1 and π (Σ g ) = Γ F g byconstruction. The reason for this is that π n for n ≥ H is contractible, so π n (Σ g )for n ≥ – 75 –¨obius transformations on the upper half plane. Using the lift to SL(2 , R ) we candefine (cid:112) ∂γ ( z ) consistently for every group element. If γ = (cid:18) a bc d (cid:19) , (D.8)then (cid:112) ∂γ ( z ) ≡ ( cz + d ) − . We can then define spinors to be automorphic forms ψ ( z ) satisfying (cid:112) ∂γ ( z ) ψ ( γ ( z )) = ψ ( z ) (D.9)on the upper half plane. This hence defines a spin bundle S . D.3 Schottky uniformization
Another type of uniformization is Schottky uniformization. It applies to all Riemannsurfaces. Here, we let a subgroup of Γ S g ⊂ PSL(2 , C ) act properly discontinuously onan open subset Ω ⊂ CP . This is a natural uniformization for the context of AdS ,since the boundary of global Euclidean AdS is CP . A Schottky group Γ S g is moreprecisely characterized by the following properties:1. Γ S g is isomorphic to a free group in g generators, whose elements are all lox-odromic PSL(2 , C ) transformations (meaning that the corresponding M¨obiustransformation has two fixed points).2. A fundamental domain of the Schottky group can be described as follows. Let C − g , . . . , C − , C , . . . , C g be 2 g circles in the complex plane bounding the discs D − g , . . . , D − , D , . . . , D g such that the discs are all disjoint. Then the gener-ators for the Schottky group are taken to be B , . . . , B g , where B I ( C − I ) = C I .Moreover, B I maps the interior of C − I to the exterior of C I . The fundamentaldomain for the Schottky group may be taken to be F = CP \ ( D − g ∪ · · · ∪ D − ∪ D ∪ · · · ∪ D g ) . (D.10)Γ S g does not act properly discontinuously on all of CP . To have a properly discon-tinuous action, one has to excise a limit set Λ ⊂ CP . Λ consists of infinitely manypoints for g ≥ CP \ Λ. The Riemann surface is then obtained as Ω / Γ S g .Every Riemann surface admits a Schottky uniformization (this is the Koebe ret-rosection theorem). The Schottky group naturally depends on 3 g complex parame-ters corresponding to the matrices B , . . . , B g . Overall conjugation of the collection B , . . . , B g again leads to the same Riemann surface and thus the parametrizationreally depends on 3 g − M g . While Fuchsian uniformization gives a natural description of the Teichm¨uller– 76 –pace T g , Schottky uniformization gives a description of an intermediate cover, theso called Schottky space S g . For g = 1, we have S = H / ( τ ∼ τ + 1) . (D.11)Schottky uniformization makes the complex structure of the moduli space manifest.However, since the action of PSL(2 , C ) on CP does not preserve a metric, it doesnot lead to a metric on the Riemann surface Σ g .Γ S g is isomorphic to a free group and as such the group cohomology of Γ S g is well-known. It follows in particular that its group cohomology H n (Γ S g , M ) is trivial for n ≥
2. Here, M is any abelian group.H (Γ S g , M ) is isomorphic to the abelianization(Γ S g ) ab ⊗ Z M ∼ = M g . Spin structure.
Using Schottky uniformization, we can describe 2 g out of the 2 g spin structures on the Riemann surface naturally. They again correspond to lifts ofthe Schottky group Γ S g ⊂ PSL(2 , C ) to SL(2 , C ). The remaining spin structures areharder to define, since its sections involve branch cuts running between the circles C I and C − I . We will not have need of these additional spin structures. D.4 Simultaneous uniformization
Finally, we discuss simultaneous uniformization of two Riemann surfaces by quasi-Fuchsian groups. This is relevant for the context of AdS for the Euclidean wormhole.The simplest case occurs when a Fuchsian group acts on CP . In this case, the limitset Λ is R ∪ {∞} . Thus, Ω = CP \ Λ = H ∪ H , where H is the lower half-plane.Consequently, Ω / Γ F g = Σ (1) g ∪ Σ (2) g is the union of two Riemann surfaces with identicalmoduli. The statement of simultaneous uniformization generalizes this statement. Aquasi-Fuchsian group Γ QF g is a discrete subgroup of PSL(2 , C ), whose limit set equalsa Jordan-curve in CP (i.e. a non-intersecting loop on CP ). The case of a Fuchsiangroup is a special case, since it preserves the Jordan curve R ∪ {∞} . In this case,Ω has always two components and so Ω / Γ QF g ∼ = Σ g ∪ Σ (cid:48) g consists of two Riemannsurfaces of the same genus, but not necessarily the same moduli. The simultaneousuniformization theorem [90] states that any two Riemann surfaces can always be uni-formized in this way and hence the space of quasi-Fuchsian groups can be identifiedwith two copies of Teichm¨uller space T g . This depends slightly on the orientation we choose. If we want the two Riemann surfaces to havethe induced orientation of the wormhole geometry, i.e. that ∂ ( H / Γ F g ) = Σ (1) g + Σ (2) g in homology,then the moduli are actually complex conjugate to each other. Sometimes this is called a quasi-Fuchsian group of the first kind. – 77 – (Branched) complex projective structures
In this Appendix, we review some facts about complex projective structures on Rie-mann surfaces. There are several equivalent definition of this. We will follow theexposition of [48, 91].
E.1 Complex projective structuresDefinition.
Let Σ g be a Riemann surface of genus g ≥
2. We choose a Fuchsianuniformization of Σ g , Σ g = H / Γ F g , where H is the upper half plane and Γ F g is aFuchsian group, see Appendix D.2. A complex projective structure is a holomorphicfunction γ : H −→ CP such that ∂γ ( z ) (cid:54) = 0 for all z ∈ H . We also requirethat ∂ ( γ ( z ) − ) (cid:54) = 0. Equivalently, γ ( z ) is locally injective. Moreover, γ ( z ) has thefollowing automorphic property: γ ( g ( z )) = ρ ( g )( γ ( z )) (E.1)for some homomorphism ρ : Γ F g −→ Γ ⊂ PSL(2 , C ) and all g ∈ Γ F g . The map γ is alsooften called the developing map. We consider two complex projective structures to beequivalent if they differ only by an overall composition with a M¨obius transformation, (cid:101) γ = M ◦ γ , ˜ ρ ( g ) = M ◦ ρ ( g ) ◦ M − (E.2)for some matrix M ∈ PSL(2 , C ). Equivalence to a projective atlas.
Often, a complex projective structure isdefined differently as follows. We first choose a coordinate covering { U α , z α } of theRiemann surface Σ g . z α are the coordinate maps z α : U α −→ V α ⊂ CP ( z α as usualare biholomorphisms).
50 51
On intersections, we have transition maps f αβ = z α ◦ z − β : z β ( U α ∩ U β ) −→ z α ( U α ∩ U β ) . (E.3)A complex projective structure is such an atlas for which all the transition functionsare projective maps.The transition maps satisfy the obvious consistency condition f αβ ◦ f βγ = f αγ , (E.4)whereever these maps are defined. Thus, f αβ define the coordinate bundle over Σ.The group of the bundle is PSL(2 , C ) and the fibre is CP . Since the transition mapsare constant (when considered as a mapping from U α ∩ U β into PSL(2 , C )) and thus We use here CP instead of the usual C because it allows us to treat cases more uniformly. Alternatively, we can define a complex projective structure on a real surface, since a complexprojective structure in particular induces a a complex structure. We take the point of view thatthe complex projective structure is subordinate to the complex structure. – 78 –he coordinate bundle is flat. Specifying a flat coordinate bundle is equivalent tospecifying the complex projective structure.This definition of complex projective structure is equivalent to the previous one.To see this, we cover a fundamental domain of the Fuchsian realization by opensubsets U α . We then simply identify z α = Γ | U α . (E.5)thus defining the coordinate maps. One can easily check that this identifies the twostructures. Examples from uniformization.
Uniformizing the Riemann surface Σ g eithervia Fuchsian, Schottky or some other uniformization leads to a complex projectivestructure (or in the case of Fuchsian uniformization even to a real projective struc-ture). As we shall see, the space of complex projective structures is however muchbigger. The Schwarzian derivative.
It is useful to look at the Schwarzian derivative ofthe developing map, S ( γ )( z ) = ∂ γ ( z ) ∂γ ( z ) − ∂ γ ( z )) ∂γ ( z )) . (E.6)Here, we view γ as map from the upper half-plane and so ∂ is the usual derivative,not a covariant derivative. Let us recall some crucial properties of the Schwarzianderivative:1. Invariance under postcomposition with M¨obius transformations: S ( g ( γ ( z ))) = S ( γ ( z )) (E.7)for g a M¨obius transformation.2. Covariance under precomposition with M¨obius transformations: S ( γ ( g ( z ))) = S ( γ ( z ))( ∂g ( z )) (E.8)for g a M¨obius transformation.3. Relation to a second order differential equation. Let f and f be two linearlyindependent solutions to the differential equation (viewed on H ) ∂ f ( z ) + 12 φ ( z ) f ( z ) = 0 . (E.9)Then the ratio γ ( z ) = f ( z ) /f ( z ) satisfies S ( γ ( z )) = φ ( z ). In fact, we maytake f ( z ) = γ ( z ) (cid:112) ∂γ ( z ) , f ( z ) = 1 (cid:112) ∂γ ( z ) (E.10)– 79 –or a choice of square root. The pair f ( z ) = ( f ( z ) , f ( z )) is a section of arank 2 vector bundle and satisfies( ∂g ( z )) − f ( g ( z )) = ρ ( g )( f ( z )) , (E.11)where ρ is a lift to SL(2 , C ) of the original homomorphism.Coming back to the Schwarzian derivative of the developing map, we see that φ ( z ) = S ( γ )( z ) is a well-defined quadratic differential on the Riemann surface Σ, thanks tothe first two properties. φ ( z ) has no poles, because ∂γ ( z ) (cid:54) = 0 for all z ∈ H .There is thus a map { complex projective structures } −→ { holomorphic quadratic differentials } (E.12)given by taking the Schwarzian of the developing map.Conversely, given a holomorphic quadratic differential φ ( z ) on Σ g , we can find acorresponding complex projective structure as follows. Essentially, one has to solvethe differential equation S ( γ )( z ) = φ ( z ) on H , which leads to the developing map γ ( z ) (unique up to composition with M¨obius transformation). The properties aboveimply that γ ( z ) has the automorphic property for some homomorphism ρ : Γ F g (cid:55)−→ PSL(2 , C ) and ∂γ ( z ) (cid:54) = 0, ∂ ( γ ( z ) − ) (cid:54) = 0. One has to work harder to show thatthe differential equation indeed always admits a solution. We refer to [47] for this.This shows that the relation between complex projective structures and quadraticdifferentials on Σ g is 1–to–1.Property 3 shows that the homomorphism ρ for a complex projective structurealways lifts to a homomorphism ρ : Γ F g −→ SL(2 , C ) . (E.13) Parameter counting.
It is useful to count the number of complex parametersthat enter these definitions. It is well-known that the dimension of quadratic dif-ferentials is 3 g −
3, see (B.15b). Thus, also the dimension of complex projectivestructures is (3 g − ρ involves 6 g − g complex matrices for the generators α , . . . , α g , β , . . . , β g of π (Σ), but they have to satisfy the relation g (cid:89) I =1 [ ρ ( α I ) , ρ ( β I )] = . (E.14)Moreover, two homomorphisms differing by an overall conjugation are consideredequivalent which accounts for another 3 parameters. This is well-defined because ∂γ ( z ) does not have zeros. Poles of ∂γ ( z ) are double poles, sincealso ∂ ( γ ( z ) − ) (cid:54) = 0. – 80 –hus, for most homomorphisms ρ , there will not be a developing map and sothey do not define a complex projective structure. From the parameter counting, wesee however that if we allow both the complex structure and the complex projectivestructure on Σ g to vary, then we get 6 g − Further properties.
Here we list further useful properties of complex projectivestructures.1. If two developing maps γ and γ lead to the same homomorphism ρ , then γ = γ . See [48, Theorem 3].2. The following conditions for the developing map γ are equivalent [48, Theorem7]:(a) γ ( H ) (cid:54) = CP .(b) γ : H −→ γ ( H ) is a covering map.(c) ρ (Γ F g ) acts properly discontinuously on the image γ ( H ).If these conditions are satisfied then, γ descends to a well-defined covering map˜ γ : Σ g −→ γ ( H ) /ρ (Γ F g ) . (E.15)These are precisely the projective structures in which we are interested in thispaper.3. A group of M¨obius transformations is elementary if its action on H has eitherone fixed point in H or one fixed in ∂ H ∼ = CP . In the first case, it isconjugate to a group of unitary transformations, whereas in the second case, itis conjugate to a group of affine transformations.Any homomorphism ρ : π (Σ) −→ PSL(2 , C ) that is liftable to SL(2 , C ) andwhose image is not an elementary group is realized by a complex structure onΣ g subordinate to some complex structure [91]. E.2 Branched complex projective structures
While complex projective structures are the relevant structure on the worldsheetfor computing partition functions, we have to turn to branched complex projectivestructures for correlation functions. – 81 – efinition.
For branched complex structures, we do not require the developingmap to be locally injective. Thus, a branched complex structure is a map γ : H −→ CP satisfying γ ( g ( z )) = ρ ( g )( γ ( z )) (E.16)for some homomorphism ρ : Γ F g −→ Γ ⊂ PSL(2 , C ) and all g ∈ Γ F g . γ is branchedover finitely many points z , . . . , z n on the Riemann surface with ramification indices w , . . . , w n , meaning that ∂ n γ ( z i ) = 0 , < n < w i (E.17)and ∂ w i γ ( z i ) (cid:54) = 0. Note that the ramification index is preserved under actions ofM¨obius transformations, so that this is well-defined.The definition in terms of a complex atlas is much less useful in this case andhence we will work only with the developing map. Relation to meromorphic quadratic differentials.
We can still define theSchwarzian in this case. The resulting quadratic differential has now quadratic dif-ferentials at z i . A straightforward computation gives S ( γ )( z ) ∼ − w i z − z i ) + O (cid:0) ( z − z i ) − (cid:1) . (E.18)Hence the Schwarzian gives us a map { branched complex projective structures with branch locus ( z , w ) , . . . , ( z n , w n ) }−→ (cid:26) meromorphic quadratic differentials with QRes z = z i φ ( z ) = 1 − w i (cid:27) , (E.19)where QRes is the quadratic residue. Contrary to the unbranched case, this map isnot 1–to–1. We can easily understand this via an example. Consider the case g = 0with four punctures, each having w i = 2. Let us take z = 0, z = 1, z = 2 and z = 3. In this case, a generic quadratic differential with the required poles takes theform φ ( z ) = − z + 72 z − z ( z − ( z − ( z − + qz ( z − z − z − . (E.20)However, we do not expect a familiy of maps γ ( z ) with simple ramifications at z i .Such a map is necessarily a branched cover of CP and as such has degree 3 by theRiemann Hurwitz formula. Hence we can write it as γ ( z ) = P ( z ) /Q ( z ) for twothird order polynomials P ( z ) and Q ( z ). Counting parameters, such a map dependson 7 parameters. Overall M¨obius transformations make 3 parameters redundant, theother 4 are fixed by requiring the ramifications at z i . Hence we do not get a family of– 82 –olutions for the Schwarzian derivative. Instead we have only two discrete solutions q = − ± √ n additional constraints on the quadratic dif-ferential in order for a solution γ to exist. As one can see from this example, thisconstraint is quite complicated. The general statement is that there exist n polyno-mials on the affine vectorspace of quadratic differentials P , . . . , P n with the requiredpoles such that the joint zero locus can be identified with space of integrable quadraticdifferentials, i.e. those that give rise to a developing map γ [47, 92].Thus, the correct statement is in this case that branched complex projectivestructures with given branch locus are in 1–to–1 correspondence with integrablemeromorphic quadratic differentials with the correct poles. For genus g ≥
2, thespace of branched complex projective structures with given ramification locus hashence again generic dimension 3 g − F Topologically twisted partition function
In this Appendix, we explain the form of topologically twisted partition functions.Our main example is the sigma-model on T that is relevant for the present paper. F.1 Topological twist
The N = 2 superconformal algebra possesses chiral supercurrents G + and G − ofweight . It also possesses an R-symmetry current J = i∂H with defining OPE J ( z ) J ( w ) ∼ c z − w ) , (F.1)where c is the central charge of the theory.We now topologically twist the theory, which corresponds to a redefinition of theVirasoro tensor ˆ T = T − ∂J . (F.2)With respect to this new Virasoro tensor, the supercharges have weight 1 ( G + ) and 2( G − ). Moreover, the new central charge vanishes. This ensures that the correlationfunctions in the twisted theory are Weyl-anomaly free. The topological twist leadsto an anomalous U(1)-current. In a correlator, the total charge with respect tothe current J has to be c (1 − g ) in order to get a non-vanishing result. Thus, thetopologically twisted partition function that we would like to compute is (cid:42) g − (cid:89) j =1 e − iH ( z j ) (cid:43) . (F.3)We suppress here as usual the right-movers. This is still not quite what we want,since this partition function vanishes identically. One way to see this is that the path– 83 –ntegral still has a zero mode: the insertion of e − iH ( z )in the correlation function isa Grassmannian 1-form that should have an a zero at z = z i . Such a one-form doesexist and its presence leads to a vanishing result.This is exactly the same issue that is present in the naive definition of thecorrelators of the N = 4 topological string [44, 45]. It is remedied in (3.7) by theinclusion of the current J . In our context, this means that we consider the followingtopologically twisted partition function: (cid:42) g − (cid:89) j =1 e − iH ( z j ) ∂H ( z ) (cid:43) . (F.4) F.2 Change of variables
We want to relate this partition function to the untwisted partition function. To doso, we follow the strategy of [93, 94]. Let us consider a complex fermion ψ and ¯ ψ (that are both spinors) and the topologically twisted versions Ψ (that is a function)and ¯Ψ (that is a 1-form). We relate the two as follows:¯Ψ = Ω ¯ ψ , Ψ = Ω − ψ . (F.5)Here Ω is a holomorphic spinor. For Ω to exist, we need it to be an element of an oddspin bundle S . Ω has g − z ∗ ,. . . , z ∗ g − and for simplicity,we are computing the correlator (cid:104) ¯Ψ( z ∗ ) · · · ¯Ψ( z ∗ g − )(Ψ ¯Ψ)( z ) (cid:105) , (F.6)which satisfies the anomalous charge conservation. When computing this correlatorvia a path integral integral approach, we simply have to analyze how the measure andthe action changes under this change of variables. In the (Ψ , ¯Ψ) variables, Ψ( z ) needsto have a first order pole at z ∗ i and ¯Ψ( z ) has a first order zero at z ∗ i because of theoperator insertions. Since also Ω( z ) has a zero at these points, this means that ( ψ, ¯ ψ )is regular at these points. Since (Ψ ¯Ψ) = ( ψ ¯ ψ ), nothing changes at the additionalpoint. The action in the (Ψ , ¯Ψ) variables translates correctly to the action in the( ψ, ¯ ψ ) variables. Finally, we have to analyze the relation between the measures. Thenorms are related according to (cid:107) δ ¯Ψ (cid:107) = (cid:107) δ ¯ ψ (cid:107) | Ω | , (cid:107) δ Ψ (cid:107) = (cid:107) δψ (cid:107) | Ω | − . (F.7)In order for this measure to coincide with the standard measure on differentials, weneed to specify the metric according to g z ¯ z = | Ω | . (F.8)Thus, we see that with this choice of metric (cid:104) ¯Ψ( z ∗ ) · · · ¯Ψ( z ∗ g − )(Ψ ¯Ψ)( z ) (cid:105) twisted = (cid:104) ( ψ ¯ ψ )( z ) (cid:105) untwisted (F.9)The right-hand side is evaluated with the fixed odd spin structure S . We see againthat the additional insertion of ( ψ ¯ ψ )( z ) is necessary to get a non-vanishing result.– 84 – eferences [1] L. Eberhardt, Partition Functions of the Tensionless String , .[2] J. M. Maldacena, The Large N limit of superconformal field theories andsupergravity , Int. J. Theor. Phys. (1999) 1113 [ hep-th/9711200 ].[3] E. Witten, Anti-de Sitter Space and Holography , Adv. Theor. Math. Phys. (1998)253 [ hep-th/9802150 ].[4] R. Jackiw, Lower Dimensional Gravity , Nucl. Phys. B (1985) 343.[5] C. Teitelboim,
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