Surface Operators and Separation of Variables
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Surface Operators and Separation of Variables
Edward Frenkel , Sergei Gukov , and J¨org Teschner University of California, Berkeley, CA 94720-3840 USA California Institute of Technology, Pasadena, CA 91125, USA Simons Center for Geometry and Physics, Stony Brook, NY 11794, USA DESY Theory, Notkestr. 85, 22603 Hamburg, Germany
Abstract:
Alday, Gaiotto, and Tachikawa conjectured relations between certain 4d N = 2 supersymmetric field theories and 2d Liouville conformal field theory. We studygeneralizations of these relations to 4d theories with surface operators. For one type ofsurface operators the corresponding 2d theory is the WZW model, and for another type –the Liouville theory with insertions of extra degenerate fields. We show that these two 4dtheories with surface operators exhibit an IR duality, which reflects the known relation (theso-called separation of variables) between the conformal blocks of the WZW model and theLiouville theory. Furthermore, we trace this IR duality to a brane creation constructionrelating systems of M5 and M2 branes in M-theory. Finally, we show that this duality maybe expressed as an explicit relation between the generating functions for the changes ofvariables between natural sets of Darboux coordinates on the Hitchin moduli space. CALT 2015-032 a r X i v : . [ h e p - t h ] J un ontents
1. Introduction 2
2. Preliminaries 7 S and AGT correspondence 72.2 Seiberg-Witten theory 82.3 Relation to the Hitchin system 92.4 Two types of surface operators 9
3. Surface operators corresponding to the codimension-2 defects 10
4. Surface operators corresponding to codimension-4 defects 18
A. Surface operators and Nahm poles 32B. Twisting of Kac-Moody conformal blocks 34
B.1 Twisted conformal blocks 34B.2 Genus zero case 35B.3 Higher genus cases 35
C. Holomorphic pictures for the Hitchin moduli spaces 36
C.1 Three models for Hitchin moduli space 37C.2 Complex-structure dependent Darboux coordinates 38C.3 Complex-structure independent Darboux coordinates 40C.4 Limit (cid:15) →
0: Recovering the Higgs pairs 41– 1 – . Classical limits of conformal field theory 42
D.1 Preparations: Insertions of degenerate fields 43D.2 Limit (cid:15) → (cid:15) → E. Explicit relation between Kac-Moody and Virasoro conformal blocks 48
E.1 SOV transformation for conformal blocks 48E.2 Reformulation as integral transformation 49E.3 Semiclassical limit 50E.4 SOV transformation in the presence of degenerate fields 51
1. Introduction
One of the most interesting phenomena in supersymmetric gauge dynamics is the appearanceof infrared (IR) duality: theories different in the ultraviolet (UV) regime may well flow to thesame IR fixed point. A prominent example is the Seiberg duality in four-dimensional N = 1super-QCD [1]. Similar dualities exist in three dimensions [2, 3] and in two dimensions [4].Moreover, it is known that certain two-dimensional dualities naturally arise on the two-dimensional world-sheets of surface operators in four-dimensional N = 2 gauge theories [5, 6].In the present paper, we propose a new IR duality between 4d N = 2 supersymmetric theorieswith two types of surface operators that we call “codimension-2” and “codimension-4” forreasons that will become clear momentarily.In general, in four dimensional gauge theory (with any amount of supersymmetry) wehave two ways of constructing non-local operators supported on a surface D ⊂ M [7]: • One can couple 4d gauge theory on M to an auxiliary 2d theory on D in such a way that the gauge group G of the 4d theory is a subgroup of the global flavorsymmetry of the 2d theory. In particular, the auxiliary 2d theory must have globalsymmetry G . • singularity: One replaces the four-dimensional space-time M with the complement M \ D so that gauge fields (and, possibly, other fields) have a prescribed singularbehavior along D . Thus, instead of introducing new degrees of freedom, one modifiesthe existing degrees of freedom.Note that both of these methods may also be used to construct other non-local operators,such as line operators (for example, Wilson operators and ’t Hooft operators, respectively).In the case of surface operators, the first of these two methods can be further subdivided into– 2 –inear and non-linear sigma-model descriptions of 2d degrees of freedom on D . However, thisdistinction will not be important in this paper.What will be important to us, however, is that sometimes these two constructions maylead to the same result. This happens when integrating out 2d degrees of freedom in the 2d-4dcoupled system leaves behind a delta-function singularity, supported on D (for the 4d fields).In particular, this is what one finds in the case of N = 4 super-Yang-Mills theory. Thus, oneobtains an equivalence of the theories with two types of surface operators, which may also bederived using brane constructions and T-dualities. Something similar may happen in certaingauge theories with less supersymmetry, e.g. free field theories, but in this paper focus on IRequivalence (or IR duality) of 4d N = 2 theories with the two types of surface operators.Surface operators in 4d N = 2 theories were first considered in [8] and later incorporatedin the framework of the Alday-Gaiotto-Tachikawa (AGT) correspondence in [9, 10] relatinga certain class of 4d N = 2 gauge theories (often called “class S ”) and 2d conformal fieldtheories on a Riemann surface C g,n of genus g with n punctures [11]. According to theseworks, there is a relation between the instanton partition functions in the 4d theories in thepresence of the two types of surface operators and conformal blocks in the WZW model for SL and the Liouville theory with extra degenerate fields, respectively. We note that for thesurface operators of the first type this relation was originally proposed by Braverman [12] andfurther analyzed in [10, 13–15].Within this framework, the IR duality between the 4d theories with two types of surfaceoperators is neatly expressed by an integral transform between the chiral partition functionsof the WZW model and the Liouville theory: Z WZ ( x, z ) = (cid:90) du K ( x, u ) Z L ( u, z ) , (1.1)This relation, which is of interest in 2d CFT, was established by Feigin, Frenkel, and Stoy-anovsky in 1995 as a generalization of the Sklyanin separation of variables for the Gaudinmodel [16] (which corresponds to the limit of the infinite central charge), see [17, 18]. Hencewe call this relation separation of variables . In this paper we present it in a more explicitform (see [19] for another presentation).One of our goals is thus to show that the relation (1.1) captures the IR duality of 4d N = 2 gauge theories of class S with surface operators. Thus, our work provides a physicalinterpretation – and perhaps a natural home – for the separation of variables (1.1) in 4dgauge theory, as well as the corresponding 6d (0 ,
2) theory on the fivebrane world-volume inM-theory.Let’s talk about the latter in more detail. In the context of the AGT correspondenceand, more broadly, in 4d N = 2 theories constructed from M-theory fivebranes wrapped onRiemann surfaces [20–23] the two types of surface operators in 4d field theories describedabove are usually represented by different types of branes / supersymmetric defects in the6d (0 ,
2) theory on the fivebrane world-volume. Codimension-4 defects that correspond tothe membrane boundaries naturally lead to the surface operators described as 2d-4d coupled– 3 –ystems. Codimension-2 defects, on the other hand, may be thought of as intersections withanother group of fivebranes and therefore they are usually characterized by a singularity forthe gauge fields at D of a specific type (described in Appendix A).Thus, altogether one has at least three different perspectives on the surface operators in4d theories corresponding to the codimension-2 and codimension-4 defects in 6d theory (thisis the reason why we will often refer to them as codimension-2 and codimension-4 surfaceoperators). Namely, the 2d CFT perspective, the 4d gauge theory perspective, and the6d fivebrane / M-theory perspective. Moreover, the 4d gauge theory perspective is furthersubdivided into UV and IR regimes. A simple way to keep track of these perspectives is tothink of a sequence of RG flows,M-theory / 6d (cid:32)
4d gauge theory UV (cid:32)
4d gauge theory IR (1.2)where arrows correspond to integrating out more and more degrees of freedom. This rela-tion between different theories is somewhat analogous to a more familiar relation between a2d gauged linear sigma-model, the corresponding non-linear sigma-model, and the Landau-Ginzburg theory that describes the IR physics of the latter.It is natural to ask whether one can see any trace of our IR equivalence in the UV,either in 4d or 6d. We answer this question in the affirmative, by showing that the braneconfigurations in M-theory that give rise to the codimension-2 and codimension-4 surfaceoperators are related by a certain non-trivial phase transition, a variant of the brane creation effect of Hanany and Witten [24] (see Figure 1 in Section 4.1). We will show that certainquantities protected by supersymmetry remain invariant under this phase transition, therebyrevealing the 6d / M-theory origin of our IR equivalence. In four dimensions, the IR dualitymanifests itself in the most direct way as a relation between instanton partition functions inthe presence of surface operators and conformal blocks in WZW/Liouville CFTs discussedabove. However, what we actually claim here is that the IR duality holds for the full physicaltheories (and not just for specific observables); that is to say, the 4d theories with two typesof surface operators become equivalent in the IR. This has many useful implications (andapplications), far beyond a mere relation between the instanton partition functions.In order to show that, we use the fact that the low-energy effective action in our theoriesis essentially determined by their respective effective twisted superpotentials (see Sections4.2 and 4.3 for more details). Hence we need to compare the twisted superpotentials arisingin our theories, and we compute them explicitly using the corresponding 2d conformal fieldtheories. The result is that the two twisted superpotentials, which we denote by (cid:102) W M5 ( a, x, τ )and (cid:102) W M2 ( a, u ( a, x, τ ) , τ ), respectively, are related by a field redefinition (cid:102) W M5 ( a, x, τ ) = (cid:102) W M2 ( a, u ( a, x, τ ) , τ ) + (cid:102) W SOV ( x, u ( a, x, τ ) , τ ) . (1.3)Here the variables x and u are parameters entering the UV-definitions of the two types ofsurface operators. The relation u = u ( a, x, τ ) extremizes the superpotential on the right of(1.3), reflecting the fact that u becomes a dynamical field in our brane creation transition. As usual, it is convenient to think of parameters as background fields [25]. – 4 –ormula (1.3) has an elegant interpretation in terms of the mathematics of the Hitchinintegrable system for the group SL . Namely, we show that the two effective twisted superpo-tentials are the generating functions for changes of variables between natural sets of Darbouxcoordinates for the Hitchin moduli space M H ( C ) of SL .There are in fact three such sets: ( x, p ), the natural coordinates on M H ( C ) arising fromits realization as a cotangent bundle; ( a, t ), the action-angle coordinates making the completeintegrability of M H ( C ) manifest; and ( u, v ), the so-called “separated variables” making theeigenvalue equations of the quantized Hitchin systems separate. We show that the twistedsuperpotentials (cid:102) W M5 ( a, x, τ ) and (cid:102) W M2 ( a, u, τ ) are the generating functions for the changes ofDarboux coordinates ( x, p ) ↔ ( a, t ) and ( u, v ) ↔ ( a, t ), respectively. The generating functionof the remaining change ( x, p ) ↔ ( u, v ) is the function (cid:102) W SOV ( x, u, τ ) appearing on the RHS ofthe relation (1.3) – it is the generating function for the separation of variables in the Hitchinintegrable system. ( x, p ) coordinates (cid:75) (cid:83) (cid:102) W SOV (cid:11) (cid:19) ( a, t ) coordinates (cid:113) (cid:121) (cid:102) W M5 (cid:49) (cid:57) (cid:101) (cid:109) (cid:102) W M2 (cid:37) (cid:45) ( u, v ) coordinatesThus, the IR duality between the 4d gauge theories with the two types of surface operatorsthat we study in this paper becomes directly reflected in the separation of variables of theHitchin integrable system.To derive the relation (1.3), we first express the twisted superpotentials (cid:102) W M5 ( a, x, τ )and (cid:102) W M2 ( a, u, τ ) as the subleading terms in the expansion of the logarithms of the instantonpartition functions in the limit of vanishing Omega-deformation [26]. Assuming that theinstanton partition function in our 4d theories are equal to the chiral partition functions inthe WZW model and the Liouville theory, respectively [10, 12–15], we express the subleadingterms of the instanton partition functions as the subleading terms of the chiral partitionfunctions in the corresponding 2d CFTs. What remains to be done then is to find a relationbetween the subleading terms of these two chiral partition functions (one from the WZWmodel and one from the Liouville theory with extra degenerate fields).This is now a problem in 2d CFT, which is in fact a non-trivial mathematical problemthat is interesting on its own right. In this paper, by refining earlier observations from [27], wecompute explicitly the subleading terms of the chiral partition functions in the WZW modeland the Liouville theory (with extra degenerate fields) and identify them as the generatingfunctions for the changes of Darboux coordinates mentioned above. In this way we obtainthe desired relation (1.3). – 5 –he details of these computations are given in the Appendices, which contain a numberof previously unpublished results that could be of independent interest. In performing thesecomputations, we addressed various points in the mathematics of the WZW model and itsrelation to the Hitchin integrable system that, as far as we know, have not been discussed inthe literature before (for example, questions concerning chiral partition functions on Riemannsurfaces of higher genus). In particular, our results make precise the sense in which Liouvilletheory and the WZW model both appear as the result of natural quantizations of the Hitchinintegrable systems using two different sets of Darboux coordinates, as was previously arguedin [27].Once we identify the subleading terms of the chiral partition functions of the two 2dCFTs with the generating functions, we obtain the relation (1.3). Alternatively, this relationalso appears in the infinite central charge limit from the separation of variables relation (1.1)between conformal blocks in the WZW and Liouville CFTs. Therefore, the relation (1.1) maybe viewed as a relation between the instanton partition functions in the 4d theories with twotypes of surface operators in non-trivial Omega-background. This suggests that these two4d theories remain IR equivalent even after we turn on the Omega-deformation. However, innon-zero Omega-background this relation is rather non-trivial, as it involves not just a changeof variables, but also an integral transform. This relation deserves further study, as does thequestion of generalizing our results from the group SL to groups of higher rank.The paper is organized as follows. In Section 2 we review class S supersymmetric gaugetheories, AGT correspondence, surface operators, and the Hitchin system. In Section 3 wediscuss the 4d theories with the surface operators obtained from codimension-2 defects in 6d,the brane construction, conformal blocks in the corresponding CFT (WZW model), and therelation to the Hitchin system. In Section 4 we consider the 4d theories with the surfaceoperators obtained from codimension-2 defects in 6d and the corresponding CFT (Liouvilletheory with degenerate fields). We also discuss general properties of the 4d theories in theIR regime and the corresponding twisted superpotentials. Anticipating the IR duality thatwe establish in this paper, we start with the brane system introduced in Section 3 (the onegiving rise to the codimension-2 defects) and deform it in such a way that the end result isa collection of codimension-4 defects. This allows us to demonstrate that the two types ofdefects preserve the same subalgebra of the supersymmetry algebra and to set the stage forthe IR duality. In the second half of Section 4, we bring together the results of the previoussections to demonstrate the IR duality of two 4d gauge theories with surface operators andthe separation of variables in conformal field theory and Hitchin system.The necessary mathematical results on surface operators, on chiral partition functions inthe WZW model and the Liouville theory, and on the separation of variables are presentedin the Appendices. There one can also find detailed computations of the chiral partitionfunctions of the WZW model and the Liouville theory and their classical limits (some ofwhich have not appeared in the literature before, as far as we know).– 6 – .1 Acknowledgments We would like to thank D. Gaiotto, K. Maruyoshi, and N. Nekrasov for useful discussionsand comments. The research of E.F. was supported by the NSF grants DMS-1160328 andDMS-1201335. The work of S.G. is funded in part by the DOE Grant DE-SC0011632 andthe Walter Burke Institute for Theoretical Physics.
2. Preliminaries
In this section we review some background and introduce the notation that will be used in ourpaper. Toward this end, we will recall the notion of class S supersymmetric gauge theoriesand review very briefly how the Seiberg-Witten theory of this class is related to the Hitchinsystem. S and AGT correspondence A lot of progress has been made in the last few years in the study of N = 2 supersymmetricfield theories in four dimensions. Highlights include exact results on the expectation valuesof observables like supersymmetric Wilson and ’t Hooft loop operators on the four-sphere S ,see [28, 29] for reviews, and [30] for a general overview containing further references.A rich class of field theories with N = 2 supersymmetry, often denoted as class S , canbe obtained by twisted compactification of the six-dimensional (2 ,
0) theory with Lie algebra g [23]. Class S theories of type g = A have Lagrangian descriptions specified by a pair ofpants decompositions of C , which is defined by cutting C along a system C = { γ , . . . , γ h } ofsimple closed curves on C [22]. In order to distinguish pants decompositions that differ byDehn twists, we will also introduce a trivalent graph Γ inside C such that each pair of pantscontains exactly one vertex of Γ, and each edge e of Γ goes through exactly one cutting curve γ e ∈ C . The pair σ = ( C , Γ) will be called a refined pants decomposition.Then, to a Riemann surface C of genus g and n punctures one may associate [22, 23]a four-dimensional gauge theory G C with N = 2 supersymmetry, gauge group (SU(2)) h , h := 3 g − n and flavor symmetry (SU(2)) n . The theories in this class are UV-finite,and therefore they are characterized by a collection of gauge coupling constants g , . . . , g h .To the k -th boundary there corresponds a flavor group SU (2) k with mass parameter M k .The hypermultiplet masses are linear combinations of the parameters m k , k = 1 , . . . , n asexplained in more detail in [11, 22].The correspondence between the data associated to the surface C and the gauge theory G C is then summarized in the table below.We place this in the context of M-theory, following the standard conventions of braneconstructions [20]. Namely, we choose x and x as local coordinates on the Riemann surface C and parametrize the four-dimensional space-time M by ( x , x , x , x ). This choice of localcoordinates can be conveniently summarized by the diagram:– 7 –rane 0 1 2 3 4 5 6 7 8 9 10 M A theorieson a four-sphere can be expressed in terms of Liouville correlation functions.Riemann surface C Gauge theory G C Cut system C + trivalent Lagrangian description withgraph Γ on C, σ = ( C , Γ ) action functional S στ cutting curve γ e vector multiplet ( A e,µ , φ e , . . . ) n boundaries n hypermultipletsGluing parameters q e = e πiτ e , UV-couplings τ = ( τ , . . . , τ h ), e = 1 , . . . , h , h := 3 g − n τ e = 4 πig e + θ e π Change of pants decomposition various dualities
The low-energy effective actions of class S theories are determined as follows. Given aquadratic differential t on C one defines the Seiberg-Witten curve Σ SW in T ∗ C as follows:Σ SW = (cid:8) ( u, v ) ∈ T ∗ C ; v + t ( u ) = 0 (cid:9) . (2.1)The curve Σ SW is a two-sheeted covering of C with genus 4 g − n . One may embed theJacobian of C into the Jacobian of Σ SW by pulling back the holomorphic differentials on C un-der the projection Σ SW → C . Let H (cid:48) (Σ SW , Z ) = H (Σ SW , Z ) /H ( C, Z ), and let us introduce acanonical basis B for H (cid:48) (Σ SW , Z ), represented by a collection of curves ( α , . . . , α h ; α D , . . . , α D h )with intersection index α k ◦ α D l = δ kl , α k ◦ α l = 0, α D k ◦ α D l = 0. The corresponding periods ofthe canonical differential on v = v ( u ) du are defined as a k = (cid:90) α k v , a D k = (cid:90) α D k v . (2.2)Using the Riemann bilinear relations, it can be shown that there exists a function F ( a ), a = ( a , . . . , a h ) such that a D k = ∂ a k F ( a ). The function F ( a ) is the prepotential determiningthe low-energy effective action associated to B .Different canonical bases B for H (cid:48) (Σ SW , Z ) are related by Sp (2 h, Z )-transformations de-scribing electric-magnetic dualities in the low-energy physics. It will be useful to note that forgiven data σ specifying UV-actions there exists a preferred class of bases B σ for H (cid:48) (Σ SW , Z )which are such that the curves α e project to the curves γ e ∈ C , e = 1 , . . . , h defining the pantsdecomposition C , respectively. – 8 – .3 Relation to the Hitchin system The Seiberg-Witten analysis of the theories G C has a well-known relation to the mathematicsof the Hitchin system [31, 32] that we will recall next.The phase space M H ( C ) of the Hitchin system for G = SL (2) is the moduli space ofpairs ( E , ϕ ), where E is a holomorphic rank 2 vector bundle with fixed determinant, and ϕ ∈ H ( C, End( E ) ⊗ K C ) is called the Higgs field. The complete integrability of the Hitchinsystem is demonstrated using the so-called Hitchin map. Given a pair ( E , ϕ ), we define thespectral curve Σ as Σ = (cid:8) ( u, v ) ∈ T ∗ C ; 2 v = tr( ϕ ( u )) (cid:9) . (2.3)To each pair ( E , ϕ ) one associates a line bundle L on Σ, the bundle of eigenlines of ϕ for agiven eigenvalue v . Conversely, given a pair (Σ , L ), where Σ ⊂ T ∗ C is a double cover of C ,and L a holomorphic line bundle on Σ, one can recover ( E , ϕ ) via( E , ϕ ) := (cid:0) π ∗ ( L ) , π ∗ ( v ) (cid:1) , (2.4)where π is the covering map Σ → C , and π ∗ is the direct image.The spectral curves Σ can be identified with the curves Σ SW determining the low-energyphysics of the theories G C on R . However, in order to give physical meaning to the fullHitchin system one needs to consider an extended set-up. One possibility is to introducesurface operators. When the 6d fivebrane world-volume is of the form M × C , where C is a Riemann surface,there are two natural ways to construct half-BPS surface operators in the four-dimensionalspace-time M where the N = 2 theory G C lives. First, one can consider codimension-2defects supported on D × C , where D ⊂ M is a two-dimensional surface (= support of a surface operator). Another, seemingly different way, is to start with codimension-4 defectssupported on D × { p } , where p ∈ C is a point on the Riemann surface.In the case of genus-1 Riemann surface C = T , both types of half-BPS surface operatorsthat we study in this paper were originally constructed using branes in [7, 33]. In thesepapers it was argued that the two types of operators are equivalent, at least for certain“supersymmetric questions”. Here we will show that for more general Riemann surfaces C the two surface operators, based on codimension-4 and codimension-2 defects, may bedifferent in the UV but become essentially the same in the IR regime. They correspondto two different ways to describe the same physical object. Mathematically, this duality ofdescriptions corresponds to the possibility of choosing different coordinates on the Hitchinmoduli space, which will be introduced shortly. At first, the equivalence of the two typesof surface operators may seem rather surprising since it is not even clear from the outsetthat they preserve the same subalgebra of the supersymmetry algebra. Moreover, the modulispaces parametrizing these surface operators appear to be different.– 9 –ndeed, one of these moduli spaces parametrizes collections of n codimension-4 defectssupported at D × { p i } ⊂ M × C , and therefore it isSym n ( C ) := C n /S n (2.5)(Here we consider only the “intrinsic” parameters of the surface operator, and not the positionof D ⊂ M , which is assumed to be fixed.) On the other hand, a surface operator constructedfrom a codimension-2 defect clearly does not depend on these parameters, since it wraps onall of C . Instead, a codimension-2 surface operator carries a global symmetry G — whichplays an important role e.g. in describing charged matter — and, as a result, its moduli spaceis the moduli of G -bundles on C , Bun G ( C ) (2.6)Therefore, it appears that in order to relate the two constructions of surface operators, onemust have a map between (2.5) and (2.6):Bun G ( C ) −→ Sym n ( C ) x (cid:55)→ u (2.7)where n = ( g −
1) dim G = dim Bun G ( C ).It turns out that even though such a map does not exist, for G = SL (2) there is a mapof the corresponding cotangent bundles, which is sufficient for our purposes. This is thecelebrated classical separation of variables . Moreover, it has a quantum version, described inSection 4.7. The separation of variables allows us to identify the 4d theories with two typesof surface operators in the IR.The unbroken SUSY makes it possible to turn on an Omega-deformation, allowing us todefine generalizations of the instanton partition functions. In the case of codimension-2 surfaceoperators it turned out that the generalized instanton partition functions are calculable bythe localization method, and in a few simple cases it was observed that the results are relatedto the conformal blocks in the SL (2)-WZW model. For codimension-4 surface operatorsone expects to find a similar relation to Liouville conformal blocks with a certain number ofdegenerate fields inserted.
3. Surface operators corresponding to the codimension-2 defects
Our goal in this paper is to establish a relation between the surface operators constructedfrom codimension-2 and codimension-4 defects. In order to do that, we must show that theypreserve the same subalgebra of the supersymmetry algebra. This will be achieved by realizingthese defects using branes in M-theory (as we already mentioned earlier). This realizationwill enable us to link the two types of defects, and it will also illuminate their features. Even though our main examples will be theories of class S , we expected our results — in particular, theIR duality — to hold more generally. – 10 –n this section we present an M-theory brane construction of the codimension-2 defectsand then discuss them from the point of view of the 4d and 2d theories. Then, in Section 4,we will deform — in a way that manifestly preserves supersymmetry — a brane system thatgives rise to the codimension-2 defects into a brane system that gives rise to codimension-4defects. Using this deformation, we will show that the two types of defects indeed preservethe same supersymmetry algebra, and furthermore, we will connect the two types of defects,and the corresponding 4d surface operators, to each other. Following [7], we denote the support (resp. the fiber of the normal bundle) of the surfaceoperator inside M by D (resp. D (cid:48) ). In fact, for the purposes of this section, we simply take M = D × D (cid:48) . Our starting point is the following “brane construction” of 4d N = 2 gaugetheory with a half-BPS surface operator supported on D ⊂ M (= D × D (cid:48) ):M5 : D × D (cid:48) × C M5 (cid:48) : D × C × D (cid:48)(cid:48) (3.1)embedded in the eleven-dimensional space-time D × D (cid:48) × T ∗ C × R × D (cid:48)(cid:48) in a natural way.For simplicity, we will assume that D ∼ = D (cid:48) ∼ = D (cid:48)(cid:48) ∼ = R and C is the only topologically non-trivial Riemann surface in the problem at hand. And, following the standard conventionsof brane constructions [20], we use the following local coordinates on various factors of theeleven-dimensional space-time: D D (cid:48) T ∗ C R D (cid:48)(cid:48) x , x x , x x , x , x , x x x , x (3.2)With these conventions, the brane configuration (3.1) may be equivalently summarized in thefollowing diagram: Brane 0 1 2 3 4 5 6 7 8 9 10 M M (cid:48) x x x x x xNote that M (cid:48) -branes wrap the same UV curve C as the M -BPS, i.e. it preserves four real supercharges out of 32. Namely, the eleven-dimensionalspace-time (without any fivebranes) breaks half of supersymmetry (since T ∗ C is a manifoldwith SU (2) holonomy), and then each set of fivebranes breaks it further by a half.In particular, thinking of T ∗ C as a non-compact Calabi-Yau 2-fold makes it clear thatcertain aspects of the system (3.1), such as the subalgebra of the supersymmetry algebrapreserved by this system, are not sensitive to the details of the support of M5 and M5 (cid:48) braneswithin T ∗ C as long as both are special Lagrangian with respect to the same K¨ahler form ω and the holomorphic 2-form Ω. Since T ∗ C is hyper-K¨ahler, it comes equipped with a sphereworth of complex structures, which are linear combinations of I , J , K , and the corresponding– 11 –¨ahler forms ω I , ω J , ω K . Without loss of generality, we can choose ω = ω I and Ω = ω J + iω K .Then, the special Lagrangian condition means that both ω I and ω K vanish when restrictedto the world-volume of M5 and M5 (cid:48) branes. As we explain below, surface operators originating from codimension-4 defects in 6d (0 , A ( r ) µ (see Appendix A for more details): A ( r ) µ dx µ ∼ (cid:18) χ ( r ) − χ ( r ) (cid:19) dθ . (3.3)Here, following our conventions (3.2), we use a local complex coordinate x + ix = r e iθ on D (cid:48) such that surface operator is located at the origin ( r = 0). A surface operator definedthis way breaks half of supersymmetry and also breaks SO (4) rotation symmetry down to SO (2) × SO (2). From the viewpoint of the 2d theory on D , the unbroken supersymmetry is N = (2 , SO (2) × SO (2) equivariant counting of instantons with a ramification along D . Theresulting instanton partition function Z M5 ( a, x, τ ; (cid:15) , (cid:15) ) , (3.4)depends on variables x = ( x , . . . , x h ) related to the parameters χ ( r ) in (3.3) via the expo-nentiation map x r = e πiτ r χ ( r ) . (3.5)The relation between the parameters χ ( r ) and the counting parameters x r appearing in theinstanton partition functions Z M5 was found in [10]. Starting from the groundbreaking work of A. Braverman [12], a number of recent studies haveproduced evidence of relations between instanton partition functions in the presence of surfaceoperators Z M5 ( a, x, τ ; (cid:15) , (cid:15) ) and conformal blocks of affine Kac-Moody algebras (cid:98) g k [10,13–15].Such relations can be viewed as natural generalizations of the AGT correspondence. In thecase of class S -theories of type A one needs to choose g = sl and k = − − (cid:15) (cid:15) , as will beassumed in what follows.The Lie algebra (cid:98) g k has generators J an , a = 0 , + , − , n ∈ Z . A large class of representationof (cid:98) g k is defined by starting from a representation R j of the zero mode subalgebra generatedfrom J a , which has Casimir eigenvalue parametrized as j ( j + 1). One may then construct arepresentation R j of (cid:98) g k as the representation induced from R j extended to the Lie subalgebra– 12 –enerated by J an , n ≥
0, such that all vectors v ∈ R j ⊂ R j satisfy J an v = 0 for n >
0. To bespecific, we shall mostly discuss in the following the case that the representations R j have alowest weight vector e j , but more general representations may also be considered, and maybe of interest in this context [34].In order to define the space of conformal blocks, let C be a compact Riemann surface and z , . . . , z n an n -tuple of points of C with local coordinates t , . . . , t n . We attach representations R r ≡ R j r of the affine Kac–Moody algebra (cid:98) g k of level k to the points z r , r = 1 , . . . , n . Thediagonal central extension of the direct sum (cid:76) nr =1 g ⊗ C (( t r )) acts on the tensor product (cid:78) nr =1 R r . Consider the Lie algebra g out = g ⊗ C [ C \{ z , . . . , z n } ]of g -valued meromorphic functions on C with poles allowed only at the points z , . . . , z n . Wehave an embedding g out (cid:44) → n (cid:77) r =1 g ⊗ C (( t r )) . (3.6)It follows from the commutation relations in (cid:98) g and the residue theorem that this embed-ding lifts to the diagonal central extension of (cid:76) nr =1 g ⊗ C (( t r )). Hence the Lie algebra g out acts on (cid:78) nr =1 R r . By definition, the corresponding space of conformal blocks is the spaceCB g ( R , . . . , R n ) of linear functionals ϕ : R [ n ] := n (cid:79) r =1 R r → C invariant under g out , i.e., such that ϕ ( η · v ) = 0 , ∀ v ∈ n (cid:79) r =1 R r , η ∈ g ⊗ C [ C \{ z , . . . , z n } ] . (3.7)The conditions (3.7) represent a reformulation of current algebra Ward identities well-knownin the physics literature. The space CB g ( R , . . . , R n ) is infinite-dimensional in general.To each ϕ ∈ CB g ( R , . . . , R n ) we may associate a chiral partition function Z ( ϕ, C ) byevaluating ϕ on the product of the lowest weight vectors, Z WZ ( ϕ, C ; k ) := ϕ ( e ⊗ . . . ⊗ e n ) . (3.8)In the physics literature one usually identifies the chiral partition functions with expectationvalues of chiral primary fields Φ r ( z r ), inserted at the points z r , Z WZ ( ϕ, C ; k ) ≡ (cid:10) Φ n ( z n ) · · · Φ ( z ) (cid:11) C,ϕ . (3.9)Considering families of Riemann surfaces C τ parametrized by local coordinates τ for theTeichm¨uller space T g,n one may regard the chiral partition functions as functions of τ , Z WZ ( ϕ, C τ ; k ) ≡ Z WZ ( ϕ, τ ; k ) . – 13 –arge families of conformal blocks and the corresponding chiral partition functions canbe constructed by the gluing construction. Given a (possibly disconnected) Riemann surface C with two marked points P i , i = 1 , D i one can constructa new Riemann surface by pairwise identifying the points in annuli A i ⊂ D i around the twomarked points, respectively. Assume we are given conformal blocks ϕ C i associated to twosurfaces C i with n i + 1 punctures P i , P i , . . . , P in i with the same representation R associatedto P i for i = 1 ,
2. Using this input one may construct a conformal block ϕ C associated tothe surface C obtained by gluing the annular neighborhoods A i of P i , i = 1 , ϕ C ( v ⊗ · · · ⊗ v n ⊗ w ⊗ · · · ⊗ w n ) == (cid:88) ν ∈I R ϕ C ( v ⊗ · · · ⊗ v n ⊗ v ν ) ϕ C ( K ( τ, x ) v ∨ ν ⊗ w ⊗ · · · ⊗ w n ) . (3.10)The vectors v ν and v ∨ ν are elements of bases for the representation R which are dual w.r.t. tothe invariant bilinear form on R . A standard choice for the twist element K ( τ, x ) ∈ End( R )appearing in this construction is K ( τ, x ) = e πiτL x J , where the operator L represents thezero mode of the energy-momentum tensor constructed from the generators J an using theSugawara construction. The parameter q ≡ e πiτ in (3.10) can be identified with the modulusof the annular regions used in the gluing construction of C . However, it is possible toconsider twist elements K ( τ, x ) constructed out a larger subset of the generators of (cid:98) g k . Therest of the notation in (3.10) is self-explanatory. The case that P i , i = 1 , C g,n can be obtained by gluing 2 g − n pairs of pants C v , , v = 1 , . . . , g − n . It is possible to construct conformal blocks for the resulting Riemannsurface from the conformal blocks associated to the pairs of pants C v , by recursive use of thegluing construction outlined above. This yields families ϕ σj,x of conformal blocks parametrizedby • the choice of a refined pants decomposition σ = ( C , Γ), • the choice of representation R j e for each of the cutting curves γ e defined by the pantsdecomposition, and • the collection of the parameters x e introduced via (3.10) for each curve γ e ∈ C .The corresponding chiral partition functions are therefore functions Z WZ σ ( j, x, τ ; k ) ≡ Z WZ ( ϕ σj,x , τ ; k ) . The variables x = ( x , . . . , x g − n ) have a geometric interpretation as parameters for fam-ilies of holomorphic G = SL (2)-bundles B . Indeed, in Appendix B it is explained how thedefinition of the conformal blocks can be modified in a way that depends on the choice ofa holomorphic bundle B , and why the effect of this modification can be described using thetwist elements K ( τ, x ) appearing in the gluing construction. It follows from the discussion in– 14 –ppendix B that changing the twist elements K ( τ, x ) amounts to a change of local coordinates( τ, x ) for the fibration of Bun G over T g,n (the moduli space of pairs: a Riemann surface anda G -bundle on it).The chiral partition functions satisfy the Knizhnik-Zamolodchikov-Bernard (KZB) equa-tions. This is a system of partial differential equations of the form − (cid:15) (cid:15) ∂∂q e Z WZ σ ( j, x, τ ; k ) = H e Z WZ σ ( j, x, τ ; k ) , (3.11)where H e is a second order differential operator containing only derivatives with respect tothe variables x e . These equations can be used to generate the expansion of Z WZ σ ( j, x ; τ ; k ) inpowers of q e and x e , Z WZ σ ( j, x, τ ; k ) (cid:39) (cid:88) n ∈ Z h + (cid:88) m ∈ Z h + Z WZ σ ( j, m , n ; k ) h (cid:89) e =1 q ∆ e + n e x j e + m e . (3.12)The notation (cid:39) used in (3.12) indicates equality up to a factor which is j -independent. Suchfactors will be not be of interest for us. The equations (3.11) determine Z WZ σ ( j, m , n ; k )uniquely in terms of Z WZ ,σ ( j ) = Z WZ ( j, , k ). It is natural to assume that the normalizationfactor Z WZ ( j ) can be represented as product over factors depending on the choices of repre-sentations associated to the three-holed spheres C v , appearing in the pants decomposition.We are now going to propose the following conjecture: There exists a choice of twistelements K e ( τ e , x e ) such that we have Z M5 σ ( a, x, τ ; (cid:15) , (cid:15) ) (cid:39) Z WZ σ ( j, x, q ; k ) , (3.13)assuming that j e = −
12 + i a e (cid:15) , k + 2 = − (cid:15) (cid:15) . (3.14)Evidence for this conjecture is provided by the computations performed in [10, 13–15] in thecases C = C , and C = C , . The relevant twist elements K ( τ, x ) were determined explicitlyin these references. As indicated by the notation (cid:39) , we expect (3.13) to hold only up to j -independent multiplicative factors. A change of the renormalization scheme used to definethe gauge theory under consideration may modify Z M5 by factors that do not depend on j .Such factors are physically irrelevant, see e.g. [35] for a discussion. On physical grounds we expect that the instanton partition functions Z M5 σ ( a, x, τ ; (cid:15) , (cid:15) ) be-have in the limit (cid:15) → (cid:15) → Z M5 σ ( a, x, τ ; (cid:15) , (cid:15) ) ∼ − (cid:15) (cid:15) F σ ( a, τ ) − (cid:15) (cid:102) W M5 σ ( a, x, τ ) . (3.15)The first term is the bulk free energy, proportional to the prepotential F σ ( a ) defined previ-ously. The second term is a contribution diverging with the area of the plane on which the– 15 –urface operator is localized. It can be identified as the effective twisted superpotential of thedegrees of freedom localized on the surface x = x = 0.The expression of the instanton partition function as a to conformal field theory (3.13)allows us to demonstrate that we indeed have an asymptotic behavior of the form (3.15). Thederivation of (3.15) described in Appendix D leads to a precise mathematical description ofthe functions (cid:102) W M5 σ ( a, x, τ ) appearing in (3.15) in terms the Hitchin integrable system that wewill describe in the rest of this subsection. It turns out that (cid:102) W M5 σ ( a, x, τ ) can be characterizedas the generating function for the change of variables between two sets of Darboux coordinatesfor M H ( C ) naturally adapted to the description in terms of Higgs pairs ( E , ϕ ) and pairs (Σ , L ),respectively.Let us pick coordinates x = ( x , . . . , x h ) for Bun G . Possible ways of doing this are brieflydescribed in Appendix C.2. One can always find coordinates p on M H ( C ) which supplementthe coordinates x to a system of Darboux coordinates ( x, p ) for M H ( C ).There exists other natural systems ( a, t ) of coordinates for M H ( C ) called action-anglecoordinates making the complete integrability of M H ( C ) manifest. The coordinates a =( a , . . . , a h ) are defined as periods of the Seiberg-Witten differential, as described previously.The coordinates t = ( t , . . . , t h ) are complex coordinates for the Jacobian of Σ parametrizingthe choices of line bundles L on Σ. The coordinates t may be chosen such that ( a, t ) furnishesa system of Darboux coordinates for M H ( C ).As the coordinates ( a, t ) are naturally associated to the description in terms of pairs(Σ , L ), one may construct the change of coordinates between the sets of Darboux coordinates( x, p ) and ( a, t ) using Hitchin’s map introduced in Section 2.3. The function (cid:102) W M5 σ ( a, x, τ )in (3.15) can then be characterized as the generating function for the change of coordinates( x, p ) ↔ ( a, t ), p r = − ∂∂x r (cid:102) W M5 σ , t r = 12 π ∂∂a r (cid:102) W M5 σ , (3.16)with periods a defined using a basis B σ corresponding to the pants decomposition σ usedto define Z M5 σ ( a, x, τ ; (cid:15) , (cid:15) ). Having defined ( x, p ) and ( a, t ), the equations (3.16) define (cid:102) W M5 σ ( a, x, τ ) up to an (inessential) additive constant. All of the integrable system gadgets introduced above seem to find natural homes in fieldtheory and string theory. In particular, N five-branes on C describe a theory that in the IRcorresponds to an M5-brane wrapped N times on C or, equivalently, wrapped on a N -foldcover Σ → C .Though in this paper we mostly consider the case N = 2 (hence a double cover Σ → C ),certain aspects have straightforward generalization to higher ranks. It is also worth notingthat we treat both SL ( N ) and GL ( N ) cases in parallel; the difference between the two isaccounted for by the “center-of-mass” tensor multiplet in 6d (0 ,
2) theory on the five-braneworld-volume. – 16 –esides the “brane constructions” used in most of this paper, the physics of 4d N = 2theories can be also described by compactification of type IIA or type IIB string theory on alocal Calabi-Yau 3-fold geometry. This approach, known as “geometric engineering” [36, 37],can be especially useful for understanding certain aspects of surface operators and is related tothe brane construction by a sequence of various dualities. Thus, a single five-brane wrappedon Σ ⊂ T ∗ C that describes the IR physics of 4d N = 2 theory is dual to type IIB stringtheory on a local CY 3-fold zw − P ( u, v ) = 0 , (3.17)where P ( u, v ) is the polynomial that defines the Seiberg-Witten curve Σ SW .It can be obtained from our original M5-brane on Σ by first reducing on one of thedimensions transversal to the five-brane (down to type IIA string theory with NS5-brane onΣ) and then performing T-duality along one of the dimensions transversal to the NS5-brane.The latter is known to turn NS5-branes to pure geometry, and supersymmetry and a fewother considerations quickly tell us that type IIB background has to be of the form (3.17).Now, let us incorporate M5 (cid:48) -brane which in the IR version of brane configuration (3.1)looks like: M5 : D × D (cid:48) × ΣM5 (cid:48) : D × Σ × D (cid:48)(cid:48) (3.18)What becomes of the M5 (cid:48) -brane upon duality to type IIB setup (3.17)?It can become any brane of type IIB string theory supported on a holomorphic subman-ifold in the local Calabi-Yau geometry (3.17). Indeed, since the chain of dualities from M-theory to type IIB does not touch the four dimensions parametrized by x , . . . , x the resultingtype IIB configuration should still describe a half-BPS surface operator in 4d Seiberg-Wittentheory on M . Moreover, since type IIB string theory contains half-BPS p -branes for oddvalues of p , with ( p + 1)-dimensional world-volume, M5 (cid:48) can become a p -brane supported on D × C p − , where C p − is a holomorphic submanifold in a local Calabi-Yau 3-fold (3.17).Depending on how one performs the reduction from M-theory to type IIA string theoryand then T-duality to type IIB, one finds different p -brane duals of the M5 (cid:48) -brane. Here, wewill be mostly interested in the case p = 3, which corresponds to the reduction and then T-duality along the coordinates x and x , cf. (3.2). Effectively, one can think of compactifyingthe M-theory setup (3.18) on D (cid:48)(cid:48) = T , and that gives precisely the type IIB setup (3.17)with extra D3-brane supported on Σ, i.e. at z = w = 0 in (3.17).A D3-brane carries a rank-1 Chan-Paton bundle L (cid:48) → Σ. Therefore, we conclude that thesurface operators made from codimension-2 defects that are obtained from the intersectionswith M5 (cid:48) -branes as described above, have an equivalent description in dual type IIB stringtheory in terms of pairs (Σ , L (cid:48) ). It seems likely that the line bundle L (cid:48) is closely related tothe line bundle L appearing in the description of the Hitchin system in terms of pairs (Σ , L ).Note, the degree of this line bundle, d ( L (cid:48) ), is equal to the induced D1-brane charge alongthe ( x , x ) directions. For completeness, we describe what it corresponds to in the dual M-– 17 –heory setup (3.18). The T-duality that relates type IIA and type IIB brane configurationsmaps D1-branes supported on ( x , x ) into D2-branes with world-volume along ( x , x , x ).Hence, we conclude d ( L (cid:48) ) = M2-brane charge along ( x , x , x ) (3.19)It seems worthwhile investigate the description of surface operators in terms of type IIB braneconfigurations in more detail.
4. Surface operators corresponding to codimension-4 defects
As we mentioned earlier, there is another way to construct surface operators in 4d N = 2theories of class S – namely, by introducing codimension-4 defects in 6d five-brane theory[20–22, 38].In this section we present this construction. The idea is to start with the brane systemwhich we used in the previous section to produce the codimension-2 defects and to deform itin such a way that the end result is a collection of codimension-4 defects. The advantage ofthis way of constructing them is that, as we will see below, this process does not change thesubalgebra of the supersymmetry algebra preserved by the defects. Therefore, it follows thatthe two types of defects in fact preserve the same subalgebra.In the next sections we will also use this link between the codimension-4 and codimension-2 defects in the 6d theory in order to establish the connection between the corresponding 4d N = 2 theories in the IR. The origin of codimension-4 defects in 6d theory and the resulting surface operators in 4d N = 2 theory are best understood via the following brane construction:Brane 0 1 2 3 4 5 6 7 8 9 10 M M N M M × C (as in Section 3.1) we have addeda number of M D × R + , where R + = { x ≥ } . Note that each ofthese M C and therefore gives rise to acodimension-4 defect in the 6d theory.One of the main goals of this paper is to show that the surface operators in 4d N = 2 the-ory corresponding to these codimension-4 defects describe in the IR the same physical objectas (3.1), up to a field transformation (which is related to a change of Darboux-coordinates inthe associated integrable system). For such an equivalence to make sense, it is necessary thatthe two types of defects preserve the same supersymmetry subalgebra. This is a non-trivialstatement that we explain presently.A simple and elegant way to analyze supersymmetry and to gain further insight into therelation between the two types of surface operators is to perform a continuous deformation– 18 – c)b)a) Figure 1:
An M5 (cid:48) -brane wrapped on the curve C can be perturbed to a curve (cid:101) C which meets C atfinitely many points u i . Then, separating the five-branes on C and (cid:101) C along the x direction results increation of M2-branes (shown in red). of one brane configuration into the other preserving the corresponding subalgebra of thesupersymmetry algebra. Starting with our original system (3.1), we keep the world-volumeof the M5-branes to be D × D (cid:48) × C , but deform the support of the M5 (cid:48) -branes to be D × (cid:101) C × D (cid:48)(cid:48) ,where (cid:101) C ⊂ T ∗ C is a deformation of the zero section C ⊂ T ∗ C , which is special Lagrangianwith respect to ω = ω I and Ω = ω J + iω K :M5 : D × D (cid:48) × C M5 (cid:48) : D × (cid:101) C × D (cid:48)(cid:48) (4.1)According to the discussion in Section 3.1, this deformation does not affect the amount ofunbroken supersymmetry, and so (4.1) preserves the same part of the supersymmetry algebraas the original system (3.1). Note that deformations of special Lagrangian submanifolds areinfinitesimally parametrized by H ( C ) and, in most cases of interest, this is a fairly largespace. However, what’s even more important is that, after the deformation, (cid:101) C meets theoriginal curve C only at finitely many points u i , as illustrated on Figure 1 b . The number ofsuch intersection points is determined by the Euler characteristic (or genus) of the curve CC · C = 2 g ( C ) − . (4.2)At low energies one may effectively represent the stack of M ⊂ T ∗ C [20]. The M5 (cid:48) -branes will be represented by a curve Σ (cid:48) related to Σ byholomorphic deformation. Using the same arguments as above one may show, first of all,that two types of IR surface operators preserve the same SUSY and, furthermore, determinesthe number of intersection points on Σ to beΣ · Σ = 2 g Σ − , (4.3) The argument presented below applies equally well to a system where the UV curve C is replaced by theIR curve Σ. In fact, the latter version, which similarly explains that IR surface operators preserve the sameSUSY is also responsible for the IR duality that underlies the separation of variables map. – 19 –here g Σ = 4 g − C has no punctures [31], as will be assumed in this section for simplicity.After the deformation, every intersection of M5 and M5 (cid:48) locally looks like a productof R with a submanifold in R , which is a union of two perpendicular 4-spaces R ∪ R ,intersecting at one point, times the real line R parametrized by the coordinate x . Indeed,M5 and M5 (cid:48) overlap along a 2-dimensional part of their world-volume, D , and the remaining4-dimensional parts of their world-volume span R = { x = 0 } . If we separate these five-branes in the x direction, they become linked in the 9-dimensional space which is the partof the space-time orthogonal to D . Then, if we make one of the five-branes pass through theother by changing the value of its position in the x direction, an M2-brane is created, asshown on Figure 1 c . The support of the M2-brane is D × I , where I is the interval along x connecting the deformations of the 4-spaces, which we denote by R a and R b (where a and b are the values of the coordinate x corresponding to these two subspaces):M5 : D × R a M5 (cid:48) : D × R b M2 : D × { a ≤ x ≤ b } (4.4)This creation of the M2-brane between two linked M5-branes is a variant of the so-calledHanany-Witten effect [24]. What this means for us is that a surface operator representedby a codimension-2 defect wrapped on D × Σ in the fivebrane theory can be equivalentlyrepresented by a collection of codimension-4 defects supported at various points u i ∈ Σ.Indeed, globally, after separating M5 and M5 (cid:48) in the x direction, the brane configuration(4.1) looks like this: M5 : D × D (cid:48) × ΣM5 (cid:48) : D × D (cid:48)(cid:48) × (cid:101) ΣM2 : D × I (4.5)Here, adding M2-branes does not break supersymmetry any further, so that (4.5) is a -BPSconfiguration for arbitrary special Lagrangian submanifolds Σ and (cid:101) Σ ⊂ T ∗ Σ. Of course, thespecial case (cid:101) Σ ≡ Σ takes us back to the original configuration (3.1), schematically shown inFigure 1 a . On the other hand, separating M5 and M5 (cid:48) farther and farther apart, we basicallyend up with the standard brane configuration, shown on Figure 2 b , that describes half-BPSsurface operator(s) built from codimension-4 defects, or M2-branes. In fact, even our choiceof space-time conventions (3.2) agrees with the standard notations used in the literature, sothat (4.5) can be viewed as M-theory lift of the following brane system in type IIA stringtheory: NS5 : 012345D4 : 0123 6NS5 (cid:48) : 01 45 89D2 : 01 7 (4.6)– 20 –onversely, reduction of (4.5) on the M-theory circle (parametrized by x ) gives the typeIIA system (4.6) shown on Figure 2 a . Figure 2:
The brane construction of a surface operator in pure N = 2 super Yang-Mills theory ( a )in type IIA string theory and ( b ) its M-theory lift. How many M2-branes are created in the configuration (4.5)? If the number of M5-branesis N and the number of M5 (cid:48) -branes is k , then each intersection point u i ∈ Σ ∩ (cid:101) Σ contributes k · N M2-branes (due to the s -rule [24]). When we multiply this by the number of intersectionpoints (4.3), we get the answer 2( g − kN . This number, however, counts how many M2-branes are created as one pulls a stack of M5 (cid:48) -branes through the stack of M5-branes bychanging their x -position from x < x >
0, while we are interested in a process thatstarts at x = 0 and then goes to either x < x > x = 0 is somewhat singular. However, as in a similar “geometricengineering” of 2d field theories with the same amount of supersymmetry [39], we shall assumethat both phases x < x > x = 0 to either x < x >
0. In fact, via a chainof dualities [40] our “brane engineering” of the 2d theory on M2-branes can be mapped tothe “geometric engineering” of [39], which therefore justifies applying the same arguments.Then, it means that the answer we are looking for is only half of 2( g − kN , i.e. g − kN (4.7)The case considered in this paper is N = k = 2, giving a number of 4 g − T ∗ C . The M5 (cid:48) -branes are supportedon a holomorphic deformation of Σ, which may be represented by a section of a line bundleof the same degree as K Σ ,deg( K Σ ) = 2 g Σ − g − − g − . (4.8)– 21 –t seems natural to assume that Σ (cid:48) is symmetric under the involution exchanging the twosheets of Σ. This implies that the projection π : Σ → C of the intersection points defines4 g − u = ( u , . . . , u g − ) on C . Following the discussion above, one expects to finda collection of M2-branes created with end-points at u r , r = 1 , . . . , g − D ⊂ M breaks translation invariance in thetransverse directions (along D (cid:48) ), it must necessarily break at least part of supersymmetryof the 4d N = 2 gauge theory on M . In addition, our analysis above shows that bothtypes of surface operators preserve the same part of supersymmetry. It is convenient toexpress the unbroken parts of 4d Lorentz symmetry and supersymmetry in 2d language.Indeed, the unbroken generators of the Lorentz symmetry (in x and x directions along D ) conveniently combine with the unbroken supercharges and the R-symmetry generators toform 2d N = (2 ,
2) supersymmetry algebra.
We now start discussing the implications of this construction for the IR physics of 4d N = 2gauge theories with surface operators.The Lagrangian of a 4d N = 2 gauge theory with surface operators may have additionalterms corresponding to 2d N = (2 ,
2) supersymmetric theories coupled to the surface opera-tors. Recall that the Lagrangian of a theory with 2d N = (2 ,
2) supersymmetry is allowed tohave a particular type of F-term called the twisted superpotential , denoted by (cid:102) W . From thepoint of view of a 4d theory, such a term is a two-dimensional feature, i.e. such terms wouldnot be present in a 4d N = 2 theory without surface operators, and it is partially protectedby the supersymmetry from quantum corrections. Moreover, in the IR, the 4d N = 2 gaugetheory with surface operators is completely determined by the prepotential F and the twistedsuperpotential (cid:102) W (see e.g. [41] for a recent review).Recall that the low-energy effective action has a four-dimensional part and a two-dimen-sional part, S = (cid:90) d xd θ F + (cid:18) (cid:90) d xd (cid:101) θ (cid:102) W + c . c . (cid:19) , (4.9)where F is the prepotential giving the low-energy effective action of the four-dimensionaltheory in the absence of a surface operator, and (cid:102) W is the holomorphic twisted superpotential.We will mostly consider F as a function F ( a, τ ), with a being a collection a = ( a , . . . , a h )of coordinates for the moduli space of vacua M vac , where h is the dimension of M vac , and τ being the collection of UV gauge coupling constants τ = ( τ , . . . , τ h ). The dependence onthe mass parameters will not be made explicit in our notations. (cid:102) W ≡ (cid:102) W ( a, κ, τ ) dependson a and τ , and may furthermore depend on a collection of parameters κ characterizing thesurface operator in the UV.The presence of surface operators implies that the abelian gauge fields A r , r = 1 , . . . , h appearing in the same vector-multiplet as the scalars a r will generically be singular at thesupport D of the surface operator. The singularity is such that the field strength F r associated– 22 –o A r has a singularity of the form ( F r ) = 2 πα r δ ( x ) δ ( x ). The parameters α r are relatedto the twisted superpotential (cid:102) W by a relation of the form t r ≡ η r + τ rs α s := 12 π ∂∂a r (cid:102) W , τ rs := ∂∂a r ∂∂a s F . (4.10)The parameters η r in (4.10) characterize the divergence of the dual gauge fields in a similarway. As indicated in (4.10), it is useful to combine the Gukov-Witten parameters α r and η r into complex variables t = ( t , . . . , t h ) which are functions of a , τ and κ .The argument of the previous subsection shows that the brane configuration (3.1) thatdescribes codimension-2 defects can be continuously deformed without changing the unbrokensupersymmetry to a brane configuration describing codimension-4 defects:M5 : D × D (cid:48) × C M2 : D × R + (4.11)This has important implications for our story. First, it means that the same type of Omega-background in both cases leads to the same kind of F-terms (appearing in the instantonpartition functions) for both types of surface operators. Namely, in the language of unbroken2d N = (2 ,
2) supersymmetry, it is the twisted superpotential (cid:102) W in both (3.15) and (4.18).Note that by itself, the existence of a continuous deformation relating surface operatorscorresponding to the codimension-2 defects to those corresponding to the codimension-4 de-fects does not necessarily imply their equivalence. Indeed, there are many physical systemsrelated by a continuous deformation which describe completely different physics, e.g. gaugetheory at different values of a coupling constant is a simple example. However, certain quan-tities may be insensitive to a change of parameter, and in fact, in the case at hand, we willshow that the twisted superpotential (cid:102) W is precisely such a quantity that does not depend onthe deformation described in the previous subsection (up to a change of variables).But the twisted superpotential (cid:102) W determines the vacuum structure and the IR physicsof the 4d theories with surface operators. Therefore if we can show that (cid:102) W is independent ofthe deformation, it will follow that the corresponding 4d theories are equivalent in the IR.So, our plan is the following. In this subsection, we show that the twisted superpotential (cid:102) W is indeed independent of the separation of M5 and M5 (cid:48) in the x direction, which was ourdeformation parameter in the brane configuration (4.5) that interpolates between (3.1) and(4.11). And then, in the next section, we will use this independence of (cid:102) W on the deformationparameter to argue that the 4d theories with the surface operators corresponding to thecodimension-2 and codimension-4 defects describe the same physics in the IR regime (inother words, they are related by an IR duality).In order to show the x -independence of (cid:102) W , we need to focus more closely on the surfaceoperators produced from codimension-4 defects and explain a few facts about the branesystems (4.5)–(4.11) that involve M2-branes. As we already pointed out earlier, the braneconfiguration (4.5) is simply an M-theory lift of the brane system (4.6) illustrated in Figure2 a . Usually, such M-theory lifts capture IR quantum physics of the original type IIA system,– 23 – f. [20]. In the present case, the relevant theory “lives” on D4-branes and D2-branes in (4.6).The theory on D4-branes is simply the 4d gauge theory on M , and describing its IR physicsvia its M-theory lift was one of the main points of [20]. The theory on D2-branes is a 2dtheory with N = (2 ,
2) supersymmetry preserved by the system (4.6), see e.g. [9,42–44]. This2d theory couples to 4d gauge theory and, hence, describes a half-BPS surface operator as acombined 2d-4d system.This has to be compared with our earlier discussion in Section 3.2, where we saw that sur-face operators constructed from codimension-2 defects naturally lead to singularities of gaugefields in the 4d gauge theory, while now we see that surface operators built from codimension-4defects naturally lead to a description via combined 2d-4d system. Furthermore, the number N of D4-branes that determines the rank of the gauge group in four dimensions is the rank ofthe flavor symmetry group from the viewpoint of 2d theory on the D2-branes. In particular,in the basic case of N = 2 each D2-branes carries a U (1) linear sigma-model with N = 2charged flavors, whose Higgs branch is simply the K¨ahler quotient C //U (1) ∼ = CP .This implies that codimension-4 defects give rise to a 2d-4d coupled system, in whichgauge theory in the bulk is coupled to the CP
2d sigma-model on D ⊂ M , which is IR-equivalent to the corresponding 2d gauged linear sigma model. Moreover, this also shows whythe deformation associated to the separation along x direction in (4.5) does not affect thecorresponding twisted superpotential. And here the identification of unbroken supersymmetryand the precise type of the F-terms in 2d becomes crucial.Namely, from the viewpoint of the D2-branes in (4.6), the separation along the x direc-tion is the gauge coupling constant of the 2d gauged linear sigma-model [9, 42–44], g = ∆ x (cid:96) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D2 (4.12)On the other hand, it is a standard fact about 2d N = (2 ,
2) supersymmetry algebra thattwisted superpotential is independent on the 2d gauge coupling constant [45].The reader may observe that the number of variables u i parametrizing the positions ofthe created M2-branes exceeds the number of parameters χ ( r ) introduced via (3.3) for surfacesof genus g >
1. At the moment it does not seem to be known how exactly one may describethe system with M5- and M5 (cid:48) -branes at an intermediate energy scale in terms of a four-dimensional quantum field theory. It seems quite possible that the resulting description willinvolve coupling one gauge field A ( r ) µ to more than one copy of the CP
2d sigma-model on D ⊂ M , in general. As we have seen in the previous subsection, regardless how different the theories with twotypes of surface operators may be in the UV, their effective descriptions in the IR have arelatively simple and uniform description. More specifically, the theories we are consideringin this paper are essentially determined in the IR by their twisted superpotentials. Hence wefocus on them. – 24 –he twisted superpotentials in the presence of codimension-2 and codimension-4 surfaceoperators will be denoted by (cid:102) W M5 and (cid:102) W M2 , respectively. The twisted superpotential (cid:102) W M5 ≡ (cid:102) W M5 ( a, x, τ ) depends besides a and τ on coordinates x for Bun G ( C ), and (cid:102) W M2 ≡ (cid:102) W M2 ( a, u, τ )on the positions of the points on C where the codimension-2 defects are located.From both (cid:102) W M5 and (cid:102) W M2 we can find the corresponding Gukov-Witten parameters t M5 ( a, x, τ ) and t M2 ( a, u, τ ) via (4.10). If the two surface operators are equivalent in thedeep IR there must in particular exist an analytic, locally invertible change of variables u = u ∗ ( x ; a, τ ) relating the Gukov-Witten parameters t and t (cid:48) as t M5 ( a, x, τ ) = t M2 ( a, u ∗ ( x ; a, τ ) , τ ) . (4.13)It follows that the twisted superpotentials (cid:102) W M5 and (cid:102) W M2 may differ only by a functionindependent of a .One may furthermore note that the variables u i are dynamical at intermediate scales, orwith non-vanishing Omega-deformation. The system obtained by separating the M5 (cid:48) -branesby some finite distance ∆ x from the M5-branes will be characterized by a superpotential (cid:102) W (cid:48) depending both on x and u , in general. We had argued above that this superpotentialdoes not depend on the separation ∆ x . Flowing deep into the IR region one expects toreach an effective description in which extremization of the superpotential determines u asfunction of x and the remaining parameters, u = u ∗ ( x, a, τ ). The result should coincide with (cid:102) W M5 ( a, x, τ ), which is possible if the resulting superpotential (cid:102) W (cid:48) differs from (cid:102) W M2 ( a, u, τ ) byaddition of a function (cid:102) W (cid:48)(cid:48) ( u, x, τ ) that is a -independent (cid:102) W (cid:48) ( a, x, u, τ ) = (cid:102) W M2 ( a, u, τ ) + (cid:102) W (cid:48)(cid:48) ( u, x, τ ) ; (4.14)the additional piece (cid:102) W (cid:48)(cid:48) ( u, x, τ ) may be attributed to the process creating the M2-branesfrom M5 (cid:48) -branes. Extremization of (cid:102) W (cid:48) implies that ∂∂u r (cid:102) W M2 ( a, u, τ ) (cid:12)(cid:12)(cid:12) u = u ∗ ( x,a,τ ) = − ∂∂u r (cid:102) W (cid:48)(cid:48) ( u, x, τ ) (cid:12)(cid:12)(cid:12) u = u ∗ ( x,a,τ ) , (4.15)and (cid:102) W (cid:48) ( a, x, u, τ ) (cid:12)(cid:12) u = u ∗ should coincide with (cid:102) W M5 ( a, x, τ ).We are now going to argue that W M5 , W M2 and (cid:102) W (cid:48)(cid:48) represent generating functions forchanges of variables relating three different sets of Darboux-coordinates for the same modulispace M locally parametrized by the variables a and x (see, for example, [46], Section 2.1,for the definition of generating functions and a discussion of their role in the Lagrangianformalism).Considering W M5 first, one may define other local coordinates for M as p r = − ∂∂x r (cid:102) W M5 ( a, x, τ ) . (4.16)Both ( x, p ) and ( a, t ), with t defined via (4.10), will generically define local coordinates for M . Having a Poisson-structure on M that makes ( x, p ) into Darboux-coordinates itfollows from (4.10) and (4.16) that ( a, t ) will also be Darboux-coordinates for M .– 25 –f x and u are related by a locally invertible change of variables u = u ∗ ( x ; a, τ ) it followsfrom (4.15) that u together with the coordinates v defined by v r = ∂∂u r (cid:102) W M2 ( a, u, τ ) , (4.17)will represent yet another set of Darboux coordinates for M . In this way one may identify W M2 and W (cid:48) as the generating functions for changes of Darboux variables ( a, t ) ↔ ( u, v ) and( u, v ) ↔ ( x, p ) for M vac , respectively.There are various ways to compute the twisted superpotential (cid:102) W . One (though not theonly one!) way is to compute the asymptotic expansion of the Nekrasov partition function [26]in the limit (cid:15) , →
0. It takes the formlog Z inst = − F (cid:15) (cid:15) − (cid:102) W (cid:15) + . . . (4.18)Here, F is the Seiberg-Witten prepotential that does not depend on the surface operator anddefines the corresponding IR 4d theory in the bulk. The next term in the expansion, (cid:102) W , iswhat determines the IR theory with the surface operator. In what follows we will use the relations of the instanton partition functions to conformalblocks to determine (cid:102) W M5 ( a, x, τ ) and (cid:102) W M2 ( a, u, τ ) via (4.18). Both functions will be identifiedas generating functions for changes of Darboux-variables ( x, p ) ↔ ( a, t ) and ( u, v ) ↔ ( a, t )for the Hitchin moduli space M H ( C ), respectively. Among other things, this will imply that (cid:102) W M5 ( a, x, τ ) and (cid:102) W M2 ( a, m ; u ) indeed satisfy a relation of the form (cid:102) W M5 ( a, x, τ ) = (cid:102) W M2 ( a, u ∗ ( x, a, τ ) , τ ) + (cid:102) W SOV ( u ∗ ( x, a, τ ) , x, τ ) . (4.19)In view of the discussion above one may view this result as nontrivial support for the conjec-tured IR duality relation between the theories with the surface operators of co-dimensions 2and 4, if we set (cid:102) W (cid:48)(cid:48) ≡ (cid:102) W SOV . We had previously observed that the twisted superpotentials (cid:102) W M5 σ ( a, x, τ ) that may be cal-culated from the instanton partition functions Z M5 σ ( a, x, τ ; (cid:15) , (cid:15) ) via (3.15) represent changesof Darboux variables for the Hitchin integrable system. We will now discuss analogous re-sults for (cid:102) W M2 σ ( a, u, τ ). To this aim we begin by describing the expected relations between theinstanton partition functions Z M2 σ ( a, x, τ ; (cid:15) , (cid:15) ) and Liouville conformal blocks.Conformal blocks for the Virasoro algebra with central charge c b = 1 + 6( b + b − ) may be defined in close analogy to the Kac-Moody conformal blocks discussed above. Ourdiscussion shall therefore be brief. Given a Riemann surface C with n punctures, we associaterepresentations V α r generated from highest weight vectors v α r to the punctures z r , r = 1 , . . . , l .The Lie algebra Vect( C \ { z , . . . , z l } ) of meromorphic vector fields on C with poles only at z r , In a system without surface operators one has (cid:102) W = 0. – 26 – = 1 , . . . , l , is naturally embedded into the direct sum of l copies of the Virasoro algebra withthe central elements identified (using the expansion of the vector fields near the punctures).Conformal blocks ϕ are then defined as linear functionals on (cid:78) lr =1 V α r that are invariantunder the action of Vect( C \ { z , . . . , z l } ). This invariance condition represents the conformalWard identities. Chiral partition functions Z F ( ϕ, C ; b ) are defined as the evaluation of ϕ onthe product of highest weight vectors (cid:78) lr =1 v α r , in the physics literature often denoted as Z F ( ϕ, C ; b ) ≡ (cid:10) e α n ϕ ( z n ) · · · e α ϕ ( z ) (cid:11) C,ϕ . (4.20)In general, the space of conformal blocks is infinite-dimensional. However, it can bedecomposed into a direct sum (or direct integral, depending on the situation) of finite-dimensional spaces (in some cases, such as that of the Liouville model, one-dimensionalspaces, so that we obtain a basis) using the gluing construction reconstructing C from itspants decompositions specified by the data σ = ( C , Γ) introduced in Section 2.1. Its elementsare labeled by representation parameters β e assigned to the cut curves γ e ∈ C . We denotethe resulting chiral partition functions by Z L ( β, τ ; b ).We shall also discuss the situation of d additional degenerate representations V − / b (sometimes called Φ , primary fields) associated to points S = { u , . . . , u d } ⊂ C that aredistinct and different from the punctures z , . . . , z l . The corresponding chiral partition func-tions then satisfy d second order differential equations resulting from the existence of degree2 null vectors in V − / b . A basis for the space of solutions can be obtained by starting froma pants decomposition σ of C . Each pair of pants C v , obtained by cutting along C containsa subset S v of S . Choosing a pants decomposition of C v , \ S v one obtains a refined pantsdecomposition (cid:98) σ that can be used to define chiral partition functions Z L (cid:98) σ,(cid:36) ( β, u, q ; b ) as before.The additional set of labels (cid:36) entering the definition of Z L (cid:98) σ,(cid:36) is constrained by the fusion rulesfor existence of conformal blocks with degenerate representations inserted, and may thereforebe represented by elements of Z d .The precise definition of the instanton partition functions Z M2 d ≡ Z M2 (cid:98) σ,(cid:36) in the presenceof d codimension 4 surface operators depends on the choice of a refined pants decomposition (cid:98) σ , decorated with certain additional discrete data collectively denoted (cid:36) , see [44]. In [9] itwas conjectured that the instanton partition functions Z M2 (cid:98) σ,(cid:36) coincide with Liouville conformalblocks with d additional degenerate fields inserted, Z M2 (cid:98) σ,(cid:36) ( a, u, τ ; (cid:15) , (cid:15) ) = Z L (cid:98) σ,(cid:36) ( β, u, τ ; b ) , (4.21)given that the parameters are related as β e = Q a e √ (cid:15) (cid:15) , b = (cid:15) (cid:15) . (4.22)Further evidence for (4.21) and some of its generalizations were discussed in [6, 44, 47, 48].Now we are ready to bring together the results of the previous sections to demonstratethe IR duality of two 4d gauge theories with surface operators and to link it to the separationof variables in CFT and Hitchin system. – 27 – .5 Relation to the Hitchin system and to the separation of variables It is shown in the Appendix D that (4.21) implies thatlog Z M2 ( a, u, τ ; (cid:15) , (cid:15) ) ∼ − (cid:15) (cid:15) F ( a, τ ) − (cid:15) (cid:102) W M2 ( a, u, τ ) , (4.23)as already proposed in [9]. The function (cid:102) W M2 ( a, u, τ ) is given as (cid:102) W M2 ( a, u, τ ) = − h (cid:88) k =1 (cid:90) u k v . (4.24)We are now going to explain that there exist other sets of natural Darboux-coordinates ( u, v )for Hitchin moduli space allowing us to identify the function (cid:102) W M2 ( a, u, τ ) defined in (4.24)as the generating function for the change of variables ( a, t ) ↔ ( u, v ).Recall from Section 2.3 that the spectral cover construction allows us to describe M H ( C )as the space of pairs (Σ , L ). The line bundle L may be characterized by a divisor of zerosof a particular section of L representing a suitably normalized eigenvector of the Higgs field ϕ ∈ H ( C, End( E ) ⊗ K C ) that we describe presently. Even though this divisor is not unique,it’s projection onto C is uniquely determined by the data of the rank two bundle B with afixed determinant and the Higgs field ϕ .Locally on C , we can trivialize the bundle B and choose a local coordinate z . Then wecan write ϕ as ϕ = (cid:18) a ( z ) b ( z ) c ( z ) − a ( z ) (cid:19) dz. We have the following explicit formula for the eigenvectors of ϕ Ψ ± = (cid:18) a ( y ) ± v ( y ) c ( z ) (cid:19) , v ( y ) = 12 tr( ϕ ( y )) . Note that for the matrix element c ( z ) dz to be well-defined globally on C and independent ofany choices, we need to represent B as an extension of two line bundles, see Appendix C.2for more details.If c ( z ) (cid:54) = 0, then Ψ (cid:54) = 0 for either branch of the square root. If c ( z ) = 0, then one of themvanishes. Now recall that the line bundle L on the double cover Σ of C is defined preciselyas the line bundle spanned by eigenvectors of ϕ (at a generic point p of C , ϕ has two distincteigenvalues, which correspond to the two points, p (cid:48) and p (cid:48)(cid:48) , of Σ that project onto p , and thefibers of L over p (cid:48) and p (cid:48)(cid:48) are the corresponding eigenvectors). Therefore, if we denote by D the divisor of zeros of c ( z ) dz on C , Ψ gives rise to a non-zero section of L outside of thepreimage of D in Σ. As explained in Appendix C.2, a natural possibility is to consider rank two bundles B whose determinantis a fixed line bundle of degree 2 g − n . The moduli space of such bundles is isomorphic to the moduli spaceof SL -bundles on C . – 28 –enerically, D is multiplicity-free and hence may be represented by a collection u =( u , . . . , u d ) of d := deg( D ) distinct points. The number number d depends on the degreesof the line bundles used to represent B as an extension, in general. It may be larger than3 g − n , the dimension of Bun G . However, fixing the determinant of B defines a collectionof constraints allowing us to determine u k , k = h + 1 , . . . , d in terms of the coordinates u i , i = 1 , . . . , u h .There are two distinct points, u (cid:48) i and u (cid:48)(cid:48) i , in Σ over each u i ∈ C . Then for each i = 1 , . . . , h ,our section has a non-zero value at one of the points, u (cid:48) i or u (cid:48)(cid:48) i , and vanishes at anotherpoint. Thus, the divisor of this section on Σ is the sum of particular preimage of the points u i , i = 1 , . . . , h , in Σ, one for each i . While there is a finite ambiguity remaining for thisdivisor, the unordered collection u = ( u , . . . , u h ) of points of C is well-defined (generically).And then for each u i we choose the eigenvalue v k ∈ T ∗ i C , for which our section provides anon-zero eigenvector. It is known that the collection ( u, v ) = (( u , v ) , . . . , ( u h , v h )) can beused to get to a system of Darboux coordinates for M H ( C ) [49, 50], see also [51] for relatedresults.It was observed in [50] that the definition of the variables ( u, v ) outlined above can be seenas a generalization of the method called separation of variables in the literature on integrablemodels [16]. A familiar example is the so-called Gaudin-model which can be identified with theHitchin integrable system associated to surfaces C of genus zero with n regular singularitiesat distinct points z , . . . , z n . The Higgs field can then be represented explicitly as ϕ = n (cid:88) r =1 A r y − z i dy, n (cid:88) r =1 A r = 0 , where A r = (cid:18) A r A + r A − r − A r (cid:19) , and the separated variables are obtained as the zeros of the lower left entry A − ( y ) dy of ϕ : A − ( y ) = u (cid:81) n − k =1 ( y − u k ) (cid:81) n − r =1 ( y − z r ) , (4.25a) v k = n − (cid:88) r =1 A r u k − z r . (4.25b)One may think of the separation of variables as a useful intermediate step in the construc-tion of the mapping from the original formulation of an integrable model to the descriptionas the Hitchin fibration in terms of action-angle coordinates ( a, t ). The remaining step fromthe separated variables ( u, v ) to the action-angle variables is then provided by the Abel map.The function (cid:102) W M2 ( a, u, τ ) is nothing but the generating function for the change of Darbouxcoordinates between ( u, v ) and ( a, t ). A few more details can be found in Appendix C.4. More precisely, we have 2 g − n choices of the preimages u (cid:48) i or u (cid:48)(cid:48) i for each i , which agrees with the numberof points in a generic Hitchin fiber corresponding to a fixed SL bundle. – 29 – .6 IR duality of surface operators from the defects of codimension 2 and 4 In this section we combine the ingredients of the brane analysis in Section 4.1 with our resultson the twisted superpotentials to show that the 4d gauge theories with the surface operatorsconstructed from codimension-2 and codimension-4 defects are equivalent in the IR.Indeed, their vacuum structures are controlled by the twisted superpotentials (cid:102) W M5 ( a, x, τ )and (cid:102) W M2 ( a, u, τ ), and we have found that they are related by a change of variables (that is,a redefinition of fields).Furthermore, when combined, the above arguments – including the brane creation uponthe change of separation in the x direction – show that two types of surface operatorsconstructed from codimension-2 and codimension-4 defects preserve the same supersymmetrysubalgebra and have the same twisted chiral rings. This is sufficient to establish theirequivalence for the purposes of instanton counting. In order to demonstrate the IR equivalenceof the full physical theories, we need to show the isomorphism between their chiral rings (andnot just the twisted chiral rings). In general, this is not guaranteed by the arguments wehave used, but the good news is that for simple types of surface operators, including theones considered here, the chiral rings are in fact trivial and, therefore, we do obtain theequivalence of the two full physical theories.As we already mentioned in the Introduction, this equivalence, or duality, between the IRphysics of 4d N = 2 gauge theories with two types of surface operators is conceptually similarto the Seiberg duality of 4d N = 1 gauge theories [1]. In fact, it would not be surprising ifthere were a more direct connection between the two phenomena since they both enjoy thesame amount of supersymmetry and in its brane realization, Seiberg’s duality involves thesame kind of “moves” as the ones described in the previous section. The relation between (cid:102) W M5 ( a, x, τ ) and (cid:102) W M2 ( a, u, τ ) has a rather nontrivial generalization inthe case of non-vanishing Omega-deformation that we will describe in this subsection. Thefact that in 2d this a variant to the separation of variables continues to hold for non-zerovalues of (cid:15) and (cid:15) suggests that the two 4d N = 2 gauge theories remain IR equivalent evenafter Omega-deformation. The possibility of such an equivalence certainly deserves furtherstudy.When we quantize the Hitchin system, the separation of variables may also be quantized.In the genus zero case, in which the quantum Hitchin system is known as the Gaudin model,this was first shown by E. Sklyanin [16]. Note that the quantization of the classical Hitchinsystem corresponds, from the 4d point of view, to “turning on” one of the parameters of the Twisted chiral rings are Jacobi rings of the twisted chiral superpotential (cid:102) W which has been our mainsubject of discussion in earlier sections. In general, 2d N = (2 ,
2) theories may have non-trivial chiral and twisted chiral rings, see for example [52].However, if we start with a 2d theory without superpotential, then, as long as chiral superfields are all massivein the IR, integrating them out leads to a theory of twisted chiral superfields with a twisted superpotential,and so the chiral ring is indeed trivial. – 30 –mega-deformation which is the case studied in [53]. It has been explained in Section 6 of [17]that one may interpret the separation of variables in the Gaudin model, as well as more generalquantum Hitchin systems, as the equivalence of two constructions of the geometric Langlandscorrespondence (Drinfeld’s “first construction” and the Beilinson–Drinfeld construction).Feigin, Frenkel, and Stoyanovsky have shown (see [18]) that in genus zero the separationof variables of the quantum Hitchin system maybe further deformed when we “turn on”both parameters of the Omega deformation. This result was subsequently generalized toget relations between non-chiral correlation functions of the WZW-model and the Liouvilletheory in genus 0 [19], and in higher genus [54]. It has furthermore been extended in [27]to larger classes of conformal blocks. From the 4d point of view, this relation amounts to arather non-trivial relation via an integral transform (a kind of “Fourier transform”) betweenthe instanton partition functions of the Omega-deformed 4d theories with surface operatorscorresponding to the defects of codimensions 2 and 4.The resulting relation has its roots in the quantum Drinfeld–Sokolov reduction. Werecall [55, 56] that locally it amounts to imposing the constraint J − ( z ) = 1 on one of thenilpotent currents of the affine Kac–Moody algebra (cid:98) sl . The resulting chiral (or vertex)algebra is the Virasoro algebra. Furthermore, if the level of (cid:98) sl is k = − − b , then the central charge of the Virasoro algebra is c = 1 + 6( b + b − ) . Globally, on a Riemann surface C , the constraint takes the form J − ( z ) dz = ω , where ω isa one-form, if we consider the trivial SL -bundle, or a section of a line bundle if we consider anon-trivial SL -bundle that is an extension of two line sub-bundles (the representation as anextension is necessary in order to specify globally and unambiguously the current J − ( z ) dz ).Generically, ω has simple zeros, which leads to the insertion at those points of the degeneratefields V − / b of the Virasoro algebra in the conformal blocks.It is important to remember that classically the separated variables u i are the zeros of aparticular component of the Higgs field ϕ . But the Higgs fields correspond to the cotangentdirections on M H ( C ), parametrized by the p -variables. After quantization, these variablesare realized as the derivatives of the coordinates along the moduli of SL -bundles (the x -variables), so we cannot directly impose this vanishing condition. Therefore, in order todefine the separated variables u in the quantum case, we must first apply the Fourier transformmaking the p -variables into functions rather than derivatives (this is already needed at thelevel of the quantum Hitchin system, see [17]). Since the Fourier transform is an integraltransform, our formulas below involve integration. Indeed, the separation of variables linkingthe chiral partition functions in the WZW-model and the Liouville model is an integraltransform. – 31 –n Appendix E it is shown that the relations described above can be used to derive thefollowing explicit integral transformation,ˇ Z WZ ( x, z ) = N J (cid:90) γ du . . . du n − K SOV ( x, u ) ˇ Z L ( u, z ) , (4.26)where ˇ Z WZ and ˇ Z L are obtained from Z WZ and Z L by taking the limit z n → ∞ , and thekernel K SOV ( x, u ) is defined as K SOV ( x, u ) := (cid:34) n − (cid:88) r =1 x r (cid:81) n − k =1 ( z r − u k ) (cid:81) n − s (cid:54) = r ( z r − z s ) (cid:35) J n − (cid:89) k Complex (co)adjoint orbits are ubiquitous in the study of both half-BPS surface operatorsand boundary conditions. This happens for a good reason, and here we present a simpleintuitive explanation of this fact. In short, it’s due to the fact that both half-BPS surfaceoperators and boundary conditions are labeled by solutions to Nahm equations. Then, thecelebrated work of Kronheimer [57] relates the latter to complex coadjoint orbits.Suppose that in our setup (3.1) we take C = S C × R and M = D × D (cid:48) ∼ = R , where D (cid:48) ∼ = R is the “cigar.” In other words, D (cid:48) is a circle fibration over the half-line, R + = { y ≥ } ,with a singular fiber at y = 0 so that asymptotically (for y → + ∞ ) D looks like a cylinder,see Figure 3. Then, the six-dimensional (2 , 0) theory on M × C with a codimension-2 defecton D × C can be reduced to five-dimensional super-Yang-Mills theory in two different ways.First, if we reduce on a circle S C , we obtain a 5d super-Yang-Mills on M × R ∼ = R with asurface operator supported on D × R ∼ = R . If we denote by r = e − y the radial coordinatein the plane transverse to the surface operator, then the supersymmetry equations take theform of Nahm’s equations: dady = [ b, c ] , dbdy = [ c, a ] , dcdy = [ a, b ] , (A.1)– 32 – upe r ⌧ Y ang ⌧ M ill s y y Figure 3: The six-dimensional (2 , 0) theory with a codimension-2 defect at the tip of the cigar reducesto 5d super-Yang-Mills theory with a non-trivial boundary condition. where we used the following ansatz for the gauge field and for the Higgs field: A = a ( r ) dθ , φ = b ( r ) drr + c ( r ) dθ . On the other hand, if we first reduce on the circle fiber S F of the cigar geometry D (cid:48) , weobtain a 5d super-Yang-Mills on R + × D × C with a non-trivial boundary conditions at y = 0 determined by the codimension-2 defect of the six-dimensional theory. Note, theseboundary conditions are also associated with solutions to Nahm’s equations (A.1) for theHiggs field (cid:126)φ = ( a, b, c ). Further dimensional reductions of these two systems yield many half-BPS boundary conditions and surface operators in lower-dimensional theories, all labeled bysolutions to Nahm’s equations.Among other things, this duality implies that Figure 4: In the presence of surface operatorand/or Omega-background line operators donot commute. similar physical and mathematical structures canbe found on surface operators as well as in thestudy of boundaries and interfaces. A prominentexample of such structure is the algebra of pa-rameter walls and interfaces, i.e. Janus-like soli-tons realized by monodromies in the space of pa-rameters. (In the case of surface operators, suchmonodromy interfaces are simply line operators,which in general form non-commutative algebraif they can’t move off the surface operator, asillustrated in Figure 4.)This description of walls, lines and interfacesas monodromies in the parameter space providesa simple and intuitive way of understanding theirnon-commutative structure and commutation relations; it is captured by the fundamentalgroup of the parameter space [7]: π ( { parameters } ) (A.2)– 33 –or instance, in the case of C = T one finds π (( T C /S N ) reg ), which is precisely the braidgroup (in the case, of type A N − ). It is generated by parameter walls / interfaces L i thatobey the standard braid group relations: L i (cid:63) L i +1 (cid:63) L i = L i +1 (cid:63) L i (cid:63) L i +1 (A.3)From 2d and 3d perspectives, these systems are often described by sigma-models based onflag target manifolds (or their cotangent bundles) where the lines/walls L i are representedby twist functors; see [8, 41] for further details and many concrete examples of braid groupactions on boundary conditions. The case of the parameter space (2.5) is qualitatively similar. B. Twisting of Kac-Moody conformal blocks This appendix collects some relevant mathematical background concerning the dependenceof Kac-Moody conformal blocks on the choice of a holomorphic bundle on C . B.1 Twisted conformal blocks A generalization of the defining invariance condition allows us to define a generalized notionof conformal blocks depending on the choice of a holomorphic G -bundle B on C . One maymodify the defining invariance condition (3.7) by replacing the elements of the Lie algebra g out by a section of g B out := Γ( C, g B ) , g B := B × G g . (B.1)Describing B in terms of a cover {U ı ; ı ∈ I} of C allows us to describe B in terms of the G -valued transition functions h ı ( z ) defined on the intersections U ı = U ı ∩ U . The sectionsof g B out are represented by families of g -valued functions η ı in U ı , with η ı and η related onthe intersections U ı by conjugation with h ı ( z ). In this way one defines B -twisted conformalblocks ϕ B depending on the choice of a G -bundle B .More concrete ways of describing the twisting of conformal blocks are obtained by choos-ing convenient covers {U ı ; ı ∈ I} . One convenient choice is the following: Let us choose discs D k around the points z k , k = 1 , . . . , n such that U out := C \ { z , . . . , z n } and U in = (cid:83) nk =1 D k form a cover of C . It is known that for G = SL (2) G -bundles B can always be trivializedin U out and U in . An arbitrary G -bundle B can then be represented by the G -valued tran-sition functions h k ( t k ) defined in the annular regions A k := U out ∩ D k modulo changes oftrivialization in U in and in U out , respectively.Introducing the dependence on the choice of B in the way described above makes it easyto see that infinitesimal variations δ of B can be represented by elements of (cid:76) ni =1 g ⊗ C (( t i )).Choosing a lift X δ to the diagonal central extension of (cid:76) ni =1 g ⊗ C (( t i )) allows us to define a(projective) action of T Bun G (cid:12)(cid:12) B on CB g ( R , . . . , R n ). This means that a differential operator δ representing an element T Bun G (cid:12)(cid:12) B can be represented on the conformal blocks in terms ofthe action of η δ on (cid:78) nr =1 R r , schematically δϕ ( e [ n ] ) = ϕ (cid:0) η δ e [ n ] (cid:1) , e [ n ] := e ⊗ · · · ⊗ e n . (B.2)– 34 –his action describes the response of a conformal block ϕ B with respect to an infinitesimalvariation of B . B.2 Genus zero case In the case of genus 0 it suffices to choose the transition functions h k ( t k ) in the annular regions A k around the points z k to be the constant nilpotent matrices h k ( t k ) = (cid:0) x k (cid:1) . The collectionof parameters x = ( x , . . . , x n ) can be used to represent the dependence on the choice of B inthis case. The action of T Bun G (cid:12)(cid:12) B on spaces of conformal blocks defined via (B.2) may thenbe represented more explicitly in terms of the differential operators J ar defined as J − r = ∂ x r , J r = x r ∂ x r − j r , J + r = − x r ∂ x r + 2 j r x r . (B.3)The Casimir operator is represented as multiplication by j r ( j r + 1).The parametrization in terms of n variables x = ( x , . . . , x n ) is of course redundant.The conformal Ward-identities (3.7) include the invariance under global sl -transformations,allowing us to eliminate three out of the n variables x , . . . , x n in the usual way.The operators H r appearing in the Knizhnik-Zamolodchikov equations (3.11) are thengiven by the formulae H r ≡ (cid:88) s (cid:54) = r J rs z r − z s , (B.4)where the differential operator J rs is defined as J rs := η aa (cid:48) J ar J a (cid:48) s := J r J s + 12 ( J + r J − s + J − r J + s ) . (B.5)The operators H r commute, and may therefore be used as Hamiltonians for generalizationsof the Gaudin models associated to more general representations of SL (2 , C ). B.3 Higher genus cases Instead of the covers considered in Subsection B.1 above one may use alternatively use coversdefined using the gluing construction. One thereby gets a cover {U ı ; ı ∈ I} with intersectionsrepresented by annuli A e between pairs of pants or connecting two legs of the same pair ofpants. Choosing constant diagonal transition functions (cid:0) x e x − e (cid:1) in the annuli A e gives us acollection of local coordinates x e , e = 1 , . . . , g − n for Bun G , G = SL (2). The resultingparameters x for Bun G are easily identified with the parameters x introduced in the gluingconstruction of conformal blocks via (3.10) provided we choose K ( τ, x ) to be e πiτL x J .In order to have a globally well-defined current J − on C one needs to represent B as anextension. Taking 0 −→ O −→ B −→ L −→ , (B.6)appears to be particularly natural. This allows us to represent J − as a section of L ⊗ K C . Asexplained in Appendix C it is natural in our case to consider fixed line bundles L of degree d (cid:48) . Let us represent L as O ( D (cid:48) ), with divisor D (cid:48) being represented by the points y , . . . y d (cid:48) .– 35 –he bundle B may be described by using a cover {U ı ; ı ∈ I} for C containing small discs D (cid:48) k around y k , k = 1 , . . . , d (cid:48) , with transition functions h (cid:48) k = (cid:18) t k (cid:19)(cid:18) x k (cid:19) , (B.7)on the annuli A (cid:48) k = D (cid:48) k \{ y k } , where t k is a coordinate on D (cid:48) k vanishing at y k . Sections of B mayalternatively be represented locally by functions that are regular outside of { y k , k = 1 , . . . , d (cid:48) } and may have poles with residue in a fixed line (cid:96) k at y k , k = 1 , . . . , d (cid:48) . Using the transitionfunctions (B.7) determines the lines (cid:96) k in terms of the parameters x k . Modifications of B thatincrease the degree d (cid:48) of L are called Hecke modifications.Using covers defined with the help of the gluing construction it appears to be natural totake d (cid:48) = 2 g − 2. In this case one may assume that there is exactly one y k contained in eachpair of pants. Kac-Moody conformal blocks associated to each pairs of pants appearing in thepants decomposition of a closed Riemann surface can then be defined using conformal blockson C , , with one insertion being the degenerate representation of the Kac-Moody algebra R k/ representing the Hecke modifications within conformal field theory [27]. If the Riemannsurface has punctures, one may use conformal blocks on C , without extra insertion of R k/ for the pairs of pants containing the punctures.It is worth remarking that d (cid:48) = 2 g − J − , being asection of K C ⊗ L , has 4 g − u i , as required by the identification of the points u i withthe end-points of the M2-branes created from the M5 (cid:48) -branes. C. Holomorphic pictures for the Hitchin moduli spaces The Hitchin space M H ( C ) was introduced in the main text as the space of pairs ( B , ϕ ).Interpreting the Higgs fields ϕ ∈ H ( C, End( E ) ⊗ K C ) as representatives of cotangent vectorsto Bun G , one may identify M H ( C ) with T ∗ Bun G , the cotangent bundle of the moduli spaceof holomorphic G -bundles on C . This description equips M H ( C ) with natural complex andsymplectic structures, leading to the definition of local sets of Darboux coordinates ( x, p )parametrizing the choices of G -bundles via coordinates x , and the choices of Higgs fields ϕ interms of holomorphic coordinates p .In order to exhibit the relation with conformal field theory we will find it, following [27,58],useful to consider a family of other models for M H ( C ). We will consider moduli spaces M (cid:15) H ( C )of pairs ( B , ∇ (cid:48) (cid:15) ) consisting of holomorphic bundles B with holomorphic (cid:15) -connections ∇ (cid:48) (cid:15) . An (cid:15) -connection is locally represented by a differential operator ∇ (cid:48) (cid:15) = ( (cid:15)∂ y + A ( y )) dy transformingas (cid:101) ∇ (cid:48) (cid:15) = g − · ∇ (cid:48) (cid:15) · g under gauge-transformations. Consideration of M (cid:15) H ( C ) will represent auseful intermediate step which helps clarifying the link between conformal field theory andthe Hitchin system. Noting that any two (cid:15) -connections ∇ (cid:48) (cid:15) and (cid:101) ∇ (cid:48) (cid:15) differ by an element of H ( C, End( E ) ⊗ K C ) one sees that M (cid:15) H ( C ) can be regarded as a twisted cotangent bundle T ∗ (cid:15) Bun G . Picking a reference connection ∇ (cid:48) (cid:15), , one may represent a generic connection as ∇ (cid:48) (cid:15) = ∇ (cid:48) (cid:15), + ϕ . – 36 –o avoid confusion let us stress that the resulting isomorphism M (cid:15) H ( C ) (cid:39) T ∗ Bun G is notcanonical, being dependent on the choice of ∇ (cid:48) (cid:15), . Instead we could use the known results ofHitchin, Donaldson, Corlette and Simpson [59–63] relating pairs ( B , ϕ ) to flat connections on C to identify the moduli spaces M H ( C ) and M (cid:15) H ( C ). The description of M (cid:15) H ( C ) as twistedcotangent bundle yields natural complex and symplectic structures which are inequivalent fordifferent values of (cid:15) . This can be used to describe the hyperk¨ahler structure on M H ( C ), with (cid:15) being the hyperk¨ahler parameter [64].However, in order to discuss the relation with conformal field theory we find it useful toadopt a different point of view. The definition of conformal blocks depends on the choice ofa G -bundle B , which may be parametrized by variables x in a way that does not depend on (cid:15) and (cid:15) . The gluing construction yields natural choices for the reference connection ∇ (cid:48) (cid:15), ,e.g. the trivial one. All dependence on the parameter (cid:15) is thereby shifted into the relationsbetween different charts U ı on M (cid:15) H ( C ) parametrized in terms of local coordinates ( x ı , p ı ) ina way that does not explicitly depend on (cid:15) .One may formally identify ϕ ∈ H ( C, End( E ) ⊗ K C ) as an (cid:15) -connection for (cid:15) = 0. Wetherefore expect that the Darboux coordinates ( x (cid:15) , p (cid:15) ) turn into the Darboux coordinates( x, p ) discussed in the main text when (cid:15) → 0. This will be further discussed below, afterhaving discussed possible choices of Darboux coordinates more concretely. C.1 Three models for Hitchin moduli space There are three models for M (cid:15) H ( C ) of interest for us:(A) As space of representations of the fundamental groupHom( π ( C ) , SL(2 , C )) / SL(2 , C ) . (C.1)(B) As space of bundles with connections ( E , ∇ (cid:48) (cid:15) ), ∇ (cid:48) (cid:15) = ( (cid:15)∂ y + A ( y )) dy , A ( y ) = (cid:18) A ( y ) A + ( y ) A − ( y ) − A ( y ) (cid:19) . (C.2)Having n punctures z , . . . , z n means that A ( y ) is allowed to have regular singularitiesat y = z r of the form A ( y ) = A r y − z r + regular . (C.3)(B’) As space of opers (cid:15) ∂ y + t ( y ), where t ( y ) has n regular singularities at y = z r , t ( y ) = δ r ( y − z r ) − H r y − z r + regular , (C.4)and d apparent singularities at y = u k , t ( y ) = − (cid:15) y − u k ) + (cid:15)v k y − u k + regular , (C.5)– 37 –aving an apparent singularity at y = u k means that the monodromy around u k istrivial in PSL(2 , C ). This is known [17, Section 3.9] to be equivalent to the fact thatthe residues H r , r = 1 , . . . , n are constrained by the linear equations v k + t k, = 0 , k = 1 , . . . , l , t ( y ) = (cid:88) l =0 t k,l ( y − u k ) l − . (C.6a)If g = 0, the parameters H s , s = 1 , . . . , n are furthermore constrained by n (cid:88) r =1 z ar ( z r H r + ( a + 1) δ r ) = 0 , a = − , , , (C.6b)ensuring regularity of t ( y ) at infinity.Models (B) and (B’) are related by singular gauge transformations which transform A ( y ) tothe form (cid:101) A ( y ) = (cid:18) − t ( y )1 0 (cid:19) . (C.7)In order to describe the relation between (B) and (B’) more concretely let us, without loss ofgenerality, assume that elements of Bun G are represented as extensions0 −→ L (cid:48) −→ B −→ L (cid:48)(cid:48) −→ . (C.8)Describing the bundles B by means of a covering U ı of C and transition functions B ı betweenpatches U ı and U , one may assume that all E ı are upper triangular, B ı = (cid:18) L (cid:48) ı L (cid:48)(cid:48) ı (cid:19)(cid:18) E ı (cid:19) . (C.9)This implies that the lower left matrix element A − ( y ) of the (cid:15) -connection (cid:15)∂ y + A ( y ) is asection of the line bundle ( L (cid:48) ) − ⊗ L (cid:48)(cid:48) ⊗ K C , with K C being the canonical line bundle. Thegauge transformation which transforms A ( y ) to the form (C.7) will be singular at the zeros u k of A − ( y ), leading to the appearance of the apparent singularities u k in (C.4). C.2 Complex-structure dependent Darboux coordinates Let us briefly discuss possible ways to introduce Darboux coordinates ( x, p ) for M (cid:15) H ( C ), andhow the passage from (cid:15) -connections to opers defines a change of Darboux coordinates from( x, p ) to ( u, v ). Genus zero In the cases of genus g = 0 we may parametrize the matrices A r in (C.2) as A r ≡ (cid:18) A r A + r A − r − A r (cid:19) ≡ (cid:18) − x r (cid:19)(cid:18) l r p r − l r (cid:19)(cid:18) x r (cid:19) , (C.10)– 38 –ssuming that ( x r , p r ) are a set of Darboux coordinates with { p r , x s } = δ r,s . Let P n be thephase space whose algebra of functions is generated by functions of ( x r , p r ), r = 1 , . . . , n .The space M flat ( C ,n ) can be described as the symplectic reduction of P n w.r.t. the global sl -constraints n (cid:88) r =1 A ar = 0 , (C.11)for a = − , , +, or, more conveniently, as the symplectic reduction of P n − w.r.t. the con-straints (C.11) for a = − , z n → ∞ . We will use the latter description.The change of ( x, p ) ↔ ( u, v ) induced by the relation between models (B) and (B’) isexplicitly described by the formulas (note that the same formulas (4.25) appear in the limit (cid:15) → A − ( y ) = u (cid:81) n − k =1 ( y − u k ) (cid:81) n − r =1 ( y − z r ) , (C.12a) v k := A ( u k ) , A ( y ) = n − (cid:88) r =1 A r y − z r . (C.12b)The resulting change of variables ( x, p ) ↔ ( u, v ) is known to be a change of Darboux coor-dinates. It is in fact the classical version of the separation of variables transformation forthe Schlesinger system [65]. In order to see this, let us consider in the model (B’) the case l = n − 3. In this case the equations (C.6) determine the H r as functions of the parameters( u, v ), u = ( u , . . . , u l ), v = ( v , . . . , v l ). The solutions H r ( u, v ; z ) to the constraints (C.6) arethe Hamiltonians of the Garnier system. The flows generated by the Hamiltonians H r ( u, v ; z )preserve the monodromy of the oper (cid:15) ∂ y + t ( y ).In the model (B) one may consider the Schlesinger Hamiltonians defined as H r ( x, p ; z ) := (cid:88) s (cid:54) = r η ab A ar A bs z r − z s ; (C.13)It is well-known that the non-autonomous Hamiltonian flows generated by the H r preservethe monodromy of the connection (cid:15)∂ y + A ( y ). The change of variables defined via (C.12)relates the Hamiltonians H r ( x, p ; z ) to the Hamiltonians H r ( u, v ; z ) of the Garnier system. Higher genus Considering the cases of higher genus one may introduce Darboux coordinates associated tothe model (B) as follows. To simplify the discussion slightly let us consider closed Riemannsurfaces, n = 0. Representing the bundles B as extensions (C.8), there are two places wherethe moduli may hide, in general: They may be hidden in the choice of the line bundles L (cid:48) , L (cid:48)(cid:48) ,as well as in the extension classes E ∈ H ( L − ), in terms of transition functions representedby the E ı in (C.9). A particularly simple case is found by choosing L (cid:48) = O and L (cid:48)(cid:48) ≡ L in(C.8), with L being a fixed line bundle of degree 2 g − 2. Fixing L is equivalent to fixing thedeterminant of B . – 39 –he dimension of the space of extension classes is then dim( H ( L − )) = g − L ) =3 g − 3. The moduli of Bun G can therefore be parametrized by the choices of extension classes.Coordinates x = ( x , . . . , x g − ) on H ( L − ) give coordinates for Bun G .Serre duality implies that the dual of H ( L − ) is the space H ( L ⊗ K C ). Recall thatthe lower left matrix element A − ( y ) of an (cid:15) -connection (cid:15)∂ y + A ( y ) is a section of the linebundle L ⊗ K C . Finding coordinates for H ( L ⊗ K C ) that are dual to the coordinates x on H ( L − ) with respect to the pairing provided by Serre duality will therefore give uscoordinates p = ( p , . . . , p g − ) that are canonically conjugate to the coordinates x on Bun G . C.3 Complex-structure independent Darboux coordinates Representing elements of M flat ( C ) in terms of the model (A) mentioned above allows one tointroduce useful Darboux coordinates which do not depend on a choice of complex structure of C as opposed to the coordinates ( u, v ) and ( x, p ) introduced before. A convenient descriptionwas given in [66] and references therein.Let us use the set-up from Section 2.1. A trivalent 34 21 Figure 5: Pants decomposition offour-holed sphere with a numbering ofboundary components. graph σ on C determines a pants decomposition de-fined by cutting along the simple closed curves γ e whichintersect the edge e of σ exactly once. For each (ori-ented) edge e we shall denote γ e,s ≡ γ e , γ e,t and γ e,u thesimple closed curves which encircle the pairs of bound-ary components ( γ e, , γ e, ), ( γ e, , γ e, ) and ( γ e, , γ e, ),respectively, with labeling of boundary components in-troduced via Figure 5. Let L e,i := tr( ρ ( γ e,i )) for i ∈{ s, t, u, , , , } . One may represent L e,s , L e,t and L e,u in terms of Darboux coordinates a e and k e which havePoisson bracket { a e , k e (cid:48) } = (cid:15) (2 π ) δ e,e (cid:48) . (C.14)The expressions are L e,s = 2 cosh(2 πa e /(cid:15) ) , (C.15a) L e,t (cid:0) ( L e,s ) − (cid:1) = 2( L e, L e, + L e, L e, ) + L e,s ( L e, L e, + L e, L e, ) (C.15b)+ 2 cosh(2 πk e /(cid:15) ) (cid:113) c ( L e,s ) c ( L e,s ) ,L e,u (cid:0) ( L e,s ) − (cid:1) = 2( L e, L e, + L e, L e, ) + L e,s ( L e, L e, + L e, L e, ) (C.15c)+ 2 cosh( π (2 k e − a e ) /(cid:15) ) (cid:113) c ( L e,s ) c ( L e,s ) , where c ij ( L s ) is defined as c ij ( L s ) = L s + L i + L j + L s L i L j − . (C.16)Restricting these Darboux coordinates to the Teichm¨uller component we recover the Fenchel-Nielsen length-twist coordinates well-known in hyperbolic geometry.– 40 – .4 Limit (cid:15) → : Recovering the Higgs pairs We now want to send (cid:15) → 0. One may note that the equation ( (cid:15)∂ y + A ( y )) ψ ( y ; x, z ) can inthe limit (cid:15) be solved to leading order in (cid:15) by an ansatz of the form ψ ( y ; x, z ) = e − (cid:15) (cid:82) y du v ( u ) χ ( y ; x, z ) , (C.17)where χ ( y ; x, z ) is an eigenvector of A ( y ) with eigenvalue v , A ( y ) χ ( y ; x, z ) = v ( y ) χ ( y ; x, z ) . (C.18)The function v ( y ) representing the eigenvalue of A ( y ) must satisfy v + t ( y ) = 0, where t ( y ) = − 12 tr( A ( y )) . (C.19)Using t ( y ) we define the Seiberg-Witten curve as usual byΣ = { ( v, u ) | v + t ( u ) = 0 } . (C.20)Two linearly independent eigenvectors of A ( y ) are given by χ ± ( y ; x, z ) = (cid:18) A ( y ) ± vA − ( y ) (cid:19) . (C.21)One of χ ± ( y ; x, z ) vanishes at the zeros u k of A − ( y ). It easily follows from these observationsthat the coordinates ( x, p ) and ( u, v ) for M (cid:15) H ( C ) turn into the coordinates for M H ( C ) usedin the main text when (cid:15) → a e and k e are in the limit (cid:15) → v on Σ. Given a canonical basis B = { α , . . . , α h ; α D , . . . , α D h } for H (cid:48) (Σ , Z ) = H (Σ , Z ) /H ( C, Z ) one may define the corresponding periods as a i = 12 π (cid:90) α i v , a D i = 12 π (cid:90) α D i v . (C.22)For given pants decomposition σ one may find a basis B σ with the following property: Foreach edge e of σ there exists an index i e ∈ { , . . . , h } such that the functions a i e and a D i e defined in (C.22) represent the limits (cid:15) → a e and k e defined via (C.15),respectively.The coordinates a = ( a , . . . , a h ) may be completed into a system of Darboux coordinates( a, t ) for M H ( C ) by introducing the coordinates t = ( t , . . . , t h ) using a variant of the Abelmap defined as t k = − d (cid:88) l =1 (cid:90) u l ω k , (C.23)where ω k , k = 1 , . . . , h are the Abelian differentials of the first kind on the spectral curveΣ which are dual to the differentials α i in the sense that (cid:82) α i ω k = δ ik . The functions t r – 41 –epresent coordinates on the Prym variety. The fact that the coordinates ( a, t ) representDarboux coordinates for M H ( C ) follows from the fact that (cid:102) W L ( a, u, z ) = − d (cid:88) l =1 (cid:90) u l v , (C.24)is a generating function for the change of coordinates ( u, v ) ↔ ( a, t ). Indeed, note that ω k := 12 π ∂∂a k v , (C.25)is an abelian differential on Σ satisfying (cid:82) α i ω k = δ ik as a consequence of (C.22). We maytherefore conclude that (cid:102) W L ( a, u, z ) satisfies12 π ∂∂a k (cid:102) W L ( a, u, z ) = t k , ∂∂u k (cid:102) W L ( a, u, z ) = − v k , (C.26)identifying (cid:102) W L ( a, u, z ) as the generating function for the change of coordinates ( u, v ) ↔ ( a, t ). D. Classical limits of conformal field theory We had in the main text introduced chiral partition functions Z L ( β, u, τ ; b ) and Z WZ ( j, x, τ ; k )in Liouville theory and the WZWN model respectively. It will be helpful to parametrize therepresentation labels β and j appearing in the arguments of the functions Z L ( β, u, τ ; b ) and Z WZ ( j, x, τ ; k ) as β e = Q a e √ (cid:15) (cid:15) , b = (cid:15) (cid:15) , (D.1) j e = − 12 + i a e (cid:15) , k + 2 = − (cid:15) (cid:15) . (D.2)Using this parametrization allows us to introduce chiral partition functions Z L ( a, u, τ ; (cid:15) , (cid:15) )and Z WZ ( a, x, τ ; (cid:15) , (cid:15) ) depending on two parameters (cid:15) and (cid:15) . We may therefore define twodifferent classical limits of Liouville theory and the SL (2)-WZW model by sending (cid:15) or (cid:15) to zero, respectively. We are interested in the limit where both (cid:15) and (cid:15) are sent to zero, butit helps to first study the limit (cid:15) → (cid:15) finite before sending (cid:15) → 0. After sending (cid:15) to zero we will find a relation to the moduli space M (cid:15) ( C ) of (cid:15) -connections.The two cases related to Virasoro and Kac-Moody algebra, respectively, can be treatedin very similar ways. In each of these cases we will show that the leading asymptotic behaviorof the chiral partition functions,log Z WZ σ ( a, x, τ ; (cid:15) , (cid:15) ) ∼ − (cid:15) Y WZ σ ( a, x, τ ; (cid:15) ) , log Z L σ ( a, u, τ ; (cid:15) , (cid:15) ) ∼ − (cid:15) Y L σ ( a, u, τ ; (cid:15) ) (D.3)– 42 –s represented by functions Y WZ ( a, x, τ ; (cid:15) ) and Y L ( a, u, τ ; (cid:15) ), which are generating functionsfor the changes of Darboux variables ( x, p ) ↔ ( a, k ) and ( u, v ) ↔ ( a, k ) for M (cid:15) ( C ), respec-tively.The dependence on the variables x (resp. u ) will be controlled by the partial differen-tial equations satisfied by Z WZ ( a, x, τ ; (cid:15) , (cid:15) ) (resp. Z L ( a, u, τ ; (cid:15) , (cid:15) )), known as Knizhnik-Zamolodchikov-Bernard (KZB) and Belavin-Polyakov-Zamolodchikov (BPZ) equations. Inorder to control the dependence on the variables a in both cases the crucial tool will be theVerlinde loop operators defined by integrating the parallel transport defined by KZB- andBPZ-equations, respectively. The Verlinde loop operators can be represented as difference op-erators acting on the a -variables. The limit (cid:15) → a -dependence of Y WZ ( a, x, τ ; (cid:15) ) and Y L ( a, u, τ ; (cid:15) ). The following discussion considerably refines the previous observations [67, 68]by supplementing the “other side of the coin” represented by the Verlinde loop operators.To simplify the exposition we will spell out the relevant arguments only in the case when C has genus zero. The dependence on the complex structure of C may then be described usingthe positions z = ( z , . . . , z n ) of the marked points. We will therefore replace the parameters τ by the variables z in the following. The generalization of this analysis to higher genusRiemann surfaces will not be too hard. D.1 Preparations: Insertions of degenerate fields It will be useful to modify the conformal blocks by inserting a variable number of m extradegenerate fields at position y = ( y , . . . , y m ). WZW model We will consider conformal blocks of the form Z WZ ( w, y ; x, z ) := (cid:10) Φ j n ( x n | z n ) . . . Φ j ( x | z ) Φ ( w m | y m ) . . . Φ ( w | y ) (cid:11) C,ϕ . (D.4)We will impose the “null vector decoupling” equation on the degenerate field Φ ( w | y ): ∂ w Φ ( w | y ) = 0 , (D.5)which means that Φ +(2 , ( w | y ) transforms in the two-dimensional representation C (cid:39) C [ w ] / ( w )of sl . It follows that Z WZ ( w, y ; x, z ) defines an element Ψ WZ ( y ; x, z ) of ( C ) ⊗ m .The corresponding chiral partition functions Ψ WZ ( y ; x, z ) satisfy additional first order dif-ferential equations governing the y -dependence which will be formulated explicitly below. Thefamily of chiral partition functions obtained in this way represents a convenient repackaging ofthe information contained in the chiral partition function Z WZ ( x, z ) without extra degeneratefields ( m = 0). The chiral partition functions Z WZ ( x, z ) essentially represent the bound-ary conditions for the integration of the differential equations governing the y -dependence of– 43 – WZ ( y ; x, z ). One may recover Z WZ ( x, z ) from the family of Ψ WZ ( y ; x, z ) by taking suitablelimits. The presence of extra degenerate fields modifies the KZ-equations as − (cid:15) (cid:15) ∂∂z r Ψ WZ ( y ; x, z ) = n (cid:88) r (cid:48) =1 r (cid:48) (cid:54) = r η aa (cid:48) J ar J a (cid:48) r (cid:48) z r − z r (cid:48) Ψ WZ ( y ; x, z ) + m (cid:88) s =1 η aa (cid:48) J ar t a (cid:48) s z r − y s Ψ WZ ( y ; x, z ) , (D.6)where t as denote the matrices representing sl on the s-th tensor factor of ( C ) ⊗ m , and J ar arethe differential operators introduced in (B.3). In addition we get the following m differentialequations: − (cid:15) (cid:15) ∂∂y s Ψ WZ ( y ; x, z ) = n (cid:88) r =1 η aa (cid:48) t as J a (cid:48) r y s − z r Ψ WZ ( y ; x, z ) + n (cid:88) s =1 s (cid:48) (cid:54) = s η aa (cid:48) t as t a (cid:48) s (cid:48) z s − z s (cid:48) Ψ WZ ( y ; x, z ) , (D.7)The space of solutions to the equations (D.7) is determined by the space of conformal blockswithout extra degenerate fields m = 0. This follows from the fact that one may regard thepartition function Z WZ ( x, z ) as initial values for the solution of (D.7). One may, on the otherhand, recover the partition functions Z WZ ( x, z ) by considering even m and taking a limitwhere the insertion points y s collide pairwise. Liouville theory The situation is similar in the case of Liouville theory. In the presence of m degenerate fieldsof weight − − b − and l degenerate fields of weight − − b the chiral partition functionswill satisfy l BPZ equations (D.9a) We shall consider the Liouville conformal blocks Z L ( y ; u, z ) ≡ (cid:42) n (cid:89) r =1 e α r φ ( z r ) m (cid:89) s =1 e − bφ ( y s ) l (cid:89) k =1 e − b φ ( u k ) (cid:43) C,ϕ . (D.8)The conformal blocks (D.8) satisfy the null vector decoupling equations (cid:32) b ∂ ∂u k + n (cid:88) r =1 (cid:18) ∆ r ( u k − z r ) + 1 u k − z r ∂∂z r (cid:19) − m (cid:88) s =1 (cid:18) b + 24( u k − y s ) − u k − y s ∂∂y s (cid:19) − l (cid:88) k (cid:48) =1 k (cid:48) (cid:54) = k (cid:18) b − + 24( u k − u k (cid:48) ) − u k − u k (cid:48) ∂∂u k (cid:48) (cid:19) (cid:33) Z L ( y ; u, z ) = 0 , (D.9a) (cid:32) b ∂ ∂y s + n (cid:88) s =1 (cid:18) ∆ r ( y s − z r ) + 1 y s − z r ∂∂z r (cid:19) − l (cid:88) k =1 (cid:18) b − + 24( y s − u k ) − y s − u k ∂∂u k (cid:19) − m (cid:88) s (cid:48) =1 s (cid:48) (cid:54) = s (cid:18) b + 24( y s − y s (cid:48) ) − y s − y s (cid:48) ∂∂u s (cid:48) (cid:19) (cid:33) Z L ( y ; u, z ) = 0 . (D.9b)Equations (D.9) imply the fusion rules[ V − b/ ] · [ V α ] ∼ [ V α − b/ ] + [ V α − b/ ] , (D.10)[ V − / b ] · [ V α ] ∼ [ V α − / b ] + [ V α − / b ] . (D.11)– 44 – .2 Limit (cid:15) → (cid:15) → WZW-model In order to study the limit (cid:15) → (cid:15) and (D.6) by (cid:15) . Onemay solve the system of equation (D.6) and (D.7) with the following ansatz,Ψ WZ ( y ; x, z ) = e − (cid:15) Y WZ ( x,z ) n (cid:79) s =1 ψ ( y s ; x, z ) (cid:0) O ( (cid:15) ) (cid:1) , (D.12)which will yield a solution to (D.7) provided ψ ( y ; x, z ) and Y WZ ( x, z ) satisfy the followingsystem of equations: (cid:18) (cid:15) ∂∂y + A ( y ) (cid:19) ψ ( y ; x, z ) = 0 , (D.13a)where A ( y ) = n (cid:88) r =1 η aa (cid:48) t as A a (cid:48) r y − z r , A r = (cid:18) x r p r − l r l r x r − x r p r p r l r − x r p r (cid:19) (D.13b) p r = − ∂∂x r Y WZ ( x, z ) . (D.13c)We recognize model (B) for the flat connections. The limit of (D.6) yields in addition H r := (cid:15) ∂∂z r Y WZ ( x, z ) = n (cid:88) r (cid:48) =1 r (cid:48) (cid:54) = r η aa (cid:48) A ar A a (cid:48) r (cid:48) z r − z r (cid:48) . (D.13d)These equations characterize the Hamiltonians of the Schlesinger system. We have therebyreproduced results of [67, 68]. Liouville theory In order to study the limit (cid:15) → (cid:15) (cid:15) . Onemay solve the system of equation (D.9a) and (D.9b) with the following ansatz,Ψ L ( y ; u, z ) = e − (cid:15) Y L ( u,z ) n (cid:89) s =1 ψ L ( y ; u, z ) (cid:0) O ( (cid:15) ) (cid:1) , (D.14)which will yield a solution (D.9b) provided ψ L ( y ; u, z ) and Y L ( u, z ) satisfy the following systemof equations: (cid:18) (cid:15) ∂ ∂y s + t ( y s ) (cid:19) ψ L ( y ; u, z ) = 0 , (D.15a)– 45 –here t ( y ) = n (cid:88) s =1 (cid:18) δ r ( y s − z r ) − H r y s − z r (cid:19) − (cid:15) l (cid:88) k =1 (cid:18) (cid:15) y s − u k ) − v k y s − u k (cid:19) , (D.15b) v k = − ∂∂u k Y L ( u, z ) , δ r = (cid:15) (cid:15) ∆ r , (D.15c)The equations (D.9a) yield in addition v k + t k, = 0 , t ( y ) = ∞ (cid:88) l =0 t k,l ( y − u k ) l − , (D.16a) H r = (cid:15) ∂∂z r Y L ( u, z ) , (D.16b)These equations define the Hamiltonians of the Garnier system. D.3 Verlinde loop operators The dependence of the chiral partition function on the variables a is controlled by the Verlindeloop operators. They are defined by modifying a conformal block by inserting the vacuumrepresentation in the form of a pair of degenerate fields, calculating the monodromy of oneof them along a closed curve γ on C , and projecting back to the vacuum representation,see [9, 69] for more details. A generating set is identified using pants decompositions.The calculation of the Verlinde loop operators is almost a straightforward extensionof what has been done in the literature. The necessary results have been obtained in [9,69] for Liouville theory without extra insertions of degenerate fields V − b/ ( y ). It would bestraightforward to generalize these observations to the cases of our interest. For the caseof Kac-Moody conformal blocks one could assemble the results from the known fusion andbraiding matrices of an extra degenerate field Φ ( w, y ). As a shortcut let us note, however,that the results relevant for the problem of our interest, the limit (cid:15) → 0, can be obtained ina simpler way.One may start on the Liouville side. The key observation to be made is the fact that thepresence of extra degenerate fields V − / b ( y ) modifies the monodromies of V − b/ ( y ) only byoverall signs, as the monodromy of V − b/ ( y ) around V − / b ( u k ) is equal to minus the identity.It is useful to observe (see Appendix E.4) that the separation of variables transformation mapsthe degenerate field Φ ( w, y ) to the degenerate field V − b/ ( y ). It follows that the monodromiesof Φ ( w, y ) must coincide with the monodromies of V − b/ ( y ) up to signs. Using the resultsof [9, 69] we conclude that( π V ( γ e,s ) Z WZ )( a, u, z ) = ν e,s L e,s · Z WZ ( a, u, z ) , ( π V ( γ e,t ) Z WZ )( a, u, z ) = ν e,t L e,t · Z WZ ( a, u, z ) , (D.17)– 46 –here ν e,s ∈ {± } and ν e,r ∈ {± } , while the explicit expressions for the difference operators L e,s , L e,t are L e,s = 2 cosh(2 π a e /(cid:15) ) . (D.18a) L e,t = 2 cos( π(cid:15) /(cid:15) )( L e, L e, + L e, L e, ) + L e,s ( L e, L e, + L e, L e, )2 sinh (cid:0) π(cid:15) ( a e + i2 (cid:15) ) (cid:1) (cid:0) π(cid:15) ( a e − i2 (cid:15) ) (cid:1) (D.18b)+ (cid:88) ξ = ± (cid:112) π a e /(cid:15) ) e πξ k e /(cid:15) (cid:112) c ( L r,s ) c ( L r,s )2 sinh(2 π a e /(cid:15) ) e πξ k e /(cid:15) (cid:112) π a e /(cid:15) ) , using the notation c ij ( L e,s ) = L e,s + L e,i + L e,j + L e,s L e,i L e,j − 4, and k e = (cid:15) (cid:15) π i ∂∂a e . (D.19)As the KZB-equations (D.7) turn into the horizontality condition (D.13a), the Verlindeloop operators will turn into trace functions when (cid:15) → 0. The limit of the left hand side of(D.17) is therefore found by replacing π V ( γ e,s ) and π V ( γ e,t ) with the expressions in (C.15),calculated from the connection A ( y ) appearing in (D.13a). Note that the connection A ( y ) isthereby defined as a function of the parameters x and a . The limit (cid:15) → Y WZ with respect to the variable a . In this way one finds thatthe the limit (cid:15) → k e ( a, u ) = (cid:15) i2 π ∂∂a e Y WZ ( a, u, z ) . (D.20)Equation (D.20) identifies Y WZ ( a, u, z ) as the generating function for the change of variables( x, p ) ↔ ( a, k ). The analysis in the Liouville case is very similar. D.4 Limit (cid:15) → (cid:15) → Y WZ ( a, x, z ; (cid:15) ) and Y L ( a, u, z ; (cid:15) ).We claim that in the two cases we find a behavior of the form Y WZ ( a, x, z ) ∼ (cid:15) F WZ ( a, z ) + (cid:102) W WZ ( a, x, z ) + . . . , (D.21) Y L ( a, u, z ) ∼ (cid:15) F L ( a, z ) + (cid:102) W L ( a, u, z ) + . . . , (D.22)where F WZ ( a, z ) = F L ( a, z ), while (cid:102) W WZ ( a, x, z ) and (cid:102) W L ( a, x, z ) are the generating functionsfor the changes of variables ( x, p ) ↔ ( a, t ) and ( u, v ) ↔ ( a, t ), respectively.We begin by considering (D.22). The equation (D.15) can be solved to leading order bya WKB-ansatz ψ L ( y ; u, z ) (cid:16) e − (cid:15) (cid:82) y du v ( u ) , ( v ( y )) + t ( y ) = 0 . (D.23)– 47 –he asymptotics of the generating function Y L ( a, u ; z ) which coincides with the classicalLiouville conformal blocks will be of the form Y L ( a, u, z ) ∼ (cid:15) F L ( a, z ) + (cid:102) W L ( a, u, z ) + . . . , . (D.24)Indeed, an expansion of the form will satisfy (D.16) and (D.20) if F L ( a, z ) satisfies ∂∂z r F L ( a, z ) = H r , i2 π ∂∂a e F L ( a, z ) = a D e , (D.25)identifying F ( a, z ) as the prepotential, and if furthermore ∂∂u k (cid:102) W L ( a, u, z ) = − v k . (D.26)This means that (cid:102) W L ( a, u ; z ) = − d (cid:88) l =1 (cid:90) u l v . Following the discussion in Appendix C.4 we may identify (cid:102) W L ( a, u ; z ) as the generatingfunction of the standard change of Darboux variables ( u, v ) ↔ ( a, t ) which is defined by theAbel map.The corresponding statement for (cid:102) W WZ ( a, x ; z ) now follows easily from (D.13c), and thefact that Y WZ ( a, x, z ) and Y L ( a, u, z ) differ only by the generating function Y SOV ( x ; u, z ) forthe change of Darboux variables ( x, p ) ↔ ( u, v ) which does not depend on a . E. Explicit relation between Kac-Moody and Virasoro conformal blocks We will explain in this appendix how to obtain an explicit integral transformation betweenthe conformal blocks in Liouville theory and in the WZW model using the observations madein Section 4.7. This is the separation of variables (SOV) relation (1.1) which we discussed inthe Introduction. E.1 SOV transformation for conformal blocks In order to partially fix the global sl -constraints we shall send z n → ∞ and x n → ∞ ,defining the reduced conformal blocks ˇ Z WZ ( x, z ) which depend on x = ( x , . . . , x n − ) and z = ( z , . . . , z n − ). Let (cid:101) Z WZ ( µ, z ) be the Fourier-transformation of the reduced conformalblock ˇ Z WZ ( x, z ) of the WZW model w.r.t. the variables x . It depends on µ = ( µ , . . . , µ n − )subject to (cid:80) n − r =1 µ r = 0. There then exists a solution Z L ( y, z ) to the BPZ-equations D BPZ u k · Z L = 0 , ∀ k = 1 , . . . , l, (E.1)with differential operators D BPZ u k given as D BPZ u k = b ∂ ∂u k + n (cid:88) r =1 (cid:18) ∆ r ( u k − z r ) + 1 u k − z r ∂∂z r (cid:19) − l (cid:88) k (cid:48) =1 k (cid:48) (cid:54) = k (cid:18) b − + 24( u k − u k (cid:48) ) − u k − u k (cid:48) ∂∂u k (cid:48) (cid:19) , – 48 –uch that the following relation holds (cid:101) Z WZ ( µ, z ) = u δ (cid:0)(cid:80) n − i =1 µ i (cid:1) Θ n ( y, z ) Z L ( y, z ) . (E.2)The function Θ n ( y, z ) that appears in this relation is defined asΘ n ( y, z ) = (cid:89) r We want to write the expression for ˇ Z WZ ( x, z )ˇ Z WZ ( x, z ) = (cid:90) dµ µ . . . dµ n − µ n − δ (cid:0)(cid:80) n − r =1 µ r (cid:1) Θ n ( u, x ) Z L ( u, z ) n − (cid:89) r =1 µ − j r r e iµ r x r , (E.6)as explicitly as possible. To this aim let us note first that µ r ( u ) = u λ r ( u ) , λ r ( u ) := (cid:81) n − k =1 ( z r − u k ) (cid:81) n − s (cid:54) = r ( z r − z s ) , (E.7)and furthermore dµ µ . . . dµ n − µ n − δ (cid:0)(cid:80) n − r =1 µ r (cid:1) Θ n ( u | x ) = du u dν ( u ) , (E.8) dν ( u ) := du . . . du n − n − (cid:89) r (cid:54) = s ( z r − z s ) b n − (cid:89) r =1 n − (cid:89) k =1 ( z r − u k ) − − b n − (cid:89) k Now we consider semiclassical limit (cid:15) , (cid:15) → 0, setting α r = ( (cid:15) (cid:15) ) − l r . (E.13)We have then log K SOV ( x, u ) = (cid:15) − (cid:102) W SOV ( x, u ) + O ( (cid:15) ) , (E.14)– 50 –ith (cid:102) W SOV ( x, u ) = κ log (cid:34) n − (cid:88) r =1 x r (cid:81) n − k =1 ( z r − u k ) (cid:81) n − s (cid:54) = r ( z r − z s ) (cid:35) (E.15)+ n − (cid:88) r =1 l r (cid:34) n − (cid:88) s (cid:54) = r log( z r − z s ) − n − (cid:88) k =1 log( z r − u k ) (cid:35) . We have denoted κ := − l n + (cid:80) n − r =1 l r . If we send only (cid:15) → 0, we get a modified result:log K SOV ( x, u ) = (cid:15) − (cid:102) W SOV ( x, u ; (cid:15) ) + O ( (cid:15) ) (E.16)with (cid:102) W SOV ( x, u ; (cid:15) ) = κ log (cid:34) n − (cid:88) r =1 x r (cid:81) n − k =1 ( z r − u k ) (cid:81) n − s (cid:54) = r ( z r − z s ) (cid:35) + (cid:15) n − (cid:88) k We also use the version of this correspondence in the presence of the fields Φ ( w | y ), asappear in (D.4). This is kind of interesting. Note that the Fourier-transformation of the nullvector equations ∂ w Φ ( w | y ) = 0 gives µ (cid:101) Φ ( µ | y ) = 0. This indicates that conformal blockscontaining (cid:101) Φ ( µ | y ) must be understood as distributions with support at µ = 0. If we send µ r → t = z r on the left hand side.This means that one u k must approach z r in order to cancel the pole at t = z r on the righthand side of (E.4). 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