Non-Perturbative Topological Strings And Conformal Blocks
IITFA-2010-21
Non-Perturbative Topological StringsAnd Conformal Blocks
Miranda C. N. Cheng (cid:91),(cid:92) , Robbert Dijkgraaf (cid:93) and Cumrun Vafa ♦∗ (cid:91) Department of Mathematics, Harvard University,Cambridge, MA 02138, USA (cid:92)
Department of Physics, Harvard University,Cambridge, MA 02138, USA (cid:93)
Institute for Theoretical Physics & KdV Institute for Mathematics,University of Amsterdam, The Netherlands ♦ Center for Theoretical Physics, MIT,Cambridge, MA 02139, USA
Abstract
We give a non-perturbative completion of a class of closed topo-logical string theories in terms of building blocks of dual open strings.In the specific case where the open string is given by a matrix modelthese blocks correspond to a choice of integration contour. We thenapply this definition to the AGT setup where the dual matrix modelhas logarithmic potential and is conjecturally equivalent to Liouvilleconformal field theory. By studying the natural contours of these ma-trix integrals and their monodromy properties, we propose a precisemap between topological string blocks and Liouville conformal blocks.Remarkably, this description makes use of the light-cone diagrams ofclosed string field theory, where the critical points of the matrix po-tential correspond to string interaction points. ∗ On leave from Harvard University. a r X i v : . [ h e p - t h ] O c t ontents Topological string amplitudes have played an important role in understand-ing quantum protected amplitudes in string theory and supersymmetric fieldtheories. More precisely, they capture the F-terms of the corresponding su-persymmetric theories. Topological string amplitudes are indexed by thegenus of the worldsheet, and amplitudes of different genera compute differ-ent physical quantities. For example, the genus zero amplitudes correspondto F-terms in supersymmetric field theories, such as the Yukawa couplingsand the superpotential terms. The higher genus amplitudes, on the otherhand, involve gravitational corrections. This raises the following question:Does topological string theory have a non-perturbative meaning that goesbeyong this term by term identification? And, if so, what is the physicalquantity that the full partition function computes?Various dualities [1, 2, 3] relate the A- and B-model topological string am-plitudes to quantities in complex Chern-Simons theories and matrix models1espectively. A natural question is hence whether these alternative formula-tions can help us to find a non-perturbative definition of topological strings.Indeed, both matrix models [4, 5] and Chern-Simons amplitudes [6, 7, 8]do admit non-perturbative definitions, and it is thus natural to ask what themeanings of these non-perturbative completions are in the context of topolog-ical string and superstring theories. One feature of such definitions is that, fora given perturbative definition there does not exist a unique non-perturbativecompletion. Instead there are discrete choices, which we will denote collec-tively by A , to be made. For each such choice A there is a non-perturbativelydefined partition function Z top A and we are interested in the meaning of theseamplitudes Z top A in the superstring theory. Recalling that topological stringamplitudes are interpreted as computing partition functions of superstringtheories in specific backgrounds which involve the Taub-NUT spacetime or a2-dimensional subspace of it, we argue that the choice of A translates into thechoice of boundary condition for this non-compact spacetime, i.e. a choiceof spacetime branes.One main motivation for revisiting this question is related to the AGTcorrespondence [9] that connects the Nekrasov partition function of four-dimensional N = 2 theories to the conformal blocks of Liouville (and moregenerally Toda [10]) theories. These conformal blocks Z CFT B are labeled byparameters specifying the intermediate channels that we denote collectivelyby B . On the other hand, using geometric engineering of the N = 2 theoriesin the superstring setup, it was shown [11] that these same amplitudes shouldalso correspond to topological string amplitudes represented by matrix modelwith Penner-like logarithmic potentials. A non-perturbative completion oftopological string amplitudes can hence be applied to this special case andshould make precise the connection to the conformal blocks of the Liouvilletheory. In particular, a question to be answered is how the choices A of thenon-perturbative topological blocks should be mapped to the choices B ofconformal blocks.In this paper we argue that the conformal blocks Z CFT B are in fact a linearcombination of the non-perturbative string partition functions Z top A : Z CFT B = (cid:88) A C B , A Z top A , where C B , A are certain constants. By giving such a map, we specify the pre-cise relationship between the Liouville conformal blocks and non-perturbativematrix model amplitudes. 2his relation between the conformal blocks of Liouville theory and theperiods of the Penner-like matrix integral is one of the interesting by-productsof the AGT correspondence. It has been discussed among others in [12, 13, 14,15]. Recently it was also considered in the case of genus one surfaces [16, 17].The gauge theory aspect of the relation has been further investigated in, forexample, [18, 19]. Of course, this problem is directly related to a classicalsubject in conformal field theory: the contour prescription of the screeningcharges for minimal models `a la Dotsenko-Fateev, see e.g. [20, 21].As an interesting aspect of defining the topological string blocks Z top A forPenner-like matrix models, we find that the eigenvalues of the matrix arebest represented as points on a light-cone parametrization of a genus zeroworldsheet. In this light-cone diagram, the incoming and outgoing tubescorrespond to the positions of impurities in the Penner-like potential. Thelight-cone time X + is identified with the (real part of the) matrix modelpotential. In particular, the critical points of the matrix model potential aremapped to the interaction points in the light-cone diagram and vice versa.The basic integration contours of the matrix model are given by straightlines on the light-cone diagram, emanating from the interaction points andgoing backwards in the light-cone time X + . This connection with light-cone string diagrams is very intriguing and suggests a potentially importantconnection between light-cone string field theory and N = 2 amplitudes infour dimensions.The organization of this paper is as follows: In section 2 we formulatea non-perturbative completion of topological string and its interpretation interms of superstring amplitudes. This non-perturbative definition can beapplied to all topological string theories with a dual open string description.In section 3 we apply this definition to specific topological string theoriesthat are dual to matrix models with Penner-like logarithmic potential. Insection 4 we briefly review the content of the AGT correspondence and itsrelation to the matrix model, and in section 5 we give a dictionary betweenthe parameters in the matrix model and those of Liouville conformal blocks.In this context, in section 6 we propose a map between the non-perturbativeblocks of topological strings and the blocks of the 2d Liouville CFT. In section7 we give examples which illustrate this correspondence. In section 8 weconclude with some discussion on future directions. In the Appendix weexplicitly demonstrate the relation between degenerate four-point conformalblocks and the corresponding matrix model expressions.3 Non-Perturbative Topological String Blocks
Topological string amplitudes have an interpretation as computing certainphysical amplitudes in superstring theories in the presence of branes. Thesimplest situation where this relation is realized is the following: the A-modeltopological string amplitude computes the partition function of M-theory inthe background X × ( S × T N ) q , where X is a Calabi-Yau threefold and ( S × T N ) q denotes the space obtainedby rotating the circle symmetry of the Taub-NUT space by an angle q = exp(2 πig s )as one goes around S [22]. We can extend this dictionary between the phys-ical and the topological theory to the open topological sectors by introducingA-branes wrapping a Lagrangian subspace L ⊂ X . In the M-theory setup,adding a topological A-brane corresponds to adding an M5 brane wrappingthe subspace L × ( S × C ) q ⊂ X × ( S × T N ) q , where C is the two-dimensional cigar subspace of T N [23, 24, 25]. Thegeometry ( S × C ) q = M C q is the so-called ‘Melvin Cigar’ in [24].Reducing on the S , we can view the M-theory system from the per-spective of type IIA superstring theory and obtain a two-dimensional theoryon C with N = 2 supersymmetry. The chiral degrees of freedom for thistwo-dimensional N = 2 theory are associated with the gauge connection A on the Lagrangian submanifold L . Moreover, the superpotential of thetwo-dimensional theory is simply given by the Chern-Simons action, up toworldsheet instanton corrections: W ( A ) /g s = 1 g s (cid:90) L Tr(
AdA + 23 A ) + instanton corrections . In other words, we have Z top = (cid:90) DA exp( W ( A ) /g s ) . Naively there may appear to be some tension between the following twofacts: in type IIA superstring theory W is the superpotential, whereas here4t appears as the action in the topological theory! However, it is known thatin order to preserve supersymmetries in two-dimensional N = 2 theories withboundaries, we need to add to the action boundary terms which are exactlygiven by the integral of the superpotential W evaluated at the boundary [26].In the context of the cigar geometry this has been discussed in [27]. Theboundary of the cigar C is a circle, and the path integral localizes on thethe field configurations that are constant on it. Hence we arrive at the sameformula as above for the physical partition function: Z top = Z IIA . By mirror symmetry we expect a similar relation between the physicaland the topological quantities to hold for the case of the B-model topologicalstring. In this case we have D3-branes wrapping holomorphic curves andfilling the cigar subspace of the Taub-NUT geometry. In particular, we canconsider a local Calabi-Yau geometry given by a hypersurface in C of theform y + W (cid:48) ( x ) + uv = 0with D3-branes wrapped on the holomorphic curves that are described bythe conifold-type geometries localized at critical points of W ( x ). In this caseit is known [3] that the B-model reduces to a matrix model with potential W (Φ): Z top = Z IIB = (cid:90) D Φ e Tr W (Φ) /g s . (2.1)Under this correspondence the rank N of the matrix Φ is equal to the totalnumber of branes.We will be interested in an expansion near the saddle point in the large N semi-classical limit. A saddle point is specified by a distribution of the eigen-values among the critical points p , · · · , p n of the potential W , labeled by thefilling fractions N , · · · , N n satisfying (cid:80) N (cid:96) = N . This distribution has theinterpretation as the number of the branes populating different holomorphiccycles.However, there is an intrinsic incompleteness in the above statement. Theparameters of the superpotential W (Φ) are complex and the integration overΦ is holomorphic. Hence, even though the above recipe specifies a pertur-bative expansion, it does not give an unambiguous non-perturbative answer.To remedy this problem, one should specify a contour for the matrix inte-gral. In fact, this ambiguity is matched by a corresponding ambiguity in the5 = 2 theories in two dimensions. In that context, the ambiguity lies in thechoice of a supersymmetric boundary condition. As noted in [27], to specifya boundary condition for the N = 2 theory in two dimensions, we need tochoose a Lagrangian subspace in the field space. The allowed Lagrangiansubmanifolds should satisfy the following conditions. First, the projectionsof the submanifolds in the field space onto the W -plane are straight linesemanating from the critical values W ( p (cid:96) ). Moreover, to ensure the integralis well-defined when the submanifolds have boundaries, we require that | exp( W/g s ) | → Re(
W/g s ) ··· W ( p ) W ( p n ) W Figure 1:
A natural set of contours ˜ C , · · · , ˜ C n is given by the pre-images of the straightlines on the W -plane. The slope of the straight lines is given by the downward gradientflow of Re ( W/g s ) . A way to visualize these straight lines on the original Riemann surface isvia the following quasi-conformal mapping ds = | dw | = | φ zz | | dz | , associated with a so-called Jenkins-Strebel holomorphic quadratic differential φ zz ( z )( dz ) given in terms of the matrix potential as φ zz ( dz ) = ( dw ) = (cid:0) g s dW (cid:1) . Clearly, such a quasi-conformal mapping defines a metric that is flat every-where on the Riemann surface except at the zeros and poles of φ zz , wherethe curvature is singular. By definition, the zeros are exactly the critical6oints of the matrix potential W ( z ) and their images under the W -map arethe points where the Lagrangian submanifolds emanate from.The choice of the Lagrangian submanifold makes precise the way the com-plex integral over the chiral fields should be performed, and as a result theanswer for the partition function depends on this choice. In other words, onboth the topological and physical sides, a choice of a Lagrangian subman-ifold is required to define the amplitude unambiguously. In the first case,the Lagrangian submanifold plays the role of the integration contour, whilein the latter case it completes the definition of the theory by providing asupersymmetric boundary condition.Let us summarize this main point that is crucial to the rest of the pa-per: Both the topological and physical amplitudes are uniquely defined once achoice of the Lagrangian submanifold is made. It is hence natural to identifythis choice of the Lagrangian submanifold as a choice of a non-perturbativecompletion of the topological string amplitudes.
In other words, for a given choice of the topological string coupling con-stant g s and the parameters of the superpotential W , the data of branedistribution N (cid:96) that characterizes the matrix model saddle points also leadsto a non-perturbative answer for the topological amplitude Z ( N , . . . , N n ) = non-perturbative topological block labeled by { N , . . . , N n } . The fact that the matrix model integrals with complex potentials can bedefined using such complex contours has been known for a long time [4,5]. More recently they were discussed in the context of a non-perturbativeformulation of topological strings in [28, 29, 30]. In the context of complexChern-Simons, the idea of associating a contour to a given classical solutionhas appeared in [6]. Moreover, similar Lagrangian contours, which are infinitedimensional in this context, also feature in the discussion on the complexChern-Simons path integral [7].However, such a ‘topological string block’ Z ( N , . . . , N n ) is not a contin-uous function of the parameters of the superpotential. In particular, as canbe seen in Figure 1, it jumps at special values of the parameters such thatthere are two critical points p (cid:96) , p (cid:96) withIm (cid:18) W ( p (cid:96) ) − W ( p (cid:96) ) g s (cid:19) = 0 . These jumps in the corresponding cycles result in the Stokes’ phenomenon forthe integral described by the Picard-Lefschetz theory, and cause non-trivial7onodromy transformations when the parameters move around in the com-plex plane. This property is of course reminiscent of that of conformal blocksin two-dimensional conformal theory. In this paper we will demonstrate howthis analogy, when suitably interpreted, can be made into an equality in thecontext of the Liouville conformal theory interpreted as topological stringtheory [11].
In the previous section we have given a definition of non-perturbative topo-logical string blocks through matrix integrals. In this section we would like toapply such a definition to Penner-like matrix models with logarithmic poten-tials. As we will review later, this type of matrix models is of special interestdue to its relation to the Liouville conformal blocks. Hence an understandingof the corresponding topological string blocks will be a prerequisite for a con-crete realization of [11], which connects certain topological string amplitudesto Liouville conformal blocks in the context of the AGT correspondence [9].To keep the discussion explicit we will focus on the genus zero conformalblocks of the SL (2) Liouville theory. Similar consideration should also applyto the more general situations of higher genus surfaces and higher rank gaugegroups discussed in [11].We are interested in the matrix models with the following type of loga-rithmic potential W ( z ) /g s = n +1 (cid:88) (cid:96) =1 m (cid:96) log( z − z (cid:96) ) , (3.1)corresponding to the matrix integral Z = (cid:90) · · · (cid:90) N (cid:89) i =1 du i (cid:89) ≤ i W/g s ), asshown in Figure 2. Such a genus zero light-cone diagram is specified by thefollowing data: the external momenta { m , . . . , m n +1 } , the times { τ , . . . , τ n } of interaction, and the twisting angles { θ , . . . , θ n − } with which an interme-diate closed string is twisted before rejoining with the rest of the diagram[32]. Fixing the initial time to be τ = 0, we see that the τ ’s and θ ’s togethergive the ( n − 1) complex variables corresponding to the locations of the poles z , . . . , z n +1 , z n +2 = ∞ up to SL (2 , C ) equivalence.In order to obtain an unambiguous and uniform description of the contourfor the matrix model integral (3.2), we will consider the potential (3.1) withall momenta m (cid:96) being real and positive. The answer for the integral can thenbe analytically continued to other ranges of the parameters. For such cases,the quasi-conformal map described above leads to light-cone diagrams of thekind depicted in Figure 2. In other words, we consider scattering diagramswith n +1 incoming and one outgoing closed string. As a result, in our matrixmodel discussion, the pole of dW at the infinity of the genus zero curve is insome sense more special than all the other poles at z , . . . , z n +1 .10 z z z = ∞ Re( W/g s ) N N n ··· ··· z = ∞ N N p p p p p n z n +1 z z , α z , α z , α z , α z n +1 , α n +1 a a a θ θ θ τ τ · · · · · · Figure 2: An example of a light-cone diagram for a given matrix potential W . Thelocations z (cid:96) of the poles of dW determine the twist angles θ (cid:96) and the interaction times τ (cid:96) ,while the residues m (cid:96) determine the size of the closed strings. Such a light-cone diagramgives a pants decomposition of the genus zero Riemann surface and thereby determinesa tree structure. Together with the data of the distribution { N , . . . , N n } of the matrixeigenvalues among the critical points { p , . . . , p n } of the matrix potential, this tree specifiesa corresponding conformal block. As mentioned in section 2, we would like to associate a saddle point, la-beled by the eigenvalue distribution { N , . . . , N n } , a matrix model integral(3.2) with a specific contour. In the language of the light-cone diagram (seeFigure 2), this means we should associate a class of contour to each interac-tion point of the strings. As discussed in section 2, there is a very naturalway to specify these contours from the matrix model point of view: they aregiven by the gradient flow of Re( W/g s ). In the present case they are simplygiven by the downward-flowing (backward in time) straight lines emanatingfrom the interaction points on the light-cone diagram. In particular, thecorresponding contours are not closed cycles.Concretely, for a given potential W of the form (3.1) and thereby a givenlight-cone diagram, label the poles z , . . . , z n +1 and zeros p , . . . , p n of thecorresponding quadratic differential in the order depicted in Figure 2. Thenthe contour associated in the abovementioned way to the (cid:96) -th critical point isthe line segment going between the point z (cid:96) and z (cid:96) +1 and passing through the11ritical point p (cid:96) . We choose the orientation and denote by ˜ C (cid:96) the line segmentgoing from z (cid:96) +1 to z (cid:96) . As a result, to a given distribution of eigenvalues N (cid:96) among the critical points p (cid:96) , we can associate a matrix integral (3.2) withthe following contour ⊗ n(cid:96) =1 ˜ C ⊗ N (cid:96) (cid:96) , ˜ C (cid:96) = [ z (cid:96) +1 , z (cid:96) ] . (3.3)As we will discuss in more details later, when the momenta m ’s are not allintegers and when we consider the β -ensembles generalization of the matrixmodel, the genus zero curve on which the matrix eigenvalues live is in factmulti-sheeted. The matrix integrals with the above prescription of contourstransform non-trivially when changing from sheet to sheet and in fact mixwith one another under such monodromy actions. One might hence deem itnatural to use an alternative basis for matrix integrals on which the mon-odromy transformations act diagonally. In fact this property is mandatoryfrom the point of view of the conformal blocks. We will construct such analternative basis and relate the corresponding matrix integrals to conformalblocks in section 6. In this section we will briefly review the relation between certain 2d conformalblocks and the Nekrasov partition functions for certain N = 2, d = 4 super-conformal field theories [9], and how this relation can be understood throughtheir connections to topological string theory and matrix model [11].In [9], the authors consider a class of N = 2, d = 4 superconformal fieldtheories, which can be obtained from compactifying the six-dimensional M5brane N = (2 , 0) superconformal field theories on punctured Riemann sur-faces Σ [33]. These theories admit an A-D-E classification. These authorspropose that, for the case of SU (2) theories corresponding to two M5 branes,the Nekrasov partition functions in the Ω-background are essentially given bycertain conformal blocks of the Liouville conformal field theory. Underlyingboth objects is the genus g Riemann surface Σ with n + 2 punctures, andits pants decomposition determines a weakly coupled Lagrangian descriptionon the gauge theory side and a specific channel on the 2d CFT side. Morespecifically, the sewing parameter of a neck connecting two pairs of pantsis related to the gauge coupling constant of the corresponding SU (2) gauge12roup. Moreover, the external momenta of the Liouville conformal block cor-respond to the hypermultiplets masses of the gauge theory, while the internalmomenta map to the Coulomb parameters. Finally, the two real numbers (cid:15) , (cid:15) characterizing the Ω-background fix the parameter b of the Liouvilletheory as b = (cid:15) /(cid:15) . Later this proposal was generalized to the A r 4d SCFT/2d Toda theory for r > β -deformation) to compute theNekrasov partition function, and at the same time also has a description interms of matrix model amplitude via the open/closed geometric transition[3]. More concretely, in the genus zero case, from the brane insertion pictureit was clear that a logarithmic matrix model with potential of the form (3.1)should be the relevant one for connecting the 2d and 4d quantities, and theparameters m (cid:96) ’s should be related to the hypermultiplet mass parameters onthe 4d SCFT side, or the external momenta on the 2d conformal block side.Moreover, to describe a general Ω-background, corresponding to a refinedtopological string theory, we should consider the β -ensemble deformation ofthe matrix model, turning the measure in the matrix integral from (3.2) into Z = (cid:90) · · · (cid:90) N (cid:89) i =1 du i (cid:89) ≤ i The momentum violation at each vertex is given by the number of eigenvaluesat the corresponding critical point, directed towards increasing light-cone time. As a result, the n − a (cid:96) =1 , ,...,n − are determined bythe following momenta violation condition at each vertex vertices of the con-formal block ˜ α (1) (cid:96) + ˜ α (2) (cid:96) + ˜ α (3) (cid:96) = Q − N (cid:96) b . (5.1)Here ˜ α (1 , , (cid:96) denote the three momenta coming into the (cid:96) -th vertex of theconformal block. Equivelently, when choosing ˜ α (1 , (cid:96) to be incoming and ˜ α (3) ∗ (cid:96) to be outgoing at the (cid:96) -th vertex, we have˜ α (3) ∗ (cid:96) = ˜ α (1) (cid:96) + ˜ α (2) (cid:96) + N (cid:96) b as shown in Figure 3. Recall that flipping the orientation of a momentum α in the presence of the background charge amounts to taking α ∗ = Q − α , which leaves the conformal dimension∆ α = ∆ α ∗ = α ( Q − α )of the corresponding vertex operator V α invariant. For example, when thedirection of all momenta in Figure 2 is chosen to flow forward in time to thevertex z = ∞ , the first three internal momenta are a = α + α + N b , a = α + α + N b , a = a + a + N b . α n +2 of the operator inserted at z n +2 = ∞ is given by m ,...,n +1 and the rank of the matrix as n +2 (cid:88) (cid:96) =1 α (cid:96) = Q − N b . This completes the map between the parameters on the matrix model andon the conformal block sides. In this section, by studying the monodromy transformations of different ma-trix model blocks, we will determine a family of contours appropriate for thepurpose of computing CFT conformal blocks. See also [20] and [21] for someearlier discussions on the integral representation of conformal blocks in thecontext of minimal models.As alluded to earlier, the different ways of gluing the pairs of pants intro-duce extra subtleties when defining the contours. When the parameters m (cid:96) and β are not integral, the contour ˜ C (cid:96) = [ z (cid:96) +1 , z (cid:96) ] which cuts through a gluingcurve along which different pants are glued together will not be invariant un-der the corresponding Dehn twist operation. This is because the presence ofthe branch cuts has rendered the genus zero Riemann surface multi-sheeted.As a result, the basis (3.3) for the matrix integral with a given potential isin general not a diagonal basis under the monodromy transformation of thevarious parameters in the matrix model potential, in particular the locations z (cid:96) of the poles.Therefore, in order to unambiguously specify a contour for the matrixintegral with a given potential W and eigenvalue distribution { N , . . . , N n } ,it seems natural to look for an alternative basis on which the monodromyoperations act diagonally. More concretely, we would like to combine thecontour ˜ C (cid:96) and its images under the monodromy operations, in such a waythat the combined contour C (cid:96) furnishes such a diagonal basis. We will seethat there exists a unique (up to overall multiplicative factors) contour C (cid:96) satisfying this condition.We are now ready to analyze the action of the Dehn twist operation onour contours ˜ C (cid:96) defined in section 2 and 3. As is suggested by the picture(see, for example, Figure 4), the contour ˜ C (cid:96) can be seen as the union of17 p p p p p p p ˜ C C Re( W/g s ) z z z z z z z z z z z z z z z z z z M M ˜ C Figure 4: (a) The contour ˜ C given by the pre-images of straight line in the W plane.(b) Its image under monodromy transformation M that takes the location z of the polearound z in a counterclockwise orientation. (c) The combination C of the two contoursthat is invariant under the monodromy transformation. two line segments connecting the critical point p (cid:96) to the poles z (cid:96) and z (cid:96) +1 respectively. Depending on the light-cone diagram, none, one, or both ofthese two line segments will cut through some gluing curves. In the firstcase, there is no Dehn twist to be performed and the contour ˜ C (cid:96) we definedearlier is also the unambiguous contour C (cid:96) = ˜ C (cid:96) we are looking for. See Figure 6 for such an example.To discuss the other cases, let us first look at the simple example shown inFigure 4. The contour ˜ C cuts through the gluing curve along which the tubesextending from z , z are glued to the rest of the light-cone diagram. Hence,as can be seen from the Figure 4, our original contour ˜ C is not invariantunder the monodromy M that takes the point z around the point z in18he counter-clockwise orientation before returning to the original position.To find the invariant contour, let us first observe that the image of ˜ C underthis monodromy is M ˜ C = ˜ C + L , where L = ( e φ − e φ + φ ) ˜ C is a loop going through z and circling around z , and e φ (cid:96) is the phase incurred by passing counter-clockwise through thebrach cut connecting the point z (cid:96) and infinity. More precisely, in the basisgiven by the contours { ˜ C , ˜ C } , the monodromy matrix reads M (cid:18) ˜ C ˜ C (cid:19) = (cid:18) e φ + φ e φ − e φ + φ (cid:19) (cid:18) ˜ C ˜ C (cid:19) . Hence we have( M ) k ˜ C = ˜ C + (1 + q + . . . + q k − ) L , q = e φ + φ . We would like to take a combination (cid:80) k c k ( M ) k ˜ C of the images of thecontour ˜ C that is invariant under the monodromy. It is not hard to see thatthe contour C = ( M − q ) ˜ C , which can be combined into a loop passing through the point z and circlingthe points z , z , is the unique invariant combination up to a multiplicativefactor. Hence we arrive at the contour C shown in Figure 4 which has thedesired property under monodromy transformation.The above conclusion can be extended to the cases when the straightline from a zero p (cid:96) to a nearby pole z (cid:96) cuts through a larger gluing curveconnecting more than two tubes to the rest of the light-cone diagram. In thiscase the contour C (cid:96) is a closed loop passing through z (cid:96) +1 and enclosing all thepoles enclosed in the gluing curve. Finally, obviously the same result applieswhen we swap left and right and consider a situation where the segment from p (cid:96) to a nearby pole z (cid:96) +1 cuts through a gluing curve. An example of suchcontours for computing the five-point conformal blocks is given in Figure 5.The final case to consider is when both of the line segments emanatingfrom the zero p (cid:96) to the pole z (cid:96) and to z (cid:96) +1 cut through some gluing curves.Obviously, the corresponding Dehn twists on the left- and the right-handside commute with each other and we can simply apply the above argumentseparately on them and combine the final results.19 p z z z z z z z , α z , α z , α N N z p z , α z = ∞ , α N z C C C Figure 5: An example of the contours for computing a five-point conformal block. Theordering of the contour is given by the time-ordering of the interaction points on the light-cone diagram. Hence one should first perform the integration over the eigenvalue alongthe contours C , then C , and finally C . Moreover, we would like to bring the readers’ attention to a furthersubtlety regarding the ordering of the contours. When considering the β -ensemble deformation of the matrix model when β is not integral, an in-spection of the integral formula (4.1) shows that the phase e φ (cid:96) incurred bycrossing the brach cut connecting the point z (cid:96) to infinity might in fact dependon the pre-existing contours. Hence in this case the contours will no longer besimple tensor products of the same cycles but will have to be ordered instead.There is a completely natural way how this can be done. Namely, the time-ordering of the critical points on the light-cone diagram provides a naturalordering among distinct classes of contours corresponding to distinct criticalpoints. We should hence perform the integration along the integration cycle C (cid:96) corresponding to the earliest interaction point p (cid:96) , and move forward intime. See Figure 5 for an example. This ordering property of the contoursshould be regarded as a part of the definition for the non-perturbative blocksand will be illustrated in the examples discussed in section 7.20inally let us comment on the contours relevant for the comparison withthe conformal blocks described in section 5. By definition, with fixed ex-ternal momenta, conformal blocks in a given channel with different internalmomenta furnish a diagonal basis for the monodromy transformation of thepunctures on the Riemann surface that are equivalent to the Dehn twists inthe corresponding light-cone diagram. Hence it is natural to conjecture thatthe matrix integral (4.1) with the monodromy-diagonal contours describedabove, is the same as the conformal block described in section 5 up to over-all (leg) factors. In the next section we will show some examples of suchequivalence. In the last section we have provided a prescription for the contours of thematrix integrals, and proposed its relation to the Liouville conformal blocks.In this section, we will illustrate and test our prescription by studying a fewexamples in detail. The simplest non-trivial example is given a pair of pants, which is the light-cone diagram corresponding to the potential W ( z ) /g s = m log( z − z ) + m log( z − z ) . According to the dictionary given in section 5, the β -ensemble matrixmodel partition function of a rank N matrix with the above potential cor-responds to the chiral half of the three-point function, with the momenta ofthe inserted operators V α ( z ), V α ( z ) and V α ( z = ∞ ) given by α , = − b m , , α = Q − N b − α − α . (7.1)To keep the notation uniform we also define m = − bα = − m − m − N − β − . Now we shall apply the contour prescription described in previous sectionsto this simple case. As can be seen in Figure 6, the contour given by thesimple gradient line on the W -plane does not cut through any gluing curve21 z N ˜ C z z z z = ∞ z z ··· γ γ N Re( W/g s ) Figure 6: The simplest light-cone diagram is given by a single pair of pants. In this casethe contour is simply ˜ C given by the downward gradient flow of Re ( W/g s ) emanating fromthe unique critical point of the potential. The N contour lines are ordered in such a waythat they do not intersect with each other. The corresponding quantity on the 2d CFT sideis the chiral half of the three-point function, with the three momenta determined by thematrix potential W and the rank of the matrix N . in this case and is therefore free from any Dehn twist ambiguity. Therefore,we shall use the contour ˜ C = [ z , z ] in our matrix integral. The discussionin section 5 then suggests that the matrix integral Z ( z , z ) = z m m / β × (7.2) (cid:90) z z du N · · · (cid:90) z z du N (cid:89) i =1 ( u i − z ) m ( u i − z ) m (cid:89) ≤ i 0) = (cid:90) dt N · · · (cid:90) dt N (cid:89) i =1 t m i ( t i − m (cid:89) ≤ i 0) = ( − m N N ! N − (cid:89) j =0 Γ(( j + 1) β )Γ(1 + m + jβ )Γ(1 + m + jβ )Γ( β )Γ( − m − jβ ) . (7.3)To put Selberg’s formula in a form closer to that of the the DOZZ for-mula, we use the following relation between the Γ-function and the DoubleΓ-function Γ( x ) = √ πω / − x Γ ( ω x | ω , ω )Γ ( ω x + ω | ω , ω ) . Moreover, eventually we would like to analytically continue the blocks ob-tained from matrix integrals to general values of momenta α , α , α notnecessarily corresponding to integral units of the screening charges. To facil-itate this, we would like to write the final answer in terms of the momenta α , , alone by eliminating the rank N of the matrix from the final answer23sing the relation N = ( Q − (cid:80) i =1 α i ) /b . By the same token we will nowadopt a new notation F α ,α ,α = Z (1 , . Our proposed relation between the matrix integral and the conformal blocksthen states that the DOZZ three-point function C α ,α ,α is given by C α ,α ,α ∼ F α ,α ,α F α ∗ ,α ∗ ,α ∗ , where “ ∼ ” signifies equality up to leg factors and overall normalization fac-tors.Using the shorthand notation Γ b ( x ) = Γ ( x | b, b − ), we get F α ,α ,α ∼ Γ b ( α + α + α − Q )Γ b (0) Γ b ( α + α − α )Γ b ( Q − α ) × Γ b ( α + α − α )Γ b ( Q − α ) Γ b ( Q + α − α − α )Γ b (2 α ) , which can be shown to satisfy the above relation. In fact, the leg factors canalso be determined if we impose the following two simple conditions relatedto the two-point function:lim (cid:15) → C α,(cid:15),Q − α = lim (cid:15) → C (cid:15),α,Q − α ∼ (cid:15) , lim (cid:15) → C α,(cid:15),α = lim (cid:15) → C (cid:15),α,α ∼ (cid:15) S ( α ) , where S ( α ) is the so-called reflection amplitude given by S ( α ) ∼ Γ b (2 Q − α )Γ b (2 α − Q )Γ b (2 α )Γ b ( Q − α ) . From the above requirement, we conclude that the leg factors are givenby f ( α ) = Γ b (2 α − Q )Γ b (2 α )for the incoming strings in our light-cone diagram and g ( α ) = f ( α ∗ ) for theoutgoing ones. Incorporating these leg factors, we finally get f ( α ) f ( α ) f ( α ∗ ) F α ,α ,α F α ∗ ,α ∗ ,α ∗ ∼ Υ (cid:48) Υ(2 α )Υ(2 α )Υ(2 α )Υ( α + α + α − Q )Υ( α + α − α )Υ( α + α − α )Υ( α + α − α ) , x ) = 1Γ b ( x )Γ b ( Q − x ) . This is precisely the DOZZ formula [36, 37] apart from the extra normaliza-tion factor (cid:18) πµ Γ( b )Γ(1 − b ) b − b (cid:19) ( Q − (cid:80) (cid:96) =1 α (cid:96) ) /b . Notice that this normalization factor depends on the coefficient µ of theexponential term (cid:82) d z µ e bϕ in the Liouville action which can be shiftedaway by shifting φ → φ + const, and is therefore clearly scheme-depedent.Hence, we see that in the case of the three-point function, the matrix modelindeed captures the chiral information. The relation between the matrixintegral and the DOZZ formula was also discussed in [12].Finally, note that our matrix integral F α ,α ,α ∼ N − (cid:89) j =0 Γ(( j + 1) β )Γ(1 + m + jβ )Γ(1 + m + jβ )Γ( β )Γ( − m − jβ )is not the only possible chiral half of the DOZZ formula C α ,α ,α . Thereare a few discrete choices that can be made here. Obviously, some otherpossibilities are given by permuting α , α , α in the above formula. Moregenerally, it is not hard to see that by replacing the factor Γ(1 + m i + jβ )with 1 / Γ( − m i − jβ ) in the product or another way around, we obtain anotherobject which squares to C α ,α ,α . In particular, another possibility is givenby ˆ F α ,α ,α ∼ N − (cid:89) j =0 Γ(( j + 1) β )Γ(1 + m + jβ )Γ( β )Γ( − m − jβ )Γ( − m − jβ ) , satisfying ˆ F α ,α ,α ˆ F α ∗ ,α ∗ ,α ∗ ∼ C α ,α ,α . (7.4)This property of the three-point function will be useful later in our discussionof the four-point conformal blocks. Now we consider the four-point function corresponding to the following ma-trix potential W ( z ) /g s = m log( z − z ) + m log( z − z ) + m log( z − z ) . (7.5)25 p z z z z z z ······ γ γ N ··· γ N +1 γ N z , α z , α z , α N N z = ∞ , α a Figure 7: On the left is a light-cone diagram corresponding to the potential (7.5). To-gether with the eigenvalue distribution N , N = N − N , it maps to the s -channel conformalblock shown on the right, with the internal momenta given by a = α + α + N b . Depending on the parameters m , , and z , , , the corresponding light-conediagram might take a different shape corresponding to a different pair of pantsdecomposition. In Figure 7 we show one of the possibilities which correspondsto the s -channel conformal blocks. According to the prescription in section6, the corresponding contour of integration is the one shown in Figure 7.The SL (2 , C ) invariance dictates the following form for the four-pointfunction (cid:104) V α ( z ) V α ( z ) V α ( z ) V α ( z ) (cid:105) = | z | +∆ − ∆ − ∆ ) | z | − | z | +∆ − ∆ − ∆ ) | z | − ∆ − ∆ − ∆ ) × G α ,α ,α ,α ( z, ¯ z )where z ij = z j − z i and z is the cross ratio of the four points z = z z z z . In particular, all the information of interest is contained in the followingobject G α ,α ,α ,α ( z, ¯ z ) = lim z →∞ | z | (cid:104) V α ( z ) V α (1) V α ( z ) V α (0) (cid:105) . s -channel sewing procedure, using the operator alge-bra one can see that it has the following decomposition into conformal blockswith different internal momenta aG α ,α ,α ,α ( z, ¯ z ) = (cid:88) α C α ,α ,a C a ∗ ,α ,α (cid:12)(cid:12)(cid:12)(cid:12) F sa (cid:20) α α α α ; z (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) , where the prefactors, given in terms of the DOZZ three-point functions, arechosen such that the conformal blocks have the following normalization F sa (cid:20) α α α α ; z = 0 (cid:21) = 1 . In this subsection we would like to verify the relation between the matrixintegrals and the above four-point conformal blocks. First we would like tosee that the matrix integrals indeed take the above form. From the discus-sion in section 5, we expect the four-point conformal block with operatorinsertions at locations z , z , z , z = ∞ to be given by the matrix integral (cid:89) ≤ k<(cid:96) ≤ ( z (cid:96) − z k ) m (cid:96) m k / β (cid:90) · · · (cid:90) N (cid:89) i =1 du i (cid:89) ≤ i We would like to thank J. Teschner for valuable discussions. M.C. wouldlike to thank the University of Amsterdam, LPTHE Jussieu and LPTENS inParis, Max Planck Institut f¨ur Physik in Munich, and KITP Santa Barbara,for kind hospitality during the process of this project. R.D. and C.V. alsothank the Simons Workshop in Mathematics and Physics 2010 for providinga stimulating research environment as well as for its warm hospitality. Theresearch of M.C. is supported in part by DOE grant DE-FG02-91ER40654.The research of R.D. was supported by a NWO Spinoza grant and the FOMprogram String Theory and Quantum Gravity . The research of C.V. wassupported in part by NSF grant PHY-0244821. A Degenerate Four-Point Conformal Blocks In this appendix we explicitly show that the expected relation between thefour-point conformal blocks and the matrix model blocks (7.6) indeed holdsfor the special case with degenerate insertions. When one of the insertionshas degenerate momenta α = − nb/ − m/ b for non-negative integers n and m , the corresponding Virasoro representations are reducible and the four-point functions satisfy linear differential equations [40]. We will now focuson the simplest case with α = − b/ 2. In this case, the differential equationdictates the conformal blocks to be given in terms of the hypergeometric29unctions F (cid:2) A, BC ; z (cid:3) as [40] F sa = α − b/ (cid:20) α − b/ α α ; z (cid:21) = z ∆ a − ∆ − ∆ (1 − z ) ∆ α α − ∆ − ∆ × F (cid:104) N β, m + ( N − β − m + β ; z (cid:105) F sa = α + b/ (cid:20) α − b/ α α ; z (cid:21) = z ∆ a − ∆ − ∆ (1 − z ) ∆ α α − ∆ − ∆ (A.1) × F (cid:104) m + ( N − β, − m − ( N − β m − β ; z (cid:105) and all other conformal blocks with internal momentum a (cid:54) = α ± b/ F ( A, B ; C ; z ) is analytic at theorigin z = 0. More precisely, it is normalized such that F (cid:104) A, BC ; z = 0 (cid:105) = 1 . Together with z − C F (cid:104) A + 1 − C, B + 1 − C − C ; z (cid:105) it spans the space of solutions to the following ODE z (1 − z ) f (cid:48)(cid:48) + [ C − (1 + A + B ) z ] f (cid:48) − ABf = 0 . (A.2)Now we would like to compare the matrix integral with the above knownresult. In the present case, we have m = − β and the relevant matrix modelquantity (7.6) is z − m / (1 − z ) − m / (cid:90) ˆ C a N (cid:89) i =1 du i u m i ( u i − z ) − β ( u i − m (cid:89) ≤ i 31e then get Z a = α − b/ ( z ) = z − m / (1 − z ) − m / × (cid:90) ˆ C α = α − b/ N (cid:89) i =1 du i u m i ( u i − z ) − β ( u i − m (cid:89) ≤ i 1. A similar calculation shows (cid:90) ˆ C a = α b/ N (cid:89) i =1 du i u m i ( u i − z ) − β ( u i − m (cid:89) ≤ i Adv.Theor.Math.Phys. (Adv.Theor.Math.Phys.2:413-442,1998) 413–442, hep-th/9802016 .[2] R. Gopakumar and C. Vafa, “On the Gauge Theory/GeometryCorrespondence,” Adv.Theor.Math.Phys. (1999) 1415–1443, hep-th/9811131 .[3] R. Dijkgraaf and C. Vafa, “Matrix Models, Topological Strings, andSupersymmetric Gauge Theories,” Nucl.Phys. B (2002) 3–20, hep-th/0206255 .[4] F. David, “Phases of the large N matrix model and nonperturbativeeffects in 2-d gravity,” Nucl. Phys. B348 (1991) 507–524.[5] F. David, “Non-Perturbative Effects in Matrix Models and Vacua ofTwo Dimensional Gravity,” Phys.Lett. B (1993) 403–410, hep-th/9212106 . 336] T. Dimofte, S. Gukov, J. Lenells, and D. Zagier, “Exact results forperturbative chern-simons theory with complex gauge group,” Commun.Num.Theor.Phys. (2009) .[7] E. Witten, “Analytic Continuation Of Chern-Simons Theory,” .[8] E. Witten, “A New Look At The Path Integral Of QuantumMechanics,” .[9] L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville CorrelationFunctions from Four-dimensional Gauge Theories,” .[10] N. Wyllard, “ A N − conformal Toda field theory correlation functionsfrom conformal N=2 SU(N) quiver gauge theories,” JHEP (July, 2009) , .[11] R. Dijkgraaf and C. Vafa, “Toda Theories, Matrix Models, TopologicalStrings, and N=2 Gauge Systems,” .[12] R. Schiappa and N. Wyllard, “An A r threesome: Matrix models, 2dCFTs and 4d N=2 gauge theories,” .[13] A. Mironov, A. Morozov, and S. Shakirov, “Conformal blocks asDotsenko-Fateev Integral Discriminants,” .[14] A. Mironov, A. Morozov, and A. Morozov, “Matrix model version ofAGT conjecture and generalized Selberg integrals,” .[15] A. Morozov and S. Shakirov, “The matrix model version of AGTconjecture and CIV-DV prepotential,” .[16] K. Maruyoshi and F. Yagi, “Seiberg-Witten curve via generalizedmatrix model,” .[17] A. Mironov, A. Morozov, and S. Shakirov, “On Dotsenko-Fateevrepresentation of the toric conformal blocks,” .[18] T. Eguchi and K. Maruyoshi, “Penner Type Matrix Model andSeiberg-Witten Theory,” .3419] T. Eguchi and K. Maruyoshi, “Seiberg-Witten theory, matrix modeland AGT relation,” .[20] V. S. Dotsenko and V. A. Fateev, “Four Point Correlation Functionsand the Operator Algebra in the Two-Dimensional ConformalInvariant Theories with the Central Charge c < Nucl. Phys. B251 (1985) 691.[21] G. Felder, “BRST Approach to Minimal Methods,” Nucl. Phys. B317 (1989) 215.[22] R. Dijkgraaf, C. Vafa, and E. Verlinde, “M-theory and a TopologicalString Duality,” hep-th/0602087 .[23] R. Dijkgraaf, P. Sulkowski, and C. Vafa, “unpublished,”.[24] S. Cecotti, A. Neitzke, and C. Vafa, “R-Twisting and 4d/2dCorrespondences,” .[25] M. Aganagic and M. Yamazaki, “Open BPS Wall Crossing andM-theory,” Nucl.Phys.B (Nov., 2009) , .[26] N. P. Warner, “Supersymmetry in Boundary Integrable Models,” Nucl.Phys.B (Nucl.Phys.B450:663-694,1995) , hep-th/9506064 .[27] K. Hori, A. Iqbal, and C. Vafa, “D-Branes And Mirror Symmetry,” hep-th/0005247 .[28] M. Marino, R. Schiappa, and M. Weiss, “Nonperturbative Effects andthe Large-Order Behavior of Matrix Models and Topological Strings,” .[29] M. Marino, “Nonperturbative effects and nonperturbative definitionsin matrix models and topological strings,” JHEP (May,2008) , .[30] B. Eynard and M. Marino, “A holomorphic and backgroundindependent partition function for matrix models and topologicalstrings,” . 3531] S. B. Giddings and E. J. Martinec, “Conformal Geometry and StringField Theory,” Nucl. Phys. B278 (1986) 91.[32] S. B. Giddings and S. A. Wolpert, “A TRIANGULATION OFMODULI SPACE FROM LIGHT CONE STRING THEORY,” Commun. Math. Phys. (1987) 177.[33] D. Gaiotto, “N=2 dualities,” .[34] A. Marshakov, A. Mironov, and A. Morozov, “Generalized matrixmodels as conformal field theories: Discrete case,” Phys. Lett. B265 (1991) 99–107.[35] I. K. Kostov, “Conformal field theory techniques in random matrixmodels,” hep-th/9907060 .[36] A. B. Zamolodchikov and A. B. Zamolodchikov, “Structure Constantsand Conformal Bootstrap in Liouville Field Theory,” Nucl.Phys. B (1996) 577–605, hep-th/9506136 .[37] H. Dorn and H. J. Otto, “Two and three-point functions in Liouvilletheory,” Nucl.Phys. B (1994) 375–388, hep-th/9403141 .[38] J. Teschner, “Liouville theory revisited,” Class.Quant.Grav. (2001)R153–R222, hep-th/0104158 .[39] V. Pestun, “Localization of gauge theory on a four-sphere andsupersymmetric Wilson loops,” .[40] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infiniteconformal symmetry in two-dimensional quantum field theory,” Nucl.Phys. B241 (1984) 333–380.[41] J. Kaneko, “Selberg Integrals and Hypergeometric FunctionsAssociated with Jack Polynomials,” Siam. J. Math. Anal. 24, No. 424, No. 4