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Non-perturbative topological strings and conformal blocks

Cheng, M.C.N.; Dijkgraaf, R.; Vafa, C. DOI 10.1007/JHEP09(2011)022 Publication date 2011 Document Version Final published version Published in The Journal of High Energy Physics

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Citation for published version (APA): Cheng, M. C. N., Dijkgraaf, R., & Vafa, C. (2011). Non-perturbative topological strings and conformal blocks. The Journal of High Energy Physics, 2011(9), 022. [22]. https://doi.org/10.1007/JHEP09(2011)022

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Download date:02 Oct 2021 JHEP09(2011)022 Springer July 4, 2011 : August 22, 2011 : pological string September 5, 2011 Theories : Received he dual matrix model , 1 10.1007/JHEP09(2011)022 Accepted d, als and their monodromy Published the specific case where the d to a choice of integration doi: otential correspond to string ring blocks and Liouville con- e light-cone diagrams of closed to Liouville conformal field the- matics, University of Amsterdam, f Techology, Published for SISSA by and Cumrun Vafa c [email protected] , Robbert Dijkgraaf a,b 1010.4573 Topological Strings, Matrix Models, Duality in Gauge Field We give a non-perturbative completion of a class of closed to [email protected] On leave from Harvard University. 1 E-mail: Cambridge, MA 02138, U.S.A. Institute for Theoretical Physics &Amsterdam, The KdV Netherlands Institute for Mathe Center for Theoretical Physics, MassachusettsCambridge, Institute MA o 02139, U.S.A. Department of Mathematics, HarvardCambridge, University, MA 02138, U.S.A. Department of Physics, Harvard University, [email protected] c b d a Open Access Abstract: Miranda C.N. Cheng, string field theory, whereinteraction the points. critical points of the matrix p Keywords: open string is givencontour. by a We matrix then modelhas apply these logarithmic this blocks potential definition correspon and toory. is the conjecturally AGT By equivalent setup studyingproperties, where the we t propose natural a contoursformal precise of blocks. map between Remarkably, these this topological description matrix st makes integr use of th theories in terms of building blocks of dual open strings. In Non-perturbative topological strings andblocks conformal ArXiv ePrint: JHEP09(2011)022 1 3 6 9 16 20 23 11 13 16 22 ] and 10 , 9 and, involve gravitational odels respectively. A nat- plitudes of different genera tification? And, if so, what wa couplings and the super- eld theories. More precisely, e in understanding quantum ompletions are in the context ic theories. Topological string us zero amplitudes correspond th matrix models [ ons can help us to find a non- e of such definitions is that, for mputes? logical string theory have a non- – 1 – ] do admit non-perturbative definitions, and it is thus 79 , ] relate the A- and B-model topological string amplitudes 78 34 , , 18 33 , 15 Various dualities [ 7.1 Three-point functions 7.2 Four-point functions potential terms. Thecorrections. higher This genus raises amplitudes, theperturbative following on meaning question: the that Does other goes topo beyong h is this the term physical quantity by that term the iden full partition function co protected amplitudes in stringthey theory capture and the F-terms supersymmetric of fi amplitudes the corresponding are supersymmetr indexed by thecompute genus different of physical the quantities. worldsheet, Forto and example, F-terms am the in gen supersymmetric field theories, such as the Yuka 1 Introduction Topological string amplitudes have played an important rol 8 Discussions A Degenerate four-point conformal blocks 6 Topological string blocks versus conformal7 blocks Examples 3 Application to Penner-type matrix4 models Logarithmic matrix models and5 the AGT correspondence From matrix models to conformal blocks Contents 1 Introduction 2 Non-perturbative topological string blocks ural question is henceperturbative whether definition these of alternativeChern-Simons formulati topological amplitudes [ strings.natural Indeed, to ask bo what theof meanings topological of these string non-perturbative and c superstring theories. One featur to quantities in complex Chern-Simons theories and matrix m JHEP09(2011)022 = 2 N and we nels that top A = 2 theories Z for Penner-like N d to the choices ecise relationship , to be made. For top . This connection A A + Z X ]) theories. These con- ensional to the AGT correspon- ]. est represented as points ise the connection to the 80 locks ory and the periods of the 25 a [ mpurities in the Penner-like translates into the choice of , prescription of the screening non-perturbative completion. e time one diagram, emanating from topological string amplitudes hoice of spacetime branes. In this light-cone diagram, the uestion to be answered is how the Taub-NUT spacetime or a 20 are in fact a linear combination A ve matrix model amplitudes. lectively by omputing partition functions of . The basic integration contours y matrix model with Penner-like ]. Recently it was also considered in the superstring theory. Recall- , model potential are mapped to the oducts of the AGT correspondence. CFT : B 71 Z , top top A A top A Z Z 58 Z ] that these same amplitudes should also A , , B 13 54 C , ]. Of course, this problem is directly related 53 A – 2 – 23 X , , = 41 ]. The gauge theory aspect of the relation has been 22 55 is identified with the (real part of the) matrix model , CFT B + Z 52 X . On the other hand, using geometric engineering of the B are labeled by parameters specifying the intermediate chan there is a non-perturbatively defined partition function A CFT B Z of the non-perturbative topological blocks should be mappe are certain constants. By giving such a map, we specify the pr A A , B ] that connects the Nekrasov partition function of four-dim 5 C This relation between the conformal blocks of Liouville the One main motivation for revisiting this question is related In this paper we argue that the conformal blocks As an interesting aspect of defining the topological string b of conformal blocks. Penner-like matrix integral is one of theIt interesting by-pr has been discussed among others in [ between the Liouville conformal blocks and non-perturbati where theories in the superstring setup, it was shown [ are interested in the meaning of these amplitudes each such choice boundary condition for this non-compact spacetime, i.e. a c dence [ B of the non-perturbative string partition functions we denote collectively by to the conformal blocks of Liouville (and more generally Tod Instead there are discrete choices, which we will denote col a given perturbative definition there does not exist a unique 2-dimensional subspace of it, we argue that the choice of ing that topological stringsuperstring amplitudes are theories interpreted in as specific c backgrounds which involve to a classical subjectcharges in for conformal minimal field models `ala theory: Dotsenko-Fateev, see the e.g. contour [ in the case of genusfurther one surfaces investigated [ in, for example, [ conformal blocks of thethe Liouville choices theory. In particular, a q correspond to topological string amplitudeslogarithmic represented potentials. b Acan non-perturbative hence be completion applied of to this special case and should make prec formal blocks potential. The light-cone time matrix models, we findon that a the light-cone parametrization eigenvalues of ofincoming a the and genus matrix zero outgoing worldsheet. are tubes b correspond to the positions of i potential. In particular, the criticalinteraction points points of in the the matrix light-coneof diagram the and matrix vice model versa arethe given interaction by points straight lines and on going the backwards light-c in the light-con JHEP09(2011)022 s = we q IIA ) 6 C × 1 S e appendix we . = 2 supersymmetry. we formulate a non- N 2 ith Penner-like logarithmic , with ons action, up to worldsheet ]. The geometry ( ests a potentially important simplest situation where this gical strings and the blocks of q troducing A-branes wrapping In this context, in section d to all topological string the- ) C 12 ur-point conformal blocks and = 2 theory are associated with mputing certain physical ampli- , . Moreover, the superpotential of 8 rpretation in terms of superstring cal string amplitude computes the N TN = 2 amplitudes in four dimensions. L , we apply this definition to specific 3 [ × 3 , N 1 q denotes the space obtained by rotating ) ) + instanton corrections S s q ( TN )  3 × TN πig A we give a dictionary between the parameters TN × X 2 3 5 1 × – 3 – ⊂ + S 1 ]. ( q S 8 ) = exp(2 × C q AdA X × we give examples which illustrate this correspondence.  1 7 Tr S . In the M-theory setup, adding a topological A-brane ( L Z X × s ]. We can extend this dictionary between the physical and 1 L g ⊂ 17 [ L = on the Lagrangian submanifold we briefly review the content of the AGT correspondence and it , we can view the M-theory system from the perspective of type s 1 1 4 A S S /g ) A ( W we conclude with some discussion on future directions. In th 8 is a Calabi-Yau threefold and ( is the two-dimensional cigar subspace of X C is the so-called ‘Melvin Cigar’ in [ The organization of this paper is as follows: In section Reducing on the q the gauge connection the two-dimensional theory is simplyinstanton given corrections: by the Chern-Sim The chiral degrees of freedom for this two-dimensional superstring theory and obtain a two-dimensional theory on connection between light-cone string field theory and with light-cone string diagrams is very intriguing and sugg the topological theory toa the Lagrangian open subspace topological sectors by in Topological string amplitudes have antudes interpretation in as co superstring theoriesrelation in is the realized presence is the ofpartition following: branes. function the The of A-model topologi M-theory in the background In section explicitly demonstrate the relationthe corresponding between matrix degenerate model fo expressions. 2 Non-perturbative topological string blocks topological string theoriespotential. that are In dual section torelation to matrix the matrix models model,in w and the in matrix section model and thosepropose of a Liouville map conformal between blocks. thethe non-perturbative blocks 2d of Liouville topolo CFT. In section perturbative completion of topological stringamplitudes. and its This inte non-perturbative definitionories can with be a applie dual open string description. In section MC the circle symmetry of the Taub-NUT space by an angle corresponds to adding an M5 brane wrapping the subspace where where as one goes around JHEP09(2011)022 in d- nce, (2.1) satisfying ] that the n 15 evaluated at semi-classical ,N W N nfigurations that ··· bspace in the field , 1 ]. The boundary of he branes populating N hysical and the topo- 37 y lies in the choice of a n [ scribed by the conifold- wing two facts: in type the large potential preserve supersymmetries Yau geometry given by a tatement. The parameters e for the physical partition igar subspace of the Taub- envalues among the critical ne should specify a contour gical string. In this case we . ion, it does not give an unam- s . y a corresponding ambiguity in ) /g s (Φ) /g ) (Φ): W ], to specify a boundary condition for = 0 A ( W Tr . ). In this case it is known [ 37 e x uv W ( Φ IIA + W Z D 2 ) exp( = x Z – 4 – ( of the matrix Φ is equal to the total number ′ = top DA W N Z IIB Z + , labeled by the filling fractions Z 2 = y W = top Z top Z is the superpotential, whereas here it appears as the action (Φ) are complex and the integration over Φ is holomorphic. He W = 2 theories with boundaries, we need to add to the action boun W N of the form 4 ]. of the potential C n 77 , p is a circle, and the path integral localizes on the the field co . This distribution has the interpretation as the number of t ··· C , N 1 p = 2 theory in two dimensions, we need to choose a Lagrangian su = 2 theories in two dimensions. In that context, the ambiguit = ℓ In the context of the cigar geometry this has been discussed i We will be interested in an expansion near the saddle point in By mirror symmetry we expect a similar relation between the p However, there is an intrinsic incompleteness in the above s N N N points limit. A saddle point is specified by a distribution of the eig function: are constant on it. Hence we arrive at the same formula as abov P In other words, we have biguous non-perturbative answer. Tofor remedy the matrix this integral. problem, Inthe o fact, this ambiguity is matched b even though the above recipe specifies a perturbative expans of branes. with D3-branes wrapped ontype geometries the localized holomorphic at curves critical that points are of de the B-model reduces to a matrix model with potential the cigar have D3-branes wrapping holomorphicNUT curves geometry. and filling In thehypersurface particular, c in we can consider a local Calabi- logical quantities to hold for the case of the B-model topolo supersymmetric boundary condition. As noted in [ different holomorphic cycles. of the superpotential the topological theory!in However, two-dimensional it is known that in order to Under this correspondence the rank Naively there may appearIIA to superstring be theory some tension between the follo the boundary [ ary terms which are exactly given by the integral of the super JHEP09(2011)022 2 . ) ) s dz )( W/g z te from. ( Both the zz ) and their φ z ( W the paper: nn surface is via the . 1 ay the complex integral -plane are straight lines answer for the partition W t is flat everywhere on the rd gradient flow of Re( providing a supersymmetric e opological and physical sides, he integration contour, while amplitude unambiguously. In . W ce a choice of the Lagrangian ice of the Lagrangian subman- ological string amplitudes. atrix potential the following conditions. First, 2 n figure uadratic differential  , 2 | ) s dW , where the curvature is singular. By dz 0 s 1 zz g | | W/g φ  zz | → Re( φ ) | = s 2 is given by the pre-images of the straight lines on = ) ) n 2 n – 5 – p | W/g ˜ ( dw C ). Moreover, to ensure the integral is well-defined , ℓ W dw | p ) = ( ( exp( ··· 1 , | 2 p =

··· 1 ( ) W ˜ 2 C W dz ds ( zz φ -map are the points where the Lagrangian submanifolds emana W . A natural set of contours -plane. The slope of the straight lines is given by the downwa Let us summarize this main point that is crucial to the rest of The choice of the Lagrangian submanifold makes precise the w A way to visualize these straight lines on the original Riema W topological and physical amplitudessubmanifold are is uniquely made. defined It on ifold is as hence a natural to choice of identify a this cho non-perturbative completion of the top given in terms of the matrix potential as associated with a so-called Jenkins-Strebel holomorphic q boundary condition. over the chiral fieldsfunction depends should on be this choice. performed,a In choice and other of words, a as on Lagrangian both a submanifoldthe the is result first t required to case, the define the the in Lagrangian submanifold the plays latter the case role it of completes t the definition of the theory by images under the the definition, the zeros are exactly the critical points of the m emanating from the critical values Figure 1 following quasi-conformal mapping the projections of the submanifolds in the field space onto th space. The allowed Lagrangian submanifolds should satisfy when the submanifolds have boundaries, we require that Clearly, such a quasi-conformalRiemann mapping surface defines except at a the metric zeros tha and poles of at these boundaries. These two conditions are illustrated i JHEP09(2011)022 e ]. In (3.1) ]. To 5 . 49 and the } with , s n , it jumps 2 g 47 ℓ 1 , , p 1 24 ℓ p ,...,N 1 that characterizes N ℓ { N ]. 13 his property is of course e of logarithmic potential interpreted as topological urbative topological string rpreted, can be made into oupling constant gical string blocks will be a itical points onformal blocks of the SL(2) ials can be defined using such ke to apply such a definition As we will review later, this ive answer for the topological d in [ conformal theory. In this pa- ’ phenomenon for the integral ) is not a continuous function on to the Liouville conformal ivial monodromy transforma- ical strings in [ , g a contour to a given classical on the complex Chern-Simons n . an be seen in figure he AGT correspondence [ ) ℓ to the more general situations of z = 0 ]. More recently they were discussed −  z ,...,N 10 ) 1 , 2 ℓ 9 N p log( ( ( ℓ Z ], which connects certain topological string W m s 13 − g – 6 – +1 =1 ) ℓ X n 1 ℓ = p ( s , the data of brane distribution W /g ) W  z ( Im W ]. Moreover, similar Lagrangian contours, which are infinit 18 ) = non-perturbative topological block labeled by n ]. ]. 13 78 ,...,N 1 N ( Z We are interested in the matrix models with the following typ However, such a ‘topological string block’ In other words, for a given choice of the topological string c type of matrix modelsblocks. is of Hence special anprerequisite interest understanding due for of to the a its corresponding concrete relati topolo realization of [ 3 Application to Penner-typeIn matrix the models previous sectionblocks we through have matrix given integrals.to a In definition Penner-like this matrix of section non-pert models we with would logarithmic li potentials. Liouville theory. Similar considerationhigher should also genus surfaces apply and higher rank gauge groups discusse amplitudes to Liouville conformal blockskeep in the the context discussion of explicit t we will focus on the genus zero c dimensional in this context,path also integral feature [ in the discussion at special values of the parameters such that there are two cr in the context ofthe a context non-perturbative of complex formulation Chern-Simons, ofsolution the topolog has idea of appeared associatin in [ reminiscent of that ofper conformal we blocks will in two-dimensional demonstratean how equality this in analogy,string the when theory suitably context [ inte of the Liouville conformal theory These jumps in the correspondingdescribed cycles by result the in the Picard-Lefschetztions Stokes theory, when and the cause parameters move non-tr around in the complex plane. T of the parameters of the superpotential. In particular, as c parameters of the superpotential the matrix model saddle pointsamplitude also leads to a non-perturbat The fact that the matrix modelcomplex contours integrals has with been complex known potent for a long time [ JHEP09(2011)022 . N × (3.2) N 2 ‘net’ − g ce with respect . , or more precisely ℓ 2 m ) e gradient flows of of the matrix Φ ) s ℓ N z by ]. To see this, let us first , u − ). There is a particularly w i dW/g 28 u ··· conformal transformation is ( , 1 = ( ge [ +1 , u =1 2 Riemann surface as a scattering Y ℓ n Riemann surface has 2 dz N =1 i Y zz g describe the incoming and outgoing 2 φ . → ±∞ ) , i ω, . ω ) 2 ℓ u ℓ )

ℓ z z ℓ z z ω ω. ( I − − m C z − j dW = W I z = 0, together with u s z Z s 1 ( ℓ g 1 ℓ g

∼ N – 7 – m = ℓ m ≤ ) log( z = = ℓ z ℓ w ( 2 Y i

, the times +1 n correspond to an interaction point where two incoming } n W/g once, the coordinate ]. Fixing the initial time to be together with an extra pole at dz ℓ ω +1 1) complex variables corresponding to the locations of the dW , we would like to associate a saddle point, labeled by the p n 30 2 up to SL(2 +1 − of + 1 incoming and one outgoing closed string. As a result, in , . . . , z ), we will consider the potential ( ,...,N n ℓ 1 1 ∞ n p ,...,n z 1 3.2 N = z { -th critical point is the line segment going between the poin , . . . , m ℓ 1 with which an intermediate closed string is twisted before r = critical points +2 m } n z { 1 n , z − n +1 n encircles the zero z , . . . , θ 1 θ , . . . , z . Such a genus zero light-cone diagram is specified by the foll { 1 2 z ’s together give the ( θ As mentioned in section Concretely, for a given potential In order to obtain an unambiguous and uniform description of Similarly, the zeroes The arrow of time of the light-cone is given by increasing Re( In the present case of genus zero, the poles of the one-form on the light-cone diagram. In particular, the correspondin downward-flowing (backward in time) straight lines emanati matrix model integral ( our matrix model discussion,some the sense pole more of special than all the other poles at differential in the order depicted in figure diagram, label the poles mentioned way to the eigenvalue distribution In the language ofclass the of light-cone contour diagram to (see eacha figure interaction very point natural way of to the specifygiven strings. these by As contours the from gradient the flow matrix of m Re( leads to light-cone diagramsscattering of the diagrams kind with depicted in figure being real and positive.other The ranges answer of for the the parameters. integral can For then such cases, the quasi-co external momenta and poles Clearly if angles this implies that closed strings merge into anones. outgoing Let one, us or recall one why incoming this st is the case. If with the rest of the diagram [ figure surplus angle that comes with the conical singularity of a pa at the finite values are located at the JHEP09(2011)022 ℓ ℓ ], of of ˜ ro 5 C m ℓ } z n (3.3) +1 ∞ n = α , z , . . . , p 1 +1 p n

n {

z

N ··· ··· 4 . The locations α ) with the following , actions. One might 4 W , while the residues ’s are not all integers z ℓ 3.2 τ ial m 3 tions to topological string . α ntegrals on which the mon- 6 3 ernative basis and relate the , 3 heeted. The matrix integrals 2 N rtain 2d conformal blocks and z . a ially when changing from sheet ty is mandatory from the point structure. Together with the data 2 ] ion 3 ℓ α a 2 , , z N 2 iagram gives a pants decomposition of z = 4 superconformal field theories [ 1 nformal block. +1 ℓ a d z 1 N = [ ) = 2, ℓ s 1 ˜ α C . We choose the orientation and denote by , ℓ N 1 and the interaction times p W/g – 9 – z , ℓ θ ℓ Re( N . As a result, to a given distribution of eigenvalues ℓ ⊗ ℓ z ˜ +1 C n n z to p =1 n ℓ , we can associate a matrix integral ( +1 ⊗ ℓ of the matrix eigenvalues among the critical points ℓ p z ]. } -ensembles generalization of the matrix model, the genus ze n β 13 ······ ∞ 4 p = 4 ,...,N z z 1 2 3 θ N p 3 { 2 z p determine the twist angles 3 2 θ z 1 p 1 dW θ 1 z . An example of a light-cone diagram for a given matrix potent 1 2 and passing through the critical point τ τ As we will discuss in more details later, when the momenta among the critical points ℓ +1 ℓ Figure 2 and when we consider the In this section we willthe briefly Nekrasov partition review functions the for relationand certain between how ce this relationtheory can and be matrix model understood [ through their connec 4 Logarithmic matrix models and the AGT correspondence contour odromy transformations act diagonally. Inof fact view of this the proper conformalcorresponding blocks. matrix We integrals will to construct conformal such blocks an in alt sect curve on which thewith matrix the eigenvalues above live prescription is ofto in contours sheet fact transform multi-s and non-triv inhence fact deem it mix natural with to one use another an under alternative basis such for monodromy matrix i the poles of the genus zero Riemannof surface the and distribution thereby determines a tree determine the size of the closed strings. Such a light-cone d the matrix potential, this tree specifies a corresponding co z the line segment going from N JHEP09(2011)022 ]. 80 (4.1) 1 in [ r > 0) superconfor- , , ℓ m ) = (2 ℓ wn that the collective z ’s should be related to ) into ℓ N e described by a chiral iouville-like chiral CFT, tion picture it was clear ]. In this way of under- − ) should be the relevant role of a bridge between m 3.2 ackground are essentially i ]. On the other hand, the 13 f the corresponding SU(2) e external momenta on the meter of a neck connecting of matrix model amplitude u rator. Hence, the data of 3.1 rs ( U(2) theories corresponding rane ackground, corresponding to 51 m ( e conformal block correspond -ensemble deformation of the internal momenta map to the auge theory side and a specific +1 , =1 l field theory. Underlying both β Y ℓ n . characterizing the Ω-background 45 l Calabi-Yau geometry which en- 2 N =1 ]. These theories admit an A-D-E 2 i Y ǫ 1 , ǫ 27 ǫ β 1 2 ǫ ) − i =4 superconformal field theories, which . u = d 2 2 s +2 punctures, and its pants decomposition − /ǫ 4d SCFT/2d Toda theory for j n 1 u r ǫ ]. =2, ( , g A N = – 10 – 15 N 2 -deformation) to compute the Nekrasov partition ≤ 2 β /ǫ b 1 Y i

which has the desired property under monodromy = ( P 2 +1 = 4 ℓ C z + (1 + ! . By definition, with fixed external momenta, conformal 2 1 2 5 ˜ for an example. This ordering property of the contours ˜ ˜ C C C corresponding to the earliest interaction point 5

= ℓ cut through some gluing curves. Obviously, the correspondi 2 C 2;1 ˜ C +1 k ℓ M ) z cuts through a larger gluing curve connecting more than two t 2;1 ℓ z shown in figure M ( 2 to a nearby pole and to C ℓ is not integral, an inspection of the integral formula ( ℓ p z and enclosing all the poles enclosed in the gluing curve. Fin β +1 incurred by crossing the brach cut connecting the point ℓ z ℓ φ e , is the unique invariant combination up to a multiplicative Finally let us comment on the contours relevant for the compa The final case to consider is when both of the line segments ema The above conclusion can be extended to the cases when the str Moreover, we would like to bring the readers’ attention to a f 2 to a nearby pole to the pole , z ℓ ℓ 1 through the same result appliessegment when from we swap left and right and consid contours for computing the five-point conformal blocks is gi matrix reads to the rest of the light-cone diagram. In this case the contou and infinity. More precisely, in the basis given by the contou should be regarded asillustrated a in part the of examples the discussed definition in for section the non-pert mal blocks described inblocks section in a given channel with different internal momenta fur phase p p transformation. Hence we have the ordering of the contours. When considering the Dehn twists on thesimply apply left- the and above the argument separately right-hand on side them commute and combi with We would like to take a combination model when be simple tensor products ofa the completely same natural cycles way but howpoints will this on have can to the be b done.contours light-cone corresponding Namely, diagram to the distinct provides critical tim points.along a the We natural sho integration cycle orderin fact depend on the pre-existing contours. Hence in this case is invariant under the monodromy. It is not hard to see that th forward in time. See figure at the contour z which can be combined into a loop passing through the point JHEP09(2011)022 , 1 C lf of 5 ) and α 2 , z ( , then 4 2 ∞ 2 α α C , V = 4 5 z ), 3 z 1 4 α z z ( , 1 3 α z 3 V N 2 mal block. The ordering . 2 α 3 ) , N z 2 2 rface that are equivalent to z z l. ich is the light-cone diagram 2 − perators nce it is natural to conjecture ntours of the matrix integrals, C 1 z N . In this section, we will illustrate 2 -ensemble matrix model partition 1 up to overall (leg) factors. In the on points on the light-cone diagram. value along the contours z log( β α 1 3 5 , 2 C C 1 1 z m z , the 5 ) + 1 z − – 16 – z log( 1 m = 4 z s ) with the monodromy-diagonal contours described above, is /g ) 3 z 4.1 ( p 3 matrix with the above potential corresponds to the chiral ha z W 2 p N 2 z 1 p . 3 C as the conformal block described in section 1 . An example of the contours for computing a five-point confor z According to the dictionary given in section of the contour isHence given one should by first the perform time-ordering the of integration over the the interacti eigen Figure 5 function of a rank 7.1 Three-point functions The simplest non-trivial examplecorresponding is given to a the pair potential of pants, wh 7 Examples In the last section weand have proposed provided its a relation prescription to for theand Liouville the test conformal co blocks our prescription by studying a few examples in detai the same next section we will show some examples of such equivalence. monodromy transformation of the punctures onthe the Dehn Riemann twists su in thethat corresponding the light-cone matrix diagram. integral He ( and finally the three-point function, with the momenta of the inserted o JHEP09(2011)022 (7.1) (7.2) β 2 ) j and the rank u , W ] in our matrix − 1 2 i 13 z u , z ( 2 2∆ N | z ) invariance dictates s therefore free from ≤ . pants. In this case the C 13 vious sections to this , i z = [ Y . | i

··· z γ γ 2 V Q ,α h ) 2 − m 3 s 1 i = ,α z − 1 u 3 4∆ ( α 1 – 17 – W/g − | C N =1 m , α 3 . Hence all information of the three-point function , the contour given by the simple gradient line on i Y z Re( j 6 − = 2 1 | , then suggests that the matrix integral 1 i contour lines are ordered in such a way that they do not 6= = ) du 1 5 m 1 2 z →∞ 3 2 z N b 3 z z 1 ( z 2 i, k Z 1 bα α 2 − 1 = lim 6= V ∞ ··· p − ) 3 k = N 2 = ˜ ,α , z C = 2 2 , j ( 3 du 1 3 2 1 z ,α ∆ z α 2 1 α m z α 1 V − Z ) z C i 3 β 2 z given by the downward gradient flow of Re( ∆ / ( 2 3 ˜ − C α m 1 k V . h m 21 N z ) given by = ∆ ∞ . The simplest light-cone diagram is given by a single pair of ij ) = 2 = -plane does not cut through any gluing curve in this case and i 3 , z z 1 To see the validity of the above claim, first recall that the SL Now we shall apply the contour prescription described in pre W ( z 3 ( α Z is encoded in the object V where ∆ of the matrix To keep the notation uniform we also define Figure 6 contour is simply three-point function, with the three momenta determined by critical point of theintersect with potential. each other. The The corresponding quantity on th the the three-point functions to have the following form gives the chiral part of the three-pointleg-factors function which up only to norma depend on individual momenta but not any Dehn twist ambiguity. Therefore, we shall use the contou integral. The discussion in section simple case. As can be seen in figure JHEP09(2011)022 . 21 z (7.3) πi ) into 2 e 7.2 → . . ) β 21 2 z ) jβ j β t 2 + ) , j − 2 t ) ) transformation to i 2 t m C rix integral ( Z formula, we use the ( − ion factors. α ) , i 1 N t − ) ≤ α ( ) s 1 . 1 N ) − nformal blocks then states α Y α i

··· ··· ··· V z γ m . . a 1 b +1 er in our discussion of the four- N 3 1 1 2(∆ (1) N -channel conformal block shown on γ 1 | N γ 3 1 ,α s ) + γ N z 2 α 2 N 31 z V + 1 z ,α | ) 1 α 2 ) 4 − , α 4 α z 1 ( z C ∆ z . 4 + − α 1 ∼ 1 21 31 V ∆ α log( z z 3 ∗ h , the corresponding light-cone diagram might − 4 2 ,α 43 42 = 2 3 2 ∗ , z z 2 m 4∆ a , ,α | – 20 – , the corresponding contour of integration is the 1 +∆ 1 ∗ , it maps to the 4 = 3 6 z 1 α z | ) + ˆ z N F 1 2(∆ 3 z | − and →∞ ,α 41 − 4 2 3 , N z z 3 | 2 z ,α z , 2 1 = 1 α i 2 4∆ ) ˆ ) = lim m 2 F log( N 1 ¯ z − p | z , 1 ( 2 1 z, 42 1 z is the cross ratio of the four points ( m N z α 4 | 1 z ) V = ,α p 4 ) 3 2 s ∆ . z ,α 7 − ( 2 and 1 /g 3 2 ) z i ,α α ∆ z 1 z ( V ) invariance dictates the following form for the four-point − α ) 2 − C 3 G W , z j ( +∆ z 1 3 α = V . On the left is a light-cone diagram corresponding to the pot 2(∆ ) | ij 4 z z 43 ( z | 4 The SL(2 In particular, all the information of interest is contained α = V h where According to the prescription in section the right, with the internal momenta given by we show one of the possibilities which corresponds to the Figure 7 7.2 Four-point functions Now we consider the four-point function corresponding to th This property of the three-pointpoint function conformal will blocks. be useful lat one shown in figure the eigenvalue distribution satisfying Depending on the parameters take a different shape corresponding to a different pair of pan JHEP09(2011)022 an (7.6) ℓ , we then m ) 6 ) transfor- ℓ C z β , 2 − ) i i , . , u u ( β 31 to be given by the 2 2 −

) 3 =1 i j /z # Y ℓ ∞ u u z 21 and section ( ; nder SL(2 z N =1 = − N i Y we will show the validity 2 4 5 j ≤ 4 α α = β functions, are chosen such A u , we expect the four-point en the matrix integrals and 2 ( z 1 3 , z 5 . ) Y locks with different internal i

j , z # 2 ≤ 1 u corresponding to the given internal z 4 1) 1 ( 3 Nb z ,α ; 1 a ,α m N 2 4 3 − α ˆ − C , ≤ α α i 1) ,α C β ∗ u Q 2 1 3 # ( Y a i