aa r X i v : . [ m a t h - ph ] M a r Open intersection numbers, matrix models and MKPhierarchy
A. Alexandrov a Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, Germany &Mathematics Institute, University of Freiburg, Germany &ITEP, Moscow, Russia
ITEP/TH-32/14
In this paper we conjecture that the generating function of the intersection numbers onthe moduli spaces of Riemann surfaces with boundary, constructed recently by R. Pand-haripande, J. Solomon and R. Tessler and extended by A. Buryak, is a tau-function of theKP integrable hierarchy. Moreover, it is given by a simple modification of the Kontsevichmatrix integral so that the generating functions of open and closed intersection numbers aredescribed by the MKP integrable hierarchy. Virasoro constraints for the open intersectionnumbers naturally follow from the matrix integral representation.
Keywords: enumerative geometry, matrix models, tau-functions, KP hierarchy, Virasoro con-straints a E-mail: alexandrovsash at gmail.com ntroduction Intersection numbers on the moduli spaces of Riemann surfaces is a challenging and complicatedsubject of enumerative geometry. While for closed Riemann surfaces an effective description isknown for more than twenty years [1, 2], a similar description of open intersection numbers wasnot available. Recently, in the paper [3] the generating function of open intersection numberswas described by the Virasoro constraints and an infinite hierarchy of PDE’s, called there the“open KdV hierarchy.” This important development makes it possible to apply to the subjectthe theory of matrix models, a power tool of modern mathematical physics. In this paper wepresent a simple and natural description of the generating function of the open intersectionnumbers.Namely, let us consider a family of the Kontsevich-Penner models τ N = det(Λ) N C − Z [ d Φ] exp (cid:18) − Tr (cid:18) Φ − Λ Φ2 + N log Φ (cid:19)(cid:19) . (1)From the Kontsevich proof of Witten’s conjecture [1, 2] we know that intersection theory on themoduli spaces of closed Riemann surfaces is governed by a representative of this family with N = 0: τ KW = τ , (2)which is a tau-function of the integrable KdV hierarchy. The main observation of this work isthat N = 1 case corresponds to the open intersection theory. Namely, the extended generatingfunction (which includes descendants of the boundary points), introduced and studied in [4, 5],coincides with (1) for N = 1: τ o = τ . (3)Then, from matrix model theory it immediately follows that the extended open generatingfunction is a tau-function of the KP hierarchy. Moreover, the variable N in (1) plays the role ofthe discrete time. Thus, (1) describes a solution of the modified KP (MKP) hierarchy, and, inaddition to the KP (KdV) equations the tau-functions τ o and τ KW satisfy the bilinear identity I ∞ e ξ ( t − t ′ ,z ) z τ o ( t − [ z − ]) τ KW ( t ′ + [ z − ]) dz = 0 . (4)We claim that the open KdV hierarchy equations, as well as other PDE’s, obtained or conjecturedin [3–5] for the generating function of open intersection numbers follow from the equations ofthe MKP integrable hierarchy.Virasoro constraints is a natural property of the matrix integrals. The Virasoro constrains,obtained for the tau-function τ , are equivalent to the extended Virasoro constraints, derivedin [5]. An advantage of our version of the Virasoro constraints is that they belong to thesymmetry algebra of the integrable hierarchy, thus, they are natural from the point of view ofintegrability.The present paper is organized as follows. Section 1 contains material on the Kontsevich–Witten tau-function. In Section 2 we establish a relation between the generating function of the2pen intersection numbers and the matrix model (1) for N = 1. In Section 3 we describe somegeneral properties of the matrix integral (1) and identify the equations of the MKP hierarchywith the equations, obtained in [3–5]. Section 4 is devoted to concluding remarks. For the sakeof simplicity in this paper we omit the genus expansion parameter (denoted by u in [3–5]), sinceit can be easily restored by rescaling of times. The closed intersection theory is governed by the Kontsevich–Witten tau-function. Let M p ; n bethe Deligne–Mumford compactification of the moduli space of genus p complex curves X with n marked points x , . . . , x n . Let us associate with a marked point a line bundle L i whose fiberat a moduli point ( X ; x , . . . , x n ) is the cotangent line to X at x i . Intersection numbers of thefirst Chern classes of these holomorphic line bundles Z M p ; n ψ m ψ m . . . ψ m n n = h σ m σ m . . . σ m n i (5)are rational numbers, not equal to zero only if n X i =1 ( m i −
1) = 3 p − . (6)Their generating function F c ( t ) = * exp ∞ X m =0 (2 m + 1)!! t m +1 σ m !+ (7)is given by the Kontsevich–Witten tau-function which is a formal series in odd times t k +1 withrational coefficients: τ KW ( t ) = exp ( F c ( t ))= 1 + 16 t + 18 t + 172 t + 2548 t t + 25128 t + 58 t t + 11296 t + 49576 t t + 1225768 t t + 3548 t t + 12253072 t + 24564 t t t + 3516 t t + 105128 t + . . . (8)In the Miwa parametrization it is equal to the asymptotic expansion of the Kontsevich matrixintegral [1, 2, 6–11] over the Hermitian matrix Φ: τ KW ([Λ]) = Z [ d Φ] exp (cid:18) − Tr (cid:18) Φ
3! + ΛΦ (cid:19)(cid:19)Z [ d Φ] exp (cid:18) − Tr ΛΦ (cid:19) . (9) We use the variables t k , which are the standard variables of the KP hierarchy. From (7) it is clear that theydo not coincide with the variables, natural for the intersection theory. The difference between two families ofvariables is given by the factor (2 m + 1)!! (see also footnote 2). t k are given by the Miwa transform of this matrix: t k = 1 k Tr Λ − k . (10)All t k can be considered as independent variables as the size of the matrices tends to infinityand in this limit (9) gives the Kontsevich–Witten tau-function. After the shift of the integrationvariable Φ = X − Λ (11)one has τ KW ([Λ]) = C − Z [ dX ] exp (cid:18) − Tr (cid:18) X − Λ X (cid:19)(cid:19) , (12)where C = e Tr Λ33 Z [ d Φ] exp (cid:18) − Tr ΛΦ (cid:19) . (13)The Harish-Chandra–Itzykson–Zuber formula [12, 13] allows us to reduce the r.h.s. of (9) tothe ratio of determinants [6, 7] τ KW ([Λ]) = det Ni,j =1 Φ KWi ( λ j )∆ ( λ ) . (14)Here λ j are the eigenvalues of the matrix Λ and∆( z ) = Y i KWk ( z ) = r z π e − z Z ∞−∞ dy y k − exp (cid:18) − y 3! + yz (cid:19) = r z π Z ∞−∞ dy ( y + z ) k − exp (cid:18) − y − y z (cid:19) . (16)The coefficients of the basis vectors can be found explicitly, in particular [7, 11]Φ KW ( z ) = ∞ X k =0 k Γ (cid:0) k + (cid:1) k (2 k )! Γ (cid:0) (cid:1) z − k , Φ KW ( z ) = − ∞ X k =0 k + 16 k − k Γ (cid:0) k + (cid:1) k (2 k )! Γ (cid:0) (cid:1) z − k . (17)4he first line of (16) allows us to find the Kac–Schwarz operators of the KW tau-function[14, 15]. Indeed, we have:Φ KWk +1 ( z ) = r z π e − z (cid:18) z ∂∂z (cid:19) Z ∞−∞ dy y k − exp (cid:18) − y 3! + yz (cid:19) = a KW Φ KWk ( z ) , (18)where a KW = 1 z ∂∂z + z − z . (19)Thus, the operator a KW preserves the subspace spanned by the vectors Φ KWi a KW (cid:8) Φ KW (cid:9) ⊂ (cid:8) Φ KW (cid:9) (20)and it is the Kac–Schwarz operator (see, e.g., [16] for more details).To construct another Kac–Schwarz operator we use the identity (cid:0) a KW − z (cid:1) Φ KW ( z ) = 0 . (21)From this identity and the recursion relation (18) it follows that z Φ KWk = Φ KWk +2 − k − KWk − . (22)Thus, b KW = z (23)is also the Kac–Schwarz operator. The Kac–Schwarz operators (19) and (23) satisfy the canonicalcommutation relation [ a KW , b KW ] = 2 (24)and generate an algebra of the Kac–Schwarz operators for the KW tau-function.Given a Kac–Schwarz operator, there is an explicit formula for an operator from the algebra W ∞ such that the corresponding tau-function is an eigenfunction of this operator [16]. Inparticular, the above given Kac–Schwarz operators allow us to construct two infinite series ofoperators, which annihilate the tau-function. One of them guarantees that the tau-function doesnot depend on even times b J KWk = ∂∂t k , k ≥ , (25)so that it is a tau-function of the KdV hierarchy.The second series is given by the Virasoro operators b L KWk = 12 b L k − ∂∂t k +3 + 116 δ k, , k ≥ − , (26)5hich correspond to the Kac–Schwarz operators l KWk = − (cid:16) ( b KW ) k +1 a KW + a KW ( b KW ) k +1 (cid:17) . (27)The operators b L m = 12 X a + b = − m abt a t b + ∞ X k =1 kt k ∂∂t k + m + 12 X a + b = m ∂ ∂t a ∂t b (28)in (26) generate a family of the Virasoro operators from the W ∞ algebra of the symmetriesof the KP hierarchy.To find corresponding eigenvalues it is enough to check that the operators (25) and (26)satisfy the commutation relations: h b J KWk , b J KWm i = 0 , k, m ≥ hb L KWk , b J KWm i = − m b J KWk + m , k ≥ − , m ≥ , hb L KWk , b L KWm i = ( k − m ) b L KWk + m , k, m ≥ − . (29)Since all generators of the algebra can be obtained as commutators of some other generators,the eigenvalues of all of them are equal to zero: b J KWm τ KW = 0 , m ≥ b L KWm τ KW = 0 , m ≥ − . (31)Then, for any function Z depending only on odd times t m +1 , we have b L k Z = (cid:18)b L k + 18 δ k, (cid:19) Z, k ≥ − , (32)where the operators b L m = ∞ X k =1 (2 k + 1) t k +1 ∂∂t k +2 m +1 + 12 m − X k =0 ∂ ∂t k +1 ∂t m − k − + t δ m, − + 18 δ m, , m ≥ − h b L n , b L m i = 2( n − m ) b L n + m , k, m ≥ − . (34)6hus, the Virasoro constraints (31) are equivalent to the standard Virasoro constraints for theKW tau-function b L m τ KW = ∂∂t m +3 τ KW , m ≥ − , (35)which follow from the invariance of the Kontsevich matrix integral. In [3] the intersection theory on the moduli spaces of Riemann surfaces with boundary wasinvestigated. In particular, open intersection numbers in the genus zero were constructed, andthe generalization to all higher genera was conjectured. This conjectural all-genera generatingfunction is uniquely specified by the so called open KdV equations and the Virasoro constraints.In [4] it is proved that the open KdV equations and the corresponding Virasoro constraints giveequivalent descriptions of the (conjectural) intersection theory on the moduli space of Riemannsurfaces with boundary. In [5] the generating function introduced in [3] was generalized todescribe also the descendants on the boundary, and the Virasoro constrains for this conjecturalgeneralized (or extended) generating function were proved. Namely, the generating function ofopen intersection numbers with descendants τ o = exp( F o + F c ) , (36)where F c = log( τ KW ), satisfy the linear equations b L n + 2 ∞ X k =0 k t k ∂∂t k +2 n + 3 n + 32 ∂∂t n + 2 t δ n, − + 32 δ n, − ∂∂t n +3 ! τ o = 0 (37)for n ≥ − 1. These Virasoro operators can not be represented as a linear combination of theoperators (28) and the operators t k , ∂∂t k , thus it is obvious that they do not belong to the W ∞ symmetry algebra of the KP hierarchy.According to [5] the open generating function is related to the KW tau-function by theresidue formula τ o ( t ) = 12 πi I dzz D ( z ) τ KW ( t − (cid:2) z − (cid:3) ) exp( ξ ( t , z )) , (38)where ξ ( t , z ) = P ∞ k =1 t k z k and we use the standard notation t ± (cid:2) z − (cid:3) = (cid:26) t ± z , t ± z , t ± z , . . . (cid:27) . (39)The series D ( z ) = 1 + X k =1 d k z k = 1 + 4124 z − + 92411152 z − + 507522582944 z − + 51530089457962624 z − + . . . (40) The function F o here is the extended open potential F o,ext of [5]. Below we continue to use the variables,natural from the point of view of the integrable hierarchies and matrix models. Thus, the variables from [5] arerelated to our variables as t Bk = (2 k + 1)!! t k +1 , s Bk = 2 k +1 ( k + 1)! t k +2 . 7s uniquely defined by the equation a KW (cid:18) z D ( z ) (cid:19) = Φ KW ( z ) , (41)where a KW is the Kac–Schwarz operator for the KW tau-function (19), thus (18) allows us toidentify Φ KW ( z ) = 1 z D ( z ) . (42)One can easily recover the integral representation for this series. Namely, it is given by thesteepest descent expansion of the integral D ( z ) = z / √ π e − z Z C d y y − exp (cid:18) − y 3! + yz (cid:19) , (43)with a properly chosen contour C .Let us consider the Kontsevich matrix integral (9) with ( M + 1) × ( M + 1) matrix Λ =diag ( y , y , . . . , y M , − z ). Then, in the Miwa variables the relation (38) yields τ o ([ Y ]) = 12 πi I dzz D ( z ) τ KW ([Λ]) det (cid:18) YY − z (cid:19) , (44)where Y = diag ( y , y , . . . , y M ). In particular, for M = 1 we have τ o ([ y ]) = D ( y ) Φ KW ( − y )Φ KW ( y ) − Φ KW ( y )Φ KW ( − y )2 y . (45)Since Φ KW ( − y )Φ KW ( y ) − Φ KW ( y )Φ KW ( − y ) = 2 y, (49)we have τ o ([ y ]) = D ( y ) = y Φ KW ( y ) . (50) This relation is valid for the basis vectors of any KdV tau-function. Indeed, consider a KdV tau-function τ ( t )in the Miwa parametrization with 2 × y, − y ). In this parametrization all odd times(10) vanishes t k +1 = 12 k + 1 (cid:18) y k +1 − y k +1 (cid:19) = 0 (46)and the tau-function is identically equal to 1. On the other hand, the same tau-function in this parametrizationcan be represented in terms of the basis vectors as a ratio of determinants of the form (14) τ ([Λ]) = Φ ( y )Φ ( − y ) − Φ ( − y )Φ ( y )( − y ) (47)so that for any tau-function independent of even times the basis vectors satisfyΦ ( − y )Φ ( y ) − Φ ( y )Φ ( − y ) = 2 y. (48) 8e conjecture, that the extended generating function of open intersection numbers τ o is aKP tau-function, fixed by the set of basis vectors:Φ oj ( z ) = z Φ KWj − ( z ) = z / √ π e − z Z d y y j − exp (cid:18) − y 3! + yz (cid:19) , j = 1 , , , . . . (51)so that it is given by the matrix integral τ o ([Λ]) = C − det(Λ) Z [ d Φ] exp (cid:18) − Tr (cid:18) Φ − Λ Φ2 + log Φ (cid:19)(cid:19) , (52)where C is given by (13). This matrix integral belongs to the family of the generalized Kontsevichmodels [6, 7, 10, 11, 17, 18], and, for M (size of the matrix Φ) large enough, has the followingexpansion τ o = 1 + 138 t + 2 t t + 16 t +8 t + 172 t + 43 t + 3748 t t + 13 t t + 2 t t + 374 t t t + 4 t t + 8 t t + 481128 t + 658 t t + 45516 t t + 61576 t t + 2257768 t t + 9548 t t + 7665128 t + 396564 t t t + 11296 t + 293413072 t + 149 t t + 13 t t + 136 t t + 83 t t + 32 t t + 64 t t + 61 t t + 283 t t +60 t t + 30 t t + 23 t t + 616 t t + 2454 t t + 64 t t t + 1254 t t t + 32 t t t + 283 t t t + 61 t t t + 612 t t t + 225764 t t t + 6124 t t t + 614 t t t + . . . , (53)which coincides with the expansion of (38).The Kac–Schwarz operator for the tau-function (52) is a o = z a KW z − = 1 z ∂∂z − z + z, (54)so that Φ ok +1 ( z ) = a o Φ ok ( z ) . (55)Let us stress that, contrary to the case of the KW tau-function, this tau-function depends bothon odd and even times, since z is not a Kac–Schwarz operator anymore: z Φ o ( z ) / ∈ { Φ o ( z ) } . (56)Nevertheless, from (55) it immediately follows that the operators l ok = − z k +2 a o = − z k +2 (cid:18) z ∂∂z − z + z (cid:19) (57)9or k ≥ − l ok satisfy the Virasoro commu-tation relations (with the trivial central charge):[ l ok , l om ] = 2( k − m ) l k + m . (58)Then, from the general properties of the Kac–Schwarz operators [16] it follows that the tau-function τ o is an eigenfunction of the corresponding operators: b L o − = b L − − ∂∂t + 2 t , b L o = b L − ∂∂t + 18 + 32 , b L ok = b L k − ∂∂t k +3 + ( k + 2) ∂∂t k , k > , (59)where the operators b L k are given by (28). These operators satisfy the commutation relation ofthe Virasoro algebra hb L ok , b L om i = 2( k − m ) b L ok + m , k, m ≥ − . (60)From these commutation relations it follows that for k ≥ − b L ok τ o = 0 , k ≥ − . (61)The Virasoro constraints (61) can be reduced to the constraints (37) with the help of relations ∂∂t k τ o = ∂ k ∂t k τ o (62)proved in [5]. Thus, we see that up to the relations (62) the tau-function, given by the matrixintegral (52) and an extended generating function of [5] satisfy the same Virasoro constraints.The KP hierarchy for the generating function τ o which, in particular, immediately followsfrom the conjectural matrix integral representation (52), is described by the bilinear identity I ∞ e ξ ( t − t ′ ,z ) τ o ( t − [ z − ]) τ o ( t ′ + [ z − ]) dz = 0 . (63)The first non-trivial Hirota equation contained here is( D + 3 D − D D ) τ o · τ o = 0 . (64)We use the symbols D i for the “Hirota derivatives” defined by P ( D ) f ( t ) · g ( t ) := P ( ∂ X )( f ( t + X ) g ( t − X )) | X =0 , (65)10here P ( D ) is any polynomial in D i , so that (64) yields the KP equation in terms of thetau-function τ o ∂ τ o ∂t − ∂τ o ∂t ∂ τ o ∂t + 3 (cid:18) ∂ τ o ∂t (cid:19) + 3 τ o ∂ τ o ∂t − (cid:18) ∂τ o ∂t (cid:19) − τ o ∂ τ o ∂t ∂t + 4 ∂τ o ∂t ∂τ o ∂t = 0 . (66)In the next section we will consider a more general integrable structure, equations of which aredirectly related to the equations derived in [3–5]. Let us consider a family of the Kontsevich-Penner models [2, 19] τ N = det(Λ) N C − Z [ d Φ] exp (cid:18) − Tr (cid:18) Φ − Λ Φ2 + N log(Φ) (cid:19)(cid:19) , (67)which for N = 0 corresponds to the closed intersections and for N = 1 according to ourconjecture corresponds to the open ones. Here N is the independent parameter, which hasnothing to do with the size of the matrices.Corresponding basis vectorsΦ Nj ( z ) = z N Φ KWj − N ( z ) = z N +1 / √ π e − z Z d y y j − − N exp (cid:18) − y 3! + yz (cid:19) , j = 1 , , , . . . (68)satisfy the recursive relation a N Φ Nj = Φ Nj +1 , (69)where a N = z N a KW z − N = 1 z ∂∂z − (cid:18) N + 12 (cid:19) z + z (70)is the Kac–Schwarz operator for τ N . Thus, from the general relation between the Kac–Schwarzoperators and the operators from the algebra W ∞ it immediately follows that τ N satisfies thestring equation (cid:18)b L − − ∂∂t + 2 N t (cid:19) τ N = 0 . (71)Moreover, it is straightforward to check that the operators z a N and z a N − N − z arealso the Kac–Schwarz operators so that the tau-function satisfy the equations b L k τ N = 0 , k = − , , , (72)11here b L − = b L − − ∂∂t + 2 N t , b L = b L − ∂∂t + 18 + 3 N , b L = b L − ∂∂t + 3 N ∂∂t , (73)and these three operators satisfy the commutation relations hb L i , b L j i = 2( i − j ) b L i + j . (74)A complete family of the Virasoro and W-constraints can also be obtained by variation of thematrix integral [17, 20, 21] and is derived in [22].The functions of the family (67) with different N are related with each other by the differential-difference equations of the KP/Toda type [17]. In particular, the tau-functions τ and τ satisfythe MKP integrable hierarchy. It is given by the bilinear identity I ∞ e ξ ( t − t ′ ,z ) z τ o ( t − [ z − ]) τ KW ( t ′ + [ z − ]) dz = 0 (75)valid for all t , t ′ . This bilinear identity is equivalent to an infinite series of PDE’s, the simplestof which is (cid:0) D − D (cid:1) τ o · τ KW = 0 . (76)Since τ KW = exp( F c ) does not depend on even times, from the definition of the “Hirota deriva-tives” (65) it immediately follows that all operators D k in our case can be substituted by ∂∂t k .Then, equation (76) is equivalent to ∂F o ∂t = 2 ∂ F c ∂t + ∂ F o ∂t + (cid:18) ∂F o ∂t (cid:19) , (77)which was derived in [4]. A combination of this equation and the next equation of the MKPhierarchy (cid:0) D − D + 3 D D (cid:1) τ o · τ KW = 0 (78)leads to ∂F o ∂t = ∂F o ∂t ∂F o ∂t + ∂ F c ∂t ∂F o ∂t + ∂ F o ∂t ∂t − ∂ F o ∂t . (79)This is the first equation of the open KdV hierarchy of [3]. For more details on MKP hierarchy see, e.g., [23, 24] and references therein. 12n the next level we have two equations (cid:0) D − D D + 3 D − D (cid:1) τ o · τ KW = 0 , (cid:0) D − D − D D (cid:1) τ o · τ KW = 0 , (80)from which, in particular, it immediately follows the first equation of (62) ∂∂t τ o = ∂ ∂t τ o . (81)We conjecture that other equations of the open KdV hierarchy and other equations of [3–5] alsofollow from the bilinear identity (75) of the MKP hierarchy. In this paper we present a description of the open intersection numbers by means of the gen-eralized Kontsevich model. It is more then natural to look for other elements of the modernmatrix model theory in this case and to apply these elements to the investigation of the openintersection theory. These elements, in particular, include: • Cut-and-join type operator • Givental decomposition • (Quantum) spectral curve • Topological recursionThe generating function of open intersection numbers should also describe the model of the opentopological string with the simplest possible target-space (a point). It would also be interestingto establish the meaning of the whole family (67) from the point of view of enumerative geometry.We also expect that other families of the generalized Kontsevich models should be related to r -spin versions of the open intersection numbers. Some of the above mentioned topics areconsidered in [22]. Acknowledgments The author is grateful to A.Morozov for useful discussions and an anonymous referee for manyhelpful suggestions. This work was supported in part by RFBR grant 14-01-00547, NSh-1500.2014.2 and by Federal Agency for Science and Innovations of Russian Federation. 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