Open intersection numbers, matrix models and MKP hierarchy

In this paper we conjecture that the generating function of the intersection numbers on the moduli spaces of Riemann surfaces with boundary, constructed recently by R. Pandharipande, J. Solomon and R. Tessler and extended by A. Buryak, is a tau-function of the KP integrable hierarchy. Moreover, it is given by a simple modification of the Kontsevich matrix integral so that the generating functions of open and closed intersection numbers are described by the MKP integrable hierarchy. Virasoro constraints for the open intersection numbers naturally follow from the matrix integral representation.


Introduction
Intersection numbers on the moduli spaces of Riemann surfaces is a challenging and complicated subject of enumerative geometry. While for closed Riemann surfaces an effective description is known for more than twenty years [1,2], a similar description of the open intersection numbers was not available. Recently, in the paper [3] the generating function of open intersection numbers was 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 subject the theory of matrix models, a power tool of modern mathematical physics. In this paper we present a simple and natural description of the generating function of the open intersection numbers.
Namely, let us consider a family of the Kontsevich-Penner models From the Kontsevich proof of Witten's conjecture [1,2] we know that intersection theory on the moduli spaces of closed Riemann surfaces is governed by a representative of this family with N = 0: which is a tau-function of the integrable KdV hierarchy. The main observation of this work is that N = 1 case corresponds to the open intersection theory. Namely, the extended generating function (which includes descendants of the boundary points), introduced and studied in [4,5], coincides with (1) for N = 1: Then, from matrix model theory it immediately follows that the extended open generating function is a tau-function of the KP hierarchy. Moreover, the variable N in (1) plays the role of the discrete time. Thus, (1) describes a solution of the modified KP (MKP) hierarchy, and, in addition to the KP (KdV) equations the tau-functions τ o and τ KW satisfy the bilinear identity We claim that the open KdV hierarchy equations, as well as other PDE's, obtained in [3][4][5] follow from the equations of the MKP integrable hierarchy. Virasoro constraints is a natural property of the matrix integrals. The Virasoro constrains, obtained for the tau-function τ 1 , are equivalent to the extended Virasoro constraints, derived in [5]. An advantage of our version of the Virasoro constraints is that they belong to the symmetry algebra of the integrable hierarchy, thus, they are natural from the point of view of integrability.
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 the open intersection numbers and the matrix model (1) for N = 1. In Section 3 we describe some general properties of the matrix integral (1) and identify the equations of the MKP hierarchy with the equations, obtained in [3][4][5]. Section 4 is devoted to concluding remarks. For the sake of simplicity in this paper we omit the genus expansion parameter (denoted by u in [3][4][5]), since it can be easily restored by rescaling of times.

Kontsevich-Witten tau-function
The closed intersection theory is governed by the Kontsevich-Witten tau-function, which is given by a formal series in times with rational coefficients: In the Miwa parametrization it is equal to the Kontsevich matrix integral [1, 2, 6-9] over the Hermitian matrix Φ: This integral depends on the external matrix Λ, which is assumed to be a positive defined diagonal matrix. The times t k are given by the Miwa transform of this matrix: All t k can be considered as independent variables as the size of the matrices tends to infinity and in this limit (6) gives the Kontsevich-Witten tau-function. After the shift of the integration variable one has where C = e Tr Λ 3 The Itzykson-Zuber formula allows us to reduce the r.h.s. of (6) to the ratio of determinants where the basis vectors are given by the integrals The coefficients of the basis vectors can be found explicitly, in particular The first line of (12) allows us to find the Kac-Schwarz operators of the KW tau-function [10,11]. Indeed, we have: where Thus, and the operator a KW is the Kac-Schwarz operator.
To construct another Kac-Schwarz operator we use the identity From this identity and the recursion relation (14) it follows that Thus, is also the Kac-Schwarz operator. The Kac-Schwarz operators (15) and (19) satisfy the canonical commutation relation and generate an algebra of the Kac-Schwarz operators for the KW tau-function. The above constructed Kac-Schwarz operators allow us to find two infinite series of operators, which annihilate the tau-function. One of them guarantees that the tau-function does not depend on even times so that it is a tau-function of the KdV hierarchy. From the general properties of the Kac-Schwarz operators it follows that the KW tau-function is an eigenfunction of the operators (21). The same is true for the Virasoro operators which correspond to the Kac-Schwarz operators in (22) belong to the W 1+∞ algebra of the symmetries of the KP hierarchy.
To find corresponding eigenvalues it is enough to check that these operators satisfy the commutation relations: 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: and Then, for any function Z depending only on odd times t 2m+1 , we have where the operators (29) constitute the same subalgebra of the Virasoro algebra: Thus, the Virasoro constraints (27) are equivalent to the standard Virasoro constraints for the KW tau-function, which follow from the invariance of the Kontsevich matrix integral

Open generating function
In [3] the intersection theory on the moduli spaces of Riemann surfaces with boundary was investigated, and the corresponding Virasoro constraints were introduced. These Virasoro constraints are proved in [4]. In [5] the generating function introduced in [3] was generalized, and the Virasoro constrains for this generalized (or extended) generating function were proved. Namely, the generating function of the open intersection numbers with descendants 1 where F c = log(τ KW ), satisfy the linear equations for n ≥ −1. It is obvious that these Virasoro operators do not belong to the W 1+∞ symmetry algebra of the KP hierarchy. According to [5] the open generating function is related to the KW tau-function by the residue formula where ξ(t, z) = ∞ k=1 t k z k and we use the standard notation 1 The function F o here is the extended open potential F o,ext of [5]. Below we use the variables, natural from the point of view of the integrable hierarchies and matrix models. Thus, the variables from [5] are related to our variables as t B k = (2k + 1)!!t 2k+1 , s B k = 2 k+1 (k + 1)!t 2k+2 .
The series is uniquely defined by the equation where a KW is the Kac-Schwarz operator for the KW tau-function (15). One can easily recover the integral representation for this series. Namely, it is given by the steepest descent expansion of the integral with a properly chosen contour C. Let us consider the Kontsevich matrix integral (6) with (M + 1) × (M + 1) matrix Λ = diag (y 1 , y 2 , . . . , y M , −z). Then, in the Miwa variables the relation (34) yields where Y = diag (y 1 , y 2 , . . . , y M ). In particular, for M = 1 we have we have We claim, that τ o is a KP tau-function, fixed by the set of basis vectors: so that it is given by the matrix integral where C is given by (10). This matrix integral belong to the family of the generalized Kontsevich models [6,9,12,13], and, for M (size of the matrix Φ) large enough, has the following expansion which coincides with the expansion of (34).
The Kac-Schwarz operator for the tau-function (44) is so that Let us stress that, contrary to the case of the KW tau-function, this tau-function depends both on odd and even times, since z 2 is not a Kac-Schwarz operator anymore: Nevertheless, from (47) it immediately follows that the operators for k ≥ −1 belong to the Kac-Schwarz algebra. The operators l o k satisfy the Virasoro commutation relations (with the trivial central charge): Then, from the general properties of the Kac-Schwarz operators [14] it follows that the tau-function τ o is an eigenvalue of the corresponding operators: where the operators L k are given by (24). These operators satisfy the commutation relation of the Virasoro algebra From these commutation relations it follows that for k ≥ −1 the eigenvalues of these operators are equal to zero, thus The Virasoro constraints (53) can be reduced to the constraints (33) with the help of relations proved in [5]. Thus, we see that up to the relations (54) the tau-function, given by the matrix integral (44) and an extended generating function of [5] satisfy the same Virasoro constraints. From the representation (44) it immediately follows that τ o satisfies the KP hierarchy, The first non-trivial equation contained here is We use the symbols D i for the "Hirota derivatives" defined by where P (D) is any polynomial in D i , so that (56) yields In the next section we will consider a more general integrable structure, equations of which are directly related to the equations derived in [3][4][5].

MKP hierarchy
Let us consider a family of the Kontsevich-Penner models [2,15].
which for N = 0 corresponds to the closed intersections and for N = 1 corresponds to the open ones. Here N is the independent parameter, which has nothing to do with the size of the matrices. Corresponding basis vectors satisfy the recursive relation is the Kac-Schwarz operator for τ N . Thus, from the general relation between the Kac-Schwarz operators and the operators from the W 1+∞ algebra it immediately follows that τ N satisfies the string equation Moreover, it is straightforward to check that the operators z 2 a N and z 4 a N − 2(N − 1)z 2 are also the Kac-Schwarz operators so that the tau-function satisfy the equations where and these three operators satisfy the commutation relations A complete family of the Virasoro and W-constraints can also be derived by variation of the matrix integral [12,16,17] and will be presented elsewhere. The functions of the family (59) with different N are related with each other by the differentialdifference equations of the KP/Toda type [12]. In particular, the tau-functions τ 0 and τ 1 satisfy the MKP integrable hierarchy. 2 It is given by the bilinear identity valid for all t, t ′ . This bilinear identity is equivalent to an infinite series of PDE's, the simplest of which is Since τ KW = exp(F c ) does not depend on even times, from the definition of the "Hirota derivatives" (57) it immediately follows that all operators D 2k in our case can be substituted by ∂ ∂t 2k . Then, equation (68) is equivalent to which was derived in [4]. A combination of this equation and the next equation of the MKP hierarchy leads to This is the first equation of the open KdV hierarchy of [3]. On the next level we have two equations from which, in particular, it immediately follows the first equation of (54) We claim that other equations of the open KdV hierarchy and other equations of [3][4][5] also follow from the bilinear identity (67) of the MKP hierarchy.

Concluding remarks
In this paper we present a description of the open intersection numbers by means of the generalized Kontsevich model. It is more then natural to look for other elements of the modern matrix model theory in this case and to apply these elements to the investigation of the open intersection theory. These elements, in particular, include: • Cut-and-join type operator 3 • Givental decomposition • (Quantum) spectral curve

• Topological recursion
It would also be interesting to establish the meaning of the whole family (59) 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. We are going to return to this subject in future publications [21].