Open intersection numbers, Kontsevich-Penner model and cut-and-join operators

We continue our investigation of the Kontsevich--Penner model, which describes intersection theory on moduli spaces both for open and closed curves. In particular, we show how Buryak's residue formula, which connects two generating functions of intersection numbers, appears in the general context of matrix models and tau-functions. This allows us to prove that the Kontsevich--Penner matrix integral indeed describes open intersection numbers. For arbitrary $N$ we show that the string and dilaton equations completely specify the solution of the KP hierarchy. We derive a complete family of the Virasoro and W-constraints, and using these constraints, we construct the cut-and-join operators. The case $N=1$, corresponding to open intersection numbers, is particularly interesting: for this case we obtain two different families of the Virasoro constraints, so that the difference between them describes the dependence of the tau-function on even times.


Introduction
In our previous paper [1] we have shown that the Kontsevich-Penner model [2][3][4][5][6] is directly related to the intersections on the moduli spaces. Namely, we clamed that in addition to the wellknown description of the intersections on the moduli spaces of the closed Riemann surfaces [?,7,8] it also describes intersection theory on the moduli spaces of the Riemann surfaces with boundary. This intersection theory has been recently constructed and investigated by R. Pandharipande, J. Solomon, R. Tessler and A. Buryak [10][11][12] (see also [13,14]).
The Kontsevich-Penner matrix model for N = 0 coincides with the famous Kontsevich matrix model, which is known to describe the intersections on the moduli spaces of closed Riemann surfaces. In this paper we prove that for N = 1 this integral indeed can be identified with the generating function of open intersection numbers of R. Pandharipande, J. Solomon, R. Tessler and A. Buryak. For this purpose, in particular, we prove that Buryak's residue formula [12], which describes a relation between open and closed intersection numbers, follows from the matrix integral representation (1). Moreover, we show how a generalization of Buryak's formula appears in the general context of the Grassmannian description of the KP/Toda-type integrable hierarchies [15,16].
In this paper we also draw attention to the properties of the tau-function (1) for general N . Using the Sato Grassmannian description we derive the full family of the Virasoro and W-constraints, which completely specify the partition function of the Kontsevich-Penner model for arbitrary N . In particular, (1) satisfies the string equation and the dilaton equation Contrary to the constraints for the generalized Kontsevich model with the monomial potential [3,17,18] our constraints for general N do not belong to the W 1+∞ algebra of symmetries of the integrable hierarchy. Obtained constraints allowed us to construct the cut-and-join type operator, which yields an explicit expression for the tau-function (1). 1 The coefficients of the series expansion of (1) depend of the parameter N in a relatively simple way. Namely, they are polynomials in N . This property allows us to consider N as a continuous parameter. As we have already seen, at least for two values of N the Kontsevich-Penner matrix integral corresponds to interesting problems of enumerative geometry. The case N = 0, which describes the Kontsevich-Witten tau-function of the KdV hierarchy, is very well studied. In particular, to completely specify the generating function in this case we do not need higher W-constraints, and the cut-and-join operator can be derived from the the Virasoro constraints [19]. The properties of the generating functions of open intersection numbers (N = 1) are quite different. It appears that in this case we have two different families of the Virasoro constraints. The difference between them describes the dependence of the tau-function of the even times This relation for the generating function of the open intersection numbers was established in [12]. We claim that for a positive integer N the tau-function (1) is also related to interesting enumerative geometry and topological string theory models. In this paper we describe in some details the case N = 2. For this case we have two families of the cubic W-operators. The difference between them describes a dependence of the even times t 2k for k > 2 and yields an analog of the relation (4). It is well known that there exists a unique KdV tau-function, satisfying the string equation, namely, the Kontsevich-Witten tau-function [17,20]. We found an analogous description for the Kontsevich-Penner model (1). Namely, we prove that for arbitrary N there is a unique tau-function of the KP hierarchy, satisfying both the string equation (2) and dilaton equation (3).
The present paper is organized as follows. In Section 1 we briefly remind the reader the action of the w 1+∞ algebra on Sato's Grassmannian and show, how one can describe the acton of the simple operators from the universal enveloping algebra of W 1+∞ on the tau-functions. In Section 2 we prove that the tau-function (1) for N = 1 is given by Buryak's formula, thus proving the matrix integral representation of the generating function of open intersection numbers. Section 3 contains the derivation of the finite number of Virasoro and cubic W-constraints for general N , which follow from the existence of the Kac-Schwars operators and belong to the W 1+∞ algebra. In Section 4 we derive the complete (infinite) family of the Virasoro and cubic W-constraints, which allow us to construct the cut-and-join operator in Section 5. Sections 6 and 7 are devoted to the case N = 1, which corresponds to the open intersection numbers. In Section 8 we briefly describe the tau-function (1) for integer N > 1, in particular, we investigate the dependence on the even times for the next interesting case (N = 2). In Appendix A we give the first terms of the series expansion of the tau-function τ N and the corresponding free energy.

W 1+∞ algebra and the Sato Grassmannian
In this section we give a brief reminder of some important properties of the algebra w 1+∞ and its central extension, the algebra W 1+∞ . They describe the symmetries of the KP integrable hierarchy and play a central role in our construction. For more details see, i.e., [15][16][17]21] and references therein.
The KP hierarchy can be described by the bilinear equation, satisfied by the tau-function where ξ(t, z) = ∞ k=1 t k z k and we use the standard notation From the free fermion description of the KP hierarchy it immediately follows that the operators correspond to the bilinear combinations of fermions and span the algebra W 1+∞ of symmetries of the KP hierarchy. 2 Here J(z) is the so-called bosonic current where The normal ordering for bosonic operators * * . . . * * puts all operators J k with positive k to the right of all J k with negative k.
The Virasoro subalgebra of W 1+∞ is generated by the operators, bilinear in J k namely it is spanned by the operators The operators from the W (3) algebra, are generated by The operators J k , L k , and M k satisfy the following commutation relations A commutator of two operators from W (3) contains the terms of fourth power of the current components J m , so it can not be represented as a linear combination of J k , L k , and M k . The description of the integrable hierarchies in terms of the Grassmannian [15,16] allows us to consider the operators from the algebra w 1+∞ (the differential operators in one variable, which describe diffeomorphisms of the circle) instead of the operators from W 1+∞ . This significantly simplifies the calculations in some cases. In this paper we consider the tau-functions, given by matrix models of the Kontsevich type. Thus, we use the following Miwa parametrization for a diagonal matrix Z = diag (z 1 , z 2 , . . . , z M ). A tau-function of the KP hierarchy in this parametrization is given by where ∆(z) is the Vandermonde determinant and we use a natural generalization of the notation (6), For the parametrization (15) a relation between the algebras w 1+∞ and W 1+∞ is as follows. The algebra w 1+∞ is spanned by the operators (−∂ z ) m z −k which are identified with the operators [21] from W 1+∞ : When the size M of the auxiliary matrix Z in (16) tends to infinity, we have an infinite number of Miwa parameters: Then, for any operator b ∈ w 1+∞ and the corresponding operator Let us denote the determinant in the numerator by where Φ i is an infinite column Then, the family of the group elements, corresponding to the Y b ∈ W 1+∞ act as follows: Here ǫ is an arbitrary parameter and it is assumed that in the jth row the operator b acts on the variable z j . The first two terms of expansion of this identity in ǫ give respectively and

Buryak's residue formula
In this section we prove that the tau-function τ 1 indeed describes the open intersection theory. In particular, we show that the residue formula for the generation function, proved in [12], follows from the determinant expression (16) for the tau-functions of the KP hierarchy. Moreover, this type of relations appears to be universal for tau-functions.
Any KP tau-function can be expanded in the Schur polynomials Then, the sum restricted to Young diagrams with at most n non-zero lines is also a KP tau-function for all n > 0. This type of tau-functions often appears in matrix integrals [22,23]. Assume we have an expression for the tau-function in the Miwa parametrization τ Z −1 . Then, the orthogonality of the Schur functions allows us to find the tau-function, dependent on an infinite number of times: where Z = diag (z 1 , z 2 , . . . , z M ). On substitution the determinant representation (16) we obtain Here we use the antisymmetry of the Vandermonde determinant. Let us extract the integral over z 1 : where Here (30) and (32) we see that Z(t; z 1 ) can be identified with the tau-function When M tends, to infinity we get a relation Thus, we proved the following statement: for any tau-function τ and arbitrary series Φ 1 (z) = 1 + O(z −1 ) the residue (35) gives a tau-function of the KP hierarchy. Moreover, it is easy to see that the resulting tau-function satisfies the MKP hierarchy equation The relation is a particular case of a more general relation between tau-functions. Namely, in the same way it is easy to show that for any tau-functions τ and τ * the function τ defined by is a tau-function. The corresponding point of the Sato Grassmannian is given by For the tau-function τ N of the Kontsevich-Penner model considered in Section 3 relation (35) reduces to In particular, for N = 1 we have Since τ 0 = τ KW , the r.h.s. coincides with the expression for the generating function of the open intersection numbers, derived by A. Buryak in [12]. Thus, we proved that the generating function of open intersection numbers is given by the Kontsevich-Penner model for N = 1 3 Kac-Schwarz operators and corresponding constraints for general N As we have established in our previous work [1], an operator is the KS operator for the tau-function, corresponding to the Kontsevich-Penner model Namely, for the basis vectors with a properly chosen contour C, we have a relation The basis vectors (44) have an expansion where p = k − N . Using the integral representation (44) it is easy to see that an operator of multiplication by z 2 acts as follows: This operator is not the KS operator for N = 0, because However, is straightforward to check that the operators are the KS operators for any N [1]. For example, from (47) we see that for l 1 it is enough to check the condition (18) only for Φ N 1 . A constant term in the operator l 0 is chosen in such a way that the following commutation relations hold: This algebra can be extended to the full semi-infinite Virasoro algebra of the KS operators l k = z 2k+2 a N + . . . with k ≥ −1 only for N = 0 (the KW tau-function, both a 0 and z 2 are the KS operators, so that any of their combinations is also the KS operator) and for N = 1 (the open intersection numbers of [10][11][12], any operator z 2k a k for k ≥ 0 is the KS operator).
The relation (19) shows that the operators (49) correspond to the following operators from the W 1+∞ The operators L i satisfy the commutation relation of the subalgebra of the Virasoro algebra so that the constrains are satisfied. In what follows we call the equations with k = −1 and k = 0 the string equation and the dilaton equations. 3 Let us show that the string and dilaton equations uniquely specify the solution of the KP hierarchy (in the same way as the string equation specifies the KW tau-function of the KdV hierarchy [17,20]). We follow the approach of [24], namely, we prove that the corresponding KS operators l −1 and l 0 specify a point of the Sato Grassmannian (let us note that these operators, however, do not generate the KS algebra for τ N ). Indeed, the operator l −1 = a N allows us to find all higher basis vectors via (45) if the first basis vector is known. Thus, it remains to show that the first basis vector is completely defined by the KS operators l −1 and l 0 . Indeed, from the definition of the KS operators, it follows that the series l 0 Φ N 1 = z 3 + . . . should be a combination of the basis vectors: On substitution of the anzats into this equation, we immediately obtain an expression for the coefficients α k : are the KS operators. Of course, these operators are not unique KS operators with the leading terms z 2k−4 a 2 N . Namely, one can add to them a combination of the operators (49) and a constant. Our choice corresponds to the commutation relations The correspondence (19) for the operators (58) yields so that in general we can write For k = −1, 0, 1 and m = −2, −1, 0, 1, 2 we have the following commutation relations However, equations (53) and (63) without referring to the integrability have more then one solution. In the next section we will construct an infinite family of constraints, which completely specify partition function of the Kontsevich-Penner model. These operators, in general, do not correspond to any KS operators, thus, they do not belong to the algebra W 1+∞ .
while for N = 0 it can be reduced to the second order equation In this sense the derivation of the constraints for the Kontsevich-Penner model (43) is similar to the calculations for the generalized Kontsevich model with the quartic potential, performed in [25]. This is why we take a different route and develop here a new approach based on the correspondence between the W 1+∞ and w 1+∞ (19). Let us show that the operator annihilates the tau-function (43). This operator, because of the term ∂ 2 ∂t 2 2 does not belong to the W 1+∞ algebra, thus, it does not directly correspond to any KS operator.
First of all, let us consider an operator which, via the identification (19), corresponds to the operator This operator belongs to the W 1+∞ algebra, thus, its action can be easily considered on the level of the basis vectors. Indeed, from (45) and (47) it immediately follows that only two terms in the r.h.s. of (25) survives: For the operator q = z 2 the corresponding operator from the algebra W 1+∞ is Y q = ∂ ∂t 2 and from (26) we have Again, in the first sum only terms with l = 1 and l = 2 survive while in the double sum only a term with k = 1 and l = 2 survives Combining (69)-(72) and an expression for the Virasoro operator , we obtain To find a full algebra of the Virasoro constraints it is enough to consider nested commutators of the operators L 2 with L −1 . The resulting operators (74) constitute an extension of the algebra (51) to an infinite subalgebra of the Virasoro algebra and annihilate the tau-function Let us now construct the operators M k for k > 2. One can find all higher M k operators assuming that the commutation relation (62) holds for k = −1, 0, 1 and arbitrary l. Then, a commutation relation allows us to find Then, a commutation relation between the Virasoro and W-operators A straightforward calculation shows that the following commutation relations between the Virasoro and W-operators hold for k ≥ −1 and l ≥ −2, so that Of course, one can choose another basis in the space of constraints. Let us consider the operators which also annihilate the tau-function These operators satisfy the commutation relations Here a combination of the Kronecker symbols c kl = 8 (δ k,−1 (1 − δ l,−1 ) − (k + 2)δ l,−2 (1 − δ k,0 )) guarantees that in the r.s.h. there appear operators L k only with k ≥ −1. In particular, we have (86) At the end of this section let us describe a simple Sugawara construction of the Virasoro constraints (74). For this purpose we introduce the bosonic operators and the corresponding bosonic current with the dilaton shiftt k = t k − 1 3 δ k,3 . 5 Then 5 Cut-and-join operator for general N Following the idea of [26] in this section we construct the cut-and-join operator for the taufunction τ N . Let us introduce the gradation deg t k = k 3 such, that and the degree operator Then, the operators have the degree −2k/3, and the operators consist of the terms with degree −2k/3 and −2k/3 − 1. From the Virasoro and W-constraints it immediately follows that An operator in the r.h.s. is a sum of the operators W 1 and W 2 : such that From (94) it is clear that τ N is a sum of components with integer degree: where deg τ (k) N = k. Let us introduce a variable q which counts the degree: Then the operator D acts as a derivative q ∂ ∂q , so that τ N (q) satisfies the cut-and-join type equation For the commuting operators W 1 and W 2 the solution would be but it is easy to check that Thus, the solution can be represented in terms of an ordered exponential, and the operators (95) define a recursion with the initial conditions τ Explicite expression for these three terms can be found in Appendix A. From our construction it is clear that the operators W 1,2 are not unique. In particular, if one substitute the operator M * k in (94) with an operator M * k + β k L k for arbitrary constant β k 's the equation remains valid. This gives the following change in the operators: However, it is easy to see that this freedom is not enough to make the operators W 1 and W 2 commuting with each other. However, one can consider more general transformations, generated by the operators ∞ j=−1 β k,j J k−j L j with some constant matrix β k,j .

Open intersection numbers: Virasoro and W-constraints
In this section we consider the case N = 1 which, as we proved in Section 2, describes the open intersection numbers of [10][11][12].
As we have established in [1], an operator is the KS operator for the tau-function τ 1 . Moreover, in this case we have a family of the KS operators which constitute a subalgebra of the Virasoro algebra and guarantee [1] that the tau-function satisfies the Virasoro constraints Here Thus, for N = 1 we have two sets of the Virasoro constraints,namely (107) and (73), which do not coincide for k > 1. The difference is .
From the constraints we have the relations, which effectively describe the dependence on the even times t 2k for k > 1 6 This property of the generating function of the open intersection numbers has been established in [12]. This equation describes a dynamics with respect to the times t 2k for k > 1: Thus, there is a one-parametric family of the constraints where we assume that O k = 0 for k = −1, 0, 1. These operators satisfy the Virasoro commutation relations and annihilate the tau-function τ 1 . The Virasoro constraints obtained in [12] correspond to α = 1/2. In addition to the Virasoro constraints we have infinitely many higher W-constraints. Let us consider the KS operators They satisfy the following commutation relations while commutators [w k , w l ] contain terms with a 3 1 . Using the correspondence (19) we construct the following operators from W 1+∞ : The constant term in M o 0 is chosen in such a way that the following commutation relations hold These commutation relations guarantee that

Cut-and-join operator for open intersection numbers
In Section 5 we have already obtained the cut-ant-join operator description of the Kontsevich-Penner model, which is valid for arbitrary N . In particular, it is valid for the case N = 1, corresponding to open intersection numbers. However, as we have seen in the previous section, for this case the Virasoro and W-constraints have additional parameters, thus, we have a vast family of the cut-and-join type operators. Here we construct a representative of this family. We will directly follow the construction of Section 5. Let us introduce the shifted operators and so that Then Again, the operator in the r.h.s. is a sum of two operators (2k + 1)(k + 1)t 2k+1 ∂ ∂t 2k−2 + 39 8 t 3 + 6t 1 t 2 , such that For the expansion of the tau-function where deg τ such that τ (1) These three terms give the expansion of τ o , presented in [1].

Other interesting values of N
In general, for integer N > 0, operators z 2k (a N ) N +m with non-negative m and k are the KS operators for the tau-function τ N . Indeed, an operator (a N ) N maps the basis vectors (44) into the set, proportional to the basis vectors of the KW model and, on this space, operators z 2 and a N are the KS operators. Thus, for integer N > 1 we have additional constraints, which follow from the existence of two different families of the W (N +1) constraints. These constraints allow us to restore the dependence of the tau-function on the even times t 2k for k > N + 1 from the dependence of the times t 2 , . . . , t 2N . For example, for N = 2 the operators z 4+2k a 2 N for k ≥ −2 are the KS operators. Thus, for N = 2 in addition to the constraints (82) we have the operators M ′ k = M 2k + 2(k + 4) L 2k − 2 J 2k+3 + J 2k+6 + 4 3 k 2 + 8k + 179 12 J 2k − L 2k+3 + 17 δ k,0 , k ≥ −2, which annihilate the tau-function The simplest equation, which follows from the existence of two different families of the constraints, is ∂ ∂t 6 τ 2 (t) = P 3 τ 2 (t), where This equation describes a dynamics with respect to the time t 6 : τ 2 (t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 7 , . . . ) = exp t 6 P 1 τ 2 (t 1 , t 2 , t 3 , t 4 , t 5 , 0, t 7 , . . . ).
In general, for k > 2 we have ∂ ∂t 2k τ 2 (t) = P k τ 2 (t), where P k = 3 2(k − 1) describes the dependence on the time t 2k (and, for arbitrary k > 2, the operator P k can be reduced to the operator, acting only on the times t 2 and t 4 ). This dependence follows from the integral representation dz 1 2πiz 1 dz 2 2πiz 2 (138) which is a particular case of (37).