Open KdV hierarchy and minimal gravity on disk

We show that the minimal gravity of Lee-Yang series on disk is a solution to the open KdV hierarchy proposed for the intersection theory on the moduli space of Riemann surfaces with boundary.


Introduction
The generating function of the intersection theory on the moduli space of Riemann surfaces had been conjectured to satisfy the KdV hierarchy together with the string equation [1]. It was shown in [2,3] that Witten's conjecture is equivalent to the Virasoro constraints. On the other hand, minimal gravity is 2-dimensional quantum gravity coupled with minimal conformal matter so that the resulting theory still remains conformal and topological (c = 0) [4,5,6], and also obeys the KdV hierarchy [7,8]. As a result, KdV hierarchy plays an important role in understanding intersection theory as well as topological gravity.
Recently, the generating function for intersection theory on the moduli space of Riemann surfaces with boundary has been conjectured to satisfy the so-called open KdV hierarchy which contains the KdV flow parameters as well as new flow parameters related with the boundary [9]. It has also been checked that the open KdV hierarchy can be represented in modified form of Virasoro constraints [10]. It is naturally expected that the open KdV hierarchy also describes minimal gravity on a disk. We check this expectation in this letter.
In section 2, we summarize minimal gravity on sphere and its connection with KdV hierarchy and in section 3, open KdV hierarchy is explicitly checked for the free energy of minimal gravity on a disk. Section 4 is the conclusion.

Minimal gravity on sphere and KdV hierarchy
Minimal gravity M (2, 2p + 1) of Lee-Yang series is represented in terms of one-matrix model and its free energy on a sphere F sphere is given as [11] F sphere = 1 2 The polynomial equation P(u) = 0 is called the string equation. u 0 is one of the solutions of the string equation and is related to the free energy as u 0 = ∂ 2 F sphere /∂t 2 0 .
1 abawane@sogang.ac.kr 2 hmuraki@sogang.ac.kr 3 rimpine@sogang.ac.kr The free energy contains the set of parameters {t m } and becomes the generating function because derivatives with respect to {t m }'s evaluated on-shell provide correlations of the corresponding operators. By on-shell we mean that t i = 0 for all i, except t p−1 and t p+1 . We normalize t p+1 to 1, and t p−1 is proportional to the cosmological constant µ. Multi-correlation on sphere is given as Computing the above quantity on-shell gives us the physical correlation. For example, the two-point cor- where the symbol * stands for on-shell value. One can also see that the free energy in (1) satisfies the KdV hierarchy on the sphere The matrix model representation is shown to be equivalent to Liouville minimal gravity if one takes care of the resonance transformation [11]. The KdV hierarchy in general is given by where n ≥ 1. The string equation is The free energy has the genus expansion where F c (0) = F sphere and λ is a formal expansion parameter. The sphere KdV and string equations can be deduced by considering the dominant part of the general equations. (The equation P(u) = 0 is obtained by a combination of (5) and (6) for g = 0.) Witten's conjecture for intersection theory was proved by Kontsevich [12] using the one-matrix model. The generating function for minimal gravity on g = 0, 1, 2 has also been constructed using the KdV hierarchy [13,14], and the resulting correlations (but off-shell, i.e., with arbitrary t k parameters) have been shown to obey the recursion relations of topological gravity, as suggested by Witten. (In the string equation, t 1 is to be shifted by 1 for the comparison of the two cases.)

Minimal gravity on a disk and KdV hierarchy
A similar KdV hierarchy ("open KdV hierarchy") has been proposed for intersection theory on the moduli space of Riemann surfaces with boundary, using an additional flow parameter s. The flow along t n is given as [9] 2n + 1 2 The open string equation is given by The open KdV together with the string equation is shown to be equivalent to the Virasoro constraints [10].
In this section, we explicitly check that the open KdV equations are satisfied by the free energy on a disk. The free energy with a (1, 1) boundary is given by [15,16] where µ B is the boundary cosmological constant. In (9), u satisfies the string equation and KdV on a sphere and is, therefore, u = u(x, t n>0 ) a function of x and t n 's (n > 0) but independent of t 0 . The genus expansion of the free energy with a boundary shows that F o (0) satisfies the open KdV at this order: The one-point correlation on a disk is non-trivial. The correlation is found from (9) by differentiating with respect to t n , using the KdV on sphere (but staying off-shell): where u 0 = u(x = t 0 , t n>0 ) and is the same as the one in (1). Since u has a gravitational dimension, one may set u = u 0 ξ so that u 0 is dimensionful while ξ is dimensionless. The integration over u results in the incomplete gamma function. However, it is more convenient to rewrite the monomial ξ n as a linear combination of Legendre polynomials P k : ξ n = k=n,n−2,···≥0 (2k + 1) n! a n,k P k (ξ) (13) where a n,k = 1 2 (n−k)/2 ((n − k)/2)!(n + k + 1)!! .
Then the integration over ξ is given as the modified Bessel function K n of the second kind: Further, its integration over l is performed (after analytic continuation if necessary) to give where we put 4 µ B /u 0 = cosh(τ ). Therefore, the correlation number is given as follows in terms of the Chebyshev polynomial T n (cosh(x)) = cosh(nx): k=n,n−2,···≥0 a n,k T 2k+1 (cosh(τ /2)).
If one uses the identity of Chebyshev polynomials T n+2 = 2T 2 T n − T |n−2| and the two-point correlation on sphere O 0 O n−1 sphere = u n 0 /n!, one has the following recursive relation Comparing the recursion in (18) with the open KdV on disk in (11), one concludes that the free energy on a disk follows the open KdV and the flow along s reads: Considering (11), (18) and (19) we arrive at the conclusion that F o (0) and F disk are related by the Legendre transformation We note that the above calculations are done off-shell and therefore, (19) holds off-shell also. Using this result, one can prove that the open KdV on the disk holds for F disk (and multi-correlations) using just the integral representation of F disk in (9) and the fact −µ B e −lµ B = ∂e −lµ B ∂l . It is worth pointing out that the result of [9,10] still holds, 5 essentially because of the Chebyshev polynomial identity T 2 = 2T 2 1 − 1.

Conclusion
We demonstrate that the open KdV hierarchy holds for the matrix models of the Lee-Yang series of the minimal gravity on disk. In order to prove this, the fact that the s flow of the generating function on a disk is governed by the boundary cosmological constant µ B is essential. This result can be compared with [17,18], where intersection theory with boundary is investigated using a Penner-type matrix model, but has no boundary parameter. One could investigate more general boundary conditions and boundary correlations on the disk, and the corresponding KdV hierarchies. It might be interesting to investigate if the s n flows of [10,17,19] have an interpretation in the Lee-Yang series, which would be related to the other boundary condition than the simple (1,1) boundary condition considered in the text. The minimal gravity M (2, 2p + 1) is described by one variable u which is the coordinate of A 1 Frobenius manifold [20]. It is known that its dual A 2p Frobenius manifold (which has 2p number of coordinates {u a }) can describe the same correlation on sphere. However, the correlation on disk is not fully understood in 4 The boundary parameter µB is independent of KdV parameters but τ depends on KdV parameters due to u0. 5 To make it match exactly, we may rescale F disk → cF disk with c 2 = −1/(2π 2 ). terms of the dual manifold due to the non-canonical nature of the integral representation of the free energy [21,22]. The open KdV hierarchy can be a guide to define the free energy on disk whose details will be considered in a separate paper.