Open KdV hierarchy of 2d minimal gravity of Lee-Yang series

We present the open KdV hierarchy of 2d minimal gravity of Lee-Yang series which uses the boundary cosmological constant as a flow parameter. The boundary cosmological constant is a conjugate variable to the boundary flow parameter used in the open KdV hierarchy of the intersection numbers on the moduli space of Riemann surfaces with boundaries. The two generating functions are related through the Laplace transform.


Introduction
Minimal gravity is a 2-dimensional quantum gravity coupled with minimal conformal matter, so that the resulting theory still remains conformal and topological (c = 0) [1,2,3], and its correlation numbers obey the Gelfand-Dickey hierarchy in general [4,5,6]. The minimal gravity of the Lee-Yang series is described by one-matrix model, whose correlation numbers show the KdV hierarchy, and thus is closely related with the intersection theory on the moduli space of Riemann surfaces. In fact, by Witten, the generating function of the intersection theory had also been conjectured to satisfy the KdV hierarchy together with the string equation [7]. Witten's conjecture for intersection theory was proved by Kontsevich using one-matrix model [8]. The generating functions for minimal gravity of Lee-Yang series on g = 0, 1, 2 have also been constructed using the KdV hierarchy [9,10], and the resulting correlations (but off-shell, i.e., with arbitrary t k parameters but t 1 → t 1 + 1) have been shown to obey the recursion relations of topological gravity, as suggested by Witten. Recently, the intersection theory is extended to the moduli space of Riemann surfaces with boundaries [11,12]. This extension is based on an extension of the KdV structure, where the KdV hierarchy for closed surfaces has an equivalent description by the Virasoro constraints as shown in [13,14]. With boundaries, the generating function is described by the so-called open KdV hierarchy which contains the KdV flow parameters as well as a new boundary flow parameter s indicating the boundary contribution.
It is naturally expected that the open KdV hierarchy also describes minimal gravity of Lee-Yang series on a disk. In our previous paper [15], we check this expectation using the free energy on a disk, where the boundary cosmological constant µ B is introduced as an additional boundary parameter that has a similar role of the boundary flow parameter s in the intersection theory [16,17,18,19].
The parameter s looks natural in the the intersection theory. However, from the viewpoint of the 2d minimal gravity, the boundary cosmological constant µ B appears rather naturally and it is desired to have the open KdV hierarchy in terms of µ B . In this paper, we present the open KdV hierarchy using the boundary cosmological constant µ B .
In section 2, we summarize the KdV hierarchy of the minimal gravity of the Lee-Yang series on the closed Riemann surfaces. It is confirmed that the free energy known in the minimal gravity satisfies the KdV hierarchy along with the string equation.
In section 3, we investigate the KdV hierarchy on Riemann surfaces with boundaries. We use the known free energy of the minimal gravity on a disk to check the open KdV hierarchy and the open string equation. The free energy is a function of the boundary cosmological constant µ B rather than the s-parameter. Therefore, one needs to find the relation of the hierarchy of the intersection theory with that of the minimal gravity. It turns out that s and µ B are conjugate to each other. We present the Virasoro constraints of the µ B -representation as their Laplace transforms of the s-representation and check those equivalence. The open hierarchy allows one to find the free energy with genus g ≥ 1 using the free energy with genus 0 (disk). We demonstrate it for g = 1.
Section 4 is devoted to the summary and discussion. In appendices, useful identities used in the text can be found.

Lee-Yang series on a closed Riemann surface
Minimal gravity of Lee-Yang series M(2, 2p + 1) is described by one-matrix model. At the continuum limit, the matrix variable is described by a differential operatorQ 2 = ∂ 2 x + u(x) whose dispersionless limit (neglecting derivatives) reduces to a second order polynomial Q 2 = y 2 + u(x). The polynomial Q 2 defines a one-dimensional Frobenius manifold A 1 , u being its coordinate. This section describes how u behaves according to the flow equation on a Riemann surface without boundaries.

Lee-Yang series on a sphere
The Frobenius manifold allows flat coordinates and the one-dimensional coordinate u is trivial and obviously regarded as the flat one, which will be renamed as v. The coordinate is a function of x. The variable x can be identified with one of the variable of the generating function of the minimal gravity through the Douglas equation, which obtains from the least action principle. The string action describing the minimal gravity of Lee-Yang series M(2, 2p + 1) on a sphere is given by where θ 1,n is defined by The parameters t n are called the KdV parameters with the gravitational scaling dimensions (gcd) (defined as the power of the cosmological constant µ): where t 0 has the highest gcd. It is noted that t p−1 has a special role in the minimal gravity since [t p−1 ] = 1 and is identified as the cosmological constant µ. It is also useful to remember [Q 2 ] = 1/2 so that [v] = 1/2 and [y] = 1/4] in the Lee-Yang series. The action principle results in the string equation: Its derivative with respect to v vanishes: where we use the convention t p+1 = 1 and t p = 0. The defining relation of x in (2.5) shows that v is the function of x and t n : v = v(x, {t n =0 }). The variable x has the same gcd as that of t 0 and is identified with t 0 if one applies the string equation (2.4).
To distinguish the A 1 coordinate v from v({t n }), the KdV parameter dependent variable, we will use separate notations: v for the original A 1 coordinate and w for v(x, {t n =0 }). Then, w satisfies the flow equations, consistent with the KdV hierarchy on a sphere: The free energy on a sphere F sphere is known to have the form [20] F sphere ({t m }) = 1 2 w * is one of the solutions of the string equation P(w * ) = 0, (2.8) so that w * = w(x = t 0 , {t n =0 }) which reduces to √ −t p−1 when t n =(p−1,p+1) → 0. Since the only difference between w * and w is the parameter t 0 or x, we will omit the star if there is no confusion.
The free energy (2.7) shows that the two-point correlation ∂ 2 F sphere /∂t 2 0 = w. Multi-correlation is given in terms the derivatives of F sphere with respect to {t m }: Especially, two-point correlation 3 is given as (2.10) One may evaluate the correlation on-shell if t n is set to be 0 for all n, except t p−1 and t p+1 = 1.

Hierarchy on a closed Riemann surface
Beyond the sphere one may generalize the KdV hierarchy appearing in the intersection numbers [7] 1 λ 2 2n + 1 2 with the string equation Note that the free energy F c has the genus expansion where λ is a formal expansion parameter and F c (0) = F sphere . If one has the genus 0 part of (2.11) and (2.12), one compares with the KdV hierarchy (2.6) and the string equation (2.8) in the Lee-Yang series, one finds t 1 is to be shifted by 1 (t 1 → t 1 + 1) [9,10]. Therefore, the hierarchy on a closed Riemann surface has the form (2.11) but the string equation for the Lee-Yang series has to be modified (2.14)

Lee-Yang series on a torus
The hierarchy and string equation have the form at genus 1: One may simplify (2.15) if one multiplies (2.15) with t n and sums over n ≥ 1: Here we use the string equation at genus 0 (2.14) and (2.16). The free energy on a torus is known in [9,10] F c Here P ′ (w) stands for ∂P (w)/∂w with t n 's fixed. The coefficient a is a constant and is found a = −1/24 (see appendix B for detailed derivations 4 ). One can check that the free energy (2.19) satisfies (2.16) and (2.18). The proof goes as follows. Note that P ′ (w) is the function of w n 's and t n 's: where w is again the function of t n 's through the KdV equation. For example, where () a means differentiation with a fixed. One can check the string equation (2.16), using the relation where we used the second derivative of (2.14) and an identity P ′′ = n≥1 t n+1 w n−1 (n−1)! . The simplified hierarchy (2.18) is evaluated as follows: Note that using (2.8) and (2.20). The first term of the right hand side is written as In addition, we have the null equality 3 Open KdV hierarchy of minimal gravity with boundaries

Open KdV in the intersection theory
A similar KdV hierarchy ("open KdV hierarchy") has been proposed for intersection theory on the moduli space of Riemann surfaces with boundaries, using an additional flow parameter s. The flow along t n is given as [11] 2n + 1 2 The open string equation is given by In The free energy is expanded in the genus expansion whose lowest order (g = 0) gives

Free energy on a disk and hierarchy
One may conjecture that the free energy F disk on a disk [21,22] can be a solution of the hierarchy (3.5) [15]. It is noted that the free energy F disk corresponds to the continuum limit of the trace of log(M + µ B ) where the one-matrix element is replaced by Q and µ B is the boundary cosmological constant. The integral representation is given as follows: Here we assume a proper regularization (subtracting the infinity) in the integration limit at l → 0. In fact, F disk satisfies a similar equation as in (3.5) 2n + 1 2 (3.10)

Open hierarchy with µ B
The open KdV formula in (3.9) obtained using the free energy on a disk is to be compared with the one in (3.5) from the intersection theory. The result is On the other hand, if the string equation (3.6) is used, the parameter s turns out which is identified as the loop operator which is the continuum limit of the resolvent, trace of 1/(M + µ B ). This identification raises a problem: s identified in (3.12) depends on the KdV variables {t n } and spoils the property of s that s should be independent of {t n } as conjectured in the intersection theory.
To cure this problem, one notes that s and µ B are conjugate according to (3.11) and (3.12). Therefore, one may use either s or µ B , not both. According to the free energy of minimal gravity, it is desirable to put the hierarchy in terms of µ B as given in (3.9). One may follow the same steps in [12] using the half Burger-KdV hierarchy except the two changes: One is ∂F o (0) /∂s → −µ B using the fact (3.11) and the other is s → ∂F disk (µ B )/∂µ B as noted in (3.12). Then one has µ B -representation.
The open KdV hierarchy on a disk is given as (3.5). The relation of the boundary parameter is given as (3.10). Finally, the open string equation (after t 1 → t 1 + 1) is modified as According to this µ B -representation, one may have the open KdV hierarchy 2n + 1 2 the open string equation 15) and the boundary parameter constraint (3.16)

Virasoro constraints
The µ B -representation is obtained by replacing s → ∂/∂µ B and ∂/∂s → −µ B . The hierarchy is equivalent to put (the half set of) the Virasoro generators of form L n is the Virasoro generator of the closed surface 5 : which imposes the Virasoro constraints on the tau function of intersection theory of closed surfaces One may show that the partition function exp(F o + F c ) is constrained by the Virasoro generator (3.17): The µ B -representation of L n is obtained if one uses the Laplace transformation from the original srepresentation of the partition function with an appropriate integration contour so that the integration converges. The Laplace transform ensures the replacements: s → ∂/∂µ B and ∂/∂s → −µ B . Therefore, one may expect that the Virasoro constraint in , and, by successive use of inductive relation (3.23), one has L 1 τ = 0, and then L 2 τ = 0. 5 We use the original t1 before shifted by 1

Free energy on a cylinder
In this section, we present a solution to the open hierarchy with genus 1. The genus expansions of (3.14), (3.15) and (3.16), respectively, have the forms at each order of g: where, needless to say, the term involving F c (g ′ /2) is absent when g ′ /2 is not an integer. The free energy on a disk satisfies the lowest order (g = 0) hierarchy. Therefore, we have The next order (g = 1) has the following equations: the open KdV hierarchy The constraint equation (3.31) hints at the solution of the form where f is a t 0 -independent function. One can easily check that F o (1) and F o (1) + cF o (0) , for any constant c, obey the same hierarchy (3.29) since F o (0) satisfies the (g = 0)-hierarchy (3.9). Thus f can be a function of F o (0) , but which contradicts the fact that f should be independent of t 0 . A natural choice is f = 0. Therefore, we conclude that the free energy at g = 1 has the form One can check that F o (1) in (3.33) satisfies (3.29), noting that (3.29) reduces to the t 0 derivative of (3.9): Finally, it is obvious that

Summary and discussions
We consider the open KdV hierarchy for 2d minimal gravity of Lee-Yang series. The hierarchy is given to have the boundary cosmological constant µ B rather than the boundary flow parameter s appearing in the intersection theory of open surfaces with boundaries. It is noted that µ B and s are conjugate variables and give rise to the Laplace transform of the free energy of 2d minimal gravity, resulting in rephrasing the Virasoro constraint equations in terms of µ B . It is noted, however, that the Laplace transform of the partition function does not need to produce the result of the intersection theory since the initial conditions of the two approaches are different.
The explicit form of the free energy at each order of genus g can be obtained according to the closed and open KdV hierarchies starting with the free energy of g = 0. As an example, we present the free energy at g = 1 in the text.
The Lee-Yang series M(2, 2p + 1) is obtained from the one-matrix model. The dual picture of the Lee-Yang series (weak strong duality b ↔ 1/b in the Liouville gravity) is described by the A 2p Frobenius manifold [23,24]. We expect the dual picture will show very different behavior and shall be worth studying. This is because the KdV hierarchy of the one-matrix model in this dual picture does not work anymore and is to be replaced by the Gelfand-Dickey hierarchy. In addition, the flow parameter of the dual theory coincides with the conformal parameter in the Liouville minimal gravity [24], which is in contrast with the original A 1 description since the resonance transformation among those parameters plays a central role in general [20,25,26]. As a result, the open hierarchy in the dual picture will show very different behavior unlike the open KdV hierarchy. We will provide this new feature in a separate paper.
has the (off-shell) one-point correlation where w * is the value of w at x = t 0 . On the other hand, the correlation multiplied by the boundary cosmological constant has the form if one uses the identity −µ B e −lµ B = ∂e −lµ B /∂l and integrates by part with respect to l Here the boundary term at l → 0 vanishes due to the proper regularization.
What we trying to prove is the difference between O n disk and µ B O n−1 disk . This can be done using the following two identities. One is the total derivative with respect to y whose integrated value is zero: The other one is total derivative with respect to w: where f (w) is an arbitrary polynomial. Then, one may find the linear relation with coefficients c and α such that In addition, one may eliminate the y-dependent term in (4.3) by fixing c and α: This results in the desired the recursive relation (3.9): where we use the result for two-point correlation on a sphere (2.10).

Appendix B
We may check that the coefficient a in (2.19) turns out to be −1/24. As a in (2.19) is independent of n, here we demonstrate it using the case for n = 2 of (2.11) (though the case for n = 1 is much simpler). Note that the KdV hierarchy (2.11) can be rephrased in the form whose (λ −2 )-order terms of λ-expansion give in particular for n = 2 where P n stands for n-th derivative of P with respect to w. Then we have 1 a 5 2 giving the left hand side of (4.7): , while the right hand side: , which gives us Therefore we finally obtain 1 a ∂ ∂t 0 5 2 by which the relation (4.7) is reduced to