Genus expansion of open free energy in 2d topological gravity

We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy, the generating function for the open intersection numbers, obeys the open KdV equations and Buryak's differential equation and is related by a formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierarchy. Using these properties we study the genus expansion of the free energy in detail. We construct explicitly the genus zero part of the free energy. We then formulate a method of computing higher genus corrections by solving Buryak's equation and obtain them up to high order. This method is much more efficient than our previous approach based on the saddle point calculation. Along the way we show that the higher genus corrections are polynomials in variables that are expressed in terms of genus zero quantities only, generalizing the constitutive relation of closed topological gravity.


Introduction
Two-dimensional gravity is one of the simplest models of quantum gravity which has been intensively studied for quite some time. In [1][2][3][4] it was found that two-dimensional gravity is described by a certain double-scaled random matrix model (see [5] for a review). Mathematically, two-dimensional gravity corresponds to an intersection theory on the moduli space of closed Riemann surfaces, as first conjectured by Witten [6] and proved by Kontsevich [7]. It is known that the free energy of two-dimensional gravity on closed Riemann surfaces satisfies the KdV equations [6][7][8] and the Virasoro constraints [9,10]. Recently, it is realized that this story holds for Jackiw-Teitelboim (JT) gravity as well; Saad, Shenker and Stanford [11] showed that JT gravity is described by a doubled-scaled matrix model and it corresponds to a particular background of Witten-Kontsevich topological gravity [12][13][14].
Recently, Pandharipande, Solomon and Tessler [15] initiated the study of open topological gravity, i.e. the intersection theory on the moduli space of Riemann surfaces with boundary. See also [16][17][18][19][20][21] for related works. It is conjectured in [15] and proved in [18] that the open free energy F o (s), 1 or the generating function of the open intersection numbers, satisfies the open version of the KdV equations and the Virasoro constraints. As explained in [13], open topological gravity is physically realized by adding vector degrees of freedom to the matrix model of two-dimensional gravity. After integrating out the vector degrees of freedom, this amounts to the insertion of the determinant operator det(ξ −M ) to the matrix integral, where ξ is a parameter and M is the random matrix. The expectation value of this determinant operator corresponds to the wavefunction of the FZZT brane [23,24]. Here V (ξ) is the matrix model potential and g s is the genus counting parameter (denoted as u in [15][16][17][18]). ψ(ξ) is also identified as the Baker-Akhiezer (BA) function of the KdV hierarchy [25]. It is known that the exponential of the open free energy e F o (s) and the BA function ψ(ξ) are related by the formal Fourier transformation [13,18]  2gs V eff (ξ) is given by the so-called effective potential V eff (ξ). In our previous paper [22], we obtained the explicit form of V eff (ξ) for arbitrary background couplings {t n }. Then the leading term F o 0 (s) in (1.3) is given by the Legendre transform of V eff (ξ). One can in principle continue this saddle point computation for the higher order corrections in g s , but the computation becomes very cumbersome as the order of g s increases.
It turns out that the small g s expansion of F o (s) can be computed systematically by recursively solving Buryak's equation [17], which is understood as the Fourier transform of the Schrödinger equation of the KdV hierarchy. This is based on the fact that F õ g≥2 (s) in (1.3) is written as a polynomial in variables which are expressed in terms of genus zero quantities only. This is similar to the situation in original Witten-Kontsevich topological gravity, where the genus-g(≥ 2) closed free energy F c g is written as a polynomial in a certain basis [26,27]. This paper is organized as follows. In section 2 we briefly review closed and open topological gravities. We also explain how the open KdV equations and Buryak's equation are derived from the KdV hierarchy. In section 3 we study the genus expansion of the open free energy. We first compute it from the genus expansion of the BA function by the saddle point calculation. We next derive an explicit expression of the genus zero open free energy. We then formulate a method of computing the genus expansion by solving Buryak's equation. We conclude in section 4 with discussions on the future directions. Some details of the calculations are relegated to the appendices A and B.
In this section we will briefly review the basics and known results about closed and open topological gravities. We will also explain how the open KdV equations and Buryak's equation, which will be the main tools of our study of the open free energy, are derived from the KdV hierarchy.

Witten-Kontsevich closed topological gravity
In Witten-Kontsevich topological gravity [6,7] (see also [13]) observables are made up of the intersection numbers They are defined on a closed Riemann surface Σ of genus g with n marked points p 1 , . . . , p n . We let M g,n denote the moduli space of Σ and M g,n the Deligne-Mumford compactification of M g,n . Here τ d i = ψ d i i and ψ i is the first Chern class of the complex line bundle whose fiber is the cotangent space to p i . The intersection numbers (2.1) obey the selection rule τ d 1 · · · τ dn g,n = 0 unless d 1 + · · · + d n = 3g − 3 + n. (2. 2) The generating function for the above intersection numbers is defined as We will call F c the closed free energy. It was conjectured [6] and proved [7] that e F c is a tau function for the KdV hierarchy. This means that satisfies the KdV equations where R k are the Gelfand-Dikii differential polynomials of u Here we have introduced the notation For k = 1, (2.5) gives the traditional KdV equation Integrating (2.5) once in t 0 we have It is well known (see e.g. [28]) that the KdV equations (2.5) are obtained as the compatibility condition of the Schrödinger equation Qψ = ξψ (2.10) and the KdV flow equations (2.13) The wave function ψ that satisfies (2.10) and (2.11) is known as the Baker-Akhiezer function.
Another important constraint that the closed free energy F c obeys is the string equation [8]. For Witten-Kontsevich gravity it is written as (2.14) The genus zero part of this string equation is written as where u 0 is the genus-zero part of u and we have introduced the Itzykson-Zuber variables [26] Throughout this paper I n without specifying its arguments should always be understood as It is also convenient to introduce the variable It was conjectured [26] and proved [27,29] that F g ({t k }) (g ≥ 2) are polynomials in I n≥2 and t −1 . This fact significantly helps us to compute higher genus free energy F c g . An efficient way to compute F c g is as follows. (See [14] for a more detailed explanation.) Let us expand u as u g can be computed by recursively solving the KdV equation (2.8). To do this, let us regard t k≥2 as parameters and consider the change of variables from (t 0 , t 1 ) to (u 0 , t). The differentials ∂ 0,1 are then written in the new variables as 3 By expanding both sides of the equation in g s (2.8) is written as the recursion relation This is easily solved with the help of (2.21). First few of u g are As explained in [14] one can easily integrate u g twice in t 0 and obtain the well-known results [26] (2.24)

Pandharipande-Solomon-Tessler open topological gravity
Pandharipande, Solomon and Tessler proposed an open analog of Witten-Kontsevich topological gravity [15]. They introduced the open intersection numbers The new insertion σ corresponds to the addition of a boundary marking and the power k of σ specifies the number of boundary markings. In [15] a natural lift e(E, s) of the Euler class e(E) = ψ d 1 1 · · · ψ dn n is defined. Mg ,k,n denotes a suitable compactification of the moduli space Mg ,k,n of Riemann surfaces with boundary of doubled genusg with k boundary markings and n interior markings. The open intersection numbers (2.25) obey the selection rule The generating function for the open intersection numbers is defined as (2.27) We will call F o the open free energy. It was conjectured [15] and then proved [18] that F o satisfies the open KdV equations In fact it is known [17] that F o is fully determined by the above system of equations with the initial condition given the closed free energy F c . Buryak proved that F o further satisfies another differential equation [17] These equations play a crucial role in the study of F o in this paper.
In [16] Buryak constructed an explicit expression for e F o in terms of F c . In this sense an explicit form of F o is known. For many purposes, however, it is still useful to express F o in the form of genus expansion (2.27) and construct an explicit, closed expression of F õ g (s, {t k }) for fixedg. This is our primary goal in this paper.

Open partition function as Fourier transform of BA function
It is known [13,18] (

Genus expansion of open free energy
In this section we will study the genus expansion of the open free energy. We will first compute it from the genus expansion of the BA function by the saddle point calculation.
We will next derive a fully explicit expression of the genus zero open free energy. Finally, we will formulate a method of computing the genus expansion by solving Buryak's equation, which turns out to be much more efficient than the saddle point calculation.

Genus expansion of BA function
We saw in [22] that the BA function ψ admits the following expansion where we have introduced 5 Ag can be computed up to any order by solving the recursion relation for vg : In [22] we performed this computation with special values of t k corresponding to the case of JT gravity, but as advertised in [31] it can be generalized without any effort to the case of general values of t k , as we have seen above.  As in [14,22] one can calculate F õ g by the saddle point method. The saddle point ξ * is given by the condition In [22] the constant part of A1 is fixed so that it fits well with the convention of closed topological gravity. In this paper we will use this degree of freedom later for compensating the difference of the normalizations of e F o and ψ, so that we can avoid putting an inessential normalization factor in (1.2). 5 Ag, vg and z in this paper are related to those in our previous paper [22] by A herẽ This is equivalent to where By using the Lagrange inversion theorem this is inverted as (see Appendix B) (3.9) As in [14] let us introduce a new variable φ as (3.10) The integral (3.5) is then written as By expanding the integrand in g s , the integral in φ can be performed order by order as a Gaussian integral. In fact, we did essentially the same calculation in [14] up to the order of g 1 s . We thus immediately obtain where we have introduced the notation In the last equality in (3.12) we have used (3.14) We have fixed the constant part of A 1 in (3.2) in such a way that the initial condition (2.29) is satisfied. Using this method one can in principle calculate F õ g up to any order. However, this calculation gets quickly involved asg increases. We will propose an alternative, much more efficient method of computing F õ g in the following subsections. An advantage of the above calculation is that we can prove the polynomial structure of the higher genus free energies F õ g≥2 . The expansion (3.11) implies that F õ g≥2 are polynomials in A g≥2 * and φ 2m (m ≥ 1). On the other hand, by using the polynomial structure of the closed free energy reviewed in section 2.1, it is easy to show that Ag ≥2 are polynomials in t −1 , I k≥2 and z −1 . Combining these two lemmas we arrive at the conclusion that F õ g≥2 are polynomials in the variables t −1 , I k≥2 , z −1 * , ξ . It is well known that closed topological gravity exhibits the constitutive relation [32], i.e. higher genus quantities are expressed in terms of genus zero quantities only. In the case of Witten-Kontsevich gravity F c 1 is given as in (2.24) and F c g≥2 are expressed as polynomials in the variables t −1 and I k≥2 , as we saw in section 2.1. These variables are expressed explicitly in terms of genus zero quantities ∂ n 0 u 0 (n ≥ 1) [33]. Since z * and ξ

Genus zero open free energy
In the last subsection we have obtained an explicit expression of F o 0 : By plugging (3.8) into the second line of (3.12) we have with z * (s) given in (3.9). As we will see below, we can write down a more direct expression for F o 0 by using the relations among F o 0 , ξ * and z * which follow from the system of equations (2.30) and (2.28).
Buryak's equation (2.30) at the order of g −1 Note also that the second line of (3.12) gives By using (2.21), (3.16), (3.18) and (2.16) this becomes Applying ∂ s to both sides of the equation and using again (3.16) and (3.18) one obtains Hence, differentiating (3.9) once in t and then integrating it twice in s, one obtains (3.23) The integration constants have been fixed accordingly so that (3.23) matches with (3.15). We verified by series expansion in s that (3.23) and (3.15) are indeed in perfect agreement. Note that when t k≥1 = 0, we have t 0 = u 0 , t = 1, I k≥2 = 0 and thus the above F o 0 becomes This is consistent with the initial condition (2.29).

Recursion relation
(3.25) 7 Note that z * is the uniformization coordinate on the spectral curve. In the context of minimal string theory, it is known that z * is given by the t0-derivative of the disk amplitude F o 0 [25].
where we have introduced the differential operator (3.25) can be viewed as a recursion relation: one can recursively compute F õ g if one is able to perform the integration D −1 on the l.h.s. of (3.25). This is indeed feasible, as we will see below.
To do this, let us first study the operator D, which has in fact several interesting properties. For instance, one can show that (3.27) The first line of (3.27) follows from and On the other hand, to evaluate the r.h.s. of (3.25) it is convenient to use Again using (3.32) one can express these quantities as polynomials in t −1 , z −1 * and ξ (n≥1) * . Hence, by using the low genus results (3.12) and the polynomial structure of F õ g≥2 derived in section 3.2, it is easy to see that all quantities appearing in (3.25) are expressed as polynomials in the variables t −1 , I k≥2 , z −1 * , ξ

Higher genus open free energy
We are now in a position to solve the recursion relation (3.25) and compute the higher genus free energy F õ g . To begin with, we verified that F õ g withg = 0, 1, 2 given in (3.12) indeed satisfy the recursion relation (3.25) forg = 1, 2. This is easily done by using various identities derived in the last two subsections. Moreover, based on the polynomial structure discussed above, one can perform the integration D −1 completely and determine F õ g unambiguously forg ≥ 2. The algorithm to solve (3.25) and obtain F õ g from the data of {F õ g }g <g is as follows: (i) Compute the r.h.s. of (3.25) using (3.35) and express it as a polynomial in the variables t −1 , I k≥2 , z −1 * , ξ (3.36) Therefore subtract this from the obtained expression.
(iii) Repeat the procedure (ii) down to m = 3. Then all the terms of order t −2 automatically disappear and the remaining terms are of order t −1 or t 0 . Note also that the expression does not contain any I k .
(iv) In the result of (iii), collect all the terms of order t −1 and let t −1 ∂ z * g(z * , ξ (n) * ) denote the sum of them. This part arises from Dg(z * , ξ (n) * ).
(3.37) Therefore subtract this from the result of (iii). The remainder turns out to be independent of t.
(v) In the obtained expression, let denote the part which is of order z −2 * as well as of the lowest order in ξ (3.39) Therefore subtract this from the obtained expression.
(vi) Repeat the procedure (v) until the resulting expression vanishes.
(vii) By summing up all the above obtained primitive functions we obtain F õ g .
Using this algorithm we computed F õ g forg ≤ 15. 8 We verified that F o 2 computed by this algorithm reproduces the result (3.12) of our saddle point calculation. Forg = 3 we obtain (3.40)

Conclusions and outlook
In this paper we have studied the small g s expansion (1.3) of the open free energy F o (s) of topological gravity. We have obtained the explicit form (3.23) of the genus zero part F o 0 of the free energy. We have then argued that the higher order corrections F õ g can be computed systematically by solving Buryak's equation recursively. We have demonstrated this computation explicitly for the first few orders. We have also elucidated the polynomial structure of F õ g≥2 . We emphasize that our result of F õ g holds for arbitrary value of the couplings {t n }. It is interesting that F õ g is written as a combination of genus-zero quantities only, which can be thought of as an open analog of the constitutive relation for closed topological gravity [26,27].
There are several interesting open questions. In general, the small g s expansion of F o in (1.3) is an asymptotic series and we expect that F o receives non-perturbative corrections in g s . Such corrections are physically interpreted as the effect of the so-called ZZ-branes [34]. It would be interesting to find the general structure of the effect of ZZ-branes for the arbitrary background {t n }. It is known that [35] some of the background {t n } exhibits a non-perturbative instability and it does not lead to a well-defined theory. It would be interesting to find the map of the "swampland" in the space of all two-dimensional topological gravities {t n }. In particular, it is argued that the JT gravity matrix model suffers from such a non-perturbative instability [11]. It is important to see if JT gravity is non-perturbatively well-defined or not. where the second sum is taken over all sequences j 1 , j 2 , . . . , j n−k of non-negative integers such that j 1 + j 2 + · · · + j n−k = k, j 1 + 2j 2 + · · · + (n − k)j n−k = n − 1. Therefore

Acknowledgments
By rewriting j 2a as j a this gives (3.9).