Open topological recursion relations in genus $1$ and integrable systems

The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov-Witten invariants in genus 1. We find an explicit formula for any solution analogous to the Dijkgraaf-Witten formula for a descendent Gromov-Witten potential in genus 1. We then prove that at the approximation up to genus 1 the exponent of an open descendent potential satisfies a system of explicitly constructed linear evolutionary PDEs with one spatial variable.


Introduction
Total descendent potentials, also called formal Gromov-Witten potentials, are certain formal power series of the form where N ≥ 1, and t α a , 1 ≤ α ≤ N, a ≥ 0, and ε are formal variables, appearing in various curve counting theories in algebraic geometry including Gromov-Witten theory, Fan-Jarvis-Ruan-Witten theory, and the more recent theory of Gauged Linear Sigma Models. The number N is often called the rank. Typically, the coefficients of total descendent potentials are the integrals of certain cohomology classes over moduli spaces of closed Riemann surfaces with additional structures. The function F g controls the integrals over the moduli spaces of Riemann surfaces of genus g. The simplest example of a total descendent potential is the Witten generating series F W (t 0 , t 1 , t 2 , . . . , ε) of intersection numbers on the moduli space of stable Riemann surfaces of genus g with n marked points M g,n . Note that here and below we omit the upper indices in the t-variables when the rank is 1.
There is a unified approach to total descendent potentials using the notion of a cohomological field theory (CohFT) [KM94] and the Givental group action [Giv01a,Giv01b,Giv04]. Briefly speaking, the generating series of correlators of CohFTs form the space of total ancestor potentials, and then using the lower-triangular Givental group action one gets the whole space of total descendent potentials (see e.g. [Sha09, Section 2] and [FSZ10]).
There is a remarkable and deep relation between total descendent potentials and the theory of nonlinear PDEs. One of its manifestations is the following system of PDEs for the descendent potential in genus 0 (see e.g. [Sha09, Section 2] and [FSZ10,Corollary 4.13]): , 1 ≤ α, β, γ ≤ N, a, b, c ≥ 0, (1.2) called the string equation and the topological recursion relations in genus 0, respectively. Here (η αβ ) = η is an N × N symmetric nondegenerate matrix with complex coefficients, the constants η αβ are defined by (η αβ ) := η −1 , and we use the Einstein summation convention for repeated upper and lower Greek indices. Note that the system of equations (1.2) can be equivalently written as where d (·) denotes the full differential.
There are equations similar to (1.2) in genus 1 (see e.g. [EGX00, Equation (1.7)]): They are called the topological recursion relations in genus 1. These equations imply that , and . Equations similar to (1.2) and (1.3) exist in all genera, but their complexity grow very rapidly with the genus (see e.g. [Liu07] for some results in genus 2).
One can see that equations (1.1), (1.2), (1.3) are universal, meaning that they do not depend on a total descendent potential. On the other hand, there is a rich theory [DZ01] of hierarchies of evolutionary PDEs with one spatial variable associated to total descendent potentials and containing the full information about these potentials. Conjecturally, for any total descendent potential F there exists a unique system of PDEs of the form where w 1 , . . . , w N ∈ C[[t * * , ε]], P α β,b are differential polynomials in w 1 , . . . , w N , i.e., P α β,b are formal power series in ε with the coefficients that a polynomials in w γ x , w γ xx , . . . (we identify x = t 1 0 ) whose coefficients are formal power series in w γ , such that a unique solution of the system (1.5) specified by the condition w α | t β b =δ β,1 δ b,0 x = δ α,1 x is given by w α = η αµ ∂ 2 F ∂t µ 0 ∂t 1 0 . This system of PDEs (if it exists) is called the Dubrovin-Zhang hierarchy or the hierarchy of topological type. The conjecture is proved at the approximation up to ε 2 [DZ98] and in the case when the Dubrovin-Frobenius manifold associated to the total descendent potential is semisimple [BPS12a,BPS12b]. The Dubrovin-Zhang hierarchy corresponding to the Witten potential F W is the Korteweg-de Vries (KdV) hierarchy This statement is equivalent to Witten's conjecture [Wit91], proved by Kontsevich [Kon92].
A more recent and less developed field of research is the study of the intersection theory on various moduli spaces of Riemann surfaces with boundary. Such a moduli space always comes with an associated moduli space of closed Riemann surfaces, and, thus, there is the corresponding total descendent potential F (t * * , ε) = g≥0 ε 2g F g (t * * ) of some rank N. There is a large class of examples [PST14, BCT18, ST19, Che18, CZ18, CZ19, Zin20] where the intersection numbers on the corresponding moduli space of Riemann surfaces with boundary of genus 0 are described by a formal power series F o 0 (t * * , s * ) ∈ C[[t * * , s * ]] depending on an additional sequence of formal variable s a , a ≥ 0, and satisfying the relations Regarding higher genera, much less is known. However, conjecturally, the intersection theory on moduli spaces of Riemann surfaces with boundary of genus 1 is controlled by formal power called the open topological recursion relations in genus 1. In the case of the intersection theory on the moduli spaces of Riemann surfaces with boundary of genus g with k boundary marked points and l internal marked points M g,k,l , these relations were conjectured by the authors of [PST14] and proved in [BCT18, Section 6. An analog of the theory of Dubrovin-Zhang hierarchies for solutions of the system (1.6)-(1.8) was developed in [BB19]. Regarding higher genera, a very promising direction was opened by the series of papers [PST14, Tes15, Bur15, Bur16, BT17] (see also [ABT17]), where the authors studied the intersection numbers on the moduli spaces of Riemann surfaces with boundary of genus g with k boundary marked points and l internal marked points M g,k,l . The main result of these works is the proof [BT17] of the Pandharipande-Solomon-Tessler conjecture [PST14] saying that the generating series of the intersection numbers satisfies the following system of PDEs: Remark 1.2. To be precise, we have presented a version of the Pandharipande-Solomon-Tessler conjecture, which is slightly different from the original one in two aspects. First of all, in [PST14] the authors considered a function F o,P ST related to our function F o,P ST by The function F o,P ST can be reconstructed from the function F o,P ST using the system of PDEs Second, the system of PDEs from [PST14, Conjecture 2] determining the function F o,P ST does not have the form of a system of evolutionary PDEs with one spatial variable. The fact that the presented version of the Pandharipande-Solomon-Tessler conjecture is equivalent to the original one was observed in [Bur16].
In this paper we study solutions of the open topological recursion relations in genus 1. First, we find an analog of formula (1.4). Then, using this formula, we construct a system of linear PDEs of the form similar to (1.9) and (1.10) such that the function exp(F o 0 + εF o 1 ) satisfies it at the approximation up to ε. An expectation in higher genera and a relation with a Lax description of the Dubrovin-Zhang hierarchies are also discussed. We are grateful to Oleg Chalykh for valuable remarks about the preliminary version of the paper.

Closed and open descendent potentials in genus 0
In this section we recall the definitions of closed and open descendent potentials in genus 0 and the construction of associated to them systems of PDEs.
2.1. Differential polynomials. Consider formal variables v α i , α = 1, . . . , N, i = 0, 1, . . .. Following [DZ01] (see also [Ros17]) we define the ring of differential polynomials A v 1 ,...,v N in the variables v 1 , . . . , v N as the ring of polynomials in the variables v α i , i > 0, with coefficients in the ring of formal power series in the variables v α := v α 0 : The standard gradation on A v 1 ,...,v N , which we denote by deg, is introduced by deg It increases the standard degree by 1.

Consider the extension
] of the space A v 1 ,...,v N with a new variable ε of standard degree deg ε := −1. Let A v 1 ,...,v N ;d denote the subspace of degree d of A. Abusing the terminology we still call elements of the space A v 1 ,...,v N differential polynomials.
2.2. Closed descendent potentials in genus 0. Let us fix N ≥ 1, an N × N symmetric nondegenerate complex matrix η = (η αβ ), and an N-tuple of complex numbers (A 1 , . . . , A N ), not all equal to zero. We will use the notation is called a descendent potential in genus 0 if it satisfies the following system of PDEs: We will sometimes call a descendent potential in genus 0 a closed descendent potential in genus 0 in order to distinguish it from an open analog that we will discuss below.
Remark 2.4. For any total descendent potential F = g≥0 ε 2g F g the function F 0 is a descendent potential in genus 0. However, describing precisely which descendent potentials in genus 0 can be extended to total descendent potentials is an interesting open problem.
Define differential polynomials Ω Then we have (see e.g. [BPS12b, Proposition 3]) This implies that the N-tuple of functions (v top ) α | t γ 0 →t γ 0 +A γ x is a solution of the following system of PDEs: which is called the principal hierarchy associated to the potential F 0 .
2.3. Open descendent potentials in genus 0. Let us fix a closed descendent potential in genus 0 F 0 .
5. An open descendent potential in genus 0 F o 0 ∈ C[[t * * , s * ]] is a solution of the following system of PDEs: Consider a new formal variable φ. Similarly to the differential polynomials Ω α,a;β,b , let us introduce differential polynomials Γ This implies that the (N + 1)-tuple of functions (v top ) 1 , . . . , (v top ) N , φ top t γ 0 →t γ 0 +A γ x satisfies the following system of PDEs: which we call the extended principal hierarchy associated to the pair of potentials (F 0 , F o 0 ).

Open descendent potentials in genus 1
Here we introduce the notion of an open descendent potential in genus 1 and prove two main results of our paper: Theorems 3.2 and 3.5.  Consider an open descendent potential in genus 1 F o 1 . Define a formal power series Theorem 3.2. We have Proof. Note that Equation (2.5) implies that Therefore, the logarithm log ∂t 1 1 0 ∂s 0 is a well-defined formal power series in the variables t * * and s * . Also, Equations (2.1) and (2.5) imply that 3) is true when t * ≥1 = s ≥1 = 0. Using the linear differential operators Equations (3.1) and (3.2) can be equivalently written as ∂s a ∂s , a ≥ 0.
3.2. Differential operators and PDEs. Consider a differential operator L of the form Let f be a formal variable and consider the PDE where Q i ∈ A f can be recursively computed by the relation Remark 3.3. Note that Q i does not depend on f and is a polynomial in the derivatives f x , f xx , . . . and ε. Moreover, if we introduce a new formal variable ψ and substitute f i+1 = ψ i , i ≥ 0, then Q i becomes a differential polynomial of degree 0.
We see that PDE (3.6) is equivalent to the following PDE: Let us look at this PDE in more details at the approximation up to ε.
The formula is clearly true for i = 0. We proceed by induction: Consider the expansion We see that Equation (3.7) has the form Theorem 3.5. The formal power series at the approximation up to ε.

Proof. Abusing notations let us denote the formal powers series
, v α , and φ, respectively. We can then write the statement of Theorem 3.2 as Let us prove Equation (3.9) at the approximation up to ε. We have Therefore, by Equation (3.8), we have to check that Using the definition of L which proves Equation (3.11) and, hence, Equation (3.9) at the approximation up to ε.
The proof of Equation (3.10) is analogous.

Expectation in higher genera
Consider a total descendent potential F (t * * , ε) = g≥0 ε 2g F g (t * * ) of some rank N.  Suppose that there exists a Dubrovin-Zhang hierarchy corresponding to our total descendent potential F (this is true when, for example, the associated Dubrovin-Frobenius manifold is semisimple). It is easy to show that if Expectation 4.1 is true, then the flows ∂