Extended r-spin theory in all genera and the discrete KdV hierarchy

Abstract

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's r-spin classes. They are parameterized by a phase space which has one extra dimension and in genus 0 they correspond to the extended r-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the r-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus 0 it corresponds to a flat F-manifold. For $r=2$ we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.

Introduction

The moduli space of r-spin curves, parameterizing stable curves C with n marked points $p_1,\dots ,p_n\in C$ together with an r-th root, $r\ge 2$, of the twisted canonical bundle ${\omega }_C({\sum }_{i=1}^n(1-{\alpha }_i)p_i)$, ${\alpha }_i\in \mathbb{Z}$, has a rich geometric structure and has recently proved to be a central tool in the study of the cohomology of the moduli space of stable curves ${\bar{\mathcal{M}}}_{g,n}$.

Witten's r-spin classes [38], [33], [14], [30], [19] are cohomological field theories on ${\bar{\mathcal{M}}}_{g,n}$, constructed out of the geometry of r-spin moduli spaces, and their intersection theory was shown in [18] to be controlled by the Gelfand-Dickey $(r-1)$-KdV hierarchy, as previously conjectured in [38].

As an example of the richness of such objects, in [31], 3-spin classes were used to produce and prove a large system of relations in the tautological subring of the cohomology ring of ${\bar{\mathcal{M}}}_{g,n}$. This system contains all other previously known relations between the generators of the tautological ring and is in fact conjectured to be a complete system, thus yielding an explicit description of the tautological ring itself.

The present paper starts with the observation that, in genus 0, from results of [24] and [7], the construction of r-spin cohomological field theories can be extended to systems of cohomology classes on ${\bar{\mathcal{M}}}_{0,n}$ with phase space 1-dimensionally bigger and satisfying a system of axioms corresponding to the structure of a flat F-manifold [28], as opposed to the usual Frobenius manifold associated to genus 0 CohFTs.

Given also the relevance of such genus 0 extended r-spin classes in describing the intersection theory of moduli spaces of Riemann surfaces with boundary [32], [35], [36], first observed in [7], we were motivated to look for a higher genus generalization of extended r-spin classes.

As we said, already in genus 0, the extended r-spin theory is not quite a cohomological field theory/Frobenius manifold. However the notion of flat F-manifold can be naturally extended to higher genus as a system of cohomology classes on ${\bar{\mathcal{M}}}_{g,n}$ satisfying a set of axioms that is correspondingly weaker than those of a CohFT. We introduce this general notion in Section 2 and we call it an F-cohomological field theory.

We then proceed to construct a specific (homogeneous) F-CohFT that reduces, in genus 0, to the extended r-spin classes. The construction is based on a generalization of a formula of J. Guéré [22]. Guéré considers two elements in the K-theory of the moduli space of r-spin curves: the derived push-forward of the universal r-th root bundle from the universal curve to the moduli space and the Hodge bundle. Then he constructs a certain combination of characteristic classes of such two objects, depending on a parameter and well defined in the K-theory, and shows that, in a certain limit of the parameter and after push-forward to ${\bar{\mathcal{M}}}_{g,n}$, this explicit formula recovers the product of the r-spin class by the top Chern class of the Hodge bundle.

It turns out that such construction can be generalized to the extended phase space of extended r-spin theory giving, for generic value of the parameter, an actual cohomological field theory which reduces, in the above limit, to a homogeneous (with respect to a natural extension of the grading for r-spin theory) F-CohFT generalizing to all genera the extended r-spin theory. When restricted to the usual phase space, this F-CohFT gives, as prescribed by Guéré's formula, the product of the top Chern class of the Hodge bundle by the r-spin class, which is of course a partial cohomological field theory. However such factorization does not happen on the extra component of the extended phase space.

We then study the problem of explicit description of the intersection theory with our F-CohFT in the case $r=2$. Note that the CohFT, depending on a parameter, discussed above, is semisimple, which means that, by the results of [12], the corresponding Dubrovin-Zhang hierarchy [16] exists and that, in the limit, it will become a homogeneous system of evolutionary PDEs. The corresponding Hamiltonian structure, however, degenerates in the limit. Using homogeneity, we manage to completely identify such system of PDEs as an extension of the discrete (or q-difference) KdV hierarchy of [20] thereby proving a Witten-Kontsevich type result for the F-CohFT: the partition function of the extended 2-spin theory satisfies the extended discrete KdV hierarchy. This effectively computes all intersection numbers of the F-CohFT with psi-classes.

We remark here that the discrete KdV hierarchy also appeared in [4], [3], together with its bigraded generalizations, as the natural candidate for the Dubrovin-Zhang hierarchy of the equivariant Gromov-Witten theory of local ${\mathbb{P}}^1$-orbifolds. It would be natural to investigate the relation between our construction of the extended r-spin classes and such Gromov-Witten theory.

Finally we prove that the double ramification hierarchy construction and results of [6], [13] can be generalized to F-CohFTs and that, for $r=2$ the DR hierarchy also corresponds to the extended discrete KdV hierarchy, the two incarnations being related by a Miura transformation, providing yet another example of DR/DZ equivalence along the lines of what was conjectured and investigated in [6], [8], [9], [11], but in the more general context of F-CohFTs.

We conclude with some remarks and a conjecture about the possible relation of our extended r-spin theory with open Hodge integrals, i.e. intersection numbers of psi-classes with Hodge classes on the moduli space of open Riemann surfaces, generalizing the genus 0 results of [7].

We would like to thank Andrea Brini, Guido Carlet, Oleg Chalykh, Allan Fordy, Paolo Lorenzoni, Alexander Mikhailov, Jake Solomon, Ran Tessler and Dimitri Zvonkine for useful discussions.

A.B. was supported by the grants RFBR-20-01-00579 and RFBR-16-01-00409.