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Theta classes: generalized topological recursion, integrability and $\mathcal{W}$-constraints

Vincent Bouchard, Nitin K. Chidambaram, Alessandro Giacchetto, Sergey Shadrin

TL;DR

The paper proves that the generalized topological recursion on the (r,s) spectral curves computes the descendant integrals of the Θ^{r,s}-classes, linking intersection theory to integrable hierarchies. It shows the Θ^{r,s}-descendant potential Z^{r,s} is a tau function of the $r$-KdV hierarchy, with initial conditions expressed through explicit Θ-integrals, and derives explicit $\\mathcal{W}$-constraints via twist-field realizations of the $\\mathcal{W}(\\mathfrak{gl}_r)$-algebra. A second determinantal/ wave-function formalism is developed to obtain loop equations, which in special cases (notably s = r−1 and s = 1) determine Z^{r,s} uniquely through (shifted) Airy-structure frameworks; for other cases, a reduced potential remains unfixed, illustrating a nuanced landscape of integrability and constraints. The results extend known correspondences for $r$-spin, Θ-classes, and Norbury Θ-classes, and provide a unified setting for understanding how topological recursion, intersection theory, and integrable hierarchies intertwine in the (r,s) family.

Abstract

We study the intersection theory of the $Θ^{r,s}$-classes, where $r \geq 2$ and $1 \le s \le r-1$, which are cohomological field theories obtained as the top degrees of Chiodo classes. We show that the recently introduced generalized topological recursion on the $(r,s)$ spectral curves computes the descendant integrals of the $Θ^{r,s}$-classes. As a consequence, we deduce that the descendant potential of the $Θ^{r,s}$-classes is a tau function of the $r$-KdV hierarchy, generalizing the Brézin--Gross--Witten tau function (the special case $r=2$, $s=1$). We also explicitly compute the $\mathcal{W}$-constraints satisfied by the descendant potential, obtained as differential representations of the $\mathcal{W}(\mathfrak{gl}_r)$-algebra at self-dual level. This work extends previously known results on the Witten $r$-spin class, the $r$-spin $Θ$-classes (the case $s=r-1$), and the Norbury $Θ$-classes (the special case $r=2$, $s=1$).

Theta classes: generalized topological recursion, integrability and $\mathcal{W}$-constraints

TL;DR

The paper proves that the generalized topological recursion on the (r,s) spectral curves computes the descendant integrals of the Θ^{r,s}-classes, linking intersection theory to integrable hierarchies. It shows the Θ^{r,s}-descendant potential Z^{r,s} is a tau function of the -KdV hierarchy, with initial conditions expressed through explicit Θ-integrals, and derives explicit -constraints via twist-field realizations of the -algebra. A second determinantal/ wave-function formalism is developed to obtain loop equations, which in special cases (notably s = r−1 and s = 1) determine Z^{r,s} uniquely through (shifted) Airy-structure frameworks; for other cases, a reduced potential remains unfixed, illustrating a nuanced landscape of integrability and constraints. The results extend known correspondences for -spin, Θ-classes, and Norbury Θ-classes, and provide a unified setting for understanding how topological recursion, intersection theory, and integrable hierarchies intertwine in the (r,s) family.

Abstract

We study the intersection theory of the -classes, where and , which are cohomological field theories obtained as the top degrees of Chiodo classes. We show that the recently introduced generalized topological recursion on the spectral curves computes the descendant integrals of the -classes. As a consequence, we deduce that the descendant potential of the -classes is a tau function of the -KdV hierarchy, generalizing the Brézin--Gross--Witten tau function (the special case , ). We also explicitly compute the -constraints satisfied by the descendant potential, obtained as differential representations of the -algebra at self-dual level. This work extends previously known results on the Witten -spin class, the -spin -classes (the case ), and the Norbury -classes (the special case , ).
Paper Structure (38 sections, 32 theorems, 154 equations)

This paper contains 38 sections, 32 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Generalized topological recursion and determinantal formulas
  3. Chiodo classes, topological recursion and determinantal formulas
  4. Chiodo classes
  5. Chiodo classes and topological recursion
  6. Chiodo classes, generalized topological recursion and determinantal formulas
  7. Theta-classes, generalized topological recursion and determinantal formulas
  8. Theta-classes
  9. The t to zero limit
  10. Theta-classes and generalized topological recursion
  11. Integrability
  12. r-KdV integrability
  13. The initial conditions
  14. Loop equations
  15. A second determinantal formula
  16. ...and 23 more sections

Key Result

Theorem A

The correlators $\omega_{g,n}$ produced by generalized topological recursion on the $(r,s)$ spectral curve calculate the descendant integrals of the $\Theta^{r,s}$-classes. More precisely, for $g \ge 0$, $n \ge 1$ such that $2g-2+n>0$, we have where $d \xi_{k,a}(z) = (rk+a)!^{(r)} \frac{dz}{z^{rk+a+1}}$.

Theorems & Definitions (74)

  • Theorem A: Generalized topological recursion for $\Theta^{r,s}$
  • Theorem B: Integrability
  • Theorem C: $\mathcal{W}$-constraints
  • Definition 2.1
  • Proposition 2.2: SSZ15LPSZ17Gia21
  • Remark 2.3
  • Proposition 2.4: ABDKS24-logTR
  • proof
  • Definition 2.5
  • Proposition 2.6
  • ...and 64 more