[Paper Review] Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

Tyler J. Jarvis, Takashi Kimura+1 more|arXiv (Cornell University)|May 5, 1999

Algebraic Geometry and Number TheoryMathematics17 references21 citations

TL;DR

This paper proves the genus-zero part of the generalized Witten conjecture by showing that intersection numbers on the moduli space of $r$-spin curves form a generating function that solves the semiclassical limit of the $\mathrm{KdV}_r$ hierarchy. It constructs a cohomological field theory of rank $r-1$ via a virtual class, yielding a Frobenius manifold isomorphic to the base of the versal deformation of the $A_{r-1}$ singularity.

ABSTRACT

We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV_r equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank $r-1$ in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the A_{r-1} singularity. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov-Witten invariants and quantum cohomology.

Motivation & Objective

• To formulate and prove the genus-zero part of the generalized Witten conjecture relating higher spin curves to $\mathrm{KdV}_r$ integrable hierarchies.
• To define a virtual cohomology class on the moduli space of $r$-spin curves that satisfies axioms for a cohomological field theory (CohFT) of rank $r-1$.
• To establish that the resulting Frobenius manifold structure on the state space is isomorphic to that of the versal deformation base of the $A_{r-1}$ singularity.
• To verify the puncture, dilaton, and topological recursion relations for the $r$-spin CohFT by analogy with Gromov-Witten theory.

Proposed method

• Define a virtual class $c^{1/r}$ on $\overline{\mathcal{M}}_{g,n}^{1/r}$ as the top Chern class of a tautological bundle in genus zero, satisfying CohFT axioms.
• Construct the large phase space potential function $\Phi_{0}(\mathbf{t})$ from integrals of $c^{1/r}$ over $\overline{\mathcal{M}}_{0,n}^{1/r}$, using $\psi$-classes and $\mu$-classes.
• Prove that $\Phi_{0}(\mathbf{t})$ satisfies the string equation and WDVV equations, ensuring uniqueness of the solution.
• Show that the potential $\Phi_{0}(\mathbf{t})$ matches the semiclassical limit of the $\mathrm{KdV}_r$ hierarchy via uniqueness theorems.
• Use the isomorphism of Frobenius manifolds to link the $r$-spin CohFT to the $A_{r-1}$ singularity’s versal deformation space.
• Verify the $r=2$ case by showing the potential reduces to Kontsevich’s $\tau$-function for the KdV hierarchy.

Experimental results

Research questions

• RQ1 Does the generating function of intersection numbers on $\overline{\mathcal{M}}_{g,n}^{1/r}$ yield a solution to the semiclassical limit of the $\mathrm{KdV}_r$ hierarchy?
• RQ2 Can a cohomological field theory of rank $r-1$ be constructed on the moduli space of $r$-spin curves via a virtual class?
• RQ3 Is the Frobenius manifold structure on the $r$-spin state space isomorphic to that of the $A_{r-1}$ singularity’s versal deformation?
• RQ4 Do the puncture, dilaton, and topological recursion relations hold for the $r$-spin CohFT, analogous to Gromov-Witten theory?

Key findings

• The genus-zero potential $\Phi_{0}(\mathbf{t})$ of the $r$-spin CohFT satisfies the string equation and WDVV equations, ensuring uniqueness as a solution.
• For $r=2$, the $r$-spin CohFT potential coincides with Kontsevich’s $\tau$-function, confirming the original Witten conjecture in genus zero.
• For $g=0$ and arbitrary $r$, the potential $\Phi_{0}(\mathbf{t})$ matches the semiclassical limit of the $\mathrm{KdV}_r$ hierarchy, proving the genus-zero generalized Witten conjecture.
• The Frobenius manifold associated to the $r$-spin CohFT is isomorphic to the Frobenius manifold of the versal deformation base of the $A_{r-1}$ singularity.
• The virtual class $c^{1/r}$ is realized as the top Chern class of a tautological bundle on $\overline{\mathcal{M}}_{0,n}^{1/r}$, satisfying all required CohFT axioms.
• The $r$-spin CohFT potential is independent of variables $t_n^{r-1}$, consistent with the vanishing axiom of the virtual class.