Tyler J. Jarvis

Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of higher 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 KdVr 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 singularity Ar−1. 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.

GEOMETRY OF THE MODULI OF HIGHER SPIN CURVES

This article treats various aspects of the geometry of the moduli of r-spin curves and its compactification . Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdVr) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology. For any line bundle on the universal curve over the stack of stable curves, there is a smooth stack of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism . In the special case that is the relative dualizing sheaf, then is the stack of r-spin curves. We construct a smooth compactification of the stack , describe the geometric meaning of its points, and prove that its coarse moduli is projective. We also prove that when r is odd and g>1, the compactified stack of spin curves and its coarse moduli space are irreducible, and when r is even and is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Wittens conjecture, and also generalize to the n-pointed case the classical fact that when is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [15], and also in the study of the cohomological field theory related to Wittens conjecture [16, 34].

Moduli of twisted spin curves

In this note we give a new, natural construction of a compactification of the stack of smooth r-spin curves, which we call the stack of stable twisted r-spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible G m -spaces and Q-line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves. We construct representable morphisms from the stacks of stable twisted r-spin curves to the stacks of stable r-spin curves and show that they are isomorphisms. Many delicate features of r-spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the ∂-operator of Seeley and Singer and Wittens cohomology class go through without complications in the setting of twisted spin curves.

Stringy K-theory and the Chern character

We construct two new G-equivariant rings:

Torsion-Free Sheaves and Moduli of Generalized Spin Curves

\mathcal{K}(X,G)

A mathematical theory of the gauged linear sigma model

, called the stringy K-theory of the G-variety X, and

Pointed admissible G -covers and G -equivariant cohomological field theories

\mathcal{H}(X,G)

Landau-Ginzburg Mirror Symmetry for Orbifolded Frobenius Algebras

, called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack

Witten’s D 4 integrable hierarchies conjecture

\mathcal{X}

Spin Gromov-Witten Invariants

, we also construct a new ring