/pdf/compactifying-the-space-of-stable-maps-10odrhuhej.pdf

Compactifying the space of stable maps

Given a smooth complex Deligne-Mumford stack ${\cal X}$ with a projective coarse moduli space, we introduce Gromov-Witten invariants of ${\cal X}$ and prove some of their basic properties, including the WDVV equation.

Gromov-Witten theory of Deligne-Mumford stacks

For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological eld theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity Ar 1. We also resolve two outstanding conjectures of Witten. The rst conjecture is that ADE-singularities are self-dual, and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

http://annals.math.princeton.edu/wp-content/uploads/annals-v178-n1-p01-p.pdf

The Witten equation, mirror symmetry, and quantum singularity theory

I. Introduction: Transcending Nationalism: The European Community 1958-2000

We study the structure of the stacks of twisted stable maps to the classifying stack of a finite group G—which we call the stack of twisted G-covers, or twisted G-bundles. For a suitable group Gwe show that the substack corresponding to admissible G-covers is a smooth projective fine moduli space. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Twisted Bundles and Admissible Covers

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.

/pdf/moduli-spaces-of-higher-spin-curves-and-integrable-3ez3d03pmc.pdf

Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

We study a system of nonlinear elliptic PDEs associated with a quasi-homogeneous polynomial. These equations were proposed by Witten as the replacement for the Cauchy-Riemann equation in the singularity (Landau-Ginzburg) setting. We introduce a perturbation to the equation and construct a virtual cycle for the moduli space of its solutions. Then, we study the wall-crossing of the deformation of the virtual cycle under perturbation and match it to classical Picard-Lefschetz theory. An extended virtual cycle is obtained for the original equation. Finally, we prove that the extended virtual cycle satisfies a set of axioms similar to those of Gromov-Witten theory and r-spin theory.

The Witten equation and its virtual fundamental cycle

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 Witten's 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 Witten's conjecture [16, 34].

Geometry of the moduli of higher 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 Witten's cohomology class go through without complications in the setting of twisted spin curves.

https://www.ams.org/proc/2003-131-03/S0002-9939-02-06562-0/S0002-9939-02-06562-0.pdf

Tyler J. Jarvis

Papers

The Witten equation, mirror symmetry, and quantum singularity theory

Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

The Witten equation and its virtual fundamental cycle

Geometry of the moduli of higher spin curves

Moduli of twisted spin curves