Introduction

In the past few years it has become quite clear that the theory of r-spin curves

and the theory of Gromov-Witten invariants have much in common. That connec-

tion has been made even more definite by the recent developments in the theory

of orbicurves and orbifold stable maps, as developed by W. Chen, Y. Ruan, D.

Abramovich, A. Vistoli and their collaborators. That theory has not only simpli-

fied the r-spin constructions considerably, but has also shown the spin theory to be

very similar to orbifold Gromov-Witten theory.

The conference in San Francisco was intended to bring together researchers

working on a wide variety of different aspects of these theories in the hope of

further illuminating these connections. The conference was very successful and the

papers here go even farther than the conference in bringing together many formerly

disparate aspects of the theory of spin curves and orbifold Gromov-Witten theory.

The first three papers, those of A. Polishchuk, A. Chiodo, and Y.-P. Lee, are

directly connected to Witten's Conjecture for r-spin curves, which was the initial

motivation for much of the interest in r-spin curve theory. Polishchuk and Chiodo's

papers are closely tied to issues of the virtual class on the moduli space or r-spin

curves, whereas Y.-P. Lee's paper shows that, from the axiomatic properties of

that class, and using Givental's potentials with known relations of Dubrovin, Zhag,

Getzler, and X. Liu, the Witten conjecture holds in genus 1 and 2.

The relations developed by X. Liu play an important role in Y.-P. Lee's proof,

and they also help illuminate other essential structures in general Gromov-Witten

theory and other cohomological field theories. Liu's paper included here, on rela-

tions in the large phase space, is even more strongly connected to r-spin theory,

since the descent axiom of that theory, coupled with the theory of stable spin maps,

illustrates a surprising connection between the large phase space of usuftl Gromov-

Witten theory and the r-spin correlators.

Another important aspect of the theory of r-spin curves and its generalizations

is the relation it has to singularities, especially as a sort of Landau-Ginzberg A-

model. The bridge from spin theories to singularities is through the orbifolding of

Frobenius algebras (and cohomological field theories), as described in the paper of

Kaufmann.

Orbifold cohomology is also closely tied to singularity theory, as detailed in the

various results and conjectures described in Y. Ruan's paper on crepant resolutions.

And just as Ruan's work illuminates the connections between orbifold cohomology

of a quotient singularity and the ordinary cohomology of a crepant resolution, so also

the paper of Lupercio and Uribe shows that other cohomology theories on orbifolds

are related, in particular the Beilinson-Deligne cohomology and the Cheeger-Simons

cohomology of a global quotient orbifold are canonically isomorphic.

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