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
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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