Moduli of spin curves and spin quantum cohomology
We review recent progress on the generalized Witten conjecture that
relates intersection theory on the moduli spaces of higher spin curves
and the Gelfand-Dickey integrable hierarchies. We discuss this conjecture
and its main ingredients from the point of view of cohomological field
theories. Then we introduce the moduli spaces of so-called stable spin
maps into a variety V and the spin Gromov-Witten classes associated
to them. Restricting to genus zero, this gives the spin quantum cohomology
of V whose Frobenius structure is isomorphic to the tensor product
of the Frobenius manifolds associated to ordinary quantum cohomology of
V
and the Gelfand-Dickey hierarchy (or, equivalently, the simple singularity
of type Ar).
OPEN PROBLEMS RELATED TO MODULI OF HIGHER SPIN CURVES AND SPIN QUANTUM
COHOMOLOGY
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Prove the generalized Witten's conjecture relating the intersection theory
on the moduli spaces of r-spin curves and the r-th Gelfand-Dickey
integrable hierarchy.
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Quantum cohomology and Gromov-Witten invariants of a variety V have an
interpretation from the point of view of enumerative geometry - they "count"
curves on V satisfying certain conditions. Find an enumerative interpretation
of the spin quantum cohomology and spin Gromov-Witten invariants of V.
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Another conjecture of Witten asserts that the generating function F
of the descendant Gromov-Witten invariants of V (the so-called partition
function of the gravitational quantum cohomology of V) is annihilated
by certain differential operators Li satisfying the commutation
relations of the Virasoro algebra. The (still conjectural) explicit form
of this operators was suggested by Eguchi, Hori, Xiong and Katz. This Virasoro
conjecture has been proved only when V is a point, and also for
more general classes of V for genus zero and one invariants.
The corresponding partition function in the case of the r-spin theory
should be annihilated by a larger collection of operators satisfying the
relations of the so-called Wr-algebra. Explicit formulas
for these operators are known in the case when V is a point (and
the corresponding Virasoro conjecture is equivalent to the generalized
Witten conjecture of Problem 1).
Find the r-spin analogs of the Eguchi-Hori-Xiong-Katz formula and
prove the corresponding Wr - conjecture (at least for
genus zero and one invariants).
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The generalized Witten's conjecture and theory of spin quantum cohomology
are related to a physical model of two-dimensional gravity with the gauge
group of type Ar.
Find analogs of r-spin curves and the (conjectural) formulas connecting
the corresponding moduli spaces with the Drinfeld-Sokolov generalizations
of the Gelfand-Dickey equations for series D and E.
Fall 2000 meeting of the PNGS