1,042 research outputs found
A topological view of Gromov-Witten theory
We study relative Gromov-Witten theory via universal relations provided by
the interaction of degeneration and localization. We find relative
Gromov-Witten theory is completely determined by absolute Gromov-Witten theory.
The relationship between the relative and absolute theories is guided by a
strong analogy to classical topology.
As an outcome, we present a mathematical determination of the Gromov-Witten
invariants (in all genera) of the Calabi-Yau quintic 3-fold in terms of known
theories.Comment: 43 pages, revised & new surface calculation adde
Curves on K3 surfaces and modular forms
We study the virtual geometry of the moduli spaces of curves and sheaves on
K3 surfaces in primitive classes. Equivalences relating the reduced
Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable
pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa
conjecture evaluating integrals (in all genera) in terms of
explicit modular forms. Indeed, all K3 invariants in primitive classes are
shown to be governed by modular forms.
The method of proof is by degeneration to elliptically fibered rational
surfaces. New formulas relating reduced virtual classes on K3 surfaces to
standard virtual classes after degeneration are needed for both maps and
sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds.
Our approach uses a result of Kiem and Li to produce reduced classes. In
Appendix A, we answer a number of questions about the relationship between the
Kiem-Li approach, traditional virtual cycles, and symmetric obstruction
theories.
The interplay between the boundary geometry of the moduli spaces of curves,
K3 surfaces, and modular forms is explored in Appendix B by A. Pixton.Comment: An incorrect example in Appendix A, pointed out to us by Dominic
Joyce, has been replaced by a reference to a new paper arXiv:1204.3958
containing a corrected exampl
Sheaf counting on local K3 surfaces
There are two natural ways to count stable pairs or Joyce–Song pairs on X=K3×C; one via weighted Euler characteristic and the other by virtual localisation of the reduced virtual class. Since X is noncompact these need not be the same. We show their generating series are related by an exponential. As applications we prove two conjectures of Toda, and a conjecture of Tanaka–Thomas defining Vafa–Witten invariants in the semistable case
Quantum cohomology of the Hilbert scheme of points on A_n-resolutions
We determine the two-point invariants of the equivariant quantum cohomology
of the Hilbert scheme of points of surface resolutions associated to type A_n
singularities. The operators encoding these invariants are expressed in terms
of the action of the affine Lie algebra \hat{gl}(n+1) on its basic
representation. Assuming a certain nondegeneracy conjecture, these operators
determine the full structure of the quantum cohomology ring. A relationship is
proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas
theories of A_n x P^1. We close with a discussion of the monodromy properties
of the associated quantum differential equation and a generalization to
singularities of type D and E.Comment: 37 pages, 2 figures; typos are correcte
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