Rahul Pandharipande

Gromov–Witten theory and Donaldson–Thomas theory, I

We conjecture an equivalence between the Gromov–Witten theory of 3-folds and the holomorphic Chern–Simons theory of Donaldson and Thomas. For Calabi–Yau 3-folds, the equivalence is defined by the change of variables

Curve counting via stable pairs in the derived category

e^{iu}=-q

The local Gromov-Witten theory of curves

, where

Relative maps and tautological classes

u

Stable pairs and BPS invariants

is the genus parameter of Gromov–Witten theory and

Virasoro constraints and the Chern classes of the Hodge bundle

q

Disk enumeration on the quintic 3-fold

is the Euler characteristic parameter of Donaldson–Thomas theory. The conjecture is proven for local Calabi–Yau toric surfaces.

Hodge Integrals and Degenerate Contributions

For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category.Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.

Curves on K3 surfaces and modular forms

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evalu- ation of basic integrals in the local Gromov-Witten theory of P 1 . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a compete and effective solution. The local Gromov-Witten theory of curves is equivalent to the lo- cal Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of C 2 , and the orbifold quantum cohomol- ogy the symmetric product of C 2 . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.

The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere

The moduli space of stable relative maps to the projective line combines features of stable maps and admissible covers. We prove all standard Gromov-Witten classes on these moduli spaces of stable relative maps have tautological push-forwards to the moduli space of curves. In particular, the fundamental classes of all moduli spaces of admissible covers push-forward to tautological classes. Consequences for the tautological rings of the moduli spaces of curves include methods for generating new relations, uniform derivations of the socle and vanishing statements of the Gorenstein conjectures for the complete, compact type, and rational tail cases, tautological boundary terms for Ionels, Looijengas, and Getzlers vanishings, and applications to Gromov-Witten theory.