The local Gromov-Witten theory of curves

Abstract

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

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

“…Similar issues were already addressed in [33]. The chiral expansion of q-deformed Yang-Mills theory could in general lead to a better understanding of the relevant Calabi-Yau threefolds.…”

“…The symbol |R| is the total number of boxes of the Young tableau of the SU (N ) representation R. The chiral block Z qYM,+ R (1) ,R (2) (t; p) agrees exactly with the perturbative topological string amplitude on X p [33] with two stacks of D-branes inserted in the fiber. It depends explicitly on the choice of two arbitrary Young tableaux which correspond to the boundary degrees of freedom of the fiber D-branes.…”

“…Any continuous mapping from a Riemann surface to P 1 is, up to homotopy, the composition of a pinch map (collapsing regions of the surface to a single point) and a branched covering map. The pinch maps are responsible for the Ω-point singularities and are related to appearance of multiple Hurwitz numbers (ramified over more than one point than just ∞ on P 1 ) in the computation of the Gromov-Witten invariants of the original threefold X p [33]. Physically, they are directly related to the insertions of the fiber Dbranes in the topological string theory on X p .…”

“…In the large N limit, this theory factorizes into a chiral and antichiral part, like its undeformed cousin QCD 2 [30]- [32], corresponding to the holomorphic and antiholomorphic structure of |Z top | 2 . The topological string amplitude Z top on these geometries has been computed recently in [33].…”

Topological Strings on Local Curves

“…The Gromov-Witten theory for these spaces was developed by Bryan and Pandharipande [6] and can be reproduced, in the equivariant case, by the topological vertex of [1]. The total partition sum in the A model is a sum over partitions.…”