Aspects of N_{T}\geq 2 Topological Gauge Theories and D-Branes

TL;DR

This paper explores various aspects of topological gauge theories with N_{T} or higher symmetry, including their constructions, relations to moduli space invariants, and connections to D-branes and string theory compactifications.

Contribution

It provides new insights into the construction, classification, and physical interpretation of topological gauge theories with extended supersymmetry, linking them to D-branes and string theory.

Findings

Equivalence of Vafa-Witten and Dijkgraaf-Moore constructions to superfield approach.

Relation between topological theories and Euler number sums of moduli spaces.

Classification of topological twists of N=8 d=3 Yang-Mills and their string theory realizations.

Abstract

We comment on various aspects of topological gauge theories possessing N_{T}\geq 2 topological symmetry: (1) We show that the construction of Vafa-Witten and Dijkgraaf-Moore of `balanced' topological field theories is equivalent to an earlier construction in terms of N_{T}=2 superfields inspired by Susy QM. (2) We explain the relation between topological field theories calculating signed and unsigned sums of Euler numbers of moduli spaces. (3) We show that the topological twist of N=4 d=4 Yang-Mills theory recently constructed by Marcus is formally a deformation of four-dimensional super-BF theory. (4) We construct a novel N_{T}=2 topological twist of N=4 d=3 Yang-Mills theory, a `mirror' of the Casson invariant model, with some unusual features. (5) We give a complete classification of the topological twists of N=8 d=3 Yang-Mills theory and show that they are realised as world-volume theories of Dirichlet two-brane instantons wrapping supersymmetric three-cycles of Calabi-Yau three-folds and G_{2}-holonomy Joyce manifolds. (6) We describe the topological gauge theories associated to D-string instantons on holomorphic curves in K3s and Calabi-Yau 3-folds.