DUALITIES IN QUANTUM FIELD THEORIES WITH TOPOLOGICAL AND TOPOLOGICAL-HOLOMORPHIC TWISTS

Abstract

This dissertation consists of three different projects involving twists of supersymmetric quantum field theories in various dimensions.

In the first part, we revisit the low-energy effective U(1) action of topologically twisted N=2 SYM theory with gauge group of rank one on a smooth four-manifold with nontrivial fundamental group. We compute the path integral for the product ruled surfaces for the reduction on either factor and compare the results with existing literature, and make numerical predictions of Gromov-Witten invariants in terms of modular data.

In the second part, we revisit Vafa-Witten theory and physically derive a Vafa-Witten invariant, their relation to Gromov-Witten invariants, Vafa-Witten Floer homology, Vafa-Witten Atiyah-Floer correspondence, a proof and generalization of a conjecture by Abouzaid-Manolescu, and a Langlands duality of these invariants. We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification.

In the final part, we perform a topological-holomorphic twist of the above theories, and unravel their mathematical implications. In particular, we consider different combinations of scalar supercharges which in turn allow us to derive a physical realization of the theory of chiral differentials operators.