Roman Mauch: Topics in Cohomological Field Theory: Let's Twist Again!

Abstract

This thesis explores different aspects of cohomologically twisted supersymmetric field theories. Compared to their untwisted cousins, these theories are characterised by the existence of a BRST-like fermionic symmetry, and observables are considered elements in its cohomology. The partition and correlation functions of such theories can be computed exactly by applying supersymmetric localisation. Employing this formalism, in the first part we compute the partition function of 4d N=2 pure gauge theory on a large class of manifolds with non-trivial two-cycles, accounting for flux contributions. The main idea is to obtain this theory from the 5d N=1 pure gauge theory as a limit of finite quotients along a free circle. We repeat this analysis, mutatis mutandis, for a case where the circle action is only locally free, starting from a resolved branched five-sphere. The result suggests a close relation with a corresponding theory on weighted projective space. In the second part, we study an equivariant deformation of the B model on the sphere. As a main feature, this model allows for new, position-dependent observables that have no counterpart in the standard B model. Two such observables—position-dependent superpotentials and complex structure deformations—are studied in detail.