Aspects of superconformal and topological quantum field theories

Abstract

We discuss four new problems in the subjects of superconformal field theories (SCFTs) and topological quantum field theories (TQFTs). In superconformal field theories, our focus is on N = 2 theories in four dimensions, where in the first two problems, we further narrow down to the case of “theories of class S”. First, we show that the previously known invariants used to classify theories of class S fail to distinguish many pairs of SCFTs in the ([doublestruck Z]₂-twisted and untwisted) D-sector. We propose a new invariant, the global form of the flavor symmetry group, and show that it successfully distinguishes these pairs of theories. Next, we study the classification of SCFTs in the D₄ sector of class S with nonabelian outer-automorphism twists around various cycles of the surface. We propose an extension of previous formulae for the superconformal index to cover this case and classify the SCFTs corresponding to fixtures (3-punctured spheres). We then go on to study families of SCFTs corresponding to once-punctured tori and 4-punctured spheres. We show that these families of SCFTs exhibit new behaviours, not seen in previous investigations. In particular, the generic theory with 4 punctures on the sphere from non-commuting [doublestruck Z]₂ twisted sectors has six distinct weakly-coupled descriptions. In our third problem, we shift our focus to arbitrary N = 2 theories in 4d (i.e. not necessarily of class S). We show that if a 4d N = 2 is equipped with an N = (2, 2) supersymmetric surface defect, a marginal perturbation of the bulk theory induces a complex structure deformation of the defect moduli space. We describe a concrete way of computing this deformation using the bulk-defect OPE. For the fourth problem, we turn to the subject of topological quantum field theories. Here we study generalized discrete symmetries of two-dimensional semisimple TQFTs. We show that, in a special basis where the fusion rules of the TQFT are diagonalized, the 0-form symmetries act as permutations while 1-form symmetries act by phases. This leads to an explicit description of the gauging of these symmetries. We use these results to study the equivariant Verlinde formula for general simple Lie groups. These formulae leads to many predictions for the geometry of Hitchin moduli spaces, which we explicitly check in special cases with low genus and SO(3) gauge group.