Aspects of Supersymmetric Gauge Theories and Conformal Field Theories in Five and Three Dimensions

• Main Author: Eckhard, Julius
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2020

Supersymmetric gauge theories and superconformal field theories have long played a central role in string theory. In this thesis, which is split into two parts, we explore multiple facets of this rich class of theories. In the first part we establish a convenient way to describe 5d SCFTs and gauge theories — generalised toric polygons (GTPs). Based on insights from (p, q) 5-brane-webs we derive a dictionary between a 5d theory TP and its associated GTP P. We provide an algorithm to describe the Higgs branch of TP by computing its magnetic quiver. Furthermore, we identify the Hasse diagram and thereby the entire foliation structure of the Higgs branch. We apply this technique to a large class of 5d SCFTs with a weakly coupled SU(N) gauge theory description. In this way we both recover known results and derive a series of new magnetic quivers. In the second part we turn to the 3d-3d correspondence, that relates observables of supersymmetric 3d gauge theories T[M3] to partition functions of topological field theories on M3. We study the sensitivity of this setup to the gauging of higher-form symmetries. We find a refinement of the 3d-3d correspondence for the Witten index, associated to the global structure of the gauge group of T[M3]. For M3 a Seifert manifold and gauge algebra g = su(2) we verify this explicitly by counting the solutions to the resulting Bethe equations. We complement this analysis by a refined counting of the flat connections on M3, accounting for the higher-form symmetries. Finally, we investigate the so far unknown 3d-3d correspondences for theories with N = 1 supersymmetry. We study the theories TN =1[M3] and analyse their connection to the geometry of G2-manifolds. On the complementary side of the 3d-3d correspondence we determine the topological field theories whose partition functions compute the Witten index and the S 3 -partition function of TN =1[M3]. In the process we point out the relevance of a new generalisation of 3d Seiberg-Witten equations.