On Symmetries, Anomalies and Their Generalizations

Abstract

This dissertation explores various generalizations of global symmetries and 't Hooft anomalies. Chapter two is based on work with Po-Shen Hsin and Nathan Seiberg [1]. It is dedicated to the study of one-form global symmetries in three and four dimensions. We investigate their physical implications, classify their 't Hooft anomalies and analyze their gauging. Chapter three is based on the work with Pranay Gorantla, Nathan Seiberg and Shu-Heng Shao [2]. It focuses on exotic theories with subsystem symmetries including theories of fractons. We reformulate these theories on a Euclidean spacetime lattice in a modified Villain formulation. This provides a rigorous treatment of the continuum theories and their singularities while preserving some of their essential properties including 't Hooft anomalies and dualities. Chapter four is based on work with Clay Cordova, Dan Freed and Nathan Seiberg [3, 4]. It extends the notion of 't Hooft anomalies to anomalies in the space of coupling constants. We demonstrate through examples in diverse dimensions that these generalized anomalies can constrain the phase diagram of the theories and their defects associated with space-dependent coupling constants.