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Topological Defects and Generalized Symmetries in Quantum Field Theory
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Topological Defects and Generalized Symmetries in Quantum Field Theory
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Dissertation
Topological Defects and Generalized Symmetries in Quantum Field Theory
2022
Overview
We explore topological defects and boundaries of quantum field theories in various dimensions. In particular, we study non-invertible topological defects that generate non-invertible symmetries. These generalized symmetries lead to powerful dynamical constraints that we explore in this dissertation.In the first part, we study two-dimensional Quantum Chromodynamics (QCD), which serves as a toy model for the four-dimensional QCD that describes the strong nuclear force in nature. In particular, we study two-dimensional QCD with a fermion in the adjoint representation of the gauge group. We discover an exponential number of non-invertible symmetries in this theory. Using these generalized symmetries, we solve the confinement problem for massless and massive adjoint fermion and compute the tension of confining strings analytically among the other results we obtain.Next, we study boundaries of topological quantum field theories (TQFTs) in 2+1-dimensions. TQFTs characterize the low energy behavior of gapped quantum systems with only topological degrees of freedom. Such theories, despite being gapped in the bulk, sometimes carry interesting gapless \"edge modes\" on their boundaries. We find new obstructions to having a gapped boundary, which necessitates gapless edge modes, in 2+1-dimensions known as higher central charges. We obtain these new invariants using non-invertible symmetries and their generalized gauging.Finally, we construct non-invertible symmetries in arbitrary spacetime dimensions known as condensation defects. The construction is by gauging a higher-form symmetry on a higher-codimension submanifold in spacetime, which we denote as higher gauging. We find the fusion rules of condensation surface defects and emphasize that the fusion \"coefficients\" in these non-invertible fusion rules are generally not numbers but rather 1+1d TQFTs.
