Deep Learning and Non-Invertible Symmetries in Gauge Theories

This thesis investigates gauge theories by combining theoretical analysis of non-invertible symmetries with computational simulations using neural network quantum states. We first investigate (3+1)d ZN gauge theories with non-invertible duality symmetries that arise from gauging one-form symmetries. These symmetries impose strict constraints on renormalization group flows, often prohibiting symmetry-preserving gapped phases. Our proof demonstrates that a self-dual theory with ZN one-form symmetry must either remain gapless or break self-duality, except when N = k²ℓ with –1 as a quadratic residue modulo ℓ. Lattice gauge theories provide concrete realizations of these constraints. We then focus on the development of neural network quantum states for studying lattice gauge theories. By implementing gauge-invariant neural networks, we compute ground-state wavefunctions for ZN lattice gauge theories in (2+1)d. For Z2 gauge theory, we characterize the continuous confinement transition and identify critical exponents consistent with the Ising universality class. In the Z3 case we observe a weakly first-order transition between the confined and deconfined phases. For both theories, we compute order and disorder parameters and verify the linear confining potential in the confining phase. These results demonstrate that neural network methods provide a powerful tool for studying quantum field theories computationally. By combining these simulations with symmetry-based analysis, this work uncovers new insights into the nonperturbative dynamics and phase structure of gauge theories.