Matrix models and phase transitions in gauge theories

Aspects of gauge theories in two, three and five dimensions are investigated using matrix models. Specifically, we consider pure Yang–Mills theory and its deformations in two dimensions, and supersymmetric Yang–Mills and Chern–Simons-matter theories in three and five dimensions. The random matrix approach allows us to explore a vast range of features of the gauge theories, including phase transitions, one-form symmetries and integrability. Partition functions and Wilson loops are studied in these setups by exploiting their matrix model presentation derived by localization. Two main lines of research are pursued: the computation of exact results at fixed and the quest for quantum phase transitions at large . The partition functions of several three-dimensional quiver Chern–Simons-matter theories are computed exactly using Mordell integrals, and we put forward a character expansion in terms of Schur polynomials, with coefficients given by topological invariants. A correspondence between two matrix models is provided as well, one computing topological invariants in pure Chern–Simons theory and the other arising from a two-dimensional, noncommutative scalar field theory. The correspondence is extended to supermatrix models, with ABJ(M) theory replacing topological Chern–Simons theory in this case. Partition functions and Wilson loop expectation values in three-dimensional = 4 gauge theories are also computed, uncovering a relation with Calogero–Moser integrable systems. Furthermore, we apply localization to five-dimensional supersymmetric Yang–Mills theory on compact product manifolds 3 × Σ, where Σ is a closed oriented Riemann surface, and introduce in this way a novel, “squashed” deformation of q-deformed Yang–Mills theory on Σ. Proceeding in the study of deformations of two-dimensional Yang–Mills theory, we analyze their perturbation by the operator ⊤₸ and prove that Abelianization still holds, although other characteristic properties such as factorization of the partition function break down. The analysis of large quantum phase transitions in matrix models and gauge theories constitutes the core of the thesis. We present a systematic study and classification of phase transitions for supersymmetric gauge theories on three- and five-dimensional spheres of large radius. The transitions are always third order for gauge theories connected to a known superconformal point, but are second order for generic five-dimensional () theories. Several multi-parameter families of unitary matrix models are also considered and their phase diagrams are established. Finally, we show how the Douglas–Kazakov transition of two-dimensional Yang–Mills on the sphere extends to its newly derived deformations. When both ⊤₸ and q-deformations are turned on, the two effects compete, and the system has two phases in the most part of the parameter space, but the weak coupling phase is removed in the regime of strong ⊤₸ -deformation, whereas the strong coupling phase is removed in the strong q-deformation regime