A natural question about Quantum Field Theory is whether there is a deformation to a trivial gapped phase. If the underlying theory has an anomaly, then symmetric deformations can never lead to a trivial phase. We discuss such discrete anomalies in Abelian Higgs models in 1+1 and 2+1 dimensions. We emphasize the role of charge conjugation symmetry in these anomalies; for example, we obtain nontrivial constraints on the degrees of freedom that live on a domain wall in the VBS phase of the Abelian Higgs model in 2+1 dimensions. In addition, as a byproduct of our analysis, we show that in 1+1 dimensions the Abelian Higgs model is dual to the Ising model. We also study variations of the Abelian Higgs model in 1+1 and 2+1 dimensions where there is no dynamical particle of unit charge. These models have a center symmetry and additional discrete anomalies. In the absence of a dynamical unit charge particle, the Ising transition in the 1+1 dimensional Abelian Higgs model is removed. These models without a unit charge particle exhibit a remarkably persistent order: we prove that the system cannot be disordered by either quantum or thermal fluctuations. Equivalently, when these theories are studied on a circle, no matter how small or large the circle is, the ground state is non-trivial.

We study the dynamics of 2+1 dimensional theories with N = 1 supersymmetry. In these theories the supersymmetric ground states behave discontinuously at codimension one walls in the space of couplings, with new vacua coming in from infinity in field space. We show that the dynamics near these walls is calculable: the two-loop effective potential yields exact results about the ground states near the walls. Far away from the walls the ground states can be inferred by decoupling arguments. In this way, we are able to follow the ground states of N = 1 theories in 2+1 dimensions and construct the infrared phases of these theories. We study two examples in detail: Adjoint SQCD and SQCD with one fundamental quark. In Adjoint QCD we show that for sufficiently small Chern-Simons level the theory has a non-perturbative metastable supersymmetry-breaking ground state. We also briefly discuss the critical points of this theory. For SQCD with one quark we establish an infrared duality between a U(N) gauge theory and an SU(N) gauge theory. The duality crucially involves the vacua that appear from infinity near the walls.

We study gauge theories with N = 1 supersymmetry in 2+1 dimensions. We start by calculating the 1-loop effective superpotential for matter in an arbitrary representation. We then restrict ourselves to gauge theories with fundamental matter. Using the 1-loop superpotential, we find a universal form for the phase diagrams of many such gauge theories, which is proven to persist to all orders in perturbation theory using a symmetry argument. This allows us to conjecture new dualities for N = 1 gauge theories with fundamental matter. We also show that these dualities are related to results in N = 2 supersymmetric gauge theories, which provides further evidence for them.

We consider the sphere free energy F(b; mI) in $$ \mathcal{N} $$ N = 6 ABJ(M) theory deformed by both three real masses mI and the squashing parameter b, which has been computed in terms of an N dimensional matrix model integral using supersymmetric localization. We show that setting $$ {m}_3=i\frac{b-{b}^{-1}}{2} $$ m 3 = i b − b − 1 2 relates F(b; mI) to the round sphere free energy, which implies infinite relations between mI and b derivatives of F(b; mI) evaluated at mI = 0 and b = 1. For $$ \mathcal{N} $$ N = 8 ABJ(M) theory, these relations fix all fourth order and some fifth order derivatives in terms of derivatives of m1, m2, which were previously computed to all orders in 1/N using the Fermi gas method. This allows us to compute $$ {\partial}_b^4F\left|{}_{b=1}\right. $$ ∂ b 4 F b = 1 and $$ {\partial}_b^5F\left|{}_{b=1}\right. $$ ∂ b 5 F b = 1 to all orders in 1/N, which we precisely match to a recent prediction to sub-leading order in 1/N from the holographically dual AdS4 bulk theory.

The partition function of a 3d N = 4 gauge theory with rank N can be computed using supersymmetric localization in terms of a matrix model, which often can be formulated as an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show how OPE coefficients of protected operators correspond in this formalism to averages of n-body operators in the Fermi gas, which can be computed to all orders in 1/N using the WKB expansion. We use this formalism to compute OPE coefficients in the U(N) k × U(N) −k ABJM theory as well as the U(N) theory with one adjoint and N f fundamental hypermultiplets, both of which have weakly coupled M-theory duals in the large N and finite k or N f regimes. For ABJM we reproduce known results, while for the N f theory we compute the all orders in 1/N dependence at finite N f for the coefficient c T of the stress tensor two-point function.