Brian Willett

3d dualities from 4d dualities

A bstractMany examples of low-energy dualities have been found in supersymmetric gauge theories with four supercharges, both in four and in three space-time dimensions. In these dualities, two theories that are different at high energies have the same low-energy limit. In this paper we clarify the relation between the dualities in four and in three dimensions. We show that every four dimensional duality gives rise to a three dimensional duality between theories that are similar, but not identical, to the dimensional reductions of the four dimensional dual gauge theories to three dimensions. From these specific three dimensional dualities one can flow to many other low-energy dualities, including known three dimensional dualities and many new ones. We discuss in detail the case of three dimensional SU(Nc) supersymmetric QCD theories, showing how to derive new duals for these theories from the four dimensional duality.

Generalized Global Symmetries

A bstractA q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. Wilson lines, surface defects, etc., and the charged excitations have q spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries (q = 0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a sub-group). They can also have ’t Hooft anomalies, which prevent us from gauging them, but lead to ’t Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.

3d dualities from 4d dualities for orthogonal groups

A bstractWe extend recent work on the relation of 4d and 3d IR dualities of supersymmetric gauge theories with four supercharges to the case of orthogonal gauge groups. The distinction between different SO(N) gauge theories in 4d plays an important role in this relation. We show that the 4d duality leads to a 3d duality between an SO(Nc) gauge theory with Nf flavors and an SO(Nf − Nc + 2) theory with Nf flavors and extra singlets, and we derive its generalization in the presence of Chern-Simons terms. There are two different O(N) theories in 3d, which we denote by O(N)±, and we also show that the O(Nc)− gauge theory is dual to a Spin(Nf − Nc + 2) theory, and derive from 4d the known duality between O(Nc)+ and O(Nf − Nc + 2)+. We verify the consistency of these 3d dualities by various methods, including index computations.

Global Properties of Supersymmetric Theories and the Lens Space

We compute the supersymmetric partition function on

Down the rabbit hole with theories of class \( \mathcal{S} \)

{L(r,1)\times \mathbb{S}^1}

A "Lagrangian" for a non-Lagrangian theory

L(r,1)×S1 , the lens space index, for 4d gauge theories related by supersymmetric dualities and involving non simply-connected groups. This computation is sensitive to the global properties of the underlying gauge group and to discrete theta angle parameters and thus distinguishes versions of dualities differing by such. We explicitly discuss

The long flow to freedom

{\mathcal{N}=1}