One-loop multicollinear limits from 2-point amplitudes on self-dual backgrounds

For scattering amplitudes in strong background fields, it is — at least in principle — possible to perturbatively expand the background to obtain higher-point vacuum amplitudes. In the case of self-dual plane wave backgrounds we consider this expansion for two-point, one-loop amplitudes in pure Yang-Mills, QED and QCD. This enables us to obtain multicollinear limits of 1-loop vacuum amplitudes; the resulting helicity configurations are surprisingly restricted, with only the all-plus helicity amplitude surviving. These results are shown to be consistent with well-known vacuum amplitudes. We also show that for both abelian and non-abelian supersymmetric gauge theories, there is no helicity flip (and hence no vacuum birefringence) on any plane wave background, generalising a result previously known in the Euler-Heisenberg limit of super-QED.

Infinite-dimensional fermionic symmetry in supersymmetric gauge theories

We establish the existence of an infinite-dimensional fermionic symmetry in four-dimensional supersymmetric gauge theories by analyzing semiclassical photino dynamics in abelian $$ \mathcal{N} $$ N = 1 theories with charged matter. The symmetry is parametrized by a spinor-valued function on an asymptotic S2 at null infinity. It is not manifest at the level of the Lagrangian, but acts non-trivially on physical states, and its Ward identity is the soft photino theorem. The infinite-dimensional fermionic symmetry resides in the same $$ \mathcal{N} $$ N = 1 supermultiplet as the physically non-trivial large gauge symmetries associated with the soft photon theorem.

Phases of five-dimensional supersymmetric gauge theories

Five-dimensional $$ \mathcal{N} $$ N = 1 theories with gauge group U(N), SU(N), USp(2N) and SO(N) are studied at large rank through localization on a large sphere. The phase diagram of theories with fundamental hypermultiplets is universal and characterized by third order phase transitions, with the exception of U(N), that shows both second and third order transitions. The phase diagram of theories with adjoint or (anti-)symmetric hypermultiplets is also determined and found to be universal. Moreover, Wilson loops in fundamental and antisymmetric representations of any rank are analyzed in this limit. Quiver theories are discussed as well. All the results substantiate the ℱ-theorem.

Notes on two-dimensional pure supersymmetric gauge theories

In this note we study IR limits of pure two-dimensional supersymmetric gauge theories with semisimple non-simply-connected gauge groups including SU(k)/ℤk, SO(2k)/ℤ2, Sp(2k)/ℤ2, E6/ℤ3, and E7/ℤ2 for various discrete theta angles, both directly in the gauge theory and also in nonabelian mirrors, extending a classification begun in previous work. We find in each case that there are supersymmetric vacua for precisely one value of the discrete theta angle, and no supersymmetric vacua for other values, hence supersymmetry is broken in the IR for most discrete theta angles. Furthermore, for the one distinguished value of the discrete theta angle for which supersymmetry is unbroken, the theory has as many twisted chiral multiplet degrees of freedom in the IR as the rank. We take this opportunity to further develop the technology of nonabelian mirrors to discuss how the mirror to a G gauge theory differs from the mirror to a G/K gauge theory for K a subgroup of the center of G. In particular, the discrete theta angles in these cases are considerably more intricate than those of the pure gauge theories studied in previous papers, so we discuss the realization of these more complex discrete theta angles in the mirror construction. We find that discrete theta angles, both in the original gauge theory and their mirrors, are intimately related to the description of centers of universal covering groups as quotients of weight lattices by root sublattices. We perform numerous consistency checks, comparing results against basic group-theoretic relations as well as with decomposition, which describes how two-dimensional theories with one-form symmetries (such as pure gauge theories with nontrivial centers) decompose into disjoint unions, in this case of pure gauge theories with quotiented gauge groups and discrete theta angles.

Singular BPS boundary conditions in $$ \mathcal{N} $$ = (2, 2) supersymmetric gauge theories

We derive general BPS boundary conditions in two-dimensional $$ \mathcal{N} $$ N = (2, 2) supersymmetric gauge theories. We analyze the solutions of these boundary conditions, and in particular those that allow the bulk fields to have poles at the boundary. We also present the brane configurations for the half- and quarter-BPS boundary conditions of the $$ \mathcal{N} $$ N = (2, 2) supersymmetric gauge theories in terms of branes in Type IIA string theory. We find that both A-type and B-type brane configurations are lifted to M-theory as a system of M2-branes ending on an M5-brane wrapped on a product of a holomorphic curve in ℂ2 with a special Lagrangian 3-cycle in ℂ3.

Quantum spin systems and supersymmetric gauge theories. Part I

The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the N site $$ \mathfrak{sl} $$ sl 2X X X spin chain (for infinite dimensional complex spin representations), as well as the SLN Gaudin system, which reduces, in a limiting case, to that of the N-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture.

The Cat’s Cradle: deforming the higher rank E1 and $$ {\tilde{E}}_1 $$ theories

We use 5-brane webs to study the two-dimensional space of supersymmetric mass deformations of higher rank generalizations of the 5d E1 and $$ {\tilde{E}}_1 $$ E ˜ 1 theories. Some of the resulting IR phases are described by IR free supersymmetric gauge theories, while others correspond to interacting fixed points. The number of different phases appears to grow with the rank. The space of deformations is qualitatively different for the even and odd rank cases, but that of the even (odd) rank E1 theory is similar to that of the odd (even) rank $$ {\tilde{E}}_1 $$ E ˜ 1 theory. One result of our analysis predicts that the supersymmetric SU(N) theory with CS level k = $$ \frac{N}{2} $$ N 2 + 4 and a single massless antisymmetric hypermultiplet exhibits an enhanced global symmetry at the UV fixed point, given by SU(2) × SU(2) if N is even, and SU(2) × U(1) if N is odd.

Chiral rings, Futaki invariants, plethystics, and Gröbner bases

We study chiral rings of 4d $$ \mathcal{N} $$ N = 1 supersymmetric gauge theories via the notion of K-stability. We show that when using Hilbert series to perform the computations of Futaki invariants, it is not enough to only include the test symmetry information in the former’s denominator. We discuss a way to modify the numerator so that K-stability can be correctly determined, and a rescaling method is also applied to simplify the calculations involving test configurations. All of these are illustrated with a host of examples, by considering vacuum moduli spaces of various theories. Using Gröbner basis and plethystic techniques, many non-complete intersections can also be addressed, thus expanding the list of known theories in the literature.