Quantum Fields and Strings

I will talk about 4d N=2 gauge theories with a co-dimension-two full surface operator, which exhibit a fascinating interplay of supersymmetric gauge theories, equivariant Gromov-Witten theory and geometric representation theory. For pure Yang-Mills and N=2* theory, a full surface operator can be described as the 4d gauge theory coupled to a 2d N=(2,2) gauge theory. By supersymmetric localizations, we present the exact partition functions of both 4d and 2d theories which satisfy integrable equations. In addition, I will show the validity of the orbifold method in one-loop computations when a full surface operator is inserted, and the form of the structure constants with a semi-degenerate field in SL(N,R) WZNW model is predicted from one-loop determinants.

We exactly evaluate the partition function (index) of N=4 supersymmetric quiver quantum mechanics in the Higgs phase by using the localization techniques. We show that the path integral is localized at the fixed points, which are obtained by solving the BRST equations, and D-term and F-term conditions. We turn on background gauge fields of R-symmetries for the chiral multiplets corresponding to the arrows between quiver nodes, but the partition function does not depend on these R-charges. We give explicit examples of the quiver theory including a non-coprime dimension vector. The partition functions completely agree with the mathematical formulae of the Poincare polynomials and the wall crossing for the quiver moduli spaces. We also discuss exact computation of the expectation values of supersymmetric (Q-closed) Wilson loops in the quiver theory.

 I will discuss various aspects of non-relativistic field theories on a curved, background spacetime. First things first, we need to know what sort of geometry these theories couple to, as well as the symmetries we ought to impose. I will argue that Galilean-invariant theories should be coupled to a form of Newton-Cartan geometry in which one enforces a one-form shift symmetry, which amounts to a covariant version of invariance under Galilean boosts. I will focus on two main applications of this result, namely consequences of these symmetries at nonzero temperature and the relation to warped CFTs.

We discuss a topological description of the confining phase of (Super-)Yang-Mills theories with gauge group SU(N) which encodes all the Aharonov-Bohm phases of configurations of non-local operators. This topological action shows an additional 1-form gauge symmetry. After the introduction of domain walls, this 1-form gauge symmetry demands the appearance of new fields on the worldvolume of the wall. These new fields have a topological Chern-Simons action at level N, also suggested by string theory constructions.