Mathematical Physics

I will review the geometric approach to the description of Coulomb branches and Chern-Simons terms of gauge theories coming from compactifications of M-theory on elliptically fibered Calabi-Yau threefolds. Mathematically, this involves finding all the crepant resolutions of a given Weierstrass model and understanding the network of flops connecting them together with computing certain topological invariants. I will further check that the uplifted theory in 6d is anomaly-free using Green-Schwartz mechanism. I will give examples on the theory with semi-simple gauge groups such as SU(2)xG2, which plays a major role in the classification of 6d superconformal theories, and SU(2)xSU(3), which describes the non-abelian sector of the standard model.

The Heisenberg algebra plays an important role in many areas of mathematics and physics. Khovanov constructed a categorical analogue of this algebra which emphasizes its connections to representation theory and combinatorics. Recently, Brundan, Savage, and Webster have shown that the Grothendieck group of this category is isomorphic to the Heisenberg algebra. However, applying an alternative decategorification functor called the trace to the Heisenberg category yields a richer structure: a W-algebra, an infinite-dimensional Lie algebra related to conformal field theory. In this talk, we will describe recent work extending this latter result to a generalized version of the Heisenberg category associated to an arbitrary graded Frobenius algebra F. The decategorifications of this Frobenius Heisenberg category conjecturally provide connections to alternative Heisenberg algebras and W-algebras associated to the algebra F. This is joint work with Alistair Savage (University of Ottawa).

This seminar will be a continued discussion on the topic of last week's seminar: 

I will describe some general mathematical structures expected to arise from field theories with boundary conditions in terms of factorization algebras, and outline some results and future directions in the study of boundary chiral algebras for 3d N=4 theories following the work of Costello and Gaiotto.

Given by Dylan Butson.