This series consists of talks in the area of 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.
Continued discussion to the two informal talks given by Dylan Butson on January 21st and 28th.
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.
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.
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.
Bernstein operators are vertex operators that create and annihilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category.
I will discuss in this talk joint work with Robert Laugwitz on a new mechanism for producing braided commutative algebras in braided monoidal categories. Namely, we construct braided commutative algebras in relative monoidal centers (in the sense of Laugwitz), which generalizes work of Davydov. Similar to how monoidal centers include representation categories of Drinfel'd doubles of Hopf algebras, an example of a relative monoidal center is a suitable category of modules over a quantized enveloping algebra.
A 3d N=4 gauge theory admits two topological twists, which we'll simply call A and B. The two twists are exchanged by 3d mirror symmetry. It is known that local operators in the A (resp. B) twist include the Coulomb-branch (resp. Higgs-branch) chiral rings. In this talk I will discuss the *line* operators preserved by the two twists, which in each case should have the structure of a braided tensor category.
The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory. I will explain how string theory unifies these phenomena. This is based on my work with Kevin Costello [arXiv:1810.01970].