This series consists of talks in the area of Mathematical Physics.
Principal bundles and their moduli have been important in various aspects of physics and geometry for many decades. It is perhaps not so well-known that a substantial portion of the original motivation for studying them came from number theory, namely the study of Diophantine equations. I will describe a bit of this history and some recent developments.
I discuss a geometric interpretation of the twisted indexes of 3d (softly broken) $\cN=4$ gauge theories on $S^1 \times \Sigma$ where $\Sigma$ is a closed genus $g$ Riemann surface, mainly focussing on quivers with unitary gauge groups. The path integral localises to a moduli space of solutions to generalised vortex equations on $\Sigma$, which can be understood algebraically as quasi-maps to the Higgs branch. I demonstrate that the twisted indexes computed in previous work reproduce the virtual Euler characteristic of the moduli spaces of twisted quasi-maps.
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.