This series consists of talks in the area of Mathematical Physics.
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].
Positive geometries are real semialgebraic spaces that are
equipped with a meromorphic ``canonical form" whose residues reflect
the boundary structure of the space. Familiar examples include
polytopes and the positive parts of toric varieties. A central, but
conjectural, example is the amplituhedron of Arkani-Hamed and Trnka.
In this case, the canonical form should essentially be the tree
amplitude of N=4 super Yang-Mills.
One can associate to any finite graph Q the skew-symmetic Kac-Moody Lie algebra g_Q. While this algebra is always infinite, unless Q is a Dynkin diagram of type ADE, g_Q shares a lot of the nice features of a semisimple Lie algebra. In particular, the cohomology of Nakajima quiver varieties associated to Q gives a geometric representations of g_Q. Encouraged by this story, one could hope to define the Yangian of g_Q, for general Q, as a subalgebra of the algebra of endomorphisms of cohomology of quiver varieties.
Work of Bezrukavnikov on local geometric Langlands correspondence and works of Gorsky, Neguţ, Rasmussen and Oblomkov, Rozansky on knot homology and matrix factorizations suggest that there should be a categorical version of a certain natural homomorphism from the affine Hecke algebra to the finite Hecke algebra in type A, sending basis lattice elements on the affine side to Jucys-Murphy elements on the finite side.