Triangle Seminars

Abstract:
This talk will describe some work on the bouncing of particle-like (“kink”) solutions to a nonlinear wave equation, called the sine-Gordon equation, against a fixed boundary. Away from the boundary, this equation has a property known as integrability, making the scattering of the kinks particularly simple. However, if this integrability is broken at the boundary, then the scattering becomes surprisingly complicated, in ways that will be outlined in the talk with the help of some movies.

Abstract:
In this talk I will present recent results on the Bethe/Gauge correspondence obtained together with Mathew Bullimore and Hee-Cheol Kim. I will describe new ingredients of the Bethe/Gauge dictionary between the XXX Heisenberg spin chain and 2d N = (2,2) supersymmetric gauge theories. In particular, I will show how to construct off-shell Bethe states as orbifold defects in the A-twisted supersymmetric gauge theory and study their correlation functions. It will allow us to include aspects of algebraic Bethe ansatz in the correspondence. In particular, I will show how to interpret spin chain R-matrices as correlation functions of Janus interfaces for mass parameters.

Abstract:
In this talk, I will discuss the generalised and basic fuzzy 4-sphere in the context of the IKKT matrix model. These spaces arise as SO(5)-equivariant projections of quantised SO(6) coadjoint orbits and exhibit full SO(5) covariance. I will sketch how (basic and generalised) 4-sphere arise as solutions in a Yang-Mills matrix model, such that the fluctuations on the 4-sphere lead to a higher-spin gauge theory.

Abstract:
These lectures will be focused on aspects of combinatorics relevant to gauge-string duality (holography).

The physical theories we will discuss include two dimensional Yang Mills
theory, N=4 super Yang Mills
theory with U(N) gauge group, Matrix and tensor models.

The key mathematical concepts include : Schur Weyl-duality,
permutation equivalence classes and associated discrete Fourier
transforms as an approach to counting problems and, branched covers and
Hurwitz spaces. Schur-Weyl duality is a powerful relation between
representations of U(N) and representations of symmetric groups.
Representation theory of symmetric groups offers a method to
define nice bases for functions on equivalence classes of permutations.

These bases are useful in counting gauge invariant functions of
matrices or tensors, as well as computing their correlators in physical
theories. In AdS/CFT these bases have proved useful in identifying
local operators in gauge-theory dual to giant gravitons in AdS.
In the simplest cases of gauge-string duality, the known mathematics
of branched covers and Hurwitz spaces provide the mechanism for the
holographic correspondence between gauge invariants and stringy geometry.


(Lecture 1: Two dimensional Yang Mills theory. Exact solution. Large N expansion. Role of Schur-Weyl duality - relation between representation theory of symmetric groups and unitary groups. Hurwitz spaces and string interpretation of the large N expansion.)

Abstract:
Discussion of "A spacetime derivation of the Lorentzian OPE inversion formula” by Simmons-Duffin, Stanford and Witten. [1711.03816]