Holographic combinatorics : 2d Yang Mills theory to tensor models via AdS/CFT


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, four-dimensional 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.

Lecture 2: Local gauge invariant operators and Hilbert space of CFTs. Young diagrams and Brane geometries. Half-BPS and quarter-BPS. Counting, construction and correlators in group theoretic combinatorics.

Lecture 3: Hermitian matrix model. Tensor models and Permutation centralizer al- gebras. Using permutation equivalences to count matrix/tensor invariants and compute correlators. Relations to covering spaces.

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