Permutation Centralizer Algebras and Multi-Matrix Invariants

Abstract

We introduce a class of permutation centralizer algebras which underly the combinatorics of multi-matrix gauge invariant observables. One family of such non-commutative algebras is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of 2-matrix models. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The centre of the algebra allows efficient computation of a sector of multi-matrix correlators. These generate the counting of a certain class of bi-coloured ribbon graphs with arbitrary genus.

45 pages, 5 figures

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• BPS operators in $\mathcal{N}=4$ $SO(N)$ super Yang-Mills theory: plethysms, dominoes and words
• Quarter-BPS states, multi-symmetric functions and set partitions
• Coherent state operators, giant gravitons, and gauge-gravity correspondence
• The quantum detection of projectors in finite-dimensional algebras and holography
• All-orders asymptotics of tensor model observables from symmetries of restricted partitions
• Quantum Information Processing and Composite Quantum Fields
• Hidden symmetries and Large N factorisation for permutation invariant matrix observables
• Gauge Invariants and Correlators in Flavoured Quiver Gauge Theories
• Noncommutative Frobenius algebras and open-closed duality
• Three-point Functions in $\mathcal{N}=4$ SYM at Finite $N_c$ and Background Independence
• Matrix and tensor witnesses of hidden symmetry algebras
• Kronecker coefficients from algebras of bi-partite ribbon graphs
• Rotating Restricted Schur Polynomials
• Backgrounds from Tensor Models: A Proposal
• $\mathcal{N}=4$ SYM, (super)-polynomial rings and emergent quantum mechanical symmetries