Finite-dimensional algebras, gauge-string duality and thermodynamics

Gauge-invariant polynomial functions of matrix and tensor variables capture combinatorial structures of gauge-string duality, which can be usefully organised using finite-dimensional associative algebras. I review recent work on eigenvalue systems using these algebras as state spaces, which provide efficient computational algorithms for the construction of orthogonal bases in the multi-matrix case. Algebraic counting formulae in matrix and tensor systems with $U(N)$ as well as $S_N$ symmetry have led to gauged quantum mechanical models which display a negative branch of specific heat capacity in the micro-canonical ensemble followed by positive specific heat capacity at larger energies measured by a polynomial degree parameter $n$. The negative branch is associated with near-exponential or factorial growth of degeneracies for $n \gg 1$ in a region of large $N$ stability, while the positive branch occurs when the finite $N$ reduction of degrees of freedom takes over as $n$ becomes sufficiently large compared to $N$.