Finite-dimensional algebras, gauge-string duality and thermodynamics

TL;DR

This paper explores how finite-dimensional associative algebras can organize gauge-string duality structures, enabling efficient computations and revealing thermodynamic behaviors such as negative specific heat in certain quantum models.

Contribution

It introduces algebraic counting methods for matrix and tensor systems with symmetries, leading to new insights into thermodynamics and stability in gauge-string duality models.

Findings

Efficient algorithms for orthogonal basis construction in multi-matrix systems

Identification of negative specific heat branch in gauged quantum models

Growth of degeneracies linked to energy and size parameters

Abstract

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 $N$.