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Finite-dimensional algebras, gauge-string duality and thermodynamics

Sanjaye Ramgoolam

TL;DR

The work develops finite-dimensional permutation algebras to organize gauge-invariant polynomials in matrix and tensor variables, linking representation theory to finite-$N$ structure and holographic thermodynamics. It introduces representation-theoretic bases and efficient eigenvalue algorithms for multi-matrix and tensor invariants via algebras such as $ ext{A}(m,n)$ and $K(n)$, exploiting Schur-Weyl duality and central elements. Counting formulas in these algebras imply a microcanonical negative specific heat branch at low energies, with a crossover to positive specific heat at higher energies, and similar phenomena persist under $S_N$ gauge symmetry using partition algebras. Collectively, the framework connects combinatorics, representation theory, and thermodynamics to illuminate finite-$N$ effects and emergent gravitational physics in gauge/string duality, while providing practical computational tools for constructing orthogonal operator bases.

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 $N$.

Finite-dimensional algebras, gauge-string duality and thermodynamics

TL;DR

The work develops finite-dimensional permutation algebras to organize gauge-invariant polynomials in matrix and tensor variables, linking representation theory to finite- structure and holographic thermodynamics. It introduces representation-theoretic bases and efficient eigenvalue algorithms for multi-matrix and tensor invariants via algebras such as and , exploiting Schur-Weyl duality and central elements. Counting formulas in these algebras imply a microcanonical negative specific heat branch at low energies, with a crossover to positive specific heat at higher energies, and similar phenomena persist under gauge symmetry using partition algebras. Collectively, the framework connects combinatorics, representation theory, and thermodynamics to illuminate finite- effects and emergent gravitational physics in gauge/string duality, while providing practical computational tools for constructing orthogonal operator bases.

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 as well as 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 . The negative branch is associated with near-exponential or factorial growth of degeneracies for in a region of large stability, while the positive branch occurs when the finite reduction of degrees of freedom takes over as becomes sufficiently large compared to .
Paper Structure (14 sections, 47 equations, 3 figures)

This paper contains 14 sections, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Permutation algebras for matrix invariants
  3. Index contractions and permutation equivalence
  4. Representation-theoretic bases
  5. 2-matrix case : contraction permutations, equivalences and algebras
  6. Rep theory bases : 2-matrix case
  7. Algorithms for orthogonal bases
  8. The eigenvalue equations in $\mathcal{A} ( m ,n )$
  9. Efficiency of algorithms and minimal generating sets for $\mathcal{Z} ( \mathbb{C} ( S_n) )$.
  10. Representation bases, counting and thermodynamics
  11. Micro-canonical negative specific heat capacity in a sector of the 2-matrix system
  12. Finite algebras and thermodynamics for tensor invariants
  13. Partition functions and phase structure in matrix and tensor models with $S_N$ gauge symmetry
  14. Summary

Figures (3)

  • Figure 1: Micro-canonical energy versus temperature for zero charge complex matrix system $N = 13$ with $k$ equals $3$ to $18$ using the symmetric $D_{ {\rm sym}}$ discrete derivative.
  • Figure 2: Micro-canonical energy versus temperature for $3$-index tensor $N = 4$ with $k$ equals $3$ to $12$ using the symmetric $D_{ {\rm sym}}$ discrete derivative.
  • Figure 3: Plot of micro-canonical energy $E={k \over N^2 }$ versus micro-canonical temperature at $N =15$ for $k_{ min } = 4 , k_{ max} = 100$ - using descending derivative -- produces consistent negative SHC trend below critical E