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<p>Recently the algebraic structure of gauge-invariant operators in multi-matrix quantum mechanics has been clarified: this space forms a module over a freely generated ring. The ring is generated by a set of primary invariants, while the module structure is determined by a finite set of secondary invariants. In this work, we show that the number of primary invariants can be computed by performing a complete gauge fixing, which identifies the number of independent physical degrees of freedom. We then compare this result to a complementary counting based on the restricted Schur polynomial basis. This comparison allows us to argue that the number of secondary invariants must exhibit exponential growth of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({e}^{c{N}^{2}}\)</EquationSource> </InlineEquation> at large <i>N</i>, with <i>c</i> a constant.</p>

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From symmetry to structure: gauge-invariant operators in multi-matrix quantum mechanics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月5日

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  • Robert de Mello Koch,
  • Minkyoo Kim,
  • Hendrik J. R. Van Zyl

摘要

Recently the algebraic structure of gauge-invariant operators in multi-matrix quantum mechanics has been clarified: this space forms a module over a freely generated ring. The ring is generated by a set of primary invariants, while the module structure is determined by a finite set of secondary invariants. In this work, we show that the number of primary invariants can be computed by performing a complete gauge fixing, which identifies the number of independent physical degrees of freedom. We then compare this result to a complementary counting based on the restricted Schur polynomial basis. This comparison allows us to argue that the number of secondary invariants must exhibit exponential growth of the form \({e}^{c{N}^{2}}\) at large N, with c a constant.