Abstract
The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of SN symmetric group elements U acting as X → UXUT. Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with U(N) symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of N of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.
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Acknowledgments
DO’C is supported by the Irish Research Council and Science Foundation Ireland under grant SFI-IRC-21/PATH-S/9391. SR is supported by the Science and Technology Facilities Council (STFC) Consolidated Grants ST/P000754/1 “String theory, gauge theory and duality” and ST/T000686/1 “Amplitudes, strings and duality”. SR thanks the Dublin Institute for Advanced Studies for hospitality while part of this work was being done. We thank Yuhma Asano, George Barnes, David Berenstein, Veselin Filev, Robert de Mello Koch, Antal Jevicki, Adrian Padellaro for discussions related to the subject of this paper.
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O’Connor, D., Ramgoolam, S. Gauged permutation invariant matrix quantum mechanics: partition functions. J. High Energ. Phys. 2024, 152 (2024). https://doi.org/10.1007/JHEP07(2024)152
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DOI: https://doi.org/10.1007/JHEP07(2024)152