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Free particles from Brauer algebras in complex matrix models

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Abstract

The gauge invariant degrees of freedom of matrix models based on an N × N complex matrix, with U(N) gauge symmetry, contain hidden free particle structures. These are exhibited using triangular matrix variables via the Schur decomposition. The Brauer algebra basis for complex matrix models developed earlier is useful in projecting to a sector which matches the state counting of N free fermions on a circle. The Brauer algebra projection is characterized by the vanishing of a scale invariant laplacian constructed from the complex matrix. The special case of N = 2 is studied in detail: the ring of gauge invariant functions as well as a ring of scale and gauge invariant differential operators are characterized completely. The orthonormal basis of wavefunctions in this special case is completely characterized by a set of five commuting Hamiltonians, which display free particle structures. Applications to the reduced matrix quantum mechanics coming from radial quantization in \( \mathcal{N} = 4 \) SYM are described. We propose that the string dual of the complex matrix harmonic oscillator quantum mechanics has an interpretation in terms of strings and branes in 2 + 1 dimensions.

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  1. Queen Mary University of London, Centre for Research in String Theory, Department of Physics, Mile End Road, London, E1 4NS, U.K.

    Yusuke Kimura, Sanjaye Ramgoolam & David Turton

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  1. Yusuke Kimura
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  2. Sanjaye Ramgoolam
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ArXiv ePrint: 0911.4408

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Kimura, Y., Ramgoolam, S. & Turton, D. Free particles from Brauer algebras in complex matrix models. J. High Energ. Phys. 2010, 52 (2010). https://doi.org/10.1007/JHEP05(2010)052

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