Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials

Tierz, Miguel

doi:10.1007/jhep04(2016)168

Cited by 9 publications

(22 citation statements)

References 54 publications

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“…They showed that the thermal partition function is given by

0true {\tilde{Z}}_{L \times N} = scriptQ \int \prod_{i < j} {sinh}^{2} ((), \frac{μ_{i} - μ_{j}}{2}) \prod_{i = 1}^{N} {cosh}^{L} ((), \frac{μ_{i}}{2}) e^{- L \overset{γ}{true} μ_{i}^{2}} \prod_{i = 1}^{N} d μ_{i},

where

\tilde{γ} > 0

is a positive parameter and the normalization constant

scriptQ

0true scriptQ = 2^{- L} \int \prod_{i < j} {sinh}^{2} ((), \frac{μ_{i} - μ_{j}}{2}) \prod_{i = 1}^{N} e^{- L \overset{γ}{true} μ_{i}^{2}} \prod_{i = 1}^{N} d μ_{i} .

It is interesting to point out that the sinh term in the above integrals also appears in the study of matrix models in Chern‐Simons‐matter theories; for example, see Refs. and .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

2018

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In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes‐Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q‐polynomials.

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“…They showed that the thermal partition function is given by

0true {\tilde{Z}}_{L \times N} = scriptQ \int \prod_{i < j} {sinh}^{2} ((), \frac{μ_{i} - μ_{j}}{2}) \prod_{i = 1}^{N} {cosh}^{L} ((), \frac{μ_{i}}{2}) e^{- L \overset{γ}{true} μ_{i}^{2}} \prod_{i = 1}^{N} d μ_{i},

where

\tilde{γ} > 0

is a positive parameter and the normalization constant

scriptQ

0true scriptQ = 2^{- L} \int \prod_{i < j} {sinh}^{2} ((), \frac{μ_{i} - μ_{j}}{2}) \prod_{i = 1}^{N} e^{- L \overset{γ}{true} μ_{i}^{2}} \prod_{i = 1}^{N} d μ_{i} .

It is interesting to point out that the sinh term in the above integrals also appears in the study of matrix models in Chern‐Simons‐matter theories; for example, see Refs. and .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

2018

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“…In particular, the tools used in [15] for Chern-Simons-matter (CSM) matrix models directly apply here, and they do so in a much simpler fashion, since the term cosh L (µ i /2) can be understood as a characteristic polynomial insertion in the matrix model (2.3), which is a Stieltjes-Wigert ensemble [14]. This insertion, in contrast to the ones in [15], is in the numerator and hence the solution is directly in terms of Stieltjes-Wigert polynomials, and not their Cauchy transform. As a matter of fact, after a simple rewriting, some of the relevant computations are contained in much earlier work [23], since to study averages of Schur polynomials in the ensemble (2.3), we computed first the averages of correlation functions of characteristic polynomials in the ensemble (2.3).…”

Section: The Modelmentioning

confidence: 99%

“…where S N (λ) denotes the monic Stieltjes-Wigert polynomial [14,23,15] of degree N and S (k) N (λ) its k-th derivative. The explicit expression for the orthonormal SW polynomials [23,15] is…”

Section: Exact Solutionmentioning

confidence: 99%

“…Interestingly, the fermionic matrix model in [5], which is to be presented and succinctly described below, admits also a description in terms of a (bosonic) matrix model, which, we show in this work, can be analyzed using standard random matrix theory tools. The matrix model turns out to be of the q-deformed type of matrix models which appears in Chern-Simons theory [14,15]. As we shall see, a consequence is that the partition function, in the simplest case, can be analytically characterized, for arbitrary N , in terms of a q-orthogonal polynomial, the Stieltjes-Wigert polynomial [14,15], which can be understood as the polynomial part of a q-oscillator wavefunction.…”

Section: Introductionmentioning

confidence: 96%

“…The matrix model turns out to be of the q-deformed type of matrix models which appears in Chern-Simons theory [14,15]. As we shall see, a consequence is that the partition function, in the simplest case, can be analytically characterized, for arbitrary N , in terms of a q-orthogonal polynomial, the Stieltjes-Wigert polynomial [14,15], which can be understood as the polynomial part of a q-oscillator wavefunction. These polynomials are also central in Chern-Simons theory but they are used, as we explain below, in a different way in the characterization of the fermionic matrix model of [5].…”

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

Polynomial solution of quantum Grassmann matrices

Tierz¹

2017

J. Stat. Mech.

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Abstract. We study a model of quantum mechanical fermions with matrix-like index structure (with indices N and L) and quartic interactions, recently introduced by Anninos and Silva. We compute the partition function exactly with q-deformed orthogonal polynomials (StieltjesWigert polynomials), for different values of L and arbitrary N . From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given L, the number of states of different energy is quadratic in N , which implies an exponential degeneracy of the energy levels. We also show that at high-temperature we have a Gaussian matrix model, which implies a symmetry that swaps N and L, together with a Wick rotation of the spectral parameter. In this limit, we also write the partition function, for generic L and N, in terms of a single generalized Hermite polynomial.

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Exact results and Schur expansions in quiver Chern-Simons-matter theories

Santilli

Tierz

2020

J. High Energ. Phys.

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We study several quiver Chern-Simons-matter theories on the three-sphere, combining the matrix model formulation with a systematic use of Mordell’s integral, computing partition functions and checking dualities. We also consider Wilson loops in ABJ(M) theories, distinguishing between typical (long) and atypical (short) representations and focusing on the former. Using the Berele-Regev factorization of supersymmetric Schur polynomials, we express the expectation value of the Wilson loops in terms of sums of observables of two factorized copies of U(N ) pure Chern-Simons theory on the sphere. Then, we use the Cauchy identity to study the partition functions of a number of quiver Chern-Simons-matter models and the result is interpreted as a perturbative expansion in the parameters tj = −e2πmj , where mj are the masses. Through the paper, we incorporate different generalizations, such as deformations by real masses and/or Fayet-Iliopoulos parameters, the consideration of a Romans mass in the gravity dual, and adjoint matter.

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Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials

Cited by 9 publications

References 54 publications

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

Polynomial solution of quantum Grassmann matrices

Exact results and Schur expansions in quiver Chern-Simons-matter theories

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