Reduced unitary matrix models and the hierarchy of τ-functions

Bowick, Mark J.; Морозов, А.; Shevitz, Daniel

doi:10.1016/0550-3213(91)90365-5

Cited by 38 publications

(45 citation statements)

References 13 publications

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“…При V(X) =X K+l /(K+ 1) модель Концевича описывает двойной скейлинговый непрерывный предел K матричной дискретной модели см. [184] о наиболее важном случае эрмитовых матриц M (а также [265] о модели с унитарными M, [266 -268] -о модификации модели для случая комплексных квадратных матриц M); рассматриваются также модели с ор тогональными, вещественными и даже векторными M (т.е. матрицами размера Эрмитова дискретная 1 матричная модель может быть также записана в форме МОДЕЛИ КОНЦЕВИЧА [186] при целых N могут быть найдены, если рассмотреть новую систему, состоящую из N копий (реплик) исходной.…”

Section: словарь терминовunclassified

String theory: what is it?

Морозов¹

1992

Успехи физических наук

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Section: словарь терминовunclassified

String theory: what is it?

Морозов¹

1992

Успехи физических наук

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“…Thus we obtain the final result 19) where in the last line the integration contour passes through the ith cut. If we go back to the w-plane (2.6), the contour runs from a point infinitely close to w = 0 to another to w = ∞.…”

Section: The Special Geometry Relationmentioning

confidence: 99%

“…Section 4 and 5 are devoted to the applications of the duality transformation. In Section 4 we use it in the unitary matrix formulation of the N = 2 pure SU(2) gauge theory and prove that it correctly reproduces 3 It was noted in the same paper ( [19], Appendix D) that the question of whether the partition function of the full unitary model was a τ function of any integrable hierarchy remained open. 4 In [26] there was also mentioned another possibility U = e iΦ [27], which was recently exploited by Tierz [28] in the study of the Chern-Simons matrix model.…”

mentioning

confidence: 99%

“…This transformation was used in [19] to derive the Virasoro constraints [25] in unitary 1-matrix models, although the fact that it implied an exact duality does not seem to have been fully recognized 3 . In [26] the possibility of using the transformation (1.1) was mentioned in the study of matrix models coupled to an external field, with a conclusion that it did not have "a particularly attractive expansion" for their purposes 4 .…”

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confidence: 99%

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On unitary/Hermitian duality in matrix models

Mizoguchi¹

2005

Nuclear Physics B

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Unitary 1-matrix models are shown to be exactly equivalent to hermitian 1-matrix models coupled to 2N vectors with appropriate potentials, to all orders in the 1/N expansion. This fact allows us to use all the techniques developed and results obtained in hermitian 1-matrix models to investigate unitary as well as other 1-matrix models with the Haar measure on the unitary group. We demonstrate the use of this duality in various examples, including:(1) an explicit confirmation that the unitary matrix formulation of the N = 2 pure SU (2) gauge theory correctly reproduces the genus-1 topological string amplitude (2) derivations of the special geometry relations in unitary as well as the Chern-Simons matrix models.

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“…With the exponent in (6.1) replaced by V = ∞ k=−∞ t k Tr U k , the studies [5,25] deduced the Schwinger-Dyson equations L ± n Z = 0, where L ± n are the Virasoro operators…”

Section: Jhep02(2015)173mentioning

confidence: 99%

Loop equation analysis of the circular β ensembles

Witte

Forrester

2015

J. High Energ. Phys.

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We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures Re β > 0 and number of eigenvalues N . Using matching arguments for the resolvent functions of linear statistics f (ζ) = (ζ + z)/(ζ − z) in a particular asymptotic regime, the global regime, we systematically develop the corresponding large N expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment m k to an equivalent length. The leading large N , large k, k/N fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low k moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of β = 1, 2, 4, general N and even, positive β, N = 2, 3.

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Reduced unitary matrix models and the hierarchy of τ-functions

Cited by 38 publications

References 13 publications

String theory: what is it?

String theory: what is it?

On unitary/Hermitian duality in matrix models

Loop equation analysis of the circular β ensembles

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