Estimate for Picard iterations of a Hermitian matrix operator

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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The planar limit of $$ \mathcal{N} $$ = 2 chiral correlators

Journal of High Energy Physics ◽

10.1007/jhep08(2021)032 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Bartomeu Fiol ◽

Alan Rios Fukelman

Keyword(s):

Free Energy ◽

Infinite Number ◽

Correlation Functions ◽

Hermitian Matrix ◽

Hooft Coupling ◽

Planar Limit ◽

Single Trace ◽

Hermitian Matrix Model ◽

Combinatorial Expression ◽

Do So

Abstract We derive the planar limit of 2- and 3-point functions of single-trace chiral primary operators of $$ \mathcal{N} $$ N = 2 SQCD on S4, to all orders in the ’t Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts. We then use these results to evaluate the analogous planar correlation functions on ℝ4. Specifically, we compute all the terms with a single value of the ζ function for a few planar 2- and 3-point functions, and conjecture general formulas for these terms for all 2- and 3-point functions on ℝ4.

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A Schur-Cohn theorem for matrix polynomials

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500004806 ◽

1990 ◽

Vol 33 (3) ◽

pp. 337-366 ◽

Cited By ~ 6

Author(s):

Harry Dym ◽

Nicholas Young

Keyword(s):

Unit Circle ◽

Matrix Polynomial ◽

Hermitian Matrix ◽

Krein Space ◽

Gram Matrix ◽

Natural Basis ◽

Test Matrix ◽

Matrix Polynomials ◽

Rational Vector ◽

Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

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