scholarly journals $${\mathcal {N}}=1$$ super topological recursion

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Vincent Bouchard ◽  
Kento Osuga
Keyword(s):  
Correlation Functions ◽  
Spectral Curve ◽  
Topological Recursion

AbstractWe introduce the notion of $${\mathcal {N}}=1$$ N = 1 abstract super loop equations and provide two equivalent ways of solving them. The first approach is a recursive formalism that can be thought of as a supersymmetric generalization of the Eynard–Orantin topological recursion, based on the geometry of a local super spectral curve. The second approach is based on the framework of super Airy structures. The resulting recursive formalism can be applied to compute correlation functions for a variety of examples related to 2d supergravity.

Random matrices and loop equations

2018 ◽  
Author(s):  
Bertrand Eynard
Keyword(s):  
Random Matrix ◽  
Matrix Theory ◽  
Spectral Curve ◽  
Coulomb Gas ◽  
Algebraic Equations ◽  
Saddle Point Approximation ◽  
Large N ◽  
Topological Recursion ◽  
Random Matrix Ensembles ◽  
Coulomb Gas Method

This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.

scholarly journals The sine-law gap probability, Painlevé 5, and asymptotic expansion by the topological recursion

2014 ◽  
Vol 03 (03) ◽  
pp. 1450013 ◽  
Author(s):  
O. Marchal ◽  
B. Eynard ◽  
M. Bergère
Keyword(s):  
Asymptotic Expansions ◽  
Correlation Functions ◽  
Hermitian Matrix ◽  
Regular Point ◽  
Recursion Method ◽  
Large N ◽  
Topological Recursion ◽  
Gap Probability ◽  
Sine Law ◽  
Hermitian Matrix Model

The goal of this paper is to rederive the connection between the Painlevé 5 integrable system and the universal eigenvalues correlation functions of double-scaled Hermitian matrix models, through the topological recursion method. More specifically we prove, to all orders, that the WKB asymptotic expansions of the τ-function as well as of determinantal formulas arising from the Painlevé 5 Lax pair are identical to the large N double scaling asymptotic expansions of the partition function and correlation functions of any Hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to O(N-5).

scholarly journals DUBROVIN’S SUPERPOTENTIAL AS A GLOBAL SPECTRAL CURVE

2017 ◽  
Vol 18 (3) ◽  
pp. 449-497 ◽  
Author(s):  
P. Dunin-Barkowski ◽  
P. Norbury ◽  
N. Orantin ◽  
A. Popolitov ◽  
S. Shadrin
Keyword(s):  
Field Theory ◽  
Spectral Curve ◽  
Frobenius Manifold ◽  
Simple Point ◽  
Topological Recursion ◽  
Cohomological Field Theory

We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

Two-matrix models and biorthogonal polynomials

2018 ◽  
Author(s):  
Giovanni Cicuta ◽  
Luca Molinari
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Correlation Functions ◽  
Spectral Curve ◽  
Biorthogonal Polynomials ◽  
The One

This article considers two cases of two-matrix models that are amenable to biorthogonal polynomials: Itzykson-Zuber interaction and Cauchy interaction. The features and applications of the biorthogonal polynomials relevant to either case are discussed, but first the article provides an overview of chain-matrix models. It then describes the Itzykson-Zuber Hermitian two-matrix model and the Christoffel–Darboux identities, along with the spectral curve. It also examines the so-called mixed correlation functions that are involved in the combinatorial applications of the two-matrix model before concluding with an analysis of the Cauchy two-matrix model, which in a ‘complication scale’ turns out to lie in between the one-matrix model and the Itzykson-Zuber two-matrix model.

scholarly journals Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure

2014 ◽  
Vol 328 (2) ◽  
pp. 669-700 ◽  
Author(s):  
P. Dunin-Barkowski ◽  
N. Orantin ◽  
S. Shadrin ◽  
L. Spitz
Keyword(s):  
Spectral Curve ◽  
Topological Recursion

scholarly journals Topological recursion with hard edges

2019 ◽  
Vol 30 (03) ◽  
pp. 1950014
Author(s):  
Leonid Chekhov ◽  
Paul Norbury
Keyword(s):  
Matrix Model ◽  
Spectral Curve ◽  
Partition Functions ◽  
Tau Function ◽  
Spectral Curves ◽  
Topological Recursion ◽  
The Matrix ◽  
Type Decomposition

We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich–Witten KdV tau function arise out of regular spectral curves and copies of the Brezin–Gross–Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve [Formula: see text].

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