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To any differential system $dPsi=PhiPsi$ where $Psi$ belongs to a Lie group (a fiber of a principal bundle) and $Phi$ is a Lie algebra $mathfrak g$ valued 1-form on a Riemann surface $Sigma$, is associated an infinite sequence of correlators $W_n$ that are symmetric $n$-forms on $Sigma^n$. The goal of this article is to prove that these correlators always satisfy loop equations, the same equations satisfied by correlation functions in random matrix models, or the same equations as Virasoro or W-algebra constraints in CFT.
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For a given polynomial $V(x)in mathbb C[x]$, a random matrix eigenvalues measure is a measure $prod_{1leq i<jleq N}(x_i-x_j)^2 prod_{i=1}^N e^{-V(x_i)}dx_i$ on $gamma^N$. Hermitian matrices have real eigenvalues $gamma=mathbb R$, which generalize to $gamma$ a complex Jordan arc, or actually a linear combination of homotopy classes of Jordan arcs, chosen such that integrals are absolutely convergent. Polynomial moments of such measure satisfy a set of linear equations called loop equations. We prove that every solution of loop equations are necessarily polynomial moments of some random matrix measure for some choice of arcs. There is an isomorphism between the homology space of integrable arcs and the set of solutions of loop equations. We also generalize this to a 2-matrix model and to the chain of matrices, and to cases where $V$ is not a polynomial but $V(x)in mathbb C(x)$.
This paper is a continuation of our previous work Six-vertex model and non-linear differential equations I. Spectral problem in which we have put forward a method for studying the spectrum of the six-vertex model based on non-linear differential equations. Here we intend to elaborate on that approach and also discuss properties of the spectrum unveiled by the aforementioned differential formulation of the transfer matrixs eigenvalue problem. In particular, we intend to demonstrate how this differential approach allows one to study continuous symmetries of the transfer matrixs spectrum through the Lie groups method.
We consider abelian twisted loop Toda equations associated with the complex general linear groups. The Dodd--Bullough--Mikhailov equation is a simplest particular case of the equations under consideration. We construct new soliton solutions of these Toda equations using the method of rational dressing.
We formulate a notion of abstract loop equations, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the O(n) model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SU(N) Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.
Starting from a $dtimes d$ rational Lax pair system of the form $hbar partial_x Psi= LPsi$ and $hbar partial_t Psi=RPsi$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfies the topological type property. A consequence is that the formal $hbar$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painleve systems.
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