Loop equations for the semiclassical 2-matrix model with hard edges

By B. Eynard. 2005

277 citations recorded by the source corpus.

The 2-matrix model can be defined in a setting more general than polynomial potentials, namely, the semiclassical matrix model. In this case, the potentials are such that their derivatives are rational functions, and the integration paths for eigenvalues are arbitrary homology classes of paths for which the integral is convergent. This choice includes in particular the case where the integration path has fixed endpoints, called hard edges. The hard edges induce boundary contributions in the loop equations. The purpose of this paper is to give the loop equations in that semi-classical setting.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6442908091c8797f7468c55c62e848e95af53339

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Citations in the corpus

Free energy topological expansion for the 2-matrix model (2006, result): “Let us also mention that we have considered here only polynomial potentials, but it is clear that the whole method should extend easily to the more general class of semi-classical potentials [6,7], whose loop equations are very similar [ 23 ].”
Les Houches lectures on matrix models and topological strings (2004, methodology): “Although we will not need it in this review, there are well-developed techniques to obtain the higher genus Fg(t) as systematic corrections to the saddle-point result F0(t) [5, 32].”
Weighted Hurwitz Numbers and Topological Recursion (2018, methodology): “This approach, originally inspired by results arising naturally in random matrix theory [27], has been shown applicable to many enumerative geometry problems, such as the counting of maps [28] or the computation of Gromov-Witten invariants [20].”
Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture (2012, methodology): “That method is based on the ”topological recursion” of [14, 19] and on mirror symmetry (this will be described in more details in section 4 below).”
Invariants of algebraic curves and topological expansion (2007, unknown): “[12, 16,36].”
Matrix eigenvalue model: Feynman graph technique for all genera (2006, methodology): “…notion is again the Bergmann kernel,2 which is the unique bi-differential on a Riemann surface Σg that is symmetrical in its arguments P,Q ∈ Σg and has the only singularity (a double pole) at the coinciding arguments where, in any local coordinate τ , it has the behavior (see [24], [23]) B(P,Q)…”
The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers (2009, background): “und of our work. Independent of the recent geometric and combinatorial works [17, 35, 36, 43], a theory of topological recursions has been developed in the matrix model/random matrix theory community [9, 13]. Its culmination is the topological recursion formula established in [13]. There are three i”
Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings (2007, background): “This was first made clear in [37] and later culminated in the geometric formalism of [38, 39, 17].”
Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings (2012, background): “This was first made clear in [28, 9, 8] and later culminated in the geometric formalism of [54, 36, 57].”