Topological expansion and boundary conditions

By B. Eynard, N. Orantin. 2007

639 citations recorded by the source corpus.

In this article, we compute the topological expansion of all possible mixed-traces in a hermitian two matrix model. In other words give a recipe to compute the number of discrete surfaces of given genus, carrying an Ising model, and with all possible given boundary conditions. The method is recursive, and amounts to recursively cutting surfaces along interfaces. The result is best represented in a diagrammatic way, and is thus rather simple to use.

Subfields: Condensed Matter Physics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f69f9659af7356c2168601ba76cefad454a9a770

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Citations in the corpus

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings (2008, background): “6.1 General picture 41 6.2 A toy model 42 6.3 Holographic description and nonperturbative effects 44”
Torus Knots and Mirror Symmetry (2011, methodology): “Following [19], we recursively define a doubly infinite sequence of meromorphic differentials ωg,h(U1, .”
Liouville quantum gravity and KPZ (2008, background): “More recently, a purely combinatorial approach to discretized quantum gravity has been successful [12, 15–19, 21–28, 33, 47, 75, 106–108, 113–115, 117, 128], as well as the so-called topological expansion involving higher-genus random surfaces [35, 36, 38, 69, 70 , 72].”
Random matrices. (2015, background): “Particular expressions for these quantities can be found in [51].”
A& B model approaches to surface operators and Toda thoeries (2010, methodology): “The topological recursion is the natural formalism to generate multiple differentials on algebraic curves [14, 35].”
Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings (2012, methodology): “3 Large N instantons from orthogonal polynomials So far we have analyzed the “perturbative” sector of the one-cut matrix model from the point of view of the spectral curve: the procedure explained in [36, 57] gives a systematic way to compute the “perturbative” 1/N corrections in terms of data of th”
Polynomial recursion formula for linear Hodge integrals (2009, methodology): “The same topological structure appears in other recursion formulas such as those discussed in [5, 8, 9, 14, 25, 26, 27, 28].”
Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture (2012, methodology): “Those invariants Fg and ωg,n’s have many fascinating properties, in particular related to integrability, to modular forms, and to special geometry, and we refer the reader to [19, 21].”
JT gravity and the ensembles of random matrix theory (2019, background): “1) is the correspondence [36] between Mirzakhani’s recursion relation for the volumes of moduli space [37] and Eynard-Orantin “topological recursion” [38] which gives the genus expansion of a Hermitian matrix integral [39].”
Quantum geometry of refined topological strings (2011, background): “10) is now the spectral curve of the matrix model, from which the higher gs correction of the matrix model partition function can be computed [36].”
From Weak to Strong Coupling in ABJM Theory (2010, methodology): “1), or equivalently from the formalism of [59] (appropriately modified as in [60,61]).”
Weighted Hurwitz Numbers and Topological Recursion (2018, methodology): “[30].”
The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral (2020, methodology): “lvent of the double-scaled integral is not convergent, but we can instead dene W 0;1(z) = 2zy(z). The main result we will use is the deformation theorem derived by Eynard and Orantin, Theorem 5.1 of [38] (see also section 4.3.2 of [52] and for a summary of other interesting properties of the topolog”
Equivalence of ELSV and Bouchard–Mariño conjectures for $$r$$r-spin Hurwitz numbers (2013, background): “In [5, 18] Chekhov, Eynard, and Orantin assigned to every plane analytic complex curve C a series of invariants Wg,n(C) called correlation n-forms.”
JT gravity as a matrix integral (2019, methodology): “And, in fact, this is a practical tool: the procedure of implementing the loop equations was streamlined by Eynard [15] into simple recursion relation, which is part of the more general theory of “topological recursion” [16].”