Hurwitz numbers, matrix models and enumerative geometry

Bouchard, Vincent; Mariño, Marcos

doi:10.1090/pspum/078/2483754

Cited by 131 publications

(218 citation statements)

References 29 publications

(15 reference statements)

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“…. , z n ) in a suitable basis of differential forms turn out to be equal to the intersection indices ("volumes" of a moduli space) of surfaces of genus g with n boundaries: we obtain the Weil-Petersson volumes of M g,n [6], the intersection numbers of ψ and κ classes on M g,n [7], [8], the open Gromov-Witten invariants of toric Calabi-Yau 3-folds [9], [10], the stationary Gromov-Witten invariants of P 1 , the simple Hurwitz numbers [11]- [13], and so on. The list will probably grow in the near future.…”

Section: Topological Recursionmentioning

confidence: 99%

Blobbed topological recursion

Borot

2015

Theor Math Phys

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We describe the formalism and the properties of the blobbed topological recursion, which provides the general solution of a set of abstract loop equations. This procedure extends the topological recursion by introducing extra terms (blobs) in the initial conditions for each multidifferential ωg,n. We apply this formalism to topological expansion of formal Hermitian matrix models (the blobs are necessary to include arbitrary interactions) and pose some open questions.This paper is a short, preliminary version of the extended electronic article [1]. Axiomatic definitionFunctional equations are encountered in the enumerative geometry of surfaces: Virasoro constraints (in matrix models and intersection theory over M g,n ), Tutte's equations (in map enumeration), cut-andjoin equations (in covering enumerations), etc. They have very similar natures, which has to do with the recursive construction of orientable surfaces. We introduced abstract loop equations in [2] as an attempt to write a universal set of equations, to which the abovementioned functional equations can be reduced, maybe after some problem-specific technical steps. Abstract loop equations.Let U i , i ∈ I, be a finite collection of open sets that are complex one-dimensional manifolds with a distinguished point α i ∈ U i . We can take an arbitrary small neighborhood of α i as U i because our secret purpose is to speak of germs of functions and differential forms at α i . We assume that U i comes with a covering {x : U i → V i ⊆ C} that has a ramification point at α i . We also assume that all ramification points are simple, i.e., α i is a simple zero of dx, although the theory can be formulated for higher-order ramifications following [3], [4]. We then have a deck transformation ι : U i → U i , i.e., a holomorphic map such that x(ι(z)) = x(z) but ι = id (see Fig.

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Section: Topological Recursionmentioning

confidence: 99%

Blobbed topological recursion

Borot

2015

Theor Math Phys

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“…in terms of intrinsic quantities of the hyperelliptic surfaces (14) and (15) is due to Eynard. For details of this formalism we refer to the original work [1] and [2,6].…”

Section: Hermitean 1-matrix Modelsmentioning

confidence: 99%

“…It has moreover been shown in [6] that one may abstract from the framework of matrix models to the construction of new invariants attached to more general Riemann surfaces (mimicking the topological expansion of matrix models). This brings into focus [6] new subjects as Kontsevich's matrix model for intersection numbers [7], topological string theory [8][9][10][11], as well as further applications to mathematical subjects, as the recursive determination of Weil-Petersson volumes [12,13] and a conjectural matrix model representation of Hurwitz numbers [14].…”

Section: Introductionmentioning

confidence: 99%

A Lagrangean formalism for Hermitean matrix models

Flume

Grossehelweg²,

Klitz

2009

Nuclear Physics B

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“…This construction allows us to derive linear constraints The constraints for the Kontsevich-Witten tau-function are well known, namely, in this case equations (0.2) describe a reduction from KP to KdV, and equations (0.3) are the Virasoro constraints. Two other families of constraints (for the Hurwitz and Hodge tau-functions) are obtained explicitly for the first time (see, however, [2,3]). Although constraints for all three tau-functions satisfy the same commutation relations, they are quite different in their form.…”

Section: Introductionmentioning

confidence: 99%