37 research outputs found
Chain of matrices, loop equations and topological recursion
Random matrices are used in fields as different as the study of
multi-orthogonal polynomials or the enumeration of discrete surfaces. Both of
them are based on the study of a matrix integral. However, this term can be
confusing since the definition of a matrix integral in these two applications
is not the same. These two definitions, perturbative and non-perturbative, are
discussed in this chapter as well as their relation. The so-called loop
equations satisfied by integrals over random matrices coupled in chain is
discussed as well as their recursive solution in the perturbative case when the
matrices are Hermitean.Comment: 28 pages, 1 figure, contribution to The Oxford Handbook of Random
Matrix Theor
CFT and topological recursion
We study the quasiclassical expansion associated with a complex curve. In a
more specific context this is the 1/N expansion in U(N)-invariant matrix
integrals. We compare two approaches, the CFT approach and the topological
recursion, and show their equivalence. The CFT approach reformulates the
problem in terms of a conformal field theory on a Riemann surface, while the
topological recursion is based on a recurrence equation for the observables
representing symplectic invariants on the complex curve. The two approaches
lead to two different graph expansions, one of which can be obtained as a
partial resummation of the other.Comment: Minor correction
Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture
The BKMP conjecture (2006-2008), proposed a new method to compute closed and
open Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through a
topological recursion based on mirror symmetry. So far, this conjecture had
been verified to low genus for several toric CY3folds, and proved to all genus
only for C^3. In this article we prove the general case. Our proof is based on
the fact that both sides of the conjecture can be naturally written in terms of
combinatorial sums of weighted graphs: on the A-model side this is the
localization formula, and on the B-model side the graphs encode the recursive
algorithm of the topological recursion. One can slightly reorganize the set of
graphs obtained in the B-side, so that it coincides with the one obtained by
localization in the A-model.Then it suffices to compare the weights of vertices
and edges of graphs on each side, which is done in 2 steps: the weights
coincide in the large radius limit, due to the fact that the toric graph is the
tropical limit of the mirror curve. Then the derivatives with respect to
K\"ahler radius coincide due to special geometry property implied by the
topological recursion.Comment: Pdf Latex, 66 pages+30 pages of appendix, about 30 figures. Revised
version: improvement in the presentation of mirror ma
Algebraic methods in random matrices and enumerative geometry
We review the method of symplectic invariants recently introduced to solve
matrix models loop equations, and further extended beyond the context of matrix
models. For any given spectral curve, one defined a sequence of differential
forms, and a sequence of complex numbers Fg . We recall the definition of the
invariants Fg, and we explain their main properties, in particular symplectic
invariance, integrability, modularity,... Then, we give several example of
applications, in particular matrix models, enumeration of discrete surfaces
(maps), algebraic geometry and topological strings, non-intersecting brownian
motions,...Comment: review article, Latex, 139 pages, many figure
Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
We compute expectation values of mixed traces containing both matrices in a
two matrix model, i.e. generating function for counting bicolored discrete
surfaces with non uniform boundary conditions. As an application, we prove the
symmetry of the algebraic curve invariants introduced in math-ph/0702045.Comment: 37 pages, late
Invariants of algebraic curves and topological expansion
For any arbitrary algebraic curve, we define an infinite sequence of
invariants. We study their properties, in particular their variation under a
variation of the curve, and their modular properties. We also study their
limits when the curve becomes singular. In addition we find that they can be
used to define a formal series, which satisfies formally an Hirota equation,
and we thus obtain a new way of constructing a tau function attached to an
algebraic curve. These invariants are constructed in order to coincide with the
topological expansion of a matrix formal integral, when the algebraic curve is
chosen as the large N limit of the matrix model's spectral curve. Surprisingly,
we find that the same invariants also give the topological expansion of other
models, in particular the matrix model with an external field, and the
so-called double scaling limit of matrix models, i.e. the (p,q) minimal models
of conformal field theory. As an example to illustrate the efficiency of our
method, we apply it to the Kontsevitch integral, and we give a new and
extremely easy proof that Kontsevitch integral depends only on odd times, and
that it is a KdV tau-function.Comment: 92 pages, LaTex, 33 figures, many misprints corrected, small
modifications, additional figure
Geometric recursion
We propose a general theory to construct functorial assignments for a large class of functors
from a certain category of bordered surfaces to a suitable target category of
topological vector spaces. The construction proceeds by successive excisions of
homotopy classes of embedded pairs of pants, and thus by induction on the Euler
characteristic. We provide sufficient conditions to guarantee the infinite sums
appearing in this construction converge. In particular, we can generate mapping
class group invariant vectors . The initial data
for the recursion encode the cases when is a pair of pants or a torus
with one boundary, as well as the "recursion kernels" used for glueing. We give
this construction the name of Geometric Recursion (GR). As a first application,
we demonstrate that our formalism produce a large class of measurable functions
on the moduli space of bordered Riemann surfaces. Under certain conditions, the
functions produced by the geometric recursion can be integrated with respect to
the Weil--Petersson measure on moduli spaces with fixed boundary lengths, and
we show that the integrals satisfy a topological recursion (TR) generalizing
the one of Eynard and Orantin. We establish a generalization of
Mirzakhani--McShane identities, namely that multiplicative statistics of
hyperbolic lengths of multicurves can be computed by GR, and thus their
integrals satisfy TR. As a corollary, we find an interpretation of the
intersection indices of the Chern character of bundles of conformal blocks in
terms of the aforementioned statistics. The theory has however a wider scope
than functions on Teichm\"uller space, which will be explored in subsequent
papers; one expects that many functorial objects in low-dimensional geometry
could be constructed by variants of our new geometric recursion.Comment: 97 pages, 21 figures. v2: misprint corrected. v3: revised and
abridged version, 66 page
Modular functors, cohomological field theories and topological recursion
Given a topological modular functor in the sense of Walker
\cite{Walker}, we construct vector bundles over ,
whose Chern classes define semi-simple cohomological field theories. This
construction depends on a determination of the logarithm of the eigenvalues of
the Dehn twist and central element actions. We show that the intersection of
the Chern class with the -classes in is
computed by the topological recursion of \cite{EOFg}, for a local spectral
curve that we describe. In particular, we show how the Verlinde formula for the
dimensions is retrieved from the
topological recursion. We analyze the consequences of our result on two
examples: modular functors associated to a finite group (for which
enumerates certain -principle
bundles over a genus surface with boundary conditions specified by
), and the modular functor obtained from Wess-Zumino-Witten
conformal field theory associated to a simple, simply-connected Lie group
(for which is the Verlinde
bundle).Comment: 50 pages, 2 figures. v2: typos corrected and clarification about the
use of ordered pairs of points for glueing. v3: unitarity assumption waived +
discussion of families index interpretation of the correlation functions for
Wess-Zumino-Witten theorie
Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
Latex, 9 pagesWe show that Mirzakhani's recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix loop equations, and therefore we confirm again that Kontsevitch's integral is a generating function for those volumes. As an application, we propose a formula for the Weil-Petersson volume Vol(M_{g,0})