4,312 research outputs found
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
Photospheric constraints, current uncertainties in models of stellar atmospheres, and spectroscopic surveys
We summarize here the discussions around photospheric constraints, current
uncertainties in models of stellar atmospheres, and reports on ongoing
spectroscopic surveys. Rather than a panorama of the state of the art, we chose
to present a list of open questions that should be investigated in order to
improve future analyses.Comment: Proc. of the workshop "Asteroseismology of stellar populations in the
Milky Way" (Sesto, 22-26 July 2013), Astrophysics and Space Science
Proceedings, (eds. A. Miglio, L. Girardi, P. Eggenberger, J. Montalban
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
Dominating surface group representations by Fuchsian ones
We prove that a representation from the fundamental group of a closed surface
of negative Euler characteristic with values in the isometry group of a
Riemannian manifold of sectional curvature bounded by -1 can be dominated by a
Fuchsian representation. Moreover, we prove that the domination can be made
strict, unless the representation is discrete and faithful in restriction to an
invariant totally geodesic 2-plane of curvature -1. When applied to
representations into PSL(2,R) of non-extremal Euler class, our result is a step
forward in understanding the space of closed anti-de Sitter 3-manifolds.Comment: Added details in lemma 2.3. Corrected a mistake about the link with
Toledo's theorem. Removed a superfluous assumption in theorem F and added a
last section about "perspectives in higher rank
- …