Skip to Main content Skip to Navigation
Theses

Du développement topologique des modèles de matrices à la théorie des cordes topologiques:
combinatoire de surfaces par la géométrie algébrique.

Abstract : The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links and extend them beyond the matrix models, following my work's evolution. First, I take care to define properly the hermitian 2 matrix model which gives rise to generating functions of discrete surfaces equipped with a spin structure. Then, I show how to compute all the terms in the topological expansion of any observable by using algebraic geometry tools. They are obtained as differential forms on an algebraic curve associated to the model: the spectral curve. In a second part, I show how to define such differentials on any algebraic curve even if it does not come from a matrix model. I then study their numerous symmetry properties under deformations of the algebraic curve. In particular, I show that these objects coincide with the topological expansion of the observable of a matrix model if the algebraic curve is the spectral curve of this model. Finally, I show that fine tuning the parameters ensure that these objects can be promoted to modular invariants and satisfy the holomorphic anomaly equation of the Kodaira-Spencer theory. This gives a new hint that the Dijkgraaf-Vafa conjecture is correct.
Document type :
Theses
Complete list of metadata

Cited literature [86 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-00173162
Contributor : Nicolas Orantin <>
Submitted on : Wednesday, September 19, 2007 - 11:06:54 AM
Last modification on : Wednesday, December 9, 2020 - 3:16:25 PM
Long-term archiving on: : Thursday, April 8, 2010 - 8:17:19 PM

Files

Identifiers

  • HAL Id : tel-00173162, version 1

Citation

Nicolas Orantin. Du développement topologique des modèles de matrices à la théorie des cordes topologiques:
combinatoire de surfaces par la géométrie algébrique.. Physique mathématique [math-ph]. Université Pierre et Marie Curie - Paris VI, 2007. Français. ⟨tel-00173162⟩

Share

Metrics

Record views

594

Files downloads

2084