# POLYAKOV'S FORMULATION OF 2d BOSONIC STRING THEORY

Abstract : Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus $g\geq 2$ and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic string theory [Po] (also called 2d bosonic string theory) and to Liouville quantum gravity. More specifically, we show the convergence of Polyakov's partition function over the moduli space of Riemann surfaces in genus $g\geq 2$ in the case of D=1 boson. This is done by performing a careful analysis of the behaviour of the partition function at the boundary of moduli space. An essential feature of our approach is that it is probabilistic and non perturbative. The interest of our result is twofold. First, to the best of our knowledge, this is the first mathematical result about convergence of string theories. Second, our construction describes conjecturally the scaling limit of higher genus random planar maps weighted by Conformal Field Theories: we make precise conjectures about this statement at the end of the paper.
Document type :
Journal articles
Domain :

https://hal-upec-upem.archives-ouvertes.fr/hal-01587093
Contributor : Rémi Rhodes <>
Submitted on : Wednesday, September 13, 2017 - 4:14:30 PM
Last modification on : Wednesday, December 4, 2019 - 12:24:29 AM
Long-term archiving on: Thursday, December 14, 2017 - 4:05:52 PM

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### Identifiers

• HAL Id : hal-01587093, version 1
• ARXIV : 1607.08467

### Citation

Colin Guillarmou, Rémi Rhodes, Vincent Vargas. POLYAKOV'S FORMULATION OF 2d BOSONIC STRING THEORY. Publications mathematiques de l' IHES, 2019, 130 (1), pp.111-185. ⟨hal-01587093⟩

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