Maslov index, Lagrangians, Mapping Class Groups and TQFT

TL;DR

This paper explores the relationship between Maslov indices, Lagrangian subspaces, and the mapping class group of surfaces, introducing a new integer invariant and its applications to topological quantum field theory (TQFT).

Contribution

It defines an integer n_{lambda}(f) to describe a universal central extension of the mapping class group using Maslov indices, with topological and homological descriptions.

Findings

Introduces a new integer invariant n_{lambda}(f) for mapping class groups.

Provides two descriptions of a subgroup related to the Maslov index.

Applications to TQFT phase factors and skein theory representations.

Abstract

Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.