Models for Modules

TL;DR

This paper explores the structure of indecomposable sl(2) modules within category O, showing their realization as quantized phase spaces in physical models and connecting algebraic representations to path integral formulations and string theory concepts.

Contribution

It introduces a path integral approach to realize indecomposable sl(2) modules and details tensor product decompositions, linking algebraic structures to physical models and string theory.

Findings

Modules can be realized as quantized phase spaces.

Explicit tensor product decomposition rules are provided.

Connections to brane quantization and string theory are discussed.

Abstract

We recall the structure of the indecomposable sl(2) modules in the Bernstein-Gelfand-Gelfand category O. We show that all these modules can arise as quantized phase spaces of physical models. In particular, we demonstrate in a path integral discretization how a redefined action of the sl(2) algebra over the complex numbers can glue finite dimensional and infinite dimensional highest weight representations into indecomposable wholes. Furthermore, we discuss how projective cover representations arise in the tensor product of finite dimensional and Verma modules and give explicit tensor product decomposition rules. The tensor product spaces can be realized in terms of product path integrals. Finally, we discuss relations of our results to brane quantization and cohomological calculations in string theory.