The Spectral Action Principle in Noncommutative Geometry and the Superstring

TL;DR

This paper explores the application of noncommutative geometry to supersymmetric string theories, deriving a spectral action for string fluctuations using the spectral action principle in a two-dimensional supersymmetric setting.

Contribution

It introduces a novel approach to modeling superstring fluctuations through noncommutative geometry and derives a spectral action for string modes in this framework.

Findings

Derived a generalized loop space Dirac operator for supersymmetric models

Formulated a spectral action principle applicable to superstring fluctuations

Connected noncommutative geometry tools with superstring theory analysis

Abstract

A supersymmetric theory in two-dimensions has enough data to define a noncommutative space thus making it possible to use all tools of noncommutative geometry. In particular, we apply this to the N=1 supersymmetric non-linear sigma model and derive an expression for the generalized loop space Dirac operator, in presence of a general background, using canonical quantization. The spectral action principle is then used to determine a spectral action valid for the fluctuations of the string modes.