Strings, Noncommutative Geometry and the Size of the Target Space

TL;DR

This paper explores how antisymmetric tensor fields in string theory affect the perceived size of the target space, linking gauge non invariance, duality symmetries, and noncommutative geometry to explain how low-energy observers perceive uncompactified dimensions.

Contribution

It introduces a framework connecting noncommutative geometry and spectral action principles to the gauge non invariance of target space size in string theory.

Findings

Target space size becomes gauge non invariant due to torsion.

Duality transformations can make all low eigenvalues momentum modes.

Observers with low energy eigenvalues perceive an uncompactified space.

Abstract

We describe how the presence of the antisymmetric tensor (torsion) on the world sheet action of string theory renders the size of the target space a gauge non invariant quantity. This generalizes the R <--> 1/R symmetry in which momenta and windings are exchanged, to the whole O(d,d,Z). The crucial point is that, with a transformation, it is possible always to have all of the lowest eigenvalues of the Hamiltonian to be momentum modes. We interpret this in the framework of noncommutative geometry, in which algebras take the place of point spaces, and of the spectral action principle for which the eigenvalues of the Dirac operator are the fundamental objects, out of which the theory is constructed. A quantum observer, in the presence of many low energy eigenvalues of the Dirac operator (and hence of the Hamiltonian) will always interpreted the target space of the string theory as effectively uncompactified.