Duality in Non-Trivially Compactified Heterotic Strings

TL;DR

This paper investigates how duality symmetry affects the analyticity of the partition function in heterotic string theories with non-trivial compactifications, revealing a singularity at the self-dual radius linked to degrees of freedom loss.

Contribution

It introduces a physical method to compute the Helmholtz free energy for heterotic strings with non-trivial compactifications and analyzes the singularity at the self-dual radius due to duality invariance.

Findings

Partition function is invariant under duality transformation.

Identifies a singularity at the self-dual radius similar to the Hagedorn transition.

Shows a loss of degrees of freedom below the self-dual radius.

Abstract

We study the implications of duality symmetry on the analyticity properties of the partition function as it depends upon the compactification length. In order to obtain non-trivial compactifications, we give a physical prescription to get the Helmholtz free energy for any heterotic string supersymmetric or not. After proving that the free energy is always invariant under the duality transformation $R\rightarrow \alpha^{'}/(4R)$ and getting the zero temperature theory whose partition function corresponds to the Helmholtz potential, we show that the self-dual point $R_{0}=\sqrt{\alpha^{'}}/2$ is a generic singularity as the Hagedorn one. The main difference between these two critical compactification radii is that the term producing the singularity at the self-dual point is finite for any $R \neq R_{0}$. We see that this behavior at $R_{0}$ actually implies a loss of degrees of freedom below that point.