On the invariance of the string partition function under duality

TL;DR

This paper investigates how the partition function of the $N=1$ supersymmetric sigma-model transforms under target-space duality, revealing that it is generally not invariant due to fermionic Jacobians affecting the equivalence of dual theories.

Contribution

It demonstrates that the supersymmetric sigma-model's partition function is not invariant under duality because of fermionic Jacobian factors related to spin structure parity.

Findings

Partition function not invariant under duality

Fermionic Jacobian affects duality invariance

Dual theories are generally not equivalent

Abstract

We consider the $N\!=\!1$ supersymmetric $\sigma$-model and we examine the transformation properties of the partition function under target-space duality. Contrary to what one would expect, we find that it is not, in general, invariant. In fact, besides the dilaton shift emerging from the Jacobian of the duality transformation of the bosonic part, there also exist a Jacobian for the fermionic part since fermions are also transform under the duality process. The latter is just the parity of the spin structure of the word sheet and since it cannot be compensated, the dual theory is not equvalent to the original one.