Notes on conformal theory with nonprimary energy-momentum tensor that applies to the Nambu-Goto string

TL;DR

This paper explores a higher-derivative conformal theory distinguishing Nambu-Goto and Polyakov strings, introducing a new technique to handle nonprimary energy-momentum tensors and analyzing their impact on conformal invariance and string susceptibility.

Contribution

It develops a novel method to incorporate quantum equations of motion for nonprimary energy-momentum tensors in conformal theories, revealing their algebraic structure and divergence behavior.

Findings

Conserved, traceless energy-momentum tensor not in the conformal family of the identity.

Conformal transformations form a Lie algebra with a central extension.

Logarithmic divergence in the central charge cancels in string susceptibility, matching known deviations.

Abstract

I investigate the higher-derivative conformal theory which shows how the Nambu-Goto and Polyakov strings can be told apart. Its energy-momentum tensor is conserved, traceless but does not belong to the conformal family of the unit operator. To implement conformal invariance in this case, I develop the new technique that explicitly accounts for the quantum equation of motion and results in singular products. I show that the conformal transformations generated by such a nonprimary energy-momentum tensor form a Lie algebra with a central extension which in the path-integral formalism gives a logarithmically divergent contribution to the central charge. I demonstrate how the logarithmic divergence is canceled in the string susceptibility and reproduce the previously obtained deviation from KPZ-DDK at one loop.