A general BRST approach to string theories with zeta function regularizations

TL;DR

The paper presents a general BRST quantization framework for string and string-like models using a universal zeta-function regularization of basic commutators, enabling rigorous operator analysis via analytic continuation to finite results. Applying the method to the ordinary bosonic string, the bosonic tensionless string, and the bosonic conformal string, it shows that nilpotency and BRST invariance depend sensitively on the chosen state space, yielding the conventional critical dimension $d=26$ for the bosonic string and revealing dimension-specific vacua (e.g., $d=4$ or $d=2$) for tensionless and conformal-tensionless cases. The work demonstrates how regulator choices interact with extended constraint algebras to determine consistency, while also highlighting potential issues with negative-norm states in certain vacua. Overall, the approach provides a rigorous, regulator-based path to BRST quantization beyond standard CFT methods and opens directions for fermionic extensions and brane theories, with careful attention to the physical viability of the resulting state spaces.

Abstract

We propose a new general BRST approach to string and string-like theories which have a wider range of applicability than e g the conventional conformal field theory method. The method involves a simple general regularization of all basic commutators which makes all divergent sums to be expressible in terms of zeta functions from which finite values then may be extracted in a rigorous manner. The method is particular useful in order to investigate possible state space representations to a given model. The method is applied to three string models: The ordinary bosonic string, the tensionless string and the conformal tensionless string. We also investigate different state spaces for these models. The tensionless string models are treated in details. Although we mostly rederive known results they appear in a new fashion which deepens our understanding of these models. Furthermore, we believe that our treatment is more rigorous than most of the previous ones. In the case of the conformal tensionless string we find a new solution for d=4.