On the Central Charge of Spacetime Current Algebras and Correlators in String Theory on AdS3

TL;DR

This paper addresses issues in the algebraic structure of string theory on AdS3 by proposing a Legendre transform approach to define free energy, resolving state-dependence and cluster decomposition problems.

Contribution

It introduces a new method to define the boundary CFT free energy as a Legendre transform, fixing the number of strings and removing algebraic and correlator inconsistencies.

Findings

Central charge becomes state-independent after Legendre transform.

Cluster decomposition property is restored in the modified framework.

The approach clarifies the role of long strings in the algebraic structure.

Abstract

Spacetime Virasoro and affine Lie algebras for strings propagating in AdS3 are known to all orders in $\alpha'$. The central extension of such algebras is a string vertex, whose expectation value can depend on the number of long strings present in the background but should be otherwise state-independent. In hep-th/0106004, on the other hand, a state-dependent expectation value was found. Another puzzling feature of the theory is lack of cluster decomposition property in certain connected correlators. This note shows that both problems can be removed by defining the free energy of the spacetime boundary conformal field theory as the Legendre transform of the formula proposed in the literature. This corresponds to pass from a canonical ensemble, where the number of fundamental strings that create the background can fluctuate, to a microcanonical one, where it is fixed.