Nearby CFT's in the operator formalism: The role of a connection

TL;DR

This work shows that two complementary strategies for studying nearby conformal field theories—NCW-style deformations within a fixed state space and a bundle picture with a distinct state space for each background—are linked by a connection $\Gamma_{KZ}$ on the background bundle $V_B$. For first-order perturbations in the deformation parameter, parallel transport via $\Gamma_{KZ}$ preserves the CFT structure, establishing an equivalence between the deformed theory in $H_E$ and the theory at a neighboring background $E'$. Moreover, projecting the KZ connection onto the marginal subspace reproduces the Zamolodchikov metric's affine geometry on the base space $B$, i.e., $P\Gamma_{KZ}=\Gamma^Z$. These results suggest a coherent, background-aware framework for near-CFTs that could inform background-independent string field theory, while higher-order corrections and finite deformations remain to be fully understood.

Abstract

There are two methods to study families of conformal theories in the operator formalism. In the first method we begin with a theory and a family of deformed theories is defined in the state space of the original theory. In the other there is a distinct state space for each theory in the family, with the collection of spaces forming a vector bundle. This paper establishes the equivalence of a deformed theory with that in a nearby state space in the bundle via a connection that defines maps between nearby state spaces. We find that an appropriate connection for establishing equivalence is one that arose in a recent paper by Kugo and Zwiebach. We discuss the affine geometry induced on the space of backgrounds by this connection. This geometry is the same as the one obtained from the Zamolodchikov metric.