K. Ranganathan | Searcharxiv

arXiv1993

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

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.

hep-th↗

arXiv1993

Connections on the State-Space over Conformal Field Theories

Motivated by the problem of background independence of closed string field theory we study geometry on the infinite vector bundle of local fields over the space of conformal field theories (CFT's). With any connection we can associate an excluded domain $D$ for the integral of marginal operators, and an operator one-form $ω_μ$. The pair $(D, ω_μ)$ determines the covariant derivative of any correlator of local fields. We obtain interesting classes of connections in which $ω_μ$'s can be written in terms of CFT data. For these connections we compute their curvatures in terms of four-point correlators, $D$, and $ω_μ$. Among these connections three are of particular interest. A flat, metric compatible connection $\HG$, and connections $c$ and $\bar c$ having non-vanishing curvature, with $\bar c$ being metric compatible. The flat connection cannot be used to do parallel transport over a finite distance. Parallel transport with either $c$ or $\bar c$, however, allows us to construct a CFT in the state space of another CFT a finite distance away. The construction is given in the form of perturbation theory manifestly free of divergences.

hep-th↗