Decoding the geometry of conformal field theories

TL;DR

This paper explores methods to extract geometric information from conformal field theories using limits, degenerations, and noncommutative geometry, aiming to reverse the usual string theory-to-CFT correspondence.

Contribution

It introduces new techniques for recovering geometrical data from conformal field theories through limits and degenerations, supported by calculations of the Virasoro zero mode.

Findings

Established a framework for limits and degenerations of CFTs

Applied noncommutative geometry techniques to CFTs

Confirmed the interpretation of the Virasoro zero mode behavior

Abstract

To certain geometries, string theory associates conformal field theories. We discuss techniques to perform the reverse procedure: To recover geometrical data from abstractly defined conformal field theories. This is done by introducing appropriate notions of limits of conformal field theories and their degenerations, and by applying techniques from noncommutative geometry. This note is a summary of our work hep-th/0308143 , aimed to be less technical than the original paper, along with some new calculations confirming our interpretation of the rescaled limiting zero mode of the Virasoro field.