Logarithmic Conformal Field Theory - or - How to Compute a Torus Amplitude on the Sphere

TL;DR

This paper explores the connection between logarithmic conformal field theories and geometric computations on Riemann surfaces, proposing a method to compute torus amplitudes via sphere correlators with vertex insertions.

Contribution

It introduces a framework linking logarithmic CFTs to geometric surface computations, especially relating sphere correlators with branch points to higher genus amplitudes.

Findings

Logarithmic operators relate to branch points in Riemann surfaces.

Correlation functions on the sphere can encode higher genus surface data.

Application to Seiberg-Witten theory demonstrates practical computations.

Abstract

We review some aspects of logarithmic conformal field theories which might shed some light on the geometrical meaning of logarithmic operators. We consider an approach, put forward by V. Knizhnik, where computation of correlation functions on higher genus Riemann surfaces can be replaced by computations on the sphere under certain circumstances. We show that this proposal naturally leads to logarithmic conformal field theories, when the additional vertex operator insertions, which simulate the branch points of a ramified covering of the sphere, are viewed as dynamical objects in the theory. We study the Seiberg-Witten solution of supersymmetric low energy effective field theory as an example where physically interesting quantities, the periods of a meromorphic one-form, can effectively be computed within this conformal field theory setting. We comment on the relation between correlation functions computed on the plane, but with insertions of twist fields, and torus vacuum amplitudes.