Logarithmic torus amplitudes

Abstract

AbstractFor the example of the logarithmic triplet theory at c = −2 the chiral vacuumtorus amplitudes are analysed. It is found that the space of these torus amplitudes isspanned by the characters of the irreducible representations, as well as a function thatcan be associated to the logarithmic extension of the vacuum representation. A fewimplications and generalisations of this result are discussed. 1. Introduction. During the last twenty years much has been understood about the struc-ture of rational conformal field theories. Rational conformal field theories are characterisedby the property that they have only finitely many irreducible highest weight representationsof the chiral algebra (or vertex operator algebra), and that every highest weight represen-tation is completely decomposable into irreducible representations. The structure of thesetheories is well understood: in particular, the characters of the irreducible representationstransform into one another under modular transformations [1] (see also [2]), and the modularS-matrix determines the fusion rules via the Verlinde formula [3]. (A general proof for thishas only recently been given in [4].)On the other hand, it is clear that rational conformal field theories are rather special, andit is therefore important to understand the structure of more general classes of conformalfield theories. One such class are the (rational) logarithmic theories that possess only finitelymany indecomposable representations, but for which not all highest weight representationsare completely decomposable. The name ‘logarithmic’ comes from the fact that their chiralcorrelation functions typically have logarithmic branch cuts. The first example of a (non-rational) logarithmic conformal field theory was found in [5] (see also [6]), and the firstrational example (that shall also concern us in this paper) was constructed in [7]; for somerecent reviews see [8, 9, 10]. From a physics point of view, logarithmic conformal fieldtheories appear naturally in various models of statistical physics, for example in the theory