Gradient Flow in Logarithmic Conformal Field Theory

We establish conditions under which the worldsheet β-functions of logarithmic conformal field theories can be derived as the gradient of some scalar function on the moduli space of running coupling constants. We derive a renormalization group invariant version of this function and relate it to the usual Zamolodchikov C-function expressed in terms of correlation functions of the worldsheet energy-momentum tensor. The results are applied to the example of D-brane recoil in string theory. 1PPARC Advanced Fellow (U.K.). E-mail: [email protected] 2Work supported in part by PPARC (U.K.). E-mail: [email protected] Logarithmic conformal field theories [1] are extensions of conventional conformal field theories which have in recent years emerged in a number of interesting physical problems in condensed matter physics [2], string theory [3]–[6], and nonlinear dynamical systems [7]. They are characterized by the fact that their Virasoro operator L0 is not diagonalizable, but rather admits a Jordan cell structure whose basis yields the set of logarithmic operators of the theory. The non-trivial mixing between these operators leads to logarithmic divergences in their correlation functions. Such theories therefore exhibit logarithmic scaling violations on the worldsheet. Nonetheless, it is possible to deal rigorously with the Jordan cell structures of these theories and classify their conformal blocks to some extent as in ordinary conformal field theories. Theories involving logarithmic operators lie on the border between conformal field theories and generic two-dimensional quantum field theories. They can be viewed as deformations of conformally-invariant theories whose critical exponents are in general not scale invariant. Being rather subtle perturbations away from the fixed points in the space of two-dimensional field theories, it is natural to ask to what extent the conventional formalisms for studying the properties of conformal field theories can be extended when logarithmic operators emerge. In this letter we shall discuss the rôle played by logarithmic conformal field theories in the geometry of the moduli space of two-dimensional renormalizable field theories. In particular, we examine in these cases an analog of the Zamolodchikov C-function [8] which interpolates among two-dimensional quantum field theories along the trajectories of the renormalization group and whose stationary points coincide with the fixed points of these flows. At these points the C-function is the central charge of the resulting conformal field theory. For a generalized σ-model on a Riemann surface Σ with worldsheet energy-momentum tensor T , the Zamolodchikov C-function [8]