Gravity Duals for Logarithmic Conformal Field Theories

Logarithmic conformal field theories with vanishing central charge describe systems with quenched disorder, percolation or dilute self-avoiding polymers. In these theories the energy momentum tensor acquires a logarithmic partner. In this talk we address the construction of possible gravity duals for these logarithmic conformal field theories and present two viable candidates for such duals, namely theories of massive gravity in three dimensions at a chiral point. Outline This talk is organized as follows. In section 1 we recall salient features of 2-dimensional conformal field theories. In section 2 we review a specific class of logarithmic conformal field theories where the energy momentum tensor acquires a logarithmic partner. In section 3 we present a wish-list for gravity duals to logarithmic conformal field theories. In section 4 we discuss two examples of massive gravity theories that comply with all the items on that list. In section 5 we address possible applications of an Anti-deSitter/logarithmic conformal field theory correspondence in condensed matter physics. 1. Conformal field theory distillate Conformal field theories (CFTs) are quantum field theories that exhibit invariance under angle preserving transformations: translations, rotations, boosts, dilatations and special conformal transformations. In two dimensions the conformal algebra is infinite dimensional, and thus two-dimensional CFTs exhibit a particularly rich structure. They arise in various contexts in physics, including string theory, statistical mechanics and condensed matter physics, see e.g. [1]. The main observables in any field theory are correlation functions between gauge invariant operators. There exist powerful tools to calculate these correlators in a CFT. The operator content of various CFTs may differ, but all CFTs contain at least an energy momentum tensor Tμν . Conformal invariance requires the energy momentum tensor to be traceless, T μ μ = 0, in addition to its conservation, ∂μT μν = 0. In lightcone gauge for the Minkowski metric, ds2 = 2dz dz̄, these equations take a particularly simple form: Tzz̄ = 0, Tzz = Tzz(z) := OL(z) and Tz̄z̄ = Tz̄z̄(z̄) := OR(z̄). Conformal Ward identities determine essentially uniquely the form of 2and 3-point correlators between the flux components OL/R of the energy momentum tensor: 〈O(z̄)O(0)〉 = cR 2z̄4 (1a) 〈O(z)O(0)〉 = cL 2z4 (1b) 〈O(z)O(0)〉 = 0 (1c) 〈O(z̄)O(z̄)O(0)〉 = cR z̄2z̄′ 2(z̄ − z̄′)2 (1d) 〈O(z)O(z)O(0)〉 = cL z2z′ 2(z − z′)2 (1e) 〈O(z)O(z̄)O(0)〉 = 0 (1f) 〈O(z)O(z)O(0)〉 = 0 (1g) The real numbers cL, cR are the left and right central charges, which determine key properties of the CFT. We have omitted terms that are less divergent in the near coincidence limit z, z̄ → 0 as well as contact terms, i.e., contributions that are localized (δ-functions and derivatives thereof). If someone provides us with a traceless energy momentum tensor and gives us a prescription how to calculate correlators,1 but does not reveal whether the underlying field theory is a CFT, then we can perform the following check. We calculate all 2and 3-point correlators of the energy momentum tensor with itself, and if at least one of the correlators does not match precisely with the corresponding correlator in (1) then we know that the field theory in question cannot be a CFT. On the other hand, if all the correlators match with corresponding ones in (1) we have non-trivial evidence that the field theory in question might be a CFT. Let us keep this stringent check in mind for later purposes, but switch gears now and consider a specific class of CFTs, namely logarithmic CFTs (LCFTs). 2. Logarithmic CFTs with an energetic partner LCFTs were introduced in physics by Gurarie [2]. We focus now on some properties of LCFTs and postpone a physics discussion until the end of the talk, see [3,4] for reviews. There are two conceptually different, but mathematically equivalent, ways to define LCFTs. In both versions there exists at least one operator that acquires a logarithmic partner, which we denote by Olog. We focus in this talk exclusively on theories where one (or both) of the energy momentum tensor flux components is the operator acquiring such a partner, for instance OL. We discuss now briefly both ways of defining LCFTs. According to the first definition “acquiring a logarithmic partner” means that the Hamiltonian H cannot be diagonalized. For example