Affine Jordan cells, logarithmic correlators, and Hamiltonian reduction

Abstract

We study a particular type of logarithmic extension of SL(2, R) Wess-Zumino-Witten models. It is based on the introduction of affine Jordan cells constructed as multiplets of quasi-primary fields organized in indecomposable representations of the Lie algebra sl(2). We solve the simultaneously imposed set of conformal and SL(2, R) Ward identities for two-and three-point chiral blocks. These correlators will in general involve logarithmic terms and may be represented compactly by considering spins with nilpotent parts. The chiral blocks are found to exhibit hierarchical structures revealed by computing derivatives with respect to the spins. We modify the Knizhnik-Zamolodchikov equations to cover affine Jordan cells and show that our chiral blocks satisfy these equations. It is also demonstrated that a simple and well-established prescription for hamiltonian reduction at the level of ordinary correlators extends straightforwardly to the logarithmic correlators as the latter then reduce to the known results for twoand three-point conformal blocks in logarithmic conformal field theory.

The logarithmic triplet theory with boundary

“…In particular, 0 = µ(x+iy) N(iy) 34) which implies that C 1 = 0, independent of the choice of boundary condition b. Similarly, we have 35) where in the last step we used that C 1 = 0. We thus get the additional condition C 2 = C 3 .…”

On logarithmic solutions to the conformal Ward identities

“…The general expansion of A reads (13) Explicit relations similar to the one between A 1 and A 2 represented by the delta function in (12) will be omitted in the following. As indicated above, the solution (9) would have been lost if one were to set θ 1 = θ 2 = 0 in (12), whereas the first two solutions in (12) neatly follow from the last solution in (12) if one sets θ 2 = 0 or θ 1 = 0, respectively.…”

Towards the construction of local logarithmic conformal field theories

“…Among the findings which were up to now stated for chiral LCFTs only, the following topics are of central interest: Correlation functions are calculated for instance in [2,3,4,5,6,7,8,9,10,11], fusion rules are investigated, among others, in [12,13,14,15,16] and some studies on null vectors can be found in [17,18]. Enlarging their scope to non-chiral theories and implementing the constraints of locality is a task of elementary importance, as only local theories have physical interpretation.…”