Conformal field theory at central charge c=0: a measure of the indecomposability (b) parameters

A good understanding of conformal field theory (CFT) at c=0 is vital to the physics of disordered systems, as well as geometrical problems such as polymers and percolation. Steady progress has shown that these CFTs should be logarithmic, with indecomposable operator product expansions, and indecomposable representations of the Virasoro algebra. In one of the earliest papers on the subject, V. Gurarie introduced a single parameter b to quantify this indecomposability in terms of the logarithmic partner t of the stress energy tensor T. He and A. Ludwig conjectured further that b=-5/8 for polymers and b=5/6 for percolation. While a lot of physics may be hidden behind this parameter - which has also given rise to a lot of discussions - it had remained very elusive up to now, due to the lack of available methods to measure it experimentally or numerically, in contrast say with the central charge. We show in this paper how to overcome the many difficulties in trying to measure b. This requires control of a lattice scalar product, lattice Jordan cells, together with a precise construction of the state L_{-2}|0>. The final result is that b=5/6 for polymers. For percolation, we find that b=-5/8 within an XXZ or supersymmetric representation. In the geometrical representation, we do not find a Jordan cell for L_0 at level two (finite-size Hamiltonian and transfer matrices are fully diagonalizable), so there is no b in this case.Comment: 24 pages, 5 figure