Modular transformation and boundary states in logarithmic conformal field theory

Abstract

We study the c = −2 model of logarithmic conformal field theory in the presence of a boundary using symplectic fermions. We find boundary states with consistent modular properties. A peculiar feature of this model is that the vacuum representation corresponding to the identity operator is a sub-representation of a "reducible but indecomposable" larger representation. This leads to unusual properties, such as the failure of the Verlinde formula. Despite such complexities in the structure of modules, our results suggest that logarithmic conformal field theories admit bona fide boundary states.

Boundary states in c=−2 logarithmic conformal field theory

“…In R 0 we choose the parameter c ΩΩ = −d to be convenient for our further discussion. Remember that d is a structure constant fixed in the metric (22). Analogously, for R 1 ⊗ R 1 one derives:…”

“…I. I. Kogan and J. F. Wheater [21] were one of the first to discuss the question of boundary states and their effects in a c = −2 logarithmic conformal field theory. In a more recent work, S. Kawai and J. F. Wheater [22] studied boundary states of the same model by use of symplectic fermions. Further studies on c = −2 were conducted by Y. Ishimoto [23].…”

Logarithmic Conformal Field Theory With Boundary

“…On the other hand, since c = −2 theory is expected to model statistical systems such as polymers, it is not conceivable that this theory has no consistent boundary states. In this subsection we see that such boundary states with consistent modular properties can be found if we use the symplectic fermion representation of the triplet model [35]. Our starting point is to notice that the right hand side of the duality condition,…”

The ABC (in any D) of logarithmic CFT

“…In spite of these developments, it is fair to say that 2d logCFTs are significantly less understood than their non-logarithmic counterparts. In particular, the computation of non-chiral (also known as bulk or local) correlation functions remains a difficult problem [28][29][30][31][32][33][34][35][36]. The references given above can serve as a starting point for the reader.…”