Two-Point Functions and Logarithmic Boundary Operators in Boundary Logarithmic Conformal Field Theories

Abstract

Amongst conformal field theories, there exist logarithmic conformal field theories such as c p,1 models. We have investigated c p,q models with a boundary in search of logarithmic theories and have found logarithmic solutions of two-point functions in the context of the Coulomb gas picture. We have also found the relations between coefficients in the two-point functions and correlation functions of logarithmic boundary operators, and have confirmed the solutions in [6]. Other two-point functions and boundary operators have also been studied in the free boson construction of boundary CFT with SU (2) k symmetry in regard to logarithmic theories. This paper is based on a part of D. Phil. Thesis [1].

Minimal string theory is logarithmic

“…As already implied, our discussion is not restricted to the unitary minimal matters of (p = q +1, q ≥ 3), but also holds for generic integer values of (p, q), except for few cases. The only restriction on (p, q) comes from no-pole condition A ∈ Z, which is intrinsically the same as (1 + s) q p ∈ Z in [14]. Nonetheless, this is not so restrictive in our case, because the bound, 1 ≤ t ≤ p − 3, with coprime (p, q) automatically satisfies the condition.…”

“…In ordinary free field realisations of CFT, there should be conditions for logarithms which are not necessarily necessary and sufficient. For example, as shown in [14], in the Coulomb gas picture of the minimal models, there are a necessary condition and a necessary and sufficient condition on (r, t) and (p, q) for logarithms in a certain correlation function. The free boson realisation of SU(2) k WZNW models also possesses restrictive conditions for logarithms.…”

“…Logarithmic conformal field theory (LCFT) is a class of conformal field theories (CFTs) which was first found in a c = −2 theory in [6] from four-point functions with logarithms [7]. The main feature of this theory is a reducible but indecomposable representation, or Jordan cell structure, and its corresponding logarithmic fields [8][9][10][11][12][13][14]. Soon after [6], another example of LCFT was shown to be a gravitationally dressed CFT, where fermionic four-point functions had logarithmic terms in the vicinity of ξ = 1 [11].…”

“…It has been suggested in [16] that Liouville correlation functions serve four-point functions of SL(2, R) WZNW models. On the other hand, one of the authors has calculated chiral fourpoint functions and conditions for logarithms in the Coulomb gas picture with a boundary in [14].…”

Applicability of Insensitive Properties of Measured Equation of Invariance Coefficients for Modified Scattering Objects

“…Boundary logarithmic conformal field theories have been investigated from several points of view, for example starting from an underlying lattice realization [1-3, 6, 9, 19, 22-27], from supergroup WZW models [28][29][30][31][32][33] or from logarithmic extensions of Virasoro minimal models [7,16,[34][35][36][37][38][39][40][41][42]. The work of most direct relevance to our purposes is [6], where the fusion rules of the W 2,3 model are analysed via the boundary theory (on a lattice) under the assumption that one can read off the fusion rules from the open string spectra as in Cardy's analysis [49].…”