Extended chiral algebras in the SU(2)₀WZNW model

Abstract

We investigate the W-algebras generated by the integer dimension chiral primary operators of the SU (2) 0 WZNW model. These have a form almost identical to that found in the c = −2 model but have, in addition, an extended affine Lie algebraic structure. Moreover on Hamiltonian reduction these SU (2) 0 W -algebras exactly reduce to those found in c = −2. We explicitly find the free field representations for the chiral j = 2 and j = 3 operators which have respectively a fermionic doublet and bosonic triplet nature. The correlation functions of these operators accounts for the rational solutions of the Knizhnik-Zamolodchikov equation that we find. We explicitly compute the full algebra of the j = 2 operators and find that the associativity of the algebra is only guaranteed if certain null vectors decouple from the theory. We conjecture that these algebras may produce a quasi-rational conformal field theory. 1 a.nichols1@physics.ox.ac.uk 2 Extended Chiral Algebras Free field representationWe shall use the standard Wakimoto representation [70] for the affine Lie algebra SU(2) specialised to the case k = 0. Our currents are:where β and γ obey the standard free field relations:

Logarithmic Conformal Field Theory With Boundary

“…By now various models, e.g. c = −2 model [3,5,6,7], gravitationally dressed CFTs [8,9], WZNW models with fractional k [4,10] and k = 0 [11,12,13,14] have been studied, and a number of applications, including critical polymers [3,15,16], percolation [17,18], quantum Hall effect [19,20], disordered systems [11,12,21,22,23], sandpile model [24], turbulence [25,26,27], MHD [28], D-brane recoil [29,30], etc. have been discussed.…”

The origin of multiplets of chiral fields inSU(2)kat rational level

“…One intriguing aspect is that the extending algebras in all the c p,1 cases involved multiplets of chiral operators with extra SU(2) quantum numbers. We have studied many examples of the appearance of such extra multiplet structure in the chiral fields of the c p,q models and also SU(2) k models (in particular SU(2) 0 ) [30][31][32][33]. The first indications of a general pattern were found by studying the appearance of rational correlation functions and their properties under crossing -in effect calculating the braid matrices.…”

“…. Using (31) one can easily derive the formula: S ± (r 0 +pr 1 ) = S ± (r 0 ) (S ± ) (p) r 1 r 1 ! (90)…”

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

“…Thus far, many studies have been devoted to this subject and have found the same sort of logarithmic behaviours in various models. For example, the gravitationally dressed CFT and WZNW models at different levels or on different groups [26,[29][30][31][32][33][34][35][36][37][38][39][40][41][42]. c p,1 and non-minimal c p,q models, including the c 2,1 = −2 model [2, 21-24, 27, 28, 30, 43-58].…”