The quantum group structure of 2D gravity and minimal models I

Abstract

Abstract. On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universal R-matrix of the quantum group SL(2)q. This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra of SL (2)q. The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of "q-deformed" factorials and binomial coefficients.

The quantum group structure of 2D gravity and minimal models II: The genus-zero chiral bootstrap

“…The formulae just written allow us to deduce the F and B matrices of the ξ fields from those of the V fields derived in the previous section. Indeed, it was already shown in ref [19] that the braiding matrix of the ξ field coincides with the universal R-matrix of U q (sl (2)). Concerning the fusing matrices, we shall establish an explicit connection later on, by first relating the coefficients |J, ̟) m M to a limit of q-Clebsch-Gordan coefficients.…”

“…The basic concepts are recalled in appendix A for completeness. The starting point is the differential equation [21,19], satisfied by the two spin 1/2 fields V ±1/2 (x), which is recalled in appendix A (Eq.A.8). In the BPZ framework it expresses the vanishing of the Virasoro null fields at the second level for operators of the type (1, 2) (Untill section 5 we concentrate on operators of the (1, 2J + 1) type, that is on the family with one of the two quantum group parameters).…”

“…These are the basic equations for establishing the operator algebra. They were already used [19] to determine the operator-product algebra (OPA) to leading order in the singularity. This showed that there are operators V (J) m , with 2J a positive integer, and −J ≤ m ≤ J, (also denoted V µ, ν , with 2J = µ + ν, and 2m = −µ + ν) and conformal weights ∆ J = −hJ(J + 1)/π − J.…”

“…In the coming section, and completing the results of refs. [13,19], we will change basis to the holomorphic operators ξ which are such that these ̟ dependences of the fusing and braiding matrices disappear. After the transformation, one is in the same situation as for quantum group, and its structure becomes more transparent.…”

“…) is the universal R-matrix R (resp.R), following the conventions of ref [19] (see Eq.3.22). We only deal in details with the case of the R-matrix R. We compute its matrix element from its universal form given in Eq.3.23 28) for n ≡ m ′ 2 − m 2 ≥ 0, and 0 otherwise.…”