General branching functions of affine Lie algebras

Explicit expressions are presented for general branching functions for cosets of affine Lie algebras ĝ with respect to subalgebras ĝ′ for the cases where the corresponding finite dimensional algebras g and g′ are such that g is simple and g′ is either simple or sums of u(1) terms. A special case of the latter yields the string functions. Our derivation is purely algebraical and has its origin in the results on the BRST cohomology presented by us earlier. We will here give an independent and simple proof of the validity our results. The method presented here generalizes in a straightforward way to more complicated g and g′ such as e g sums of simple and u(1) terms. [email protected] [email protected] Since the discovery of Kac-Moody algebras [1],[2] and the centrally extended affine ones, known as affine Lie algebras (of which a first example was given in [3]), affine Lie algebras have played an increasingly important rôle in physics. In the framework of conformal field theories, affine Lie algebras appear in the study of string theory as well as critical phenomena in solid state theory. In particular, the so-called coset construction is crucial for the description of known conformal field theories using affine Lie algebras. Examples of this construction first appeared in [3] although a general form of the stress-energy tensor was first given by [4]. It is in this connection important to find the partition function describing the different cosets. In string theory they are nothing but the zero-point one-loop amplitudes. A slightly more general problem is to find the complete branching functions of the different cosets, since from the knowledge of the latter the partition functions are easily derived. The purpose of this note is to present general expressions of the branching functions of ĝ with respect to a subalgebra ĝ′ in the cases when the finite dimensional algebras g and g′ are such that g is simple and g′ is either simple or a sum of u(1) terms. In the latter case we can specialize to the Cartan subalgebra of g. Then the branching functions are essentially the string functions. The technique which we will use here was first given in [5] and the present note is a rather straightforward application of this. However, we feel that the importance of the branching and string functions motivates a derivation of the explicit form for these. We will here also present an alternative and very direct proof of the correctness of the general expression. The proof in [5] is based on the computation of the cohomology of the BRST operator and, although correct, it is not very transparent for the present application. The explicit expressions for the branching and string functions presented here are restricted solely to the cases of integrable representations. The basic tool which we use apart from the construction in [5] is the Weyl-Kac formula [6] of the character of an integrable representation of an affine Lie algebra. Expressions of branching and string functions in special cases have appeared previously in the literature in particular in [7] and [8] (most of the results are reviewed in [9]). The general form of the string function for the case of ŝu(2)k was first derived in [7]. In ref.[10] expressions for the string functions of simply laced algebras as well as branching functions for several coset constructions was presented. The basic assumption in this work is the existence of a free field resolution of the irreducible affine module. Such 1 a resolution has only been proven to exist for the case of ŝu(2) [11]. The expressions derived here for the string functions will confirm the results given in [10]. In ref. [12] a general formula for branching functions in terms of string functions was derived 3. Using the explicit formulas for the branching as well as the string functions given here, one may verify these relations. We will, however, use our methods to give a simple derivation of the branching functions directly in terms of string functions. These expressions do not directly coincide with ref.[12], but we expect that some straightforward algebraic rearrangements will make them equal. One may consider more general coset constructions than the ones discussed here. The cases where instead of a simple subalgebra g′ one has several simple terms g′(1), g′(2), . . ., yield branching functions which are easily derived using the methods presented here. This is true also in the case where one or more of the simple terms are replaced with u(1) terms. By taking sums of several simple algebras g(i), one may have many different cases. In [13] a number of such cases will be presented: ĝ = ĝk1⊕ĝk2 and ĝ ′ = ĝk1+k2, ĝ and ĝ′ with rankg=rankg′, ĝ = (⊕a=1ĝka) and ĝ ′ = ĝ( ∑n i=1 ki) , ĝ = ĝk1 ⊕ ĝ ′′ k2 and ĝ′ = ĝ′′ k1+k2, where g ′′ ⊆ g. Let g be a simple finite dimensional Lie algebra and g′ be a subalgebra of g. Their ranks are r and r′. We denote by ĝ and ĝ′ the corresponding affine Lie algebras. The levels k and k′ of ĝ and ĝ′ are taken to be non-negative integers (k = k′ for g′ simple and g simply laced). The set of roots of the finite algebra g are α ∈ ∆g and α′ ∈ ∆g′ for g ′. The corresponding affine roots are α̂ ∈ ∆ĝ etc. The highest root of g is denoted by ψg and its length is taken to be one. The restrictions to positive roots are denoted ∆g and ∆ + g′ . The number of elements in these sets are |∆g |, |∆ + g′ |, respectively. We also define ∆ + g,g′ = {α | α ∈ ∆ + g , α 6∈ ∆ + g′} with the number of elements |∆+g,g′ | = |∆ + g | − |∆ + g′ |. The weight and root lattices of g and g′ are denoted by Γw,g,Γr,g,Γw,g′ and Γr,g′ etc for ĝ and ĝ ′. Define ρ ∈ Γr,g by {ρ | αi · ρ = α 2 i , for simple roots αi ∈ ∆g}. ρ ′, ρ̂ and ρ̂′ are the corresponding vectors for g′, ĝ and ĝ′. For the finite dimensional algebras ρ is just the sum of positive roots. Let P ĝ and P + ĝ′ be the set of integrable highest weight representations of ĝ and ĝ′ with respective highest weights λ̂ ∈ Γw,ĝ and λ̂ ′ ∈ Γw,ĝ′. Then P + ĝ = {λ̂ | α̂i · λ̂ ≥ 0 for simple roots α̂i ∈ ∆ĝ}, or equivalently P + ĝ = {λ | αi · λ ≥ We thank V.G. Kac for this reference.