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Lecture 20: Wess-Zumino-Witten CFT and Fusion Algebras

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Lecture 20: Wess-Zumino-Witten CFT and Fusion Algebras Review: Affine Kac-Moody Lie Algebras WZW Theories Lecture 20: Wess-Zumino-Witten CFT and Fusion Algebras Daniel Bump January 1, 2020 Review: Affine Kac-Moody Lie Algebras WZW Theories Overview We have so far emphasized the BPZ minimal models, in which c takes particular values < 1. These are special in that the Hilbert space H decomposes into a finite number of Vir ⊕ Vir modules. For c > 1, in order to obtain such a finite decomposition it is necessary to enlarge Vir. If this can be done, we speak of a rational CFT: see Lecture 19 for a more careful definition of this concept, due to Moore and Seiberg. One natural such extension enlarges Vir to include an affine Lie algebra. This leads to the Wess-Zumino-Witten (WZW) models. A further generalization of this theory, the coset models of Goddard-Kent-Olive ([GKO]) contains both the minimal models and the WZW models. Review: Affine Kac-Moody Lie Algebras WZW Theories Lecture 6 review: A note on notation References: [DMS] Chapter 14 Kac, Infinite-Dimensional Lie algebras, Chapters 6,7,12,13 Kac and Peterson, Adv. in Math. 53 (1984). We begin by reviewing the construction of affine Lie algebras from Lecture 6, changing the notation slightly. Let g be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra h. We will define the untwisted affine Lie algebra bg by first making a central extension, then adjoining a derivation d. Finally we will make a semidirect product of bg with Vir in which d is identified with -L0. Review: Affine Kac-Moody Lie Algebras WZW Theories Review: central extensions Let g be a Lie algebra and a an abelian Lie algebra. A bilinear map σ : g × g ! a is called a 2-cocycle if it is skew-symmetric and satisfies σ([X, Y], Z) + σ([X, Y], Z) + σ([X, Y], Z) = 0, X, Y, Z 2 a. In this case we may define a Lie algebra structure on g ⊕ a by [(X, a), (Y, b)] = ([X, Y], σ(X, Y)). Denoting this Lie algebra g0 we have a central extension 0 −! a −! g0 −! g −! 0. Review: Affine Kac-Moody Lie Algebras WZW Theories Review: Adding a derivation Suppose we have a derivation d of a Lie algebra g. This means d([x, y]) = [dx, y] + [x, dy]. 0 We may then construct a Lie algebra g = g ⊕ Cd in which [d, x] = d(x) for x 2 g. This is a special case of a more general construction, the semidirect product. Review: Affine Kac-Moody Lie Algebras WZW Theories Review: Affine Lie algebras as Central Extensions Although the affine Lie algebra bg may be constructed from its Cartan matrix, another construction described in Chapter 7 of Kac begins with the finite-dimensional simple Lie algebra g of rank `. Tensoring with the Laurent polynomial ring gives the - loop Lie algebra g ⊗ C[t, t 1]. Then one may make a central extension: 0 -1 0 ! C · K ! bg ! C[t, t ] ⊗ g ! 0. After that, one usually adjoins another basis element, which 0 acts on bg as a derivation d. This gives the full affine Lie algebra bg. Review: Affine Kac-Moody Lie Algebras WZW Theories The dual Coxeter number The dual Coxeter number, denoted h_ will appear frequently in formulas. We omit the definition, for which see Kac, Infinite-dimensional Lie algebras Section 6.1. But here are the values as a function of the Cartan type of g. Cartan Type h_ A` l + 1 B` 2l - 1 C` l + 1 D` 2l - 2 E6 12 E7 18 E8 30 F4 9 G2 4 Review: Affine Kac-Moody Lie Algebras WZW Theories The root system and Weyl group The root system is the set of nonzero elements of bh∗ that occur in the adjoint representation of bh on g. Particular roots are as follows. _ ∗ There exist αi 2 bh and αi 2 bh called simple coroots and simple roots respectively. For i = 0, ··· , r (with r the rank of g there is simple reflection acting on bh∗ defined by _ si(λ) = λ - λ(αi )αi and the group generated by the si is the affine Weyl group W. Review: Affine Kac-Moody Lie Algebras WZW Theories Real and imaginary roots As usual, roots may be partitioned into positive and negative. A root is called real if it is in the W-orbit of a simple root. A root is imaginary if it is not real. There is a minimal positive imaginary root δ. The roots of g are all roots of bg. Let ∆ be the finite root system of g. The real roots of g are fα + nδjα 2 ∆, n 2 Zg. The imaginary roots are nδ with 0 6= n 2 Z. Review: Affine Kac-Moody Lie Algebras WZW Theories Other data The Cartan subalgebra bh of bg is h with the central element K and the derivation d adjoined. Other data that are described in Kac, Infinite-Dimensional Lie _ Algebras, Chapters 6 and 12. There are coroots αi 2 bh for i = 0, 1, ··· , r, where r is the rank of g. There are positive _ integers ai dual to the ai that we will call comarks such that r r _ _ _ . K = ai αi , h = ai i=0 i=0 X X _ The numbers ai (sometimes called comarks) are all 1 if _ g = slr+1, and for all g at least a0 = 1. Review: Affine Kac-Moody Lie Algebras WZW Theories Enlargement by Vir Instead of (or more precisely, in addition to) enlarging g0 by the derivation d we may form the semidirect product of g0 by the entire Virasoro algebra 1 [L , L ] = (i - j)L + (i3 - i)δ · C i j i+j 12 i,-j i+1 where Li acs as the derivation -t d=dt. This semidirect 0 product contains the full affine Lie algebra g = g ⊕ Cd with d = -L0. It is also possible to embed Vir in the universal enveloping algebra of bg. This is the Sugawara construction which we will make precise later. Review: Affine Kac-Moody Lie Algebras WZW Theories Integrable highest weight representations ∗ Let Λi 2 bh be the fundamental weights characterized by _ Λi(αj ) = δij and Λi(d) = 0. The lattice P spanned by the Λi is called the weight lattice. If λ = i ciΛi with ci nonnegative integers, then λ is called dominant. P Given λ 2 bh it is possible to construct a Verma module M(λ) that is highest weight representation of bg. It has a unique irreducible quotient L(λ). Let P+ be the cone of dominant weights. If λ 2 P+, then L(λ) is integrable, namely exponentiates to a representation of the loop group. The character is therefore invariant under the affine Weyl group. Review: Affine Kac-Moody Lie Algebras WZW Theories String functions Kac and Peterson proved that the characters of integrable highest weight representations are modular forms. Let λ be a dominant weight and let µ be a weight. Let mλ(µ) be the multiplicity of µ in L(λ). The function m(µ - nδ) is monotone, and zero outside the cone (λ + ρjλ + ρ)-(µ + ρ, µ + ρ) > 0. Therefore we may find a smallest n such that m(µ - nδ) is nonzero. We call µ - nδ a maximal weight. Define the string function λ -kδ bµ = m(µ - kδ)e . k X replacing µ by µ - nδ just multiplies the series by ekδ so there is no loss of generality in assuming that µ is maximal. Review: Affine Kac-Moody Lie Algebras WZW Theories String functions (continued) The character λ χλ = bµ. µ maximalX Moreover using the fact that δ = w(δ) for all w 2 W we may λ λ λ have w(bµ) = bwµ and therefore we may rewrite bµ by finding a Weyl group element such that wµ is dominant. There are only a finite number of dominant maximal weights. Then wµ λ χλ = e bµ. µ maximal,X dominant w2XW=Wµ (To avoid overcounting Wµ is the stabilizer of the dominant maximal weight µ.) Review: Affine Kac-Moody Lie Algebras WZW Theories Modular characteristics and the character Define the modular characteristic (modular anamoly) jλ + ρj2 jρj2 s = - , λ 2(k + h_) 2k We are using the notation ( j ) for an invariant inner product on Λ. Here h_ is the dual Coxeter number and k = (δjλ). Also jµj2 s (µ) = s - . λ λ 2k Now define λ -sλ(µ)δ λ -sλ(µ)δ -nδ cµ = e bµ = e m(λ - nδ)e . n X>0 These modified string functions are theta functions which are modular forms, as was proved by Kac and Peterson. Because of this, the character χλ, which is the sum of all string functions, is a modular form. Review: Affine Kac-Moody Lie Algebras WZW Theories The level k alcove The level of a weight λ is _ _ k = hλ, Ki = ai λ(αi ). X There are a finite number of dominant weights of level k, and these may be visualized as follows. We may identify the fundamental weights Λ1, ··· , Λr with the weights of the finite–dimensional Lie algebra g. Then the weights of level k are identified with weights of g by projecting on the span of Λ1, ··· , Λr. The positive Weyl chamber for g is _ the set of weights λ such that λ(αi ) > 0 for 1 6 i 6 r. The level k alcove then adds one inequality, hλ, Ki 6 k, where θ is the highest root of g.
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