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

Review: Afﬁne Kac-Moody Lie Algebras WZW Theories

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 ∈ 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 ∈ 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 ∈ bh and αi ∈ 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

{α + nδ|α ∈ ∆, n ∈ Z}.

The imaginary roots are nδ with 0 6= n ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ 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 + ρ|λ + 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 ∈ 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 w∈XW/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) |λ + ρ|2 |ρ|2 s = − , λ 2(k + h∨) 2k We are using the notation ( | ) for an invariant inner product on Λ. Here h∨ is the dual Coxeter number and k = (δ|λ). Also |µ|2 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. Review: Affine Kac-Moody Lie Algebras WZW Theories

The level 3 alcove for slb3

For example, if g = sl3, here is the level 3 alcove inside the positive Weyl chamber, showing that gˆ has 10 positive weights of level 3:

The SL(3) case Review: Afﬁne Kac-Moody Lie Algebras WZW Theories

The afﬁne Weyl group (continued)

k = 3 k = 2 k = 1 k = 0

The affine Weyl group fixes the origin. But parallel to the level 0 plane, it induces affine linear maps on each level (except level 0). The fundamental alcove on the k level set is the intersection with the shaded area, consisting of dominant weights. Review: Affine Kac-Moody Lie Algebras WZW Theories

The Hilbert space

To define a WZW CFT we pick a finite-dimensional Lie algebra g with corresponding affine Lie algebra g and a level k. In place of the Virasoro algebra we have the semidirect product A = gˆ ⊕ Vir. This acts on L(λ) for any dominant weight λ, due to the Sugawara construction. The Hilbert space of the model is M L(λ) ⊗ L(λ). λ of level k As usual, any field can be decomposed into a sum of fields φ(z) ⊗ φ (z) where φ is holomorphic and φ is antiholomorphic. The holomorphic fields are called chiral. Review: Affine Kac-Moody Lie Algebras WZW Theories

Afﬁne primary ﬁelds and currents

Let Φλ be the highest weight element of L(λ), identified with a chiral field. The field Φλ(z)Φλ (z) in L(λ) ⊗ L(λ) is called an affine primary field. We will concentrate on the chiral field Φλ(z), which we will also call affine primary fields. Review: Affine Kac-Moody Lie Algebras WZW Theories

Currents

I addition to the primary ﬁelds and their descendents, there are other ﬁelds called currents. Let J ∈ g. The current J(z) is the formal expression

−n−1 n J(z) = z Jn, Jn = J ⊗ t ∈ gˆ. ∞ n=− X Currents had their origin∞ in particle physics where they were introduced by Gell-Mann in 1964. Review: Afﬁne Kac-Moody Lie Algebras WZW Theories

The Sugawara construction

Another ﬁeld is the energy-momentum tensor, which is made explicit through the Sugawara construction. Indeed the Sugawara construction which embeds Vir in the universal enveloping algebra of gˆ can be written succinctly as 1 T(z) = z−n−2L = : JaJa :(z) n 2(k + h∨) n∈ a XZ X where the sum is over an orthonormal basis of g with respect to the inner product given by the Killing form divided by 2h∨. Review: Afﬁne Kac-Moody Lie Algebras WZW Theories

Summary

This gives a conformal ﬁeld theory with central charge

k dim(g) c = . k + h∨ Through the OPE expansion, the primary ﬁelds acquire a composition Φλ ? Φµ making a Fusion ring or Verlinde algebra. This is discussed further in two lectures from the previous course on quantum groups:

Lecture 15 The Racah-Speiser algorithm. The Kac-Walton formula. Lecture 17 Afﬁne Lie algebras. The fusion ring. Modularity considerations are closely associated with the fusion rules, as we mention at the end of Lecture 17.