University of Alberta
Fixed Point Factorization and NIM-reps for the Affine Kac-Moody Algebras, and the Non-unitary W3 Minimal Models
by
Elaine Myra Beltaos
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Pure Mathematics
Mathematical and Statistical Sciences
©Elaine Myra Beltaos Fall 2009 Edmonton, Alberta
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms.
The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission. Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 OttawaONK1A0N4 Canada Canada
Your file Votre reference ISBN: 978-0-494-55802-7 Our file Notre reference ISBN: 978-0-494-55802-7
NOTICE: AVIS:
The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduce, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats.
The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission.
In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these.
While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis.
1+1 Canada Examining Committee
Terry Gannon, Mathematical and Statistical Sciences
Gerald Cliff, Mathematical and Statistical Sciences
Nicolas Guay, Mathematical and Statistical Sciences
Marc de Montigny, Faculte Saint-Jean
Mark Walton, Physics and Astronomy, University of Lethbridge I dedicate this thesis to my siblings, Andrew, Angela and Terry, and my parents, Lillian and Spyros. ABSTRACT
In this thesis, we find the modular invariants for the non-unitary W3 minimal models (which is the first nontrivial non-unitary modular invariant classification), and the fixed point factorizations and NIM-reps for the B, C and D afiine Kac- Moody algebras (the only other known case is for the yl-series, [26] and [15] respec tively). Using the fixed point factorizations, we are able to find remarkably simple expressions for the corresponding NIM-reps.
Keywords: affine Kac-Moody algebras, modular data, NIM-rep, modular invari ants ACKNOWLEDGEMENTS
I would like to first of all thank my supervisor Dr. Terry Gannon for his guidance and support (both mathematical and financial) throughout my research and the writing of this thesis. I wish to thank Dr. Gannon also more generally for his ongoing support during my graduate studies, and for generously sharing his mathematical and professional wisdom, for which I am very grateful. I am very grateful to my family for their support, which has come in many forms over the years and during the writing of this thesis, and especially thank you to my mother for her many great suggestions and strategic wisdom. Thank you to my brother Andrew and David Churchill for valuable help with LaTex, and to Hongtao Yang, the creator of the template which I used to write this thesis. I would also like to thank the Department of Mathematical and Statistical Sci ences at the University of Alberta for financial support during my graduate studies. Table of Contents
List of Tables
List of Figures
List of Symbols 1
1 Introduction 3
2 Background 6 2.1 Some Lie theory 6 2.1.1 Lie algebras 6 2.1.2 Lie algebra representations 7 2.1.3 Lie algebra characters 8 2.1.4 Affine Lie algebras 9 2.2 Conformal field theory 10 2.2.1 Quantum field theories 10 2.2.2 Modular data 11 2.2.3 Properties of modular data and the Galois symmetry 12 2.2.4 The modular invariant partition function 13 2.2.5 The S-matrix and Lie algebra characters 14 2.2.6 The fusion ring and simple-currents 15 2.3 Data for the algebras 17 2.3.1 The data 17 2.3.2 Proofs of Lemmas 2.1, 2.2 and 2.3 23
3 The Ws modular invariant classification 35 3.1 Modular invariant classifications 36
3.2 The modular invariants of {A2 © A2)W 37 1] 3.2.1 The A2 classification 37 {1) 3.2.2 The (A2 © A2) modular invariants 39 3.3 The modular invariants of W$ 41 3.4 The (A2 © A2)^ classification 45 3.4.1 The permutation invariants 46 3.4.2 The vacuum couplings 47 3.4.3 The simple-current extensions 48 3.4.4 The classification when J has no fixed points 50 3.4.5 The classification when J has a fixed point 51 3.4.6 The exceptional invariants 52
3.4.7 q = 8, 3 \ p, and HR = 1IL = {pp, pp") 52
3.4.8 q = 8,p=3l, and KR = JR(pp)UjR{pp") and KL = JL(pp)U MPP") 54 3.4.9 q = 8, 3 \ p, and 11= {{p,p),(p,p")} 55 3.4.10 q = 8, 3|p, and ft = {(JV,p), (JV,//') :i = 0,1,2} 57 3.4.11 The exceptionals at (12, q) and (24, q) 58
4 Fixed point factorization for the affine Kac-Moody algebras 59 4.1 The problem 59
4.2 The Ar ' fixed point factorization 60 4.2.1 The special case of the fundamental weights 61 4.3 The Cr fixed point factorization 62 4.3.1 The r even case 62 4.3.2 The r odd case 64 4.3.3 The special case of the fundamental weights 66 4.4 The Br ' fixed point factorization 69
4.5 The Dr ' fixed point factorization 72 4.5.1 The simple-current Jv 72 4.5.2 The simple-current Js when r is even 74 4.5.3 The simple-current Js when r is odd 81 4.5.4 The simple-current JVJS, r even 84 4.5.5 The full simple-current group (Jv, Js) when r is even 85
5 Applications of fixed point factorization to NIM-reps 86 5.1 TheNIM-rep 86 ( 1] 5.2 The NIM-reps for A r 88 1] 5.3 The NIM-reps for Cr 89 5.3.1 The r even case 89 5.3.2 The r odd case 93 5.3.3 Examples of B and £ 93 5.4 The NIM-reps for B^ 94 5.4.1 The case k odd 94 5.4.2 The case k even 95 5.5 The NIM-reps for D^ 95 5.5.1 The simple-current Jv 95 5.5.2 The simple-current Js when r is even 95 5.5.3 The simple-current Js when r is odd 97 5.5.4 The full simple-current group {JV,JS) when r is even 98 5.6 Using the Pieri rules: an example 99 6 Concluding remarks 102 6.1 Summary and discussion 102 6.1.1 The non-unitary W3 minimal models 102 6.1.2 Fixed point factorization 103 6.1.3 NIM-reps 105 6.2 Future work 106 6.2.1 Work related to fixed point factorization 106 6.2.2 Work related to the NIM-rep formulas 107
Bibliography 109 List of Tables
2.1 Simple currents and Qj, rj for the classical affine algebras 16
6.1 Fixed point factorization algebras for the classical affine algebras . . 106 List of Figures
2.1 The Coxeter-Dynkin diagrams for the affine Kac-Moody algebras 9 List of Symbols
Q Lie algebra 6
Xr simple finite-dimensional Lie algebra of rank r 7 P+(Q) dominant integral weights of g 7 L(X) highest weight module 7 Vp weight space 8 O weights of a representation 8 ch\ character of L{\) 8 p Weyl vector 8 R+ set of positive roots 8 K central element of g'1^ 9 Q^ affinization of g 9
Xr affinization of simple rank r algebra Xr 9 P\ set of level k highest weights 10 (R)CFT (rational) conformal field theory 10 $ set of primary fields 11 S, T S-matrix, T-matrix 12 t complex conjugate transpose 12 C charge-conjugation 12 * complex conjugation 12 ^ob> ^x,ti fusion coefficients 12 a Galois automorphism 13 ea(a), eCT(A) parity of a, A 13 Z(T) modular invariant partition function 13 M modular invariant 14 A Xr highest weight 14 Aj fundamental weight (usually affine) 14
Aj fundamental weight for Xr 14 XxilA characters at elements of finite order 14 mx(f3) multiplicity of (3 in L{\) 14 W(P) Weyl orbit of /? 14 fi+(A) dominant weights of A 15 Fus fusion ring 15 Ch character ring 15 j, J simple-current 15 Qj, rj rational numbers obeying simple-current symmetries 15
1 M\J] simple-current matrix 16 <5z(x) 1 if x e Z, 0 else 16 X[i], A+ [i] orthogonal coordinates (non-shifted and shifted) 17 t(X) n-ality (triality if n = 3) of A 17
(ai,..., ar) r-tuple with respect to orthonormal basis of W 23 0 the empty set 29 x =m y x = y (mod m) 31 n,p,q height of a representation 37
P++ p-shifted height n highest weights 37 0(A) simple-current orbit of A 38 o minimal primary 41 [Xfi] W% primary 41 m\n m does not divide n 42 m\n m divides n 44 T^L, T^R, TZ pp-couplings 47 JL, JR, J simple-currents coupling to pp 48 VL, VR, V set of pairs Xp, that couple to some KV 48 (f fixed point 59 21 \r 2 exactly divides r 75 Af NIM-rep 86 \I> matrix that diagonalizes the NIM-rep 87 £(M) exponents of a modular invariant 87 £ (Af) exponents of a NIM-rep 87 ord(A) order of A 88 B boundary states 88 mult(A) multiplicity of A 88 P+ set of fixed points of J 88 A^( X level t fusion product 99
2 Chapter 1 Introduction
The problems in this thesis arise from the study of the modular data of rational conformal field theories (RCFTs), although an understanding of conformal field theory (apart from a general familiarity) is not needed to understand the problems and obtain solutions. Conformal field theory has been deeply influential to many areas of mathematics since the mid-1980s (although it has been in existence since the 1950s), and one of those areas is algebra, as is the case of this thesis. A conformal field theory (CFT) is a highly symmetrical quantum field theory, and a rational CFT (RCFT) obeys a further finiteness condition. An RCFT attaches assorted data to each Riemann surface. One fundamental ingredient of an RCFT is a pair of matrices containing a great deal of information about the theory - these constitute the modular data of the theory. The modular data form a representation of SL2 (Z), the modular group of the torus, and the characters of RCFT are modular functions for SL2CI') corresponding the modular data (see equation (2.2.2)). A well-known class of RCFTs, and an important one to us, are the Wess-Zumino-Witten (WZW) models. These are RCFTs that arise from the study of string theory on compact Lie groups and have the mathematically-desirable property that all of their quantities can be expressed Lie-theoretically. We will discuss CFT in more detail in Chapter 2. For more on CFT, see [9]. We present three main results in this thesis. The first is the W3 minimal model classification, which is the first nontrivial non-unitary RCFT (modular invariant) classification. This focuses on the data the RCFT associates to tori. The second is the fixed point factorization for the affine algebras B)- , Cr , and D\- , which is, as of yet, an unexplained feature of their modular data. Our third result is using the fixed point factorizations to find remarkably simple expressions for the affine algebra NIM-reps, the data RCFT associates to the cylinder. In Chapter 2, we provide background material in the areas of Lie theory, confor mal field theory (in particular modular data), and relevant information about the classical affine algebras. In addition, we find formulas for the multiplicities of the fundamental weights which are needed for the fixed point factorizations. The proofs of the multiplicities use Freudenthal's formula. We will refer to Chapter 2 often - it is meant to be a reference chapter for this thesis. In Chapter 3, we find the modular invariants for the non-unitary minimal models.
3 The most famous modular invariant classification is that of A\ ' [6]; these modular invariants were found to fall into the well-known A-D-E pattern (and we will see that the general notation for modular invariants is influenced by this fact: they are said to fall into A, T>, and £-series). To find the W3 modular invariants, we use a method called the Galois shuffle [22] to link the W3 modular invariant classification with that of the affine algebra (A2 ® ^2)^, which is known1 [2]. The only previous nonunitary classification was the W2 minimal models by Cappelli-Itzykson-Zuber [6] which collapses to the unitary A\ ' classification for elementary reasons. All other nonunitary classifications, in particular the W3 minimal models, are much more subtle than the unitary ones, and introduce significant complications. This is our motivation for pursuing this classification. Fixed point factorization is a dramatic simplification of modular data at fixed points of simple-currents (Coxeter-Dynkin diagram symmetries). Our main result of this thesis is the fixed point factorizations for the affine Kac-Moody algebras in Chapter 4. Apart from this thesis, the only known fixed point factorization is for the A-series [26], and it was not known whether this was a feature of the >l-series only (the ^-series behaves differently in many ways from the B, C, and D-algebras (one example is their simple-currents)). We find a fixed point factorization expression at the fundamental weights (which is enough to show that it exists for all weights) for the B2 and .D-algebras and explic itly at all weights for the C-series. The reason for focusing on the the fundamental weights is that this is what we need for the applications to NIM-rep coefficients in Chapter 5 and for further work in this direction (for example, finding D-brane charges). The existence of the fixed point factorizations for all the classical affine algebras hints that something similar may happen for other RCFTs and is a starting point for further exploration of this so-far mysterious phenomenon. One direction of interest to us is that of RCFTs arising from finite groups as these have modular data that looks quite different from the affine Kac-Moody scenario (we discuss this more in Chapter 6). Finding out whether a fixed point factorization occurs here and how this case is different from the affine Kac-Moody fixed point factorizations may yield important clues about the meaning behind fixed point factorization. The NIM-rep is a basic quantity of an RCFT. The NIM-reps associated to WZW models on simply connected groups are easy: they are the fusion rings. In Chapter 5, we apply the results of Chapter 4 to find expressions for the NIM-rep coefficients for the WZW models on non-simply connected groups. To do this, we first find expressions for the so-called ^-matrix that diagonalizes the NIM-rep, and to our knowledge, this is the only place the ^-matrix is given explicitly in the literature. Even knowing \I>, explicit expressions for NIM-rep coefficients are still very difficult to find, and apart from this thesis, have been found only for the ^4-series [15], also using fixed point factorization. Our formulas turn out to be surprisingly simple and involve the fusion coefficients of the algebra, which are easy to calculate (the fusion coefficients being nonnegative integers that are closely related to tensor product
*at level (p,p') such that p and p' are coprime 2 at odd level k
4 coefficients through the Kac-Walton formula [30, 36]), and are intrinsically Lie- theoretic. They are so simple that they suggest that something systematic and universal may be occurring. The NIM-rep coefficients are conjectured in the literature to be nonnegative integers, and by the structure of our formulas, we find that they are at least half- integers!3 We investigate this further for a C-series example by using Pieri rules (or more precisely, the affine version, coming from the Kac-Walton formula) to calculate the fusion coefficients in the NIM-rep formula - this is worked out in the last section of Chapter 5. Future work would be to try to generalize this to all the fundamental weights to confirm positivity and integrality. An exciting and natural next step to the work of Chapter 5 is to find the D-brane charges for the WZW models on non-simply connected groups. Our NIM-rep formulas should interpret these as dimensions of modules of smaller rank algebras. We discuss this in more detail in our concluding remarks in Chapter 6, along with other directions for further work based on the results obtained in this thesis.
3In low-rank low-level experiments (using Maple 13), our formulas yield nonnegative integers.
5 Chapter 2 Background
2.1 Some Lie theory
Lie algebras originally arose through the study of Lie groups, which are manifolds with a compatible group structure, so that multiplication and inversion are differ- entiable maps. Since a Lie group is a manifold, we can take tangent spaces: denote by TeG the tangent space to G at e (the identity in G). What is special about TeG is that it has the structure of a Lie algebra. Furthermore, all of the local structure of G is contained in TeG. We are interested in Lie algebra theory associated with compact Lie groups; their Lie algebras are called reductive, which means they are the direct sum of an abelian Lie algebra with simple ones (simple is defined be low). Specifically we are most interested in the affine Kac-Moody algebras, which are the infinite-dimensional cousins of the complexifications of those simple Lie al gebras. The affine Kac-Moody algebras are a special class of infinite-dimensional Lie algebras (namely the Kac-Moody algebras) whose theory is mostly analogous to that of the finite-dimensional simple Lie algebras1. The association of Lie al gebras with CFT was first known through the Virasoro algebra [31], which is an infinite-dimensional Lie algebra that captures the conformal symmetry of a CFT. The association of simple Lie algebras and their affinizations with CFT is due to Witten [27] and comes from string theory on compact Lie groups (the so-called Wess-Zumino-Witten theories).
2.1.1 Lie algebras A Lie algebra Q is a vector space over a field F with a bilinear multiplication operation called a bracket [•], satisfying
[xy] + \yx] = 0; [x[yz\] + \y\zx}} + [z\xy]} = 0, (2.1.1) for all x,y,z € 9. The above identities are called anti-commutativity and anti- associativity (or the Jacobi identity), respectively (this definition assumes the char-
'There are more general infinite-dimensional Lie algebras (with more complicated theory), but these do not play a large role in this thesis.
6 acteristic of F is not 2). From now on, all Lie algebras in this report will be over C. The most important example is sfaiC), the 2x2 trace-0 matrices with complex entries, with Lie bracket the commutator [XY] = XY — YX for any X,Y £ shiC). The commutator bracket plays an important role in the theory. A subspace h of g such that [gh] C h is called an ideal, and a Lie algebra of dimension > 1 that has no proper ideals is called simple. A semi-simple Lie algebra is one for which [gg] = g; at least in finite dimension, a semi-simple Lie algebra is the direct sum of simple ones. The simple finite-dimensional Lie algebras are the building blocks for all the finite-dimensional Lie algebras. They were classified by Killing and Cartan and fall into the following four infinite families and five exceptionals: Ar for r > 1, Br for r > 3, Cr for r > 2, Dr for r > 4, Ee, E-?, Eg, F4 and G2. The nonexceptional algebras are isomorphic to easily described matrix algebras with the commutator bracket: Ar = s[r+i(C), Br = S02r+i(C), Cr = sp2r(C), and Dr — S02r(C). The subscript r denotes the rank of the algebra, which is the number of copies of 0(2 needed to generate the algebra (specifically there are 3r generators for a rank r algebra because sb is three dimensional). A presentation of each of these algebras in terms of the 3r generators is captured by its Coxeter-Dynkin diagram, a graph with r nodes. We will denote a general simple finite-dimensional Lie algebra of rank r by Xr.
2.1.2 Lie algebra representations A representation of a Lie algebra g is a Lie algebra homomorphism II : g —> gl(V), where V is a vector space. In other words, II is a matrix-valued linear map such that H[xy] = [JI(x)U(y)}, where the second bracket is the commutator bracket. The dimension of LT is the dimension of V. Equivalently, V is a '(left) g-module', where g acts on V by x.v = Tl(x)v, for x € g, v € V. This action obeys [xy].v = x.(y.v) — y.(x.v), and conversely any such g-module admits a representation of g. A representation II is called unitary if its associated module V has a Hermitian form (u,v) £ C such that (u,H(x)v) = (H(wx)u,v), where UJ is an anti-linear map on g, which is also positive-definite, that is, (u,u) > 0 for all a^O, An important example of a representation is the adjoint representation: define ad(x) : g —* g by ad(x)(y) = \xy\. Then ad : g —> gl(g) makes g into a g-module. A submodule (or subrepresentation) of a g-module V is a subspace U of V such that Q.U C U, and an irreducible module is one for which {0} and V are its only submodules. The following theorem describes how finite-dimensional Lie algebra representations decompose into irreducible modules. Theorem 2.1. Let g be a finite-dimensional semi-simple complex Lie algebra of rank r. Then any finite-dimensional module of g is completely reducible into a direct sum of irreducible modules. For each r-tuple A = (Ai,..., Ar), of nonnegative integers, there exists a unique unitary irreducible module L(X), and all irreducible modules are of that form. Let -P+(g) denote the set of these r-tuples A of nonnegative integers. The modules L(A) in Theorem 2.1 are called highest weight modules. In Section 2.1.4, we will find
7 that the characters of their infinite-dimensional analogues are related to RCFT. The definition of a Lie algebra character (see next section) is borrowed from the finite group character definition and is due to Weyl.
2.1.3 Lie algebra characters To define a Lie algebra character, we first introduce the Cartan subalgebra. Let g be a semi-simple Lie algebra. A Cartan subalgebra f) of g is a maximal subalgebra of g such that for every h £ h, ad(h) is diagonalizable. A Cartan subalgebra f) is ""notrstraii unique: as in choosing a basis, there are uncountably many different but equivalent ones for any semi-simple g. Let V be a finite-dimensional g-module, where g is a complex finite-dimensional semi-simple Lie algebra, and let f) be a Cartan subalgebra of g. The idea is that \) can be simultaneously diagonalized in any V. That is, given any functional j3 G [}*, its weight space is the simultaneous eigenspace
Vp = {v G V I h.v = (3{h)v V/i £ I)}, (2.1.2) so p fixes the eigenvalues of each h G f). Then V decomposes into a sum
V = ®0en(n)V0 , (2.1.3) where 0(11) is the set of all weights of n, that is all /3 £ t)* with Vp ^ 0. The Aj of Theorem 2.1 are the coordinates of a weight A G fi(II) with respect to a certain basis Aj of \)*\ the Aj are called the fundamental weights. The character chy of a Lie algebra representation II is a function on the Cartan subalgebra which maps h G f)* to a complex number. It is defined as
W chv(h)= Yl dim(^/3)e , (2.1.4) /3€f2(II) where h G f). If two representations have the same character, then they are equiv alent. Character arithmetic also works easily: if V and W are g-modules, then chv®w — chy + chw and chv®w = chychw- The Weyl character formula expresses the character of L(X) as an alternating sum over the Weyl group (a symmetry of the weights) divided by a product over half of the roots (the roots are the weights of the adjoint representation). Then
where z G i), p = Yll=i ^-i ls tne Weyl vector, R+ is the set of positive roots (these are given for each algebra in Section 2.3.2), and L(X) is the highest weight module with highest weight A. Figure 2.1: The Coxeter-Dynkin diagrams for the affine Kac-Moody algebras
2.1.4 Affine Lie algebras The Lie theory introduced thus far generalizes to the affine Kac-Moody algebras, which is our main interest. Twisted affine algebras also arise in this thesis, however here we talk about the nontwisted affine Kac-Moody algebras only (everything here has an analogue for the twisted ones) and discuss the differences for the twisted algebras as they appear. Let g be a finite-dimensional simple Lie algebra. We can construct an infinite- dimensional Lie algebra from it by taking its loop algebra £po(y(fl)) which is the x n algebra consisting of all Laurent polynomials ^2nez nt , where all xn G 0, and t is some variable. We want only finite sums, so we require that all but finitely n m m+n many xn are 0. Its Lie bracket is given by [xt , yt ]ioop = [x,y]gt . The loop algebra of g is infinite-dimensional, and has many projective representations (that is, representations up to a factor). In order to obtain Verma modules and highest weight modules analogous to those for the finite-dimensional simple Lie algebras, we add one more element K to the basis such that K commutes with everything in £poiy{9) - that is, [K, xtm] = [xtm,K] = 0 for all a; G g, m G Z. In other words, we are taking the (unique nontrivial) one-dimensional central extension of £poiy(s) to convert those projective representations into (true) representations. Finally, to have for example a nondegenerate invariant inner product, it is standard to (noncentrally) extend the algebra one more time by adding a derivation £o := t-^, which also commutes with g (a derivation is a linear map that satisfies the familiar product rule of derivatives). Thus we define
S(1) := -W0) © CK © Ci° , (2-1-6)
where the superscript '^'' specifies that the algebra is 'twisted' by an automorphism of order 1, that is it is nontwisted. The Coxeter-Dynkin diagrams of the affine Kac-
Moody algebras Xr ' are nearly identical to their Xr counterparts: one node (the 0th node) is added in the affine case. They are shown in Figure 2.1. The symmetries of these diagrams play a special role in RCFT as we will see in
9 Section 2.2. The affine Kac-Moody algebras also have highest weight modules M labelled by highest weights which are now (r + l)-tuples A = (Ao; Ai,..., Ar), and for every such representation, K is sent to some multiple k of the identity. The constant k is called the level of the representation - if the level of the representation is known, the Ao term is redundant and can be omitted. The highest weight representations here are not finite-dimensional. The unitary highest weight modules are those for which Aj € N: we let P+{Q^) denote the finite set of level k unitary highest weight modules.
Define the character chL^ := ch\ of the affine algebra module L(\), for A € P*(gW), exactly as in (2.1.4), except now the sum will be infinite. The Weyl-Kac character formula expresses this as an alternating sum over the affine Weyl group, divided by an infinite product over roots. The affine Weyl group consists of the finite Weyl group of g together with translations in a lattice, and because of this that alternating sum can be written in terms of classical theta functions. Theta functions are examples of modular forms, and indeed one of the coordinates of the Cartan subalgebra of g^1) plays the role of modular parameter T (see Section 2.2.2 below). The denominator is likewise a modular form, except that it is missing a 2mTC 2i simple factor. If that simple factor is included (as a function of T it is e l where c is a number called the central charge, depending only on k), then the resulting 'renormalized character' will be a modular function for some finite index subgroup of 5Z/2(Z). More important to us, the span of the characters will be invariant under the action of the modular group SL2 (Z).
2.2 Conformal field theory
A conformal field theory (CFT) is a quantum field theory (QFT) that is invariant under the conformal (angle-preserving) transformations; we normally study two- dimensional CFT as the space of conformal transformations in E2 = C is infinite- dimensional, and thus these CFTs are especially symmetrical. We begin this section with a short discussion of quantum field theories and then specialize to CFT.
2.2.1 Quantum field theories A physical theory contains states (which are descriptions of.a physical system at a time), and observables, which are physically measurable quantities, that is, what can be measured as the result of an experiment. In a quantum theory, the states form a Hilbert space Ti, and the observables are Hermitian operators that act on the state space H. The observables form a Lie algebra with the commutator bracket. The energy operator, or Hamiltonian H, plays a special role: the commutator of H with another observable A describes how A evolves with time. A classical field is a function of space and time, for example the gravitational field. A quantum field
10 which we locally identify with C. For each n G Z, putting f(z) = zn~x yields an operator
2.2.2 Modular data The chiral algebra V of holomorphic fields for an RCFT has an Lo-grading (where Lo is the energy operator H), and has finitely many irreducible modules A. The irreducible modules are labelled by primary fields, the set of which is denoted by $. The characters of those modules are functions of a complex number T £ H, the upper half plane of complex numbers with positive imaginary part:
c 24 L chA{r) = q- ' TrAq \ (2.2.1) where q = e2mT, and c is the central charge. The central charge describes how the conformal symmetry is softly broken by the CFT; algebraically how the chiral algebra carries a projective representation of the conformal symmetry (that is a true representation of the Virasoro algebra). V defines a module over itself; it is called the vacuum (which we denote by 0), and plays a special role. RCFT will exist on any Riemann surface: for each choice of genus g, there will be a finite-dimensional space of chiral blocks which build up all quantities that RCFT associates to that surface. These chiral blocks will be holomorphic, many-valued functions on the moduli space at genus g.
11 The most important example is genus 1 - the torus. Here the most interesting RCFT quantity is the partition function, which we describe in Section 2.2.4. The
chiral blocks are the characters chA of (2.2.1). They are holomorphic functions on H, and they carry a representation of 5Z/2(Z), the modular group of the torus, consisting of 2 x 2 matrices with integer entries and determinant 1. More precisely, define matrices S and T by the relations
s cM-1 A) = Yl ABchB(r); chA(l + T) = £ TABchB(T). (2.2.2) B B The matrices S and T generate a representation of SL2CZ); they are called the modular data of the theory. Modular data satisfy many properties which we will discuss below. Fortunately, neither RCFT nor VOAs are needed to understand the definition, properties, and most important examples of modular data. The matrices in (2.2.2) are the main example of modular data. A rich source of modular data comes from the affine Kac-Moody algebras at a fixed level k, where $ = P+(Xr '•) - this will be the most important example for us.
2.2.3 Properties of modular data and the Galois symmetry The entries of S and T are complex numbers. S is unitary (SS^ = I, where t denotes complex conjugate transpose) and symmetric, and T is diagonal and of finite order. 52 is a permutation matrix C called charge-conjugation; it associates a V-module to its dual. S and T satisfy
Sca,b = Sa£b = S*b (2.2.3a)
Tca,Ca = Taa , (2.2.3b) where the * denotes complex conjugation. In a unitary RCFT, we have
Soa >S00>0 (2.2.4) for all a e $. Equality occurs for primaries called simple-currents (more on this in Section 2.2.6); they also correspond to permutations of $.
The most important property of modular data is that the numbers N£b denned by Verlinde's formula
Kb = £ ^f^ (2.2.5) £l Sod are nonnegative integers. The N£b are called fusion coefficients. We can define fusion matrices Na with coefficients
(Na)bc := Kb • (2.2.6)
12 they multiply together in the following way:
NaNb = Y,N^Nc. (2.2.7)
Verlinde's formula tells us that the fusion matrices Na are simultaneously diagonal- izable and have eigenvector the 6th column of S with eigenvalues ^. One consequence of the integrality of the fusion coefficients (2.2.5) is a powerful Galois symmetry of the 5-matrix [7]. The entries of S lie in some cyclotomic field th Q[£nL where £n is an n root of unity (n can be taken to be the order of T). Let a be any Galois automorphism in GaZ(Q[£n]/Q). Then for any a,fee$,
°{Sah) = ea(a)Saa {±1}, and a i—> aa is some permutation of $. The quantity ea(a) is called the parity of a. For example, for a being complex conjugation, (2.2.3a) says that aa = Ca and all parities are +1. The symmetry (2.2.8) plays a major role in modular invariant partition function classifications (the modular invariant partition function will be denned next section).
2.2.4 The modular invariant partition function Let r € H. Z + TZ is a two-dimensional lattice, and the quotient C/(Z + rZ) defines a torus with one special point (namely the identity [0] = Z + TZ of the quotient group). Any torus is conformally equivalent to one of this form, and T,T' € H define conformally equivalent tori exactly when r' = ~^ for ( J 6 SZ,2(Z). With this in mind, define an action of SL2(Z) on r to be A.T = ^r^ for all
a b A=( c d)eSL2(Z). The modular invariant partition function
Z[T) = Y, MABchA(r)ch*B(r), (2.2.9) A,B for the torus is a fundamental quantity because it describes how the state space Ti of the RCFT decomposes into a sum (BA,BMA,B-A
13 of coefficients. M is indexed by the primary fields $ and should satisfy
Moo = 1, Mab > 0 for all a,5e$, MS = SM;MT = TM. (2.2.11)
The third condition is modular invariance and is (2.2.10), in the same order. An M satisfying (2.2.11) is called a modular invariant. The Galois symmetry of the S-matrix in (2.2.8), along with integrality of the Mab, yields the symmetry
MXll = ee(\)ee(fi)Mae{x)adfl) (2.2.12) of the modular invariant. This is a key tool in modular invariant classifications.
2.2.5 The .S-matrix and Lie algebra characters
Since we are dealing with both the simple finite-dimensional algebra Xr and its affinization Xr in this thesis, we introduce bar notation for when we need to distinguish them. Bars will be used in the following senses. If A = (Ao; Ai,..., Ar) € 1} P+(X^ ), we denote by A the weight (Ai,..., Ar) e P+(Xr). We will also use the notation A on its own to mean an Xr weight. More generally, we will use bars to denote finite-dimensional Xr quantities such as W to mean the Xr Weyl group as opposed to W for the Xr ' Weyl group. We denote by Aj, i = 0,..., r the Xi- ' fundamental weights and by A,, i = l,...,r the Xr fundamental weights. For convenience, we define Ao := (0,..., 0). A useful expression for the 5-matrix of a nontwisted affine Kac-Moody algebra X?] is due to Kac-Peterson [29]. Let A,/J€ P^X^). Then
^w{\ + p)-{ii + p) Sxp = K r'2s 2_] (detw)exp 2 (2.2.13) wew where W is the Xr Weyl group, ~p = (1,..., 1) is the Weyl vector, and K and s are constants depending on r and k which will be given for each algebra in Section 2.3. The 5-matrix is closely related to the Xr characters chj evaluated at elements of finite order. Comparing (2.2.13) with (2.1.5), we see that
Xx{») := ch-x (-2*i £±2) = |^ . (2.2.14)
Let L(A) be a highest weight module, and /3 be any weight. Define the multiplicity mj(P) of ~fi in L(X) to be the dimension of the module Vg in (2.1.2). Then (2.1.4) becomes m ex X\(IJ)= Yl x(^) P -2„^te) (2.2.15)
Define the Weyl orbit of a weight (3 to be W((3) := {w(3 \ w £ W}. An important fact is that the multiplicities mj(/3) are constant along Weyl orbits, that is, rnj((3) — mj{w(l) for all w G W. Thus we can write (2.2.15) more (conceptually) compactly
14 as 7 • (M + P) XA(AO = 51 mx(^) 51 exP -2TTZ (2.2.16) where Q+(A) is the set of dominant weights of A, that is,
n+(X) := {/? G 0(A) | 0 G NAi + • • • + NAr}. (2.2.17)
2.2.6 The fusion ring and simple-currents Let $ be the set of primary fields of an RCFT. For each a € $, formally define the symbol xa and the operation
%a * %b abxc (2.2.18)
where the N%b are the fusion coefficients (2.2.5). This defines an associative, com mutative ring with unity XQ (where 0 is the vacuum), called the fusion ring of the RCFT2. We will denote the fusion ring by Fus. Notice that the fusion matrices (2.2.6) satisfy (2.2.18), and so form a representation of a fusion ring. For an affine algebra X}- , <& is the set P+(Xr ) of level k highest weights. The characters (2.1.5) of a Lie algebra form a ring, called the character ring; we will denote it by Ch. It is generated by the characters at the fundamental weights, that is, for any A € P+(Xr),
c/lX = PA (<%!>• ••><%)> (2.2.19) where Pj is a polynomial. Equations (2.2.5) and (2.2.14) imply that the assignment chj i—> x\ extends linearly to a ring homomorphism F : Ch >—> Fus. Then Fus = Ch/I as rings, where I = ker(F). For example, for the algebra A)- ' at level fc, / is = the ideal generated by all chj such that A = YA=I ^i^i satisfies XX=i ^» k + I. Therefore, the fusion ring of I, is a direct homomorphic image of the character ring of Xr, and so we also have
x\ = PX(XA1,...,XAT) (2.2.20) for all x\ G Fus. Define a simple-current to be a primary j€$ for which there exists a permu tation J of $ such that Nla = 6b,Ja (2.2.21) with j = JO. The simple-currents are precisely those j for which SJO = Soo in (2.2.4). For each simple-current j such that j = JO, there exist a rational numbers Qj(a) for each a G $, and rj such that
2 J Sja,b = e ^ ^Sab (2.2.22a)
2For the axiomatic definition of a fusion ring see e.g. [23].
15 TjjToo = exp[27mv(n - 1)]. (2.2.22b) The numbers Qj and rj are given for each simple-current J for each classical affine Kac-Moody algebra in Table 2.1; they can be found in e.g. [13]. The simple-currents for the affine Kac-Moody algebras were classified by Fuchs in [10], and with the exception of Eg at level 2, they correspond to graph symmetries of the affine Coxeter-Dynkin diagrams in Figure 2.1 (the charge-conjugations also correspond to diagram symmetries). The simple-currents for a level k affine algebra form an abelian group. Remark 2.1. We refer to both the primary j and the permutation J as a simple- current, and for the remainder of this thesis, the use of the term 'simple-current' will refer exclusively to the permutation J. We define the order of a simple-current J to be the smallest positive integer n such that J™ A = A for all A £ P+(Xr '). For example, the diagram for A"2 ' can be considered to be an equilateral triangle, and the simple-currents form the cyclic group (J) = Z3, where J is the order 3 rotation of 27r/3 radians about the origin. The symmetry that reflects the graph about the 0th node is charge-conjugation C. The simple-currents will be given for each algebra in Section 2.3.1. Let J be a simple-current of order n. Define a matrix M\J] by its entries
n M[JU = E
9 J rj d kd J\f J (r -d) lA 2(r+l) r+1 2^=1 k An HT J 2 2 kr Of J 4 kZUii*
k Ar_i+Ar DP Jy 2 kr 2 - i V" i AlA- r-2 - D^ r even Js 8 2 2_,i=i i kr 1 v-^r-2 • x r \ D™ r even Jc ^Ar 8 2 2^i=l lAi ~ 1Ar-\ kr iA r-2 — A D^ r odd Js 8 2 53i=l i 4 Ar-1 l r
Table 2.1: Simple currents and Qj, rj for the classical affine algebras
16 2.3 Data for the algebras
In this section we include some information about the classical algebras involved and some results needed for their fixed point factorizations. In particular, we need the weight multiplicities at the fundamental weights, in Lemmas 2.1, 2.2, 2.3 below. Although weight multiplicities have been well-studied, we were unable to locate a proof of them in the literature, so we identify and prove them in Section 2.3.2 (for the fundamental weights). In many instances, we will be using the orthogonal coordinates of a weight A rather than its Dynkin labels: these are coordinates of A with respect to an orthonormal basis of W (W+1 in the case of the .A-series). We will also find it useful to work with the orthogonal coordinates of X+p. We denote the ith orthogonal coordinate of A by X[i], and the ith orthogonal coordinate of (A + p) (the 'p-shifted', or 'shifted' orthogonal coordinate) by \+[i\.
2.3.1 The data The algebra AP
A level k highest weight for A;-', r > 1, is an (r + l)-tuple A = (Ao; Ai,..., Ar) such that
A0 + Ai + --- + Ar = k. (2.3.1)
The Ar Weyl group acts on a weight by permuting its orthogonal coordinates; it is isomorphic to Sr+i. Define the n-ality (where n := r + 1) of a weight A to be
r t(A):=^iAi, (2.3.2) i=l and define orthogonal coordinates and shifted orthogonal coordinates by
r X[i] = Y1X* (2.3.3a)
A+[*]=r + l-t + A[i] (2.3.3b) for i = 1,... ,r + 1. Note that (as we will see shortly from the definition of the fundamental weights) these coordinates do not precisely describe A, but rather the part of A that is not invariant under the Weyl group - this is done for convenience. The fundamental weights are A» = Yl)=i ej~f+T Hi=i ej> f°r ea°h * = 1, • • •, r+1 (these can be found in [4]), and so we find the inner product on W+1 to be A • /Z =
17 £i=i WMi] - ^fr1- Then the S-matrix in (2.2.13) becomes r+l
5V - „-i. E <*<->«PI-*«E"^ - «(r +» l) i
(A PMM+ ) 1 (Ao; Ai,..., Ar) H-> (Ar; Ao,..., Ar_i). (2.3.6) The algebra B^ A level k highest weight for Br , r > 3, is an (r + l)-tuple A = (Ao; Ai,..., Ar) satisfying A0 + Ai + 2(A2 + • • • + Ar_x) + Ar = k. (2.3.7) The orthogonal coordinates (non-shifted and shifted) are given by r_1 A A[*] = Z>+y (2-3-8a) e=i \+[i]=r+±-i + \[i] (2.3.8b) for i = 1,... ,r. The Br inner product is A • /i = Yli=i ^M/4*]> K = k + 2r — 1, •r2 and s = ^j". The JBr Weyl group acts on a weight A by permuting its orthogonal coordinates X\i] and changing an arbitrary number of signs; it has order 2r • r\. Lemma 2.1 gives the dominant weights (and their multiplicities) of the highest weight representations L(Ai) (the fundamental representations). Lemma 2.1. Let 0 < n < r — 1. The dominant weights of the Br fundamental weight An are ,I,_nA„,A„.1,...,Ai,A0} »/0 and the weight multiplicities are: 18 (a) for any 0 < n < r: m-^ (Ar) = 1, (b) If n is even: r-2a m K (2.3.10a) An( 2a) •• — a forO By a similar calculation to (2.3.4), the S-matrix (2.2.13) becomes + + r r .A [i]M [j], SXfl = K~ lH -\ sin(27r- (2.3.12) I l J : (Ao; Ai, A2,..., Ar) H-> (AI; A0, A2,..., Ar). (2.3.13) The algebra C^ A Cr ' level k highest weight, for r > 3, is an (r + l)-tuple A = (Ao; Ai,..., Ar) of nonnegative integers A, such that Ao + Ai + h \r = k. (2.3.14) The inner product is A • /z = \ 531= 1 -M*]M*]> K — k + r + 1, and s = ^75. The orthogonal coordinates are AM = X> (2.3.15a) X+\i] =r+l-i + \[i] (2.3.15b) for 1 < i < r. The Cr Weyl group acts on a weight by permuting orthogonal coordinates and changing an arbitrary number of signs; it has order 2r • r\. Lemma 2.2 gives the dominant weights and multiplicities for the fundamental weights. 19 Lemma 2.2. Let 0 < n < r. The dominant weights of the Cr fundamental weight A„ are _ _ _ fi+(An) = ( {A.,An-2,...,A0} ifn is even +v n (2.3.16) > \ {An,A„_2,...,Ai} ifn is odd and the weight multiplicities are: (a) for any n, 0 < n ,x , 2(r-n + l) / r-2a ™-An(A2a)= n_2a (f_a_1 (2.3.17) for 0 < a < § - 1, (c) ifn is odd: , 2(r-n+l)/r-(2a + l) T (2.3.18) mIn(A2a+1) = n-(2„_,„„_,_/o + ,l)\Y-«- n_i 1 for 0 < a < ra-1 -1. The 5-matrix (2.2.13) becomes 2 \ 2 2 .A+[#+b1 sin hr (2.3.19) J l There is one nontrivial simple-current, J, which acts on a weight A as •^(Ao; Ai,..., Ar) = (Ar; Ar_i,..., Ao). (2.3.20) The algebra D^ A Dr level k highest weight, for r > 4, is an (r + l)-tuple A = (Ao; Ai, .., Ar) of nonnegative integers Aj such that A0 + Aj + 2(A2 + • • • + Ar_2) + Ar_i + Xr = k. (2.3.21) The orthogonal coordinates (non-shifted and shifted) are r-\ Af Ay—\ A[i] = ^A,+ (2.3.22a) A+[t]=r-t + A[i] (2.3.22b) for i = 1,..., r. The inner product is A • fi = YA=I -Mi]/^], K = k + 2r — 2, and • r(r—l) s = J—2—• The Weyl group acts on a weight by permuting orthogonal coordinates and changing an even number of signs; its order is 2r_1 • r!. The dominant weights and multiplicities of the fundamental weights are given in the following lemma. 20 Lemma 2.3. Let 0 < n < r. The dominant weights of the Dr fundamental weight An are {A„, A„_2, - - -, Ao} if n is even and 0 < n < r — 2 fi+(A„) = ^ {An, An_2,.. •, Ai} if n is odd and I < n fr-(2a+l) mJn(A2a+i)=[ n=1_a (2.3.25) forO The D^] 5-matrix (2.2.13) is A+[i]/u+[i] 5AM = a < det cos I 2ir l T 2 r r where a = K I2 i . The simple-current group is generated by Jv, which is of order 2 and acts on a weight A as Jv '• (Ao; Ai,..., Ar_i, Xr) I—> (Ai; Ao, A2,..., Ar_2, Ar, Ar_i), (2.3.27) and Js, which is of order 2 if r even and order 4 if r is odd, and acts as (Ar; Ar_i,..., AI, Ao) if r is even Js : (AQ; Ai,..., Ar_i, Ar) H-> . (2.3.28) (Ar_i; Ar, Ar_2,..., Aj, AQ) if r is odd We also have the simple-current Jc := Jv + Js when r is even. The conjugation (a conjugation is a graph symmetry fixing the 0th node) C\ acts on a weight A as C\ : (AQ; Ai,..., Ar_i, Ar) 1—> (AQ; AI,. .., Ar_2, Ar, Ar_i). (2.3.29) When r is odd, this is the charge-conjugation C in (2.2.3) (charge-conjugation for r even is trivial). 21 ,(2) The algebra A2r A level k highest weight for the twisted algebra A2J is an (r + l)-tuple A = (Ao; Ai,..., Ar) such that A0 + 2(Ai + • • • + Ar) = k. (2.3.30) The orthogonal coordinates of A are r (2.3.31a) \+[i] = r + 1 — z + A[i] (2.3.31b) for 1 < i < r. A formula for the S'-matrix is [16] + + T(T — 1) r .A [i]M [j] 5AM = (-l)"^2 K-5det sin I 2TT (2.3.32) l 2 The algebra A2 )_x l(2) The 5-matrix for the twisted algebra A2r-i at level k is not symmetric. The rows and the columns of S are indexed by (r + l)-tuples A = (Ao; Ai,..., Ar) and /x = (no; m,..., fj,r) respectively, such that A0 + Ai+2(A2 + --- + A7.) = k (2.3.33a) Ho + 2(/xi H 1- Hr-\) + P-r = k. (2.3.33b) The shifted orthogonal coordinates are given by r \+[i]=r + l-i + J2Xe (2.3.34a) e=i r-\ (2.3.34b) e=j The 5-matrix entries are[16] Sin 7T (2.3.35) l 22 2.3.2 Proofs of Lemmas 2.1, 2.2 and 2.3 Remark 2.2. Note that in this section, we will not use bar notation as throughout this section, we are dealing only with finite-dimensional algebras. In this section, we use FreudenthaFs multiplicity formula (see e.g. [14]) X fca ka mA A fca mA(A) = )2-(\ + )2 J2 J2( + ' ) ( + )' (2-3.36) <• P> ( + P) aeR+k>i which defines the multiplicity of a weight recursively in terms of the multiplicities of the 'higher' weights3. The ordering of the weights is as follows: denote the set + of positive roots of Xr by R , and the simple roots by on. We define an ordering >- on the roots to be A >- A if and only if A — A £ R+, that is, A — A is a linear combination Yli aiai °f simple roots such that each a, £ N. The roots, positive roots and fundamental weights can be found in Bourbaki [4]; they will be given for each algebra. We will let {e, | 1 < i < r} be an orthonormal basis for Er, and we use angle bracket notation to denote an r-tuple with respect to this basis, that is, a e (ai,...,ar) := Yn=i i i- The B-series There are r2 positive roots; they are R+ = {ei | 1 ei-ei+1 ifl The fundamental weights are v 1 5(ei + --- + er) if £ = r ' First suppose that A = Ao = (0,..., 0) in orthonormal coordinates. From + (2.3.38), we can see that R 6 W{\i) U W(X2). By Freudenthal's formula, we find that mAi(A0) = {A.+p^_{p)i (rmAi(Ai) + 2(2)mAi(A2)), for all 2 < i < r - 1. + In general, A, + R C W(Ai+2) U W(Ai+\) because each positive root a has ei ther one or two Ts in it, or a '1' and a ± '1' in it, so adding i Ts gives us Weyl orbits of the fundamental weights A^+i and Aj+2. We are interested in the set {A^ + ka | a £ R+, k > 1} for each A^. However, we need not consider any k > 2 as then A( + ka £ W(Aa) for any fundamental weight Aa and so the multiplicity in equation (2.3.36) would be 0 (recall from Lemma 2.1 that the dominant weights of a fundamental weight must also be fundamental weight). We are using the notation (A,JJ) here to denote the inner product A • fi in order to make the formulas easier to read 23 e + Let 0 < £ < r - 2. Define R := (At + R ) n (W(Ae+1) U W(Ae+2)). Then ^ _ f {Af + e, | £ + 1 < i < r} U {A r 2 2 (Ae + p) = £(A+[i]) i=l i=l i=e+l £ r = Y,{{r + \-i? + 2{r + l-i))+ £ (r + i - i)2 i=l i=t+l I r = 2^> + l-z) + £>+--i)2. Now let 1 < £ < r - 1 and 1< a < £. Then 2 2 (Ae + p) -(Aa+p) = 2 £ (r + l-i) i=a+l = 2^2(r + l-i-a) t=i = 2^(r + l-a)-2^i i=i t=i = 2(r+l-a)(£-a)-{£-a)(£-a + l) = (£-a){2r + l-£-a), 24 so 9 2 2 2 K (A/ + p) - (Aa + p) (e-a){2r + l-e-ay '' ' Let 1 < £ < r — 1, and let 0 < a < £ — 1. Then we have that the Freudenthal formula (2.3.36) at the fundamental weights is o mA,(Aa) = -—-————— £ (a,a)mA,(Aa + a), (2.3.43) V A a ' aefl -Aa where we write (a, a) instead of (Aa + a, a) since (Aa, a) = 0 ((Aj, aj) = <^j for all fundamental weights and 'co-roots' aY = ,2°^ •,). We can simplify (2.3.43) further. Define ^a := (i_a)(2r+i-i-a) • Let 0 — ^ — r — 2, and let 0 < a < £ — 1 (we will do the case £ = r — 1 separately). Then m,Ae(Ae) = 1, and «M£(Aa) = 7 = lt,a < ^ (ei,ei)mA£(Aa-|-ej) U=a+1 a+l = le,a I Yl mAe(Aa + ei) + 2 ^ 2mAf(Aa + e» + e^) I j=a+l a+l m A = Ha\ 5Z A«( a+i) + 4 53 mA£(Aa+2) I «=a+l o+l 25 and we have mAr_:i(Ar_2) = - Y2 (a,a)mAr_1(A7._2 + a) r 2 aeR - -Ar-2 = ^i{^r-i,er-i) + (er.,er))mAr_1(Ar_i) Now we are ready to prove the multiplicity formulas of Lemma 2.1 using (2.3.44). Let 0 < £ < r — 1, and let a < £ — 2. We know that the multiplicity mA£(A^) = mA((AM) = 1. Now let 0 < £ < r - 1, and let 0 < fc < £ - 2. Suppose that for all i > fc, m\e(Ai) is given by equations (2.3.10) and (2.3.11). Depending on the parity of £, k, the formulas differ slightly, we will assume first that both £ and k are even and prove the other cases afterwards. By (2.3.44), (2.3.10b) and (2.3.10a), we have mAe{Ak) = Jt,k ((r- fc)mAj,(Afc+i) +4( )mAe(Ak+2)) = (f - fc)(22r + 1^- ^ - /c) ((J - $ -J + 2(r - ^ - 1} (| - § -2l)) (r-fc)! l + 2(r-k-*=*) (^)!(r_fc_M)l 2r + l-£-k r — fc\ L _ k )' 2 2/ which is (2.3.10a) with £ = n and A; = 2a. Now suppose that £ is still even, but k is odd. We have the calculation ,» N 2(r-fc) ffr-k-\\ „. , ,,/ r-fc-2 mA^(Afc) = (/-fc)(2r + l-/-ib)U|-^J+2(r" ^U-^-l (r fc)! " V <^+*> - fc)(2r + 1 - £ - fc) (1=1=1 _ i)!(r _ fc _ i _ (t^il)! £ - fc - 1 2 (r-fc)! / t-k (£ - k)(2r + 1 - £ - fc) (r _ fc _ i _ (Iz^ll _ i)i(*=|=il _ i), \£ - k - 1 2 (r - fc)! 27+1^7^1 (r _ k _ ! _ (£z^i))!(l=|=i)i (r-fc)! (r _ k - ££il)!(£=4=i)! r — fc\ l-fc-i )' 2 / which is (2.3.10b) with £ = n and fc = 26 + 1. 26 If £ is odd, £ < r — 2 and A; is even, then m^hk) = «-fc)(2r + l-/-fc)U ^ J+2(r-*-1\*=§=I-l 2 (r - it)! (l + (^_fc_l)) (£- fc)(2r + 1 -*- fc)( r _ k _ i _ (^11)1(^1)! 1 (r-fc)! r + 1=|=4 (r - Jfe - 1 - i^zi))!(l=|=i)! (r - fc)! r — fc" l-fc-i 2 / which is (2.3.11a) with £ = n and k = 2a. Finally suppose £ is odd, £ < r — 2, and fc is odd. Then 2(r-k) ((r-k-\\ n, , fr-k-2 m Ak) +2(r fe 1) ^ = (€-fc)(2r + l-f-fe)U^-i; - - C^-l ( fc)! ^ (l+2(r-fc-^*))) (* - fc)(2r + 1 - £ - k) {r _fc _ M)!(^ - 1)! (r - fc)! r — fcN 2 which is (2.3.11b) with £ = n and k = 2b + 1. Therefore, we have proven the multiplicity formulas in Lemma 2.1. It remains to show that the dominant weights of A^ are as in equation (2.3.9). To show that £}+(Ar) = {Ar}, suppose that A G fi+(Ar). Then Ar — A G Nai + • • • + Nar, where the Dynkin labels Aj of A are nonnegative integers (we are looking for the dominant weights of Ar), and the a« are the simple roots given in equation (2.3.38) and are a basis for all roots. Therefore for A G $l+(Ar), we must have Ar-A=(i-A[l],...,i-A[r]), which must equal a\a\ + • • • arar = {a\,a2 — 01,03 — Q4,... ,ar — ar_i), with Oj G N for all i = 1,... , r. Since a\ G N, we must have \ — A[l] G N, which implies that A[l] = Ai + • • • 4- Ar_j + 4f < \. The only possibility is Ai = • • • = Ar_i = 0 and Ar = 1. We cannot have all Dynkin labels including Ar, equal to 0, because then \ - A[l] = \ 0 Z. Therefore, 0+(Ar) = {Ar}. 27 Now suppose n < r — 1, and that A € fi+(Ara). Then we must have A„ — A = (oi, a-2 — a\,...,ar — ar_i). In other words, we have the equations 1 - X\i] = The C-series There are r2 positive roots; they are R+ = {lei I 1 < i < r}\J{ei-ej \ 1 < i < j < r}U{erfe,- | 1 < i < j < r}, (2.3.47) and the simple roots are (ei-ei+1 if 1 < z < r - 1 1 K \ 2er tfi = r. ' The fundamental weights are Ai = ei + • • • + eu (2.3.49) for each i = 1,... ,r. As in the 5-series argument above, we can see that taking k > 2 in (2.3.36) will yield a set of weights such that none is in any Weyl orbit of a fundamental weight (and adding a fundamental weight to those roots will make them even further from fundamental weights), so we consider only k = 1. Therefore, (2.3.36) becomes o a a m mAl(Aa) = + 2 _ + 2 Yl ( ' ) A((A« + o), (2.3.50) where we take the inner product (a, a) instead of (Aa + a, a) because Aa is orthog onal to a for all fundamental weights Aa and roots a. We can see that adding a 28 + fundamental weight Aa to each element of R , and then discarding those weights that are not in the Weyl orbit of any fundamental weight will yield a set of weights that is a subset of W(Aa+2). More precisely, define a + R :=(Aa + R )DW(Aa+1). Then for each a = 0,..., r — 2, we cannot add Aa to roots a that have a '2' anywhere, or that have at least one '1' anywhere in the first to atb component, and still have Aa+a in the Weyl orbit of Aa+2- Because for every root ej+ej such that Aa+ei+ej e r a W(Aa+2), Aa + ei — ej is also in W(Aa+2), there are 2( ~^ ) — (r — a)(r — a—l) such roots. Note that i?r_1 = Rr = 0. To calculate the denominator in (2.3.36), we write down the p-shifted orthogonal coordinates of the fundamental weights: they are A + r-i , • A r-i f r + 2-i if i < I A+[*]=r + l-t + A/[t] = | r + 1 i if i > and so (Ae + P? = £(A,+ [i])2 = £(r + 2-i)2+ ]T(r + l-z)2 I r = £(l + 2(r + l-i)) + 5> + l-i)2 i=\ i=l e t 2 = £(2r + 3)-2^ + (A0+p) 2 = £(2r + 2 - £) + (A0 + p) . Then we have (A/ + pf - (Aa + pf = (£ - o)(2r + 2 - a - £). Define 7^a := (i-a\(2H-2-a-i) • Then at the fundamental weights Ae, I < r - 2, (2.3.36) becomes mAe(Aa) = 7f,Q ^2 (^,a)mAe(Aa+a) a aeR -Aa = 2-ye,a(r - a){r - a - l)mhl(Aa+2) We will now prove the multiplicity formulas (2.3.17) and (2.3.18) by induction. 29 Assume that the dominant weights of a fundamental weight are given by equation (2.3.16) (we will prove that this is the case afterwards). We know that m\e(Ae) — 1. For the base case, suppose a = £ — 2. Then by (2.3.51), mKl{k<,_2) = 2{r-£ + 2)(r-£ + l)ye,e-2mAe(Ae) = r-i + 1, and according to equations (2.3.17) and (2.3.18), we have that ,A ^ / S^Ci"2) if* is even "»A«(A,-2) = (24^+3J if£.sodd = r-i+1. (2.3.52) Therefore we have the base case. Now suppose that for all k such that a + 2 < k < £ — 2, mA,(Ajt) is given by the formulas (2.3.17) or (2.3.18), depending on whether or not £ is even. Suppose first that £ is even. Then by (2.3.51) and (2.3.17), (r — a)(r — a — 1) nr — £+lfr — a — 2 mAe(Aa) - 4(£_a)(2r + 2_a_f)-2£_a_2^_2 2 _ r-i + 1 (r-a)\ 2r + 2-a-£(^a)!(r._a_(£^))! r-e+1 (r-a)\ 2 {r + l-a- {^){£ - a) (^ _ !)!(,- _ - £=°2)! 2—A4- "•; V 2 V:V' a" 2 r — £+ 1 / r — a 2? £-a V^-l which is (2.3.17). Similarly, for £ odd, we get ... „ 30 a\,..., ar e N. In orthogonal coordinates, this becomes the system of equations 1 - X[i\ = at - aj_i (2.3.53) -X[j] = aj - aj-i (2.3.54) -A[r] = 2ar - ar_i (2.3.55) for 1 < i < £ < j < r, and where ao := 0. We proceed similarly as we did for the S-series. Equation (2.3.53) implies that A[l] £ {0,1}. If A[l] = 0, then Ai = • • • = Xr = 0, so A = Ao. Then all orthogonal coordinates A[l] = • • • = A[r] = 0, so (2.3.53) becomes the system 1 = Oj - di-i 0 = a,j — Oj-i 0 = 2ar — ar_i for l Continuing this reasoning, we find that An £ Q+(Ae) if and only if £ =2 n (where by x =m y we mean x = y (mod m)). If n = r, then equation (2.3.55) implies that A[r] = 1, so A = Ar. Therefore we have shown that the dominant weights of A( are as in equation (2.3.16), and so Lemma 2.2 is proven. D The .D-series The simple roots are e, - ei+i if 1 < i < r - 2 er_i + er if i = r — 1 (2.3.56) ( er_i + er if i = r There are r(r — 1) positive roots; they are R+ = {et-ej I l The fundamental weights are ei H Yei if 1 < % < r - 2 { (2.3.58) |(ei H |-er_i-er) if i = r - 1 |(ei H |-er) ifi = r 31 To find a simplified Preudenthal's formula for m\t(h.a) at the fundamental weights Ag such that 2 < t < r — 2 and £ =2 a, we first calculate the factor ,. \2^/A + 12- The shifted orthogonal coordinates of A^ such that £ < r — 2 are given by A|H = {r-i + 1 ^f^ • (2-3.59) 1 I r — 1 n£+l 2 {At + pf = X>+[i]) i=i i=l i=C+l r e e = 5>-i)2 + (2r + l)£-2£i i=\ i=l i=l = (A0 + pf + £(2r + 1) - 1(1 + 1) 2 = (A0 + p) +e(2r-£), and so : 2 2 = lLa (Ae + p) -(Aa + p) (2r - £ - a)(t - a) a As usual, we take only k = 1 in (2.3.36). Define i? := (Aa + r+) n VF(Aa+2) for eachO mA£(Aa) = 7^>a ^2 (a,a)mAe(Aa + a) a a€R -Aa = 1li,a(r - a)(r - a - 1 )mA<(Aa+2) 4(r - a)(r — a — 1) m (A ). (2.3.60) (£-a)(2r-£-a) A£ a For any weight A, m\(X) = 1, so suppose A^ is such that 2 < £ < r — 2, and 0 < a < I — 2. We will show that mAt(Aa) is as given in equations (2.3.24), (2.3.25), by induction. We will assume that the dominant weights of At are given in (2.3.23); the proof follows the proof of the multiplicity formulas. First suppose that a = I — 2. Then mA£(Aa) = mAe{Ae-2) = (?-£ + 2)mAe{Ae) =r-£ + 2, which satisfies (2.3.24) if t is even and (2.3.25) if I is odd (in either case we find +2 that mAe(Ae==2) = C~i ))- Now suppose that (2.3.24), (2.3.25) are true for all 32 a + 2 < % < £. Then in either case, (2.3.60) implies that /. x 4(r - a)(r - a - 1) (r — a- 2 mAe(Aa) = (l - a)(2r - t - a)'I \ ^ - 1 (r - a)(r - a - 1) (r-a-2)! (*?)(»•-^) (r-a-1-^)1(^-1)! (r - a)! r — a\ e-a J' 2 / which is (2.3.24), (2.3.25). To find the dominant weights of the fundamental weights, consider first the cases of Ar and Ar_i (given in equations (2.3.58)). If A G fi+(Ar), then we have the system of equations - - X[i] = di-di-i (2.3.61) -— A[r — lj = ar + ar-i — ar_2 (2.3.62) for 1 < i < r — 2, and i = r, where the orthogonal coordinates A[i] are given in (2.3.22a). Equation (2.3.61) with i = 1 implies that A[l] is a half-integer such that A[l] < ^. Since all the Dynkin labels Aj are nonnegative, \[i] > 0 for all 1 < i < r — 1. Therefore, the only possibility is A[l] = 5, which implies that Ai = • • • = Ar_2 = 0, and one of Ar_i, Ar is 1, and the other is 0. If Ar = 1 and Ar_i = 0, then A = Ar, which clearly is a dominant weight of Ar with all coefficients a^ equal to 0. Suppose Ar_i = 1. Then A[l] = • • • = X[r - 1] = \, and X[r] = -\. Then equations (2.3.61) become a\ = • • • = ar_2 = 0, ar 4- ar-i = 0, and ar — ar-\. This is a contradiction since ar must be a nonnegative integer. Therefore, J7+(Ar) = {Ar}. Similarly, we find that A G Ar_i implies that A e {Aj._i,Ar}. We have a system of equations similar to (2.3.61) except that the case i = r becomes —\ — X[r] = ar — ar_i. If A = Ar, then X\i] = | for all 1 < i < r, and equations (2.3.61) yield a^ = 0 for all 1 < i < r — 2, ar + or_i = 0, and ar — ar-\ = —1, which is a contradiction. Therefore, fi+(Ar_i) = {Ar_j}. Now suppose that A( is a fundamental weight such that 1 < £ < r — 3. Then, for A € n+(A(), we have the system of equations 1 - X[i] = at- ai_i -X{j] = aj-aj-i —X[r — 1] = ar_i — ar-2 + o.r —X[r] = ar — ar-\ for 1 < i < £ < j < r — 2, and j = r. If A( is the £lh fundamental weight such that 33 1 < £ < r — 2, then a similar argument to the C-series one yields dominant weights in (2.3.23). Therefore we have proven Lemma 2.3. D 34 Chapter 3 The W3 modular invariant classification In this chapter, we find the modular invariants for the non-unitary W3 minimal models (the unitary W3 modular invariants were found by Gannon-Walton in [25]). The WN minimal models (N > 2) comprise a reasonably accessible class of chiral algebras for RCFTs - the best-known being those at N = 2 (the so-called Virasoro minimal models [6]). The WN models were introduced by A.B. Zamolodchikov [37] and can be realized by the Goddard-Kent-Olive (GKO) coset construction [28], or the quantum DrinfeFd-Sokolov reduction [5]. Prom the VOA point of view, the GKO coset construction is taking the commutant of a subVOA in the parent VOA - it is explained in detail in [33] (Remark 6.6.24). Most attention in the literature has been paid to unitary RCFT, but unitarity is not required to have physicality. For instance, the statistical mechanical realization of RCFTs emphasizes that the requirement of unitarity has no physical justification [9]. This is the first nontrivial non-unitary classification, as the W2 classification reduces to the unitary case. In a non-unitary RCFT, the vacuum column of the S'-matrix is no longer positive. There is, however, a positive S-matrix column (by Perron-Frobenius). We denote the primary such that this is the case by o, that is Soa > 0 for all a £ $. The properties that are possessed by the vacuum primary in the unitary case are now distributed between the vacuum 0 and this primary o. The modular invariant classifications focus on the vacuum and rely heavily on the fact that Moo = l1; m a non-unitary modular invariant classification, about half of those tools are lost unless we also know that M00 = 1. The usual proofs break down here, and we can have infinitely many modular invariants for a fixed modular data. This is why so little is known about non-unitary classifications, and this is our motivation for pursuing the W3 (non-unitary) classification. The WN algebra is a vertex operator algebra generated by the vectors Wi, i = 2,..., TV, of conformal weight i (w 1The proof that there are finitely many modular invariants for a fixed modular data for example, relies on this fact. 35 appearance of the Virasoro algebra). The detailed construction is not important here - what matters to us is the modular data for WN (which for N = 3 is given explicitly in Section 3.3 below). By 'minimal models', we mean levels p, q for which the W3 symmetry suffices to have rationality (typical W3 non-minimal models won't be RCFT). In general, the modular data for WM modular data looks very much like that for (AN-\ © yljv-i)^\ and the fact that the latter classification is known for N = 3 at coprime levels [2] is what allows us to find the W3 minimal models. The two main tools here are the Galois shuffle [22], which essentially unitarizes the problem, and Lemma 3.2 below, which identifies a W3 modular invariant with an (A2 © A^)^ modular invariant. 3.1 Modular invariant classifications Recall the definition of modular invariants in Section 2.2.4. The modular invariants 1] (1) 1] l of for example A[ ', [Ax © ,4i) , A2 , and (for coprime levels) (A2 © A2p > are known. Surprisingly, for these affine algebras almost every modular invariant is built up in natural ways from the Dynkin diagram symmetries (recall that these include simple-currents). The general method which has evolved over the years for their classification follows these basic steps: Step 1. It is most convenient to first find all M obeying the condition M0x = Sox for all A 6 $ . (3.1.1) The M in (3.1.1) are necessarily permutation matrices: they have exactly one '1' in each row and column. For that reason, the M of (3.1.1) are called permutation invariants; their importance is that multiplying by them sends modular invariants to modular invariants. Step 2 (a). The vacuum row and column of a modular invariant M are heavily constrained, most significantly by (3.2.3) and (3.2.4). In this step we solve those constraints. It turns out that (usually) these A are just simple-currents - we call any levels where the vacuum row or column of a modular invariant involves pri maries other than the simple-currents exceptional. The modular invariants at the exceptional levels that do not obey (3.1.2) (see Step 2 (b)) are called exceptional invariants and are handled in Step 3. Step 2 (b). Find all modular invariants M obeying the condition MQ\ ^ 0 or M\o =£ 0 => A is a simple-current . (3.1.2) The M satisfying (3.1.2) correspond to simple-current extensions of the chiral alge bra. The point is that the modular invariants of (3.1.2) (and (3.1.1)) are the most accessible, and that they exhaust almost all modular invariants. Step 3. At small levels, modular invariants that do not obey (3.1.2) occur. These are precisely those M for which M$\ ^ 0 or M\o ^ 0 for some primary A that are 36 not related to the vacuum by a simple current. These exceptional invariants must be classified separately. (1) 3.2 The modular invariants of (A2 0 A>) 1} 3.2.1 The A2 classification As mentioned above, we approach the W3 algebra through the (A2 © ^2)'^ affine algebra. The (A2 © ^2)^- data.is built, up from the A2 data in a straightforward way, hence we begin with the problem for A2 • This was done in [20]; we briefly review the result here. Let n := k + 3 (this is the K of Chapter 2); we will refer to n as the height of the algebra. The set of level k A2 highest weights can be identified with the set P?+ = {A = (Ai,A2) G Z|0 : 0 < Ai, A2, Aj + A2 < n}, (3.2.1) where we have shifted the weight (Ai, A2) by the Weyl vector in order to simplify calculations in this chapter. In this notation, the weight p = (1,1) corresponds to the vacuum 0. The T matrix is given by n ' .A? + AiA2 + A^-n T-"( ) Zni <5A, • (3.2.2) 3n M The explicit formula for the S matrix S^ can be found in, for example, [20]. The condition that M commutes with the diagonal matrix T is easy to solve: it becomes the selection rule Mv ^ 0 =» Txx = Tm . (3.2.3) Commutation of 5 (see (2.2.13) for the S'-matrix of an affine Kac-Moody algebra) with M in addition to the Galois symmetry (2.2.8) yields the parity rule MXil / 0 =• e/(A) = e/(/x) . (3.2.4) The order 3 simple-current J is given in (2.3.6). It acts on the p-shifted weight A G ft+(4X)) M follows: J(A1,A2) = (n-A1-A2,A1), (3.2.5) and the order 2 charge-conjugation acts as C(A1,A2) = (A2,A1). (3.2.6) The simple-current and charge-conjugation obey equations (2.2.3a) and (2.2.3b), as 37 well as the symmetries q~i(n) 2 (n) 1 a a exp J x,J fi ~ — (a n-at(X)) T A/i (3.2.7a) 2m An) ?(n) J«A,J»M = 6XP (bt(X) + at(fi) + nab) 'A/i (3.2.7b) where triality t(X) := Ai — A2. The quantities Qj and rj in Table 2.1 are thus Qj(X) = 2^, and ry = ^. A simple calculation shows that a t(J X)=3na + t(X). (3.2.8) The Galois parity n in (2.2.8) satisfies WA^W^^I +1 if {tM}n + {£X }n < n 2 (3.2.9) H Wet W-\ _i if {Al}n + {a2}n > n for any I coprime to 3n, where {x}m is uniquely defined by 0 < {x}m < m and x =m {x}m. Lemma 3.1 below plays a major role in the W3 classification. Part (a) is the more important statement, as it provides the solution to Step 2 (a) of Section (3.1). 3 3 Lemma 3.1. (a) [20] Suppose X e F™+ satisfies T^f = T$ and, for all I coprime to 3n, e'f'(X)M)i = q\p).»/ Define OX := {A, JA,J2A}. Then: (i) forn=4 1,2,3, n ± 18: Ae Op, (ii) forn^iO, n/ 12,24,60: AeOpUO^,^), (tit) n = 12: A e Op U 0(3,3) U 0(5,5), n = 18: Ae OpU 0(1,4) U 0(4,1), n = 24: Ae 0pU 0(5,5) U 0(7,7) U 0(11,11), n = 60: XeOpU 0(11,11) U 0(19,19) U 0(29,29). (b) [32] Suppose n is coprime to 6. Then e™ (A) = e™ (K) for all I coprime to 3n, iffneOX. The modular invariants for A2 were classified in [18, 20], and are building blocks for those of both the (A2 © A2)^ and W3 algebras. The A2 modular invariants consist of M — I (denoted An), M = C (denoted -4*), the simple-current invariant M[J] in (2.2.23) denoted by T>n and given by (AOAji 0fi,JntWX (3.2.10) 38 when n is coprime to 3 and 1 if 11 = J* A for some i, t(X) =3 0, and A ^ (|, |) (Dn)Xil={ 3 if A = /i=(f,f) (3.2.11) 0 otherwise when 3 divides n, the matrix product CT>n =: £>*, and the exceptionals (given in the partition function notation of (2.2.9)) 2 2 2 £8 = \chp + c/i(3i3) | + \ch{h3) + ch{4t3) | + |c/i(3)i) + ch{3A) | 2 2 2 + 1^(3,2) + Cfyl,6)| + |c/l(4,l) + c/l(i;4)| + |c/l(2,3) + c/l(6,i)| 2 £l2 = \chp + c/l(ljl0) + c/l(10,l) + c/l(5>5) + c/l(5i2) + c/l(2,5) | 2 +2 |c/i(3i3) + ch{3}6) + c/i(6i3)| 2 2 ^12 = \chp + c/l(io,l) + c/l(lii0)| + |c/l(3,3) + C/1(3,6) + ch(6,3)\ + lcfyl,4) + cty7,l) + c^(4,7)|2 + lc^(4,l) + CV,7) + c^(7,4)|2 2 2 + |c/l(5,5) + c/l(5,2) + c/l(2l5)| + 2|c/l(4]4) | + (c/l(2,2) + c/l(2,8) + c/l(8,2))c/l(4)4) + c/l(4)4) (c/l(2i2) + c/l(2,8) + c/l(8]2)) ^24 = |c/lp + c/l(5j5) + c/l(7i7) + c/l(11>11) + c/l(22,l) + c/l(lill) 2 + c/i(i4]5) + C/l(5,i4) + c/l(11)2) + Ch(2,ll) + C/1(10,7) + ch(7yl0)\ + \ch(16,7) + Ch(j^6) + Ch(ie,l) + c/l(l,16) + c/l(nj8) + c/l(8ill) 2 + c/l(n,5) + C/I(5>1i) + C/1(8)5) + c/l(5)8) + Cft(7)1) + c/l(lj7)| : an( : at heights n = 8,12,12,24, as well as the matrix products C£8 = £| ^ C^'12 = c- 12- {1) 3.2.2 The (^42 © M) modular invariants The (A2 ©^2)'^ modular invariant classification at coprime levels (p, q) was found in [2]2. In this section, we very briefly review the argument and state the results of that classification. Let p, q > 4 be integers (we shortly will require them to be coprime). We w l abbreviate \A2 © A2) at height (p,q)' by (A2 © A2)^\ A highest weight for (A2 © A2yPtl is a pair (A,/i) =: A/i, where A e P++ and /x € P++ - let P++ denote l the set of these A/i. There are nine (A2 © A2)^ simple-currents, namely J K^ in obvious notation. ( The (^2 © A2) pl modular data, which we will denote by S^ and T^\ are given by the matrix tensor products of the A2 ^ and A^ J data: S^l = 5«S 2Some of the exceptional invariants were not found, but see Section 3.4.6 39 From (3.2.12) we find the Galois parities e^'9)(A//) = ef\\) e^\n). Because of (3.2.12), (3.2.13), some of the modular invariants for (y^©^)^ come from those of A2 . In particular, if M^,M^ are modular invariants for A2p, A2', respectively, then the tensor product M^ ® M^ will be one for (A2 © ^2)^,9 • This construction does not exhaust all of them, however: the simple-current invariants M\JK±1] for example are not in that factorized form. Indeed, the modular invariant classification for (yb ©^2)^ at arbitrary height (p, q) would be difficult to obtain (and probably not terribly interesting). Instead, we restrict to the much easier case where p, q are coprime. The main reason for this is that it is all we need to do the W3 minimal model classification (given next section), which is our main interest. Let M be any modular invariant of (A2 © ^b))*,? with p, q coprime. Then the Chinese Remainder Theorem implies that the parity rule (3.2.4) is equivalent to P) MA/li(0, ^ 0 => both e^(A) = 4 (K) and 4?V) = #\") , (3-2.14) for all £ coprime to 3p and all £' coprime to 3q. A slightly weaker version of (3.2.3) is 3 MV,K[/ ^ 0 =» both (T^f = (T^) and {T$f = (T^f . (3.2.15) Hence Lemma 3.1 applies directly, and this is the key to the (A2 © A2)p,q classifica tion accomplished in [2]. The resulting (A2 © ^2)p,9 classification contains nothing unexpected: q 1 (i) (i) the tensor products M^®M^ \ where M ^,!^^, resp., are arbitrary .4^, A2\ 9 modular invariants, (ii) the two simple-current invariants M(JK±l), and their conjugations (C% (Hi) matrix products of (i), (ii) by the permutation invariants (for example charge- conjugations), (iv) when p =3 1 and q = 8, the exceptional ch / . Ji\K±*p + chjiXK±i(3t3) xePl+ 2 i=0 ±l = M[JK }(Ap®£8) ±l and its conjugation M\JK ] {A*p ® E8). When instead p 8 and q =3 1, ±l these become M^K] (£8 ® Aq) and M{J K\ (£8 ® A*). 40 3.3 The modular invariants of W3 In this section, we introduce the W3 problem and discuss its relationship with the (A2 © A2)^ classification. We then find a formula (Lemma 3.2) that associates an {A2 © A^)^ modular invariant to a W3 modular invariant. The W3 minimal models have primaries [A/i] = {(A, /i), (J\, Kn), (J2\, K2fi)}, (1) where A £ P£+, /i £ P\+. That is, a W3 primary is a 'JK-orbit' of an (A2 ©J42) primary. Since gcd(p, q) — 1, we can assume that one of them, say q, is not divisible by 3, and because of equation (3.2.8), we can assume that a W3 primary [A/i] has t(n) =3 0. Thus the subset q P^:={(X,^)£Pl' +\t(„)^30} (3.3.1) of (A2 © ^2)^ weights can be put into one-to-one correspondence with the set of W3 primaries by associating A/i £ Pfy9 with the W3 primary [A/i]. The Wz modular data is given by the formulas _27ri*W*(") + W(K)' S[XII],[KU] = apq ^P S^MSWP) (3.3.2) 3 .(pX-qn)2' (3.3.3) pq fjl q n 1(1) where apq and (3pq are constants, and S / ' is the S' ' matrix for A2 n formally evaluated at the fractional height n = 2. W3 minimal models are unitary if and only if \p — q\ =1. Define the conjugation Cfy , for a, b £ {0,1}, to act on [A/i] £ Pfy? by C^([A/x]):=[CaA,CV] (3.3.4) where C is the A2 charge-conjugation (3.2.6). These C^> define W3 permutation invariants in the usual way, and thus multiplying by them sends a modular invariant to another modular invariant. The most important difference between the non-unitary W3 and (A2@A2)^ data is that the W3 S-matrix does not have a strictly positive vacuum column, that is, S does not satisfy S[\^[pp] > Sipp]ipp] > 0 for all primaries [A/i]. This is what tells us that unitarity fails. However, there is a unique strictly positive column of S, namely 3 S0|, where o is called the minimal primary . The properties normally possessed by the vacuum in a unitary theory are now distributed between the vacuum and the minimal primary. In particular, M00 = 1. To get around this complication, we use a powerful tool called the Galois shuffle developed by Gannon in [22], which permutes the entries of the 5-matrix (3.3.2) in a simple way to create a new matrix This is the primary of minimal conformal weight 41 with a strictly positive vacuum column. The Galois shuffle works as follows: Let 0 denote the vacuum primary (in our W3 notation, 0 = \pp]). Then there is a simple-current J0 and a Galois automorphism a0 (not necessarily unique) such that o = Joao0 (recall the discussion of Galois in Section 2.2.3. An RCFT for which such a permutation exists is said to possess the GS property. In [22] it is shown that the WN minimal models possess the GS property for all N, and furthermore, the simple-current J0 is trivial when N is odd. For N = 3, the Galois shuffle is as follows. We have p, q such that both 3 \ q (that is, 3 does not divide q) and gcd(3p, q) = 1. Choose t so that both £p=q 1 , and lq =3p 1 (3.3.5) hold. Then o — a^O (this was verified in [22]). Define new matrices S and T by -S[A/4[H := e e S[AM]M = dpp) Mo^pg) y/i S^v , (3.3.8) TMM = ee(pp)ae(Ppq)exp[^}T^l. (3.3.9) Thus the 'unitarized' W3 modular data S, T is, up to multiplication by an irrelevant constant, the (A2 © ^2)^ data. We can now associate the (A2 © ^2)^ modular invariants with the W3 ones as in the following lemma. Lemma 3.2. Let m := \\P^9\\, and let M be anmxm matrix indexed by P™ • Let M be the 3m x 3m matrix indexed by P++ with entries ~ _ f M{XjaA{Kjav] ift(p)=3t(u) M^-\0 ift{p)^t{v) ' (^JU) where a is the unique integer € {0,1,2} such that t(Jap) =3 t(Jav) =3 0 when ( t(p) =3 t(u). Then M is a W3 modular invariant if and only if M is an A 2 © A\ modular invariant. to Lemma 3.2 maps each entry Myxfi\{Kv\ the block M^^i^h, where 73 is the 3x3 identity matrix. Proof. Clearly M\^K„ € N for all Xp,nu e P™, and MPP,pp = M[pp][pp] = 1. To check S^ and T^o) -invariance, first note that since M is a W3 modular invariant, 42 we have S and T-invariance, so we have the equations 22S[\li}la0\M[a0\[ia>] = 2j%|M^jS]H' (3.3.11) [a/3] [a/3] T[X^][X^]M[X^[KU] = %]HV]H (3.3.12) for all [A/i], [KI/] € P++- We need to show that (P 9) (MS^VK, = (5 ' M)V;K, (3.3.13) T ^i^.- = ^%Mx^v (3.3.14) for all A^, KI/ s •?+'+• First consider (3.3.13). Let \p,Kv € -P++, and choose integers a, 6 G {0,1,2} such that a b t(K p) =3 t{K i>) =3 0 . (3.3.15) Note that since A/AMIK„ is 0 unless t{p) =3 t{u), the left-hand side (LHS) of (3.3.13) is __ K^ t(/3)=3t(M) which is Zl M[AK» [ai^/3] ^^ t(/?)=3t(/0 by the definition of M. By (3.2.7b) and (3.3.8), this becomes e-2m(bt(f3)+at(v)+qab)/3 ^ M a Z_] ;—:—: r-/=— \XK^^){aK^i3\S\aK 0\\K.K>>v} • (3.3.16) ^ ee(pp)ae(apq)V3 t(/3) = 3t(M) Finally, putting /?' := Ka/3 and using the fact that *(/?) =3 £(/z), this becomes - / N / TTf 51 ^[A/C>][a/3']5'[Q/3/]|^6l/]. (3.3.17) Q(PP)<^(apg)v3 a/3 t(/3')=30 Equation (3.3.17) is the left-hand side of (3.3.13). Similar calculations show that the right-hand side of (3.3.13) is p-2-Ki(bt(iJ,)+at(v)+qab)/3 c (nn\„ fTTTT 5^ £[AK>][a/3']^W][,cK t(0')=3O That (3.3.17) = (3.3.18) follows from equation (3.3.11). To show that (3.3.14) is true, first notice that if i(/x) #3 t(v), then both sides of 43 a (3.3.14) are 0. So suppose that t{p) =3 t{v) with t{K p) =3 t(/s:V) =3 0. Then the left-haleft-hanr d side of (3.3.14) is T^^M[XK a a an (3.3.9), n][K,K v]i d now by equations (3.2.7a) and ,2iri ., >,, ,— 47TJ, LHS of (3.3.14) = exp[^{a r 2ap-at(/x)}]exp[—^le^p^T^a^^Ka^M^a^^a,,], and similarly, i?#S of (3.3.14) = exp[—{a2p-ai(!y)}]exp[—^—Je^p^M^a^^a^T^a^^a,,]. That L#S = AtfS follows from (3.3.12) and the fact that t(p.) =3 t(v). D Now that we have Lemma 3.2, it remains only to check which (A2 © ^2)^ modular invariants are of the form M as in (3.3.10). Since ^f[pp],[pp] = 1, the entries MpPtPp, MpKp,pKp and MpK2p Theorem 3.1. The complete list ofW^ modular invariants, up to left-multiplication4 by a charge-conjugation (3.3-4), is Permutation Invariants A,i = Yl ch\Mch\xtA (3.3.19) [AH t(/i)=30 V c c/ PI = E Vl *[./ 4 It suffices to multiply by a conjugation on the left, so there are four modular invariants for each one listed. 44 Simple-Current Invariants Vrw = H \chM + ch\J^\ + ch\J2^]\2 + 3 H c/lMc/l[w] (p =3 °) [*M1 *(M)=30 t(A)=3t(rt=30 (3.3.21) Exceptional Invariants 4,8 = E lc/i[Ap]+CV(3,3)]|2 + |cV(l,4)]+^[A(4,l)]|2(P^3 0) (3.3.22) A|t(A)=30 ( 2 2 5Z lc/l[JiV]+cft[^A(3,3)]|2+ ]C lCV^(l,4)]+c/l[ja(4,l)]l i,j=0 i,j=0 2 2 + 3 \ch[ipp] + ch[ip(3t3)]\ + 3\chMl>4)] + ch[lf,{4A)]\ (p =3 0)(3.3.23) £ = c/l c/l c/l Pfi E ( [Ap] + [A(3,3)])(c/l|'j2t(A)Aj2t(A)p] + *j2t(A)Ai72t(A)(3i3)j) A|t(A)=30 ch c/l c/i : ch + ( [X(l,4)] + [A(4,l)])( | j2t(A)Aj2t(A)(li4)] + *j2tMXj2tW(4ti)]) (p ^3 0) (3.3.24) = ch C c/l c c 2 ^12,9 z2 \ W] + fy(10,l)H + c/l[(l,l0)M] + [(5,5)/j] + fy(2,5),*] + fy(5,2)^] | t(/i)=30 c/l 2 + 2 |c/i[(3,3)M] + c/i[(6j3)M] + [(3,6)Hl (3.3.25) c/l c/l c 2 C c/l ^12,g = Yl l [H + [(10,l)M] + ty(l,10)/i] I + l fy(3,3)/x] + [(6,3)M] t(/*)=30 c/l 2 c C c/l + [(3,6)^]| + l fy(5,5)^] + %2,5)M] + [(5,2)M]l c c/l c/l c/l + |c/l[(l,4)M] + fy(7,l)M] + [(4,7)/i] I + |c/l[(4,l)M] + [(7,4)M] + [(l,7)^] I 2 c l c/l + 2 |c/l[(4j4)M] | + (c/l[(2,2)M] + cty(8,2)/i] + ' [(2,8)M]) f(414)/i] c/l c/l c/l + [(4,4)M](c/lf(2,2)^]+ f(8,2)M]+ f(2,8)rf) (3.3.26) ^24,q = 22 \ch[ptl] + C/1[(5)5)M] + c/l[(7i7)M] + c/l[(lul)/(] + c/l[(22,l)/J] + c/l[(1)22)M] t(/j)=30 + c/l[(14i5)M] + c/l[(5il4)M] + c/l[(lli2)p] + c/l[(2)ll)M] + c/l^iQ^)/,] + c/l[(7)i0)M]| C/I + l<%16,7);«] + [(7,16)M] + c/l[(l,16)M] + C/1[(16,1)M] + c/i[(8,ll)M] + c/l[(11>8)^] c/l c/l c/l c/l + <%5,11)M] + [(ll,5)/i] + ((5,8)M] + [(8,5)^] + [(l,7)H + c/l[(7)1)/i](|S.3.27) (1) 3.4 The {A2 0 ^2) classification In this section, we sketch the proof of the (A2 © ^2)'^ classification for coprime levels. This was done in [2]. Some of the exceptional invariants were not worked out there; we work them out completely here (Section 3.4.6). 45 3.4.1 The permutation invariants The permutation invariants of (Ari ©• • -©A-J^, at arbitrary levels k\,..., ks, were classified in [19]; the following is a specialization of [19] to the case (A2 ffi ^2)^. This is Step 1, (3.1.1), of Section 2. We are looking for all modular invariants M such that for each A/i £ P+ g ++> M\^Kl/ ^ 0 for exactly one choice of nu and M\^tKV = 1 for this KV. Thus we can define a permutation •n on P^ and a matrix AP such that M% := 6KUtW(Xn)- If the resulting M1 is a modular invariant, then TT and M" are both called permutation invariants. By uniqueness of vacuum, we must have n{pp) = pp. By S and T invariance, IT is a permutation invariant if and only if n satisfies both of the following symmetries: c(p,9) _ C(P, 7">(P.9) _ rp{p,q) /q ^ r>\ Equations (3.4.1) and (3.4.2) show that inverses and compositions of permutation invariants are also permutation invariants. The charge-conjugations C^a'b' := CaCb, a b where the C , C are A2 charge-conjugations at level p, q respectively, are permu tation invariants. For another type of permutation invariant [19], built up from simple-currents, let a := (an, 021,012,022) be a quadruple of integers such that 2 2au + pa\x + qa a =6 3pan + 3qai2 , (3.4.3) aij + aji + paaaji + Q 7Ta : F%% > PH, (A,/i) .-* (J^itW+^ltMX^Ka12t(X)+a22t(fi)fiy (345) The 7ra's are permutation invariants, and they are the following: when 3 \ p, q =3 — p: all permutations generated by the 7ra with a £ {(p, 0,0,0), (0,0,0,g)}; when 3 \ p, q =3 p: all permutations generated by the 7ra with a £ {(p, 0,0,0), (0,0,0,p), (— p, — p, — p,— p)}; when 3|p, 3 \ q: all permutations generated by the 7ra with a € {(0,0,0, q), (q, q, q, q)}. The charge-conjugations and the ira exhaust all permutation invariants. The proof of this is roughly as follows: define a new permutation it' of P++ that has the charge-conjugations and 7ra's multiplied out. Then show that n' is the identity. To do this, we first find it' at certain 'small' weights, which are essentially the fundamental weights (1,2) and (2,1) (in our shifted language). We extend this to all of P+1 using a character argument and the unitarity of S^p'q^ (see equations (3.4.8) - (3.4.10)). The reason for this approach is that the Weyl character of every other A2 weight A can be written as a polynomial P\ in the characters of the fundamental weights, as follows: Let A e P^+(A^). Then c/iA_p(-27rt^) = i^(c/i(1,o)(-27rt^),c/i(0il)(-27ri^)) . (3.4.6) 46 Equation (3.4.6) is significant because the characters of the A2 weights are related to the entries of the 5-matrix (recall (2.2.14)) Suppose we knew that TT'(XP) = (Xp) and n'(pX) = (pX), for A G {(1,2), (2,1)} . (3.4.7) Then, for KV := TT'(KV), C(P) / c(P) e>(p) \ / o(P) O(P) \ c(p) a J J V_p (l,2),« ^(2,1),^ \ p / (1,2),K' (2,l),«' \ 5^, A A o(9) I q and a similar calculation applies to Sj?J /Sp J, so for any A/x, KV G -P++, we have c(P.9) C(P)C(P) c(p) c(«) c(P.9) ' V • • / Jpp,KV OpK Dpv OpK, Dpl/I >3ppjK'v> Multiplying both sides by S^Kl/ and summing over all A/x G P++, we have, by unitarityof 5(p'«), (3.4.10) C(P,Q) MP,Q) Jpp,KU °pp,K'v' Therefore, we must have K'V' = KV; ie, -K'(KV) = KV. This shows that the charge- conjugations and 7ra are all of the permutation invariants. The proof of (3.4.7) involves g-dimension arguments (^-dimensions are ratios ^£); for details see [20]. 3.4.2 The vacuum couplings Let A/x, KV G P++ such that M\^KV ^ 0. We say that 'A// couples to KV (on the left)', and 'KV couples to A/x (on the right)'. The hardest step in modular invariant classifications is finding which weights couple to pp. In the case of (A% © ^2)pJ when p and q are coprime integers, this step reduces to Lemma 3.1(a) and equations (3.2.14), (3.2.15). The result is Lemma 3.3. Let p and q be coprime integers greater than or equal to 4, and suppose q that 3 | q- Define TLR = {Xp e P% + \ MppM / 0} and 1ZL = {A/x G P*% | MA/t,„p / 0}, and p" := (3,3)- Then M\p}PP G {0,1}, and the choices for 7lR and TZL are (a) Any p and q that are coprime: TZR = IZL = {(p, p)}; 2 2 (b) 3 f P andp + q =3 0: UR = KL = {(p, p), (Jp, Kp), (J p, K p)}, or 2 2 TlR = nL = {(p, p), (Jp, K p), (J p, Kp)}, or 2 2 2 2 TlR = {(p,p),(JPKp),(J p,K p)} andKL = {(p, p), (Jp, K p), (J p, Kp)}, or 2 2 2 2 KR = {(pp), (Jp, K p), (J p, Kp)} and UL = {(p, p), (Jp, Kp), (J p, K p)}; 47 2 (c) 3\p, p ± 24: nR = TZL = {(p,p), {Jp,p), (J p,p)}; (d) p = 12: TlR = nL = {(JV,p),(jy,p) I % = 0,1,2}; fej p = 24: nR = KL = {(JV,P), (-/V.p), (^V",P), (-/V",p) : » = 0,1,2}, where p'=(5,5),p'" = (7,7). (j) (i) q = 8: TlR = nL = {(p, p), (p, p")}; (ii) p =3 1 and q = 8: ft* = TlL = {(JV, tf V), (J>, #V') | i = 0,1,2}, or i i i i ft* = KL = {(J p,K- p),(J p,K- p") | t = 0,1,2}, or nR = {(JV,^V),(^V,^V') I * = o,i,2} i i i i KL = {(J p,K- p),(J p,K- p")\i = 0,1,2}, or l i TlR = {(J p,K-*p),(J*p,K- p")\ i = 0,l,2} i i i TlL = {{J*p,K p),{J p,K p")\i = 0,1,2}. (in) 3|p and q = 8: KR = KL = {(J*p, p), (J>, p") | t = 0,1,2}. The only difficult part of the proof is excluding orbits involving p", when 4 divides p or q (and p,q =/= 8,12,24). This is done using Lemma 3.5(a) below. 3.4.3 The simple-current extensions In this section, we complete Step 2 of the (A2® ^2)p,9 classification. We assume that 3 10, p ^ 12,24, and that neither p nor q is 8 (these are the exceptional heights and a b will be dealt with in Section 6). Let JL {JR) denote the set of simple-currents J K such that MjapKbppp ^ 0 {MppJapKbp ^ 0). We also assume that ||JH|| = \\JL\\ = 3 - otherwise M is a permutation invariant. A corollary to Lemma 3.4 is that JR and JL are abelian groups. Lemmas 3.4 and 3.5 below are critical to this section. Let M be a modular invariant, and define the set VL{M) = {Xp \ M\PtKV / 0 for some nv € -P++}, and we define VR(M) similarly. We will usually write VL, VR unless there is ambiguity in doing so. Any modular invariant M is a square matrix, so simultaneously reordering by rows and columns, we can write it as a direct sum of indecomposable blocks ® Mi, where we take Mi to mean the block containing MPPtPp. By B(l,m) we mean the m x m block containing all Is, and we denote the largest real eigenvalue of Mj by r(Mi) Lemma 3.4. [20] 48 M S and (a) For each Xfi G P£'|, define sL{X,p.) = ^K„ W,PP ™L> M SR(X,H) = T,KV PP,KUS^I„. Then sL(X,fj.),sR(X,n) > 0, and sL(A,/x) > 0 <=>• A/z G VL and SR(X,'H) > 0 <=4> A^i € Pfl. a b c d (b) MJapKbp!pp = 1 /or all J K G JL> and MmJCpKdp = 1, /or aU J K G J*. fa) For all a,b,c,d such that MjapKbpJcpKdp = 1, MJaXKbpJcKKdv = M\^KV, for all \H,KV G P£'|. In particular, MjaXKb^KU = MX^KU = MXpJcKKdv, for all a b c d J K GjL,J K GjR. a b c d (d) J K G JL «=> at{\) + bt(n) =3 0 for all A/z G VL, and J K G JR 4=> ct(n) + dt(v) =3 0, for all KV G VR. (e) Suppose that M\PtPp =/= 0 <=>• A/i G JL(PP) and MPPiKV ^ 0 ^==> KV G a b JR(pp). T/ien PL' = {A// G P™ | at(X) + bt(ji) =3 0,' VJ K G JL} and c d Pfi = {KU G P™ I ct(«) + dt(i/) =3 0, VJ tf G JL}. Lemma 3.5. [20] (a) Suppose M has Mi = B(l,m) for some m. Then for each i, either Mi = (0) or r(Mi) = m. Also, for each (A,/z) G P™, £KJ/ Mj^ < pfg^y- fa) A^ow suppose 1ZL = JL{PP) and TZR = JR{PP), and suppose that MAM)K„ ^ 0. Then M WJLW ""'" ~ VII^(A/z)|| \\JR{KV)\\ ' // m addition, (A,/z) is no< a /ized point of JL (that is, JaKb G Jh,JaKb / J0/^0 implies a b (J X,K fj,) ^ (X,fx)), and also (K,V) is not a fixed point of JR, then M\^KU = 1. Moreover, M\p^a$ =£ 0 <=$• a(3 G JR(K,V), and Map^KV ^ 0 4=> a/3 G JL(A/Z). For Xfi G PL, define (A/Z)L to be the J7i,—orbit {J(A/z) | J G Ji,}, and for /t^ G P/j, (KV)R := {J(/w) | J G JH}- We denote the set of all JL —orbits (A/i) by PL/JL, and the set of all JR—orbits similarly. By Lemma 3.4(c), Mv^']Kv = M\p^KV for all A'// G JL, KV G JK, SO we write M,e. w . to mean any representative A^A'/*',KV such that A'// € JL(XH), K'V1 G JR(KU). We also define CVM> := H ^v? ; AV'e(V) then the partition function associated to M can be written as (A/i>, 49 Because 3 { q, we can always choose A/i to be the representative of (A/x) such that t(p) =3 0. The Weyl character of a triality-0 Ai weight can be written as a polynomial in the characters of the weights (1,1), (0,3), (3,0) (the proof is an easy induction) - these play the role of the fundamental weights (0,1) and (1,0) in Section 3.4.1. As before, we show that, after multiplying out by permutation invariants and charge-conjugations, any permutation n of the orbits (•) is the identity on the orbits (p\) and (Xp), where A € {(2,2), (1,4), (4,1)}. This will be sufficient to show that 7r is the identity on all orbits. When J has fixed points, then the 'orbit' is in fact just one weight; therefore there are two cases to consider in this step of the classification. When 3 { p, there are no fixed points of J, but when 3|p, then the weights ((|, |), p) are fixed points of J. 3.4.4 The classification when J has no fixed points In this case, every ^7-orbit contains three weights. Thus, if TT is the identity on the ^7-orbits, we get 3x3 blocks of nonzero M entries, just as in the vacuum block (Lemma 3.5(b)). In this case, all nonzero entries are 1. We include the details for Theorem 3.3(b). The other cases are the same. Here, 1 1 2 2 X 2 2 1 JL and JR are either {J°K°, J ^ , J K } or {J°K°, J K , J K }, and VL,VR = {Xp € P++ | t(X) ± t(p) =3 0}, where the plus or minus sign depends on which JL we choose. When we deal with orbits (Xp) however, we may suppose that t(X) =3 0, so in either case we can assume that t(X) = t(p) =3 0 whenever -Mf. w > ^ 0 for some (KV). For any choice of JL and JR, define a permutation n : VL/JL —* "PRI JR such e that M( v) S^t = $&v - (3-4-13) for A/x any representative of (\p) G VL/JL, KV any representative of {KV) € VL/JL, any X'fi' £ TT{XP) and any K'V' € •K(KV). We are now ready to check what happens to the small weights, using the follow ing: Claim 3.1. 7r(p,(2,2)) = (p,(2,2)) and ir{p, (1,4)) = (p,Ca(l,4)), for some a 6 {0,1}. 50 The proof of Claim 3.1 [20] also uses (^-dimension arguments. Claim 3.1 equally applies to ((2,2),p) and ((l,4),p), so 7r((2,2),p) = ((2,2),p) and n((l,4),p) = b (C q(\,A),p) for some b G {0,1}. Letting TT' := C^ on, we have 7r'(a/3) = («£} for af3 G {(p,(2,2)),(p,(l,4)),((2,2),p), ((l,4),p)}. Since (1,4) = C(4,l) and 7r' commutes with C^l,l\ we also have ir'(p, (4,1)) = (p, (4,1)) and 7r'((4, l),p) = {(4, l),p). In matrix terms, this amounts to multiplying our modular invariant M by one of the charge-conjugations CI, CIC, Ic. Multiplying a modular invariant by a permutation invariant gives another modular invariant, so the product M' denned by 7r' is a modular invariant. We will show that any TT fixing the fundamental weights is the identity. Choose any Xp,K,v G VL- Let X'p' G 7r(Ap) and K'V' G TT(KI/). By 5-invariance, we have the equations o(P) qiP) O(P) C(P) C(P) 3 (M)A' ' (4,1)A ^(1,4)A' . (2,2)A °(2,2)A' (3.4.14) qiP) ' o(p) ' o(p) q(P) ' o(p) 5pA' SM 5(9) c(9) o(9) S(9) 5(9) 0 D 5 5 UA)n °(1,4)M' (4,1)M ^(4,1 V . ' (2,2)M ' (2,2)/i' (3.4.15) 0(1) c(9) ' oil) o(9) ' o( Z c/l c/i = H 3.4.5 The classification when J has a fixed point This is the most difficult part of the classification, and we will only sketch the argument here. The details can be found in [25] or [2]. The fixed points are the weights ((|,|),p), for any p G P++. In all cases involving fixed points, the couplings are symmetric, so put J ~ JL = JR. and V := VL = VR. From Lemma 3.4 (e), VL = VR = {Xp G P£| | t(X) =3 0}. Let 51 The usual argument (with some modifications) now shows that for Xp not a fixed point, n{Xp) = (Xp), and that the value of M is 1 at those weights. 3.4.6 The exceptional invariants This is Step 3 from the (A2 © ^2)^ classification. We give more details here, because this was not fully worked out in [2]. The exceptional invariants occur at heights (p, q) when one of p or q is 8, 12, or 24. In all of the cases except for (p, 8) when 3|p, we can use Lemma 3.1(b). Another important lemma from the (A2 © A2Y1' classification is Lemma 3.4. It was used heavily in the simple-current invariant classification, because it gives useful simple-current symmetries, but we need it for the exceptionals as well. We will do the exceptionals at (p, 8) in the most detail (the other exceptional heights use similar arguments). Before beginning the proof, we can use a Galois automorphism on P++ to achieve useful symmetries. Recall the Galois automorphism 3.4.7 q = 8, 3 \ p, and TlR = KL = {pp, pp"} Here, MPPiPP = MPPtPP" = Mpp»tPP = 1 (and Mpp»tPP» = 1 by Galois), and M\v,pp = MPPM = ° for a11 V £ ^.- Suppose M\^KV 7^ 0. Then A and K must satisfy the parity rule (3.2.4), so by Lemma 3.1, K G OX. Evaluating SL(X,P) > 0 (Lemma 3.4(a)), we find that V = {Xp I A e P++,^ e OpUOp" U 0(1,4) U 0(4,1)}. Then by (3.2.15), we have M^KV ^0^/i,i/6O/jU Op" or p, v G 0(1,4) U 0(4,1). Evaluating MS^'8' = S^&M at (JpKp,pp), we get a,b=0 c,d=0 But Spp^/Sf), = 3 - 2v/2, so S$ and 5^, are linearly independent over Q. There fore, equating coefficients, we find that MJpKpJapKbp = 1 for exactly one choice of a and b. By T-invariance (3.2.3), we must have either a — b = 0, or both a and b nonzero; however, (Jp, Kp) is not a pp-coupling, so we cannot have a = b = 0. Therefore {Jap,Kbp) G {{Jp,Kp),(J2p,Kp),(Jp,K2p),(J2p,K2p)}. These are all conjugations of each other, so multiplying by the appropriate conjugation matrix if necessary, we may assume that a = b = 1; ie, MJPKP,JPKP = 1- Then by Lemma 3.4 (c), MJ\KPJKKV = M\p^v for all Xp, KV G P++, and t(X)+t(p)=3t(K)+t(u) (3.4.17) 8 whenever M^,K„ 7^ 0. Evaluating MS^ ^ = S^&M at (pp,pKp), we get that 52 a b = r M„Kp,j PK p 1 f° exactly one choice of a and b. Then (3.4.17) and T-invariance yields either a = 0 and b = 1 (for any p), or o = 1 and 6 = 0 (for p =3 2). Case 1: a = 0, b = 1. Then MpKp,PKp = 1 , (3-4.18) yielding the simple-current symmetries M\KiilKKiv — M\ptKV and by Lemma 3.4(c), equations (3.4.17) and (3.4.19), the selection rules t(n) =3 t{y) (3.4.19) t(A) =3 t(/c), (3.4.20) whenever M\^KV ^ 0. Suppose M^iKI/ ^ 0, where p s 0pU 0(3,3). It will be enough to do the case p = p, due to the Galois and simple-current symmetries. Evaluating MS^P'8^ = S^&M at (Xp,pp), we find that MXpCajbXp = MXptCcjdX(itz) = 1 for exactly one choice of a, b, c, d. By (3.4.19) and (3.4.20), either a = 6 = 0ora = l and b = pt(X) (and similarly for c and d). Thus MXp,Kl/ + 0 =• MVl(0, = 1, for KV 6 {A'p, A"p} , (3.4.21) where A',A" G {A,CJP*(AU}. We would like to show that A' = A". To do this, specialize to the weight ((2,1), p). Since M(2il)PjKJ/ 7^ 0 => M^i)PtKU = 1 for KV £ {(2, l)'p, (2,1)"(3,3)}, then evaluat 8 ing MS&& = 5 5 5 Therefor £)'P = S)"P- e, c/i(2,1)(2,1)' = cfc(2il)(2,1)" and cfc(1,2)(2,1)' = ch{lfl)(2,1)", and thus by the character argument in Section 3.4.1, (2,1)' = (2,1)". Multiplying by a permutation invariant if necessary, we may assume that (2,1)' = (2,1)" = (2,1). Thus M(2,i)p,(2,i)p = M(2,i)p,(2,i)(3,3) = 1, and M{2,i)p,Kv = 0 for all other KV. NOW let A be any P++ weight. Then 5-invariance at (Ap, (2, l)p) and (Ap, (2,1)(2,1)) shows that A' = A" = A as in equation (3.4.21). Therefore, MXp,KV ^ 0 if and only if M\p^KV = 1 and KV e {Ap, Ap"}. Now suppose Mx^Kl/ / 0, for p 6 0(4,1) U 0(1,4). It will be enough to do the case p = (4,1). From 5-invariance at (A(4, l),pp) and (3.4.19), 12 12 / , / , ^(4,l),CJf»A(4,l) + 2L 2^ M\(i,l),CaK<>\{lA) = 2- (3.4.22) a=0 6=0 a=0 6=0 Therefore, either each of the sums is 1, or one of them is 2 and the other one is 0. From S-invariance at (A(4, l),p(2,1)), 53 M a + M ^^ HiA),c Kn(4^)S{4tl){2,i) '(4,1X2,1)+ z2z2Z^Z^M4,i),C'-K>>x(i,4)S- ilA)(21) - 0, (3.4.23) a=0 b=0 a=0 b=0 which shows that each sum is 1, since 5(1 4w2 ^ = ~S,41^,2 ^. Therefore, MA(4,1),A'(4,1) = -WA(4,1),A''(1,4) = 1 (3.4.24) for exactly one choice of A', A" € {X,CJpt^X}. We would like to show that A' = A" = A. Setting A = (2,1), we have M(2,i)(41i),(2,i)'(4,i) = M(2,i)(4,i),(2,i)"(i,4) = 1, for (2,1)', (2,1)" 6 {(2,1), CJPW}. Prom 5-invariance at ((2,1)(4,1), (2,1)(2,1)), we find that (2,1)' = (2,1)", and 5-invariance at ((2,1)(4,1), (2, l)p) implies that they are both (2,1). Thus M(2ll)(4,l),(2,l)(4,l) = M(2il)(4il)i(2,1)(M) = 1. Now consider any A 6 P++. Equation (3.4.24) and 5-invariance at (A(4,1), (2,1)(2,1)), yields A' = A", and from 5-invariance at (A(4,1), (2, l)p), they are both equal to A. (1, 1ta B We have found the invariants M := Ap ® £%, C °)M M', where M* is the permutation invariant given by ira : (A,/x) i—• (J^^A,^), and their conjugations. r(l) _ V- |y(P.8) , (P.8) ,2 , | (P,8) , (P,8) ,2 , I (P,8) (pfi C IX + + + + + P,S — Z^ Ap *A(3,3)I 1^(1,3) *A(4,3)I l*A(3,l) *A(3; 4)1 A€P?+ 8) 2 + 8) 2 .+ !*$& + X&V + l*Sfi) + * 3)" 1^5, + X& 2)| (3-4.25) (forg^30,p = 8) Case 2: a = 1 and 6 = 0 (for £> =3 2). In this case, we have MPKPIJPP = 1 and the selection rules that follow from it. The matrix that follows the selection rules however, fails 5-invariance, and so is not a modular invariant. 3.4.8 q = 8, p =3 1, and TlR = JR(pp) U JR{pp") and KL = JL{pp) U MPP") The second type of exceptional occurs when TZR and TZi involve J-orbits of the weights pp and p(3,3). The possibilities for IZR and 'R.L are any combination of i i i i i i i nR,nL = {{j p,K p),{j p,K p")},{{j p,K^p),{j P,K- p")}. By Lemma 3.4, MJL{XII)IKI/ = MX^JR{KI/) = MXpnKV whenever MX^KV ^ 0, so this argument collapses down to the previous one. 54 We find the invariant 2 (P 8) (P,8) £(2)_l y- y^y(P.8) + Y 2 2 (p 8) 8 (P 8) (P 8) + Vfv ' 4-v(P- ) ^V(Y ' * +V ' * ^ H4 2fi1 3.4.9 g = 8, 3 | p, and K = {(p, p), (p, ft')} When 3|p' we can show that M\^KV /0=^K€ OA using T-invariance, Galois, and the arguments in Chapter 6 of [2]. There are two choices for 1ZL and IZR: either TlR = nL = {(p,p),(p,p")}, or KR = TZL = {(JV,P),(JW) : * = 0,1,2}. We will start with the case 1ZR = 1ZL = {(p,p),(p,p")}, and do this one in the most detail. Here, MPPtPP = MpPiPpn = Mppnpp = 1. Define KV Evaluating sL(A,p) > 0, we find that V = {Ap G P£+,p G OpU Op" U 0(1,3) U 0(3,1)}. Furthermore, we know that M\PtKl/ ^ 0 => K G 0A, and by T-invariance, either p, v G Op U Op", or p, z/ G 0(1,4). Evaluating MS = SM at (JpKp, pp) shows that MJpKpJapKbp = 1 for exactly- one choice of a and b. Along with T-invariance, we have the following choices for a and b: {a,b) G {(1,1), (1,2), (2,1), (2,2), (0,1), (0,2)}. The first four choices are all charge-conjugations of each other, as are the last two. Therefore, multiplying out by a charge-conjugation if necessary, we can consider the cases (a,b) = (1,1) and (a, b) = (0,1), and they will give us two different modular invariants, up to charge-conj ugation. Case 1: MjpKP,jPKp = 1- By Lemma 3.4, we have M^KV / 0 =• t{X) + i(p) =3 *(«) + t{v). (3.4.27) Evaluating MS = SM at (pp,pKp) yields MJapKbp^pKp = 1 for exactly one choice of a and b. We cannot have both a and b nonzero (otherwise, M or one of its charge-conjugations would have MjPKP,PKp = 1 and we would be in Case 2: (a, b) = (0,1)), and by (3.4.27), b = 1, so we must have MpjPtPKP = 1. Therefore, Mto / 0 =• t(X) =3 <(K). (3.4.28) Equations (3.4.27) and (3.4.28) then imply MXlttKV ? 0 =» <(p) =s t(i/). (3.4.29) Notice that, when i(A) =3 ±1, equation (3.4.28) implies K G {«TA | i = 0,1,2}. Suppose that M\PtK„ ^ 0, where p G Op U Op", and for now suppose that 55 t(X) =3 ±1 (we will do the case t(X) =3 0 shortly). Let p = p (the argument for the other weights in Op LI Op" will follow from Galois and simple-current symmetries). Then v € Opl) Op" and K G {J* A : i = 0,1,2}. By (3.4.29), p and p" are the only possibilities for v, so put (K,V) — (JlX,p) and (J-?A,p"), where i,j S {0,1,2}. Evaluating M5 = SM at (Xp, pp), we get that MxPtja\p = MXpjb\pn = 1 for exactly one choice of a and b, and T-invariance then implies that a = b = 0. Now suppose that M\PiKV 7^ 0, where t(X) =3 0. Then, as before, v can be either p or p", and K is Jl0X, for at least one choice of i,j £ {0,1,2}. Evaluating MS = SM at (A/o, pp), we get that a b ^XP,J C Xp = MXpJcCd\p» = 1, for exactly one choice of a,b,c,d. Let A' := JaCbX and A" := JcCdX. Then evalu ating MS = SM at (Xp, (2, l)p), we get C(P) 0(8) , O(P) 9(8) _ C(P) r 9(8) , 0(8) , ,0 4 o0l and MS = SM at (Ap, (2,1)(2,1)) implies 5A'(2,1) = ^A"(2,l)- (3.4.31) If we replace (2,1) with (1,2), we get similar equations, and clearly Sy = Syt , so we have q(p) q(p) O(P) c(P) ^'(2,1) _ ^"(2,1). ^'(1,2) _ ^"(1,2) (3.4.32) C(P) C(P) ' C(P) Q(P) ^A'p °A"p °A'p °\"p Equations (3.4.32) imply A' = A", so putting that into (3.4.30), we have SyL^(P) = Sw2iy Now the same argument we just used to show that A' = A" shows that A = A'. Therefore, A' = A" = A. Therefore, we have shown that for p e OpUO(3,3), M\p,KV 7^ 0 implies M\^KV = 1, for KV = Xp and Xp', where {p,p'} is one of the pairs {(Jap, Jap")} for a = 0,1,2. Now suppose that M\^KV ^ 0, where p e 0(1,3). By (3.4.27), we have the following choices for {^,1/}:'{(1,3), (4,3)}, {(3,4), (3,1)}, and {(1,4), (4,1)}. First suppose that p = (1,3), so that M^i3))K„ ^ 0 (the other cases when p € C(l,3) are similar). Evaluating MS — SM at (A(l, 3), pp) yields the equation 21 21 / „ / , ^A(l,3),^CJ'A(l,3) + ^ 2_^ •MA(l,3),JiCJA(4,3) = 2- (3.4.33) Let Ei denote the double sum on the left, and £2 the double sum on the right. Then evaluating MS = SM at (A(l,3),p(l,2)), we have SXp {5(l,3)(l,2)Sl + 5(4,3)(1,2)S2} = ° 56 ( ( since (p, (1,2)) g P. But S( J3)(li2j = -S( *3)(li2) and SJJ > 0 for all A e ff+, so Ei = E2. Then by (3.4.33), M\(1,3),C*J»\(1,3) = MA(l,3),C">'jb'A(4,3) = * for exactly one choice of a,b,a',b'. If t(X) ^3 0, then a = a' = 0 by (3.4.28), and a b a b by T-invariance, b = b' = 0. If t(A) s3 0, then let A' := C J X and A" := C 'j '\. Evaluating MS = SM at (A(l, 3), (2,1)(1,3)) gives the equation C(P) 0(8) , q(p) 0(8) _ Case 2: MjPKP,PKp = 1- This also represents the case MjpKppKip = 1, multi plying by the conjugation C^0'1' if necessary. Then we have MV,K„ ^ 0 =» t(A) + t(p) =3 t(v) • (3-4.34) Evaluating MS = SM at (pKp, JpKp) and using T-invariance we get MjPPtjapp = 1 for exactly one choice of a € {1,2}. Multiplying by the conjugation C^1,0) if necessary, we can assume a = 1, and so MV,K„ / 0 => t(A) =3 *(K). (3.4.35) Using equation (3.4.35), T-invariance, and S-invariance at several weights yields the modular invariant M (and its conjugations) defined by the rules: 2t(A MA/i>ltv ? 0 => MV,KI/ = 1, for KZ, = (J*W+*<">A, J V) and (J^+^A, jW)^ > (3.4.36) MK[/iV / 0 => MK„,V = 1, for KV = (j2*(A)+2t(^)Aj ^(A)^ and (J2t(A)+2t(M)Aj ^(A)^) ; (3.4.37) where (p, //) G {(p, p"), (Jp, A"/), (J2P, ^2(3,3)), ((1,3), (4,3)), ((3,4), (3,1)), ((1,4), (4,1))}. This is the modular invariant M*b(Ap®£8), where Mnb is the permutation invariant 2 2t A defined by wb{X,fi) = (J *W+*MA, J < V). 3.4.10 q = 8, 3|p, and ft = {(Jip,p),(Jip,p") :t = 0,l,2} The third type of exceptional invariant for (p, 8) when 3|p occurs when the pp- couplings of M are anything in the set {(Jlp, p), (Jlp, p") | i = 0,1,2}. In this case, Mppjipp = Mjipppp = 1 for all i = 0,1,2, and we have to consider the fixed point 57 ip = (§,§) such that JV = 0 and Sfl(p, (2,2)) > 0, we also have Mppj%ppn = Mjippupp = 1 for all % = 0,1,2. The fact that Mjpp^pp = MPp^Pp = 1 gives us the rule MA/I,K„ £ 0 =*• i(A) =3 *(K) =3 0. (3.4.38) Thus V = {Xfi G P£'+ : i(A) =3 0 and /i G OpU Op" U 0(1,3)}, and as usual, p. and z/ are either both in 0p U 0p" or 0(1,3). By Lemma 4.2 of [2], we also know that Mji^jiKU = M\^Kl/ for all i, j = 0,1,2, 2 so we can define an orbit (A/i) := {(X,p), (JA,/x), (J A,p;)} for each A/U G P++ with \ + p. Evaluating MS = SM at (pp,pKp), we get that MpKP,PKp = 1 (up to multipli cation by a charge-conjugation), and so here we also have the rule MA/Ma/^0=>t(*i) =3 *(»/). (3.4.39) Using equations (3.4.38) and (3.4.39) and S and T-invariance, we find the excep tional Vp ® £% and its conjugations. 3.4.11 The exceptionals at (12, q) and (24,g) The arguments here are similar to the (p, 8) when 3 \ p ones (see [2] for details). For (12, q), we have M and MnaM and their conjugations, for the permutation invariant na : (\,n) >-> (X,K^^fi), where M is either £$ ® Aq or ^ 58 Chapter 4 Fixed point factorization for the affine Kac-Moody algebras 4.1 The problem The S-matrix of an RCFT is a fundamental quantity of the theory, as the matrix governing the modular transformation T ^> — of the characters (2.2.2), and through Verlinde's formula (2.2.5), expressing the fusion coefficients. In this chapter, we consider entries Sx^ of the S-matrix corresponding to an affine Kac-Moody algebra where one of A or fi is a fixed point. By 'fixed point', we mean fixed by nontrivial simple-current symmetries, which will be explained in more detail below (the simple- currents correspond directly with the centre of the Lie group). The reason we are interested in the entries at fixed points, is that fixed points complicate many phenomena (see for example the discussion of NIM-reps in Section 5.1). We find that these entries unexpectedly reduce to entries of S-matrices of smaller rank affine Kac- Moody algebras (or more specifically, simple formulas involving those smaller rank S-matrix entries). 'Fixed point factorization' refers to this reduction. The only other fixed point factorization was found in [26] and concerns the Ar' algebras. In that case, the S-matrix factorized into S-matrix entries of smaller rank A^> algebras. However, this is the only case where the smaller rank algebras involved in the factorization are of the same family as the original algebra. From our point of view, the main difference with the ^4-series is that its simple-currents have an order growing with the rank (the centre of SU(n) is Z„), whereas the simple-currents of the other affine algebras have order bounded by 4. Let Xr be the affine Kac-Moody algebra corresponding to a classical simple finite-dimensional Lie algebra Xr of rank r, and let S be the S-matrix for Xr at level k. Recall the definition of simple-current given in Section 2.2.6. Let A = (Ao; Ai,..., Ar) e P+(Xr '). Label each node of the corresponding Coxeter-Dynkin diagram by a Dynkin label by labelling the extended node Ao and the remaining nodes in sequence by the remaining labels Aj. A symmetry of the graph will permute the Dynkin labels. Its fixed points are weights 59 yields the symmetry Sw = Sj^v = exp[2niQj(n)]Svll , (4-1.1) where Qj(X) is given in Table 2.1. Equation (4.1.1) yields the useful fact: Fact 4.1. Sm = 0 whenever Qj{fi) $ Z. Fact 4.1 will give us immediate zeroes in the fixed point factorizations. We will specify these for each algebra in the relevant sections. The simple-currents of a rational conformal field theory form an abelian group. The group of Ar simple-currents is isomorphic to Zr+i; the Br and Cr simple- currents each form Z2, and the Dr ' simple-currents are isomorphic to Z2 x Z2 and Z4 for r even and odd respectively. Let J be a simple-current. Recall that the order m of J is the smallest positive integer m such that J X = X for all A £ P+(Xr '). For example, consider A§ . Its Coxeter-Dynkin diagram is a regular hexagon. Let J be the rotation of TT/3 radians about the hexagon's centre. The group of simple-currents is (J) = ZQ. There are three types of fixed points: 2) is a fixed point for J3. We will use the orthogonal coordinates \[i] throughout this chapter. They are given for each algebra in Section 2.3. In particular, the orthogonal coordinates \+[i] of the p-shifted weights will be used most often. By equation (2.2.19), the character XA(AO °f a weight (i G P+(Xr ') is a polyno mial in the characters XA^M) at the fundamental weights. Therefore, to establish a fixed point factorization, it is enough to know that there is one at the fundamental weights. Furthermore, for the applications of fixed point factorization to NIM-reps in Chapter 5, we need only the fixed point factorizations at the fundamental weights. For the symplectic (C) algebras however, we find a fixed point factorization at all weights. As this argument is considerably different from the argument for the B and D algebras, we put this one first, following the overview of the >l-series fixed point factorization. We point out here that we have noticed that in all cases but Cr ' such that r is even, the smaller-rank algebras involved in the fundamental weight fixed point factorizations correspond to the orbit Lie algebras of [11]. We discuss this in the Concluding Remarks (Chaptre 6) in Section ??, along with a table of the 'fixed point factorization algebras' for the simple-current group generators, and their levels. 4.2 The Ar ' fixed point factorization The Ar ' fixed point factorization was found in [26]. For completeness, we include the result here, but see [26] for details. Let n := r + 1. The Coxeter-Dynkin diagram for Ar is a regular n-gon. Let J be the rotation of "^ radians: the group of simple-currents is the group (J) = Zn generated by J. There is a fixed point factorization for Jd where d\n, in which case 60 the order of Jd is §. The case d = n is trivial (J = id), and the case d = 1 is degenerate (the fixed point factorization formulas work with S' = x' — !)• The j4-series is the only case where the order of the simple-current group grows with the rank of the algebra. Suppose d\n, and let ip £ P+(Ar ) be a fixed point for kd /- \ d an< J . Define the truncated weight ip' := (tpo; Lemma 4.1. [26] Xxiv) is nonzero only if there are exactly d integers 1 < (\ < • • • < if < n such that \+[ef] = -i (mod §;. kd Let ^.^1-^.L. {4.2,, J n/d for j = 1,... ,d— 1, and where 7r is a permutation. Then the fixed point factorization (given for both the S-matrix and the characters x) ls 1 V Sxv = sgn^-l)^ ^) (£) S'y^r • • • S'^^, (4.2.2a) t 1 , XA(V) =sgn(7r)e(-l) ^ -H)x^(1)(¥/)•••x'v,9,(v ), (4.2.2b) where £ is a ^ root of unity (£ and the permutation IT are given in [26]). The primes denote Ad_^ level ^ quantities. The formulas (4.2.2) apply when d = 1 as well, in which case the S's and x's on the right-hand sides of (4.2.2a) and (4.2.2b) are all taken to be 1. 4.2.1 The special case of the fundamental weights The fixed point factorization for the ^4-series at the fundamental weights was used in [15] to find NIM-rep coefficients for the simple-current modular invariants M[Jd]. When A is a fundamental weight A^, we have the fixed point factorization ^-{> «uf (42'3) where primes denote Ad\ level ~ quantities. 61 4.3 The Cr fixed point factorization The simple-current J is given in (2.3.20). Its fixed points depend on whether r is even or odd: we treat each of these separately. 4.3.1 The r even case When r is even, there is a fixed point factorization at all levels k. The fixed points of J are weights £_i,¥>r, 2{iPo + ---+ipr_1) + iP^=k. (4.3.2) By Fact 4.1, we immediately have that Sw = 0 whenever yui + /^3 H h /xr-i 0 2Z. In Lemma 4.2, we find complete conditions on fi such that Svli ^ 0. We first find a symmetry of the determinant in (2.3.19). By (2.3.15b), the orthogonal coordinates of if are i-\ i-1 smnp^^l = (-l^^sin^1^1-^1, (4.3.4) k+r+1 v ' k + r + l v ' for all i — 1,...,|, in the entries of the determinant (2.3.19). Define s{x) := 1 — 2 2 r r sin(7r^). Then the entry Sw is a constant (namely (^) i ~ from (2.3.19)) times the determinant + + M+Ipl+1 + + (-iy W+i8((p [r]n+[l]) ••• (-l) *(v H/i [r]) t 1 1 + + -l)"" "I l+ s(V>+[§ + 1]M [1]) (-1)" M+IS(^+[£ + l]/x+[r]) + s(^[§ + l]^[l]) S(^+[§ + l]M [r]) + + + S( From (4.3.5), we now find conditions on \x such that SVI1 ^ 0. Lemma 4.2. Suppose r is even. Let if be a Cr level k fixed point as in equation (4-3.1). Then SV)i = 0 unless there are precisely | integers (\,..., £i, 1 < i\ < ... < 62 £L < r such that n+[£j] =2 1, and | integers mi,..., mr, 1 < mi < ... < mi < r, such that fi+[rrij\ =2 0. Therefore half of the (-1)^+W+1 are +1 and half are — 1. We move all the + signs to the left using a permutation: define -n to be the permutation on (1,..., r) that re orders the set {/z+[l],... ,/x+[r]} into the set {//+[£i],... ,^+[£r],/u+[mi],... ,^+[mr]}, that is, *ti)=tj, 7r(^+-?') =mJ (4-3-6) for 1 < j < §. Through elementary row operations, y+[r + l-i]n+[lj m S : +1 and Then + for all 1 < i < §. Let i>'+[j} := H+\£j], where there are precisely § integers l\,..., It 1 < Ij < |, such that fi+[^j] =2 1, for all j = 1,..., §, and define tp' by + + ,,,/ _ M [^]-M [^+i) 1 (4.3.10) ^ 2 for j = 1,...,§ - 1. For i/ie double primed quantities. Let y>" := (^;^.i,...,%). (4.3.11) Then by (4.3.2), y?o + 2<^i + • • • + 2 63 b so tp" e P+(A{2(Z)) y (2-3.30). By (2.3.31b), ¥>"+[*] = ¥>+[£+i] (4-3.13) for all i = l,...,lj. Let ^"+[j] := M g } , where there are precisely | integers 1 < rrij < | such that /x+Kj] =2 0. Then solving for ip", we must have + ^ = /i [mJ]-^[mJ+1]_1 (4314) 2 2 for all j = 1,..., § — 1. With the above assignments, we have the fixed point factorization: Proposition 4.1. Let r be even, and let ip be a fixed point of the simple-current J as in (4-3.1). Let ip' ,tp" ,%l)' ,\j)" be as above. We have the fixed point factorization Sw = V^i-iy-^^-rsgn^S^S';^,, , (4.3.15) a eve an where S' and S" are the S-matrices for the algebras A2(T)-i ^ ^ ^ ^ +1 ^ -^(L) at level k respectively, and 11 satisfies Lemma J^.2. Remark 4.1. The special case when 11 is a fundamental weight is important to us and is given in Corollary 4.1 of Section 4.3.3. 4.3.2 The r odd case There are fixed points of J when k is even. They are { t . P= {w, Pi,---,Vi=±iVr=±,---,P\,Vo), (4.3.16) 2 2 where by (2.3.14), k = 2{ z-1 ip+\i] = r + 1 -i-^^ipe »-i ip+[r + l-i] = i + ^2w (4.3.18) e=o K + r-\ for i = 1,..., £=i - 1. We still have the symmetry (4.3.4) for 1 < I < ^ in the I1 1 determinant for Sw, about the ( ^-)" (centre) row. This row consists of 0s and 64 ±ls. By Fact 4.1, we know that 5W = 0 whenever /zi + /X3 + • • • + //r ^ 2Z, and analogous to Lemma 4.2, we now find Lemma 4.3. Let r be odd, and let ip be a Cr fixed point as in Equation (4-3.16). r Then Svtl — 0 unless there are precisely ^- integers £\,...,£r±±, 1 < l\ < ... < lr+i < r, such that u+[lj] =2 1, awrf ^^1 integers mi,... .IHM, 1 < mi < ... < 2 ^ 2 iriM < r, such that n+[mj] =2 0. As in the r even case, define a permutation n of {1,2,..., r} such that ' r + 1 ?r(i) = £;, 7T ( —— +j (4.3.19) m•3 ' where 1 < i < ^ and 1 < j < ^. We will now make assignments for the fixed point factorization: define A ,., level fc weights For £/ie primed quantities. Let ifi' := { Then for all 1 < i < ^, r-\ Define a weight ip by its Dynkin labels as follows: = /X+I^-M+^ + I] J 2 + (4.3.21) tf'i=i+1 = M [^+i]-l for all j = 1,..., ^, so that foraUj = l,...,E±l. For i/ie double primed quantities. Let : ? < 9 : (4.3.22) v" = (v !=i;¥'£=i_i,yi^i_2'---'2 2 * 2 * < i>v '°)' so that r-1 ip [t\=ip' •t + 1 65 for alii = 1... ^V^, and define ip" by its Dynkin labels $ = ^Kl-^K+i],! (4.3.23) u+[mr-il 1>'Li = —V^-i for all 1 < j < ^f1 - 1, so that ^+b'] = /*+K-] for all j = 1,..., ^2^. With the above assignments, we have the fixed point factorization: Proposition 4.2. Let r be even, and let Svll = 1^-^-1)=^^ sgni^^S'^, (4.3.24) where S' is the A 'r+l S-matrix at level k, and S" is the C\.Ji S-matrix at level k 2' Remark 4.2. The special case when /i is a fundamental weight is given in Corollary 4.2. 4.3.3 The special case of the fundamental weights For the applications of fixed point factorization to NIM-reps (Chapter 5), we will need the fixed point factorization at the fundamental weights, so it will be useful to explicitly state this special case. For any 1 < n < r, Ag~[i] = A+[i] - An[i], or A+[i] = AQ[I\ + An[i]. Notice that An[i] = ^2^=i{An)e = 1 if i < n and 0 if i > n, so the shifted orthogonal coordinates of A„ are .+,., fr + 2-t ifl 66 Claim 4.1. ***m) = { [l^i $\Zll- (4^6) Proof. First we consider TTQ, which is the permutation 1 2 ••• I | + 1 §+2 ••• r (4.3.27) 2 4 ••• r 1 3 ••• r-1 To do this, we will bring all the even numbers to the left, starting with 2, then 4, ..., up to r by transposing each even number with its left neighbour until it is in the proper position before moving on to the next highest even number. What we are doing is the following: let n be an even number, 1 < n < r. To bring n to the left in this manner, we take the composition ((n — 1) n)((n — 2) (n — 1)) • • • (| (§ + 1)) of transpositions. By doing this, we will not have affected the order of the odd numbers, so when we are finished (that is, when r is moved to the (|)th position), the odd numbers will be in their proper position. Thus, we can write TTO = (12) (34)(23) (56)(45)(34) ((r - 1) r) • • • (£ (£ + 1)). The number of transpositions in (4.3.28) below is just 2-\— which we can see will be even if | =4 0,3 and odd if | =4 1,2. In other words, s^)={-i ill I:?;2 • /4-3-28) In orthogonal coordinates, Ao = (r, r — 1,..., 1). Consider A2. In orthogonal coor dinates, A2 = {r + 1, r, r — 2,..., 1). That is, we have (in addition to increasing the first two orthogonal coordinates) switched the parities of the first two coordinates in Ao, but otherwise kept the same order of the odds with respect to themselves and the evens with respect to themselves. Switching the first two coordinates will then yield the permutation (4.3.27), that is, TTI = (12)7ro. In the same way, we see that 7T2 = (34)7Ti, and in general, 7rm = ((2m — 1) 2m)7Tm_i for all 1 < m < |. Thus, 7rm = ((2m - 1) 2m) • • • (34)(12)TT0 , (4.3.29) which, together with (4.3.28) yields (4.3.26). • To find tp' and ip" in Proposition 4.1 when /i = Ao, we observe that (£\,... ,£r) = + + + (2,4,..., r) and 0i [*i], A* fo], • • •, M [^D = (r-l,r-3,..., 1). Similarly, (mi, m2, + + + = (1,3,... ,r - 1) and (/u [m1],^ [m2],... ,/x [mr]) = (r,r - 2,... ,2). Therefore, using formulas (4.3.10) and (4.3.14), ip'j = ip'J = 0 for all j = 1,..., §, so ip' and V" are also the vacuum for ALt, : at level k + 1 and A2X, at level k respectively. Now consider A2m, 0 < m < r. To find V' and ip" when /x = A2m, note that by (4.3.25), the shifted orthogonal coordinates A^fj] decrease by 1 with each consecutive j except from j = 2m to j — 2m + 1, where we have a difference of 2, and where both the orthogonal coordinates are even. Therefore the ordered set of 67 differences of the odd orthogonal coordinates (fJ,+ [£j] — fji+[£j+i\ \ j = 1,..., § — 1) = (2,..., 2,4,2,..., 2), where the 4 is in the mth spot, and the ordered set of differences of the even orthogonal coordinates (fi+\mj]— fj,+ [mj+i] | j = 1,..., §-1) = (2,..., 2). We then have that and ^" = 2 for all j = 1,..., §. Therefore, ip' = A'm, and ip" = A0', that is, the A^X, vacuum. If m — 0, then the previous paragraph applies, and if m = r, then the ordered set {n+[£j\ - fJ,+ [£j+i] \ j = 1, •.., § — 1) = (2,..., 2) for the odd orthogonal coordinates as well, and we get that the last orthogonal coordinate ip't = 1, so 2 ip' = A'm here too. Putting these ip' and ijj" into the fixed point factorization (4.3.15), and taking the ratio, we have the fixed point factorization at the fundamental weights for even rank Cr , level k: Corollary 4.1. (to Proposition 4-1) Let r be even, and let m £ {0,1,... §}. Then 25£ m XA2mM = % = (-l) %^ - (-irx^( where ip' is given in equation (4-3.8), and primes denote A)^,L>._V level k + 1 quan tities. The r odd case When r is odd, r + 1 is even and r + 2 is odd, so in order to have the orthogonal coordinates (4.3.25) satisfying Lemma 4.3, we must have n even. Therefore, there is a t fixed point factorization only for the fundamental weights A2m, where 0 < m < ^-. Let A2m be such a weight. The permutation TT in equation (4.3.19) has sign . j (-l)m if ^i =4 0,3 sgn(7r) = | (_1)m+1 ifr|i=4l;2 • (4-3.32) Substituting A2m into equations (4.3.21) and (4.3.23), we find that the weights ip' and ip" are Ag (the vacuum) and A'^ respectively. Now substituting in A2m and Ao into equation (4.3.24), using (4.3.32), and taking the S-ratios, we have the fixed point factorization at the fundamental weights Corollary 4.2. (to Proposition 4-2) Let r be odd, and let m G {0,..., r^-}. Then m XA2JV) = ^ = (-I) x'^( where ip" is given in (4.3.22) and double primes denote C\l1 level | quantities. 2 Remark 4.3. When r is odd, C^-i is the orbit Lie algebra for Cr . 68 4.4 The Br fixed point factorization Remark 4.4. We prove the fixed point factorization formula for k odd only. The formula for k even holds in low-level, low-rank experiments, but is only a conjecture at this point. The simple-current Js is given in (2.3.13), with fixed points By equation (2.3.7), 2( k + 2r - 1 K / J (¥V;v?r-i,---,¥>i) iffciseven ,.,,, 4 4j ^ \ (^;¥>P_i,...,¥>i) if A; is odd • ^ ' Then by (4.4.2), ip'0 + 2( ( where the primed quantities denote C).2i level -ji (k odd) or A 2,'r_^ level k (any k) quantities. Suppose fc is odd. By Table 2.1, Xx(f) = 0 whenever Ar ^2 0. The fundamental th weight Ar has r component equal to 1, so XAr( 69 7 S W(An). The set of all possible inner products 7 • (// + p) is the set {7-{fi + p) I 76 W(An)}= (4.4.6) Define variables j/ttj. := cos I 2-7T Then by (4.4.6) and (4.5.11), we have the Fact 4.2. e V exp[-2?ri ] = 2 V yaiya2•••yat *•—' K i—' 7€lV(A£) l Now we specialize to p, = ip. Define variables y>'+[i] &« == cos [ n—^y- We claim that Claim 4.2. ~f-(ip + p) yj exp [—2-ivi— = (-2)M E i/ir-•»«!_!+ E »«r-V«« l Proof. Define : V I?] = J + E ^' for all j = 1,..., r - 1. (4.4.7) £=i We have the relationships: ¥>+[!] exp 2ni cos ( 2TT^-^ ] = -1 .^[J'-1] 2TU 2?ri (4.4.8) exp K exp K 1] for 2 < j < r. Define yaj := cos hir^"^ ). Then the right-hand side of Claim 4.2 70 is equal to VlVa Va Va 2^ £ * •••Vat+ "^ £ *--- i 2 = (-l)V^ ]T ya2---yae+ £ Vai-'-Vaif, I 2 and making the substitution bj-\ := aj — 1, we have that it is equal to (-1)' 2' I Yl cos(^[b1})...cos(^[be-1}) + £ cos (^6iU" •cos (—^N ) f • (4-4-9) Observing that 1 < j < r — 1 4=> 1 < r — j < r — I, tp\j] = ip'+ [r — j], and n' = |, (4.4.9) becomes the right-hand side of Claim 4.2. • Now we can substitute the right-hand side of Claim 4.2 into equation (2.2.16) to write the Br ' character of a fundamental weight as n 1 2 m 2m XA„M = E(- )' '{ An(^)- An(^+l)} £ y'al---y'at , £=0 l Now we find an expression for the C\.2\, level ^^ characters. Let A'n, 0 < n < th r — 1 be the n C^jj fundamental weight. The dominant weights of An are given in (2.3.16), and analogous to Fact 4.2, we'find Y exp 2^-W + rt =2* £ »£,..<,• (4.4.11) 7GW(A£) l which is proved in the same manner as Fact 4.2. Using equation (4.4.11), we have n n 2em (K (-i) (x'A^')+x'Kj n (-1)VA„(A*) - 2mXn(A£+1)) = (-l) mK(A'e) , (4.4.13) for all £ = 0,..., n. Equation (4.4.13) is a straightforward consequence of Lemmas 2.1 and 2.2, so Proposition 4.3 is proved. 71 4.5 The Dr fixed point factorization There are fixed points ip € P+{Dr ') when k is even. The nontrivial simple-currents are Jv, Js and JVJS. Their definitions will be given in each case. As we will see, the group of simple-currents is isomorphic to Z2 x Z2 when r is even, and Z4 when r is odd. The weight spaces of the fundamental weights are given in Lemma 2.3. When r is even, there are fixed points at all even levels. When r is odd, there are fixed points only when A\k. 4.5.1 The simple-current Jv The simple-current Jv is given in (2.3.27) and has fixed points r-2, 2{tp1+ip2 + --- + th Proposition 4.4. Let An be the n D)- fundamental weight, 1 < n { XA>A (f') ~ XAX v (' ) J if0 s where x\ * defined in equation (2.2.14), primes denote C£_2 level | quantities, ip' is given in (4.5.5), and where XA-i(v) *s interpreted as 0. By (2.3.22b), the orthogonal coordinates of the fixed point so we Fact 4.1 implies xx(f) =0 whenever Ar ^2 K-i, know immediately that = = XAr-iC'/') XAT{ k .(1); By (4.5.2) and (2.3.14), k V3'+[i]=r-(i + l)+ 2^, (4.5.6) 72 for each 1 < i < r — 2. Then we have the relationship for all 2 < i < r - 1. _ ib Let 0 < n < r — 2, and consider the n fundamental weight An = e\ +e2-\ \-en see (2.3.58). Recall from Section 2.3 the Dr Weyl group action. Since at least two of the orthogonal coordinates of An are 0, we permute the orthogonal coordinates and change any number of (or 0) signs. Therefore the Weyl orbit of A„, 1 < n < r — 2, is W(Kn) = < 5Z si% I 1 < »i < • • • < *n < r, s,- G {±1} > , (4.5.8) and W(Ao) = {Ao} (that is, the Weyl orbit of A„ consists of all r-tuples of r — n zeroes and n nonzero components € {±1})- Let 7 € W(An), and let /Z e P+(Dr). Then from (4.5.8) we see that the set of all possible inner products 7 • (fi + p) is / (r(/i + /))|7eW }= ^s/fe] I l th Similarly, letting W denote the Cr-2 Weyl group, An the n Cr_2 fundamental k weight, 1 < n < r — 2, and V € P+ (Cr_2), we have + {7 • (17+7) I 7 G W(X)} = J i X^s^ fe] I K ii < • • • < in < r,8j € {±1} 1 (4.5.10) By a simple induction argument, we have the useful identity exp ^ {s\ai H h snan) 2™cos(ai)---cos(a„). (4.5.11) («1. —.»n) \ Sl,...,«„6{±l} / Define variables + + ,3:=cos(2^" ^=cosfy ^ for all 1 < j < r — 2, and let (4.5.12) \ 73 Fact 4.3. + p) J2 exp[-2iri* ' ^ ] = 2*{S/ - S^_2} (4.5.13) 76WA,> for £ = n,n — 2,..., 0(1) «/ n is even (odd). Then Fact 4.3 yields n , +2 XAM = £ Vm-Anfri) - 2 ' Sn, (4.5.14) J=0(1) for each even (odd) n = 0(1),..., r — 2. Rearranging, we have n J 2 XA„(¥>)= E (2 mXn(Aj)-2^ mXri(AJ+2))Ej. (4.5.15) J = 0(1) j even (odd) Now we consider the characters of the Cy_2 level | fundamental weights A^, -i(i) eve 0 < ^ < r — 2. Plugging the relevant Cyr-rl22 l ^ f information into (2.2.16), they are X'M= E mA'(^) E exp[2^7-(/X + ^], (4.5.16) . J=0(/),,, 7eW(A^) j even (odd) J and so XA;>') - XA;_>') = E •2*(mx;(A;)-nX_a(A;))Ei. (4.5.17) J=0(1) j even (odd) Equating coefficients in (4.5.16) and (4.5.17), we see that equation (4.5.3) is true if and only if m 4 5 18 ™An(^) - 4^(^+2) = njtjXj) ~ A'n_2(tj)- ( - - ) When j = n, (4.5.18) reduces to m-^ (An) = rriji (An), which is clear since both sides are then 1. When j < n, (4.5.18) follows from the multiplicities in Lemma 2.3 and Lemma 2.2. Thus Proposition 4.4 is true. • 4.5.2 The simple-current Js when r is even The order 2 simple-current Js acts on a weight A by Fixed points of Js are of the form ip = {(p0; 2c/?0 + 2<^i + 4(^2 + • • • + <^r_i) + 2 74 The shifted orthogonal coordinates (2.3.22b) of + * M = r-i + U^ZB_gw (4.5.21) for 1 < i < |, which implies that exp[-2^^+[r-(t-1)]] = - exp[2*i^] (4.5.22) for all 1 < i< \. By Fact 4.1, x\( Then by (4.5.20), ' G pj^i15). Let k' := |; then «/ := § + r- 1 = §. The shifted orthogonal coordinates of ip' are ^'+ W = ^^t1"^ - £ ^ + ^ • (4.5.24) e=i Assume first that n is an even integer, 0 < n < r — 2, and we will do the cases r XAr-i 1 XAr f° the appropriate r values separately. Recall that the dominant weights of A„ (see equation (2.3.23)) consists of all other even fundamental weights Am such that m 1 ( P + Em:= Yl exp{-2iri ' '' ri} (4.5.25) 7eW( Am) and variables yi := cos 27r for all 1 < i < |. Note that we are defining Em slightly differently here than we did in Section 4.5.1. We claim that 75 Claim 4.3. Sm= n-D^K,,..^^ £..•»?, if mi, even ^^ [ 0 ifm is odd Proof. First we consider the case that in is even. If m = 0, then it is clear that Eo = 1, which satisfies (4.5.26). Now consider the case m = 2, which is illustrative of the general case. Let £j := exp [-2wi^]. Then by (4.5.22), Zi = -£-+!-,• , (4.5.27) and l >a >r+l—a >a ^a Then by (4.5.11), we have (4.5.26) for m = 2. th Now suppose that m > 4 is an arbitrary even number, and consider the (i\,... ,im) term (or more precisely the 2m terms when we consider all the combinations of ±) : 1 1 1 of the sum Em, namely ^ T ^ • • •£,- . Suppose that there exists among the ij a pair of fixed integers a and b, 1 < a < 6 < | such that r + l-67^0, and without loss of generality, suppose that i\ = a and it = b. Then by the above argument, we can see that there will be four terms in the 'orbit' of ^f1^1^1 • • • C^1 such that all terms in the orbit cancel each other. Therefore, the only terms that will survive the summation are those of the form /^V*1 ...f±i<±i _ r_nf f±2... m 0 = XA„(¥>) = £ An(^)E,- , (4.5.30) j odd 76 which can be true if and only if Ej — 0 for all odd j € {1,... , n}, since all the sums Ej and multiplicities are nonnegative. If m is odd, choose any odd n> m and then m will be among the j. Therefore, Em = 0 for all odd m, and so Claim 4.3 is proven. • Define variables ^:=cos[27r^l] (4.5.31) and sums and let Then we claim that Claim 4.4. ) E2^ = (-2)^{afE^ + a^1S^_1+--- + a^ }. (4.5.34) Proof. Let a, := 2TT^ for all 1 < i < §. Then 2 2 y?i • • • Vie = cos (o^) • • • cos (aie) = ^(l+cos(2ail))---(l + cos(2ait)) by (4.5.31) and the trigonometric identity cos2(x) = |(1 + cos(2x)). Now we claim that { e) e) ( (1+ oi) • • • (1 + at) = s 0 + s[ + • • • + s P , a a an( we ut := can rove where Sn' := J2i k k )+ fc 1 ) (l+ai)..-(l+afc+1) = E4* ° + E4* j'=o i=o k) k) fe) = 1 + (s[ + ak+1) + (4 + afc+isi ) + • • • ,/•(*;), (fe) \ i (fc) - 1 + s1 H + sfc+1 , 77 where we used the identity CO CO V^ V^ a a a s) +ak+isj_1 = 2_ ail---aii+ 2_^ h '' • ii-i k+\ l l for j = 1,..., k, and ah+isk ' = a,\ • • • a^ah+i = sk+1 . Therefore, £21 (-2Y l E 1 + E E *4 l + E E %'X + ---+ E y'h • • • y'ie 1<*1< —<**<3 l th The 0 sum Ei E E y'iav'iby'ic • \ E E ylyiMc = re_3) E riMh- Of course, there is also nothing particular about taking the third sum, as we did: we find that the jth sum is equal to I - j\ ST ,' ' £ - n I 2-^1 «1 ' ' ' yH ' J' l 'This restriction applies only to the example immediately following. 78 that is, (|_J)S^. Hence we have Claim 4.4. • We can now calculate m A 7 ( +p) XA» = E An( ^) 5Z exp[-27ri ' ^ ] ^=0 7eW(A2«) 2 m E2f = H An(^) n 2 f ] - Z^ ^An(A2^)(-2) {a? E'e + afl^ + ••• + a^}, £=0 where the last equality is by Claim 4.4. Collecting like terms gives us the expression XAM = J21 EmA"(A*>)(-2)Xb) I^ (4-5-35) for 0 < n < r — 2 an even integer. Now consider the fundamental weights Ar_i and Ar. In orthogonal coordinates, Ar_! = i(l,l,...,l,-l), and Ar = A(l,l,...,l) (see (2.3.58)). Since all of the coordinates are nonzero in these fundamental weights, it matters here that the Weyl group changes an even number of signs. Their Weyl orbits are W(Ar-\) = < - (s\,... ,sr) | Sj• £ {±l},Sj = —1 for an odd number of j > W(Ar) = i-{s\,...,sr)\sj£ {±l},Sj = —1 for an even number of j >. Then calculating the character (2.2.16), we find the following expressions for the and characters XAr_i(v) XAr(f)'- ( -25 (Ei + E'3 + • • • + E'r) if n = r - 1 and 2||r 2 4 5 36 XAn( Define coefficients fe ' EtJmXn(A26)(-2) af if0 Notice that At£ = 1 and Aij = 0 when j ^2 (•• Then we can rewrite equation 79 (4.5.35) (and incorporate n = r — l,r) to get the expression m XA»(v ) = E^EJ (4.5.38) j=o for all 0 < n < r (the ceiling function appears in order to handle the case n = r — 1 when 2||r, which is the only odd n). Consider the -Br' level | characters. We denote all Br' level § quantities by 2 2 primes. Let A^, 0 < n < |, be the n,h Br ' fundamental weight. Recall that the dominant weights of A„ are all A such that 0 < j < n (see (2.3.9)). Let AeP^M1'). Then 2 (4.5.39) To make the remaining equations in this section cleaner, we relabel the sums £'• =: an( tne are B„j := mr (A •) (4.5.40) for all 0 < n < |, 0 < j < n. Note that Bjj = 1. We will express the fixed point factorization implicitly as a matrix equation. Define the upper triangular matrix / AL i At £_i %o \ I 2'2 2>2 x A:= 0 Ar_x r_, ^i-i,o (4.5.41) 2 *'2 V o M),0 / and the vectors / OL \ I 2 > I XAr-l(XAr) \ 2 x XAr-2 (4.5.42) XA2 \ o0 j V XA0 / (1) if 2\\r (4|r). Then the characters of the Dr level k fundamental weights can be expressed by the matrix equation ~\ = A~o . Also define the matrix of coefficients BL L Bl / 2'2 £-1 0 Br #§-1,0 B := (4.5.43) V o Bo,0 / 80 and the vector / *AV \ *A'r (4.5.44) Then we can express the characters of the BL level 4 fundamental weights in the 2 matrix equation B~a = x'• Finally, we can write the fixed point factorization in matrix form as Proposition 4.5. Let r be even, and let Conjecture 4.1. Suppose 2\\r and 0 < m < \ — 1. Then l J [¥J (4.5.46) XA„ ifn = r — 1 2 where primes denote BV' level § quantities. 4.5.3 The simple-current Js when r is odd There are fixed points when 4|fc. The simple-current Js is given in (2.3.28); its fixed points are 2using Maple 13 81 By (2.3.22b), the shifted orthogonal coordinates of y>+[l] = \ + r-l k i~l ,r + 1, k r — 1 ^ 4 2 i-l + [r + 1 - i] = i - 1 + ^ y?f ¥>+[r] = 0 for i = 2,..., 2^. Note that because exph27ri^W] = _x exp[_27ri^ll!l] = _exP[-27rz^t±i^l] (4.5.49) exp[-2^ L 2 J] = Ti K, , .±tp+[r], exp\-27n n L J = 1 K for alH = 2,..., £^. Define K! := j, and V' = (Vr=i5 Vi^3, - • • ,<^i)- (4.5.50) 2 2 Then a XA2» = £A$aV (4.5.51) Now consider CrJ3 at level |; as usual, we let primes denote these quantities. ~2~ 82 The weight space of the fundamental weight An is given in equation (2.3.16); it consists of every other fundamental weight in descending order beginning at n. As in the previous section, let ag := T,'e. We obtain the following expressions for the CrJ3 level k characters. (4.5.52) x'A2r>') = E4T% j=0 m 3=0 Define the vectors ( Xhr-3 \ 2 XAr_5 (TT-3 , X •-- a :- (4.5.53) XA2 V XAo / V ^o y and the matrix I \^ 2 I A I 2 J ^> \ AU ' AU2„ v L(r-3 A ~ A:= 0 A r_2 l \ (4.5.54) V 0 0 A°) / Then we can express the D\-i(D level k fundamental weight characters as the matrix equation It (v) = A"CT . (4.5.55) We also express the characters of the fundamental weights A'n, 0 < n < ^^, as a matrix equation. Define the vector / *A' \ *A'. X':= (4.5.56) We have the following set of equations for the characters at the fundamental weights. We will assume that ^-^ is odd. If ^^ is even, then the roles of o\ and CTQ are exchanged in equations (4.5.57) below (that is, the final term in the first line 83 would be the <7o term and the final term in the second line would be the ai term). J'r-3 2 ~o ^ 2 /r-3 ^\ /l—3% CT 5 CT 2 CT XA;_3 (y') = ^-i + iLi_3 ^-2 + • • • + ^0 o (4.5.57) XA<>') = *i CT XA0 (') = o Then define the matrix 5 to be the matrix of coefficients 6a in equations (4.5.57), and we have the matrix equation "X V) = <^- (4.5.58) We can express the fixed point factorization at the fundamental weights as Proposition 4.6. Let r be odd, and let 4.5.4 The simple-current JVJS, r even Recall the conjugation C\ in (2.3.29); we will denote it by C in this section. Let tps be a fixed point of Js as in (4.5.19). We have the identity JVJS = CJSC, which implies that i Cl XA2m{ P) = XA2m( Ps) = —£ = ~e — = XA2m{ where the last equality is because C acts trivially on A2m and 0. Now suppose 2||r. Then XAr( 84 simple-current Js at Ar, and the fixed point factorization for JVJS at the fundamen tal weight Ar is identical to that for the simple-current Js at Ar_i. The fixed point factorizations for Js are given in Section 4.5.2. 4.5.5 The full simple-current group (JV,JS) when r is even We consider weights that are fixed simultaneously by Jv (2.3.27) and Js (2.3.28); they are 2(¥>i + --- + y>r_1) + ^r = -. (4.5.61) Since XhnW) = XAJ>0 - X^_2(V'), (4-5-62) i t an where ip' = (ipi; apply Corollary 4.1 to each of X'A' ( is a level | + 1 weight. We thus have Corollary 4.3. (to Proposition 4-4 and Corollary 4-1) Let r be even, and suppose that ip G P+{Dr ) is a fixed point for both Jv and Js, as in (4-5.60). Let 0 < m < r-=£. Then m 4 5 64 XA2„» = (-l) (XA^(v") + XA« _>")), ( - - ) ( where double primes refer to A z-2,_, level | + 1 quantities. 85 Chapter 5 Applications of fixed point factorization to NIM-reps In this chapter, we use the fixed point factorizations from Chapter 4 to find simple expressions for the NIM-reps associated to the classical affine algebras. Physically, a NIM-rep is the partition function associated to a cylinder (just as a modular invariant Z(T) is the partition function of a torus). In-the context of this chapter (that is, simple current modular invariants for Xr ), the NIM-rep can loosely be regarded as the generalization of fusion coefficients to non-simply connected Lie groups. Calculating NIM-rep coefficients is difficult, and the only other formula was found by Gaberdiel-Gannon in [15], where they used the fixed point factorization of [26] for the case of the ^4-series. It is important however, to have nice expressions for NIM-reps in order to compute D-brane charges (we comment on this further in Section 6.2). We are interested in the NIM-reps at the fundamental weights - from these, we can find the NIM-reps at all A e P^(X^) (see (5.1.2) below). To our knowledge, the ^-matrix defined in (5.1.3) below does not appear explic itly in the literature for these NIM-reps, although it is generally implicit in the work of Fuchs-Schweigert and collaborators (see e.g. [3], [35]). The first step in finding the NIM-rep coefficients therefore is to find explicit expressions for 4". The proof that the matrices N\ defined in (5.1.3) do indeed constitute a NIM-rep is generally believed to be implicit in work by Fuchs-Schweigert and by Evans et al. [1]. The subtle point is proving the nonnegativity and integrality of the NIM-rep coefficients. A direct description of the NIM-rep coefficients (5.1.3) does not appear anywhere in the literature, and there would have been no reason to expect them to be partic ularly nice. We will find, using fixed point factorization, that missing description, and it is surprisingly simple. 5.1 The NIM-rep In Section 2.2.6, we defined the fusion ring of an RCFT. A NIM-rep A/" is a nonneg- ative integer representation of the fusion ring. Specifically, to each a £ $, we assign 86 a matrix TVa that has nonnegative integer entries and satisfies K^b = J2N^c (5.1.1) We also require that TVo = / and Mca = A/"^, where C is charge-conjugation (see Section 2.2.3) and ' denotes matrix transpose. We call two NIM-reps TV and TV' 1 equivalent if there is a permutation matrix P such that TVa = PN'aP~ for all a € $. An example of a NIM-rep is the assignment of the fusion matrix Na (see (2.2.6)) to each a £ $. In this example, the rows and columns of the matrices TVa = Na are indexed by $, but this is not the case for a general NIM-rep: the rows and columns of TVa are indexed by what are called physically the boundary states. In particular, these are the possible states of the endpoints of open strings - see the discussion in Section 6.2.2. Now we specialize to the affine Kac-Moody algebras. As a ring homomorphism, the NIM-rep is structure-preserving, so in the case of the affine Kac-Moody algebras, we have, analogous to (2.2.20), Mx = PxWA1,---MAr) (5.1.2) for all A s P+(Xr '), where P\ is a polynomial. Thus, to specify a NIM-rep, it is enough to find the NIM-reps at the fundamental weights A, for each i € {1,... , r}. The matrices J\f\ by definition will clearly commute and be normal, and thus can be simultaneously diagonalized, by a unitary matrix ^. Hence the NIM-rep coefficients are given by the equation where A € P$.{xP), x and y are boundary states, and the sum is over all expo nents \x. The exponents form a multi-set £ (TV), that is, elements of £ (TV) appear with multiplicities as eigenvalues (the boundary states and exponents will be given explicitly for each algebra). We can see that the form of (5.1.3) is quite similar to Verlinde's formula (2.2.5), but now \I> is the matrix that simultaneously diagonal- izes the NIM-rep matrices TV\ with eigenvalues g^1. Unlike in Verlinde's formula however, the rows and columns of \P are indexed by different sets respectively, and so we say that ^ is not 'truly square'. Recall the definition of modular invariant (2.2.11). Any RCFT has a modular invariant and a NIM-rep. Let M be a modular invariant for an RCFT; then we can speak of a NIM-rep associated to M as follows. Define the exponents of a NIM- rep TV to be the multi-set £ (TV) described above, and the exponents of a modular invariant M to be the multi-set £{M) consisting of primaries b € i> appearing with multiplicity M^,. We say that TV and M correspond if £ (TV) = £(M). For example, the NIM-rep corresponding to M = I (the identity) is the above-mentioned fusion matrix representation a H-> A^a. 87 Let J be a simple-current for an amne Kac-Moody algebra X}- . A NIM-rep corresponding to a simple-current modular invariant M[J] (see (2.2.23) and following remark) can be constructed as follows. Let A € P+(Xr ), and let \& be as in (5.1.3). To describe the boundary states, we introduce the order of a weight A. Suppose that J is a simple-current of order n, and let A £ P+{Xr '). Let m be the smallest positive integer such that JmX = X, and let ord(A) := ^, that is, ord(A) is the order of the stabilizer of A in (J). The boundary states are given by J-orbits of weights [X,j] = {J* A | i = 0,... ,n - 1}, 1 < j < ord(A). The multiplicity of A is M[J]XX- By_a_slight abuse of notation, we will identify £{M\J\) with the set £ := {(A, j) | A e P+(XW), l 2/ord(i/') when J has order 2 [17]. This is an elementary consequence of, for instance, the formulas for \I> given below. The interesting question then is to determine the entries •hfujll wnen both ord(i^) > 1, ord(*/) > 1. This is what we do in this chapter. Incidentally, when both [v,i\, [i>',j] have order greater than 1 (that is, they are both fixed points), (5.1.5) generalizes to [17] ord(j/) 2/ord(i/) i=l 1=1 Throughout this chapter, we will use the following shorthand notation: if ord(A) = 1, then we denote [A, 1] by [A], and if mult(A) = 1, then we denote (A, 1) by (A). We will use the notation P+ to refer to the set { 5.2 The NIM-reps for A^ Let n := r+1. The possible NIM-reps are parametrized by divisors of n, so the situation is more complicated (and different from) the other algebras. These NIM- reps fall into two categories - pathological and non-pathological (generic) - and the NIM-reps were found in the non-pathological case. See [15] for details. 88 5.3 The NIM-reps for rf1} There is one simple-current, of order 2, given in (2.3.20). The set of boundary states is B = {[\] | A ^ i^} U {[¥»,*] | v» e -P+, 1 < * < 2}. (5.3.1) The diagonal entries of (2.2.23) are z Z 5 3 2 M[J)XX = 5jX,x6 (QJ(X) + Y) + * (QJW)- • ( - - ) Thus we see that the multiplicities will depend on the factors of r and k. Let [X,i] £ B, (n,j) £ £. The entries of the ^-matrix are ^5V if A €P(,»?Pi *[A,I],(M) 0 if\$P(,n&P( ' (5-3-3^ £ (sv + (-ly+Je-^S-^) if A, /i € Pf where (5 is a rational number involving the quantities k and r (its explicit value need not be given here), and h = | or 1, if 4p is odd or even respectively. The tildes in the last line of (5.3.3) denote quantities of the algebra involved in the fixed point factorization (in Chapter 4, these are the primed quantities). We will explicitly state in each case to which algebra the tildes refer. 5.3.1 The r even case If A £ P^, then by equation (5.3.2), the multiplicity of A is Sz(Qj(X)) = 1. There fore, a non-fixed point A is an exponent if and only if Qj(X) £ Z. Furthermore, if such a A is an exponent, then mult(A) = 1. Let 2 2QJ(¥>) = fi + =2 0, and so Qj(ip) £ Z for all 2QJ{ 89 for any p € P+. The case ^ odd Let tp € P(. By (5.3.5), 2Qj{ip) € 2Z + 1, so Qj((p) £ Z. Thus by (5.3.2), = i, that is, mult( £ = {(A) | A # P(, Qj(X) e Z} U {(ip) | y> € P(}. (5.3.6) By Fact 4.1, Qj(y) £ Z for all y> € P^. In particular, Sw< = 0 for all , y>' € P^, and so by (5.3.3), the ^-matrix entries with A = ip are given by 5 3 7 *M,(,) = where S is the 5-matrix (2.3.35) for A^)r, level k + 1. Now we can calculate the entries of A/^ at entries involving fixed points. Let 90 0 < m < §, and let KfVP'J] A2m[V,«] E *[¥>.t1(M) 50/1 (V'JI(M) l C '~>A2mM 1 Q w E V2 50M v^ vc (»0€£ i+1 J+1 (-l) „-6^ AW (~1) ,67riDm06 '5,5 , e 5-te.3.8) E >/2 ^ 50^, v^ »)e£ l > E Q ^A2mH Q , / -I \i+7 V^ C - A2mV' c {»ep*(cl1'>) •>l>eP± C *^A2mM c E 5o^ v> UeP*(cD /_ j\i+j+TO _ sAmV" ; E S~_T The first equality in (5.3.8) is by equation (5.1.3), and the second is by equation (5.3.7). The third equality is by Fact 4.1, because all the /x € P+(Cr) such that M^ = 0 have Qj(/x) 0 Z. Therefore we are not adding anything to the first sum by summing over all weights \i € P+(Cr '). We used fixed point factorization in the last line: equation (4.3.31) in Corollary 4.1. The set of all ips is precisely the set jRj+1(>4?t)_i). which gives us the subscript in the tilde sum of the last line. Looking at the steps in the above calculation, we can see that if we had chosen a fundamental weight A2m+i instead of A^m, the second sum in the second line would have been 0 by Fact 4.1. Therefore, M^ , .,, and by the last line in (5.3.8), we have the formula MY*} ., = l(Nf' „ + (-l)i+i+mN? _). (5.3.9) Note that we are slightly abusing the tilde notation in the right-hand side of (5.3.9): th ( the tilde in the ATO indicates that Am is the m A )r. fundamental weight, and the tilde in JV-? _. This is addressed for m — 1 in Section 5.6. Am¥>' A2mV — Am 91 The case 4f even Suppose (f £ Pf. As we found above, Qj(f) £ Z, so the multiplicity of ip is z z M\J]W = & (Qjb) + ^+s {Qjm = 25z(Qj&))=2- Therefore, for ^ even, the exponents are ; £ = {(A) | A £ P^ Qj(\) e Z} U {(¥>, i)|^P+ , 1 < i < 2}. (5.3.10) By (5.3.3), the relevant ^-matrix entries are _ J ^Sm if fig Pi ^^''"^(V+t-l)^-6^) if/xG^ • (5-3'n) Let 0 < m < §, and let ip,ip' £ Pi- Then by (5.1.3) and (5.3.11), we have 2 /- ^ % + i E(^ + (-1)i+V6^%)%^(5^ + (-i)J+£^w KK2 - 2 2- ^ 5OM V" (M)££ 1<£<2 l 92 last line disappears. Therefore, continuing the above, calculation, we have that C*Oe£ M V-ePi = ^< 6*"^—VM + (-1) z^ bip*-jrrb? where we again used the fixed point factorization (4.3.31) to obtain the last equality. We once again have the formula (5.3.9) for the NIM-rep. 5.3.2 The r odd case Recall that since r is odd, k must be even for fixed points to exist. If k ZQjiv) =2 Pi + The case ^ odd In this case, | must be odd; hence Qj{tp) £ Z. The exponents are given in (5.3.6), and the relevant ^-matrix entries in (5.3.7). The calculation thus works the same way as in the r even case. Therefore, we have (5.3.9), where the tildes denote level | Cr_! quantities. We used the fixed point factorization (4.3.33) from Corollary ~~2~ 4.2. The case ^ even Here, Qj{ 5.3.3 Examples of B and £ Consider the algebra C\ at level /k = 2. There are ten highest weights. The fixed points are (0,0,1) and (1,1,0) (omitting the 0th Dynkin label) and have multiplicity 93 1 - hence the modular invariant M[J] is a permutation matrix. There are eight boundary states and exponents. The boundary states are the J-orbits [(0,0,0)], [(1,0,0)], [(0,1,0)], [(2,0,0)], [(0,0,1),1], [(0,0,1),2], [(1,1,0),1], [(1,1,0),2], and the exponents are (0,0,0), (2,0,0), (0,2,0), (0,0,2), (0,1,0), (1,0,1), (0,0,1), (1,1,0). At level k — 4, there are thirty five highest weights. There are three fixed points: they are (1,1,1), (2,2,0), and (0,0,2). There are twenty two boundary states and ex ponents. The boundary states are [(0,0,0)], [(1,0,0)], [(0,1,0)], [(0,0,1)], [(2,0,0)], [(1,1,0)], [(1,0,1)], [(0,2,0)], [(0,1,1)], [(3,0,0)], [(2,1,0)], [(2,0,1)], [(1,2,0)], [(0,3,0)], [(4,0,0)], [(3,1,0)], [(1,1,1),1], [(1,1,1),2], [(2,2,0), 1], [(2,2,0),2], [(0,0,2), 1], [(0,0,2), 2], and the exponents are (0,0,0), (0,1,0), (2,0,0), (1,0,1), (0,2,0), (2,1,0), (0,3,0), (4,0,0), (0,1,2), (3,0,1), (2,0,2), (1,2,1), (1,0,3), (0,4,0), (0,2,2), (0,0,4), ((1,1,1),1), ((1,1,1),2), ((2,2,0),1), ((2,2,0),2), ((0,0,2),1), ((0,0,2),2). 5.4 The NIM-reps for B? The simple-current Jv is given in (2.3.13) and has order 2. Since there is only one simple-current, we will refer to it as J for the remainder of this section. For the fixed points of J, see (4.4.1). The boundary states are given in (5.3.1), and by (2.2.23), the diagonal entries of M[J] are M[J]XX = SJX,X + SZ(QJ(X)). (5.4.1) We see that for A ^ P+, A is an exponent if and only if Ar is even, and if A ^ P^ is an exponent, then mult(A) = 1. If ip £ P+, then ip is an exponent, and 2Qj(ip) = , . f 1 if k is odd ,_ . mult = 5 4 2 ^ {2 if Ms even • < " - > 5.4.1 The case k odd By (5.4.2), the exponents are given in (5.3.6), where Qj(A) is now Ar. The ^-matrix entries involving fixed point boundary states are given in (5.3.7), where tildes denote Cl_i level ^i quantities. Let 0 < m < r. Using the fixed point factorization (4.4.5), an analogous calculation to that for the ^ odd case in Section 5.3 yields the NIM-rep formula +3+n + J^JI.] = lH<^^y ^L ^ij} ^^-i ,(5.4.3) AnM \ 0 if n = r where the tildes denote C^jj level ^^ quantities, and ip is given in (4.4.4). 94 5.4.2 The case k even By (5.4.2), the exponents are given in (5.3.10), and the relevant ^-matrix entries in (5.3.11). Using the fixed point factorization (4.4.5), we obtain the NIM-rep formula (2) (5.4.3), where the tildes denote ^(r-i) ^eve^ ^ quantities, and ip is given in (4.4.4). 5.5 The NIM-reps for D^ The simple-current group is generated by two simple-currents: Jv (see (2.3.27)) and s Js (see equations (2.3.28) and (2.3.28)). Let P£ := Pp, P| := P^ . Fixed points of Jv are given in (4.5.1), and fixed points of Js are given in (4.5.19) (if r is even), or (4.5.47) (if r is odd). Also, we define QV(X) := Qjv{\) and Qs{\) := Qjs(X). 5.5.1 The simple-current Jv Recall that in this case, we must have the level k even in order to have fixed points. If A ^ P+, then M\JV]\\ ^ 0 if and only if Xr =2 Ar_i, in which case M[Jv]xx = 1. If £ = {(A) I A 0 P;, \r =2 Ar_i} U {(if, i)\ The entries of the ^-matrix at fixed point boundary states are given in (5.3.11) with Pi = P^. Using the fixed point factorization (4.5.3) and the usual calculation (see the case Y even in Section 5.3.1), we obtain the NIM-rep formula <&-{H<'+(-1,W(i5L-*C*>} '"^;-2, (5,,, [0 if n = r — l,r eve where tildes denote Cy-2 ^ ^ f quantities, ip is given in (4.5.5), and where we take any empty quantities to be 0. 5.5.2 The simple-current Js when r is even The order of Js is 2. The multiplicity of The case ^ odd The exponents are E = {(A) I A £ P*, Q.(\) £ Z] U {( 95 In Proposition 4.5, we showed that for the even fundamental weights A2m such that 0 < m < ^, there is a fixed point factorization of the form \\2m — 2Zj=o ^™>j <^A Then we have (M)€£ ^ (v)6£ ^ as usual. Also, since all the weights fi £ P+{Dr ') such that either /i € P+ or fi # £ have Qs{y) & Z, we can take the first sum over all P+(Dr ') without changing anything. Thus, the first sum is \N% . Then we calculate the second sum: it is equal to m {-^ E % (E ^^ w) s?z (5-5-5) ^2°m,e E %XA£W^ 2 A£y ' If we take a fundamental weight A„ such that n is odd or n = r, we see that the sum in (5.5.5) will disappear, and if we take instead n = r — 1, we will have only the m term xxr (VO the second last line instead of the sum Yl^Lo (this is very similar to 5 the C-series calculation). We thus obtain the following formulas for the NIM-rep +i \ «„* + (-l)' £S,» 0 if n is odd or n = r (5.5.6) where tildes denote Br level \ quantities, and The case ^ even In this case, ^ is even. When 4|r, 2<5s(y) =2 0. When 4 \ r, we must have 4\k, and so 2QS( 1where we conjectured in Chapter 4 that the Cm,j are binomial coefficients when 2||r - see Conjecture 4.1 96 Therefore, the exponents are £ = {(A) |A £ P%, Qs{\) € Z} U {(if, i)k£P;,l The entries of the ^-matrix involving fixed points are given by j *[*AM) \ i(5w + (-l)^exp[-67ri*]5^) if /x € P} ' ^ The simple-current JVJS when r is even We state this result here as this case is closely related to the Js, r even case in Section 5.5.2. The NIM-rep coefficients are given in (5.5.6), with the only difference being that the roles of Ar_i and Ar are exchanged (see Section 4.5.4). 5.5.3 The simple-current Js when r is odd Recall that 4\k in order to have fixed points. The simple-current Js has order 4, and Jg = Jv has order 2. Let B = {[A]|A0P;,P£} U {[X,i] | AeP^,A£P£,l M[Js}xx = JZ54^ (GSM + ^) • (5-5.9) The case | odd In this case 2Qs{ip) =2 1, so (5.5.9) becomes z z M[JS}XX = 26jsX,x5 (QS(X) + Y)+ (*W + l)6 (Qs(X)) 1 if A £P£,P* 2 if AeP£,A£P|,<2s(A)€Z . 2 ifAeP£ Therefore, the exponents are £ = {(A)|A£P;,P«,QS(A)€Z) U {(A,t) I XePZ,\#P*.,Qa(\)£Z,l 97 Since Qs( tf [VMKMJ) (5.5.10) ^^exp[-67ri«]S^ if/z€P+ where the tildes denote ,C>-3 level | quantities. Let 0 < m < n^. In Section i = r some 4.5.3, we found that XA2m( P) Y^T=odm,e XA (£0> f° constants dmif defined implicitly by (4.5.59), and where yp is given in (4.5.50). We obtain the NIM-rep formula l+j M(^) = { 5( The case | even In this case, Qs{^>) G Z for all z M[JS]XX = (l + Sjvx,x + 26jsX,x)S (Qs(X)) 1 if\#P$,Pi,Qa(\)£Z 2 if Ae/»,A^P«,QS(A)€Z (5.5.12) 4 if A G Pi Therefore, the exponents are £ = {(A)|A£P^,P*,QS(A)GZ} U {(A,t) | AeP;A^P|,ft(A)eZ,l<«<2} U to,i)|^P|,l The ^-matrix entries involving fixed points are given by if M^n>-P+ * [V>«].(M»J) (5.5.13) 2^( V + (-l)'+j exp[-67T^]%) if /i G P£ We find that the NIM-rep coefficients are given in (5.5.11). 5.5.4 The full simple-current group (JV,JS) when r is even This is the simple-current Jc in Table 2.1; we will denote it by J. The fixed points of J are given in (4.5.60); they occur when k is even. Let Qc := QJc and P% := Pf. 98 The boundary states are given in (5.3.1), and the diagonal entries of (2.2.23) are Z Z M[J\XX = 8JX,X6 (QC(X) + y) + S (QC(X)). (5.5.14) For ip e P+, 2Qc(ip) =2 ^f, so we split into cases. The case ^ odd The exponents are given in (5.3.6), and the relevant ^"-matrix entries in (5.3.7), { where S is the 5-matrix for A ^!_1 at level § + 1. We find the NIM-reps V 2 ' r-2 Anlwl \ 0 if n is odd or n = r (5.5.15) where Ip is as defined in (4.5.63), and tildes refer to ALV=2I_I level | + 1 quantities. The case ^ even The exponents are given in (5.3.10), and the relevant ^-matrix entries in (5.3.11), where S is the 5-matrix for A„,r_2. , at level | + 1. We also find the NIM-reps 2(-j-)-l * (5.5.15). 5.6 Using the Pieri rules: an example As we have seen, the NIM-rep coefficients for a simple-current invariant reduce to the fusion coefficients for the fusion ring of the underlying algebra. This is very encouraging, as the fusion multiplicities - which are known to be nonnegative integers - should be accessible to us. The Kac-Walton formula [30, 36] N* = Z^M mx^W)) (5.6.1) wew gives the fusion coefficients in terms of the tensor product multiplicities, and thus provides a Lie theoretic interpretation for the former. We are interested in the coefficients N% such that A is a fundamental weight. It is not immediately obvious why the fusion coefficients in (5.3.9) say, should be always congruent modulo 2, or why the fusion coefficient on the left cannot be smaller than the one on the right. In this section, we investigate these questions by calculating the fusion products AiKl^A and A2 ^e ^ (where we are now using £ to denote the level of the representation) to prove positivity and integrality of the NIM-rep coefficients •^^1$ for Cv at odd level I in (5.3.9). As mentioned in Section 5.1, this would then imply positivity and integrality of the full matrix 7VA2. 99 Let p, G P+(Cr ). The dominant weights of A2 are Ao with multiplicity r — 1, and A2 with multiplicity 1 (see Lemma 2.2). Using (5.6.1), we find the following. The first fundamental. r Ai 8/ /* = ®Li(A« + AJ - Aj_i) + ® i=1(n - A* + Ai_i), (5.6.2) subject to the conditions: if /ij_i = 0, then we delete the ith term in the first sum, and if \n = 0, then we delete the ith term in the second sum. Put another way, if IM = 0, then we delete the (i + l)st term in the first sum, and the ith in the second sum, for all z = 0,..., r. We take the empty quantities A_i and Ar+i to be 0. Thus The second fundamental. For clarity, we will present the weights of A2^/x separately, as there are conditions on each one. Define c(/x):=||{W|^=0)t = l>...,r-l}||. Then K,n = r - 1 - c(n). (5.6.3) Now let 1 < i, j < r be such that j > i + 1. We have the following terms, each with multiplicity 1 provided the conditions in brackets are satisfied: fi + Ai- Aj_i + Aj - Aj-i (IH-1, Mj-i > 0) H + Ai - Ai-i - Aj + Aj-i (fii-i,fj,j>0) (5.6.4) \i - Aj + Aj_i - Aj + Aj_i (//j, HJ > 0) /j. - Ai + Aj_i + Aj - Aj-i (m, fij-i > 0) We also have (this is the j = i + 1 case) the following terms, each with multiplicity 1, provided the conditions in brackets are satisfied: H + Ai+i - Aj_i (fii-i > 0) fi - Ai+1 + Aj_i (/Xj+i > 0) , . y fj, + Ai+1 - 2A{ + Aj_i (/x, > 1) ' ' ' ' M - Ai+i + 2A, - Aj_i (Mi-i, IH+i > 0) Thus, <2M € {0,..., r - 1}, and iV^ € {0,1} when v ± //. Suppose r is odd. The NIM-rep coefficients for Cy at level fc reduce to (5.3.9) with tildes referring to CJ^ level | quantities. Suppose that y/ = ip in (5.3.9). 2 Then clearly ip' = ip as well, but by (5.6.2), N? _ = 0. Therefore, 1 ^fi.mAa[^,ml ] = ^2 ^2,V = 2 ^- -^)) where the first equality is by (5.3.9), and the second equality is by (5.6.3). The fixed point 100 an odd number of Dynkin labels tpi such that 1 < i < r — 1, to be 0. Therefore, c(p) =2 0, and so r — 1 — c(^) =2 0 as well, and it is clearly nonnegative. We have thus shown that .A/^j , is a nonnegative integer. Now suppose that Now let r be any odd rank, and suppose ip' = 0 (note that tpr=±_,2 ,<. = ip' = ipT_z±A0 -^ h (<^r^i_/n_n - l)A„_i + { If' = A +^ ^i + ^1=1 Ar±i + • • • + (Vrfi_n+1 - l)Ar±i+n_! + ( = ip -'Ar^l^^ + Ar=l _„ + Ar±l+„ - Ar±l+n_j , 2 which is the fourth line of (5.6.4) with i := L-^—n-\-\ and j := '£^-+n. The condition in brackets that ^i^i_n+j,^r±i ,n_1 > 0 is satisfied since The case 101 Chapter 6 Concluding remarks 6.1 Summary and discussion In this thesis, we presented three main results involving the affine Kac-Moody al gebra conformal field theories (the WZW models) - the non-unitary W3 minimal model modular invariant classification, the fixed point factorizations for the classi cal affine algebras, and, using fixed point factorization, formulas for the NIM-rep coefficients of the simple-current modular invariants. The fixed point factorization for the j4-series was done by Gannon-Walton in [26], and NIM-rep formulas for the j4-series were done by Gaberdiel-Gannon [15]. 6.1.1 The non-unitary W3 minimal models The W3 data is parametrized by pairs (p, q) of coprime integers - this theory is unitary only when q = p + 1 (this unitary classification was done by Gannon- Walton). The general case, done in this thesis, is far more complicated than that unitary special case. It used to be the case (in the early days of CFT) that a theory was required to be unitary in order to be interesting. However, non-unitary theories have been found to correspond to physically interesting systems (including in string theory), and the requirement of unitarity is no longer regarded as necessary. The loss of positivity of the vacuum column in the S-matrix of a non-unitary theory generally introduces complications, and this was our main motivation for looking at the W3 minimal models, that is, to see what happens differently in the non-unitary case. The proof relied on a trick (that is, the 'Galois shuffle'), to relate this non-unitary classification to that of the affine algebra {A^^A^)^. The Galois shuffle is a simple way to relate the vacuum with the primary of minimal conformal weight (giving the positive S'-column in a 'healthy' non-unitary theory). A theory possessing a Galois shuffle is said to have the GS property - the W3 minimal models have this. We essentially unitarized the non-unitary W3 modular data, and then found the modular invariants for this new unitarized data. We did this through Lemma 3.2, which gives a way to associate a W3 modular invariant for this unitarized data with that of (A2 © ^2)^ [2]- The classification (see Theorem 3.1) comprises modular 102 invariants falling into the A, T>, and ^-series, as happens in unitary theories. We note that a non-unitary A\ classification was done by Lu in [34]. This is a 'sick' theory in the sense that there is no strictly positive row or column of 5 (so some fusion coefficients are negative), and there are infinitely many modular invariants at each fractional level k. Because of the lack of a strictly positive S- column, we do not expect the Galois shuffle to work here, however a future project is to understand [34] from the point of view of the Galois shuffle, and see how simply we can do this classification using the ideas of [22] and this thesis. The W2 minimal models were done by Cappelli-Itzykson-Zuber in [6], by finding a basis for all complex matrices satisfying MS = SM for A\ level k. Prom this they obtained a basis for the W2 minimal models. This basis was simple enough that it was possible for them to determine which M had nonnegative integer entries and satisfied Moo = 1- This analysis did not see whether the theories were unitary or non-unitary however, and did not generalize to W3, or to any other modular invariant classifications. Although one can find a basis for all Wjy, this basis becomes very complicated (and not useful anymore as nonnegativity and integrality become difficult). New ideas were needed, even for unitary theories (see the discussion in Section 3.1 for an outline of the modern method). Tools such as the parity rule and the Galois shuffle enable us to do modular invariant classifications far faster and more elegantly. For example, see the modern A^ classification, re-done by Gannon in [21] with some of these tools. Having successful used these modern ideas, including the Galois shuffle, in the W3 non-unitary classification, we are interested in applying this method to re-do the Wi classification. 6.1.2 Fixed point factorization The fixed point factorizations that we found are the major result of this thesis. Fixed point factorizations are dramatic simplifications of S-matrix entries at certain points (namely fixed points of simple-currents). The fixed points are special in that they are the cause for complications in many formulas (recall (5.1.5) and (5.1.6)). Since fixed point factorization is indicating something about such a fundamental quantity of an RCFT (namely the 5-matrix), we expect it to have far-reaching consequences. Indeed, we found fixed point factorization to be a powerful tool in finding simple expressions for NIM-rep coefficients, which in turn, should allow us to find the corresponding D-brane charges in string theory relatively easily (see Section 6.2.2). Previously, it was known that there was a fixed point factorization for only the yl-series [26]. Because the yl-algebras behave differently than the B, C and D- algebras (and there are differences within these as well, most notably between the symplectic (C) and orthogonal (B, D) algebras), there was no reason to necessarily expect that fixed point factorization would be a feature of all affine algebras rather than something particular to the ^4-series. The most difficult part in obtaining this result was in finding the form that the fixed point factorizations would take. The formulas and their proofs (summarized in the next paragraph) turned out to be quite different than those for the ,4-series, with the B and D-algebras being the most different as their fixed point factorizations at the fundamental weights involved 103 linear combinations of smaller-rank characters rather than reducing to one smaller- rank character which was the case of the A-series (equation (4.2.3)) and the C-series at the fundamental weights (equations (4.3.31), (4.3.33)). The fixed point factorization was found for all weights for the C-series and at the fundamental weights for the B and D-algebras. In the case of the C-series, done in Section 4.3, the 5-matrix of (2.3.19) displays a symmetry at fixed points, which can be interpreted as two factors of smaller-rank algebra 5-matrices. The two factors are different and involve twisted yl-algebras - the exact factors depend on whether r is even or odd. The fixed point factorizations were given in Propositions 4.1 and 4.2 respectively. In Corollaries 4.1 and 4.2, we specialized to the fundamental weights, which is our main interest. The proofs for the B and D-series involved character arguments. We used the weight multiplicities of Lemmas 2.1, 2.2 and 2.3 to write the characters of the fundamental weights in terms of the characters of the smaller-rank algebras. The simple current group for Br is isomorphic to Z2, and we found the fixed point factorization in Proposition 4.3. In the case of the B-series, the smaller-rank algebra was a twisted >l-algebra. When the level k is odd, the 5-matrix of this twisted algebra is the same as the 5-matrix for a nontwisted C-algebra. Data for the nontwisted algebras is much more readily available, so we converted the problem into a nontwisted one. This means however, that our proof didn't carry through for even levels. Finding the relevant data for the twisted ^-algebras should not be difficult, but we did not do it here in the interest of time. This is something we plan to do in the near future in order to confirm our formula (4.4.5) for all levels k (based on low-level, low-rank computer experiments, we conjecture that (4.4.5) holds for any k with the twisted /1-algebra as the smaller-rank algebra). It is interesting to point out that when k is odd, there are two choices for the smaller-rank fixed point factorization algebra, one of which is the orbit Lie algebra (see the discussion in Section ??). The simple current group for the £>-algebras is generated by two simple currents - Jv and Js, and is isomorphic to Z2 x Z2 or Z4 depending on whether or not r is even. The former was the most difficult proof, particularly the proof for the fixed points of the Js simple current (2.3.28), in Section 4.5.2. The fixed point factorization for the simple current Jv (2.3.27) is given in Proposition 4.4 of Section 4.5.1. For both r even and odd, the fixed point factorizations for fixed points of Js were more complicated than the other cases and are given implicitly - see Propositions 4.5 and 4.6 in Sections 4.5.2 and 4.5.3 respectively. In one case, we conjectured an explicit formula (see Conjecture 4.1), although this is not needed in order to establish a fixed point factorization. The fixed point factorization for the simple current JVJS (occurs when r is even) is closely related to that for Js - we did this case in Section 4.5.4 by relating JVJS to Js through the conjugation (2.3.29). In Section 4.5.5, the fixed point factorization was done for fixed points of the full simple current group (Jv, Js) (occurs when r is even) by applying two previous fixed point factorizations (see Corollary 4.3 for the fixed point factorization). To prove the fixed point factorization formulas, we needed to know' the weight multiplicities at the fundamental weights (Lemmas 2.1, 2.2, 2.3) - we note here 104 that while these are surely well-known, we could not find a proof of them in the literature, so we found and proved formulas for these in this thesis. The reason for concentrating on the fundamental weights is twofold: first, this is what we needed to find the NIM-rep formulas and for further applications to D-branes (discussed in Section 6.2.2). Second, knowing that a fixed point factorization exists at the fundamental weights implies that there is a fixed point factorization at all weights A (and theoretically, we could find it as the characters for each A are polynomials in the characters of the fundamental weights), and so we do not need an expression for general A to know that the WZW models have this feature. We also note that the smaller-rank algebras appearing in the fixed point factor izations at the fundamental weights are, with the exception of Cy for r even, in exact correspondence with the orbit Lie algebras of [12] (see Table 6.1 below). The significance of the orbit Lie algebra is We do not know whether this superficial similarity corresponds to an actual relation, but it would be fascinating to investigate this further. We summarize the smaller-rank algebras we found in Table 6.1 below. For the purposes of the table, we will refer to the smaller rank algebra involved in a fundamental weight fixed point factorization of an affine algebra Xr ' as the 'FPF algebra'. Specifically, Table 6.1 lists the FPF algebras along with their levels for the generators of the simple-current group. As we mentioned in Chapter 4, our FPF algebras correspond to the orbit Lie algebras (these are described by Fuchs-Schellekens-Schweigert in [12]) - for all but Ci- when r is even, in which case the orbit Lie algebra is ^Lry Given a Lie algebra g, its orbit Lie algebra $j is obtained through a 'matrix-folding' or 'diagram-folding' technique. Note that g will be of smaller rank than g. It turns out that the so-called 'twining characters' of g (these are essentially characters that have been 'twisted' by an automorphism of g) at fixed points are true characters of $j. For an explanation of matrix-folding, see [12]; see also [24], Sections 1.5.4, 3.4.1, for more about twisted characters of Lie algebras. 6.1.3 NIM-reps Using fixed point factorization, we found that the NIM-rep coefficients for the simple- current modular invariants reduce to remarkably simple formulas involving the affine fusion coefficients, and hence have a direct Lie theoretic interpretation. These for mulas are given in equations (5.3.9), (5.4.3), (5.5.2), (5.5.6), (5.5.11), and (5.5.15). The NIM-reps for the simple-current modular invariants correspond to WZW models on non-simply connected Lie groups, and the fact that they are so simple is quite surprising. One consequence of this is a beautiful and unexpected relation between string theories on non-simply connected Lie groups and simply connected groups of smaller rank. This is reminiscent of the 'twining characters' of e.g. [11], and we wonder whether there is a connection. Another consequence (and the main motivation for finding them) is that with these NIM-rep formulas, finding the corre sponding D-brane charges (see (6.2.1) below) should be straightforward. Although 105 X?\ level k Simple-current FPF algebra Level d fed J\f J 4^ r+l o(l) J ,(2) k (k odd) ^2(r- i) (2) Cr , r even J A -l fc + 1 1} k C^ , r odd J 2 fc °r-2 2 1} DT , r even 4 2 1 r.(i) fc DJ. ', r odd Js W-3 4 2 Table 6.1: Fixed point factorization algebras for the classical affine algebras explicit expressions for the NIM-rep coefficients do not appear in the literature, they are generally believed to be nonnegative integers, and we proved that this is the case for an example, using the Kac-Walton formula. More specifically, in Section 5.6, we found the fusion multiplicities for the first and second fundamental weights for Cr , r odd. These yield nonnegativity and positivity oiM^ 'Pi o for fixed points 6.2 Future work There are two main directions for further research immediately arising from the results in this thesis. The first involves further studying fixed point factorization, and the second involves learning more about NIM-reps and D-brane charges. 6.2.1 Work related to fixed point factorization The existence of fixed point factorization for the WZW models (for the classical algebras, and we expect as well for the exceptional algebras) is mysterious, and sug gests the possibility of a wider phenomenon. Is it related to the twining characters of [11] which express affine algebra characters 'twisted' by Coxeter-Dynkin diagram automorphisms, in terms of untwisted characters of other affine algebras? Modular 106 data exists in many contexts in mathematics, and so it is natural to ask "does fixed point factorization occur for sources of modular data other than affine Kac-Moody algebras?". Modular data for finite groups has some significant differences with that of the WZW models, and yet is still accessible (see [8] for more on finite group mod ular data). The simple-currents for a finite group G form the group Z(G) x G/[GG]. We need to consider non-abelian G, as there are no fixed points for abelian G. The closest analogy with the affine algebra case considered here, would be finite groups of Lie type, although we would hope to find a finite group fixed point factorization in a wider context. For completeness, it would be interesting to investigate what a fixed point fac torization would look like at non-fundamental weights for the orthogonal algebras. A natural first step would be to consider multiples of the fundamental weights. This is not a priority for us however because of the reasons stated in the previous section. It would also be interesting to find the fixed point factorizations for the exceptional algebras E$ ' and E\ ' and the twisted algebras A^^-I an^ -Dn • Having developed methods for the classical affine algebra case, we expect these additional cases to now be relatively straightforward. 6.2.2 Work related to the NIM-rep formulas The second direction for future research involves further work with the NIM-rep coefficients and their applications to D-brane charges. Nice expressions for NIM- reps are invaluable for determining D-brane charges. D-branes are manifolds on which the endpoints of open strings reside (open strings topologically are like the closed interval [0, 1], as opposed to closed strings which are topologically circles). An important discovery was that D-branes are dynamical entities in their own right and as such are transformed by physical processes. D-brane charges are numerical quantities preserved by these processes, and as such play the same role in string theory that electric charges do in particle physics. In particular, irreducible D- branes are parametrized by the boundary states (the same set that parametrizes the rows and columns of the NIM-rep). D-brane charges are integers rib, one f°r eacn boundary state, such that dim{L(X))nb = ^2(^\)bcnc (modM) (6.2.1) c for some M, where N\ is the NIM-rep, L(X) is the g-module, and A € P+(g(1)). All D-brane charges for the simply-connected WZW models have been known for some time, and nM = dim(L(/l)). All possible D-brane charges for most of the NIM-reps of Ar were found in a series of papers by Gaberdiel-Gannon; the boundary states were identified with highest weights of some simple Lie algebra and the charges rib equalled the corresponding dimension. The next obvious class of NIM-reps to consider were the non-simply connected WZW models for B)- ', Cr , £v we found a Lie theoretic interpretation of those NIM-reps in Chapter 5. It should now be straightforward to find the possible charges, which again we expect to be given 107 by dimensions. As we mentioned, the NIM-rep coefficients should be nonnegative integers. Pos- itivity may be difficult to verify for all Af\, but integrality should follow from the Pieri rules as we illustrated in Section 5.6 for an example. The Pieri rules would also prove positivity of MA{ , but this alone is not enough to prove positivity of all As with the fixed point factorization, our NIM-rep formulas were found for the classical affine algebras. If fixed point factorizations were found for the other affine Kac-Moody algebras, it would be interesting to find these formulas (NIM-rep coef ficients and D-brane charges) for those algebras as well. The exciting projects to us are establishing the integrality of the NIM-rep coef ficients, finding the D-brane charges, and investigating fixed point factorization for finite group modular data. 108 Bibliography [i J. Bockenhauer and D.E. Evans. Modular invariants, graphs and a-induction for nets of subfactors II. Adv. Theor.Math.Phys., 200:57-103, 1999. [2: E. Beltaos. The modular invariants of A^ L ©^2,0 wnere gcd(p',p) = 1. Master's thesis, The University of Alberta, 2004. [3: L. Birke, J. Fuchs, and C. Schweigert. Symmetry breaking boundary conditions and WZW orbifolds. Adv.Theor.Math.Phys., 473(3):671-726, 1999. [4: N. Bourbaki. Groupes et Algebres de Lie, volume IV-VI. Hermann, 1968. [5: P. Bouwknegt and K. Schoutens. W-symmetry in conformal field theory. Phys.Rep., 223:183-276, 1993. A. Cappelli, C. Itzykson, and J.B. Zuber. The A-D-E classification of minimal and A\ ' conformal invariant theories. Commun.Math.Phys., 113:1-26, 1987. [7: A. Coste and T. Gannon. Remarks on Galois symmetry in RCFT. Phys.Lett, B323:316-321, 1994. [s: A. Coste, T. Gannon, and P. Ruelle. Finite group modular data. Nucl.Phys.B, 581:679-717, 2000. [9; P. Di Francesco, P. Mathieu, and D. Senechal. Conformal Field Theory. Springer, 1997. [10: J. Fuchs. Simple WZW currents. Commun.Math.Phys., 136:345-356, 1991. [11 J. Fuchs, V. Ray, and C. Schweigert. Some automorphisms of generalized Kac- Moody algebras. J.Algebra, 191:518-540, 1997. [12: J. Fuchs, B. Schellekens, and C. Schweigert. From Dynkin diagram symmetries to fixed point structures. Commun.Math.Phys., 180:39-97, 1996. [13: J. Fuchs, B. Schellekens, and C. Schweigert. A matrix S for all simple current extensions. Nucl.Phys.B, 473(l):323-366, 1996. [14 W. Fulton and J. Harris. Representation Theory. Springer, 1982. 109 M. Gaberdiel and T. Gannon. Twisted brane charges for non-simply connected groups. J. High Energy Phys., 035(l):30pp, 2007. M.R. Gaberdiel and T. Gannon. Boundary states for WZW models. Nucl.Phys.B, 639:471-501, 2002. M.R. Gaberdiel and T. Gannon. D-brane charges on non-simply connected groups. JHEP, 04(030), 2004. T. Gannon. The classification of affine SU(3) modular invariant partition func tions. Commun.Math.Phys., 161:233-264, 1994. T. Gannon. Symmetries of the Kac-Peterson modular matrices of affine alge bras. Invent, math., 122:341-357, 1995. T. Gannon. The classification of SU(3) modular invariants revisited. Ann.Inst.Henri Poincare:Phys.Theor., 65(l):15-55, 1996. T. Gannon. The Cappelli-Itzykson-Zuber A-D-E classification. Reviews in Mathematical Physics, 12(5):739-748, 2000. T. Gannon. Comments on non-unitary conformal field theories. Nucl. Phys., B670:335-358, 2003. T. Gannon. Modular data: the algebraic combinatorics of conformal field the ory. J. Alg. Combin., 22(2):211-250, 2005. T. Gannon. Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics. Cambridge University Press, 2006. T. Gannon and M. Walton. On the classification of diagonal coset modular invariants. Commun. Math Phys., 173:175-197, 1995. T. Gannon and M.A. Walton. On fusion algebras and modular matrices. Com- mun.Math.Phys., 206:1-22, 1999. D. Gepner and E. Witten. String theory on group manifolds. Nucl.Phys., B278(3):493-549, 1986. P. Goddard, A. Kent, and D. Olive. Unitary representations of Virasoro and super-Virasoro algebras. Commun.Math.Phys., 103:105-119, 1986. V. Kac and D. Peterson. Infinite-dimensional Lie algebras, theta functions and modular forms. Advances in Mathematics, 53:125-264, 1984. V.G. Kac. Infinite Dimensional Lie Algebras. Cambridge University Press, 1990. V.G. Kac and A.K. Raina. Highest Weight Representations of Infinite Dimen sional Lie Algebras. World Scientific, 1987. 110 [32] N. Koblitz and D. Rohrlich. Simple factors in the Jacobian of a Fermat curve. Can.J.Math., XXX:1183-1205, 1978. [33] J. Lepowsky and H. Li. Introduction to Vertex Operator Algebras and their Representations. Birkhauser, 2004. [34] S. Lu. On modular invariant partition functions in non-unitary theories. Phys.Lett, B218(l):46-50, 1989. [35] J. Walcher, J. Fuchs, and C. Schweigert. Conformal field theory, boundary conditions and applications to string theory. arXiv:hep-th., 1(0011109):45pp, 2000. [36] M.A. Walton. Algorithm for WZW fusion rules: a proof. Phys.Lett, B241(3):365-368, 1990. [37] A.B. Zamolodchikov. Infinite additional symmetries in two-dimensional confor mal quantum field theory. Theor.Math.Phys., 65:1205-1213, 1985. Ill