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. 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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