**Contemporary Mathematics**

Volume: 695;
2017;
274 pp;
Softcover

MSC: Primary 17; 18; 32; 81;

**Print ISBN: 978-1-4704-2666-8
Product Code: CONM/695**

List Price: $111.00

AMS Member Price: $88.80

MAA Member Price: $99.90

**Electronic ISBN: 978-1-4704-4196-8
Product Code: CONM/695.E**

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AMS Member Price: $88.80

MAA Member Price: $99.90

# Lie Algebras, Vertex Operator Algebras, and Related Topics

Share this page *Edited by *
*Katrina Barron; Elizabeth Jurisich; Antun Milas; Kailash Misra*

This volume contains the proceedings of the
conference on Lie Algebras, Vertex Operator Algebras, and Related
Topics, celebrating the 70th birthday of James Lepowsky and Robert
Wilson, held from August 14–18, 2015, at the University of Notre Dame,
Notre Dame, Indiana.

Since their seminal work in the 1970s, Lepowsky and Wilson, their
collaborators, their students, and those inspired by their work, have
developed an amazing body of work intertwining the fields of Lie
algebras, vertex algebras, number theory, theoretical physics, quantum
groups, the representation theory of finite simple groups, and
more. The papers presented here include recent results and
descriptions of ongoing research initiatives representing the broad
influence and deep connections brought about by the work of Lepowsky
and Wilson and include a contribution by Yi-Zhi Huang summarizing
some major open problems in these areas, in particular as they pertain
to two-dimensional conformal field theory.

#### Readership

Graduate students and research mathematicians interested in representation theory, vertex algebras, conformal field theory and related areas.

# Table of Contents

## Lie Algebras, Vertex Operator Algebras, and Related Topics

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Generalizations of 𝑄-systems and orthogonal polynomials from representation theory 18
- Some applications and constructions of intertwining operators in Logarithmic Conformal Field Theory 1522
- 1. Introduction 1522
- 2. Classification of irreducible modules for orbifold \triplet^{𝐴_{𝑚}}: A fusion rules approach 1623
- 3. Fusion rules for \singlet and a proof of Conjecture 2.3 for 𝑝=2. 1825
- 4. Deformed realization of the triplet and singlet vertex algebra 2128
- 5. Intertwining operators between typical \singlet and \triplet^{𝐴_{𝑚}}–modules: the 𝑝=2 case 2532
- Acknowledgments 2633
- References 2633

- Kac–Moody groups and automorphic forms in low dimensional supergravity theories 2936
- The Lusztig-Macdonald-Wall polynomial conjectures and 𝑞-difference equations 4148
- Uniqueness of representation–theoretic hyperbolic Kac–Moody groups over ℤ 5158
- 1. Introduction 5158
- 2. Tits’ Kac–Moody group 5259
- 3. Tits’ presentation 5360
- 4. Simply laced hyperbolic type 5360
- 5. Finitely many defining relations parametrized over 𝑅 5461
- 6. Representation–theoretic Kac–Moody groups over rings 5461
- 7. Uniqueness of representation–theoretic Kac–Moody groups over \Z 5764
- 8. The kernel of 𝜌_{𝜆} 6168
- Acknowledgement 6370
- References 6370

- Coends in conformal field theory 6572
- Remarks on 𝜙-coordinated modules for quantum vertex algebras 8390
- The classification of chiral WZW models by 𝐻⁴₊(𝐵𝐺,ℤ) 99106
- 1. Introduction 99106
- 2. Geometric quantization 103110
- 3. 𝐻⁴(𝐵𝐺) for connected Lie groups 105112
- 4. Representations of affine and Heisenberg VOAs 107114
- 5. Simple current extensions 109116
- 6. The minimal energy 110117
- 7. The classification of chiral WZW models 111118
- 8. WZW conformal nets 117124
- 9. Conclusion 119126
- References 119126

- Some open problems in mathematical two-dimensional conformal field theory 123130
- 1. Introduction 123130
- 2. The construction of rational conformal field theories satisfying the axioms of Kontsevich-Segal-Moore-Seiberg 124131
- 3. Cohomology theory for graded vertex algebras and complete reducibility of their modules 126133
- 4. The moduli space of conformal field theories 128135
- 5. The construction and study of logarithmic conformal field theories 129136
- 6. Orbifold conformal field theories 130137
- 7. The uniqueness of the moonshine module vertex operator algebra and the classification of meromorphic rational conformal field theories of central charge 24 131138
- 8. Calabi-Yau superconformal field theories 133140
- 9. The relation between the approaches of vertex operator algebras and conformal nets 134141
- References 135142

- On realization of some twisted toroidal Lie algebras 139146
- Twisted generating functions incorporating singular vectors in Verma modules and their localizations, I 149156
- Characterization of the simple Virasoro vertex operator algebras with 2 and 3-dimensional space of characters 175182
- Introduction 175182
- 1. Overview 177184
- 2. Special cases of the Kaneko-Zagier equations 181188
- 3. 3rd order modular linear differential equations 184191
- 4. Diophantus equation, central charges and conformal weights 186193
- 5. Condition for modules 188195
- 6. Characterization of the minimal models 195202
- 7. Appendix 202209
- References 203210

- Quasiconformal Teichmüller theory as an analytical foundation for two-dimensional conformal field theory 205212
- 1. Introduction 205212
- 2. Conformal field theory 208215
- 3. Quasiconformal mappings and \teich theory 213220
- 4. Teichmüller space/rigged moduli space correspondence 216223
- 5. Analytic setting for the determinant line bundle 222229
- 6. Summary: why Weil-Petersson class riggings? 234241
- References 235242

- Centralizing the centralizers 239246
- On Neeman’s gradient flows 261268
- Back Cover Back Cover1282