695

Lie Algebras,Vertex Operator Algebras,

and Related TopicsConference in Honor of J. Lepowsky and R. Wilson on

Lie Algebras, Vertex Operator Algebras, and Related TopicsAugust 14–18, 2015

University of Notre Dame, Notre Dame, IN

Katrina BarronElizabeth Jurisich

Antun MilasKailash Misra

Editors

American Mathematical Society

695

Lie Algebras,Vertex Operator Algebras,

and Related TopicsConference in Honor of J. Lepowsky and R. Wilson on

Lie Algebras, Vertex Operator Algebras, and Related TopicsAugust 14–18, 2015

University of Notre Dame, Notre Dame, IN

Katrina BarronElizabeth Jurisich

Antun MilasKailash Misra

Editors

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Dennis DeTurck, Managing Editor

Michael Loss Kailash Misra Catherine Yan

2010 Mathematics Subject Classification. Primary 17B45, 17B69, 18D10, 32G15, 81T05.

Library of Congress Cataloging-in-Publication Data

Names: Barron, Katrina, 1965– editor. | Jurisich, Elizabeth, 1963– editor. | Milas, Antun, 1974–editor. | Misra, Kailash C., 1954– editor.

Title: Lie algebras, vertex operator algebras, and related topics : a conference in honor ofJ. Lepowsky and R. Wilson, August 14–18, 2015, University of Notre Dame, Notre Dame,Indiana / Katrina Barron, Elizabeth Jurisich, Antun Milas, Kailash Misra, editors.

Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Con-temporary mathematics ; volume 695 | Includes bibliographical references.

Identifiers: LCCN 2017012864 — ISBN 9781470426668 (alk. paper)Subjects: LCSH: Lie algebras–Congresses. | Vertex operator algebras–Congresses. | Represen-

tations of algebras–Congresses. | Lepowsky, J. (James) | Wilson, Robert L., 1946- | AMS:Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Lie algebras of linearalgebraic groups. msc | Nonassociative rings and algebras – Lie algebras and Lie superal-gebras – Vertex operators; vertex operator algebras and related structures. msc | Categorytheory; homological algebra – Categories with structure – Monoidal categories (multiplicativecategories), symmetric monoidal categories, braided categories. msc | Several complex vari-ables and analytic spaces – Deformations of analytic structures – Moduli of Riemann surfaces,Teichmuller theory. msc | Quantum theory – Quantum field theory; related classical fieldtheories – Axiomatic quantum field theory; operator algebras. msc

Classification: LCC QA252.3 .L5547 2017 | DDC 512/.482–dc23 LC record available athttps://lccn.loc.gov/2017012864

DOI: http://dx.doi.org/10.1090/conm/695

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Contents

Preface v

Conference Participants vi

Generalizations of Q-systems and orthogonal polynomials from representationtheory

D. Addabbo and M. Bergvelt 1

Some applications and constructions of intertwining operators in logarithmicconformal field theory

D. Adamovic and A.Milas 15

Kac–Moody groups and automorphic forms in low dimensional supergravitytheories

L. Bao and L. Carbone 29

The Lusztig-Macdonald-Wall polynomial conjectures and q-difference equationsK. Bringmann, K. Mahlburg, and A. Milas 41

Uniqueness of representation–theoretic hyperbolic Kac–Moody groups over ZL. Carbone and F. Wagner 51

Coends in conformal field theoryJ. Fuchs and C. Schweigert 65

Remarks on φ-coordinated modules for quantum vertex algebrasH. Li 83

The classification of chiral WZW models by H4+(BG,Z)

A. Henriques 99

Some open problems in mathematical two-dimensional conformal field theoryY.-Z. Huang 123

On realization of some twisted toroidal Lie algebrasN. Jing, C. Mangum, and K. Misra 139

Twisted generating functions incorporating singular vectors in Verma modulesand their localizations, I

J. Lepowsky and J. Yang 149

Characterization of the simple Virasoro vertex operator algebras with 2 and3-dimensional space of characters

Y. Arike, K. Nagatomo, and Y. Sakai 175

iii

iv CONTENTS

Quasiconformal Teichmuller theory as an analytical foundation fortwo-dimensional conformal field theory

D. Radnell, E. Schippers, and W. Staubach 205

Centralizing the centralizersA. M. Semikhatov 239

On Neeman’s Gradient FlowsN. Wallach 261

Preface

This volume contains the proceedings of the international conference “Lie Alge-bras, Vertex Operator Algebras, and Related Topics”, a tribute to James Lepowskyand Robert Wilson, held at the University of Notre Dame, Notre Dame, Indiana,August 14–18, 2015.

Since their seminal work in the 1970s, Lepowsky and Wilson, their collabora-tors, and their students have developed a transformative body of work intertwiningthe fields of Lie algebras, vertex algebras, number theory, theoretical physics, quan-tum groups, the representation theory of finite simple groups, and more. Their re-search and mentorship in the field have been instrumental in the development of thesubject, and their influence continues through their work and the work of their stu-dents and collaborators. Jointly and individually they have supervised more than30 Ph.D. students, several who are now distinguished researchers in their fields.

This volume features one expository article with open problems, and fourteenoriginal research articles, dedicated to Jim and Robert reflecting just a bit of thebroad and deep influences they have had, and continue to have in Lie algebras,vertex operator algebras and related areas of research.

We thank everyone who participated in the conference, those who helped planand run the conference, and those who labored on this volume. The conference wasmade possible through the generous support of the National Science Foundation(NSF-DMS Conference Grant 1507305), a Participating Institutions ConferenceAward from the Institute for Mathematics and its Applications at the Universityof Minnesota, and through the generous support of the Center for Mathematics atthe University of Notre Dame.

Without the hard work of the contributors and the referees, as well as theeditorial staff of the American Mathematical Society, this volume would not havebeen feasible. Our thanks go to Christine M. Thivierge for her constant help andpatience. We also thank the technical staff for allowing us to include a group pictureof the participants.

We greatly appreciate the staff of the Mathematics Department and the Centerfor Mathematics at the University of Notre Dame for their help during the confer-ence, in particular Lisa Driver, for her efficient and dedicated work on logistics.

We are grateful to Haisheng Li for helping us at the initial stage of the con-ference. Finally, we express our great appreciation to Yi-Zhi Huang—for providinginvaluable help with the preparation of the NSF proposal, contributing to the pro-ceedings, advising us on many aspects of the conference, and initiating the confer-ence itself.

Katrina Barron, Elizabeth JurisichAntun Milas, Kailash Misra

v

Conference Participants

vi

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 in-

tertwining 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 repre-

senting 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.

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