# Elliptic Cohomology and Physics

COVID-19 information for researchers and visitors

Conference Date:
Monday, May 25, 2020 (All day) to Thursday, May 28, 2020 (All day)
Scientific Areas:
Mathematical Physics

Equivariant elliptic cohomology is a rapidly evolving field, with wideranging applications including in quantum field theory, geometric representation theory, integrable systems, singularity theory, combinatorial representation theory, homotopy theory, and group theory. Physics in particular has been present starting early in the elliptic cohomology story with Witten's 1987 recognition of the elliptic genus as a supersymmetric partition function. Moonshine and orbifold conformal field theory contributed deeply to the development of equivariant elliptic cohomology. Modern physical applications include using elliptic cohomology to see subtle torsion constraints on phases of supersymmetric field theories and to explain (mock) modularity phenomena. Future applications may include elliptic upgrades of K-theoretic methods in condensed matter. This workshop aims to bring together the emerging international community interested in this circle of ideas. The workshop will be held entirely online, with two lectures each day and ample time for discussion, with efforts made to accommodate time-zone differences. This should allow participation from people who, for reasons of distance or family, do not normally travel to such conferences.

Registration for this conference is now closed.

• Miranda Cheng, University of Amsterdam
• Davide Gaiotto, Perimeter Institute
• Fei Han, National University of Singapore
• Zhen Huan, Huazhong University of Science and Technology
• Nitu Kitchloo, Johns Hopkins University
• Gerd Laures, Ruhr-Universität Bochum
• Lennart Meier, Utrecht University
• Jack Morava, Johns Hopkins University
• Richard Rimanyi, University of North Carolina at Chapel Hill
• Arnav Tripathy, Harvard University
• Brian Williams, Northeastern University
• Kirillov Anatol, Kyoto University
• Andrew Baker, University of Glasgow
• Francis Bischoff, University of Oxford
• Tommaso Botta, ETH Zurich
• Emily Cliff, University of Sydney
• Mattia Coloma, Tor Vergata University of Rome
• Stefano D'Alesio, ETH Zurich
• Jorge Devoto, University of Buenos Aires
• Tom Dove, University of Melbourne
• Anne Dranowski, University of Toronto
• Chris Elliott, University of Massachusetts Amherst
• Giovanni Felder, ETH Zurich
• Domenico Fiorenza, Sapienza University of Rome
• Vassily Gorbounov, University of Aberdeen
• Meng Guo, Perimeter Institute
• André Henriques, University of Oxford
• Tatsuyuki Hikita, Kyoto University
• Gerald Hoehn, Kansas State University
• Michael Hopkins, Harvard University
• Mikhaiil Kapranov, Kavli IPMU
• Syu Kato, Kyoto University
• Ali Khalili Samani, University of Tehran
• Yuqin Kewang, University of Chinese Academy of Sciences
• Hitoshi Konno, Tokyo University of Marine Science and Technology
• Eugenio Landi, Roma Tre University
• Cristian Lenart, State University of New York at Albany
• Anatoly Libgober, University of Illinois at Chicago
• Gregory Moore, Rutgers University
• Benedict Morrissey, University of Pennsylvania
• Kohei Motegi, Tokyo University of Marine Science and Technology
• Apurva Nakade, University of Western Ontario
• Masaki Nakagawa, Okayama University
• Hiroshi Naruse, University of Yamanashi
• Niko Naumann, Unversity of Regensburg
• Arun Ram, University of Melbourne
• Doug Ravenel, University of Rochester
• Charles Rezk, University of Illinois At Urbana-Champaign
• Sarah Scherotzke, University of Luxembourg
• Eric Sharpe, Virginia Polytechnic Institute and State University
• Yiyan Shou, University of North Carolina at Chapel Hill
• Nicolò Sibilla, SISSA
• Matthew Spong, University of Melbourne
• Nathaniel Stapleton, University of Kentucky
• Stephan Stolz, University of Notre Dame
• Norio Suzuki, Kitami Institute of Technology
• Matt Szczesny, Boston University
• Shohei Tanaka, Fukuoka University
• Valerio Toledano Laredo, Northeastern University
• Kazushi Ueda, University of Tokyo
• Alexander Varchenko, University of North Carolina at Chapel Hill
• Michele Vergne, Mathematics Institute of Jussieu–Paris Rive Gauche
• Achal Vinod, Shiv Nadar University
• Hideya Watanabe, Kyoto University
• Andrzej Weber, University of Warsaw
• Matt Young, Max Planck Institute for Mathematics
• Kirill Zainoulline, University of Ottawa
• Valentin Zakharevich, Johns Hopkins University
• Chengjing Zhang, University of Melbourne
• Changlong Zhong, State University of New York at Albany
• Paul Zinn-Justin, University of Melbourne

Monday, May 25, 2020

 Time Event Location 8:55 – 9:00amEDT Theo Johnson-Freyd, Perimeter InstituteWelcome and Opening Remarks Virtual 9:00 – 10:00amEDT Fei Han, National University of SingaporeProjective elliptic genera and applications Virtual 7:00 – 8:00pmEDT Davide Gaiotto, Perimeter InstituteTBA Virtual

Tuesday, May 26, 2020

 Time Event Location 9:00 – 10:00am EDT Lennart Meier, Utrecht UniversityEquivariant elliptic cohomology with integral coefficients Virtual 11:00 – 12:00pm EDT Arnav Tripathy, Harvard UniversityThe de Rham model for elliptic cohomology from physics Virtual 7:00 – 8:00pmEDT Jack Morava, Johns Hopkins UniversityQuasisymmetric characteristic numbers for Hamiltonian toric manifolds Virtual

Wednesday, May 27, 2020

 Time Event Location 9:00 – 10:00amEDT Gerd Laures, Ruhr-Universität BochumCodes, vertex algebras and topological modular forms Virtual 11:00 – 12:00pmEDT Brian Williams, Northeastern UniversityTwisted superconformal algebras and representations of higher Virasoro algebras Virtual 7:00 – 8:00pmEDT Richard Rimanyi, University of North Carolina at Chapel HillElliptic characteristic classes, bow varieties, 3d mirror duality Virtual

Thursday, May 28, 2020

 Time Event Location 9:00 – 10:00amEDT Miranda Cheng, University of AmsterdamSigma-VOA correspondence Virtual 11:00 – 12:00pmEDT Zhen Huan, Huazhong University of Science and TechnologyQuasi-elliptic cohomology theory and the twisted, twisted Real theories Virtual 7:00 – 8:00pmEDT Nitu Kitchloo, Johns Hopkins UniversityConformal blocks in genus zero, and Elliptic cohomology Virtual

Miranda Cheng, University of Amsterdam

Sigma-VOA correspondence

In this talk I will discuss an interesting phenomenon, namely a correspondence between sigma models and vertex operator algebras, with the two related by their symmetry properties and by a reflection procedure, mapping the right-movers of the sigma model at a special point in the moduli space to left-movers. We will discuss the examples of N=(4,4) sigma models on $T^4$ and on $K3$. The talk will be based on joint work with Vassilis Anagiannis, John Duncan and Roberto Volpato.

Fei Han, National University of Singapore

Projective elliptic genera and applications

Projective vector bundles (or gerbe modules) are generalizations of vector bundles in the presence of a gerbe on manifolds. Given a projective vector bundle, we will first show how to use it to twist the Witten genus to get modular invariants, which we call projective elliptic genera. Then we will give two applications: (1) given any pseudodifferential operator, we will construct modular invariants generalizing the Witten genus, which corresponds to the Dirac operator; (2) we will enhance the Hori map in T-duality to the graded Hori map and show that it sends Jacobi forms to Jacobi forms. This represents our joint works with Mathai.

Zhen Huan, Huazhong University of Science and Technology

Quasi-elliptic cohomology theory and the twisted, twisted Real theories

Quasi-elliptic cohomology is closely related to Tate K-theory. It is constructed as an object both reflecting the geometric nature of elliptic curves and more practicable to study than most elliptic cohomology theories.  It can be interpreted by orbifold loop spaces and expressed in terms of equivariant K-theories. We formulate the complete power operation of this theory. Applying that we prove the finite subgroups of Tate curve can be classified by the Tate K-theory of symmetric groups modulo a certain transfer ideal.

In addition, we define twisted quasi-elliptic cohomology. They can be related to a twisted equivariant version Devoto’s elliptic cohomology via a Chern character map. Moreover, we construct twisted Real quasi-elliptic cohomology and the Chern character map in this case. This is joint work with Matthew Spong and Matthew Young.

Nitu Kitchloo, Johns Hopkins University

Conformal blocks in genus zero, and Elliptic cohomology

A fundamental theorem in the theory of Vertex algebras (known as Zhu’s theorem) demonstrates that the space generated by the characters of certain Vertex algebras is a representation of the modular group. We will cast this theorem in the language of homotopy theory using the language of conformal blocks. The goal of this talk is to justify the claim that equivariant elliptic cohomology, seen as a derived spectrum, is a homotopical analog of Zhu’s theorem in the special case of the Affine Vacuum vertex algebra at a fixed integral level. The talk will not require knowing the definition of Vertex algebras or conformal blocks.

Gerd Laures, Ruhr-Universität Bochum

Codes, vertex algebras and topological modular forms

The talk illuminates the role of codes and lattice vertex algebras in algebraic topology. These objects come up naturally in connection with string structures or topological modular forms. The talk tries to unify these different concepts in an introductory manner.

Lennart Meier, Utrecht University

Equivariant elliptic cohomology with integral coefficients

Thirteen years ago, Lurie has sketched a way to obtain equivariant elliptic cohomology and equivariant topological modular forms without the need to restrict to rational or complex coefficients. Recently, David Gepner and I have found one way to flesh out the details and and provide computations in the U(1)-equivariant case. On this work I will report.

Jack Morava, Johns Hopkins University

Quasisymmetric characteristic numbers for Hamiltonian toric manifolds.

Baker and Richter's $A_\infty$ analog of the complex cobordism spectrum provides characteristic numbers for complex-oriented toric manifolds, which generalize to define similar invariants for Hamiltonian toric dynamical systems: roughly, the completely integrable' systems of classical mechanics which (by KAM theory) possess remarkable stability properties. arXiv:1910.12609

Richard Rimanyi, University of North Carolina

Elliptic characteristic classes, bow varieties, 3d mirror duality

We study elliptic characteristic classes of natural subvarieties in some ambient spaces, namely in homogeneous spaces and in Nakajima quiver varieties. The elliptic versions of such characteristic classes display an unexpected symmetry: after switching the equivariant and the Kahler parameters, the classes of varieties in one ambient space coincide” with the classes of varieties in another ambient space. This duality gets explained as “3d mirror duality” if we regard our ambient spaces as special cases of Cherkis bow varieties. I will report on a work in progress with Y. Shou, based on earlier related works with G. Felder, A. Smirnov, V. Tarasov, A. Varchenko, A. Weber, Z. Zhou.

Arnav Tripathy, Harvard University

The de Rham model for elliptic cohomology from physics

I'll discuss elliptic cohomology from a physical perspective, indicating the importance of the Segal-Stolz-Teichner conjecture and joint work with D. Berwick-Evans on rigorously proving some of these physical predictions.

Brian Williams, Northeastern University

Twisted superconformal algebras and representations of higher Virasoro algebras

(Super)conformal algebras on two-dimensional spacetimes play a ubiquitous role in representation theory and conformal field theory. In most cases, however, superconformal algebras are finite dimensional. In this talk, we introduce refinements of certain deformations of superconformal algebras which share many facets with the ordinary (super) Virasoro algebras. Representations of these refinements include the higher dimensional Kac—Moody algebras, and many more motivated by physics. Finally, we will show how these algebras enhance to higher dimensional chiral/factorization algebras which upon `localization" deform to well-studied chiral algebras on Riemann surfaces.

Scientific Organizers:

• Daniel Berwick Evans, University of Illinois at Urbana-Champaign
• Nora Ganter, University of Melbourne
• Theo Johnson-Freyd, Perimeter Institute
• Yaping Yang, University of Melbourne
• Gufang Zhao, University of Melbourne