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Modular and mock modular forms have appeared in new and intriguing roles in relation to Mathieu and Umbral moonshine. They also appear in studies of the quantum geometry of CalabiYau manifolds, three dimensional gravity, and other aspects of string theory. This workshop will be focused on the recent developments in these intertwined directions.
This workshop is also part of the UNIFY network. Participants from the UNIFY nodes are welcome to register and participate.
Registration for this workshop is now closed.
Sponsorship for this workshop has been provided by:
 Miranda Cheng, University of Amsterdam
 John Duncan, Case Western Reserve University
 Matthias Gaberdiel, ETH Zurich
 Terry Gannon, University of Alberta
 Sarah Harrison, Harvard University
 Jeffrey Harvey, University of Chicago
 Shamit Kachru, Stanford University
 Christoph Keller, Rutgers University
 Heeyeon Kim, Perimeter Institute
 Albrecht Klemm, University of Bonn
 Ching Hung Lam, Institute of Mathematics, Academia Sinica
 Geoffrey Mason, Universityof California, Santa Cruz
 Greg Moore, Rutgers University
 Sameer Murthy, Kings College, London
 Hirosi Ooguri, California Institute of Technology
 Daniel Persson, Chalmers University of Technology
 Anne Taormina, Durham University
 Roberto Volpato, SLAC & Stanford University
 Katrin Wendland, University of Freiburg
 Nathan Benjamin, Stanford University
 Lakshya Bhardwaj, Perimeter Institute
 Christopher Brust, Perimeter Institute
 Miranda Cheng, University of Amsterdam
 Nima Doroud, University of Cambridge
 John Duncan, Case Western Reserve University
 Matthias Gaberdiel, ETH Zurich
 Terry Gannon, University of Alberta
 Jaume Gomis, Perimeter Institute
 Sarah Harrison, Harvard University
 Jeffrey Harvey, University of Chicago
 Shamit Kachru, Stanford University
 Christoph Keller, Rutgers University
 Heeyeon Kim, Perimeter Institute
 Albrecht Klemm, University of Bonn
 Peter Korotev, Perimeter Institute
 Ching Hung Lam, Institute of Mathematics, Academia Sinica
 Geoffrey Mason, Universityof California, Santa Cruz
 Greg Moore, Rutgers University
 Seyed Faroogh Moosavian, Perimeter Institute
 Sameer Murthy, Kings College, London
 Hirosi Ooguri, California Institute of Technology
 Natalie Paquette, Stanford University
 Daniel Persson, Chalmers University of Technology
 Callum Quigley, University of Alberta
 Miroslav Rapcak, Perimeter Institute
 Anne Taormina, Durham University
 Roberto Volpato, SLAC & Stanford University
 Katrin Wendland, University of Freiburg
 Daniel Whalen, Stanford University
 Jie Zhou, Perimeter Institute
Monday, April 13, 2015
Time 
Event 
Location 
9:30 – 10:00am 
Registration 
Reception 
10:00 – 10:05am 
Welcome and Opening Remarks 
Bob Room 
10:05 – 11:00am 
Jeffrey Harvey, University of Chicago 
Bob Room 
11:00 – 12:00pm 
Albrecht Klemm, University of Bonn 
Bob Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Geoffrey Mason, University of California, Santa Cruz 
Bob Room 
3:00 – 4:00pm 
Shamit Kachru, Stanford University 
Bob Room 
Tuesday, April 14, 2015
Time 
Event 
Location 
10:00 – 11:00am 
John Duncan, Case Western Reserve University 
Bob Room 
11:00 – 12:00pm 
Daniel Persson, Chalmers University of Technology 
Bob Room 
12:00 – 2:00pm 
Lunch Break 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Roberto Volpato, SLAC & Stanford University 
Bob Room 
3:00 – 4:00pm 
Miranda Cheng, University of Amsterdam 
Bob Room 
6:00pm Onwards 
Reception 
Bistro – 2^{nd} Floor 
Wednesday, April 15, 2015
Time 
Event 
Location 
10:00 – 11:00am 
Katrin Wendland, University of Freiburg 
Bob Room 
11:00 – 12:00pm 
Christoph Keller, Rutgers University 
Bob Room 
12:00 – 12:15 
Conference Photo 
TBA 
12:15 – 2:00pm 
Lunch Break 
Bistro – 2^{nd} Floor 
2:00  3:30 pm  Paul Ginsparg, Cornell University Colloquium: At a Physics/Infosci Intersection 

3:30 – 4:30pm 
Ching Hung Lam, Institute of Mathematics, Academia Sinica 
Bob Room 
Thursday, April 16, 2015
Time 
Event 
Location 
10:00 – 11:00am 
Hirosi Ooguri, California Institute of Technology 
Bob Room 
11:00 – 12:00pm 
Terry Gannon, University of Alberta 
Bob Room 
12:00 – 2:00pm 
Lunch Break 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Anne Taormina, Durham University 
Bob Room 
3:00 – 4:00pm 
Heeyeon Kim, Perimeter Institute 
Bob Room 
Friday, April 17, 2015
Time 
Event 
Location 
10:00 – 11:00am 
Greg Moore, Rutgers University 
Bob Room 
11:00 – 12:00pm 
Sarah Harrison, Harvard University 
Bob Room 
12:00 – 2:00pm 
Lunch Break 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Sameer Murthy, King's College London 
Bob Room 
3:00 – 4:00pm 
Matthias Gaberdiel, ETH Zurich 
Bob Room 
Miranda Cheng, University of Amsterdam
Optimal Jacobi Forms and Mock Theta Functions
John Duncan, Case Western Reserve University
Umbral Moonshine Modules
Umbral moonshine attaches mock modular forms and meromorphic Jacobi forms to automorphisms of the Niemeier lattices. It is now known that this association can be recovered from specific, graded modules for the Niemeier lattice automorphism groups. We will describe recent progress in a program to realize these modules explicitly.
Matthias Gaberdiel, ETH Zurich
Higher Spins & Strings
Terry Gannon, University of Alberta
Thoughts stolen from the enemy
Subfactors and VOAs should both describe CFT, but what is relatively easy in one formulation can be very difficult in the other. In my talk I'll describe lessons the VOA world can learn from the subfactor one.
Sarah Harrison, Harvard University
Umbral Moonshine and K3 Surfaces
Recently, 23 cases of umbral moonshine, relating mock modular forms and finite groups, have been discovered in the context of the 23 even unimodular Niemeier lattices. One of the 23 cases in fact coincides with the socalled Mathieu moonshine, discovered in the context of K3 nonlinear sigma models. Here we establish a uniform relation between all 23 cases of umbral moonshine and K3 sigma models, and thereby take a first step in placing umbral moonshine into a geometric and physical context. This is achieved by relating the ADE root systems of the Niemeier lattices to the ADE du Val singularities that a K3 surface can develop, and the configuration of smooth rational curves in their resolutions. A geometric interpretation of our results is given in terms of the marking of K3 surfaces by Niemeier lattices.
Jeffrey Harvey, University of Chicago
Traces of Singular Moduli and Moonshine for the Thompson group
We observe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight onehalf that appears in Zagier's work on traces of singular moduli and Borcherds products. We conjecture the existence of an infinite dimensional graded module for the Thompson group and use the observed relationship to propose a McKayThompson series for each conjugacy class of the Thompson group and then construct weakly holomorphic weight onehalf forms at higher level that coincide with the proposed McKayThompson series. We also observe a discriminant property in this conjectured moonshine for the Thompson group that is closely related to the discriminant property conjectured to exist in Umbral Moonshine.
Shamit Kachru, Stanford University
Moonshine at c=12
Christoph Keller, Rutgers University
Modular invariance and holographic CFTs
Modular invariance plays an important role in AdS3/CFT2 holography. I discuss the structure of nonholomorphic CFT partition functions, namely in what sense the light spectrum determines the heavy spectrum and how to construct example partition functions using Poincare series. This yields necessary conditions on the spectrum of holographic CFTs. Finally I will discuss permutation orbifolds as examples of such theories.
Heeyeon Kim, Perimeter Institute
Quiver Quantum Mechanics and WallCrossing
I will talk about computation of the Witten index of 1d N=4 gauged linear sigma model which describes wallcrossing of BPS states in 4d N=2 theories. In the phase where the gauge group is broken to a finite group, the index is expressed as the JKresidue integral. Using this result, I am going to examine largerank behaviour of the Kronecker quivers which describes the most simplest wallcrossing phenomena. I will also talk about how the refined Witten indices of quivers are preserved under the mutation process.
Ching Hung Lam, Institute of Mathematics, Academia Sinica
On holomorphic vertex operator algebras of central charge 24
I will talk about the recent progress on the classification of (strongly regular) holomorphic vertex operator algebras of central charge 24. In particular, I will discuss a construction of certain holomorphic vertex operator algebras of central charge 24 using orbifold construction associated to inner automorphisms. This talk is based on a joint work with Hiroki Shimakura.
Geoffrey Mason, University of California, Santa Cruz
Symplectic automorphisms of some hyperk\"ahler manifolds
Greg Moore, Rutgers University
Measuring the Elliptic Genus
This talk is based on the recent paper coauthored with N. Benjamin, M. Cheng, S. Kachru, and N. Paquette.
Sameer Murthy, King's College London
ADE Little string theories, Mock modular forms, and Umbral moonshine
Hirosi Ooguri, California Institute of Technology
Analytic Bootstrap Bounds
Daniel Persson, Chalmers University of Technology
Uduality, exotic instantons and automorphic forms on KacMoody groups
Anne Taormina, Durham University
Signatures of Mathieu Moonshine in Z_2orbifolds of Conformal Field Theories
Roberto Volpato, SLAC & Stanford University
Fricke Sduality in CHL models
We consider dual pairs of four dimensional heterotic/type IIA CHL models with 16 spacetime supersymmetries. We provide strong evidence for the existence of an Sduality acting on the heterotic axiondilaton by a Fricke involution S > 1/NS, where N is the order of the orbifold symmetry. While most models are selfdual, in some cases Sduality relates the CHL model to a compactification of type IIA on an orbifold of T^6. We provide a simple criterion to determine whether a model is selfdual or not. Finally, we argue that in selfdual CHL models the lattices of electric and magnetic charges must be Nmodular and verify this prediction.
Katrin Wendland, University of Freiburg
How does extended supersymmetry affect the elliptic genus?
How does extended supersymmetry affect the elliptic genus?
The elliptic genus of K3 and its decomposition into characters of the N=4 superconformal algebra of associated conformal field theories can be viewed as the outset of Mathieu Moonshine. Thus, extended supersymmetry induces additional properties of the elliptic genus, which so far lack a satisfactory geometric interpretation. We investigate the implications of this decomposition on geometric structures that underlie the elliptic genus.
Optimal Jacobi Forms and Mock Theta Functions
Fricke Sduality in CHL models
We consider dual pairs of four dimensional heterotic/type IIA CHL models with 16 spacetime supersymmetries. We provide strong evidence for the existence of an Sduality acting on the heterotic axiondilaton by a Fricke involution S > 1/NS, where N is the order of the orbifold symmetry. While most models are selfdual, in some cases Sduality relates the CHL model to a compactification of type IIA on an orbifold of T^6. We provide a simple criterion to determine whether a model is selfdual or not.
Uduality, exotic instantons and automorphic forms on KacMoody groups
Umbral Moonshine Modules
Umbral moonshine attaches mock modular forms and meromorphic Jacobi forms to automorphisms of the Niemeier lattices. It is now known that this association can be recovered from specific, graded modules for the Niemeier lattice automorphism groups. We will describe recent progress in a program to realize these modules explicitly.
Moonshine at c=12
Symplectic automorphisms of some hyperk\"ahler manifolds
TBA
Traces of Singular Moduli and Moonshine for the Thompson group
We observe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight onehalf that appears in Zagier's work on traces of singular moduli and Borcherds products. We conjecture the existence of an infinite dimensional graded module for the Thompson group and use the observed relationship to propose a McKayThompson series for each conjugacy class of the Thompson group and then construct weakly holomorphic weight onehalf forms at higher level that coincide with the proposed McKayThompson series.
Pages
Scientific Organizers:
 Miranda Cheng, University of Amsterdam
 Matthias Gaberdiel, ETH Zurich
 Jaume Gomis, Perimeter Institute
 Shamit Kachru, Stanford University