# Physics Around Mirror Symmetry

Conference Date:
Monday, October 21, 2013 (All day) to Friday, October 25, 2013 (All day)
Scientific Areas:
Mathematical Physics

This conference will discuss recent developments in mirror symmetry, a subject at the intersection of leading developments in mathematics and physics and that been at the genesis of many developments in mathematical physics.

The conference will bring together leading mathematicians and physicist to discuss the various very new approaches to the subject, that range from powerful physics based methods to new mathematical approaches.

Register for the conference HERE

• Gaetan Borot, Max Planck Institute for Mathematics & MIT
• Vincent Bouchard, University of Alberta
• Ricardo Couso, University of Santiago de Compostela
• Emanuel Diaconescu, Rutgers University
• David Favero, University of Alberta
• Davide Gaiotto, Perimeter Institute
• Marco Gualtieri, University of Toronto
• Kentaro Hori, Kavli IPMU
• Shamit Kachru, Stanford University
• Spiro Karigiannis, University of Waterloo
• Albrecht Klemm, University of Bonn
• Ilarion Melnikov, Albert Einstein Institute
• Takuya Okuda, University of Tokyo
• Callum Quigley, University of Alberta
• Yan Soibelman, Kansas State University
• Johannes Walcher, McGill University
• Nikolay Bobev, Perimeter Institute
• Gaetan Borot, Max Planck Institute for Mathematics & MIT
• Vincent Bouchard, University of Alberta
• Ricardo Couso, University of Santiago de Compostela
• Emanuel Diaconescu, Rutgers University
• Nima Doroud, Perimeter Institute
• David Favero, University of Alberta
• Sara Filippini, Fields Institute
• Ilmar Gahramanov, Humboldt University
• Davide Gaiotto, Perimeter Institute
• Jaume Gomis, Perimeter Institute
• Marco Gualtieri, University of Toronto
• Kentaro Hori, Kavli IPMU
• Shamit Kachru, Stanford University
• Spiro Karigiannis, University of Waterloo
• Albrecht Klemm, University of Bonn
• Peter Koroteev, Perimeter Institute
• Ilarion Melnikov, Albert Einstein Institute
• Alex Molnar, Queens University
• Ruxandra Moraru, University of Waterloo
• Takuya Okuda, University of Tokyo
• Peter Overholser, University of California, San Diego
• Andrija Perunicic, Fields Institute
• Callum Quigley, University of Alberta
• Simon Rose, Fields Institute
• Helga Ruddat, University of Mainz
• Laura Schaposnik, University of Illinois
• Sam Selmani, McGill University
• Yan Soibelman, Kansas State University
• Alan Thompson, Fields Institute
• Michel van Garrell, Fields Institute
• Johannes Walcher, McGill University
• Noriko Yui, Queens University
• Yuecheng Zhu, University of Texas at Austin

Monday, October 21st

 Time Event Location 9:30-10:00am Registration Reception 10:00-10:10am Jaume Gomis, Perimeter InstituteWelcome and Opening Remarks Space Room 10:10-11:10am Shamit Kachru, Stanford UniversitySome simple extensions of Mathieu Moonshine Space Room 11:10-12:10pm Ilarion Melnikov, Albert Einstein InstituteHybrid conformal field theories Space Room 12:10-2:30pm Lunch Bistro - 2nd Floor 2:30-3:30pm Emanuel Diaconescu, Rutgers UniversityCoisotropic branes, surface defects and mirror symmetry Space Room

Tuesday, October 22nd

 Time Event Location 10:00-11:00am Takuya Okuda, University of TokyoExact results for boundaries and domain walls in2d supersymmetric theories Space Room 11:00-12:00pm Kentaro Hori, Kavli IPMUExact Results In Two-Dimensional (2,2) SupersymmetricGauge Theories With Boundary Space Room 12:00-2:30pm Lunch Bistro - 2nd Floor 2:30-3:30pm Davide Gaiotto, Perimeter InstituteAlgebraic structures in massive (2,2) theories Space Room 3:30-3:45pm Conference Photo TBA 3:45-4:15pm Break Bistro 4:15-5:15pm Gaetan Borot, Max Planck Institute for Mathematics & MITBlobbed topological recursion Space Room

Wednesday, October 23rd

 Time Event Location 10:00-11:00am Yan Soibelman, Kansas State UniversityWall-crossing structures Space Room 11:00-12:00pm Albrecht Klemm, University of BonnOn refined stable pair invariants for del Pezzo surfacesand the 1/2 K3 Space Room 12:00-2:00pm Lunch Bistro - 2nd Floor 2:00-3:30pm Marco Gualtieri, University of TorontoColloquium:  The Stokes groupoids Theater 3:30-4:00pm Break Bistro 4:00-5:00pm Marco Gualtieri, University of TorontoA symplectic approach to generalized complex geometry Space Room 6:00pm Banquet Bistro

Thursday, October 24th

 Time Event Location 10:00-11:00am Vincent Bouchard, University of AlbertaTBA Space Room 11:00-12:00pm Ricardo Couso, University of Santiago de CompostelaResurgent transseries and the holomorphic anomaly Space Room 12:00-2:30pm Lunch Bistro - 2nd Floor 2:30-3:30pm David Favero, University of AlbertaTBA Space Room 3:30-4:00pm Break Bistro 4:00-5:00pm Callum Quigley, University of AlbertaHeterotic Flux Geometry from (0,2) Gauge Dynamics Space Room

Friday, October 25th

 Time Event Location 10:00-11:00am Spiro Karigiannis, University of WaterlooThe mathematics of G_2 conifolds for M-theory Space Room 11:00-12:00pm Johannes Walcher, McGill UniversityTBA Space Room 12:00-2:30pm Lunch Bistro - 2nd Floor

Gaetan Borot, Max Planck Institute for Mathematics & MIT

Blobbed topological recursion

Hermitian matrix models have been used since the early days of 2d quantum gravity, as generating series of discrete surfaces, and sometimes toy models for string theory. The single trace matrix models (with measure dM exp( - N Tr V(M)) have been solved in a 1/N expansion in the 90s by the moment method of Ambjorn et al. Later, Eynard showed that it can be rewritten more intrinsically in terms of algebraic geometry of the spectral curve, and formulated the so-called topological recursion.
In a similar way, we will show that double hermitian matrix models are solved by the same topological recursion, and more generally, that arbitrary hermitian matrix models are solved by a "blobbed topological recursion", whose properties still have to be investigated.

Ricardo Couso, University of Santiago de Compostela

Resurgent transseries and the holomorphic anomaly

Topological string theory is restricted enough to be solved completely in the perturbative sector, yet it is able to compute amplitudes in physical string theory and it also enjoys large N dualities. These gauge theory duals, sometimes in the form of matrix models, can be solved past perturbation theory by plugging transseries ansätze into the so called string equation. Based on the mathematics of resurgence, developed in the 80's by J. Ecalle, this approach has been recently applied with tremendous success to matrix models and their double scaling limits (Painlevé I, etc).  A natural question is if something similar can be done directly in the topological closed string sector. In this seminar I will show how the holomorphic anomaly equations of BCOV provide the starting point to derive a master equation which can be solved with a transseries ansatz. I will review the perturbative sector of the solutions, its structure, and how it generalizes for higher instanton nonperturbative sectors. Resurgence, in the guise of large order behavior of the perturbative sector, will be used to derive the holomorphicity of the instanton actions that control the asymptotics of the perturbative sector, and also to fix the holomorphic ambiguities in some cases. The example of local CP^2 will be used to illustrate these results.
This work is based on 1308.1695 and on-going research in collaboration with J.D. Edelstein, R. Schiappa and M. Vonk.

Davide Gaiotto, Perimeter Institute

Algebraic structures in massive (2,2) theories

I will review some ongoing work on the low energy properties of D-branes/boundary conditions in massive two-dimensional field theories with (2,2) supersymmetry.

Marco Gualtieri, University of Toronto

A symplectic approach to generalized complex geometry

I will describe a new method for understanding a large class of generalized complex manifolds, in which we view them as usual symplectic structures on a manifold with a kind of log structure. I will explain this structure in detail and explain how it can be used to prove a Tian-Todorov unobstructedness theorem as well as topological
obstructions for existence of nondegenerate generalized complex structures.

Shamit Kachru, Stanford University

Some simple extensions of Mathieu Moonshine

Mathieu Moonshine is a striking and unexpected relationship between the sporadic simple finite group M24 and a special Jacobi form, the elliptic genus, which arises naturally in studies of nonlinear sigma models with K3 target.  In this talk, we first discuss its predecessor (Monstrous Moonshine), then discuss the current evidence in favor of Mathieu Moonshine.  We also discuss extensions of this story involving second quantized mirror symmetry,' relating heterotic strings on K3 to type II strings on Calabi-Yau threefolds.

Ilarion Melnikov, Albert Einstein Institute

Hybrid conformal field theories

I will discuss a class of limiting points in the moduli space of d=2 (2,2) superconformal field theories.  These SCFTs arise as IR limits of "hybrid" UV theories constructed as a fibration of a Landau-Ginzburg theory over a base Kaehler geometry.  A significant generalization of Landau-Ginzburg and large radius geometric limit points, the hybrid theories can be used to probe general features of (2,2) and (0,2) SCFT moduli spaces.

Takuya Okuda, University of Tokyo

Exact results for boundaries and domain walls in 2d supersymmetric theories

We apply supersymmetric localization to N=(2,2) gauged linear sigma models on a hemisphere, with boundary conditions, i.e., D-branes, preserving B-type supersymmetries. We explain how to compute the hemisphere partition function for each object in the derived category of equivariant coherent sheaves, and argue that it depends only on its K theory class. The hemisphere partition function computes exactly the central charge of the D-brane, completing the well-known formula obtained by an anomaly inflow argument. We also formulate supersymmetric domain walls as D-branes in the product of two theories.  We exhibit domain walls that realize the sl(2) affine Hecke algebra.  Based on arXiv:1308.2217.

Callum Quigley, University of Alberta
Heterotic Flux Geometry from (0,2) Gauge Dynamics

Chiral gauge theories in two dimensions with (0,2) supersymmetry admit a much broader, and more interesting, class of vacuum solutions than their better studied (2,2) counterparts. In this talk, we will explore some of the possibilities that are offered by this additional freedom by including field-dependent theta-angles and FI parameters. The moduli spaces that will result from this procedure correspond to heterotic string backgrounds with non-trivial H-flux and NS-brane sources. Along the way, a remarkable relationship between (0,2) gauge anomalies and H-flux will emerge.

Yan Soibelman, Kansas State University

Wall-crossing structures

The concept of wall-crossing structure (WCS for short) was introduced recently in my joint work with Maxim Kontsevich. WCS appear in different disguises in the theory of Donaldson-Thomas invariants of Calabi-Yau 3-folds, quiver representations,integrable systems of Hitchin type, cluster algebras, Mirror Symmetry, etc.
I plan to discuss the definition of WCS and illustrate it in several well-known examples. If time permits I will speak about a special class of WCS called rational WCS. It gives rise to wall-crossing formulas with factors which are algebraic functions. Conjecturally such WCS appear in Hitchin integrable systems with singularities.

Albrecht Klemm, University of Bonn

On refined stable pair invariants for del Pezzo surfaces and the 1/2 K3

TBA

Kentaro Hori, Klavi IPMU

Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

We compute the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models. The result provides a general exact formula for the central charge of the D-brane placed at the boundary. It takes the form of Mellin-Barnes integral and the question of its convergence leads to the grade restriction rule concerning branes near the phase boundaries. We find expressions in various phases including the large volume formula in which a characteristic class called the Gamma class shows up. The two sphere partition function factorizes into two hemispheres glued by inverse to the annulus. The result can also be written in a form familiar in mirror symmetry, and suggests a way to find explicit mirror correspondence between branes.

Spiro Karigiannis, University of Waterloo

The mathematics of G_2 conifolds for M-theory

G_2 manifolds play the analogous role in M-theory that Calabi-Yau manifolds play in string theory. There has been work in the physics community on conjectural "mirror symmetry" in this context, and it has also been observed that singularities are necessary for a satisfactory theory. After a very brief review of these physical developments (by a mathematician who doesn't necessarily understand the physics), I will give a mathematical introduction to G_2 conifolds. I will then proceed to give a detailed survey of recent mathematical developments on G_2 conifolds, including desingularization, deformation theory, and possible constructions of G_2 conifolds. This includes separate joint works of myself with Jason Lotay and with Dominic Joyce.

## Algebraic structures in massive (2,2) theories

Tuesday Oct 22, 2013
Speaker(s):

I will review some ongoing work on the low energy properties of D-branes/boundary conditions in massive two-dimensional field theories with (2,2) supersymmetry.

Collection/Series:

## Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

Tuesday Oct 22, 2013
Speaker(s):

We compute the
partition function on the hemisphere of a class of two-dimensional (2,2)
supersymmetric field theories including gauged linear sigma models. The
result provides a general exact formula for the central charge of the
D-brane placed at the boundary. It takes the form of Mellin-Barnes
restriction rule concerning branes near the phase boundaries. We find
expressions in various phases including the large volume formula in

Collection/Series:

## Exact results for boundaries and domain walls in 2d supersymmetric theories

Tuesday Oct 22, 2013
Speaker(s):

We apply supersymmetric localization to N=(2,2) gauged linear sigma
models on a hemisphere, with boundary conditions, i.e., D-branes,
preserving B-type supersymmetries. We explain how to compute the
hemisphere partition function for each object in the derived category of
equivariant coherent sheaves, and argue that it depends only on its K
theory class. The hemisphere partition function computes exactly the
central charge of the D-brane, completing the well-known formula

Collection/Series:

## Coisotropic branes, surface defects and mirror symmetry

Monday Oct 21, 2013
Speaker(s):
Collection/Series:

## Hybrid conformal field theories

Monday Oct 21, 2013
Speaker(s):

I will discuss a class of limiting points in the moduli space of d=2
(2,2) superconformal field theories.  These SCFTs arise as IR limits of
"hybrid" UV theories constructed as a fibration of a Landau-Ginzburg
theory over a base Kaehler geometry.  A significant generalization of
Landau-Ginzburg and large radius geometric limit points, the hybrid
theories can be used to probe general features of (2,2) and (0,2) SCFT
moduli spaces.

Collection/Series:

## Some simple extensions of Mathieu Moonshine

Monday Oct 21, 2013
Speaker(s):

Mathieu Moonshine is a striking and unexpected relationship between the
sporadic simple finite group M24 and a special Jacobi form, the elliptic
genus, which arises naturally in studies of nonlinear sigma models with
K3 target.  In this talk, we first discuss its predecessor (Monstrous
Moonshine), then
discuss the current evidence in favor of Mathieu Moonshine.  We also
discuss extensions of this story involving second quantized mirror
symmetry,' relating heterotic strings on K3 to type II strings on

Collection/Series:

## Pages

Scientific Organizers

Vincent Bouchard, University of Alberta
Jaume Gomis, Perimeter Institute
Sergei Gukov, University of California, Santa Barbara
Johannes Walcher, McGill University
Shing-Tung Yau, Harvard University