The Mathematics of Quantum Theory
from
Friday, May 23, 2014 (9:00 AM)
to
Monday, May 26, 2014 (5:15 PM)
Monday, May 19, 2014
Tuesday, May 20, 2014
Wednesday, May 21, 2014
Thursday, May 22, 2014
Friday, May 23, 2014
9:15 AM
Breakfast and Registration
Breakfast and Registration
9:15 AM  10:00 AM
Room: Alpha Gamma Rho Room
10:00 AM
Is Quantization Unique?

Sergei Gukov
(
Caltech
)
Is Quantization Unique?
Sergei Gukov
(
Caltech
)
10:00 AM  11:00 AM
Room: Alpha Gamma Rho Room
Quantization of planar algebraic curves naturally enters many branches of physics and mathematics. For example, quantization of the zero locus of the Apolynomial of a knot gives the best known way to package an infinite set of colored Jones invariants of that knot into a single equation. Quantization of such Apolynomial curves is similar and, in some examples even identical, to quantization of spectral curves of trigonometric integrable systems. The latter lead to Baxter equations. Surprisingly, however, if the same planar curve appears as the Apolynomial of a knot and the spectral curve of a trigonometric integrable system, the result of its ``quantization'' in these two frameworks will be different.
11:00 AM
Coffee
Coffee
11:00 AM  11:15 AM
Room: Alpha Gamma Rho Room
11:15 AM
Categorification of quantum groups at a prime root of unity.

Mikhail Khovanov
(
Colombia
)
Categorification of quantum groups at a prime root of unity.
Mikhail Khovanov
(
Colombia
)
11:15 AM  12:00 PM
Room: Alpha Gamma Rho Room
Upon categorification quantum parameter q becomes a grading shift. Root of unity requires a tricker setup, involving pcomplexes and characteristic p. We will explain the construction, some results and conjectures for categorification of quantum groups, when the order of q is a prime number.
12:00 PM
Lunch
Lunch
12:00 PM  2:00 PM
Room: Alpha Gamma Rho Room
2:00 PM
Moduli of super Riemann surfaces

Ron Donagi
(
UPenn
)
Moduli of super Riemann surfaces
Ron Donagi
(
UPenn
)
2:00 PM  2:45 PM
Room: Alpha Gamma Rho Room
Albert Schwarz has made some of the most decisive early contributions to the theory of super Riemann surfaces and its connections with perturbative superstring theory. This subject has been revisited in recent works of Witten, and is rapidly developing in the form of super algebraic geometry. In this talk I will survey some of these recent developments. We will study various aspects of supergeometry, including obstruction, Atiyah, and superAtiyah classes. This will be applied to the geometry of the moduli space of super Riemann surfaces. We prove that for genus greater than or equal to 5, this moduli space is not projected (and in particular is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathematically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own. If time allows, we will describe some of the other new features of this space. (This is based on joint with E. Witten)
2:45 PM
Do all subfactors admit attendant conformal field theories?

Vaughan Jones
(
Berkeley
)
Do all subfactors admit attendant conformal field theories?
Vaughan Jones
(
Berkeley
)
2:45 PM  3:30 PM
Room: Alpha Gamma Rho Room
A subfactor is functional analytic object with highly combinatorial structure theory. Subfactors arise in various ways in conformal field theory via monodromy of npoint functions or more simply via commutation of local observable algebras. Subfactor technology has undergone many advances recently with a classification program for subfactors of small index. We meet subfactors that are do not arise from any currently know conformal field theory but there seems to be no reason that such CFT's do not exist, indeed Evans and Gannon give some evidence that such CFT's do exist in the context of Vertex operator algebras. We will describe some of these "exotic" subfactors and suggest ways in which CFT's might be made out of them.
3:30 PM
Coffee
Coffee
3:30 PM  4:00 PM
Room: Alpha Gamma Rho Room
4:00 PM
N=4 Super YangMills Theory on the Coulomb Branch

John Schwarz
(
Caltech
)
N=4 Super YangMills Theory on the Coulomb Branch
John Schwarz
(
Caltech
)
4:00 PM  4:45 PM
Room: Alpha Gamma Rho Room
It is conjectured that the worldvolume action of a probe D3brane in an AdS_5XS^5 background of type IIB superstring theory, with one unit of flux, can be reinterpreted as the exact effective action for U(2) N =4 super YangMills theory on the Coulomb branch. An analogous conjecture for ABJM theory is also presented. The main evidence supporting these conjectures is that the brane actions have all of the expected symmetries and dualities.
5:00 PM
Reception
Reception
5:00 PM  6:00 PM
Room: Alpha Gamma Rho Room
Saturday, May 24, 2014
9:15 AM
A mathematical approach to quantum curves

Motohico Mulase
(
U.C. Davis
)
A mathematical approach to quantum curves
Motohico Mulase
(
U.C. Davis
)
9:15 AM  10:00 AM
Room: Alpha Gamma Rho Room
A quantum curve is an hbar deformation family of Dmodules on a complex analytic curve. It takes the form of a stationary Schroedinger equation in one dimension, quantizing the spectral curve, which is a ramified covering of the starting curve. The coordinate of the starting curve is a parameter of a generating function, and the spectral curve is the Riemann surface of holomorphy of this function. The quantum curve, as a differential equation, then characterizes this function, which is a generating function of quantum topological invariants. In this talk, I will present recent mathematical developments on this subject, obtained jointly with Dumitrescu, DuninBarkowski, Norbury, Popolitov, Shadrin, and Sulkowski.
10:00 AM
Coffee
Coffee
10:00 AM  10:30 AM
Room: Alpha Gamma Rho Room
10:30 AM
AGT and Triality

Mina Aganagic
(
Berkeley
)
AGT and Triality
Mina Aganagic
(
Berkeley
)
10:30 AM  11:15 AM
Room: Alpha Gamma Rho Room
The AGT correspondence relates a class of gauge theories in four dimensions with two dimensional CFT’s. I will describe a simple proof of the correspondence when the CFT admits a free field representation. In those cases, vortex defects of the gauge theory play a crucial role, extending the correspondence to a triality.
11:15 AM
Quantum curves and the infinitedimensional Grassmannian

Albert Schwarz
(
UC Davis
)
Quantum curves and the infinitedimensional Grassmannian
Albert Schwarz
(
UC Davis
)
11:15 AM  12:00 PM
Room: Alpha Gamma Rho Room
One says that a pair $(P,Q)$ of ordinary differential operators specify a quantum curve if $[P,Q]=\hbar$. If a pair of difference operators $(K,L)$ obey the relation $KL=\lambda LK$ where $\lambda =e^{\hbar}$ we say that they specify a discrete quantum curve. This terminology is prompted by well known results about commuting differential and difference operators , relating pairs of such operators with pairs of meromorphic functions on algebraic curves obeying some conditions. Our methods are based on the interpretation of quantum curves in terms of infinitedimensional Grassmannian; in particular, it follows from this interpretation that (discrete) KPhierarchy can be used to deform a (discrete) quantum curve. The main goal is to study the moduli spaces of quantum curves. We will relate the moduli spaces for different $\hbar$. We will show how to quantize a pair of commuting differential or difference operators (i.e. to construct the corresponding quantum curve or discrete quantum curve)
12:00 PM
Lunch
Lunch
12:00 PM  2:00 PM
Room: Alpha Gamma Rho Room
2:00 PM
2functions, Lfunctions, and mirror symmetry

Johannes Walcher
(
McGill
)
2functions, Lfunctions, and mirror symmetry
Johannes Walcher
(
McGill
)
2:00 PM  2:45 PM
Room: Alpha Gamma Rho Room
I will review extended mirror symmetry, explain the notion of 2functions that we introduced (as an arithmetic generalization of dilogarithm) in recent work with Schwarz and Vologodsky, and the calculation of special values of Lfunctions that arise in this context.
2:45 PM
Degeneration of algebraic varieties and Ktheory

Vadim Vologodsky
(
University of Oregon
)
Degeneration of algebraic varieties and Ktheory
Vadim Vologodsky
(
University of Oregon
)
2:45 PM  3:30 PM
Room: Alpha Gamma Rho Room
The talk is based on our joint works with Kontsevich, Schwarz, and Walcher. Let X be a smooth proper algebraic variety over the formal punctured disk, maximally degenerated at the origin. Generalizing Mumford's construction in the case of abelian varieties I will attach to X a certain mixed Tate motive over the punctured disk and explain some application of this construction to the Mirror Symmetry.
3:30 PM
Coffee
Coffee
3:30 PM  4:00 PM
Room: Alpha Gamma Rho Room
4:00 PM
Naturalness of Slow NambuGoldstone Modes and Graph Theory

Petr Horava
(
University of California, Berkeley
)
Naturalness of Slow NambuGoldstone Modes and Graph Theory
Petr Horava
(
University of California, Berkeley
)
4:00 PM  4:45 PM
Room: Alpha Gamma Rho Room
TBA
5:00 PM
Snacks
Snacks
5:00 PM  5:45 PM
Room: Alpha Gamma Rho Room
6:00 PM
Public LectureString Theory: Past, Present and Future

John Schwarz
(
Caltech
)
Public LectureString Theory: Past, Present and Future
John Schwarz
(
Caltech
)
6:00 PM  7:00 PM
Room: 1002
￼String theory connects the microscopic quantum world of elementary particles to the largescale world of gravity and geometry. Physicists believe it may have the potential to achieve two very ambitious goals: (1) to provide a complete mathematical description of the physical laws that determine the properties of elementary particles and the forces that act on them and (2) to describe the origin and evolution of the universe. Much has been achieved, but string theory is still a work in progress. This talk will give a historical overview of the subject and discuss (without technical details) some of the problems that remain to be overcome.
7:30 PM
Speaker's Dinner
Speaker's Dinner
7:30 PM  10:30 PM
Sunday, May 25, 2014
9:15 AM
Symmetry Protected Topological Phases and Cobordisms"

Anton Kapustin
(
Caltech
)
Symmetry Protected Topological Phases and Cobordisms"
Anton Kapustin
(
Caltech
)
9:15 AM  10:00 AM
Room: Alpha Gamma Rho Room
Recently a new and rather unexpected connection between physics and algebraic topology has been noted. Namely, it appears that phases of matter with an energy gap, no longrange entanglement, and fixed symmetry can be classified using cobordism theory. I will exhibit several examples of this connection and describe a possible explanation.
10:00 AM
Coffee
Coffee
10:00 AM  10:30 AM
Room: Alpha Gamma Rho Room
10:30 AM
Mtheory and DTtheory

Andrei Okounkov
(
Colombia
)
Mtheory and DTtheory
Andrei Okounkov
(
Colombia
)
10:30 AM  11:15 AM
Room: Alpha Gamma Rho Room
This will be a report on a joint work with Nikita Nekrasov ([arXiv:1404.2323][1]), the goal of which is to find an exact match between the M2branes contributions to the Mtheory index and computations in Ktheoretic DonaldsonThomas theory of 3folds. [1]: http://arxiv.org/abs/1404.2323
11:15 AM
Open String Hodge Theory

Alexander Goncharov
(
Yale
)
Open String Hodge Theory
Alexander Goncharov
(
Yale
)
11:15 AM  12:00 PM
Room: Alpha Gamma Rho Room
TBA
12:00 PM
Lunch
Lunch
12:00 PM  2:00 PM
Room: Alpha Gamma Rho Room
2:00 PM
Nonperturbative DysonSchwinger equations and qqcharacters

Nikita Nekrasov
(
IHES
)
Nonperturbative DysonSchwinger equations and qqcharacters
Nikita Nekrasov
(
IHES
)
2:00 PM  2:45 PM
Room: Alpha Gamma Rho Room
TBA
2:45 PM
A hyperholomorphic line bundle in ${\cal N}=2$ theories

Andy Neitzke
(
U. Texas
)
A hyperholomorphic line bundle in ${\cal N}=2$ theories
Andy Neitzke
(
U. Texas
)
2:45 PM  3:30 PM
Room: Alpha Gamma Rho Room
Compactifying ${\cal N}=2$ supersymmetric field theory from four to three dimensions on a circle gives rise to a complex integrable system carrying a hyperkahler metric. In many cases this integrable system carries in addition a canonical hyperholomorphic line bundle, conjecturally related to the physics of the theory compactified on TaubNUT space. I will describe the construction of this line bundle, its connection to complex ChernSimons theory, and a closely related new smooth generating function for BPS state counts / DonaldsonThomas invariants. The bundle in some cases coincides with one introduced by Haydys and AlexandrovPerssonPioline.
3:30 PM
Coffee
Coffee
3:30 PM  4:00 PM
Room: Alpha Gamma Rho Room
4:00 PM
Towards the mathematics of AdS/CFT

Kevin Costello
(
North Western
)
Towards the mathematics of AdS/CFT
Kevin Costello
(
North Western
)
4:00 PM  4:45 PM
Room: Alpha Gamma Rho Room
I'll discuss a conjectural framework for a twisted form of the AdS/CFT correspondence, which describes the operator product of a large N limit of a twisted form of N=4 super YangMills in terms of a dual gravitational theory. The gravitational theory relevant for the twist we use is the 5complex dimensional analog of the BCOV KodairaSpencer theory.
5:30 PM
BBQ
BBQ
5:30 PM  8:00 PM
Monday, May 26, 2014
9:15 AM
The HirzebruchRiemannRoch theorem in quantum Ktheory

Alexander Givental
(
U.C. Berkeley
)
The HirzebruchRiemannRoch theorem in quantum Ktheory
Alexander Givental
(
U.C. Berkeley
)
9:15 AM  10:00 AM
Room: Alpha Gamma Rho Room
The title theorem (which is a joint result of the speaker with Valentin Tonita) expresses genus0 Ktheoretic GromovWitten invariants in terms of cohomological ones. The former are holomorphic Euler charactersistics of some interesting vector bundles over spaces of rational holomorphic curves in a given Kahler manifold, while the latter are suitable intersection indices in these spaces. The subject relies on many previous developments in GromovWitten theory, and is quite involved technically and conceptually. In this talk, we will focus on some relatively elementary aspect of the theory which, hopefully, has a general mathematical appeal. Namely, in contrast with the classical HirzebruchRiemannRoch formula, the theorem in question is not a formula, but an example of what we call "adelic characterization". That is, generating functions for Ktheoretic GromovWitten invariants (which happen to have the form of Laurent polynomials in one variable) are completely characterized by interpreting their Laurent series expansions near the poles at the roots of unity as generating functions for certain cohomological GromovWitten invariants.
10:00 AM
Coffee
Coffee
10:00 AM  10:30 AM
Room: Alpha Gamma Rho Room
10:30 AM
Convex polytopes and infrared categories.

Yan Soibelman
(
Kansas
)
Convex polytopes and infrared categories.
Yan Soibelman
(
Kansas
)
10:30 AM  11:15 AM
Room: Alpha Gamma Rho Room
In a recent work of Gaiotto,Moore and Witten the ``algebra of the infrared" for certain massive 2d theories with (2,2) supersymmetry was introduced. For LandauGinzburg models it gives a Morsetheoretical description of the corresponding $A_{\infty}$category of Abranes. It turns out that the combinatorial part of their work admits a higherdimensional generalization. I am going to discuss that generalization, its relation to GaiottoMooreWitten's work and speculate about possible applications.
11:15 AM
ChernSimons theory, Sduality, and a Tridiagonal Determinant Identity

Ori Ganor
(
Berkeley
)
ChernSimons theory, Sduality, and a Tridiagonal Determinant Identity
Ori Ganor
(
Berkeley
)
11:15 AM  12:00 PM
Room: Alpha Gamma Rho Room
An equivalence between two Hilbert spaces will be discussed: (i) the space of states of $U(1)^n$ ChernSimons theory on $T^2$ with coupling constants given by a certain class of tridiagonal matrices (with corners); and (ii) the space of ground states of strings on an associated mapping torus with $T^2$ fiber. The equality of dimensions of the two Hilbert spaces (i) and (ii) is equivalent to a known identity on determinants of tridiagonal matrices with corners. The equivalence of operator algebras acting on the two Hilbert spaces follows from a relation between the Smith normal form of the ChernSimons coupling constant matrix and the isometry group of the mapping torus, as well as the torsion part of its first homology group. The equivalence follows by studying the space of ground states of SL(2,Z)twisted circle compactifications of U(1) gauge theory, connected with a Janus configuration, and further compactified on a torus. I will also discuss generalizations to U(n) gauge theory.
12:00 PM
Lunch
Lunch
12:00 PM  2:00 PM
Room: Alpha Gamma Rho Room
2:00 PM
On regularized geometry of loop spaces.

Michael Movshev
(
Stony Brook
)
On regularized geometry of loop spaces.
Michael Movshev
(
Stony Brook
)
2:00 PM  2:45 PM
Room: Alpha Gamma Rho Room
The $O(N+1)$model or a sigma model whose target is a round $N$dimensional sphere is a well established (by physical standards) subject. It attracts attention because the theory exhibit spontaneous mass generationa feature that is also expected in a more realistic but also more complicated fourdimensional gauge theories. In addition, the $O(N+1)$model is believed to be completely integrable. In particular, an explicit formula for the mass gap is know. I will discuss mathematical aspects of quantum Hamiltonian formalism for the $O(N+1)$model such as a precise statement of the mass gap conjecture and a possible definition of the renormalization group that goes beyond perturbation theory.
2:45 PM
Topological recursion, cohomological field theories and quantization

Nicolas Orantin
(
Instituto Superior Tecnico
)
Topological recursion, cohomological field theories and quantization
Nicolas Orantin
(
Instituto Superior Tecnico
)
2:45 PM  3:30 PM
Room: Alpha Gamma Rho Room
The topological recursion method is a formalism developed in the context of random matrix theories in order to solve an associated problem of combinatorics consisting in the enumeration of discrete surfaces. This inductive procedure allows to enumerate such surfaces of arbitrary topology out of the only genus 0 data. This theory has further been formalized out of the context of random matrices and mysteriously solved many problem of enumerative geometry using a universal inductive procedure. In this talk, I will present this topological recursion procedure and explain the reason why it solves many problems of enumerative geometry at once. I will show that, given a semisimple Frobenius manifold, one can identify the formula of the ancestor GromovWitten potential derived by Givental with the correlation functions computed by a local version of the topological recursion. The role of mirror symmetry will be explained and exemplified in the computation of the GromovWitten invariants of the projective line. I will finally explain how this procedure produces a semiclassical approximation of a wave function obtained by quantizing the corresponding spectral curve. Based on joint works with Chekhov, DuninBarkowski, Eynard, Norbury, Shadrin and Spitz.
3:30 PM
Coffee
Coffee
3:30 PM  4:00 PM
Room: Alpha Gamma Rho Room
4:00 PM
Holomorphicity in QFT

Maxim Kontsevich
(
IHES
)
Holomorphicity in QFT
Maxim Kontsevich
(
IHES
)
4:00 PM  5:00 PM
Room: 1147
TBA