Department of Mathematics

University of California

Berkeley, CA 94720-3840

October 2010:

Abstract:

Donaldson-Thomas (DT) invariants of 3-dimensional Calabi-Yau manifolds are related to the (properly defined) count of various geometric objects like: special Lagrangian manifolds, semistable vector bundles, ideal sheaves, etc. It turns out that the unifying framework is the one of 3-dimensional Calabi-Yau categories endowed with Bridgeland stability condition. Then we count the ``number" of semistable objects with the fixed class in K-group. Corresponding theory was developed in a series of our papers with Maxim Kontsevich. It gives also a mathematical approach to BPS invariants (both enumerative and refined) in gauge and string theory. Similarly to geometric story, our invariants change on the real codimension one ``walls" in the space of stability conditions. Categorically, the wall-crossing formulas show how the ``motive of semistable objects" changes across the wall (this explains the term ``motivic").

There are two different approaches to the theory of motivic DT-invariants. I plan to discuss mostly a recent one based on moduli spaces of representations of quiver with potential. Applications include cluster transformations, complex integrable systems and new invariants of 3-dimensional manifolds.

Lecture Notes are available here .

January 13-15 2011:

Abstract:

These lectures are devoted to the supersymmetric (susy) vacua of two, three and four dimensional $N=2$ susy gauge theories with matter which will be shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians and other integrable quantum many body systems.

The correspondence between the Heisenberg spin chain and the two dimensional $U(N)$ theory with fundamental hypermultiplets will be reviewed in detail. The relation between twisted effective superpotential and Yang-Yang function will be explained. This correspondence extends to other spin chains, $XXZ$, $XYZ$, with any spin group, representations, boundary conditions, inhomogeneity, etc.

Then we move to the study of four dimensional $N=2$ supersymmetric gauge theory in the Omega-background with the two dimensional $N=2$ super-Poincare invariance. This gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional $N=2$ theory. We present the thermodynamic-Bethe-ansatz like formulae for these Yang-Yang function and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. Examples include the periodic Toda chain, the elliptic Calogero-Moser system (and their relativistic versions), for which we present a complete characterization of the $L^2$-spectrum. If time permits we briefly discuss the quantization of Hitchin systems in general.

Based on:

1) Quantization of Integrable Systems and Four Dimensional Gauge Theories. Nikita A. Nekrasov, Samson L. Shatashvili, 16th International Congress on Mathematical Physics, P. Exner, Editor, pp.265-289, World Scientific 2010 e-Print: arXiv:0908.4052 2) Quantum integrability and supersymmetric vacua. Nikita A. Nekrasov, Samson L. Shatashvili, Prog.Theor.Phys.Suppl.177:105-119,2009. e-Print: arXiv:0901.4748 3) Supersymmetric vacua and Bethe ansatz. Nikita A. Nekrasov, Samson L. Shatashvili, Nucl.Phys.Proc.Suppl.192-193:91-112,2009. e-Print: arXiv:0901.4744 4) Two-dimensional gauge theories and quantum integrable systems. Anton A. Gerasimov, Samson L. Shatashvili, Proceedings of Symposia in Pure Mathematics, Vol. 78, American Mathematical Society,Providence, Rhode Island, 2008; e-Print: arXiv:0711.1472 5) Higgs Bundles, Gauge Theories and Quantum Groups. Anton A. Gerasimov, Samson L. Shatashvili, Commun.Math.Phys.277:323-367,2008. e-Print: hep-th/0609024

September 27-October 3 2011:

Abstract:

These lectures summarize results of study of integrable models of the Quantum Field Theory (QFT) in two space-time dimensions.

There are more or less the only relativistic models local models in quantum field theory where exact, non-perturbative formulation exists and many quantities can be computed explicitly. In the lectures the main example will be sine-Gordon (sG) model. This model is related to the c<1 Conformal Field Theory (CFT) and to the six vertex model in statistical mechanics. This relation and it consequences will be explained.

The lectures will start with an introduction to Minkowski formulation of quantum field theory (QFT) explaining factorable S-matrices and form factors. Then we will focus on the relation between the short distance behavior of two-point correlation functions in the sG model and correlation functions in corresponding CFT.

After this we will discuss the relation between Euclidean QFT and lattice models of 2D statistical physics. Using the fermionic description of the space of local operators for the six vertex model and passing to the scaling limit new way of describing the CFT which is compatible with the integrable perturbation will be introduced. This material is based on recent works by Boos, Jimbo, Miwa, Takeyama and FS . In particular, these results allow to compute the one-point functions for sG model explicitly and solves the problem of describing the short-distance behavior of correlation functions.

Lecture Notes are available here and here .

October 17-21 2011:

Abstract:

In these lectures, we'll discuss relations between the Hitchin moduli spaces, their quantization with respect to the different symplectic structures furnished by the hyperkaehler structure, and the quantum Liouville theory. Here is the rough plan of the lectures.

The main players: Hitchin's integrable systems, Flat connections and isomonodromic deformations, Liouville theory.

Quantization of moduli spaces of flat connections: Using monodromy data; Using representation in terms of holomorphic connections; Relation to Liouville theory.

If time permits we will discuss how to obtain quantum Hitchin system from Liouville theory: Quantization conditions from single-valuedness, Reformulation in terms of Yang's potential, Relation with geometric Langlands correspondence.

January 9-16 2012:

Abstract:

The lectures will start with a short introduction to basic notions about super-manifolds, and in particular to the integration on super-manifolds. Then the lectures will focus on BV-Laplacian (Batalin-Vilkovisky). Then Faddeev-Popov and BV-integration will be discussed. The lectures are aimed at general graduate student audience.

After the lectures there will be more technical seminar where of the notion of a BV-resolution of a polynomial S (BV-action) will be introduced. Then the essential uniqueness of such a resolution will be proven and it will be shown that the corresponding BRST Poisson algebra does not depend on a choice of a resolution.

March 2012:

Abstract:

The lectures are devoted to a somewhat detailed presentation of Stochastic Schramm-Loewner Evolutions (SLE), which are Markov processes describing fractal curves or interfaces in two-dimensional critical systems. A substantial part of the lectures covers the connection between statistical mechanics and processes which, in the present context, leads to a connection between SLE and conformal field theory (CFT). These lectures aim at filling part of the gap between the mathematical and physics approaches. They are intended to be at an introductory level.

April, 2012:

DiscreteIntegrable Systems and Cluster Algebras

We discuss discrete time non-linear evolution equations arising from the study of quantum integrable spin chains in physics, and show how they fit in the cluster algebra structure of Fomin and Zelevinsky. The latter is a sort of dynamical system on formal coordinate patches that are transformed according to rational mutation rules, guaranteeing the Laurent property that each mutated coordinate is a Laurent polynomial of the variables in any other coordinate patch. We shall focus on the main open positivity conjecture of cluster algebras, stating that these Laurent polynomials have only non-negative integer coefficients.

The combinatorial content of our evolution equations is revealed by their exact solutions, which make use of their discrete integrable structure to express them as partition functions for Viennot's heaps of pieces, or alternatively for positively weighted paths on target graphs. Cluster algebra mutations are implemented by local transformations in various guises: local continued fraction rearrangements, matrix representations local moves a la Yang-Baxter, etc.

This generalizes nicely to a non-commutative setting, by use of the notion of quasi-determinants introduced by Gelfand and Retakh, and allows us to prove the Kontsevich non-commutative positivity conjecture in rank 2. In higher rank, we obtain a noncommutative version of the discrete Hirota equation. (Lectures based on work in collaboration with R. Kedem).

Lecture 1: Cluster Algebras – Tuesday April 10, 12:00-2:00, 736 Evans

1. Preamble: total positivity and networks

2. Cluster Algebras: definition and examples

3. Quantum deformations

Lecture 2: Discrete Integrable Systems I – Wednesday April 11, 12:00-2:00, 740 Evans

1. Q-systems: cluster algebra of initial data

2. Q-systems: path solutions, continued fractions and positivity

3. Non-commutative extensions

Lecture 3: Discrete Integrable Systems II – Thursday April 12, 12:00-2:00, 939 Evans

1. T-systems: cluster algebra of initial data

2. T-systems: network solution

3. Generalized Yang-Baxter equation

November 12- 16, 2012:

Automorphism groups of quantum nilpotent algebras

Automorphism groups of algebras are often large and very difficult to compute. Dixmier did this for the first Weyl algebra, but Joseph, Alev and Shestakov-Umirbaev proved that the universal enveloping algebras of 3 dimensional Lie algebras have wild automorphisms. About 10 years ago Andruskiewitsch and Dumas conjectured that the positive parts of all quantized universal enveloping algebras of simple Lie algebras are rigid, i.e. have small automorphims groups that can be described explicitly. Touching upon the second part of the title, all Kac-Moody Lie algebras have quantizations that have been at the heart of many recent developments. However, much less is known about deformations of universal enveloping algebras of nilpotent Lie algebras. The goal of the lectures is twofold. Firstly, we will describe a method for classification of automorphism groups of certain quantized universal enveloping algebras based on a rigidity theorem for automorphisms of completed quantum tori. The method has broad range of applications and in particular settles the Andruskiewitch-Dumas conjecture. Secondly, following Cauchon, Goodearl, and Letzter we will develop a general axiomatic setting for quantum nilpotent algebras. We will then treat the full family in the framework of noncommutative unique factorization domains and construct initial clusters for cluster algebra structures on all of these algebras. This will make the above mentioned rigidity technique applicable to all of them.

December 2-7, 2012:

Schur functions and Littlewood-Richardson rule from exactly solvable tiling models

In these lectures, we revisit a very classical subject, that of Schur functions and of the Littlewood-Richardson rule, using the language and tools of statistical mechanics. More precisely we identify Schur functions and Littlewood-Richardson coefficients as the partition functions of certain so-called ``exactly solvable'' models of two-dimensional statistical mechanics. There is a powerful associated machinery which allows to rederive various identities and perform analytical calculations. The plan of the three lectures should be:

Lecture 1. Tuesday 3-4, 939 Evans. Introduction to Schur functions from representation theory and geometry. Connection to the quantum mechanics of free fermions. Lattice fermions as Non-Intersecting Lattice paths and as tilings.

Lecture 2. Wednesday 4-6 891, Evans. The Littlewood-Richardson rule. Various formulations and their connection to the projection method in the theory of tilings.

Lecture 3. Thursday 13, 939 Evans. The square-triangle-rhumbus tiling model. Integrability, Yang-Baxter equation. Physical discussion of the model as one of random tilings. If time permits, various extensions will be discussed including factorial Schur functions, Q-Schur functions, etc.

Lecture notes are available here .

Feb 25-March 1, 2013:

Discrete Riemann Surfaces.

The general idea of discrete differential geometry is to find and investigate discrete models that exibit properties and structures characterisitic of the corresponding smooth geometric objects. Several structure preserving definitions of discrete holomorphic functions and Riemann surfaces are known today. Linear theories are based on discrete Cauchy-Riemann equations. Nonlinear theories are based on patterns of circles or on a discrete notion of conformal equivalence for triangulated surfaces. In these lectures we introduce discrete versions of conformal structure, holomorphic functions, period matrix, discrete conformal metrics and other notions from the classical theory. We focus on proving discrete versions of the Riemann mapping theorem, classical uniformization theorems and on computation of periods of Riemann surfaces. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps and to convex variational principles. We establish a connection between conformal geometry for triangulated surfaces and the geometry of ideal hyperbolic polyhedra. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also allows to merge the theories of discretely conformally equivalent triangulated surfaces and of circle packings. Applications in geometry processing and computer graphics will be discussed.

April 22-26, 2013:

Classical and quantum Lagrangian field theories on manifolds with boundaries.

Lecture 1. The lectures will start with presenting the program of studying classical and perturbative quantum field theories on manifolds with boundary. Then I plan to discuss classical Lagrangian field theories and to show how they lead to associating symplectic manifolds to boundary components and canonical relations to the bulk. Relevant concepts from symplectic geometry will be recalled and examples will be discussed.

Lecture 2. In the second lecture I plan to discuss the Batalin-Vilkovisky (BV) formalism, which allows to study the perturbative quantization of field theories with symmetries on closed manifolds. I will recall the relevant concepts from supergeometry and explain how the BV integral allows deforming integration domains. Next I plan to describe how this has to be modified in the case of manifolds with boundary or corners. Examples will be discussed, in particular, the so-called AKSZ topological field theories (which include Chern-Simons and BF theories and the Poisson sigma model).

Lecture 3. In the third lecture, I plan to present some preliminary results on the perturbative quantization of BV field theories on manifolds with boundary, focusing on the example of abelian BF theories.