Informal Mathematical Physics Seminar

organized by Igor Krichever and Andrei Okounkov

Mondays, 5:30, Room 507

Schedule of talks for Spring 2014:

Feb. 3
Leonid Chekhov
Poisson and quantum algebras originated from bilinear forms
Feb. 10
Petr Pushkar
Square Tiled Surfaces and Teichmuller Volumes of Moduli
Spaces of Abelian Differentials
Feb. 17
no seminar

Feb. 24
room taken
all go to the Minerva lecture
March 3
Ivan Loseu
Representation theory of quantizations of the Gieseker moduli space
March 10
Daniel Halpern-Leistner Combinatorial structures in the theory of instability
March 17
no seminar
spring break
March 24
Alexander Gorsky Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems
March 31
Vasily Pestun
Instantons and integrable systems
April 7
Semeon Artamonov
HOMFLY polynomial calculus for links and AENV conjecture
April 14
Macdonald polynomials for everyone (in preparation for Minerva lectures)
April 21
Minerva lectures
all attend
April 28
Eugene Gorsky
Floer homology of algebraic links
May 5
Mohammed Abouzaid
Hidden symmetries of symplectic manifolds


Feb. 3 

Bilinear forms A with the transformation law A->BAB^T manifest rich Poisson
and quantum algebraic structures and admit a number of Poisson reductions
among which are reductions to algebras of geodesic functions on hyperbolic
Riemann surfaces (for upper-triangular A) studied from the mathematical
side by Bondal, Dubrovin, and Ugaglia and from the physical side by
Klimyk, Gavrilik, Nelson, and Regge. We describe the algebroid of bilinear
forms, its reductions, the associated braid-group action, generalization
to the affine case (joint papers with M.Mazzocco at Advances Math. and
Comm.Math.Phys.). If time will allow, I will also present some new results
on groupoid structures consistent with the transformation laws.

March 3

Gieseker moduli spaces are especially nice and important examples of Nakajima quiver varieties.
One can quantize them using quantum Hamiltonian reduction and get associative algebras that generalize
type A Rational Cherednik algebras. The latter algebras were extensively studied in the last decade, but the
general case has been untouched until very recently. I will explain what I know about the representation
theory in the general case. This is based on my work in preparation/in progress.

March 10

In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of an point in the stack. To such a degeneration one can often associate a real number which measures "how destabilizing" it is, and in these situations one can ask the question of whether there is a "maximal destabilizing" or "canonically destabilizing" degeneration of a given unstable point.

I will discuss a framework for formulating and discussing this question which generalizes several commonly studied examples: geometric invariant theory, the moduli of bundles on a smooth curve, the moduli of Bridgeland-semistable complexes on a smooth projective variety, the moduli of K-stable varieties. The key construction assigns to any point in an algebraic stack a topological space parameterizing all possible iso-trivial degenerations of that point. When the stack is BG for a reductive G, this recovers the spherical building of G, and when the stack is X/T for a toric variety X, this recovers the support of the fan of X.

March 24
In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous 𝔤𝔩n-invariant XXX spin chain on N sites with twisted boundary conditions can be found in terms of velocities of particles in the rational N-body Ruijsenaars-Schneider model. The possible values of the velocities are to be found from intersection points of two Lagrangian submanifolds in the phase space of the classical model. One of them is the Lagrangian hyperplane corresponding to fixed coordinates of all N particles and the other one is an N-dimensional Lagrangian submanifold obtained by fixing levels of N classical Hamiltonians in involution. The latter are determined by eigenvalues of the twist matrix. To support this picture, we give a direct proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars-Schneider model, where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians, calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with certain multiplicities. We also prove a similar statement for the 𝔤𝔩n Gaudin model with N marked points (on the quantum side) and the Calogero-Moser system with N particles (on the classical side). The realization of the results obtained in terms of branes and supersymmetric gauge theories is also discussed.

April 7
Recently Aganagic, Ekholm, Ng, and Vafa conjectured a relation between the augmentation variety in the large N limit of the colored HOMFLY and quantum A-polynomials. In this talk I will describe the methods used for direct confirmation of this conjecture for certain links.

It appears that colored knot polynomials possess an internal structure (we call it Z-expansion) which behaves naturally under inclusion of the representation into the product of the fundamental ones. In particular, for some large families of links the colored HOMFLY polynomial for symmetric and anti-symmetric representations can be presented as a truncated sum of a certain q-hypergeometric series. The latter allows us to extend the formulas for the arbitrary symmetric representations and study the asymptotic of the colored HOMFLY polynomials for large symmetric representations.

In addition I will say a few words about the extension of Z-expansion beyond the symmetric representations for some simplest examples. Although for generic representation we no longer have those cute truncated q-hypergeometric series we still have some interesting structure beyond the HOMFLY and superpolynomials. In particular, the introduction of the recently developed fourth grading in all existing examples can be presented as an elegant redefinition of the constituents of Z-expansion.

April 28
Campillo, Delgado and Gusein-Zade defined an infinite collection of hyperplane arrangements for a given plane curve singularity. They proved that their Euler characteristics match the coefficients in the multi-variable Alexander polynomial of the corresponding link. I will explain the relation between their homology and the Heegaard-Floer link homology, categorifying the Alexander polynomial. The talk is based on joint works 1301.7636 and 1403.3143
with Andras Nemethi.

I will discuss the mirror of a holomorphic action of C* on a
Calabi-Yau manifold. The formulation will depend on the behaviour of
the holomorphic volume form under such an action. Whenever the
holomorphic volume form is preserved, we shall see that the action
corresponds to a class in the first cohomology of the mirror. If the
action is dilating, I will pass to the infinitesimal case, and discuss
different formulations for the mirror. This is a report on
Seidel-Solomon, Abouzaid-Smith, and Seidel.