Informal Mathematical Physics Seminar
organized by Igor Krichever and Andrei Okounkov
Mondays, 5:30, Room 507
Schedule of talks for Spring 2014:
Abstracts
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.
May5
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.