Participants - Geometry and Quantization 2007

A. N. Tyurin (1940–2002)

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Russian Academy of Sciences
Steklov Mathematical Institute
Lomonosov Moscow State University

International School-Conference
GEOMETRY and QUANTIZATION
dedicated to the memory of the outstanding mathematician
Andrei Nikolaevich Tyurin

ABSTRACTS

Quantization of integrable systems with monodromy
Michael CARL

Starting from the integral affine structure on the regular image of an integrable moment map we would like to construct and compare different representation spaces. Key words: Integrable systems, quantization, monodromy

Noncommutative unitons
Andrei DOMRIN

By the results of Uhlenbeck, every harmonic map from the Riemann sphere $S^2$ to the unitary group $U(n)$ is a product of the so-called unitons: special maps from $S^2$ to the Grassmannians $Gr_k(C^n)\subset U(n)$ satisfying certain first-order systems of differential equations. We construct a noncommutative analogue of this factorization. It holds for those solutions of the noncommutative sigma-model that are finite-dimensional perturbations of solutions of energy zero. Applications include: (1) quantization of energy of solutions, (2) an abundance of examples of non-Grassmannian solutions, (3) a detailed description of the moduli spaces of solutions of small energy.

Relativistic anti-Wick quantization of gauge fields in 3+1 dimensions
Alexander DYNIN

We show that the reduced Yang-Mills phase space has a weak Kaehler structure. Then we apply quantum white noise calculus to the Shwinger-Bogoliubov-Shirkov quantization scheme. Key words: Quantum field theory, Yang-Mills fields

Complete Ricci-flat K\"ahler metrics on the canonical bundles of toric Fano manifolds
Akito FUTAKI

We prove the existence of complete Ricci-flat K\"ahler metrics on the canonical bundles of toric Fano manifolds. This is an application of an existence result of toric Sasaki- Einstein metrics.

Introduction to deformation qunatization
Dmitry KALEDIN

I will give an introduction to the so-called deformation quantization of symplectic and poisson manifolds, from an algebraic point of view. The main topics will be: the setup, the necessary deformation theory material (Hochschild cohomology of associative algebras, star-products, Maurer-Cartan formalism), Kontsevich formality and quantization of the affine space with an arbitrary poisson bracket. If time permits, I will also touch on quantization of smooth manifolds, including Fedosov quantization.

Some dynamical property of the Ricci flow on the twistor space of quaternionic Kaehler manifolds with positive scalar curvature
Ryoichi KOBAYASHI

We consider the Ricci map on the space of Riemannian metrics on the twistor space of quaternionic Kaehler manifold with positive scalar curvature and show the existence of the subspace which is invariant under the Ricci map. Its consequences will be discussed. Key words: quaternionic Kaehler manifolds, twistor space, Ricci flow.

Singular unitarity in quantization commutes with reduction
Hui LI

We will discuss the relation between the inner products of the quantum Hilbert spaces before and after symplectic reduction. Key words: Geometric quantization, moment map, symplectic quotient.

Superconnection and family Bergman kernels
Xiaonan MA

We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection formalism as in the local family index theorem.

Geometry of G_2 orbits.
Reiko MIYAOKA

Actions of G_2 and its subgroup cause cohomogeneity one actions on vector spaces. We discuss such orbits, and some applications. Key words: G_2 orbit .

Asymptotically Holomorphic Embeddings of Presymplectic Manifolds into Projective Spaces.
Takayuki MORIYAMA

We construct asymptotically holomorphic embeddings of presymplectic manifolds in the complex projective space by Donaldson's asymptotically holomorphic techniques. Key words: Asymptotically holomorphic map, presymplectic manifold

Projective embeddings and Lagrangian fibrations of Abelian varieties and Kummer varieties
Yuichi NOHARA

The first half of this talk is a brief introduction to geometric quantization. We mainly consider the case of Abelian varieties and see a relation between natural basis of the spaces of theta functions and certain Lagrangian fibrations. In the latter half we discuss this relation from the point of view of projective embeddings of the Abelian varieties.

Gerbes and symbols
Denis OSIPOV

We consider construction of the 2-dimensional tame symbol on a 2-dimensional local field as monodromy of some combinatorial gerbe. From this construction we prove up to sign the Parshin reciprocity laws for these symbols for any algebraic surfaces over any perfect ground field.

Lie antialgebras
Valentin OVSIENKO

We introduce a new class of algebras that we call Lie antialgebras.

Commutator identities on associative algebras and integrability of nonlinear pde's
Andrei POGREBKOV

It is shown that commutator identities on associative algebras generate solutions of linearized integrable equations. Next, a special kind of the dressing procedure is suggested that in a special class of integral operators enables to associate to such commutator identity both nonlinear equation and its Lax pair. Thus problem of construction of new integrable pde's reduces to construction of commutator identities on associative algebras.

Bethe ansatz for principal nilpotent argument shift subalgebras
Leonid RYBNIKOV

For any semisimple Lie algebra g, quantum argument shift subalgebras form a family of maximal commutative subalgebras in the universal enveloping algebra U(g). This family is parameterized by regular elements g^*. The classical limits of these subalgebras were constucted by Mischenko and Fomenko, and studied later by Vinberg, Shuvalov, Tarasov and others. The quantum ones can be obtained from the center of the completed universal enveloping algebra of the corresponding affine Kac-Moody algebra via Feigin-Frenkel-Reshetikhin scheme. I will describe the action of quantum argument shift subalgebras in irreducible finite-dimensional g-modules, using an appropriate version of Bethe ansatz. The most interesting case is when the parameter of shift is the pricipal nilpotent element. In this case, the elements of the quantum argument shift subalgebra act in as lowering operators with respect to the principal grading. I will show that every finite-dimensional irreducible g-module is cyclic over this subalgebra, and describe the annihilator in terms of monodromy-free conditions on certain space of g^L-opers. This gives a natural structure of commutative Gorenstein ring on every finite-dimensional irreducible g-module.

Semiclassical quantization of Lagrangian manifolds
Andrei SHAFAREVICH

Lecture 1.
Semiclassical quantization of invariant manifolds of Hamiltonian systems. Quantization of real Lagrangian manifolds. Bohr -- Sommerfeld -- Maslov conditions and Maslov index. Semiclassical quantization of integrable systems.
Lecture 2.
Quantization of isotropic manifolds. Invariant complex vector bundles and tensor fields. Almost integrable systems. Quantization of resonant systems. Quantum selection of periodic trajectories and Liouville foliations. Quantization of singular fibers.
Lecture 3.
Quantization of complex manifolds. Integrable systems with discrete symmetry and splitting of eigenvalues for self- adjoint operators. Spectral series of non-selfadjoint operators. Spectral graphs and quantization conditions on Riemann surfaces.

Tautological relations and the master equation
Sergey SHADRIN

We are going to discuss a misterious relation between the master equation in Batalin-Vilkovisky formalism and the intersection theory of the moduli space of curves. The main result is that the master equation reflects correctly the excessive intersection of tautolological classes in the moduli space of curves. (a join work with A. Losev and I. Shneiberg, arXiv: 0704.1001)

Lax operator algebras
Oleg SHEINMAN

I would like to present a new class of gauge algebras generalizing (non-twisted) Kac-Moody and Krichever-Novikov algebras. In hep-th/0108110 I.Krichever, for the purposes of Hamiltonian theory of Lax equations, has given a general description of Lax operators with the spectral parameter on an algebraic curve. In math/0701648 we found out a Lie algebra structure on the space of such operators, and introduced their orthogonal and symplectic analogues. Every new algebra is given by a classic simple or reductive Lie algebra, a Riemann surface and a Tyurin data on it. The talk is based on the joint results with I.Krichever and M.Schlichenmaier.

Kontsevich formality and PBW algebras
Boris SHOIKHET

Let $\alpha$ be a polynomial Poisson bivector on a finite-dimensional vector space $V$ over $\mathbb{C}$. Then Kontsevich in his 1997 paper gives a formula for a quantization $f\star g$ of the algebra $S(V)^*$. We give a construction of an algebra with the PBW property defined from $\alpha$ by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra $T(V^*)$ by relations $x_i\otimes x_j-x_j\otimes x_i=R_{ij}(\hbar)$ where $R_{ij}(\hbar)\in T(V^*)\otimes\hbar \mathbb{C}[[\hbar]]$, $R_{ij}=\hbar \Sym(\alpha_{ij})+\mathcal{O}(\hbar^2)$, with one relation for each pair of $i,j=1...\dim V$. We prove that the constructed algebra obeys the PBW property, and this is a generalization of the Poincar\'{e}-Birkhoff-Witt theorem. In the case of a linear Poisson structure we get a new very conceptual proof of the PBW theorem itself, and for a quadratic Poisson structure we get an object closely related to a quantum $R$-matrix on $V$. The construction uses the Kontsevich formality. Namely, our quantities $R_{ij}(\hbar)\in T(V^*)\otimes \mathbb{C}[[\hbar]]$ are written directly in Kontsevich integrals, but in a sense of dual graphs than the graphs used in the deformation quantization. We conjecture that the relation $x_i\otimes x_j-x_j\otimes x_i=R_{ij}(\hbar)$ holds in the Kontsevich star-algebra, when we replace $\otimes$ by $\star$. This conjecture implies in particular that our algebra is isomorphic to the Kontsevich star-algebra with the same $\alpha$, but also gives a highly-nontrivial identity on Kontsevich integrals. Probably, it is a particular case of a more general duality acting on the AKSZ model on open disc, u

On a geometric analog of the Birch and Swinnerton-Dyer conjecture for a hyperbolic threefold of finite volume
Ken-ichi SUGIYAMA

Based on the philosohpy of the number theory, we will discuss a special value of the Ruelle and Selberg L-function. Key words: L-function, Franz-Reidemeister torsion

Non-commutative structures and *-product.
Dmitry Treschev

We present a construction of algebras, generalizing the well-known Heisenberg algebra of quantum observables. Such algebras can be defined over a smooth manifold $M$ and present a non-commutative analog of $C^\infty(M)$. Non-commutative analogs of vector fields and differential forms can be also naturally defined. Hence some kind of non-commutative differential geometry appears. These constructions (non-commutative structures) are strongly connected with star-products on $M$.

Moduli space of Brody curves, energy and mean dimension.
Masaki TSUKAMOTO

Brody curve is a holomorphic map from the complex line to a complex manifold with bounded derivative. Since the complex line is not compact, the moduli space of Brody curves is infinite dimensional in general. We study the Gromov mean dimension of the moduli space of Brody curves by using energy of Brody curves. This is an attempt to evaluate the dimension of infinite dimensional space. Key words: Brody curve, mean dimension, energy, the Nevanlinna theory

Quantization of the Riemann Zeta-Function
Igor VOLOVICH

Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat-Wiles and the Langlands program is also indicated. Ref: I.Ya. Aref\\\'eva, I.V.Volovich, Quantization of the Riemann Zeta-Function and Cosmology.. Key words: Quantization, Zeta-function We describe our analytic approach with Youliang Tian on the Guillemin-Sternberg geometric quantization conjecture and related topics. Key words: Analytic localization, geometric quantization conjecture

Formal punctured ribbons and two-dimensional local fields
Alexander ZHEGLOV

We shall introduce a notion of a formal punctured ribbon on a curve and talk about a one-to-one correspondence between such ribbons and certain subspases of a two-dimensional local field. This correspondence is a generalization of the Krichever-Parshin-Osipov map for a usual geometric data. Certain applications to Parshin\\\'s higher-dimensional analogues of the KP hierarchy will be discussed. The talk is based on the joint results with H.Kurke and D.Osipov