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arXiv1997

Lectures in Topological Quantum Field Theory

In these lectures we present a general introduction to topological quantum field theories. These theories are discussed in the framework of the Mathai-Quillen formalism and in the context of twisted N=2 supersymmetric theories. We discuss in detail the recent developments in Donaldson-Witten theory obtained from the application of results based on duality for N=2 supersymmetric Yang-Mills theories. This involves a description of the computation of Donaldson invariants in terms of Seiberg-Witten invariants. Generalizations of Donaldson-Witten theory are reviewed, and the structure of the vacuum expectation values of their observables is analyzed in the context of duality for the simplest case.

hep-th↗
arXiv1997

Integration over the u-plane in Donaldson theory

We analyze the u-plane contribution to Donaldson invariants of a four-manifold X. For $b_2^+(X)>1$, this contribution vanishes, but for $b_2^+=1$, the Donaldson invariants must be written as the sum of a u-plane integral and an SW contribution. The u-plane integrals are quite intricate, but can be analyzed in great detail and even calculated. By analyzing the u-plane integrals, the relation of Donaldson theory to N=2 supersymmetric Yang-Mills theory can be described much more fully, the relation of Donaldson invariants to SW theory can be generalized to four-manifolds not of simple type, and interesting formulas can be obtained for the class numbers of imaginary quadratic fields. We also show how the results generalize to extensions of Donaldson theory obtained by including hypermultiplet matter fields.

hep-th↗
arXiv1997

Non-Recursive Multiplicity Formulas for $A_N$ Lie Algebras

It is shown that there are infinitely many formulas to calculate multiplicities of weights participating in irreducible representations of $A_N$ Lie algebras. On contrary to recursive character of Kostant and Freudenthal multiplicity formulas, they provide us systems of linear algebraic equations with N-dependent polinomial coefficients. These polinomial coefficients are in fact related with polinomials which represent eigenvalues of Casimir operators.

math-ph↗
arXiv1995

Localization of $\frak{u}$-modules. II. Configuration spaces and quantum groups

This paper is a sequel to "Localization of $\frak{u}$-modules. I", hep-th/9411050. We are starting here the geometric study of the tensor category $\cal{C}$ associated with a quantum group (corresponding to a Cartan matrix of finite type) at a root of unity. The main results establish isomorphisms between homogeneous components of irreducible objects in $\cal{C}$ and spaces of vanishing cycles at the origin of certain Goresky-MacPherson sheaves on configuration spaces; establish isomorphisms of the stalks at the origin of the above GM sheaves with certain Hochschild complexes (which compute the Hochschild homology of a certain "triangular" subalgebra of our quantum group with coefficients in the coresponding irreducible representation); establish the analogous results for tensor products of irreducibles. In geometry, the tensor product of representations corresponds to a "fusion" of sheaves on configuration spaces --- operation defined using the functor of nearby cycles.

q-alg↗
arXiv1995

Is a Knot Classification possible?

The goal of this paper is to discuss the possibility of finding an algorithm that can give all distinct knots up to a desired complexity. Two such algorithms are presented, one based on projections on a plane, the other on closed self-avoiding walks.

q-alg↗
arXiv1995

Elliptic Algebras and Equivariant Elliptic Cohomology

In this paper we explain the parallelism in the classification of three different kinds of mathematical objects: (i) Classical r-matrices. (ii) Generalized cohomology theories that have Chern classes for complex vector bundles. (iii) 1-dimensional formal groups. The main point of the paper is a construction of the elliptic algebra associated to Belavin's classical elliptic r-matrix in terms of Equivariant elliptic cohomology of the Steinberg varieties associated to some partial flag manifolds.

q-alg↗
arXiv1999

Affine Algebras, Langlands Duality and Bethe Ansatz

We review various aspects of representation theory of affine algebras at the critical level, geometric Langlands correspondence, and Bethe ansatz in the Gaudin models. Geometric Langlands correspondence relates D-modules on the moduli space of G-bundles on a complex curve X and flat G^L-bundles on X. Beilinson and Drinfeld construct it by applying a localization functor to representations of affine algebras of critical level. We show that in genus zero the corresponding D-modules are closely related to the diagonalization problem in the Gaudin model associated to G. This allows us to give a new interpretation of the Bethe ansatz and Sklyanin's separation of variables in the Gaudin model in terms of Langlands correspondence.

q-alg↗
arXiv1995

Modular functor and representation theory of \hat{sl}_{2} at a rational level

We define a new modular functor based on Kac-Wakimoto admissible representations and the corresponding D-module on the moduli space of rank 2 vector bundles with parabolic structure. A new fusion functor arises which is related to representation theory of the pair "osp(1|2),sl(2)" in the same way as the fusion functor for the Virasoro algebra is related to representation theory of the pair "sl(2),sl(2)"

q-alg↗
arXiv1996

Berezin-Toeplitz Quantization of compact Kaehler manifolds

Invited lecture at the XIV-th workshop on geometric methods in physics, Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global Toeplitz operators, approximation results for the quantum operators are shown. From them it follows that the quantum operators have the correct classical limit. A star product deformation of the Poisson algebra is constructed.

q-alg↗
arXiv1996

Noninvertibility and ``Semi-'' Analogs of (Super) Manifolds, Fiber Bundles and Homotopies

Supersymmetry contains initially noninvertible objects, but it is common to deal with the invertible ones only, factorizing former in some extent. We propose to reconsider this ansatz and try to redefine such fundamental notions as supermanifolds, fiber bundles and homotopies using some weakening invertibility conditions. The prefix semi- reflects the fact that the underlying morphisms form corresponding semigroups consisting of a known group part and a new ideal noninvertible part. We found that the absence of invertibility gives us the generalization of the cocycle conditions for transition functions of supermanifolds and fiber bundles in a natural way, which can lead to construction of noninvertible analogs of Čech cocycles. We define semi-homotopies, which can be noninvertible and describe mappings into the semi-supermanifolds introduced.

q-alg↗
arXiv1996

Deformation quantization of compact Kähler manifolds via Berezin-Toeplitz operators

This talk reports on results on the deformation quantization (star products) and on approximative operator representations for quantizable compact K"ahler manifolds obtained via Berezin-Toeplitz operators. After choosing a holomorphic quantum line bundle the Berezin-Toeplitz operator associated to a differentiable function on the manifold is the operator defined by multiplying global holomorphic sections of the line bundle with this function and projecting the differentiable section back to the subspace of holomorphic sections. The results were obtained in (respectively based on) joint work with M. Bordemann and E. Meinrenken. (Talk at the XXI Int. Coll. on Group Theoretical Methods in Physics, 15-20 July, Goslar, Germany)

q-alg↗
arXiv1998

Factorizable sheaves and quantum groups

The category $\cal{C}$ (studied by Andersen-Jantzen-Soergel) of representations of a Lusztig's small quantum group at a root of unity, together with its modular structure, is defined geometrically, using configuration spaces.

q-alg↗
arXiv1997

Five Lectures on Soliton Equations

This is a self-contained review of a new approach to soliton equations of KdV type developed by the author together with B. Feigin and B. Enriquez.

q-alg↗
arXiv1997

The quantum Euler class and the quantum cohomology of the Grassmannians

The Poincare duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical ``characteristic element;'' in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the ``quantum Euler class.'' We prove that the characteristic element of a Frobenius algebra A is a unit if and only if A is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.

q-alg↗
arXiv1994

Baxterization, dynamical systems, and the symmetries of integrability

We resolve the `baxterization' problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group acting on matrices, and exists as such, {\em beyond the narrow context of strict integrability}. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the complexity of iterations.

hep-th↗
arXiv1995

On Alternative Supermatrix Reduction

We consider a nonstandard odd reduction of supermatrices (as compared with the standard even one) which arises in connection with possible extension of manifold structure group reductions. The study was initiated by consideration of the generalized noninvertible superconformal-like transformations. The features of even- and odd-reduced supermatrices are investigated on a par. They can be unified into some kind of "sandwich" semigroups. Also we define a special module over even- and odd-reduced supermatrix sets, and the generalized Cayley-Hamilton theorem is proved for them. It is shown that the odd-reduced supermatrices represent semigroup bands and Rees matrix semigroups over a unit group.

alg-geom↗