Volume 4 (2004), Number 3. Abstracts
A. Beilinson, R. Bezrukavnikov, and I. MirkoviŠ. Tilting Exercises [PDF]
We discuss tilting objects in categories of perverse sheaves smooth along some stratification. In case of the Schubert stratification we show that the Radon transform interchanges tilting, projective, and injective objects, and prove Kapranov's conjecture on the Serre functor.
Keywords. Tilting objects, perverse sheaves, Serre functor.
2000 Mathematics Subject Classification. 14F05, 18E30.
R. Bezrukavnikov and D. Kaledin. Fedosov Quantization in Algebraic Context [PDF]
We consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. We show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The assumptions are satisfied, for example, for affine and for projective varieties. We also give a classification of all possible quantizations.
Keywords. Formal geometry, quantization.
2000 Mathematics Subject Classification. 14F05.
H. Boos, V. Korepin, and F. Smirnov. New Formulae for Solutions to Quantum Knizhnik—Zamolodchikov Equations of Level −4 and Correlation Functions [PDF]
This paper is a continuation of our previous papers  and . We discuss the new form of solution to the quantum Knizhnik—Zamolodchikov equation (qKZ) of level −4 obtained in  for the Heisenberg XXX spin chain in more detail. The main advantage of this form is that it explicitly reduces to one-dimensional integrals. We believe that the basic mathematical reason for this is some special cohomologies of deformed Jacobi varieties. We apply this new form of the solution to correlation functions by using the Jimbo—Miwa conjecture . Formula (45) for correlation functions obtained in this way is in a good agreement with the ansatz for the emptiness formation probability from . Our previous conjecture describing the structure of correlation functions of the XXX model in the homogeneous limit through the Riemann zeta functions at odd arguments is a corollary to (45).
Keywords. Exactly solvable models, correlation functions, quantum Knizhnik—Zamolodchikov equations.
2000 Mathematics Subject Classification. 81T40.
V. Drinfeld. On the Notion of Geometric Realization [PDF]
We explain why geometric realization commutes with Cartesian products and why the geometric realization of a simplicial set (resp., cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment [0,1] (resp., the circle).
Keywords. Simplicial set, cyclic set, geometric realization, cyclic homology, fiber functor.
2000 Mathematics Subject Classification. 18G30, 55U10, 19D55.
P. Etingof and V. Ostrik. Finite Tensor Categories [PDF]
We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D. Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols—Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive elements in a finite dimensional Hopf algebra in zero characteristic. We also develop the theory of module categories and dual categories for not necessarily semisimple finite tensor categories; the crucial new notion here is that of an exact module category. Finally, we classify indecomposable exact module categories over the simplest finite tensor categories, such as representations of a finite group in positive characteristic, representations of a finite supergroup, and representations of the Taft Hopf algebra.
Keywords. Tensor categories, Hopf algebras.
2000 Mathematics Subject Classification. 18D10.
E. Frenkel. Opers on the Projective Line, Flag Manifolds and Bethe Ansatz [PDF]
We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called ``opers'' on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation.
The Bethe Ansatz is a procedure to construct explicitly the eigenvectors of the generalized Gaudin hamiltonians. We show that each solution of the Bethe Ansatz equations defines what we call a ``Miura oper'' on the projective line. Moreover, we show that the space of Miura opers is a union of copies of the flag manifold (of the dual group), one for each oper. This allows us to prove that all solutions of the Bethe Ansatz equations, corresponding to a fixed oper, are in one-to-one correspondence with the points of an open dense subset of the flag manifold.
The Bethe Ansatz equations can be written for an arbitrary Kac—Moody algebra, and we prove an analogue of the last result in this more general setting.
For the Lie algebras of types A, B, C similar results were obtained by other methods by I. Scherbak and A. Varchenko and by E. Mukhin and A. Varchenko.
Keywords. Gaudin model, oper, Bethe ansatz, flag manifold.
2000 Mathematics Subject Classification. 17B67, 82B23.
D. Fuchs and T. Ishkhanov. Invariants of Legendrian Knots and Decompositions of Front Diagrams [PDF]
The authors prove that the sufficient condition for the existence of an augmentation of the Chekanov—Eliashberg differential algebra of a Legendrian knot, which is contained in a recent work of the first author, is also necessary. As a by-product, the authors describe an algorithm for calculating Chekanov—Eliashberg invariants in terms of the front diagram of a Legendrian knot.
Keywords. Contact manifold, Legendrian knot, Chekanov—Eliashberg invariant, augmentation, decomposition of a front diagram.
2000 Mathematics Subject Classification. Primary: 53D10; Secindary: 57M27.
W. Gan and V. Ginzburg. Hamiltonian Reduction and Maurer—Cartan Equations [PDF]
We show that solving the Maurer—Cartan equations is, essentially, the same thing as performing the Hamiltonian reduction construction. In particular, any differential graded Lie algebra equipped with an even nondegenerate invariant bilinear form gives rise to modular stacks with symplectic structures.
Keywords. Maurer—Cartan equations, Hamiltonian reduction, L∞-algebras.
2000 Mathematics Subject Classification. Primary: 53D12, 14D20.
V. Gorbounov and F. Malikov. Vertex Algebras and the Landau—Ginzburg/Calabi—Yau Correspondence [PDF]
We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi—Yau hypersurface and whose first term is a vertex algebra closely related to the Landau—Ginzburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi—Yau hypersurfaces.
Keywords. Vertex algebra, chiral rings, polyvector fields, spectral sequence, orbifold.
2000 Mathematics Subject Classification. 14M25, 14F05, 17B65, 17B69, 17B81, 81T20, 55N34.
Moscow Mathematical Journal