Semiclassical geometry and integrability of the AdS/CFT correspondence

We discuss the semiclassical geometry and integrable systems related to the gauge—string duality. We analyze semiclassical solutions of the Bethe ansatz equations arising in the context of the AdS/CFT correspondence, comparing them to stationary phase equations for the matrix integrals. We demonstrate how the underlying geometry is related to the integrable sigma models of the dual string theory and investigate some details of this correspondence.Translated from Teoreticheskaya Matematicheskaya Fizika, Vol. 142, No. 2, pp. 265–283, February, 2005.     

Noncommutative differential geometry on the quantum SU(2), I: An algebraic viewpoint

We compute the cyclic homology of the coordinate ring A(SL_q(2)) of the quantum algebraic group SL _q (2). We observe a degeneration of the noncommutative de Rham complex. The results are also verified from the point of view of Connes' noncommutative differential geometry.     

On Quasiclassical Field Theories Invariant with Respect to a Lie Group

Mathematical Notes -     

Quantization of the N=2 supersymmetriC KdV hierarchy

We continue the study of the quantization of supersymmetric integrable KdV hierarchies. We consider the N=2 KdV model based on the sl(1)(2|1) affine algebra but with a new algebraic construction for the L-operator, different from the standard Drinfeld-Sokolov reduction. We construct the quantum monodromy matrix satisfying a special version of the reflection equation and show that in the classical limit, this object precisely gives the monodromy matrix of the N=2 supersymmetric KdV system. We also show that at both the classical and the quantum levels, the trace of the monodromy matrix (transfer matrix) is invariant under two supersymmetry transformations and the zero mode of the associated U(1) current. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 303–314, May, 2006.     

On the structure of rigid semistable sheaves on algebraic surfaces

LetS be a smooth projective surface, letK be the canonical class ofS and letH be an ample divisor such thatH • K < 0. We prove that for any rigid sheafF (Ext¹ (F, F) = 0) that is Mumford-Takemoto semistable with respect toH there exists an exceptional set (E ₁ ,..., E _n ) of sheaves onS such thatF can be constructed from {E _i } by means of a finite sequence of extensions. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 692–700, November, 1998. The author wishes to express his gratitude to S. A. Kuleshov for useful discussions and to A. N. Rudakov and A. L. Gorodentsev for their attention to the present work. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01323 and by the INTAS Foundation.     

On the order bound of one-point algebraic geometry codes

Let S={s_i}_i∈N⊆N be a numerical semigroup. For each i∈N, let ν(s_i) denote the number of pairs (s_i−s_j,s_j)∈S²: it is well-known that there exists an integer m such that the sequence {ν(s_i)}_i∈N is non-decreasing for i>m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m and in several cases we determine m explicitly. In particular we give the value of m when the Cohen-Macaulay type of the semigroup is three or when the multiplicity is less than or equal to six. When S is the Weierstrass semigroup of a family {C_i}_i∈N of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {C_i}.     

On the mapping class group action on the cohomology of the representation space of a surface

The mapping class group of a -pointed Riemann surface has a natural action on any moduli space of parabolic bundles with the marked points as the parabolic points. We prove that under some numerical conditions on the parabolic data, the induced action of the mapping class group on the cohomology algebra of the moduli space of parabolic bundles factors through the natural epimorphism of the mapping class group onto the symplectic group.

     

On the equations defining irreducible algebraic groups

Abstract

We consider five known problems concerning positive laws in groups. One of them has a counterexample, the others are open in general. It is shown that three of the problems are equivalent and all of them have a positive solution in the class of locally graded groups.     

On the Construction of K-Operators in Field Theories as Sections Along Legendre Maps

The time-evolution K-operator (or relative Hamiltonian vector field) in mechanics is a powerful tool which can be geometrically defined as a vector field along the Legendre map. It has been extensively used by several authors for studying the structure and properties of the dynamical systems (mainly the nonregular ones), such as the relation between the Lagrangian and Hamiltonian formalisms, constraints, and higher-order mechanics. This paper is devoted to defining a generalization of this operator for field theories, in a covariant formulation. In order to do this, we use sections along maps, in particular multivector fields (skew-symmetric contravariant tensor fields of order greater than 1), jet fields and connection forms along the Legendre map. As a relevant result, we use these geometrical objects to obtain the solutions of the Lagrangian and Hamiltonian field equations, and the equivalence among them (specially for nonregular field theories).     

On coverings in the lattice of all group topologies of arbitrary Abelian groups

The remainder of the completion of a topological abelian group (G, τ₀) contains a nonzero element of prime order if and only if G admits a Hausdorff group topology τ₁ that precedes the given topology and is such that (G, τ₀) has no base of closed zero neighborhoods in (G, τ₁).     

On the picard group and the brauer group of a real algebraic surface

The map of the Brauer group of a real algebraic surface to the invariant part of the Brauer group of its complexification is studied. In this study, the real cycle map of the Picard group is used. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 211–220, February, 2000.     

Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT

To study the representation category of the triplet W-algebra that is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category C _p of finite-dimensional representations of the restricted quantum group Ū _q sℓ(2) at . We fully describe the category C _p by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the -and Ū _q sℓ(2)-representation categories is conjectured for all p = 2 and proved for p = 2. The implications include identifying the quantum group center with the logarithmic conformal field theory center and the universal R-matrix with the braiding matrix. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 398–427, September, 2006.     

On the Automorphism Groups of Affine-Invariant Codes.

Using the classification of finite simple groups, we obtain a condition for a permutation group on a finite field GF(pm) to have the affine group AGL(1,pm) as a subgroup. Applying this result to the automorphism groups of non trivial affine-invariant codes, we prove that these automorphism groups are always subgroups of the general affine group AGL(m,p).     

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20     

On Certain Integral Tensor Categories and Integral TQFTs

We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over some ring of algebraic integers. We show how to obtain TQFTs over algebraic integers from these categories.     

On the Indecomposable Modules of Uq（sl（2）） Constructed via Ore-extension

Let A be a subalgebra of Uq (sl(2)) generated by K, K-1 and F and Aδ be a subalgebra of Uq (sl(2)) generated by K, K-1 (and also Fd if q is a primitive d-th root of unity with d an odd number). Given an Aδ -module M, a Uq (sl(2))-module AAδ M is constructed via the iterated Ore extension of Uq (sl(2)) in a unified framework for any q. Then all the submodules of AAδ M are determined for a fixed finite-dimensional indecomposable Aδ -module M . It turns out that for some indecomposable Aδ -module M , the Uq (sl(2))-module AAδ M is indecomposable, which is not in the BGG-categories Oq associated with quantum groups in general.     

On the Brauer group of a two-dimensional local field

The two-dimensional local field K = F _q((u))((t)), char K = p, and its Brauer group Br(K) are considered. It is proved that, if L = K(x) is the field extension for which we have x ^p ? x = ut ^?p =: h, then the condition that (y, f | h]_K = 0 for any y ε K is equivalent to the condition f ε Im(Nm(L*)).     

The n-Wave Procedure and Dimensional Regularization for the Scalar Field in a Homogeneous Isotropic Space

We obtain expressions for the vacuum expectations of the energy–momentum tensor of the scalar field with an arbitrary coupling to the curvature in an N-dimensional homogeneous isotropic space for the vacuum determined by diagonalization of the Hamiltonian. We generalize the n-wave procedure to N-dimensional homogeneous isotropic space–time. Using the dimensional regularization, we investigate the geometric structure of the terms subtracted from the vacuum energy–momentum tensor in accordance with the n-wave procedure. We show that the geometric structures of the first three subtractions in the n-wave procedure and in the effective action method coincide. We show that all the subtractions in the n-wave procedure in a four- and five-dimensional homogeneous isotropic space correspond to a renormalization of the coupling constants of the bare gravitational Lagrangian.