On AdS/CFT correspondence of EYM fields

In the compactized Minkowski space, which is equivalent to the conformal spaceM ₄, we introduced a Lorentz metric d σ² and a Yang-Mills field θ. Later, we proved that dσ² and θ together satisfy the EYM (Einstein-Yang-Mills) equation. In this paper, it is proved that θ onM ₄ (which is the boundary of the anti-de-Sitter space AdS₅) can be extended to be a Yang-Mills field [^(q)]\hat \theta on AdS₅ such that Hua’s metric ds² on AdS₅, together with [^(q)]\hat \theta satisfies the EYM equation on AdS₅.     

Global geometric properties of AdS space and the AdS/CFT correspondence

The Poisson kernels and relations between them for a massive scalar field in a unit ball Bn with Hua's metric and conformal flat metric are obtained by describing the Bn as a submanifold of an (n+1)-dimensional embedding space. Global geometric properties of the AdS space are discussed. We show that the(n+1)-dimensional AdS space AdSn+1 is isomorphic to RP1×Bn and boundary of the AdS is isomorphic to RP1×Sn-1. Bulk-boundary propagator and the AdS/CFT like correspondence are demonstrated based on these global geometric properties of the RP1×Bn.     

Hua operator on vector bundle: Application to AdS/CFT correspondence of Dirac fields

Hua’s theory of harmonic functions on classical domains is generalized to the theory on holomorphic vector bundles over classical domains and further on vector bundles over the real classical domains and quaternion classical domains. In case of the simplest quaternion classical domain there is a relation between Hua operator and Dirac operator, by which an AdS/CFT correspondence of Dirac fields is established.     

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.     

Mixed-symmetry massless gauge fields in AdS5

Using the su(2, 2) spinor language, we describe free mixed-symmetry massless bosonic and fermionic gauge fields of arbitrary spins in the AdS₅ space. We construct manifestly covariant action functionals and derive field equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 47–59, October, 2006.     

AdS/CFT duality at strong coupling

We study the strong-coupling limit of the AdS/CFT correspondence in the framework of a recently proposed fermionic formulation of the Bethe ansatz equations governing the gauge theory anomalous dimensions. We give examples of states that do not follow the Gubser-Klebanov-Polyakov law at a large ’t Hooft coupling λ, in contrast to recent results on the quantum string Bethe equations that are valid in that regime. This result indicates that the fermionic construction cannot be trusted at large λ, although it remains an efficient tool for computing the weak-coupling expansion of anomalous dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 213–224, August, 2007.     

AdS3/CFT2 on a Torus in the Sum over Geometries

We investigate the AdS₃/CFT₂ correspondence for the Euclidean AdS₃ space compactified on a solid torus with the CFT field on the regularizing boundary surface in the bulk. Correlation functions corresponding to the bulk theory at a finite temperature tend to the standard CFT correlation functions in the limit of removed regularization. In the sum over geometries in both the regular and the ℤ _N orbifold cases, the two-point correlation function for massless modes transforms into a finite sum of products of the conformal-anticonformal CFT Green's functions up to divergent terms proportional to the volume of the SL(2, ℤ)/ℤ group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 17–30, January, 2006.     

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.     

Complex structures adapted to smooth vector fields

In this paper we study the flat geometry and real dynamics of meromorphic vector fields on compact Riemann surfaces. Necessary and sufficient conditions to assert the existence of meromorphic vector fields with prescribed singularities are given. A characterization of the real dynamics of meromorphic vector fields is also given. Several explicit examples of meromorphic vector fields, using singular flat metrics, are provided. Received: 12 June 1998 / Accepted: 28 September 2000 / Published online: 18 January 2002     

A parametric approach to correspondence analysis

We compare correspondence analysis (CA) and the alternative approach using Hellinger distance (HD), for representing categorical data in a contingency table. As both methods may be appropriate, we introduce a parameter and define a generalized version of correspondence analysis (GCA) which contains CA and HD as particular cases. Comparison with alternative approaches are performed. We propose a coefficient which globally measures the similarity between CA and GCA, which can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions on the difference between both representations. Two criteria for choosing the best value of the parameter are proposed.     

Flows associated to adapted vector fields on the Wiener space

The existence and uniqueness of a flow associated to an adapted vector field ξ on the Wiener space with are proved by a modified Picard's iteration method, mainly under the conditions of exponential integrability concerning b^α as well as the first-order Malliavin gradients of and b^α. A Newton–Leibnitz type inequality for a kind of Malliavin differentiable functionals is also proved, which is the key point to prove the above result, and has an independent interest.     

The Darmon-Dasgupta units over genus fields and the Shimura correspondence

We study a special case of the Gross-Stark conjecture (Gross, 1981 [Gr]), namely over genus fields. Based on the same idea we provide evidence of the rationality conjecture of the elliptic units for real quadratic fields over genus fields, which is a refinement of the Gross-Stark conjecture given by Darmon and Dasgupta (2006) [DD]. Then a relationship between these units and the Fourier coefficients of p-adic Eisenstein series of half-integral weight is explained.     

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.