Tachyon solution in a cubic Neveu-Schwarz string field theory

We find a class of analytic solutions in a modified cubic theory of fermionic strings that includes the GSO(−) sector. This class contains a solution that involves a tachyon field from the GSO(−) sector and reproduces the correct value of the non-BPS D-brane tension. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 378–390, March, 2009.     

Multipoint correlation functions in Liouville field theory and minimal Liouville gravity

We study (n+3)-point correlation functions of exponential fields in the Liouville field theory with n degenerate and three arbitrary fields and derive an analytic expression for these correlation functions in terms of Coulomb integrals. We consider the application of these results to the minimal Liouville gravity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 536–556, March, 2008.     

Vacuum solutions in the Neveu-Schwarz field theory of a fermionic string and the zero-curvature representation in graded spaces

We demonstrate that vacuum solutions in the Neveu-Schwarz field theory of a fermionic string incorporating the GSO(−) sector are naturally related to the zero-curvature representation in a graded space of special form. We use the same representation to describe the equivalence of the cubic and nonpolynomial theories with the GSO(−) sector also taken into account. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 131–141, April, 2009.     

Geometric Properties of W-Algebras and the Toda model

The W-algebra minimal models on hyperelliptic Riemann surfaces are constructed. Using a proposal by Polyakov, we reduce the partition function of the Toda field theory on the hyperelliptic surface to a product of partition functions: one of a free field theory on the sphere with inserted Toda vertex operators and one of a free scalar field theory with antiperiodic boundary conditions with inserted twist fields.     

A Kind of Discretization of the KdV Hierarchy

We propose a method for introducing higher-order terms in the potential expansion in order to study the continuum limits of the Toda hierarchy. These higher-order terms are differential polynomials in the lower-order terms. This type of potential expansion allows using fewer equations in the Toda hierarchy to recover the KdV hierarchy by the so-called recombination method. We show that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend toward the corresponding objects of the KdV hierarchy in the continuum limit. This method gives a kind of discretization of the KdV hierarchy.     

Towards an Effectivisation of the Riemann Theorem

Let Q be a connected and simply connected domain on the Riemann sphere, not coinciding with the Riemann sphere and with the whole complex plane ℂ. Then, according to the Riemann Theorem, there exists a conformal bijection between Q and the exterior of the unit disk. In this paper, we find an explicit form of this map for a broad class of domains with analytic boundaries. Communicated by M. A. Shubin (Moscow) Mathematics Subject Classifications (2000): 30Cxx, 37Kxx.     

Cohomology of elementary states in twistor conformal field theory

In Twistor Conformal Field Theory the Riemann surfaces and holomorphic functions of two-dimensional conformal field theory are replaced by flat twistor spaces (arising from conformally-flat four-manifolds) and elements of the holomorphic first cohomology. The analogue of a Laurent Series is the expansion of a cohomology element in elementary states and we calculate the dimension of the space of these states for twistor spaces of compact hyperbolic manifolds. Our method follows the strategy used in the classical problem of calculating the number of meromorphic functions with prescribed poles on Rieiemann surface. We express the problem globally (in terms of the cohomology of a blown-up twistor space), calculate the holomorphic Euler characteristic of this blown-up space, and then use some vanishing theorems to isolate the first cohomology term.     

Geometric torsions and an Atiyah-style topological field theory

We generalize the construction of invariants of three-dimensional manifolds with a triangulated boundary that we previously proposed for the case where the boundary consists of not more than one connected component to any number of components. These invariants are based on the torsion of acyclic complexes of geometric origin. An adequate tool for studying such invariants turns out to be Berezin’s calculus of anticommuting variables; in particular, they are used to formulate our main theorem, concerning the composition of invariants under a gluing of manifolds. We show that the theory satisfies a natural modification of Atiyah’s axioms for anticommuting variables. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 405–418, March, 2009.     

Class field theory for arithmetic schemes

Let X be a regular arithmetic scheme, i.e. a regular integral separated scheme flat and of finite type over Spec . Generalising classical class field theory for number fields, we define a class group C _X and show there is a natural surjective map whose kernel is the connected component of 0.      

Quantum field theory meets Hopf algebra

This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S ‐matrix, Feynman diagrams, connected diagrams, Green functions, renormalization. The use of Hopf algebra for their definition allows for simple recursive derivations and leads to a correspondence between Feynman diagrams and semi‐standard Young tableaux. Reciprocally, these concepts are used as models to derive Hopf algebraic constructions such as a connected coregular action or a group structure on the linear maps from S (V) to V. In many cases, noncommutative analogues are derived (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quintessence scalar field in the relativistic theory of gravity

We propose a variant of the quintessence theory in order to obtain an accelerated expansion of the Friedmann universe in the framework of the relativistic theory of gravity. The scalar field of dark energy creates the substance of quintessence. We show that the function V(Φ) that factors the Lagrangian of the scalar field Φ does not influence the evolution of the universe. We find some relations that allow finding the explicit time-dependence of Φ if only the function V(Φ) is chosen. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 551–560, September, 2007.     

Descent relations and the oscillator level-truncation method

We reexamine the oscillator level-truncation method in the bosonic string field theory by calculating the descent relation . We use the fermionic vertices in the standard oscillator basis for the ghost sector. We propose two new calculation schemes: in the first, we assume that the insertion satisfies the overlap equation for the vertices; in the second, we use direct calculations. In both schemes, we obtain the correct structures of the exponent and preexponent of the vertex 〈V₂|, but we find different normalization factors . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 5–25, January, 2007.     

Topological quantum field theory with corners based on the Kauffman bracket

We describe the construction of a topological quantum field theory with corners based on the Kauffman bracket, that underlies the smooth theory of Lickorish, Blanchet, Habegger, Masbaum and Vogel. We also exhibit some properties of invariants of 3-manifolds with boundary. Received: March 4, 1996     

Inverse scattering transform for the Toda hierarchy with quasi-periodic background

We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy in the case of a quasi-periodic finite-gap background solution.

     

Integrability in String/Field Theories and Hamiltonian Flows in the Space of Physical Systems

Integrability in string/field theories is known to emerge when considering dynamics in the moduli space of physical theories. This implies that one must study the dynamics with respect to unusual time variables such as coupling constants or other quantities parameterizing the configuration space of physical theories. The dynamics given by variations of coupling constants can be considered as a canonical transformation or, infinitesimally, a Hamiltonian flow in the space of physical systems. We briefly consider an example of integrable mechanical systems. Then any function T generates a one-parameter family of integrable systems in the vicinity of a single system. For an integrable system with several coupling constants, the corresponding Hamiltonians T_i satisfy the Whitham equations and, after quantization (of the original system), become operators satisfying the zero-curvature condition in the coupling-constant space.     

TD孤子方程族的可积耦合及其哈密顿结构

用拓展谱问题方法构造TD族的可积耦合,并应用二次型恒等式寻求拓展的TD族哈密顿结构.     

Quantum inverse scattering method and (super)conformal field theory

We consider the possibility of using the quantum inverse scattering method to study the superconformal field theory and its integrable perturbations. The classical limit of the considered constructions is based on the (1|2) super-KdV hierarchy. We introduce the quantum counterpart of the monodromy matrix corresponding to the linear problem associated with the L-operator and use the explicit form of the irreducible representations of _q(1|2) to obtain the fusion relations for the transfer matrices (i.e., the traces of the monodromy matrices in different representations).Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 252–264, February, 2005.     

Anti-de Sitte:空间H_1~3中的广义弹性曲线(英文)

In this paper,the extremals of curvature energy actions on non-null Frenet curves in 3-dimensional Anti-de Sitter space are studied.We completely solve the Euler-Lagrange equation by quadratures.By using the Killing fields,we obtain existence for closed general-ized elastica fully immersed in Anti-de Sitter space H_1~3.