Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups

We considered two types of string models: on the Riemmann space of string coordinates with null torsion and on the Riemman-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and Dubrovin solutions of WDVV associativity equation to construct new integrable string equations of hydrodynamic type on the torsionless Riemmann space of chiral currents in first case. We used the invariant local chiral currents of principal chiral models for SU(n), SO(n), SP(n) groups to construct new integrable string equations of hydrodynamic type on the Riemmann space of the chiral primitive invariant currents and on the chiral non-primitive Casimir operators as Hamiltonians in second case. We also used Pohlmeyer tensor nonlocal currents to construct new nonlocal string equation

Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents

We considered two types of string models: on the Riemann space of string coordinates with null torsion and on the Riemann-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and the Dubrovin solutions of the WDVV associativity equation to construct new integrable string models of hydrodynamic type on the torsion less Riemann space of chiral currents in the first case. We used the invariant local chiral currents of principal chiral models for SU(n), SO(n), SP(n) groups to construct new integrable string models of hydrodynamic type on the Riemann-Cartan space of invariant chiral currents and on the Casimir operators, considered as the Hamiltonians, in the second case.Розглянуто два типу струнних моделей: на просторі Рімана струнних координат з нульовим скрутом та на просторі Рімана-Картана з постійним скрутом. В першому випадку, ми використали гідродинамічний підхід Дубровіна, Новікова до інтегрованих систем та розв’язок Дубровіна рівняння асоціативності ВДВВ, щоб побудувати нові інтегровані струнні моделі гідродинамічного типу на безскрутному просторі Рімана кіральних токів. У другому випадку використали інваріантні локальні кіральні токи SU(n), SO(n), SP(n)-моделі головного кірального поля, щоб побудувати нові інтегровані струнні моделі гідродинамічного типу на просторі Рімана-Картана інваріантних кіральних токів та на операторах Казіміра, розглянутих як гамільтоніани.Рассмотрены два типа струнных моделей: на пространстве Римана струнных координат с нулевым кручением и на пространстве Римана-Картана с постоянным кручением. В первом случае использовали гидродинамический подход Дубровина, Новикова к интегрированным системам и Дубровина решения ВДВВ уравнения ассоциативности, чтобы построить новые итегрированные струнные модели гидродинамического типа на пространстве Римана киральных токов с нулевым кручением. Во втором случае использовали локальные инвариантные киральные токи в модели главного кирального поля для SU(n), SO(n), SP(n)-групп, чтобы построить новые интегрированные струнные модели гидродинамического типа на Римана-Картана-пространстве инвариантных киральных токов и на операторах Казимира, рассматриваемых как гамильтонианы

Bihamiltonian approach to the gauge models: closed string model

The closed string model in the background gravity field and the antisymmetric B-field is considered as the bihamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. It is shown that bihamiltonity is origin of two types of the T-duality of the closed string models. The dual nonlocal Poisson brackets, depending of the background fields and of their derivatives, are obtained. The integrability condition is formulated as the compatibility of the bihamoltonity condition and the Jacobi identity of the dual Poisson bracket. It is shown, that the dual brackets and dual hamiltonians can be obtained from the canonical (PB) and from the initial hamiltonian by imposing of the second kind constraints on the initial dynamical system, on the closed string model in the constant background fields, as example. The closed string model in the constant background fields is considered without constraints, with the second kind constraints and with first kind constraints as the B-chiral string. The two particles discrete closed string model is considered as two relativistic particle system to show the difference between the Gupta-Bleuler method of the quantization with the first kind constraints and the quantization of the Dirac bracket with the second kind constraints

σ\sigma-models on the quantum group manifolds SLq(2,R)SL_{q}(2,R), SLq(2,R)/Uh(1)SL_{q}(2,R)/U_{h}(1), Cq(2∣0)C_{q}(2|0) and infinitesimal trasformations

The differential and variational calculus on the $SL_{q}(2,R)$ group is constructed. The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$ quantum group symmetry and in the $\sigma$-models with ${SL_{q}(2,R)/U_{h}(1)}$ ,$C_{q}(2|0)$ quantum group symmetry is considered. The Lagrangian formalism over the quantum group manifolds is discussed. The classical solution of $C_{q}(2|0)$ {$\sigma$}-model is obtained.Comment: LaTex, 7 page

Geometrical properties of Riemannian superspaces, observables and physical states

Classical and quantum aspects of physical systems that can be described by Riemannian non degenerate superspaces are analyzed from the topological and geometrical points of view. For the N=1 case the simplest supermetric introduced in [Physics Letters B \textbf{661}, (2008),186] have the correct number of degrees of freedom for the fermion fields and the super-momentum fulfil the mass shell condition, in sharp contrast with other cases in the literature where the supermetric is degenerate. This fact leads a deviation of the 4-impulse (e.g. mass constraint) that can be mechanically interpreted as a modification of the Newton's law. Quantum aspects of the physical states and the basic states and the projection relation between them, are completely described due the introduction of a new Majorana-Weyl representation of the generators of the underlying group manifold. A new oscillatory fermionic effect in the $B_{0}$ part of the vaccum solution involving the chiral and antichiral components of this Majorana bispinor is explicitly shown.Comment: 16 pags. 3 figures. To Anna Grigorievna Kartavenko and Academic Professor Alexei Norianovich Sissakian, in memoria

Worldline approach to vector and antisymmetric tensor fields

The N=2 spinning particle action describes the propagation of antisymmetric tensor fields, including vector fields as a special case. In this paper we study the path integral quantization on a one-dimensional torus of the N=2 spinning particle coupled to spacetime gravity. The action has a local N=2 worldline supersymmetry with a gauged U(1) symmetry that includes a Chern-Simons coupling. Its quantization on the torus produces the one-loop effective action for a single antisymmetric tensor. We use this worldline representation to calculate the first few Seeley-DeWitt coefficients for antisymmetric tensor fields of arbitrary rank in arbitrary dimensions. As side results we obtain the correct trace anomaly of a spin 1 particle in four dimensions as well as exact duality relations between differential form gauge fields. This approach yields a drastic simplification over standard heat-kernel methods. It contains on top of the usual proper time a new modular parameter implementing the reduction to a single tensor field. Worldline methods are generically simpler and more efficient in perturbative computations then standard QFT Feynman rules. This is particularly evident when the coupling to gravity is considered.Comment: 30 pages, 5 figures, references adde

Higher spin fields from a worldline perspective

Higher spin fields in four dimensions, and more generally conformal fields in arbitrary dimensions, can be described by spinning particle models with a gauged SO(N) extended supergravity on the worldline. We consider here the one-loop quantization of these models by studying the corresponding partition function on the one-dimensional torus. After gauge fixing the supergravity multiplet, the partition function reduces to an integral over the corresponding moduli space which is computed using orthogonal polynomial techniques. We obtain a compact formula which gives the number of physical degrees of freedom for all N in all dimensions. As an aside we compute the physical degrees of freedom of the SO(4) = SU(2)xSU(2) model with only a SU(2) factor gauged, which has attracted some interest in the literature.Comment: 21 page

U(N) spinning particles and higher spin equations on complex manifolds

Guided by a spinning particle model with U(N)-extended supergravity on the worldline we derive higher spin equations on complex manifolds. Their minimal formulation is in term of gauge fields which satisfy suitable constraints. The latter can be relaxed by introducing compensator fields. There is an obstruction to define these systems on arbitrarily curved spaces, just as in the usual theory of higher spin fields, but we show how to couple them to Kaehler manifolds of constant holomorphic curvature. Quite interestingly, the first class gauge algebra defining the U(N) particles on these manifolds is quadratic and realizes the zero mode sector of certain nonlinear U(N) superconformal algebras introduced sometimes ago by Bershadsky and Knizhnik in 2D.Comment: 26 page

The Spinning Particles as a Nonlinear Realizations of the Superworldline Reparametrization Invariance

The superdiffeomorphisms invariant description of $N$ - extended spinning particle is constructed in the framework of nonlinear realizations approach. The action is universal for all values of $N$ and describes the time evolution of $D+2$ different group elements of the superdiffeomorphisms group of the $(1,N)$ superspace. The form of this action coincides with the one-dimensional version of the gravity action, analogous to Trautman's one.Comment: 4 pages, RevTe

On Paragrassmann Differential Calculus

Explicit general constructions of paragrassmann calculus with one and many variables are given. Relations of the paragrassmann calculus to quantum groups are outlined and possible physics applications are briefly discussed. This paper is the same as the original 9210075 except added Appendix and minor changes in Acknowledgements and References. IMPORTANT NOTE: This paper bears the same title as the Dubna preprint E5-92-392 but is NOT identical to it, containing new results, extended discussions, and references.Comment: 19p