XXIX WORKSHOP ON GEOMETRIC METHODS IN PHYSICS

On Differential Expressions with δ‐Potential: Exceptional Case
View Description Hide DescriptionThe paper is devoted to the study of the formal differential expressions for arbitrary l∈N and the dimension of equal Approximations of the singular part by means of a family of rank‐one operators are constructed and resolvent convergence of this family is investigated. It is demonstrated that the results are different from the case

A Note on Gauge Systems from the Point of View of Lie Algebroids
View Description Hide DescriptionIn the context of the variational bi‐complex, we re‐explain that irreducible gauge systems define a particular example of a Lie algebroid. This is used to review some recent and not so recent results on gauge, global and asymptotic symmetries.

On Weyl Calculus in Infinitely Many Variables
View Description Hide DescriptionWe outline an abstract approach to the pseudo‐differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of representations associated with infinite‐dimensional coadjoint orbits. We illustrate the approach by the case of infinite‐dimensional Heisenberg groups. The classical Weyl‐Hörmander calculus is recovered for the Schrödinger representations of the finite‐dimensional Heisenberg groups.

Asymmetric Time Evolution and Indistinguishable Events
View Description Hide DescriptionWith a time asymmetric theory, in which quantum mechanical time evolution is given by a semigroup of operators rather than by a group, the states of open systems are represented by density operators exhibiting a branching behavior. To treat the indistinguishably of the members of experimental ensembles, we hypothesize that environmental interference occurs during events that are themselves fundamentally indistinguishable.

Addition Theorems, Formal Group Laws and Integrable Systems
View Description Hide DescriptionWe consider elliptic curves, given in the Weierstrass parametrization by the equation In Tate coordinates and the geometric addition laws on this curves correspond to the general elliptic formal group law over the ring
This formal group law is well‐known in the number theory and cryptography. One can find this law in recent works on the theory of elliptic functions and algebraic topology. In the focus of our interest are questions, important from the point of view of Hirzebruch genera and the theory of integrable systems (see references).

Deformations of Geometric Structures in Topological Sigma Models
View Description Hide DescriptionWe study a Lie algebra of formal vector fields with it application to the perturbative deformed holomorphic symplectic structure in the A‐model, and a Calabi‐Yau manifold with boundaries in the B‐model. We show that equivalent classes of deformations are described by a Hochschild cohomology of the DG‐algebra which is defined to be the cohomology of Here ∂̄ is the initial non‐deformed BRST operator while is the deformed part whose algebra is a Lie algebra of linear vector fields

On the Existence of Non‐Abelian Monopoles: the Algebro‐Geometric Approach
View Description Hide DescriptionWe develop the Atiyah‐Drinfeld‐Manin‐Hitchin‐Nahm construction to study SU (2) non‐abelian charge 3 monopoles within the algebro‐geometric method. The method starts with finding an algebraic curve, the monopole spectral curve, subject to Hitchin’s constraints. We take as the monopole curve the genus four curve that admits a symmetry, with real parameters α, β and γ. In the case we prove that the only suitable values of γ/β are (β is given below) which corresponds to the tetrahedrally symmetric solution. We then extend this result by continuity to non‐zero values of the parameter α and find finally a new one‐parameter family of monopole curves with symmetry.

Towards the Geometry of Reproducing Kernels
View Description Hide DescriptionIt is shown here how one is naturally led to consider a category whose objects are reproducing kernels of Hilbert spaces, and how in this way a differential geometry for such kernels may be settled down.

Conservation Laws for under Determined Systems of Differential Equations
View Description Hide DescriptionThis work extends the Ibragimov’s conservation theorem for partial differential equations [J. Math. Anal. Appl. 333 (2007 311–328] to under determined systems of differential equations. The concepts of adjoint equation and formal Lagrangian for a system of differential equations whose the number of equations is equal to or lower than the number of dependent variables are defined. It is proved that the system given by an equation and its adjoint is associated with a variational problem (with or without classical Lagrangian) and inherits all Lie‐point and generalized symmetries from the original equation. Accordingly, a Noether theorem for conservation laws can be formulated.

E‐Discretization of Tori of Exceptional Compact Simple Lie Groups
View Description Hide DescriptionWe consider an exceptional compact simple Lie group G, the corresponding affine Weyl group and its even subgroup. Given a positive integer M, we introduce a finite set of lattice points The even affine Weyl group determines the symmetry of the grid the number M determines its density. We present a construction of the set and explicitly count the numbers of its points for the cases of and We specify the maximal set of pairwise orthogonal E‐functions over This finite set allows us to calculate Fourier like discrete expansions of an arbitrary discrete function on

Twist Deformation of (Conformal) Quantum Mechanics
View Description Hide DescriptionWe apply the abelian Drinfel’d twist to deform the universal enveloping algebra of a Lie algebra. This Lie algebra contains, as subalgebras, both the 1D conformal algebra of quantum mechanics and the Heisenberg algebra. The Hopf structure of the deformed algebra allows, in particular, to find the spectrum of a deformed harmonic oscillator and the noncommutative structures defined on it.

One‐Side Invertibility of the Weighted Shift Operators
View Description Hide DescriptionWeighted shift operators B in generated by Morse‐Smale type mappings α: X→X are considered. A necessary and sufficient conditions for B—λI to be one‐sided invertible are obtained. These conditions use a new notion: oriented decomposition of the oriented graph G(X, α) generated by the mapping α.

Equivariant Quantization of Spin Systems
View Description Hide DescriptionWe investigate the geometric and conformally equivariant quantizations of the super‐cotangent bundle of a pseudo‐Riemannian manifold (M,g), which is a model for the phase space of a classical spin particle. This is a short review of our previous works [1, 2].

On the Plane Curves whose Curvature Depends on the Distance from the Origin
View Description Hide DescriptionHere we suggest and have exemplified a simple scheme for reconstruction of a plane curve if its curvature belongs to the class specified in the title by deriving explicit parametrization of Bernoulli’s lemniscate and newly introduced co‐lemniscate curve in terms of the Jacobian elliptic functions. The relation between them and with the Bernoulli elastica are clarified.

From Buscher Duality to Poisson‐Lie T‐Plurality on Supermanifolds
View Description Hide DescriptionIn this review paper we summarize basic concepts of T‐duality. Starting from the simplest case of abelian T‐duality, we show the techniques used for finding the dual model and summarize developments in the field of Poisson‐Lie T‐duality/plurality, which deals with non‐abelian groups. We also mention possible extension of T‐duality to supermanifolds.

Central elements of quantum deformations
View Description Hide DescriptionStructure of the center of quantum algebra is described, when q is root of unity, All Casimir elements are presented explicitly, their polynomial dependence is computed for general values of n.

Inverting the Free Evolution: a Key to Quantum Control?
View Description Hide DescriptionA possible techniques to invert the free evolution in the Schrödinger’s quantum mechanics are presented and their pertinence to more general control problems is discussed. A new sense of the traditional Strutt diagram is exhibited.

Sigma function and dispersionless hierarchies
View Description Hide DescriptionTo illustrate the use of special functions in physics, we survey the origin and the significance of the theta, sigma, and tau functions in algebraic geometry and partial differential equations. Recent results on the Jacobi inversion problem allow us to construct a new solution to the dispersionless KP equation, defined locally on the Jacobian of any curve. We pose some new questions.

Heat Kernel Short‐Time Expansion within the Scope of Feynman‐Kac Formula
View Description Hide DescriptionThe paper gives an outlook of the stochastic approach to derivation of heat kernel short‐time asymptotic expansion and to effective evaluation of corresponding coefficients.