Algebras with operator and Campbell-Hausdorff formula

We introduce some new classes of algebras and establish within these algebras Campbell-Hausdorff-like formulae. We describe the application of these constructions to the problem of the connectivity of the Feynman graphs corresponding to Green functions in quantum fields theory.     

Odd Invariant Semidensity and Divergence-like Operators on an Odd Symplectic Superspace

The divergence-like operator on an odd symplectic superspace which acts invariantly on a specially chosen odd vector field is considered. This operator is used to construct an odd invariant semidensity in a geometrically clear way. The formula for this semidensity is similar to the formula of the mean curvature of hypersurfaces in Euclidean space. Received: 19 August 1997 / Accepted: 27 March 1998     

Universal volume of groups and anomaly of Vogel’s symmetry

We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on \({ CP }^2\), which transform as two triplets under group of permutations of Vogel’s projective parameters. This substitutes expected invariance under permutations of universal parameters by more complicated covariance. We provide an analytic continuation of these functions and calculate their change (anomaly) under permutations of parameters (Vogel’s symmetry). This last relation is universal generalization, for an arbitrary simple Lie group and, moreover, to an arbitrary point in Vogel’s plane, of the Kinkelin’s reflection relation on Barnes’ \(G(1+N)\) function. Kinkelin’s relation gives asymmetry of the \(G(1+N)\) function (which is essentially reciprocal of the volume function for \({ SU }(N)\) groups) under \(N\leftrightarrow -N\) transformation (which is equivalent of the permutation of Vogel’s parameters for \({ SU }(N)\) groups), and coincides with above-mentioned anomaly of permutations at the \({ SU }(N)\) line on Vogel’s plane. Our results also give an anomaly of Vogel’s symmetry of the universal partition function of Chern–Simons theory on three-dimensional sphere. This effect is analogous to modular covariance, instead of invariance, of partition functions of appropriate gauge theories under modular transformation of couplings.     

On Odd Laplace Operators

We consider odd Laplace operators acting on densities of various weights on an odd Poisson (= Schouten) manifold M. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an 'orbit space' of volume forms. This includes earlier results for the odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on M is partitioned into orbits by the action of a natural groupoid whose arrows correspond to the solutions of the quantum Batalin–Vilkovisky equations. We compare this situation with that of Riemannian and even Poisson manifolds. In particular, we show that the square of an odd Laplace operator is a Poisson vector field defining an analog of Weinstein's 'modular class'.     

Odd Laplacians: geometrical meaning of potential and modular class

A second-order self-adjoint operator \(\Delta =S\partial ^2+U\) is uniquely defined by its principal symbol S and potential U if it acts on half-densities. We analyse the potential U as a compensating field (gauge field) in the sense that it compensates the action of coordinate transformations on the second derivatives in the same way as an affine connection compensates the action of coordinate transformations on first derivatives in the first-order operator, a covariant derivative, \(\nabla =\partial +\Gamma \). Usually a potential U is derived from other geometrical constructions such as a volume form, an affine connection, or a Riemannian structure, etc. The story is different if \(\Delta \) is an odd operator on a supermanifold. In this case, the second-order potential becomes a primary object. For example, in the case of an odd symplectic supermanifold, the compensating field of the canonical odd Laplacian depends only on this symplectic structure and can be expressed by the formula obtained by K. Bering. We also study modular classes of odd Poisson manifolds via \(\Delta \)-operators, and consider an example of a non-trivial modular class which is related with the Nijenhuis bracket.     

Double Complexes and Cohomological Hierarchy in a Space of Weakly Invariant Lagrangians of Mechanics

For a given configuration space M and a Lie algebra G acting on M, the space V _0.0 of weakly G-invariant Lagrangians, i.e., Lagrangians whose motion equations left-hand sides are G-invariant, is studied. The problem is reformulated in terms of the double complex of Lie algebra cochains with values in the complex of Lagrangians. Calculating the cohomology of this complex by the method of spectral sequences, we arrive at the hierarchy in the space V _0.0: The double filtration {V _s.}, s = 0, 1, 2, 3, 4, = 0, 1, and the homomorphisms on every space {V _s.} are constructed. These homomorphisms take values in the cohomologies of the algebra G and the configuration space M. Every space {V _s.} is the kernel of the corresponding homomorphism, while the space itself is defined by its physical properties.     

Berezinians,Exterior Powers and Recurrent Sequences

We study power expansions of the characteristic function of a linear operator A in a p|q-dimensional superspace V. We show that traces of exterior powers of A satisfy universal recurrence relations of period q. ‘Underlying’ recurrence relations hold in the Grothendieck ring of representations of GL(V). They are expressed by vanishing of certain Hankel determinants of order q+1 in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to express the Berezinian of an operator as a ratio of two polynomial invariants. We analyze the Cayley–Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer’s rule Mathematics Subject Classification (2000): 15A15, 58A50, 81R99 To the memory of Felix Alexandrovich Berezin     

Integral invariants of the Buttin bracket

We study the densities (most general objects which may be integrated over supersurfaces in superspace), invariant with respect to supercanonical transformations which do not change the Buttin bracket. The only such nontrivial object is, in a definite sense, the odd semidensity explicitly constructed here.     

πo and π+ production with polarized photons in the energy range 1–2 GeV

The asymmetry of π^o and π⁺ photoproduction from hydrogen has been measured. The π^o-mesons were detected at 130° cms with E_γ ranged from 0.9 to 1.65 GeV, and the π⁺-mesons at 40° cms with E_γ ranged from 0.9 to 1.2 GeV. The results agree with model predictions of single pion photoproduction in the resonance region using fixed-t dispersion relations.