Klaus Bering

Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent Δ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the “big bracket” of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets.

On generalized gauge fixing in the field–antifield formalism

Abstract We consider the problem of covariant gauge fixing in the most general setting of the field–antifield formalism, where the action W and the gauge-fixing part X enter symmetrically and both satisfy the quantum master equation. Analogous to the gauge-generating algebra of the action W , we analyze the possibility of having a reducible gauge-fixing algebra of X . We treat a reducible gauge-fixing algebra of the so-called first stage in full detail and generalize to arbitrary stages. The associated “square root” measure contributions are worked out from first principles, with or without the presence of antisymplectic second-class constraints. Finally, we consider an W – X alternating multi-level generalization.

Absolute instruments and perfect imaging in geometrical optics

We investigate imaging by spherically symmetric absolute instruments that provide perfect imaging in the sense of geometrical optics. We derive a number of properties of such devices, present a general method for designing them and use this method to propose several new absolute instruments, in particular a lens providing a stigmatic image of an optically homogeneous region and having a moderate refractive index range.

Odd scalar curvature in anti-Poisson geometry

Abstract Recent works have revealed that the recipe for field–antifield quantization of Lagrangian gauge theories can be considerably relaxed when it comes to choosing a path integral measure ρ if a zero-order term ν ρ is added to the Δ operator. The effects of this odd scalar term ν ρ become relevant at two-loop order. We prove that ν ρ is essentially the odd scalar curvature of an arbitrary torsion-free connection that is compatible with both the anti-Poisson structure E and the density ρ. This extends a previous result for non-degenerate antisymplectic manifolds to degenerate anti-Poisson manifolds that admit a compatible two-form.

A note on semidensities in antisymplectic geometry

We revisit Khudaverdian’s geometric construction of an odd nilpotent operator ΔE that sends semidensities to semidensities on an antisymplectic manifold. We find a local formula for the ΔE operator in arbitrary coordinates and we discuss its connection to Batalin-Vilkovisky quantization.

Semidensities, second-class constraints, and conversion in anti-Poisson geometry

We consider Khudaverdian’s geometric version of a Batalin–Vilkovisky (BV) operator ΔE in the case of a degenerate anti-Poisson manifold. The characteristic feature of such an operator (aside from being a Grassmann-odd, nilpotent, second-order differential operator) is that it sends semidensities to semidensities. We find a local formula for the ΔE operator in arbitrary coordinates. As an important application of this setup, we consider the Dirac antibracket on an antisymplectic manifold with antisymplectic second-class constraints. We show that the entire Dirac construction, including the corresponding Dirac BV operator ΔED, exactly follows from conversion of the antisymplectic second-class constraints into first-class constraints on an extended manifold.

Three Natural Generalizations of Fedosov Quantization

Fedosovs simple geometrical construction for deformation quantization of sym- plectic manifolds is generalized in three ways without introducing new variables: (1) The base manifold is allowed to be a supermanifold. (2) The star product does not have to be of Weyl/symmetric or Wick/normal type. (3) The initial geometric structures are allowed to depend on Plancks constant.

A systematic study of finite BRST-BV transformations within W–X formulation of the standard and the Sp(2)-extended field–antifield formalism

Finite BRST-BV transformations are studied systematically within the W–X formulation of the standard and the Sp(2)-extended field–antifield formalism. The finite BRST-BV transformations are introduced by formulating a new version of the Lie equations. The corresponding finite change of the gauge-fixing master action X and the corresponding Ward identity are derived.

Path Integral Formulation with Deformed Antibracket

Abstract We propose how to incorporate the Leites–Shchepochkina–Konstein–Tyutin deformed antibracket into the quantum field–antifield formalism.

Reducible gauge algebra of BRST-invariant constraints

Abstract We show that it is possible to formulate the most general first-class gauge algebra of the operator formalism by only using BRST-invariant constraints. In particular, we extend a previous construction for irreducible gauge algebras to the reducible case. The gauge algebra induces two nilpotent, Grassmann-odd, mutually anti-commuting BRST operators that bear structural similarities with BRST/anti-BRST theories but with shifted ghost number assignments. In both cases we show how the extended BRST algebra can be encoded into an operator master equation. A unitarizing Hamiltonian that respects the two BRST symmetries is constructed with the help of a gauge-fixing boson. Abelian reducible theories are shown explicitly in full detail, while non-Abelian theories are worked out for the lowest reducibility stages and ghost momentum ranks.