# On odd Laplace operators. II

@article{Khudaverdian2002OnOL, title={On odd Laplace operators. II}, author={Hovhannes M. Khudaverdian and Theodore Th. Voronov}, journal={arXiv: Differential Geometry}, year={2002} }

We analyze geometry of the second order differential operators, having in mind applications to Batalin--Vilkovisky formalism in quantum field theory. As we show, an exhaustive picture can be obtained by considering pencils of differential operators acting on densities of all weights simultaneously. The algebra of densities, which we introduce here, has a natural invariant scalar product. Using it, we prove that there is a one-to-one correspondence between second-order operators in this algebra…

## 24 Citations

### Second order operators on the algebra of densities and a groupoid of connections

- Mathematics
- 2011

This work considers the geometry of second order linear operators acting on the commutative algebra of densities on a (super)manifold and obtains operators that depend on equivalence classes of connections and a groupoid of connections such that the orbits of this groupoid are these equivalence Classes.

### OPERATOR PENCIL PASSING THROUGH A GIVEN OPERATOR

- Mathematics
- 2013

Let Δ be a linear differential operator acting on the space of densities of a given weight λ0 on a manifold M. One can consider a pencil of operators Π(Δ)={Δλ} passing through the operator Δ such…

### Kaluza-Klein theory revisited: projective structures and differential operators on algebra of densities

- Mathematics
- 2013

We consider differential operators acting on densities of arbitrary weights on manifold M identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can…

### Differential forms and odd symplectic geometry

- Mathematics
- 2006

We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {\v{S}}evera. We study…

### Differential operators on the algebra of densities and factorization of the generalized Sturm–Liouville operator

- MathematicsLetters in Mathematical Physics
- 2018

We consider factorization problem for differential operators on the commutative algebra of densities (defined either algebraically or in terms of an auxiliary extended manifold) introduced in 2004 by…

### Operator pencils on the algebra of densities

- Mathematics
- 2014

We continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role played by the geometry of the extended manifold where the algebra of…

### Odd Laplacians: geometrical meaning of potential and modular class

- Mathematics
- 2015

A second-order self-adjoint operator $$\Delta =S\partial ^2+U$$Δ=S∂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…

### On the finite dimensional BV Formalism

- Mathematics
- 2005

We analyze the finite dimensional mathematical framework for Batalin Vilkovisiky (BV) formalism also called antifield formalism as suggested in [Schw]. In a general procedure for quantization of…

### Kaluza-Klein theory revisited: projective structures and differential operators on algebra of densities

- Mathematics
- 2013

We consider differential operators acting on densities of arbitrary weights on manifold M identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can…

### Odd connections on supermanifolds: existence and relation with affine connections

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2020

The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however,…

## References

SHOWING 1-10 OF 19 REFERENCES

### On Odd Laplace Operators

- Mathematics
- 2002

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…

### Laplacians in Odd Symplectic Geometry

- Mathematics
- 2002

We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky…

### ON THE GEOMETRY OF THE BATALIN-VILKOVISKY FORMALISM

- Mathematics
- 1993

An invariant definition of the operator Δ of the Batalin-Vilkovisky formalism is proposed. It is defined as the divergence of a Hamiltonian vector field with an odd Poisson bracket (anti-bracket).…

### Semiclassical approximation in Batalin-Vilkovisky formalism

- Physics
- 1993

The geometry of supermanifolds provided with aQ-structure (i.e. with an odd vector fieldQ satisfying {Q, Q}=0), aP-structure (odd symplectic structure) and anS-structure (volume element) or with…

### Symmetry transformations in Batalin-Vilkovisky formalism

- Physics
- 1994

Let us suppose that the functionalS on an odd symplectic manifold satisfies the quantum master equation Δρes = 0. We prove that in some sense every quantum observable (i.e. every functionH obeying…

### Delta-Operator on Semidensities and Integral Invariants in the Batalin-Vilkovisky Geometry

- Mathematics
- 1999

The action of Batalin-Vilkovisky Delta-operator on semidensities in an odd symplectic superspace is defined. This is used for the construction of integral invariants on surfaces embedded in an odd…

### Closure of the gauge algebra, generalized lie equations and Feynman rules

- Physics, Mathematics
- 1984

### Geometry of Batalin-Vilkovisky quantization

- Mathematics
- 1992

The geometry ofP-manifolds (odd symplectic manifolds) andSP-manifolds (P-manifolds provided with a volume element) is studied. A complete classification of these manifolds is given. This…

### The analysis of linear partial differential operators

- Physics
- 1990

the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certainproducts. Many products that you buy can be…