Batalin-Vilkovisky algebras

In theoretical physics, Batalin-Vilkovisky (BV) formalism was developed as a method for determining the ghost structure for theories, such as gravity and supergravity, whose Hamiltonian formalism has constraints not related to a Lie algebra action. The formalism, based on a Lagrangian that contains both fields and "antifields", can be thought of as a very complicated generalization of the BRST formalism.

Batalin-Vilkovisky algebras

A Batalin-Vilkovisky algebra is a graded supercommutative algebra (with identity 1) with a second-order differential operator Δ of degree -1, with Δ2=0 and Δ(1)=0. More precisely it satisfies the identities

* |ab| = |a| + |b| (The product has degree 0)

* |Δ(a)| = 1+|a| (Δ has degree -1)

* (ab)c=a(bc), ab=(−1)|a||b|ba (the product is associative and (super) commutative)

* Δ2=0

* Δ(1)=0 (Normalization)

* Δ is second order, in other words for any a, the supercommutator [Δ,a] is a derivation.

A Batalin-Vilkovisky algebra becomes a Gerstenhaber algebra if one defines the Poisson bracket by

[a,b] = ( − 1) | a | Δ(ab) − ( − 1) | a | Δ(a)b − aΔ(b).

Master equation

The (classical) master equation for an odd degree element S of a Batalin-Vilkovisky algebra (or more generally a Lie superalgebra) is the equation

[S,S] = 0

The quantum master equation for an odd degree element S of a Batalin-Vilkovisky algebra (or more generally a Lie superalgebra with an odd derivation Δ) is the equation

[S,S] = 2Δ(S)

or equivalently

[S − Δ,S − Δ] = 0.

Examples

* If L is a Lie superalgebra, and Π is the operator exchanging the even and odd parts of a super space, then the symmetric algebra of Π(L) (the "exterior algebra" of L) is a Batalin-Vilkovisky algebra with Δ given by the usual differential used to compute Lie algebra cohomology.

* If A is a Batalin-Vilkovisky algebra, and S a solution of the quantum master equation, then changing Δ to Δ + [S, ] gives a new Batalin-Vilkovisky algebra.

See also

* analysis of flows.

References

* E. Getzler Batalin-Vilkovisky algebras and two-dimensional topological field theories, Communications in Mathematical Physics, Volume 159, Number 2 / January, 1994, Pages 265-285. doi:10.1007/BF02102639

* Steven Weinberg The Quantum Theory of Fields Vol. II ISBN 0521670543

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