In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra (i.e., the role of Lie algebra structure constants are played by more general structure functions).
The BV formalism, based on an action that contains both fields and "antifields", can be thought of as a vast generalization of the original BRST formalism for pure Yang–Mills theory to an arbitrary Lagrangian gauge theory.
It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart.
In mathematics, a Batalin–Vilkovisky algebra is a graded supercommutative algebra (with a unit 1) with a second-order nilpotent operator Δ of degree −1.
The antibracket satisfies The normalized operator is defined as It is often called the odd Laplacian, in particular in the context of odd Poisson geometry.
operator is a Hamiltonian vector field with odd Hamiltonian Δ(1) which is also known as the modular vector field.
Assuming normalization Δ(1)=0, the odd Laplacian
is just the Δ operator, and the modular vector field
as and the supercommutator [,] as for two arbitrary operators S and T, then the definition of the antibracket may be written compactly as and the second order condition for Δ may be written compactly as where it is understood that the pertinent operator acts on the unit element 1.
The classical master equation for an even degree element S (called the action) of a Batalin–Vilkovisky algebra is the equation The quantum master equation for an even degree element W of a Batalin–Vilkovisky algebra is the equation or equivalently, Assuming normalization Δ(1) = 0, the quantum master equation reads In the definition of a generalized BV algebra, one drops the second-order assumption for Δ.
is the Koszul sign of the permutation The brackets constitute a homotopy Lie algebra, also known as an
algebra, which satisfies generalized Jacobi identities The first few brackets are: In particular, the one-bracket
is defined as The Δ operator is by definition of n'th order if and only if the (n + 1)-bracket
vanishes within a BV algebra, which means that the antibracket here satisfies the Jacobi identity.
A BV 1-algebra that satisfies normalization Δ(1) = 0 is the same as a differential graded algebra (DGA) with differential Δ.
Let there be given an (n|n) supermanifold with an odd Poisson bi-vector
and denote the left and right derivative of a function f wrt.
transform as where sdet denotes the superdeterminant, also known as the Berezinian.
Then the odd Poisson bracket is defined as A Hamiltonian vector field
with Hamiltonian f can be defined as The (super-)divergence of a vector field
is defined as Recall that Hamiltonian vector fields are divergencefree in even Poisson geometry because of Liouville's Theorem.
In odd Poisson geometry the corresponding statement does not hold.
Up to a sign factor, it is defined as one half the divergence of the corresponding Hamiltonian vector field, The odd Poisson structure
are said to be compatible if the modular vector field
In that case, there exists an odd Darboux Theorem.
, of degree such that the odd Poisson bracket is on Darboux form In theoretical physics, the coordinates
are called fields and antifields, and are typically denoted
acts on the vector space of semidensities, and is a globally well-defined operator on the atlas of Darboux neighborhoods.
Nevertheless, it is technically not a BV Δ operator as the vector space of semidensities has no multiplication.
, one may construct a nilpotent BV Δ operator as whose corresponding BV algebra is the algebra of functions, or equivalently, scalars.