Batalin-Vilkovisky formalism - AbsoluteAstronomy.com

Theoretical physics

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, 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

Supergravity

In theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...

, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

(i.e., the role of Lie algebra structure constants are played by more general structure functions). The BV formalism, based on an action

Action (physics)

In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...

that contains both fields

Field (physics)

In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

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. Other names for the Batalin–Vilkovisky formalism are field-antifield formalism, Lagrangian BRST formalism, or BV-BRST formalism. It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart.

Batalin–Vilkovisky algebras

In mathematics, a Batalin–Vilkovisky algebra is a graded

Graded algebra

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....

supercommutative algebra

Supercommutative algebra

In mathematics, a supercommutative algebra is a superalgebra such that for any two homogeneous elements x, y we haveyx = ^In mathematics, a supercommutative algebra is a superalgebra In mathematics, a supercommutative algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two...

(with a unit 1) with a second-order nilpotent operator Δ of degree −1. More precisely, it satisfies the identities

|ab| = |a| + |b| (The product has degree 0)
|Δ(a)| = |a| − 1 (Δ has degree −1)
(ab)c = a(bc) (The product is associative)
ab = (−1)^|a||b|ba (The product is (super-)commutative)
Δ² = 0 (Nilpotency (of order 2))
Δ(abc) − Δ(ab)c −(−1)^|a|a Δ(bc) − (−1)^(|a|+1)|b|b Δ(ac) + Δ(a)bc + (−1)^|a|aΔ(b)c + (−1)^|a| + |b|abΔ(c) − Δ(1)abc = 0 (The Δ operator is of second order)

One often also requires normalization:

Δ(1) = 0 (normalization)

Antibracket

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

Other names for the Gerstenhaber bracket are Buttin bracket, antibracket, or odd Poisson bracket. The antibracket satisfies

|(a,b)| = |a|+|b| − 1 (The antibracket has degree -1)
(a,b) = −(−1)^{(|a|+1)(|b|+1)}(b,a) (Skewsymmetry)
(−1)^{(|a|+1)(|c|+1)}(a,(b,c)) + (−1)^{(|b|+1)(|a|+1)}(b,(c,a)) + (−1)^{(|c|+1)(|b|+1)}(c,(a,b)) = 0 (The Jacobi identity)
(ab,c) = a(b,c) + (−1)^|a||b|b(a,c) (The Poisson property;The Leibniz rule)

Odd Laplacian

The normalized operator is defined as

It is often called the odd Laplacian, in particular in the context of odd Poisson geometry. It "differentiates" the antibracket

(The operator differentiates )

The square

of the normalized

operator is a Hamiltonian vector field with odd Hamiltonian Δ(1)

(The Leibniz rule)

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

vanishes.

Compact formulation in terms of nested commutators

If one introduces the left multiplication operator

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

(The Δ operator is of second order)
where it is understood that the pertinent operator acts on the unit element 1. In other words,

is a first-order (affine) operator, and

is a zeroth-order operator.

Master equation

The classical master equation for an even degree element S (called the action

Action (physics)

) 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

Generalized BV algebras

In the definition of a generalized BV algebra, one drops the second-order assumption for Δ. One may then define an infinite hierarchy of higher brackets of degree -1

The brackets are (graded) symmetric

(Symmetric brackets)
where

is a permutation, and

is the Koszul sign of the permutation

.
The brackets constitute a homotopy Lie algebra, also known as an

algebra, which satisfies generalized Jacobi identities

(Generalized Jacobi identities)
The first few brackets are:

(The zero-bracket)
(The one-bracket)
(The two-bracket)
(The three-bracket)

In particular, the one-bracket

is the odd Laplacian, and the two-bracket

is the antibracket up to a sign. The first few generalized Jacobi identities are:

( is -closed)
( is the Hamiltonian for the modular vector field )
(The operator differentiates generalized)
(The generalized Jacobi identity)

where the Jacobiator for the two-bracket

is defined as

BV n-algebras

The Δ operator is by definition of n'th order if and only if the (n+1)-bracket

vanishes. In that case, one speaks of a BV n-algebra. Thus a BV 2-algebra is by definition just a BV algebra. The Jacobiator

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)

Differential graded algebra

In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.- Definition :...

with differential Δ. A BV 1-algebra has vanishing antibracket.

Odd Poisson Manifold with Volume Density

Let there be given an (n|n) supermanifold

Supermanifold

In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.- Physics :...

with an odd Poisson bi-vector

and a Berezin volume density

, also known as a P-structure and an S-structure, respectively. Let the local coordinates be called

. Let the derivatives

and

denote the left and right derivative of a function f wrt.

, respectively. The odd Poisson bi-vector

satisfies more precisely

(The odd Poisson structure has degree -1)
(Skewsymmetry)
(The Jacobi identity)

Under change of coordinates

the odd Poisson bi-vector

and Berezin volume density

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

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

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. The odd Laplacian

measures the failure of Liouville's Theorem. Up to a sign factor, it is defined as one half the divergence of the corresponding Hamiltonian vector field,

The odd Poisson structure

and Berezin volume density

are said to be compatible if the modular vector field

vanishes. In that case the odd Laplacian

is a BV Δ operator with normalization Δ(1)=0. The corresponding BV algebra is the algebra of functions.

Odd Symplectic Manifold

If the odd Poisson bi-vector

is invertible, one has an odd symplectic manifold. In that case, there exists an odd Darboux Theorem. That is, there exist local Darboux coordinates, i.e., coordinates

, and momenta

, of degree

such that the odd Poisson bracket is on Darboux form

In theoretical physics

Theoretical physics

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, the coordinates

and momenta

are called fields and antifields, and are typically denoted

and

, respectively. Khudaverdian's canonical operator

acts on the vector space of semidensities, and is a globally well-defined operator on the atlas of Darboux neighborhoods. Khudaverdian's

operator depends only on the P-structure. It is manifestly nilpotent

, and of degree -1. Nevertheless, it is technically not a BV Δ operator as the vector space of semidensities has no multiplication. (The product of two semidensities is a density rather than a semidensity.) Given a fixed density

, one may construct a nilpotent BV Δ operator as

,
whose corresponding BV algebra is the algebra of functions, or equivalently, scalars. The odd symplectic structure

and density

are compatible if and only if Δ(1) is an odd constant.

Examples

The Schouten-Nijenhuis bracket
Schouten-Nijenhuis bracket
In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different versions, both rather confusingly called by the same...

for multi-vector fields is an example of an antibracket.
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
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....

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
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

.