Batalin–Vilkovisky formalism

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. More precisely, it satisfies the identities

• ${\displaystyle |ab|=|a|+|b|}$ (The product has degree 0)
• ${\displaystyle |\Delta (a)|=|a|-1}$ (Δ has degree −1)
• ${\displaystyle (ab)c=a(bc)}$ (The product is associative)
• ${\displaystyle ab=(-1)^{|a||b|}ba}$ (The product is (super-)commutative)
• ${\displaystyle \Delta ^{2}=0}$ (Nilpotency (of order 2))
• ${\displaystyle \Delta (abc)-\Delta (ab)c+\Delta (a)bc-(-1)^{|a|}a\Delta (bc)-(-1)^{(|a|+1)|b|}b\Delta (ac)+(-1)^{|a|}a\Delta (b)c+(-1)^{|a|+|b|}ab\Delta (c)-\Delta (1)abc=0}$(The Δ operator is of second order)

One often also requires normalization:

• ${\displaystyle \Delta (1)=0}$ (normalization)

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

${\displaystyle (a,b):=(-1)^{\left|a\right|}\Delta (ab)-(-1)^{\left|a\right|}\Delta (a)b-a\Delta (b)+a\Delta (1)b.}$

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

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

The normalized operator is defined as

${\displaystyle {\Delta }_{\rho }:=\Delta -\Delta (1).}$

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

• ${\displaystyle {\Delta }_{\rho }(a,b)=({\Delta }_{\rho }(a),b)-(-1)^{\left|a\right|}(a,{\Delta }_{\rho }(b))}$ (The ${\displaystyle {\Delta }_{\rho }}$ operator differentiates (,))

The square ${\displaystyle {\Delta }_{\rho }^{2}=(\Delta (1),\cdot )}$ of the normalized ${\displaystyle {\Delta }_{\rho }}$ operator is a Hamiltonian vector field with odd Hamiltonian Δ(1)

• ${\displaystyle {\Delta }_{\rho }^{2}(ab)={\Delta }_{\rho }^{2}(a)b+a{\Delta }_{\rho }^{2}(b)}$ (The Leibniz rule)

which is also known as the modular vector field. Assuming normalization Δ(1)=0, the odd Laplacian ${\displaystyle {\Delta }_{\rho }}$ is just the Δ operator, and the modular vector field ${\displaystyle {\Delta }_{\rho }^{2}}$ vanishes.

If one introduces the left multiplication operator ${\displaystyle L_{a}}$ as

${\displaystyle L_{a}(b):=ab,}$

and the supercommutator [,] as

${\displaystyle [S,T]:=ST-(-1)^{\left|S\right|\left|T\right|}TS}$

for two arbitrary operators S and T, then the definition of the antibracket may be written compactly as

${\displaystyle (a,b):=(-1)^{\left|a\right|}[[\Delta ,L_{a}],L_{b}]1,}$

and the second order condition for Δ may be written compactly as

${\displaystyle [[[\Delta ,L_{a}],L_{b}],L_{c}]1=0}$ (The Δ operator is of second order)

where it is understood that the pertinent operator acts on the unit element 1. In other words, ${\displaystyle [\Delta ,L_{a}]}$ is a first-order (affine) operator, and ${\displaystyle [[\Delta ,L_{a}],L_{b}]}$ is a zeroth-order operator.

The classical master equation for an even degree element S (called the action) of a Batalin–Vilkovisky algebra is the equation

${\displaystyle (S,S)=0.}$

The quantum master equation for an even degree element W of a Batalin–Vilkovisky algebra is the equation

${\displaystyle \Delta \exp \left[{\frac {i}{\hbar }}W\right]=0,}$

or equivalently,

${\displaystyle {\frac {1}{2}}(W,W)=i\hbar {\Delta }_{\rho }(W)+\hbar ^{2}\Delta (1).}$

Assuming normalization Δ(1) = 0, the quantum master equation reads

${\displaystyle {\frac {1}{2}}(W,W)=i\hbar \Delta (W).}$

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

${\displaystyle \Phi ^{n}(a_{1},\ldots ,a_{n}):=\underbrace {[[\ldots [\Delta ,L_{a_{1}}],\ldots ],L_{a_{n}}]} _{n~{\rm {nested~commutators}}}1.}$

The brackets are (graded) symmetric

${\displaystyle \Phi ^{n}(a_{\pi (1)},\ldots ,a_{\pi (n)})=(-1)^{\left|a_{\pi }\right|}\Phi ^{n}(a_{1},\ldots ,a_{n})}$ (Symmetric brackets)

where ${\displaystyle \pi \in S_{n}}$ is a permutation, and ${\displaystyle (-1)^{\left|a_{\pi }\right|}}$ is the Koszul sign of the permutation

${\displaystyle a_{\pi (1)}\ldots a_{\pi (n)}=(-1)^{\left|a_{\pi }\right|}a_{1}\ldots a_{n}}$.

The brackets constitute a homotopy Lie algebra, also known as an ${\displaystyle L_{\infty }}$ algebra, which satisfies generalized Jacobi identities

${\displaystyle \sum _{k=0}^{n}{\frac {1}{k!(n\!-\!k)!}}\sum _{\pi \in S_{n}}(-1)^{\left|a_{\pi }\right|}\Phi ^{n-k+1}\left(\Phi ^{k}(a_{\pi (1)},\ldots ,a_{\pi (k)}),a_{\pi (k+1)},\ldots ,a_{\pi (n)}\right)=0.}$ (Generalized Jacobi identities)

The first few brackets are:

• ${\displaystyle \Phi ^{0}:=\Delta (1)}$ (The zero-bracket)
• ${\displaystyle \Phi ^{1}(a):=[\Delta ,L_{a}]1=\Delta (a)-\Delta (1)a=:{\Delta }_{\rho }(a)}$ (The one-bracket)
• ${\displaystyle \Phi ^{2}(a,b):=[[\Delta ,L_{a}],L_{b}]1=:(-1)^{\left|a\right|}(a,b)}$ (The two-bracket)
• ${\displaystyle \Phi ^{3}(a,b,c):=[[[\Delta ,L_{a}],L_{b}],L_{c}]1}$ (The three-bracket)
• ${\displaystyle \vdots }$

In particular, the one-bracket ${\displaystyle \Phi ^{1}={\Delta }_{\rho }}$ is the odd Laplacian, and the two-bracket ${\displaystyle \Phi ^{2}}$ is the antibracket up to a sign. The first few generalized Jacobi identities are:

• ${\displaystyle \Phi ^{1}(\Phi ^{0})=0}$ (${\displaystyle \Delta (1)}$ is ${\displaystyle \Delta _{\rho }}$-closed)
• ${\displaystyle \Phi ^{2}(\Phi ^{0},a)+\Phi ^{1}\left(\Phi ^{1}(a)\right)}$ (${\displaystyle \Delta (1)}$ is the Hamiltonian for the modular vector field ${\displaystyle {\Delta }_{\rho }^{2}}$)
• ${\displaystyle \Phi ^{3}(\Phi ^{0},a,b)+\Phi ^{2}\left(\Phi ^{1}(a),b\right)+(-1)^{|a|}\Phi ^{2}\left(a,\Phi ^{1}(b)\right)+\Phi ^{1}\left(\Phi ^{2}(a,b)\right)=0}$ (The ${\displaystyle {\Delta }_{\rho }}$ operator differentiates (,) generalized)
• ${\displaystyle \Phi ^{4}(\Phi ^{0},a,b,c)+{\rm {Jac}}(a,b,c)+\Phi ^{1}\left(\Phi ^{3}(a,b,c)\right)+\Phi ^{3}\left(\Phi ^{1}(a),b,c\right)+(-1)^{\left|a\right|}\Phi ^{3}\left(a,\Phi ^{1}(b),c\right)+(-1)^{\left|a\right|+\left|b\right|}\Phi ^{3}\left(a,b,\Phi ^{1}(c)\right)=0}$ (The generalized Jacobi identity)
• ${\displaystyle \vdots }$

where the Jacobiator for the two-bracket ${\displaystyle \Phi ^{2}}$ is defined as

${\displaystyle {\rm {Jac}}(a_{1},a_{2},a_{3}):={\frac {1}{2}}\sum _{\pi \in S_{3}}(-1)^{\left|a_{\pi }\right|}\Phi ^{2}\left(\Phi ^{2}(a_{\pi (1)},a_{\pi (2)}),a_{\pi (3)}\right).}$

The Δ operator is by definition of n'th order if and only if the (n + 1)-bracket ${\displaystyle \Phi ^{n+1}}$ 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 ${\displaystyle {\rm {Jac}}(a,b,c)=0}$ 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. A BV 1-algebra has vanishing antibracket.

Let there be given an (n|n) supermanifold with an odd Poisson bi-vector ${\displaystyle \pi ^{ij}}$ and a Berezin volume density ${\displaystyle \rho }$, also known as a P-structure and an S-structure, respectively. Let the local coordinates be called ${\displaystyle x^{i}}$. Let the derivatives ${\displaystyle \partial _{i}f}$ and

${\displaystyle f{\stackrel {\leftarrow }{\partial }}_{i}:=(-1)^{\left|x^{i}\right|(|f|+1)}\partial _{i}f}$

denote the left and right derivative of a function f wrt. ${\displaystyle x^{i}}$, respectively. The odd Poisson bi-vector ${\displaystyle \pi ^{ij}}$ satisfies more precisely

• ${\displaystyle \left|\pi ^{ij}\right|=\left|x^{i}\right|+\left|x^{j}\right|-1}$ (The odd Poisson structure has degree –1)
• ${\displaystyle \pi ^{ji}=-(-1)^{(\left|x^{i}\right|+1)(\left|x^{j}\right|+1)}\pi ^{ij}}$ (Skewsymmetry)
• ${\displaystyle (-1)^{(\left|x^{i}\right|+1)(\left|x^{k}\right|+1)}\pi ^{i\ell }\partial _{\ell }\pi ^{jk}+{\rm {cyclic}}(i,j,k)=0}$ (The Jacobi identity)

Under change of coordinates ${\displaystyle x^{i}\to x^{\prime i}}$ the odd Poisson bi-vector ${\displaystyle \pi ^{ij}}$ and Berezin volume density ${\displaystyle \rho }$ transform as

• ${\displaystyle \pi ^{\prime k\ell }=x^{\prime k}{\stackrel {\leftarrow }{\partial }}_{i}\pi ^{ij}\partial _{j}x^{\prime \ell }}$
• ${\displaystyle \rho ^{\prime }=\rho /{\rm {sdet}}(\partial _{i}x^{\prime j})}$

where sdet denotes the superdeterminant, also known as the Berezinian. Then the odd Poisson bracket is defined as

${\displaystyle (f,g):=f{\stackrel {\leftarrow }{\partial }}_{i}\pi ^{ij}\partial _{j}g.}$

A Hamiltonian vector field ${\displaystyle X_{f}}$ with Hamiltonian f can be defined as

${\displaystyle X_{f}[g]:=(f,g).}$

The (super-)divergence of a vector field ${\displaystyle X=X^{i}\partial _{i}}$ is defined as

${\displaystyle {\rm {div}}_{\rho }X:={\frac {(-1)^{\left|x^{i}\right|(|X|+1)}}{\rho }}\partial _{i}(\rho X^{i})}$

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 ${\displaystyle {\Delta }_{\rho }}$ 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,

${\displaystyle {\Delta }_{\rho }(f):={\frac {(-1)^{\left|f\right|}}{2}}{\rm {div}}_{\rho }X_{f}={\frac {(-1)^{\left|x^{i}\right|}}{2\rho }}\partial _{i}\rho \pi ^{ij}\partial _{j}f.}$

The odd Poisson structure ${\displaystyle \pi ^{ij}}$ and Berezin volume density ${\displaystyle \rho }$ are said to be compatible if the modular vector field ${\displaystyle {\Delta }_{\rho }^{2}}$ vanishes. In that case the odd Laplacian ${\displaystyle {\Delta }_{\rho }}$ is a BV Δ operator with normalization Δ(1)=0. The corresponding BV algebra is the algebra of functions.

If the odd Poisson bi-vector ${\displaystyle \pi ^{ij}}$ 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 ${\displaystyle q^{1},\ldots ,q^{n}}$, and momenta ${\displaystyle p_{1},\ldots ,p_{n}}$, of degree

${\displaystyle \left|q^{i}\right|+\left|p_{i}\right|=1,}$

such that the odd Poisson bracket is on Darboux form

${\displaystyle (q^{i},p_{j})=\delta _{j}^{i}.}$

In theoretical physics, the coordinates ${\displaystyle q^{i}}$ and momenta ${\displaystyle p_{j}}$ are called fields and antifields, and are typically denoted ${\displaystyle \phi ^{i}}$ and ${\displaystyle \phi _{j}^{*}}$, respectively.

${\displaystyle \Delta _{\pi }:=(-1)^{\left|q^{i}\right|}{\frac {\partial }{\partial q^{i}}}{\frac {\partial }{\partial p_{i}}}}$

acts on the vector space of semidensities, and is a globally well-defined operator on the atlas of Darboux neighborhoods. Khudaverdian's ${\displaystyle \Delta _{\pi }}$ operator depends only on the P-structure. It is manifestly nilpotent ${\displaystyle \Delta _{\pi }^{2}=0}$, 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 ${\displaystyle \rho }$, one may construct a nilpotent BV Δ operator as

${\displaystyle \Delta (f):={\frac {1}{\sqrt {\rho }}}\Delta _{\pi }({\sqrt {\rho }}f),}$

whose corresponding BV algebra is the algebra of functions, or equivalently, scalars. The odd symplectic structure ${\displaystyle \pi ^{ij}}$ and density ${\displaystyle \rho }$ are compatible if and only if Δ(1) is an odd constant.

• The Schouten–Nijenhuis bracket 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 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.

• BRST quantization
• Gerstenhaber algebra
• Supermanifold
• Analysis of flows
• Poisson manifold