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 • (ab)c = a(bc) (The product is associative) • ab = (-1)^ba (The product is (super-)commutative) • |ab| = |a| + |b| (The product has degree 0) • |\Delta(a)| = |a| - 1 (Δ has degree −1) • \Delta^2 = 0 (Nilpotency (of order 2)) • The Δ operator is of second order: :\begin{align} 0 = & \Delta(abc) \\ &-\Delta(ab)c-(-1)^a\Delta(bc)-(-1)^{(|a|+1)|b|}b\Delta(ac)\\ &+\Delta(a)bc+(-1)^a\Delta(b)c+(-1)^ab\Delta(c)\\ &-\Delta(1)abc \end{align} One often also requires normalization: • \Delta(1)=0 (normalization) ==Antibracket==

A Batalin–Vilkovisky algebra becomes a Gerstenhaber algebra if one defines the Gerstenhaber bracket by :(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 • |(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)^b(a,c) (The Poisson property; the Leibniz rule) ==Odd Laplacian==

The normalized operator is defined as : {\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 • {\Delta}_{\rho}(a,b) = ({\Delta}_{\rho}(a),b) - (-1)^{\left|a\right|}(a,{\Delta}_{\rho}(b)) (The {\Delta}_{\rho} operator differentiates (,)) The square {\Delta}_{\rho}^{2}=(\Delta(1),\cdot) of the normalized {\Delta}_{\rho} operator is a Hamiltonian vector field with odd Hamiltonian Δ(1) • {\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 {\Delta}_{\rho} is just the Δ operator, and the modular vector field {\Delta}_{\rho}^{2} vanishes. ==Compact formulation in terms of nested commutators==

If one introduces the left multiplication operator L_{a} as : L_{a}(b) := ab , and the supercommutator [,] as :[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 : (a,b) := (-1)^{\left|a\right|} \Delta,L_{a}],L_{b}]1 , and the second order condition for Δ may be written compactly as : [\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, [\Delta,L_{a}] is a first-order (affine) operator, and \Delta,L_{a}],L_{b}] is a zeroth-order operator. ==Master equation==

The classical master equation for an even degree element S (called the action) of a Batalin–Vilkovisky algebra is the equation :(S,S) = 0 . The quantum master equation for an even degree element W of a Batalin–Vilkovisky algebra is the equation : \Delta\exp \left[\frac{i}{\hbar}W\right] = 0 , or equivalently, :\frac{1}{2}(W,W) = i\hbar{\Delta}_{\rho}(W)+\hbar^{2}\Delta(1) . Assuming normalization Δ(1) = 0, the quantum master equation reads :\frac{1}{2}(W,W) = i\hbar\Delta(W) . ==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 : \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 : \Phi^{n}(a_{\pi(1)},\ldots,a_{\pi(n)}) = (-1)^{\left|a_{\pi}\right|}\Phi^{n}(a_{1},\ldots, a_{n}) (Symmetric brackets) where \pi\in S_{n} is a permutation, and (-1)^{\left|a_{\pi}\right|} is the Koszul sign of the permutation :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 L_{\infty} algebra, which satisfies generalized Jacobi identities : \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: • \Phi^{0} := \Delta(1) (The zero-bracket) • \Phi^{1}(a) := [\Delta,L_{a}]1 = \Delta(a) - \Delta(1)a =: {\Delta}_{\rho}(a) (The one-bracket) • \Phi^{2}(a,b) := \Delta,L_{a}],L_{b}]1 =: (-1)^{\left|a\right|}(a,b) (The two-bracket) • \Phi^{3}(a,b,c) := [\Delta,L_{a}],L_{b}],L_{c}]1 (The three-bracket) • \vdots In particular, the one-bracket \Phi^{1}={\Delta}_{\rho} is the odd Laplacian, and the two-bracket \Phi^{2} is the antibracket up to a sign. The first few generalized Jacobi identities are: • \Phi^{1}(\Phi^0) = 0 (\Delta(1) is \Delta_\rho-closed) • \Phi^{2}(\Phi^{0},a)+\Phi^{1}\left(\Phi^{1}(a)\right) (\Delta(1) is the Hamiltonian for the modular vector field {\Delta}_{\rho}^{2}) • \Phi^{3}(\Phi^{0},a,b) + \Phi^{2}\left(\Phi^{1}(a),b\right)+(-1)^\Phi^{2}\left(a,\Phi^{1}(b)\right) +\Phi^{1}\left(\Phi^{2}(a,b)\right) = 0 (The {\Delta}_{\rho} operator differentiates (,) generalized) • \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) • \vdots where the Jacobiator for the two-bracket \Phi^{2} is defined as : {\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) . ==BV n-algebras==

The Δ operator is by definition of '''n'th order' if and only if the (n'' + 1)-bracket \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 {\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. ==Odd Poisson manifold with volume density==

Let there be given an (n|n) supermanifold with an odd Poisson bi-vector \pi^{ij} and a Berezin volume density \rho, also known as a P-structure and an S-structure, respectively. Let the local coordinates be called x^{i}. Let the derivatives \partial_{i}f and : 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. x^{i}, respectively. The odd Poisson bi-vector \pi^{ij} satisfies more precisely • \left|\pi^{ij}\right| = \left|x^{i}\right| + \left|x^{j}\right| -1 (The odd Poisson structure has degree –1) • \pi^{ji} = -(-1)^{(\left|x^{i}\right|+1)(\left|x^{j}\right|+1)} \pi^{ij} (Skewsymmetry) • (-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 x^{i} \to x^{\prime i} the odd Poisson bi-vector \pi^{ij} and Berezin volume density \rho transform as • \pi^{\prime k\ell} = x^{\prime k}\stackrel{\leftarrow}{\partial}_{i} \pi^{ij} \partial_{j}x^{\prime \ell} • \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 : (f,g) := f\stackrel{\leftarrow}{\partial}_{i}\pi^{ij}\partial_{j}g . A Hamiltonian vector field X_{f} with Hamiltonian f can be defined as : X_{f}[g] := (f,g) . The (super-)divergence of a vector field X=X^{i}\partial_{i} is defined as : {\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 {\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, : {\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 \pi^{ij} and Berezin volume density \rho are said to be compatible if the modular vector field {\Delta}_{\rho}^{2} vanishes. In that case the odd Laplacian {\Delta}_{\rho} 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 \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 q^{1}, \ldots, q^{n} , and momenta p_{1},\ldots, p_{n} , of degree : \left|q^{i}\right|+\left|p_{i}\right|=1, such that the odd Poisson bracket is on Darboux form : (q^{i},p_{j}) = \delta^{i}_{j} . In theoretical physics, the coordinates q^{i} and momenta p_{j} are called fields and antifields, and are typically denoted \phi^{i} and \phi^{*}_{j} , respectively. :\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 \Delta_{\pi} operator depends only on the P-structure. It is manifestly nilpotent \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 \rho, one may construct a nilpotent BV Δ operator as : \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 \pi^{ij} and density \rho are compatible if and only if Δ(1) is an odd constant. ==Examples==

• 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. ==See also==