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. 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 supercommutative algebra (with a unit 1) with a second-order nilpotent operator Δ of degree −1. More precisely, it satisfies the identities
( a b ) c = a ( b c ) {\displaystyle (ab)c=a(bc)} (The product is associative)
a b = ( − 1 ) | a | | b | b a {\displaystyle ab=(-1)^{|a||b|}ba} (The product is (super-)commutative)
| a b | = | a | + | b | {\displaystyle |ab|=|a|+|b|} (The product has degree 0)
| Δ ( a ) | = | a | − 1 {\displaystyle |\Delta (a)|=|a|-1} (Δ has degree −1)
Δ 2 = 0 {\displaystyle \Delta ^{2}=0} (Nilpotency (of order 2)) The Δ operator is of second order:
0 = Δ ( a b c ) − Δ ( a b ) c − ( − 1 ) | a | a Δ ( b c ) − ( − 1 ) ( | a | + 1 ) | b | b Δ ( a c ) + Δ ( a ) b c + ( − 1 ) | a | a Δ ( b ) c + ( − 1 ) | a | + | b | a b Δ ( c ) − Δ ( 1 ) a b c {\displaystyle {\begin{aligned}0=&\Delta (abc)\\&-\Delta (ab)c-(-1)^{|a|}a\Delta (bc)-(-1)^{(|a|+1)|b|}b\Delta (ac)\\&+\Delta (a)bc+(-1)^{|a|}a\Delta (b)c+(-1)^{|a|+|b|}ab\Delta (c)\\&-\Delta (1)abc\end{aligned}}}
One often also requires normalization:
Δ ( 1 ) = 0 {\displaystyle \Delta (1)=0} (normalization)
Antibracket A Batalin–Vilkovisky algebra becomes a Gerstenhaber algebra if one defines the Gerstenhaber bracket by
( a , b ) := ( − 1 ) | a | Δ ( a b ) − ( − 1 ) | a | Δ ( a ) b − a Δ ( b ) + a Δ ( 1 ) b . {\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
| ( a , b ) | = | a | + | b | − 1 {\displaystyle |(a,b)|=|a|+|b|-1} (The antibracket (,) has degree −1)
( a , b ) = − ( − 1 ) ( | a | + 1 ) ( | b | + 1 ) ( b , a ) {\displaystyle (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 {\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)
( a b , c ) = a ( b , c ) + ( − 1 ) | a | | b | b ( a , c ) {\displaystyle (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
Δ ρ := Δ − Δ ( 1 ) . {\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
Δ ρ ( a , b ) = ( Δ ρ ( a ) , b ) − ( − 1 ) | a | ( a , Δ ρ ( b ) ) {\displaystyle {\Delta }_{\rho }(a,b)=({\Delta }_{\rho }(a),b)-(-1)^{\left|a\right|}(a,{\Delta }_{\rho }(b))} (The Δ ρ {\displaystyle {\Delta }_{\rho }} operator differentiates (,)) The square Δ ρ 2 = ( Δ ( 1 ) , ⋅ ) {\displaystyle {\Delta }_{\rho }^{2}=(\Delta (1),\cdot )} of the normalized Δ ρ {\displaystyle {\Delta }_{\rho }} operator is a Hamiltonian vector field with odd Hamiltonian Δ(1)
Δ ρ 2 ( a b ) = Δ ρ 2 ( a ) b + a Δ ρ 2 ( b ) {\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 Δ ρ {\displaystyle {\Delta }_{\rho }} is just the Δ operator, and the modular vector field Δ ρ 2 {\displaystyle {\Delta }_{\rho }^{2}} vanishes.
Compact formulation in terms of nested commutators If one introduces the left multiplication operator L a {\displaystyle L_{a}} as
L a ( b ) := a b , {\displaystyle L_{a}(b):=ab,}
and the supercommutator [,] as
[ S , T ] := S T − ( − 1 ) | S | | T | T S {\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
( a , b ) := ( − 1 ) | a | [ [ Δ , L a ] , L b ] 1 , {\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
[ [ [ Δ , L a ] , L b ] , L c ] 1 = 0 {\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, [ Δ , L a ] {\displaystyle [\Delta ,L_{a}]} is a first-order (affine) operator, and [ [ Δ , L a ] , L b ] {\displaystyle [[\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. {\displaystyle (S,S)=0.}
The quantum master equation for an even degree element W of a Batalin–Vilkovisky algebra is the equation
Δ exp [ i ℏ W ] = 0 , {\displaystyle \Delta \exp \left[{\frac {i}{\hbar }}W\right]=0,}
or equivalently,
1 2 ( W , W ) = i ℏ Δ ρ ( W ) + ℏ 2 Δ ( 1 ) . {\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
1 2 ( W , W ) = i ℏ Δ ( W ) . {\displaystyle {\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
Φ n ( a 1 , … , a n ) := [ [ … [ Δ , L a 1 ] , … ] , L a n ] ⏟ n n e s t e d c o m m u t a t o r s 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
Φ n ( a π ( 1 ) , … , a π ( n ) ) = ( − 1 ) | a π | Φ n ( a 1 , … , a n ) {\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 π ∈ S n {\displaystyle \pi \in S_{n}} is a permutation, and ( − 1 ) | a π | {\displaystyle (-1)^{\left|a_{\pi }\right|}} is the Koszul sign of the permutation
a π ( 1 ) … a π ( n ) = ( − 1 ) | a π | a 1 … a n {\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 L ∞ {\displaystyle L_{\infty }} algebra, which satisfies generalized Jacobi identities
∑ k = 0 n 1 k ! ( n − k ) ! ∑ π ∈ S n ( − 1 ) | a π | Φ n − k + 1 ( Φ k ( a π ( 1 ) , … , a π ( k ) ) , a π ( k + 1 ) , … , a π ( n ) ) = 0. {\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:
Φ 0 := Δ ( 1 ) {\displaystyle \Phi ^{0}:=\Delta (1)} (The zero-bracket)
Φ 1 ( a ) := [ Δ , L a ] 1 = Δ ( a ) − Δ ( 1 ) a =: Δ ρ ( a ) {\displaystyle \Phi ^{1}(a):=[\Delta ,L_{a}]1=\Delta (a)-\Delta (1)a=:{\Delta }_{\rho }(a)} (The one-bracket)
Φ 2 ( a , b ) := [ [ Δ , L a ] , L b ] 1 =: ( − 1 ) | a | ( a , b ) {\displaystyle \Phi ^{2}(a,b):=[[\Delta ,L_{a}],L_{b}]1=:(-1)^{\left|a\right|}(a,b)} (The two-bracket)
Φ 3 ( a , b , c ) := [ [ [ Δ , L a ] , L b ] , L c ] 1 {\displaystyle \Phi ^{3}(a,b,c):=[[[\Delta ,L_{a}],L_{b}],L_{c}]1} (The three-bracket)
⋮ {\displaystyle \vdots }
In particular, the one-bracket Φ 1 = Δ ρ {\displaystyle \Phi ^{1}={\Delta }_{\rho }} is the odd Laplacian, and the two-bracket Φ 2 {\displaystyle \Phi ^{2}} is the antibracket up to a sign. The first few generalized Jacobi identities are:
Φ 1 ( Φ 0 ) = 0 {\displaystyle \Phi ^{1}(\Phi ^{0})=0} ( Δ ( 1 ) {\displaystyle \Delta (1)} is Δ ρ {\displaystyle \Delta _{\rho }} -closed)
Φ 2 ( Φ 0 , a ) + Φ 1 ( Φ 1 ( a ) ) {\displaystyle \Phi ^{2}(\Phi ^{0},a)+\Phi ^{1}\left(\Phi ^{1}(a)\right)} ( Δ ( 1 ) {\displaystyle \Delta (1)} is the Hamiltonian for the modular vector field Δ ρ 2 {\displaystyle {\Delta }_{\rho }^{2}} )
Φ 3 ( Φ 0 , a , b ) + Φ 2 ( Φ 1 ( a ) , b ) + ( − 1 ) | a | Φ 2 ( a , Φ 1 ( b ) ) + Φ 1 ( Φ 2 ( a , b ) ) = 0 {\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 Δ ρ {\displaystyle {\Delta }_{\rho }} operator differentiates (,) generalized)
Φ 4 ( Φ 0 , a , b , c ) + J a c ( a , b , c ) + Φ 1 ( Φ 3 ( a , b , c ) ) + Φ 3 ( Φ 1 ( a ) , b , c ) + ( − 1 ) | a | Φ 3 ( a , Φ 1 ( b ) , c ) + ( − 1 ) | a | + | b | Φ 3 ( a , b , Φ 1 ( c ) ) = 0 {\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 Φ 2 {\displaystyle \Phi ^{2}} is defined as
J a c ( a 1 , a 2 , a 3 ) := 1 2 ∑ π ∈ S 3 ( − 1 ) | a π | Φ 2 ( Φ 2 ( a π ( 1 ) , a π ( 2 ) ) , a π ( 3 ) ) . {\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).}
BV n-algebras The Δ operator is by definition of n'th order if and only if the (n + 1)-bracket Φ n + 1 {\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 J a c ( a , b , c ) = 0 {\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.
Odd Poisson manifold with volume density Let there be given an (n|n) supermanifold with an odd Poisson bi-vector π i j {\displaystyle \pi ^{ij}} and a Berezin volume density ρ {\displaystyle \rho } , also known as a P-structure and an S-structure, respectively. Let the local coordinates be called x i {\displaystyle x^{i}} . Let the derivatives ∂ i f {\displaystyle \partial _{i}f} and
f ∂ ← i := ( − 1 ) | x i | ( | f | + 1 ) ∂ i f {\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. x i {\displaystyle x^{i}} , respectively. The odd Poisson bi-vector π i j {\displaystyle \pi ^{ij}} satisfies more precisely
| π i j | = | x i | + | x j | − 1 {\displaystyle \left|\pi ^{ij}\right|=\left|x^{i}\right|+\left|x^{j}\right|-1} (The odd Poisson structure has degree –1)
π j i = − ( − 1 ) ( | x i | + 1 ) ( | x j | + 1 ) π i j {\displaystyle \pi ^{ji}=-(-1)^{(\left|x^{i}\right|+1)(\left|x^{j}\right|+1)}\pi ^{ij}} (Skewsymmetry)
( − 1 ) ( | x i | + 1 ) ( | x k | + 1 ) π i ℓ ∂ ℓ π j k + c y c l i c ( i , j , k ) = 0 {\disp