View source for Batalin–Vilkovisky formalism

In [[theoretical physics]], the '''Batalin&ndash;Vilkovisky''' ('''BV''') '''formalism''' (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the [[Faddeev&ndash;Popov ghost|ghost]] structure for Lagrangian [[gauge theories]], such as gravity and [[supergravity]], whose corresponding [[Hamiltonian formalism|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 (physics)|action]] that contains both [[Field (physics)|fields]] and "antifields", can be thought of as a vast generalization of the original [[BRST formalism]] for [[Yang–Mills theory|pure Yang&ndash;Mills]] theory to an arbitrary Lagrangian gauge theory. Other names for the Batalin&ndash;Vilkovisky formalism are '''field-antifield formalism''', '''Lagrangian BRST formalism''', or '''BV–BRST formalism'''. It should not be confused with the [[Batalin&ndash;Fradkin&ndash;Vilkovisky formalism|Batalin&ndash;Fradkin&ndash;Vilkovisky (BFV) formalism]], which is the Hamiltonian counterpart.

==Batalin&ndash;Vilkovisky algebras==
In mathematics, a '''Batalin&ndash;Vilkovisky algebra''' is a [[Graded algebra|graded]] [[supercommutative algebra]] (with a unit 1) with a second-order nilpotent operator Δ of degree&nbsp;&minus;1. More precisely, it satisfies the identities
*<math>|ab| = |a| + |b| </math>   (The product has degree&nbsp;0)
*<math>|\Delta(a)| = |a| - 1    </math>  (Δ has degree&nbsp;&minus;1)
*<math>(ab)c = a(bc)     </math>  (The product is associative)
*<math>ab = (-1)^{|a||b|}ba   </math>    (The product is (super-)commutative)
*<math>\Delta^2 = 0      </math> (Nilpotency (of order 2))
*<math>\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  </math>(The Δ operator is of second order)

One often also requires normalization:

*<math>\Delta(1)=0  </math> (normalization)

==Antibracket==
A Batalin–Vilkovisky algebra becomes a [[Gerstenhaber algebra]] if one defines the '''Gerstenhaber bracket''' by
:<math>(a,b) := (-1)^{\left|a\right|}\Delta(ab) - (-1)^{\left|a\right|}\Delta(a)b - a\Delta(b)+a\Delta(1)b .</math>
Other names for the Gerstenhaber bracket are '''Buttin bracket''', '''antibracket''', or '''odd Poisson bracket'''. The antibracket satisfies
* <math>|(a,b)| = |a|+|b| - 1 </math> (The antibracket (,) has degree −1)
* <math> (a,b) = -(-1)^{(|a|+1)(|b|+1)}(b,a) </math>     (Skewsymmetry)
* <math> (-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 </math> (The Jacobi identity)
* <math> (ab,c) = a(b,c) + (-1)^{|a||b|}b(a,c)</math>     (The Poisson property;The Leibniz rule)

==Odd Laplacian==
The normalized operator is defined as
:<math> {\Delta}_{\rho} := \Delta-\Delta(1) . </math>
It is often called the '''odd Laplacian''', in particular in the context of odd Poisson geometry. It "differentiates" the antibracket
* <math>  {\Delta}_{\rho}(a,b) = ({\Delta}_{\rho}(a),b) - (-1)^{\left|a\right|}(a,{\Delta}_{\rho}(b)) </math>        (The <math>{\Delta}_{\rho}</math> operator  differentiates (,))
The square <math>{\Delta}_{\rho}^{2}=(\Delta(1),\cdot)</math> of the normalized <math>{\Delta}_{\rho}</math> operator is a Hamiltonian vector field with odd Hamiltonian Δ(1)
* <math> {\Delta}_{\rho}^{2}(ab) = {\Delta}_{\rho}^{2}(a)b+ a{\Delta}_{\rho}^{2}(b) </math>       (The Leibniz rule)
which is also known as the '''modular vector field'''. Assuming normalization Δ(1)=0, the odd Laplacian <math> {\Delta}_{\rho} </math> is just the Δ operator, and the modular vector field <math> {\Delta}_{\rho}^{2} </math> vanishes.

==Compact formulation in terms of nested commutators==
If one introduces the '''left multiplication operator''' <math>L_{a}</math> as 
:<math> L_{a}(b) := ab  ,  </math>
and the [[supercommutator]] [,] as 
:<math>[S,T]:=ST - (-1)^{\left|S\right|\left|T\right|}TS </math>
for two arbitrary operators ''S'' and ''T'', then the definition of the antibracket may be written compactly as
:<math> (a,b) := (-1)^{\left|a\right|} [[\Delta,L_{a}],L_{b}]1  , </math>
and the second order condition for Δ may be written compactly as
:<math> [[[\Delta,L_{a}],L_{b}],L_{c}]1 = 0   </math>    (The Δ operator is of second order)
where it is understood that the pertinent operator acts on the unit element 1. In other words, <math> [\Delta,L_{a}] </math> is a first-order (affine) operator, and <math> [[\Delta,L_{a}],L_{b}] </math> is a zeroth-order operator.

==Master equation==
The '''classical master equation''' for an even degree element ''S'' (called the [[Action (physics)|action]]) of a Batalin–Vilkovisky algebra is the equation
:<math>(S,S) = 0   . </math>
The '''quantum master equation''' for an even degree element ''W'' of a Batalin–Vilkovisky algebra is the equation
:<math> \Delta\exp \left[\frac{i}{\hbar}W\right] = 0 ,</math>
or equivalently,
:<math>\frac{1}{2}(W,W) = i\hbar{\Delta}_{\rho}(W)+\hbar^{2}\Delta(1) . </math>
Assuming normalization Δ(1)&nbsp;=&nbsp;0, the quantum master equation reads
:<math>\frac{1}{2}(W,W) = i\hbar\Delta(W) . </math>

==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&nbsp;−1
:<math> \Phi^{n}(a_{1},\ldots,a_{n}) :=  \underbrace{[[\ldots[\Delta,L_{a_{1}}],\ldots],L_{a_{n}}]}_{n~{\rm nested~commutators}}1  .   </math>
The brackets are (graded) symmetric
:<math> \Phi^{n}(a_{\pi(1)},\ldots,a_{\pi(n)}) = (-1)^{\left|a_{\pi}\right|}\Phi^{n}(a_{1},\ldots, a_{n})    </math>  (Symmetric brackets)
where <math>\pi\in S_{n}</math> is a permutation, and <math>(-1)^{\left|a_{\pi}\right|}</math> is the [[Koszul sign]] of the permutation 
:<math>a_{\pi(1)}\ldots a_{\pi(n)} = (-1)^{\left|a_{\pi}\right|}a_{1}\ldots a_{n}</math>.
The brackets constitute a [[homotopy Lie algebra]], also known as an <math>L_{\infty}</math> algebra, which satisfies generalized Jacobi identities 
:<math>  \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. </math>  (Generalized Jacobi identities)
The first few brackets are:
* <math> \Phi^{0} := \Delta(1) </math>      (The zero-bracket)
* <math> \Phi^{1}(a) := [\Delta,L_{a}]1 = \Delta(a) - \Delta(1)a =: {\Delta}_{\rho}(a) </math>    (The one-bracket)
* <math> \Phi^{2}(a,b) := [[\Delta,L_{a}],L_{b}]1 =: (-1)^{\left|a\right|}(a,b) </math>    (The two-bracket)
* <math> \Phi^{3}(a,b,c) := [[[\Delta,L_{a}],L_{b}],L_{c}]1 </math>    (The three-bracket)
* <math> \vdots </math>
In particular, the one-bracket <math> \Phi^{1}={\Delta}_{\rho}</math> is the odd Laplacian, and the two-bracket <math> \Phi^{2}</math> is the antibracket up to a sign. The first few generalized Jacobi identities are:
* <math> \Phi^{1}(\Phi^0) = 0 </math>  (<math>\Delta(1)</math>  is <math>\Delta_\rho</math>-closed)
* <math> \Phi^{2}(\Phi^{0},a)+\Phi^{1}\left(\Phi^{1}(a)\right)</math>  (<math>\Delta(1)</math>  is the Hamiltonian for the modular vector field <math>{\Delta}_{\rho}^{2}</math>)
* <math> \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 </math>  (The <math> {\Delta}_{\rho} </math> operator differentiates (,) generalized)
* <math> \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 </math>  (The generalized Jacobi identity)
* <math> \vdots </math>
where the [[Jacobiator]] for the two-bracket <math>\Phi^{2}</math> is defined as
:<math> {\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) .  </math>

==BV ''n''-algebras==
The Δ operator is by definition of '''n'th order''' if and only if the (''n''&nbsp;+&nbsp;1)-bracket <math> \Phi^{n+1} </math> 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 <math> {\rm Jac}(a,b,c)=0 </math> vanishes within a BV algebra, which means that the antibracket here satisfies the Jacobi identity. A '''BV 1-algebra''' that satisfies normalization Δ(1)&nbsp;=&nbsp;0 is the same as a [[differential graded algebra|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 <math> \pi^{ij}</math> and a Berezin volume density <math>\rho</math>, also known as a '''P-structure''' and an '''S-structure''', respectively. Let the local coordinates be called <math>x^{i}</math>. Let the derivatives <math> \partial_{i}f </math> and 
:<math> f\stackrel{\leftarrow}{\partial}_{i}:=(-1)^{\left|x^{i}\right|(|f|+1)}\partial_{i}f </math>
denote the [[left derivative|left]] and [[right derivative]] of a function ''f'' wrt. <math>x^{i}</math>, respectively. The odd Poisson bi-vector <math> \pi^{ij}</math> satisfies more precisely
* <math> \left|\pi^{ij}\right| = \left|x^{i}\right| + \left|x^{j}\right| -1  </math>   (The odd Poisson structure has degree&nbsp;–1)
* <math> \pi^{ji} = -(-1)^{(\left|x^{i}\right|+1)(\left|x^{j}\right|+1)} \pi^{ij} </math>       (Skewsymmetry)
* <math> (-1)^{(\left|x^{i}\right|+1)(\left|x^{k}\right|+1)}\pi^{i\ell}\partial_{\ell}\pi^{jk} + {\rm cyclic}(i,j,k) = 0 </math>    (The Jacobi identity)
Under change of coordinates <math>x^{i} \to x^{\prime i} </math> the odd Poisson bi-vector <math> \pi^{ij}</math> 
and Berezin volume density <math>\rho</math> transform as
* <math> \pi^{\prime k\ell} = x^{\prime k}\stackrel{\leftarrow}{\partial}_{i} \pi^{ij} \partial_{j}x^{\prime \ell} </math>
* <math>\rho^{\prime} = \rho/{\rm sdet}(\partial_{i}x^{\prime j}) </math>
where ''sdet'' denotes the [[superdeterminant]], also known as the Berezinian.
Then the '''odd Poisson bracket''' is defined as
:<math> (f,g) := f\stackrel{\leftarrow}{\partial}_{i}\pi^{ij}\partial_{j}g . </math>
A '''Hamiltonian vector field''' <math> X_{f}</math>  with Hamiltonian ''f'' can be defined as
:<math> X_{f}[g] := (f,g) .</math>
The (super-)[[divergence]] of a vector field <math> X=X^{i}\partial_{i} </math> is defined as
:<math> {\rm div}_{\rho} X := \frac{(-1)^{\left|x^{i}\right|(|X|+1)}}{\rho} \partial_{i}(\rho X^{i}) </math>
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''' <math> {\Delta}_{\rho}</math> 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,  
:<math> {\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.</math>
The odd Poisson structure <math> \pi^{ij}</math> and Berezin volume density <math>\rho</math> are said to be '''compatible''' if the modular vector field <math> {\Delta}_{\rho}^{2} </math> vanishes. In that case the '''odd Laplacian''' <math> {\Delta}_{\rho}</math> 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 <math> \pi^{ij}</math> is invertible, one has an odd [[Symplectic geometry|symplectic]] manifold. In that case, there exists an '''odd Darboux Theorem'''. That is, there exist local '''Darboux coordinates''', i.e., coordinates  <math> q^{1}, \ldots, q^{n} </math>, and momenta <math> p_{1},\ldots, p_{n} </math>, of degree
:<math> \left|q^{i}\right|+\left|p_{i}\right|=1, </math>
such that the odd Poisson bracket is on Darboux form
:<math> (q^{i},p_{j}) = \delta^{i}_{j} . </math>
In [[theoretical physics]], the coordinates <math>q^{i} </math> and momenta <math>p_{j} </math> are called '''fields''' and '''antifields''', and are typically denoted <math>\phi^{i} </math> and <math>\phi^{*}_{j} </math>, respectively.
:<math>\Delta_{\pi} := (-1)^{\left|q^{i}\right|}\frac{\partial}{\partial q^{i}}\frac{\partial}{\partial p_{i}}  </math>
acts on the vector space of [[semidensities]], and is a globally well-defined operator on the atlas of Darboux neighborhoods. Khudaverdian's <math>\Delta_{\pi}</math> operator depends only on the P-structure. It is manifestly nilpotent <math>\Delta_{\pi}^{2}=0</math>, and of degree&nbsp;−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 <math>\rho</math>, one may construct a nilpotent BV Δ operator as
:<math> \Delta(f) :=\frac{1}{\sqrt{\rho}}\Delta_{\pi}(\sqrt{\rho}f),</math>
whose corresponding BV algebra is the algebra of functions, or equivalently, [[scalar (physics)|scalar]]s. The odd symplectic structure <math> \pi^{ij}</math> and density <math>\rho</math> 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==
*[[BRST quantization]]
*[[Gerstenhaber algebra]]
*[[Supermanifold]]
*[[Analysis of flows]]

== References ==

*{{Cite journal |first=I. A. |last=Batalin |lastauthoramp=yes |first2=G. A. |last2=Vilkovisky |title=Gauge Algebra and Quantization |journal=[[Physics Letters|Phys. Lett. B]] |volume=102 |year=1981 |issue=1 |pages=27–31 |doi=10.1016/0370-2693(81)90205-7 |bibcode = 1981PhLB..102...27B }}
*{{Cite journal |first=I. A. |last=Batalin |first2=G. A. |last2=Vilkovisky |title=Quantization of Gauge Theories with Linearly Dependent Generators |journal=Physical Review D |volume=28 |year=1983 |issue=10 |pages=2567–2582 |doi=10.1103/PhysRevD.28.2567 |bibcode = 1983PhRvD..28.2567B }} Erratum-ibid. '''30''' (1984) 508 {{DOI|10.1103/PhysRevD.30.508}}.
*{{Cite journal |last=Getzler |first=E. |title=Batalin-Vilkovisky algebras and two-dimensional topological field theories |journal=Communications in Mathematical Physics |volume=159 |issue=2 |year=1994 |pages=265–285 |doi=10.1007/BF02102639 |arxiv = hep-th/9212043 |bibcode = 1994CMaPh.159..265G }}
*{{Citation | last1=Brandt | first1=Friedemann | last2=Barnich | first2=Glenn | last3=Henneaux | first3=Marc | title=Local BRST cohomology in gauge theories | doi=10.1016/S0370-1573(00)00049-1 | mr=1792979  | year=2000 | journal=Physics Reports. A Review Section of Physics Letters | issn=0370-1573 | volume=338 | issue=5 | pages=439–569|arxiv = hep-th/0002245 |bibcode = 2000PhR...338..439B }}
*{{Cite book |first=Steven |last=Weinberg |authorlink=Steven Weinberg |year=2005 |title=The Quantum Theory of Fields Vol. II |location=New York |publisher=Cambridge Univ. Press |isbn=0-521-67054-3 }}

{{DEFAULTSORT:Batalin-Vilkovisky Formalism}}
[[Category:Algebras]]
[[Category:Quantum field theory]]
[[Category:Symplectic geometry]]
[[Category:Theoretical physics]]