### On BV Quantization, Part IX: Antibracket and BV-Laplacian

#### Posted by Urs Schreiber

So far, in my discussion of BV-formalism (part
I,
II,
III,
IV,
V,
VI,
VII, VIII) I had concentrated on the nature and meaning of the underlying complex, without saying a word yet about the *antibracket* and the *BV-Laplacian* and the *master equation*.

I hadn’t mentioned that yet because it wasn’t clear to me yet what the big story here actually is. But now I might be getting closer.

Recall from the discussion in Transgression of $n$-Transport and $n$-Connections that

Every differential non-negatively graded commutative algebra is, essentially, the algebra $\Omega^\bullet(X)$ of differential forms onsomespace.

Now generalize this fact from the cotangent bundle $T^* X$ to the Clifford bundle $T^*X \oplus T X$ as suggested in Categorified Clifford Algebra and weak Lie n-Algebras and recently discussed again in weak Lie $\infty$-algebras:

then we want to find

A kind of algebras such that each of them is, essentially, the Clifford algebra of $T^* X \oplus T X$ onsomespace$X$.

Apparently, this kind of algebra is: *BV-algebra*.

**Definition** *A BV-algebra is a graded commutative algebra $A$ with an operator $\Delta : A \to A$ such that $\Delta^2 = 0$ and such that the “derived bracket” or “antibracket”
$[a,b] := \Delta(a b) - \Delta(a) b +(-1)^{|a|} a \Delta(b)$
is a Gerstenhaber bracket on $A$.
*

The key to seeing this is related to Clifford algebra has been noticed two decades ago in

E. Witten
*A note on the antibracket formalism*

Modern Physics Letters A, **5** 7, 487 - 494

(pdf)

The punchline is:

The BV Laplacian $\Delta$ is nothing but the exterior derivative in disguise.

The master equation $\Delta \exp(- S/\hbar) = 0$ is hence nothing but the statement that the path integral integrand $\exp(- S/\hbar)$ is a closed form.

The antibracket is precisely that bracket which makes the master action equivalent to the flatness condition $- \hbar \Delta S = [S,S] \,.$

(This last statement is no secret. The first two statements I haven’t seen emphasized much in the literature.)

From this point of view, one can also see the BV-formalism as a way to conceiving *volume forms* on spaces $X$ which are not finite-dimensional manifolds.

The crucial insight which Witten presented in that paper is simply that the BV-Laplacian is nothing but the exterior differential acting non on the standard Clifford irrep build from monomials in $\Gamma(T^* X)$, but from the co-standard Clifford irrep build from monomials in $\Gamma(T X)$.

Hence “antifields” are simply the duals to differential forms, hence are vector fields. (This I have discussed at length before.)

Most crucial seems to be the interpretation of all that:

for toy examples of physics where configuration space is a compact manifold $X$, the path integral over that manifold is really the integral over a top-dimensional form which people usually write

$e^{-S /\hbar} d\mu$

but which should best be thought of not as a 0-form times a vomule form, but just as a volume form. To emphasize that, we just $e^{-S /\hbar}$ and think of the “$d\mu$” part as absorbed in the exponential.

Then how does that generalize to the standard valilla configuration space in physics, which is generically a space of maps $\mathrm{hom}_{S^\infty}(X,Y)$ between two manifolds?

To answer that, we need

- a notion of differential forms on generalized smooth spaces

- a notion of “volume forms” for these.

The master equation of BV-formalism is apparently picking these “volume forms” on generalized smooth spaces for us.

## Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

Take a peek at M. Henneaux “Lectures on the Anti-Field-BRST Formalism for Gauge Theories” Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106. This explains some of this stuff; in particular see section 8.8. Similarly see Henneaux and Teitelboim “Quantization of Gauge Systems,” Princeton University Press 1992, Theorem 18.1 and section 18.1.3.