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

January 4, 2008

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

Posted by Urs Schreiber

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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 $Ω^{•} (X)$ of differential forms on some space.

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 $∞$ -algebras:

then we want to find

A kind of algebras such that each of them is, essentially, the Clifford algebra of $T^{*} X \oplus T X$ on some space $X$ .

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

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

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

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

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

(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 $Γ (T^{*} X)$ , but from the co-standard Clifford irrep build from monomials in $Γ (T X)$ .

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

$e^{- S / ℏ} d μ$

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 $e^{- S / ℏ}$ and think of the “ $d μ$ ” 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 ${hom}_{S^{∞}} (X, Y)$ 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.

Posted at January 4, 2008 5:09 PM UTC

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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.

Posted by: kelly on January 4, 2008 8:24 PM | Permalink | Reply to this

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

For some of this stuff from a more formally mathematical point of view, see also

“The (secret?) Homological Algebra of the Batalin-Vilkovisky Approach”,
Proceedings of the Conference on Secondary Calculus and Cohomological Physics,Moscow, August 1997, Contemporary Mathematics 219 (1998) 195-210

pdf available

and/or

“Homological (ghost) methods in mathematical physics”, in Proceedings of the First Caribbean Summer School of Mathematics and Theoretical Physics 1993:
Infinite dimensional geometry, non commutative Geometry, Operator Algebras,
Fundamental Interactions, World Scientific (1995) 242-265

Posted by: jim stasheff on January 5, 2008 3:27 PM | Permalink | Reply to this

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

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Thanks, Jim, for keeping providing help here.

I am quite aware of your The (secret?) Homological Algebra of the Batalin-Vilkovisky Approach, which I have linked to a couple of times in previous installments (I, II).

And I very much appreciate the clarifications as to the mathematical interpretation of BV formalism which you give there.

But what I haven’t seen you discuss anywhere (but that may well be just my ignorance) is what strikes me as a rather powerful observation, the one by Witten that the above entry is devoted to:

the BV-Laplacian is (best thought of as) nothing but the exterior derivative acting on a not-quite standard Clifford module for $Cl (T^{*} X \oplus T X)$ .

This I find quite remarkable. I’ll have to re-check again the standard references that “kelly” recommended, which I haven’t actually looked at since long time ago, but this is the kind of good-looking interpretation I was looking for in vain in many reviews of BV-formalism.

In particular, it explains in one stroke not only what the master equation and the antibracket are, but also why we should care!

In most treatments the only reason for “why should we care” that is given is: “look, it works powerful magic”.

Which is fine. But not fully satisfactory.

For instance at the end of your “(secret) homological algebra” you notice that the master equation looks like a Maurer-Cartan equation and then end your article by asking:

But why?

We have talked about this “But why?” before, and maybe I am misinterpreting what “why” you meant, but in Witten’s interpretation of the BV Laplacian which I talked about in my entry above, I find, personally, precisely this “But why?” answered:

Answer: we are really looking at an equation of the form $d ω = 0$ with $d$ the exterior differential, and only because we are looking at this equation not just inside $Ω^{•} (X)$ but inside the larger $Cl (T^{*} X \oplus T X)$ and because we interpret it relative to a slightly nonstandard Clifford module does

- a) the first order differential operator $d$ superficially look like a second order differential operator $Δ$

- b) does the linear $d ω$ turn into the non-linear $Δ S + {S, S}$ .

And on top of all that, it seems that we even begin to see here why this change of perspective is actually useful and important: because it comes from interpreting $d$ and volume forms on spaces which are not finite dimensional manifolds.

That’s what I like about Witten’s interpretation, which was what the above entry was focusing on.

Posted by: Urs on January 5, 2008 3:59 PM | Permalink | Reply to this

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

I’ll have to go back and look at exactly what Witten did - how a second order operator appeared as exterior derivative/

Posted by: jim stasheff on January 6, 2008 3:33 AM | Permalink | Reply to this

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

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how a second order operator appeared as exterior derivative

The idea is quickly told: the exterior derivative is like the product of a differential operator (partial derivative of functions) with a multiplication operator (wedge product of forms).

Witten points out that the BV-Laplacian is nothing but the odd Fourier transformation of the exterior differential with respect to this odd multiplication operator.

In components, let ${x^{i} : ℝ^{n} \to ℝ}$ be the canonical coordinate functions on $ℝ^{n}$ . Then the operator which acts like the exterior derivative on the $Cl (T^{*} ℝ^{n} \oplus T ℝ^{n})$ -module $Ω^{•} (X)$ on $ℝ^{n}$ can be written as

$d = ({dx}^{i}) \land \frac{\partial}{\partial x^{i}} .$

But consider any volume form $μ \in Ω^{n} (ℝ^{n})$ and the induced “odd Fourier transformed” Clifford module $\oplus_{n} \land^{n} Γ (T X) \cdot μ$ which we think of as being spanned by exterior powers of vector fields $x^{i}^{*} : = ι_{\partial_{x^{i}}}$ instead of by differential forms $d x^{i} .$

Then on this the exterior derivative acts as the opertor $d = \frac{\partial}{\partial x^{i}^{*}} \frac{\partial}{\partial x^{i}} .$

Finally one observes that under this transformation the wedge product also undergoes a funny transformation, and we instead end up with the “antibracket”.

I think it all comes down to this:

it looks like for analysis on non-finite dimensional spaces it is possibly not good to consider all differential forms $ω$ as relative to the constant 0-form 1 obtained by wedging $ω = ω \land 1$ but rather to pick something like a volume form $μ$ and regard all differential forms as obtained from that by interior products with “antifields” $ω = ι_{something} μ .$

And apparently the BV-Laplacian/antibracket formalism can be thought of as handling this.

I guess relevant are also the respective remarks which one finds in the second part of Witten’s old Supersymmetry and Morse theory, which point out that the right reference object for forms on loop space is not the constant 0-form, but some “middle form”.

This can actually be traced back also to string theory. One can see that the 0-modes of the supercharges of the type II superstring are nothing but Morse-theoretically deformed exterior derivative and its adjoint on loop space. But the worldsheet vacuum state which they act on is not the constant 0-form on loop space, but some “middle form”.

But the constant 0-form on loop space also plays an important role here: it is the “boundary state” of the space-filling D9-brane.

I once talked about that in On deformations of 2D SCFTs.

Posted by: Urs Schreiber on January 6, 2008 1:20 PM | Permalink | Reply to this

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

There’s an extensive math literature with
`semiinfinite’ it the title, e.g. semiinfinite cohomology. The analogy with the Dirac sea is quite precise and noticed early on.

Perhaps the most famous reference is:

MR0865483 (88d:17016) Frenkel, I. B.; Garland, H.; Zuckerman, G. J. Semi-infinite cohomology and string theory. Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 22, 8442–8446. (Reviewer: J. Stasheff) 17B65 (17B55 17B67 81E30 81E99)

Posted by: jim stasheff on January 6, 2008 7:54 PM | Permalink | Reply to this

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

Urs wrote:

For instance at the end of your `(secret) homological algebra’ you notice that the master equation looks like a Maurer-Cartan equation and then end your article by asking:
But why?
We have talked about this `But why? before, and maybe I am misinterpreting what you meant, but in Witten’s interpretation of the BV Laplacian which I talked about in my entry above, I find, personally, precisely this `But why?’ answered:

Answer: we are really looking at an equation of the form
d??=0
with d the exterior differential, and only because we are looking at this equation not just inside Omega(X) but inside the larger Cl(T *X oplus TX) and because we interpret it relative to a slightly nonstandard Clifford module does
- a) the first order differential operator d superficially look like a second order differential operator
- b) does the linear d turn into the non-linear \Delta S+{S,S}.

Jim: My `but why?’ was meant at a deeper level,
so let me ramble:

First of all, I misspoke when I said the master eqn terminology was becoming dominant; in fact, it’s the Maurer-Cartan or MC that’s now dominant. I take it that this is entirely on the basis of the form of the eqn, the original MC being for diff forms on a Lie group.

So at the top level: It holds for Lie groups G because TG is trivializeable
(G is parallelizeable).

Historically I’d bet that regarding it as the equation of flatness came earlier, but doesn’t provide a catchier title.

Relation to deformation theory expessed in terms of a differential d s/t d^2=0.
A deformation can be described in terms of
d+p s/t (d+p)^2 =0 which can be expanded
to [d,p]+p^2=0 since d^2=0. In char \neq 2, p^2 = 1/2 [p,p] et voila!

A particular case of that occurs for twisting cochains p - regarding p (called: perturbation) as giving a deformation of d the differential on B x F so that the differential d+p corresponds to a fibration F –> E –> B.

The distinction between geometry and homotopy theory shows up in that finding a flat connection is not always possible for a given bundle but a twisting cochain always exists.

That MC appears as the Master Equation I take it as reflecting the fact that a bundle (at least the corresponding qDGCA)
is in hte background. The quantum ME is
then related to regarding quantization as deformation.

So maybe that’s all my `WHY?’ needs as an answer

Posted by: jim stasheff on January 6, 2008 3:17 PM | Permalink | Reply to this

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

Urs wrote:

it seems that we even begin to see here why this change of perspective is actually useful and important: because it comes from interpreting d and volume forms on spaces which are not finite dimensional manifolds.

That’s what I like about Witten’s interpretation, which was what the above entry was focusing on.

JIM: Much (all?) of that can be consider as formal, cf. Gel’fand’s formal differential geometry. But that would avoid having to discuss smootheness!! Why not?

Posted by: jim stasheff on January 6, 2008 7:41 PM | Permalink | Reply to this

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

I may have mentioned this before: the BV formalism may refer to two different aspects of their work and I think both are relevant to what Urs posted:

1. BV algebra as Urs reviews is a purely algebraic definition. Notice the BV operator is asecond order diff op

2.the BV construction is a combination of the Chevalley Eilenberg cochain algebra with coefficients in the fields and the Koszul chain algebra for the ideal of the EoMs.

Posted by: jim stasheff on January 5, 2008 3:38 PM | Permalink | Reply to this

ME: the Master Equation

Once upon a time, dw = \pm [w,w]
was note as flatness
In deformation theory, aka integrability
In homotopy theory, twisting cochain

Now some powerful lobby (derived from the BV camp, I believe) has called it

The Master Equation

and the world seem to be follwoing suit.

Posted by: jim stasheff on January 5, 2008 3:59 PM | Permalink | Reply to this

Re: ME: the Master Equation

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Once upon a time, $d w = \pm [w, w]$ was note as flatness In deformation theory, aka integrability In homotopy theory, twisting cochain

Now some powerful lobby (derived from the BV camp, I believe) has called it

The Master Equation

and the world seem to be follwoing suit.

Yes, this is the attitude I am trying to follow here. I don’t like thinking of weird sounding “master equations”. Instead, I am thinking that what people find to be very important but address by funny names should actually be something essentially standard and well known appearing to us in slightly different guise.

So I want to understand: why is the flatness condition called the “master equation” the key to perturbative quantization of gauge systems?

There must be a good powerful general interpretation.

Possibly the answer is well known (maybe to you, too, but then I haven’t absorbed it from you yet, sorry) to somebody. But certainly I am not the only one puzzled about this (even though not everybody seems to express his or her puzzlement). So in any case I have the feeling it is justified to talk more about this stuff.

Which, luckily, we are indeed doing here.

Posted by: Urs Schreiber on January 5, 2008 4:11 PM | Permalink | Reply to this

Witten’s anti-bracket formalism

\begin{document}
Thanks to Urs for forcing me to go back and reread Witteen’s ` A note on the anti-bracket formalism’.
I had long since appreciated part of it and apparently forgotten what I did not apprreciate,
especially the Clifford part.

Much of the following is a translation of Witten into my language, as I think I saw it at the time.
The quadratic form in question is just a dual pairing - in this case, of tangent and cotangent vectors
on a *finite dimensional* manifold M.
Extending this to the graded comm alg they generate gives the anti-bracket, though that’s not how Witten sees it. From my point of view, $TM x_M T*M$ is a graded symplectic manifold.

Then Witten considers irreducible reps of this graded Lie algebra. To do this locally, he uses
the free GC algebras, one generated by $ dx^i$ and and the other by $\partial_x^j$. now comes the
`physics’: treat in the first case $ dx^i$ as creation operators and $\partial_x^j$ as annihilators,
while for the second case, vice versa (or in phys notation ). That is, interchange the roles of the generators. Notice that we now have representations as $\partial_{dx^i}$ and $\partial_{\partial_{x^j}}.$

Thus, depending on what we consider as the variable, these are either first order or second order,
which Witten keeps track of by pseudonysms: $z^i$ and $w_j$.

It is in establishing an isomorphism (NOT unique) that volume form appears.

He then recovers the anti-bracket (I consider it to be there already) as the deviation of $Delta$ from being a derivation - cf. Gerstenhaber.

Witten finds it `is striking’ that infinitesimal transformations have the form of gauge invariance in OSFT,
but later work has shown this is generic for Lagrangian variations.

He calls for `a Lagrangian whose variational equation would be the qME (quantum master equation)’.
I have a feeling this has been done but can’t surface a reference.

\end{document}

Posted by: jim stasheff on January 6, 2008 7:43 PM | Permalink | Reply to this

Re: Witten’s anti-bracket formalism

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Thanks a lot, Jim, for your comments! One remark: you write:

what I did not apprreciate, especially the Clifford part.

[…]

From my point of view, $TM x_{M} T * M$ is a graded symplectic manifold.

Okay, good. So maybe that clarifies much of what we had been discussing at various points, latey: I kept saying that it might be worthwhile to think of a symplectic dg-manifold (what people call symplectic $n$ -algebroids) as a kind of Clifford structure.

Anyway, have to call it quits now. More later. Thanks again for your comments.

Posted by: Urs Schreiber on January 6, 2008 10:28 PM | Permalink | Reply to this

Re: Witten’s anti-bracket formalism

Thinking of it as Clifford must `mean’ something to you or your intuitiion - but what? It motivates Witten to set up his duality, but is not needed to state the duality.

Posted by: jim stasheff on January 7, 2008 1:52 PM | Permalink | Reply to this

Re: Witten’s anti-bracket formalism

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Thinking of it as Clifford must ‘mean’ something to you or your intuitiion - but what?

One way to put it is this:

we know that every graded-commutative dga comes to us

i) as the CE-algebra of some semistrict Lie $∞$ -algebra

ii) as the dga of differential forms on some generalized space (on some presheaf).

If I know what $Ω^{•} (X)$ means for a generalized smooth space, there is nothing more natural than trying to define what $Cl (T^{*} X \oplus T X)$ means for a general smooth space. It is something we should want to look at anyway.

And then we might notice something like that

every graded Clifford dg-algebra comes to us

i)as the CE-algebra of a hemistrict Lie $∞$ -algebra

ii) as the canonical Clifford algebra $Cl (T^{*} X \oplus T X)$ over some generalized space (some presheaf).

It would be a beautiful pattern which would unify much super-thinking with much Lie thinking, in particular.

For one, if true it would immediately suggest what hemistrict $L_{∞}$ -algebras integrate to.

And it indeed seems to be at least roughly true.

Posted by: Urs Schreiber on January 7, 2008 3:40 PM | Permalink | Reply to this

Re: On BV Quantization, Part IX…

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It might be helpful for anyone who wants to reproduce the above algebra to point out that Witten’s seminal paper Mod.Phys.Lett.A5 (1990) 487 contains two misprints, which have proliferated into this blog at various places. A preprint version of Witten’s paper can be downloaded from the KEK preprint library:

http://ccdb4fs.kek.jp/cgi-bin/img_index?9004090

1) Wittens’s formula (15) should read

$(- 1)^{F} [F, G] = Δ (FG) - (Δ F) G - (- 1)^{F} F (Δ G)$

Here we have assumed that $(Δ 1) = 0$ . In general (15) contains one more term:

$(- 1)^{F} [F, G] = Δ (FG) - (Δ F) G - (- 1)^{F} F (Δ G) + (Δ 1) FG$

2) In Witten’s formula (19), which is the Euclidean version of the quantum master equation, there is missing a factor $(1 / 2)$ in front of the antibracket term. It should read

$ℏ (Δ S) = \frac{1}{2} [S, S]$

Again we have assumed that $(Δ 1) = 0$ . In general (19) contains two more terms:

$ℏ (Δ S - (Δ 1) S) = \frac{1}{2} [S, S] + ℏ^{2} (Δ 1)$

Posted by: Klaus Bering on January 7, 2008 4:29 PM | Permalink | Reply to this

Integration over Diffeological Spaces

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What’s the integration theory for diffeological spaces (aka Chen-smooth spaces etc)?

Understanding that should go a long way towards understanding path integrals.

This morning on the train, I had the following thoughts:

look at the path integral in “time slice definition” as in every standard introductory textbook and as in the the first quarter of this Wikipedia entry.

Notice: this is nothing but working with plots for the space of all paths. The $n$ -time slicing is a plot $ϕ : ℝ^{n} \to P X .$

The path integral is conceived in terms of an ordinary integral on each plot.

This is the most obvious statement ever made. But has it ever been made before?

Now consider this:

let there be a metric $g$ on $X$ . A plot

$U \to P X$

of path space is precisely a smooth map

$U \times I \to X,$

where $I = [0,1]$ is the unit interval. By pullback, we get a metric on all of these $U \times I$ .

Notice what this looks like when $U = {pt}$ :

the volume of the plot

$γ : {pt} \times I \to X,$

which happens to be just a single path, is precisely the “kinetic Nambu-Goto action” of that path

$\int_{[0,1]} \sqrt{g (γ (σ)) (γ' (σ), γ' (σ)} d σ,$

where $g$ is the metric on $X$ .

To approach a path integral, we might want to consider such pulled back metric on all plots $U \times I$ , then do the fiber integration just over $I$ – this should yield the kinetic part of the actions. The remaining part of the volume element would be the path integral measure – on that finite dimensional plot.

Then the whole subtlety is in how to, somehow, take the colimit over all plots, or something.

I am thinking: we want for plots of all dimensions a volume form. So maybe we should consider something like $\exp (g_{ij} d x^{i} \otimes d x^{j})$ with the pulled back metric

(written here in components to emphasize the tensor product structure) and then form the exponential in wedge products.

Then doing a kind of fiber integration over the $[0,1]$ -factor would, by the above observation, actually produce the kinetic action part. And even exponentiated.

Phew, I am getting too tired. I’ll say all this again more coherently and more nicely tomorrow.

But if there is anyone out there who has already heard and/or thought about integrals over diffeological spaces – please drop me a note.

Posted by: Urs Schreiber on February 5, 2008 12:21 AM | Permalink | Reply to this

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