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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 20th 2011
   • (edited Sep 21st 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe [[BV-BRST complex]] of some local Lagrangian $L$ over some manifold $\Sigma$ famously comes equipped with a canonical Poisson bracket of grade +1: the antibracket. The whole point of this construction is that it serves as a tool for computing another bracket that one is interested in: the ordinary grade-0 bracket induced from the symplectic form that is obtained by descending the canonical presymplectic form on [[covariant phase space]] (as discussed there) to the quotient by the symmetries. Recently I was beginning to entertain myself with the idea that there should be a more immediate relation between these two brackets than I see considered in the usual literature. I was beginning to think that it should be possible to regard a [[Poisson n-algebra]] as a "homotopy" between two Poisson $(n-1)$-algebras. (Does that sound reasonable to anyone?) So I started toying around with some formulas to see if I can substantiate my guess. I am not as sophisticated yet with these computations as Frédéric, who is reading here, and likely I am making some silly mistakes. But taken together this is are two good reasons to chat about my thoughts here! So for the purposes of this post here, I'll be a bit naive, following the physics literature, and say things like: _Let, locally, $\{\Phi^a(x)\}$ be a collection of generators for the fields and ghosts..._ as well as _... and let $\{\bar \Phi_a(x)\}$ be the corresponding collection of generators for the antifields and antighosts._ The point is that I want to write down the following suggestive symbols: _In this basis the symplectic form of which the antibracket is the Poisson bracket is locally_ $$ \omega_{BV} := \int_\Sigma \mathbf{d} \bar \Phi^a(x) \wedge \mathbf{d} \Phi_a(x) \,. $$ Here this expression is meant to be an object in the (Kähler-)differential forms on our BV-complex. If we think of the BV-complex as the [[Chevalley-Eilenberg algebra]] of a [[derived L-infinity algebroid]], then this expression lives in the corresponding [[Weil algebra]]. As such, the situation is very similar to that of [[symplectic Lie n-algebroid]]s. Contrary to that case, however, the 2-form here is not closed under the given differential -- here: the BV-differential -- and what I am after is precisely the question: _What is the image of this 2-form under the BV-differential?_ Notice that $\omega_{BV}$ is of "grade" -1 and that the BV-differential will send it therefore to a 2-form of grade 0, hence an ordinary 2-form. I am about to claim that this ordinary 2-form is $$ \omega_{in} - \omega_{out} \,, $$ where $\omega_{in}$ is the canonical presymplectic form on [[covariant phase space]] for the incoming boundary of $\Sigma$, and $\omega_{out}$ that for the outgoing boundary. So I am about to be claiming that $$ \omega_{out} = \omega_{in} + d_{BV} \omega_{BV} \,. $$ If true, this would formalize the idea that I mentioned at the beginning: $\omega_{BV}$ would be exhibited as a homotopy between the two choices for the canonical presymplectic form. In order to check this, I now appeal to the notation and results of the old but seminal article Gregg Zuckerman, _Action principles and global geometry_ ([pdf](http://ncatlab.org/nlab/files/ZuckermanVariation.pdf)). For the remainder of this message here I will assume that you either are familiar with this, or that you now spend five minutes with glancing over the first 10 pages, which is all I'll need. I think to make sense of my $\omega_{BV}$ I want to identify the symbol $\mathbf{d}$ there with the variational derivative that Zuckerman writes $\partial$ on p. 4 $$ \mathbf{d} = \partial $$ For the action of the BV-differential I observe that by definition $$ \begin{aligned} d_{BV} \int_\Sigma \bar \Phi_a(x) \wedge \mathbf{d} \Phi^a(x) &= \int_\Sigma (d_{BV} \bar \Phi_a(x)) \wedge \mathbf{d} \Phi^a(x) \\ & = \frac{\delta \int_\Sigma L}{\delta \Phi^a(x)}_{EL} \wedge \mathbf{d} \Phi^a(x) \\ & = \int_\Sigma E_a \wedge \mathbf{d} \Phi^a \\ & = \int_\Sigma E \,, \end{aligned} $$ where $E$ is Zuckerman's Euler-Lagrange form from page 9. With the equations there this is equivalently (still writing "$\mathbf{d}$" for Zuckerman's "$\partial$") $$ \cdots = \int_\Sigma \mathbf{d} L - \int_\Sigma D M \,, $$ where now on the right we have the boundary contribution $M$ that induces the canonical presymplectic form that I am after: this is defined as $$ \omega_{in} := \int_{\partial_{in} \Sigma} \mathbf{d} M $$ and similarly for $\omega_{out}$. Collecting all this together I get finally $$ \begin{aligned} d_{BV} \omega_{BV} & = d_{BV} \mathbf{d} \bar \Phi_a \wedge \mathbf{d} \Phi^a \\ & = - \mathbf{d} \left( d_{BV} \bar \Phi_a \wedge \mathbf{d}\Phi^a \right) \\ & = - \mathbf{d}\left( \int_\Sigma \mathbf{d} L - \int_\Sigma D M\right) \\ & = - \mathbf{d}\mathbf{d} \int_\Sigma L - \int_\Sigma D \mathbf{d} M \,. \end{aligned} $$ Here the first term vanishes, due to $\mathbf{d}^2 = 0$. To the second term the Stokes theorem applies, as on p. 10 of Zuckerman's article. Therefore we are left with $$ \begin{aligned} \cdots &= - \int_{\partial \Sigma} \mathbf{d} M \\ & = \int_{\partial_{out} \Sigma} \mathbf{d}M - \int_{\partial_{in} \Sigma} \mathbf{d}M \\ & = \omega_{out} - \omega_{in} \,. \end{aligned} $$ QED. ... Anyone who made it so far, please give me a sanity check.

   The BV-BRST complex of some local Lagrangian over some manifold famously comes equipped with a canonical Poisson bracket of grade +1: the antibracket.

   The whole point of this construction is that it serves as a tool for computing another bracket that one is interested in: the ordinary grade-0 bracket induced from the symplectic form that is obtained by descending the canonical presymplectic form on covariant phase space (as discussed there) to the quotient by the symmetries.

   Recently I was beginning to entertain myself with the idea that there should be a more immediate relation between these two brackets than I see considered in the usual literature. I was beginning to think that it should be possible to regard a Poisson n-algebra as a “homotopy” between two Poisson -algebras. (Does that sound reasonable to anyone?)

   So I started toying around with some formulas to see if I can substantiate my guess. I am not as sophisticated yet with these computations as Frédéric, who is reading here, and likely I am making some silly mistakes. But taken together this is are two good reasons to chat about my thoughts here!

   So for the purposes of this post here, I’ll be a bit naive, following the physics literature, and say things like:

   Let, locally, be a collection of generators for the fields and ghosts…

   as well as

   … and let be the corresponding collection of generators for the antifields and antighosts.

   The point is that I want to write down the following suggestive symbols:

   In this basis the symplectic form of which the antibracket is the Poisson bracket is locally

   Here this expression is meant to be an object in the (Kähler-)differential forms on our BV-complex. If we think of the BV-complex as the Chevalley-Eilenberg algebra of a derived L-infinity algebroid, then this expression lives in the corresponding Weil algebra. As such, the situation is very similar to that of symplectic Lie n-algebroids.

   Contrary to that case, however, the 2-form here is not closed under the given differential – here: the BV-differential – and what I am after is precisely the question:

   What is the image of this 2-form under the BV-differential?

   Notice that is of “grade” -1 and that the BV-differential will send it therefore to a 2-form of grade 0, hence an ordinary 2-form. I am about to claim that this ordinary 2-form is

   where is the canonical presymplectic form on covariant phase space for the incoming boundary of , and that for the outgoing boundary. So I am about to be claiming that

   If true, this would formalize the idea that I mentioned at the beginning: would be exhibited as a homotopy between the two choices for the canonical presymplectic form.

   In order to check this, I now appeal to the notation and results of the old but seminal article

   Gregg Zuckerman, Action principles and global geometry (pdf).

   For the remainder of this message here I will assume that you either are familiar with this, or that you now spend five minutes with glancing over the first 10 pages, which is all I’ll need.

   I think to make sense of my I want to identify the symbol there with the variational derivative that Zuckerman writes on p. 4

   For the action of the BV-differential I observe that by definition

   where is Zuckerman’s Euler-Lagrange form from page 9. With the equations there this is equivalently (still writing “” for Zuckerman’s “”)

   where now on the right we have the boundary contribution that induces the canonical presymplectic form that I am after: this is defined as

   and similarly for .

   Collecting all this together I get finally

   Here the first term vanishes, due to . To the second term the Stokes theorem applies, as on p. 10 of Zuckerman’s article. Therefore we are left with

   QED.

   …

   Anyone who made it so far, please give me a sanity check.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 21st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexIgor Khavkine kindly points out by private email that all this boils down to the following simple statement: let in the variational bicomplex, following [Zuckerman's notation](http://ncatlab.org/nlab/files/ZuckermanVariation.pdf): * $L \in \Omega^{dim X, 0}(ConfigSpace)$ be the local Lagrangian; * $M \in \Omega^{dim X-1, 1}(ConfigSpace)$ be the presymplectic potential density; * $U \in \Omega^{dim X, 2}(ConfigSpace)$ be the presymplectic density; and finally * $A \in \Omega^{dim X, 2}(BVSpace)$ be the density of the symplectic form corresponding to the anti-bracket; Then the statement is simply that. $$ d_{BV} \, A = D U \,. $$ Simple as it is, has this been considered anywhere before?

   Igor Khavkine kindly points out by private email that all this boils down to the following simple statement:

   let in the variational bicomplex, following Zuckerman’s notation:

   • be the local Lagrangian;

   • be the presymplectic potential density;

   • be the presymplectic density;

   and finally

   • be the density of the symplectic form corresponding to the anti-bracket;

   Then the statement is simply that.

   Simple as it is, has this been considered anywhere before?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 5th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added a discussion along the above lines to [[phase space]] in a new section [via BV](http://ncatlab.org/nlab/show/phase+space#ReducedCovariantPhaseSpaceViaBV). Since this still needs attention, maybe I should better put this on my personal web. For the moment I just left a cautionary remark. I think I'll move it tomorrow to a more suitable place.

   I have added a discussion along the above lines to phase space in a new section via BV.

   Since this still needs attention, maybe I should better put this on my personal web. For the moment I just left a cautionary remark. I think I’ll move it tomorrow to a more suitable place.