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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 $L$ 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-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 $\omega_{BV}$ 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).

   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.

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:

   • $L \in \Omega^{dim X, 0}(ConfigSpace)$ 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?

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.