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July 7, 2008

Basics of Poisson Reduction and BV, I

Posted by Urs Schreiber

In our little “internal seminar” at HIM the last two times Alejandro Cabrera gave an introduction to BV-formalism and Poisson reduction. He had some useful slides

Alejandro Cabrera
Homological BV-BRST methods: from QFT to Poisson reduction

on the BV background. Then he summarized Poisson reduction as follows below. The combination of the two is the content of his next talk.

For more on symplectic reduction see for instance

J. Butterfield
On symplectic reduction in classical mechanics

Symplectic reduction is about forming quotients by group actions of symplectic manifolds.

So consider (X,ω)(X,\omega) be a symplectic manifold with symplectic 2-form ω\omega and let GG be a compact Lie group acting on XX by symplectomorphisms:

R:X×GX. R : X \times G \to X \,.

Instead of trying to directly form the quotient X/GX/G, one forms the quotient of subsets of XX obtained as follows:

Write gg for the Lie algebra of GG. For each ξg\xi \in g we have the corresponding vector field ξ XΓ(TX)\xi_X \in \Gamma(T X) along XX ξ X(x)=R(x,) *ξ. \xi_X(x) = R(x,\cdot)_* \xi \,.

With respect to ω\omega this vector field has a Hamiltonian generating function J(ξ)C (X)J(\xi) \in C^\infty(X), which means that ι ξ Xω=d(J(ξ)). \iota_{\xi_X} \omega = d (J(\xi)) \,.

This construction is well behaved in ξ\xi so that JJ is indeed a map J:Xg * J : X \to g^* from XX to the linear dual space of gg. This is the moment map of the GG-action (since it generalizes the concept of angular momentum). This map is GG-equivariant with respect to the coadjoint action of GG on its dual Lie algebra g *g^*.

The preimage J 1(0)J^{-1}(0) of 0g *0 \in g^* under JJ is a submanifold of XX

J 1(0) i X J g *, \array{ J^{-1}(0) &\stackrel{i}{\hookrightarrow}& X &\stackrel{J}{\to}& g^* } \,, but not a symplectic one. However, it still has GG acting on it. Under some conditions (00 being a regular value of JJ and GG acting freely and properly on J 1(0)J^{-1}(0)) the quotient of J 1(0)J^{-1}(0) by GG is a symplectic manifold J 1(0) i X J g * (J 1(0)/G,ω˜) \array{ J^{-1}(0) &\stackrel{i}{\hookrightarrow}& X &\stackrel{J}{\to}& g^* \\ \downarrow \\ (J^{-1}(0)/G, \tilde \omega) } called the symplectic quotient or symplectic reduction or Marsden-Weinstein quotient.

Notice how it involves in a way quotienting by GG twice. Accordingly, the dimension of the quotient is that of XX minus twice that of GG.

The same story can be told in terms of function algebras in a way that generalizes also to Poisson reduction.

So let now (P,{,})(P,\{\cdot,\cdot\}) be a manifold PP with Poisson bracket {,}\{\cdot,\cdot\} on its algebra of functions such that (C (X),{,})(C^\infty(X), \{\cdot,\cdot\}) is a Poisson algebra.

For IC (X)I \subset C^\infty(X) an algebra ideal (not necessarily a Poisson ideal), for instance the collection of functions vanishing on some submanifold CX,C \hookrightarrow X\,, we say that II is first class if it is closed under the Poisson bracket {I,I}I. \{I,I\} \subset I \,. For a Poisson ideal we would even have {I,C (X)}I\{I, C^\infty(X)\} \subset I.

The quotient algebra C (X)/I, C^\infty(X)/I \,, which plays the role of functions on CC if II is functions that vanish on CC, is not in general itself a Poisson manifold. But if we further restrict to the algebra (C (X)/I) I \left( C^\infty(X) / I \right)^{I} of those elements in there which are invariant under the Poisson action {I,}\{I, \cdot\} by II, then this does inherit an induced Poisson structure, simply because for those elements {f+I,g+I}={g,f}+I, \{ f + I , g + I\} = \{g,f\} + I \,, by assumption. This is the reduced Poisson algebra.

Notice how it is again not just a single quotient but a two-step quotient one performs. In a way II is divided out twice.

To make the connection with symplectic reduction, assume that the Poisson structure is actually symplectic and identify the submanifold CC with the level set of the moment map C:=J 1(0) C := J^{-1}(0) and notice that then {J ξ,}=ξ X, \{J_\xi , \cdot\} = \xi_X \,, where the left hand side is a derivation which we identify with the action of the vector field ξ X\xi_X. Assume that the symplectic quotient exists as a symplectic manifold.

Then the reduced Poisson algebra is the algebra of functions on the symplectic quotient:

(C (X)/I) IC (J 1(0)/G). \left( C^\infty(X)/I \right)^I \simeq C^\infty(J^{-1}(0)/G) \,.

Next time: the BV complex giving a cohomological realization of this quotient.

Posted at July 7, 2008 4:26 PM UTC

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5 Comments & 2 Trackbacks

Re: Basics of Poisson Reduction and BV, I

Curious to see the symmetries first then the moment map - I usually do it the other way around and the constraints need not form an equivariant moment map.

I would expect what’s coming next to be BFV and not BV - the latter being for the Lagrangian version with anti-fields.
Oh, F = Fradkin

Posted by: jim stasheff on July 8, 2008 1:45 AM | Permalink | Reply to this

Re: Basics of Poisson Reduction and BV, I

And, of course, symmetries do not necessarily lead to any moment maps: consider the action of a 2-torus on itself by left translations. It preserves the standard area form, but there is no moment map. And when symmetries do lead to moment maps, the moment maps need not be equivariant. This is typical for loop group actions, for example. This is probably why Jim Stasheff prefers to start with equivariant moment maps and not with symplectic group actions.

Posted by: Eugene Lerman on July 10, 2008 1:49 AM | Permalink | Reply to this

Re: Basics of Poisson Reduction and BV, I

Jim, Eugene,

thanks for emphasizing this point.

I think the idea here was that operationally we want to start with a group action on a Poisson manifold and then ask if we can form the quotient in the world of Poisson manifolds. For that to work, a bunch of assumptions has to hold, one of them being that the group action comes from an equivariant moment map.

There are more conditions, right? One of them being that 0 is a regular value of the moment map and that GG acts freely and properly on J 1(0)J^{-1}(0).

Posted by: Urs Schreiber on July 10, 2008 12:39 PM | Permalink | Reply to this

Re: Basics of Poisson Reduction and BV, I

Eugene and Urs,
As usual, I was too telegraphic. I meant I, following BFV, start with a set of first class constraints, e.g. a map from the symplectic (or Poisson) manifold W to R^n.
I shouldn’t have referred to it as a moment map - perhaps momap? In the symplectic case,the constraints generate a Lie algebra, hence infinitesimal symmetries - no need to have group. The action of the Lie algebra on the constraint surface V gives a foliation and we are trying to reduce to the spac eof leaves of the foliation.

A major point of BFV is tht we do NOT need regular value 0 nor that the map from V to the space of leaves be a bundle. BFV compute H^0 asthe `space of funcitons’ on the homotopy quotient. The higher H^i have a geometric interprettion as de Rham along the leaves.

Posted by: jim stasheff on July 10, 2008 12:59 PM | Permalink | Reply to this

Re: Basics of Poisson Reduction and BV, I

Okay, so we need to distinguish here between the symplectic reduction, the Poisson reduction, and its further generalizations. The BFV stuff is supposed to be discussed next time.

Posted by: Urs Schreiber on July 10, 2008 1:05 PM | Permalink | Reply to this
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