### 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*

(pdf)

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*

pdf.

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

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

$R : X \times G \to X \,.$

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

Write $g$ for the Lie algebra of $G$. For each $\xi \in g$ we have the corresponding vector field $\xi_X \in \Gamma(T X)$ along $X$ $\xi_X(x) = R(x,\cdot)_* \xi \,.$

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

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

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

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

Notice how it involves in a way quotienting by $G$ *twice*. Accordingly, the dimension of the quotient is that of $X$ minus twice that of $G$.

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

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

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

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

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

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

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

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

## 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