Generalising some constructions of Poisson geomety to higher Poisson geometry

[ajb]

I would like to draw your attention to a preprint of mine, "On higher Poisson and Koszul--Schouten brackets", arxiv:math-ph/0910.1992.

 

Abstract

In this note we show how to construct a homotopy BV-algebra on the algebra of differential forms over a higher Poisson manifold. The Lie derivative along the higher Poisson structure provides the generating operator.

 

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In the preprint I generalise the construction of Koszul--Schouten brackets found in Poisson geometry to higher Poisson geometry [1].

 

Recall that a Poisson manifold is a manifold equipped with a bi-vector field that satisfies the condition that it Schouten--Nijenhuis self commutes. Associated with this structure is a Poisson algebra on the algebra of smooth functions over the manifold.

 

This is well-known and forms the mathematical basis of classical mechanics.

 

What is less well-known is that the algebra of differential forms also comes equipped with a (Grassmann odd) Poisson bracket called the Kozul--Schouten bracket. This bracket does not exactly contain the Poisson bracket, so it should not be thought of as a generalisation to differential forms. Instead, you should view it as another bracket that is built from the same information.

 

One question you can ask is "are we constrained to consider just bi-vectors?"

 

It turns out that we can consider arbitrary degree, but Grassmann even multivector fields over supermanifolds that Schouten--Nijenhuis self commute. Or in more physical language, satisfy the classical master equation.

 

The initial work in this direction comes from several sources, but the closest to what I have used comes from Voronov [2] in his discussion of higher derived brackets.

 

The interesting point is that one can generalise the notion of a Poisson algebra to a homotopy Poisson algebra. That is we leave the theory of Poisson-Lie brackets and enter the world of Sh Lie algebras. Instead of just a binary bracket one now has a series of brackets that take 0,1,2,3... arguments, generally all the way up to infinity.

 

These brackets satisfy a Leibniz rule and form a higher analogue of a Poisson algebra.

 

The question answered in the preprint is "can we generalise the Koszul--Schouten bracket?"

 

The answer is yes, but there appears to be two ways to do it. I initially take my cue from the BV-formulism and construct a series of brackets as brackets generated by an (odd) operator. The operator is the generalised Lie derivative along the higher Poisson structure.

 

This series of brackets forms a homotopy BV-algebra. There series of brackets do not satisfy the Leibniz rule and are not an odd analogue of a higher Poisson algebra. (I call then higher Schouten algebras).

 

I have shown how to "de-quantise" the higher Koszul--Schouten brackets to obtain higher Schouten brackets as first obtained in [3].

 

There are plenty of open question here. Two obvious one are

 

i) Can we use higher Poisson brackets to describe dynamics?

ii) Can one quantise the higher Poisson brackets?

 

Question ii) I doubt that one can apply known methods of quantisation. i) Possibly, but it would require a series of Hamiltonians.

 

I welcome any questions and comments.

 

 

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[1]J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in Élie Cartan et les mathématiques d'aujourd'hui, Astérisque, hors série, Soc. Math. Fr., 1985, p. 257-271

 

[2]Th. Th. Voronov. Higher derived brackets and homotopy algebras. J. Pure and Appl. Algebra, 202, 2005.

 

[3]H.M. Khudaverdian and Th. Th. Voronov. Higher Poisson Brackets and Differential Forms, 2008. arXiv:math-ph/0808.3406v2.

[self drive]

Hi your article is very good

Thanks for posting

We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manifold, and show how it can be obtained via the Hamiltonian reduction from a symplectic structure on the diffeomorphism group.

[ajb]

That sounds interesting.

 

You are considering the algebra of densities of arbitrary weights?

 

I ask this as one would want an algebra of densities and not just a vector space of densities of a given weight.

 

Do you require a symplectic structure or will a (degenerate) Poisson structure do?

 

 

Could you provide a link to an article on your work? Is this it? arXiv:0802.4439v1 [math.DG]

 

A friend of mine has considered how to extend the "antibracket" to densities, arXiv:0909.5411v1 [math.DG]. I should really get round to reading it carefully.

 

It has crossed my mind if one can extend Poisson and Schouten brackets to densities. It maybe something to work on in the near future.

Edited November 23, 2009 by ajb

[rozerdemit]

Wow really nice article

thanks for sharing this article with us

[ajb]

In the preprint I did not stress the fact that mod ghost grading and possible difficulties due to the infinite dimensionality that a higher Poisson structure can be found lying behind the classical BV formalism. (I do this in my thesis.)

 

Thus, we have a series of higher Koszul--Schouten and higher Schouten brackets on the space of field and ghost valued (functional) differential forms. In short there is an [math]L_{\infty}[/math]-algebra structure on the differential forms over the extended configuration space.

 

I am wondering if this can be useful in discussing gauge symmetries and quantum anomalies.

 

Anyway, a question to those familiar with quantum field theory. Do you know of any discussion of the role of (functional) differential forms in field theory? (I do not mean differential form valued fields like electromagnetism or Yang-Mills theory.) I am struggling to find much.

 

Any ideas and suggestions welcomed.

 

Cheers.

[jonnybleyr]

its interesting one while reading, but feeling some confusion...

[ajb]

its interesting one while reading, but feeling some confusion...

 

I am more than happy to listen to comments and take questions.

 

(It is unlikely that I will place any further versions of this work on the arXiv as I have a more general paper on similar things under consideration for publication)