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symplectic Lie n-algebroid

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\infty-Lie theory

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Symplectic geometry

Contents

Idea

A Lie n-algebroid is symplectic if it is equipped with a non-degenerate binary invariant polynomial. This generalizes the notion of a symplectic form on a symplectic manifold, to which it reduces for n=0n = 0.

Definition

Definition

A symplectic Lie nn-algebroid is a pair

(𝔞,,) (\mathfrak{a}, \langle -,- \rangle)

consisting of

Properties

The Chern-Simons element that witnesses this transgression is the Lagrangian of the corresponding AKSZ theory sigma-model with 𝔞\mathfrak{a} as its target space and the invariant polynomial ,\langle -,- \rangle as the (curvature of) its background gauge field.

Examples

n=0n=0: symplectic manifold

A 0-Lie algebroid is just a smooth manifold XX.

Its Chevalley-Eilenberg algebra is the algebra of smooth functions on XX

CE(X)=C (X). CE(X) = C^\infty(X) \,.

The Weil algebra of XX is

W(X)=Ω (X) W(X) = \Omega^\bullet(X)

the de Rham algebra of XX. A degree 2-invariant polynomial on XX is therefore a non-degenerate closed 2-form ωΩ 2(X)\omega \in \Omega^2(X), a symplectic 2-form.

A symplectic manifold, being a pair

(X,ω) (X,\;\; \omega)

consisting of a smooth manifold XX and a symplectic 2-form ω\omega, is a symplectic Lie 0-algebroid.

n=1n=1: Poisson manifold

For a Poisson manifold XX with Poisson bivector πΓ(TX)Γ(TX)\pi \in \Gamma(T X) \wedge \Gamma(T X) the Chevalley-Eilenberg algebra CE(𝔞)CE(\mathfrak{a}) of the corresponding Poisson Lie algebroid

𝔞:=𝔓(X,π) \mathfrak{a} := \mathfrak{P}(X,\pi)

is that of multi-vector fields on XX, equipped with the differential d CE(𝔞)=[π,] Schd_{CE(\mathfrak{a})} = [\pi, -]_{Sch} given by the Schouten bracket.

If we work locally in coordinates then CE(𝔞)CE(\mathfrak{a}) is generated from degree 0 elements x ix^i and degree 1 elements i\partial_i. The differential is

d CE(𝔞)=[π,] Sch. d_{CE(\mathfrak{a})} = [\pi, -]_{Sch} \,.

The Poisson tensor is ν:=π=π ij i j\nu := \pi = \pi^{i j} \partial_i \wedge \partial_j and that this is a Lie algebroid cocycle is the fact that

d CE(𝔞)π=[π,π] Sch=0. d_{CE(\mathfrak{a})} \pi = [\pi,\pi]_{Sch} = 0 \,.

By definition the Weil algebra W(𝔞)W(\mathfrak{a}) is generated from the x ix^i, the i\partial_i and their shifted partners dx i\mathbf{d}x^i and d i\mathbf{d}\partial_i. The differential here is

d W(𝔞)=[π,]+d. d_{W(\mathfrak{a})} = [\pi , - ] + \mathbf{d} \,.
Proposition

The invariant polynomial ω\omega that is in transgression with the cocycle ν=π\nu = \pi is

ω=(dx i)(d i)inv(𝔞). \omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i) \;\;\; \in inv(\mathfrak{a}) \,.
Proof

One checks directly that the element

cs ω=π ij i j+x id i cs_\omega = \pi^{i j} \partial_i \wedge \partial_j + x^i \wedge \mathbf{d} \partial_i

is a Chern-Simons transgression element for ν\nu and ω\omega,

i.e. d W(𝔞)cs(ω)=ωd_{W(\mathfrak{a})} cs(\omega) = \omega. The restriction of cs ωcs_\omega to CE(𝔞)CE(\mathfrak{a}) is evidently the Poisson tensor π\pi.

More details on this at Chern-Simons element.

For a Poisson manifold XX with Poisson tensor π=π ij i j\pi = \pi^{i j} \partial_i \wedge \partial_j, the pair

(𝔓(X,π),ω=(dx i)(d i)) (\mathfrak{P}(X,\pi), \;\;\; \omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i))

consisting of the Poisson Lie algebroid 𝔓(X,π)\mathfrak{P}(X,\pi) and of the invariant polynomial ω\omega that is in transgression with its canonical 2-cocycle ν=π\nu = \pi (the Poisson tensor) is a symplectic Lie algebroid.

n=2n=2: Courant algebroid

A 22-symplectic manifold encodes and is encoded by the structure of a Courant algebroid.

A Courant 2algebroid over the point if given by a semisimple Lie algebra with the symplectic form being the Killing form. The coresponding Poisson tensor is the canonical 3-cocycle ,[,]\langle -, [-,-] \rangle on a semisimple Lie algebra. The extension classified by this is the string Lie 2-algebra.

Relation to other concepts

To \infty-Chern-Simons theory

Since the symplectic form on a symplectic Lie nn-Algebroid may be understood Lie theoretically as an invariant polynomial on an L-∞ algebroid, every symplectic Lie nn-algebroid serves as a target space for an ∞-Chern-Simons theory: this is AKSZ theory.

We have

To multisymplectic geometry

There is also the closely related notion of multisymplectic geometry. See

for some relations of this to the above situation for n=2n = 2. Essentially multisymplectic geometry studies the higher nn-ary brackets induced from the binary graded symplectic form discussed here. The relation between these two pictures is the same as that between as studied in the context of hemistrict Lie 2-algebra?s.

An article with more details on this:

  • Chris Rogers, Courant algebroids from categorified symplectic geometry (pdf).

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)

References

The notion originates somewhere in the school of Alan Weinstein’s school of higher categorial symplectic geometry. The first published appearance of the notion at least for 0n30 \leq n \leq 3 is

  • Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv)

A good writeup of this material is in

  • Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids in Quantization, Poisson Brackets and Beyond , Theodore Voronov (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 (arXiv)

The idea for all nn was then sketched, together with many other ideas about L-infinity algebroids in the article with the nice title

  • Pavol Ševera, Some title containing the words “homotopy” and “symplectic”, e.g. this one (arXiv)

What we call nn-symplectic manifold here is called Σ n\Sigma_n-manifold there.

Warning This article here uses the term ”nn-symplectic” in a related but not identical sense to the one used here:

  • M. de Leon, D. Martin de Diego, M. Salgado, S. Vilariño, K-symplectic formalism on Lie algebroids (arXiv)

A discussion of aspects of how multisymplectic geometry related to nn-symplectic manifolds is in

  • Chris Rogers, Courant algebroids from categorified symplectic geometry (pdf)

arXiv:1001.0040v1 [math-ph]

A discussion of symplectic Lie n-algebroids from an infinity-Lie theory perspective as discussed here is in

Revised on February 8, 2013 02:02:14 by Urs Schreiber (89.204.155.59)