Higher Contact Geometry and L Algebras · Introduction: Symplectic Geometry A symplectic manifold...

Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds

Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry

Higher Contact Geometry and L∞ Algebras

Luca Vitagliano

University of Salerno, Italy

IMPAN, Warsaw, May 14, 2014

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 1 / 36



Introduction: Symplectic Geometry

A symplectic manifold is a manifold M equipped with a closed, non-degenerate differential 2-form ω.

The original motivation for symplectic geometry comes from analyticalmechanics: the phase space of many classical systems is a symplecticmanifold! Actually, symplectic geometry pervades both differentialgeometry and mathematical physics: Hamiltonian systems, Poisson ge-ometry, Lie algebroids, Courant algebroids, Kahler geometry, etc..

RemarkOne can attach an algebraic structure to any symplectic manifold(M, ω), namely a Poisson bracket −,− on the algebra C∞(M). ThePoisson bracket −,− plays a key role in numerous contexts: integra-bility of Hamiltonian systems, action of Lie groups on symplectic manifolds /moment maps, geometric quantization, etc.




Introduction: Contact Geometry

A contact manifold is a manifold M equipped with a maximally non-integrable, hyperplane distribution C.The original motivation for contact geometry comes from first order scalarPDEs: the first jet space of hypersurfaces is a contact manifold! Ac-cordingly, contact geometry has numerous applications both in dif-ferential geometry and mathematical physics: Jet spaces, control theory,geometric quantization, geometric optics, thermodynamics, etc..

Remark

One can attach an algebraic structure to any contact manifold (M, C),namely a Jacobi bracket −,− on the module Γ(TM/C). The Jacobibracket −,− plays a key role in various contexts: symmetries ofPDEs, integration by characteristics, etc.

Contact geometry can be seen as a part of symplectic geometry andthe Jacobi bracket can be derived from a suitable Poisson bracket!




Introduction: Higher Symplectic Geometry

A multisymplectic manifold is a manifold M equipped with a closed,non-degenerate, higher degree differential form ω.

The original motivation for multisymplectic geometry comes from clas-sical field theory: the phase space of many field theories is a multisym-plectic manifold!

RemarkC. Rogers and M. Zambon showed that, similarly as for symplecticmanifolds, one can attach an algebraic structure to any multisymplec-tic manifold (M, ω), namely an L∞ algebra g(M, ω), which plays asimilar role as the Poisson algebra of a symplectic manifold: action ofLie groups on multisymplectic manifolds / homotopy moment maps, geomet-ric (pre-)quantization of field theories.




Introduction: Higher Contact Geometry

A multicontact manifold is a manifold M equipped with a maximallynon-integrable distribution C of higher codimension.

The motivation for multicontact geometry comes from the geometry ofPDEs: finite jet spaces are multicontact manifolds!

RemarkSimilarly as for contact manifolds, one can attach an algebraic struc-ture to any multicontact manifold (M, C), namely an L∞ algebrag(M, C), which plays a similar role as the Jacobi line bundle of a con-tact manifold: concrete applications are still to be explored!.

Multicontact geometry can be seen as a part of multisymplectic geom-etry and the “multicontact” L∞ algebra can be derived from a suitable“multisymplectic” L∞ algebra!




Introduction: Homotopy Algebras

Let A be a type of algebra (associative, Lie. etc.). A homotopy A alge-bra structure on a chain complex is a set of operations that satisfy theaxioms of A only up to homotopy (in fact, a coherent system of higherhomotopies).

RemarkHomotopy algebras appear as a consequence of the interaction be-tween algebraic structures and homology/homotopy. For instance,homotopy Lie algebras often govern formal deformation problems of al-gebraic/geometric structures.

Remark

Homotopy algebras do also appear in geometry as higher versions ofstandard algebras. For instance

symplectic : multisymplectic = Lie algebra : homotopy Lie algebra




Outline

1 Symplectic and Contact Manifolds

2 Multisymplectic and Multicontact Manifolds

3 Strong Homotopy Lie Algebras

4 L∞ Algebras from Multisymplectic and Multicontact Geometry




Outline








Symplectic Manifolds

Let M be a smooth manifold.

Definition

A symplectic structure on M is a non-degenerate 2-form ω such thatdω = 0. The pair (M, ω) is a symplectic manifold.

Example

Let N be a manifold. M := T∗N possesses a tautological 1-form θ:

θp(ξ) := p(π∗(ξ)), ξ ∈ Tp M (π : M −→ N).

ω := dθ is a canonical symplectic structure on M.

This example plays a distinguished role in classical mechanics!




The Poisson Algebra of a Symplectic Manifold

Let (M, ω) be a symplectic manifold. There is a binary bracket −,−in C∞(M) defined as follows:

X(M) −→ Ω1(M), X 7−→ ω(X,−)

possesses an inverse

Ω1(M) −→ X(M), d f 7−→ X f .

Then f , g := ω(X f , Xg).

Remark

(M, −,−) is a Poisson manifold, i.e.−,− is a Lie bracket,−,− is a derivation in each argument.




Distributions on Manifolds


Definition

A distribution on M is a vector subbundle D ⊂ TM. The curvature formof D is the following vector bundle valued 2-form on D:

ωD : Γ(D)× Γ(D) −→ Γ(TM/D), (X, Y) 7−→ [X, Y] mod D

RemarkD is integrable iff ωD = 0. More generally, ker ωD consists of vectorfields in D whose flows preserve D, i.e. characteristic symmetries.

Proposition

The characteristic distribution KD := ker ωD is integrable.




Contact Manifolds


DefinitionA contact structure on M is an hyperplane distribution C, such thatωC is non-degenerate. The pair (M, C) is a contact manifold. The line-bundle L := TM/C is the Jacobi bundle of (M, C).

Example

Let N be a manifold, dim N = n, and M := Gr(TN, n− 1). There isa canonical line bundle L → M, whose fiber at y ∈ Gr(Tx N, n− 1) isLy := Tx N/y. Moreover, M possesses a tautological L-valued 1-form ϑ:

ϑy(ξ) := π∗(ξ) mod y, ξ ∈ Ty M (π : M −→ N).

ker ϑ is a canonical contact structure on M.

This example plays a distinguished role for first order scalar PDEs!




The Kirillov Algebroid of a Contact Manifold

Let (M, C) be a contact manifold. There is a binary bracket −,− inΓ(L) defined as follows:

0 −→ Γ(C) −→ X(M) −→ Γ(L) −→ 0

possesses a canonical splitting

Γ(L) −→ X(M), λ 7−→ Xλ.

Then λ, µ := [Xλ, Xµ] mod C.

Remark

(L, −,−) is a Kirillov algebroid, i.e.−,− is a Lie bracket,−,− is a first order differential operator in each argument.




Symplectization of a Contact Manifold

Contact geometry is actually part of symplectic geometry!

Let (M, C) be a contact manifold and L → M its Jacobi bundle. L∗

identifies with Ann C ⊂ T∗M. Put (M, ω) := (L∗ r 0, ω|L∗r0).

Remark

M is a principal R×-bundle over M. Let ∆ be the Euler vector field.

Proposition

(M, ω) is a symplectic manifold. More precisely, it is a symplectic principalR×-bundle over M. In particular, L∆ω = ω.

(M, ω) is the symplectization of (M, C)!

Example

The symplectization of Gr(N, n− 1) is T∗N r 0.




Jacobi Brackets from Poisson Brackets

Let (M, C) be a contact manifold. The Jacobi bracket −,− on L canbe reconstructed from the Poisson bracket −,−M on C∞(M)!

Remark

There is an obvious bijection λ 7→ λ between sections of L and fiber-wise homogeneous functions λ on M = L∗ r 0, i.e. functions λ suchthat L∆λ = λ.

Proposition

The Jacobi bracket −,− is the restriction of the Poisson bracket −,−Mto fiber-wise homogeneous functions, namely

λ, µ = λ, µM, λ, µ ∈ Γ(L).




Outline








Multisymplectic Manifolds


Definition

An m-plectic structure on M is a non-degenerate (m + 1)-form ω, suchthat dω = 0. The pair (M, ω) is an m-plectic manifold.

Remark1-plectic manifolds are symplectic manifolds!

Example

Let N be a manifold. M := ∧mT∗N possesses a tautological m-form θ:

θp(ξ1, . . . , ξm) := p(π∗(ξ1), . . . , π∗(ξm)), ξi ∈ Tp M (π : M −→ N).

ω := dθ is a canonical m-plectic structure on M.

m-plectic geometry underlies field theory in m space-time dimensions!




Pre-Multisymplectic Manifolds and Their Reduction

More generally, consider the following

Definition

A pre-m-plectic structure on M is a (possibly degenerate) (m + 1)-formω, such that dω = 0. Vector fields X such that iXω = 0 span thecharacteristic distribution Kω of ω. The pair (M, ω) is a pre-m-plecticmanifold.

RemarkThe characteristic distribution is integrable.

Proposition

Suppose that the leaves of Kω form a smooth manifold M and the projectionπ : M → M is a fibration. Then there is a unique m-plectic structure ω onM such that ω = π∗(ω).




Multicontact Manifolds


DefinitionAn m-contact structure on M is an m-codimensional distribution C,such that ωC is non-degenerate. The pair (M, C) is an m-contact mani-fold.

Example

Let N be a manifold, dim N = n, and M := Gr(TN, n− m). There isa canonical m-dimensional vector bundle V → M, whose fiber at y ∈Gr(Tx N, n−m) is Vy := Tx M/y. Moreover, M possesses a tautologicalV-valued 1-form ϑ:

ϑy(ξ) := π∗(ξ) mod y, ξ ∈ Ty M (π : M −→ N).

ker ϑ is a canonical m-contact structure on M. More generally, higher jetspaces with their Cartan distribution are multicontact manifolds.

These examples play a distinguished role for PDEs!Luca Vitagliano Higher Contact Geometry and L∞ Algebras 19 / 36



Pre-Multicontact Manifolds and Their Reduction

More generally, consider the following

DefinitionA pre-m-contact structure on M is an m-codimensional distribution C(possibly possessing characteristic symmetries). The pair (M, C) is apre-m-contact manifold.

RemarkDistributions are ubiquitous in Differential Geometry: a system of(non-linear) PDEs can be interpreted geometrically as a manifold witha distribution. Thus, the above definition is extremely general!

Proposition

Suppose that the leaves of KC form a smooth manifold M and the projectionπ : M→ M is a fibration. Then C := π∗(C) is an m-contact structure.




(Pre-)Multisymplectization of a (Pre-)Multicontact Manifold

Multicontact geometry is actually part of multisymplectic geometry!

Let (M, C) be a pre-m-contact manifold and L := ∧m(TM/C). Then

L∗ ' Annm C := α ∈ ∧mT∗M : iξ α = 0 ∀ξ ∈ C ⊂ ∧mT∗M.

Put (M, ω) := (L∗ r 0, ω|L∗r0).

Remark

M is a principal R×-bundle over M. Moreover, M possesses a principalflat KC-connection K.

Proposition

(M, ω) is a pre-m-plectic manifold with Kω = K. More precisely, it is apre-m-plectic principal R×-bundle over M. In particular, L∆ω = ω.

(M, ω) is the pre-m-plectization of (M, C)!




Outline








Homotopy Algebras

Consider a chain complex of vector spaces (V, δ) and letA be an alge-braic structure (Lie, associative algebra, etc.).

Rough Definition

A homotopy A-structure in (V, δ) is a set of operations in (V, δ) which1 is compatible with δ,2 is of the type A only up to coherent (higher) homotopies.

Rough Motivation

Let (A, d) be a differential algebra of typeA and f : (A, d) (V, δ) : ga pair of homotopy equivalences. The algebra structure in A can betransferred to V along ( f , g), but the transferred structure is of thetype A only up to higher homotopies. On the other hand

homotopy algebras are homotopy invariant!




Strong Homotopy Lie Algebras

Let V be a graded vector space.

Definition

An L∞ algebra structure in V is a family of degree k − 2 operations[−, . . . ,−]k : V∧k → V, such that

∑i+j=k

∑σ∈Si,j

±[[xσ(1), . . . , xσ(i)], xσ(i+1), . . . , xσ(i+j)] = 0,

for all x1, . . . , xk ∈ V, k ∈N.

Put [−] = δ.k = 1 δ2(x) = 0k = 2 δ[x, y] = [δx, y]± [x, δy]k = 3 [x, [y, z]]± [y, [z, x]]± [z, [x, y]] =

−δ[x, y, z]− [δx, y, z]± [x, δy, z]± [x, y, δz]

H(V, δ) is a genuine Lie algebra!




L∞ Morphisms and L∞ Quasi-Isomorphisms

RemarkL∞ algebras build up a category: a morphism between L∞ algebrasV, W, or L∞ morphism, is a family of degree k− 1 maps fk : V∧k → Wsatisfying suitable coherence conditions.

Definition

An L∞ morphism fk is an L∞ quasi-isomorphism if f1 induce an iso-morphism in homology.

Finally, there is a notion of homotopy between L∞ morphism. The homo-topy category of L∞ algebras is then obtained by identifying homotopicL∞ morphisms. L∞ quasi-isomorphisms are isomorphisms in the homotopycategory.




Homotopy Transfer and Massey Products

L∞ algebra structures can be transferred along contraction data. Namely,let (K, δ) be a chain complex and H := H(K, δ).

Remark

There are always contraction data, i.e. chain maps p, j and an h:

(K, δ)h%% p // (H, 0)

joo ,

such that [h, δ] = id− jp, and pj = id.

Homotopy Transfer Theorem

Let (K, δ, [−,−], . . .) be an L∞ algebra, and (p, j, h) contraction data. Then(H, 0) can be prolonged to an L∞ algebra h and j can be prolonged to an L∞quasi-isomorphism, both defined in terms of the contraction data.

h contains a full information about the homotopy type of K.




Outline








Hamiltonian Fields/Forms on Multisymplectic Manifolds

Let (M, ω) be a pre-m-plectic manifold.

Definition

An (m− 1)-form σ on (M, ω) is Hamiltonian if iXσ ω = −dσ for somevector field Xσ called Hamiltonian.

RemarkIn the 1-plectic case, every 0-form is Hamiltonian.

Remark

(σ, τ) 7−→ −iXσ iXτ ω

is a well-defined, skew-symmetric bracket on Hamiltonian forms.However, it is not a Lie bracket, in general. In the 1-plectic case, it isprecisely the Poisson bracket.




The L∞ Algebra of a Pre-Multisymplectic Manifold

Let (M, ω) be a pre-m-plectic manifold and Ωm−1Ham(M, ω) Hamiltonian

forms on it. Consider the truncated de Rham complex g(M, ω):

0←− Ωm−1Ham(M, ω)

d←− Ωm−2(M)←− · · · d←− C∞(M)←− 0.

Remark

g(M, ω) is a resolution of infinitesimal symmetries of the reduction(M, ω) up to topological obstructions.

Theorem [C. Rogers, M. Zambon]

g(M, ω) is an L∞ algebra concentrated in degrees 0, . . . , n− 1, with

[σ1, . . . , σk] =

−(−)kiXσ1

· · · iXσkω if σ1, . . . , σk ∈ Ωm−1

Ham(M, ω)

0 otherwise.




Homogeneous de Rham Complex of a Principal R×-bundle

Let (M, C) be a pre-m-contact manifold and (M, ω) its pre-m-plectiza-tion. Recall that M is a principal R×-bundle over M.

Definition

A differential form σ on a principal R×-bundle M is homogeneous ifL∆σ = σ, with ∆ the Euler vector field.

Let Ω(M) be homogenous forms on M.

Remark

The de Rham differential preserves Ω(M).

Proposition

The homogeneous de Rham complex (Ω(M), d) is acyclic.




Hamiltonian Fields/Forms on Multicontact Manifolds

Let (M, C) be a pre-m-contact manifold and (M, ω) its pre-m-plectiza-tion. In the 1-contact case, sections of the Jacobi bundle L are homogenousfunctions on M. This suggests the following

Definition

An homogeneous (m− 1)-form σ on (M, ω) is C-Hamiltonian if iXσ ω =−dσ for some projectable vector field Xσ called C-Hamiltonian.

RemarkIn the 1-contact case, all sections of L are C-Hamiltonian.

Remark

(σ, τ) 7−→ −iXσ iXτ ω

is a well-defined, skew-symmetric bracket on C-Hamiltonian forms.However, it is not a Lie bracket, in general. In the 1-contact case, it isprecisely the Jacobi bracket.




The L∞ Algebra of a Pre-Multicontact Manifold

Let (M, C) be a pre-m-contact manifold and Ωm−1,Ham(M, C) C-Hamiltonian

forms on M. Consider the truncated homogeneous de Rham complex g(M, C):

0←− Ωm−1,Ham(M, C) d←− Ωm−2

(M)←− · · · d←− C∞ (M)←− 0.

Remark

Independently of topology, g(M, C) is a resolution of infinitesimal symme-tries of the reduction (M, C).

Theorem [L. V.]

g(M, C) is an L∞ algebra concentrated in degrees 0, . . . , n− 1, with

[σ1, . . . , σk] =

−(−)kiXσ1

· · · iXσkω if σ1, . . . , σk ∈ Ωm−1

,Ham(M, C)0 otherwise

.




Conclusions

C. Rogers and M. Zambon proved that there is an L∞ algebra g(M, ω)attached to every pre-multisymplectic manifold (M, ω). g(M, ω) isan higher analogue of the Poisson structure of a symplectic mani-fold. g(M, ω) has a contact analogue. Namely, one can define a pre-multicontact manifold as a manifold with a distribution. Then

Every pre-multicontact manifold (M, C) can be prolonged to apre-multisymplectic manifold (M, ω). (M, ω) is an higher ana-logue of the symplectization of a contact manifold.There is an L∞ algebra g(M, C) attached to every pre-multicontactmanifold (M, C). g(M, C) is an higher analogue of the Jacobistructure of a contact manifold.

RemarkA system of PDEs is geometrically a pre-multicontact manifold. Oneconcludes that there is an L∞ algebra attached to every system of PDEs!




Perspectives

Remark

The multisymplectic and multicontact cases are different: g(M, ω) does notresolve infinitesimal symmetries of (M, ω) in general, and, from thehomotopic point of view, in view of the homotopy transfer theorem, con-tains more information then them. On the other hand, g(M, C) re-solves infinitesimal symmetries and contains no new information.

However, a pre-multicontact manifold (M, C) can be also understoodas an exterior differential system. As such, it can be infinitely pro-longed. There is an other L∞ algebra g∞ encoding both infinitesimalsymmetries and formal deformations of the prolongation. It would benice to explore whether or not g(M, C) and g∞ can be made interact-ing and whether or not one can extract new information about (M, C)from this interaction.




References

C. Rogers, L∞-algebras from multisymplectic geometry, Lett. Math. Phys.100 (2012) 29–50; e-print: arXiv:1005.2230.

M. Zambon, L∞-algebras and higher analogues of Dirac structures andCourant algebroids, J. Symplectic Geom. 10 (2012) 1–37; e-print: arXiv:1003.1004.

Y. Fregier, C. Rogers, and M. Zambon, Homotopy moment maps; e-print:arXiv:1304.2051.

D. Fiorenza, C. Rogers, and U Schreiber, Higher geometric prequantumtheory; e-print: arXiv:1304.0236.

D. Fiorenza, C. Rogers, and U Schreiber, L∞-algebras of local observablesfrom higher prequantum bundles; e-print: arXiv:1304.6292.

L. V., L∞-algebras from multicontact geometry; e-print: arXiv:1311.2751.




Thank you!


Higher Contact Geometry and L Algebras · Introduction: Symplectic Geometry A symplectic manifold...

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