Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ 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 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Introduction: Symplectic Geometry

A symplectic manifold is a manifold M equipped with a closed, non- degenerate differential2-form ω. The original motivation for symplectic geometry comes from analytical mechanics: the phase space of many classical systems is a symplectic manifold! Actually, symplectic geometry pervades both differential geometry and mathematical physics: Hamiltonian systems, Poisson ge- ometry, Lie algebroids, Courant algebroids, K¨ahlergeometry, etc..

Remark One can attach an algebraic structure to any symplectic manifold (M, ω), namely a Poisson bracket {−, −} on the algebra C∞(M). The Poisson 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.

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 2 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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 scalar PDEs: 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 Jacobi bracket {−, −} plays a key role in various contexts: symmetries of PDEs, integration by characteristics, etc.

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

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 3 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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!

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

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 4 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Introduction: Higher Contact Geometry

A multicontact manifold is a manifold M equipped with a maximally non-integrable distribution C of higher codimension. The motivation for multicontact geometry comes from the geometry of PDEs: finite jet spaces are multicontact manifolds!

Remark Similarly as for contact manifolds, one can attach an algebraic struc- ture to any multicontact manifold (M, C), namely an L∞ algebra g(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!

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 5 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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 the axioms of A only up to homotopy (in fact, a coherent system of higher homotopies).

Remark Homotopy 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 of standard algebras. For instance symplectic : multisymplectic = Lie algebra : homotopy Lie algebra

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 6 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Outline

1 Symplectic and Contact Manifolds

2 Multisymplectic and Multicontact Manifolds

3 Strong Homotopy Lie Algebras

4 L∞ Algebras from Multisymplectic and Multicontact Geometry

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 7 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Outline

1 Symplectic and Contact Manifolds

2 Multisymplectic and Multicontact Manifolds

3 Strong Homotopy Lie Algebras

4 L∞ Algebras from Multisymplectic and Multicontact Geometry

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 8 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Symplectic Manifolds

Let M be a smooth manifold. Definition A symplectic structure on M is a non-degenerate2-form ω such that dω = 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!

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 9 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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

1 Ω (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.

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 10 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Distributions on Manifolds

Let M be a smooth manifold. Definition A distribution on M is a vector subbundle D ⊂ TM. The curvature form of D is the following vector bundle valued2-form on D:

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

Remark

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

Proposition

The characteristic distribution KD := ker ωD is integrable.

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 11 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Contact Manifolds

Let M be a smooth manifold. Definition A contact structure on M is an hyperplane distribution C, such that ωC 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 is a canonical line bundle L → M, whose fiber at y ∈ Gr(Tx N, n − 1) is Ly := Tx N/y. Moreover, M possesses a tautologicalL -valued1-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!

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 12 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry The Kirillov Algebroid of a Contact Manifold

Let (M, C) be a contact manifold. There is a binary bracket {−, −} in Γ(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.

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 13 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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∗ ∗ ∗ 0 ∗ identifies with Ann C ⊂ T M. Put (Me , ωe) := (L r , ω|L r0). Remark Me is a principal R×-bundle overM . Let ∆ be the Euler vector field. Proposition

(Me , ωe) is a symplectic manifold. More precisely, it is a symplectic principal × R -bundle overM. In particular,L ∆ωe = ωe.

(Me , ωe) is the symplectization of (M, C)! Example ∗ The symplectization of Gr(N, n − 1) is T N r 0.

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 14 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Jacobi Brackets from Poisson Brackets

Let (M, C) be a contact manifold. The Jacobi bracket {−, −} on L can {− −} C∞(M) be reconstructed from the Poisson bracket , Me on e ! Remark There is an obvious bijection λ 7→ eλ between sections of L and fiber- ∗ wise homogeneous functions eλ on Me = L r 0, i.e. functions eλ such that L∆eλ = eλ.

Proposition The Jacobi bracket {− −} is the restriction of the Poisson bracket {− −} , , Me to fiber-wise homogeneous functions, namely

{^λ µ} = {λ µ} λ µ ∈ (L) , e, e Me , , Γ .

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 15 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Outline

1 Symplectic and Contact Manifolds

2 Multisymplectic and Multicontact Manifolds

3 Strong Homotopy Lie Algebras

4 L∞ Algebras from Multisymplectic and Multicontact Geometry

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 16 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Multisymplectic Manifolds

Let M be a smooth manifold. Definition An m-plectic structure on M is a non-degenerate (m + 1)-form ω, such that dω = 0. The pair (M, ω) is an m-plectic manifold.

Remark 1-plectic manifolds are symplectic manifolds!

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

θp(ξ1,..., ξm) := p(π∗(ξ1),..., π∗(ξm)), ξi ∈ Tp M (π : M −→ N). ω := dθ is a canonicalm-plectic structure on M.

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

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 17 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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)-form ω, such that dω = 0. Vector fields X such that iXω = 0 span the characteristic distributionK ω of ω. The pair (M, ω) is a pre-m-plectic manifold.

Remark The characteristic distribution is integrable.

Proposition

Suppose that the leaves ofK ω form a smooth manifoldM and the projection π : M → M is a fibration. Then there is a uniquem-plectic structure ω on M such that ω = π∗(ω).

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 18 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Multicontact Manifolds Let M be a smooth manifold. Definition An 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 is a 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 tautological V-valued1-form ϑ:

ϑy(ξ) := π∗(ξ) mod y, ξ ∈ Ty M (π : M −→ N). ker ϑ is a canonicalm-contact structure on M. More generally, higher jet spaces 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 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Pre-Multicontact Manifolds and Their Reduction

More generally, consider the following

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

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

Proposition

Suppose that the leaves ofK C form a smooth manifoldM and the projection π : M → M is a fibration. ThenC := π∗(C) is anm-contact structure.

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 20 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry (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

∗ m ∗ m ∗ L ' Annm C := {α ∈ ∧ T M : iξ α = 0 ∀ξ ∈ C}⊂∧ T M.

∗ 0 ∗ Put (Me , ωe) := (L r , ω|L r0). Remark Me is a principal R×-bundle overM . Moreover, Me possesses a principal flatK C-connection Ke. Proposition

(Me , ωe) is a pre-m-plectic manifold withK ω = Ke. More precisely, it is a × e pre-m-plectic principal R -bundle overM. In particular,L ∆ωe = ωe.

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

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1 Symplectic and Contact Manifolds

2 Multisymplectic and Multicontact Manifolds

3 Strong Homotopy Lie Algebras

4 L∞ Algebras from Multisymplectic and Multicontact Geometry

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 22 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Homotopy Algebras

Consider a chain complex of vector spaces (V, δ) and let A 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) which 1 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 type A and f : (A, d)  (V, δ) : g a pair of homotopy equivalences. The algebra structure in A can be transferred to V along ( f , g), but the transferred structure is of the type A only up to higher homotopies. On the other hand homotopy algebras are homotopy invariant!

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 23 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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 [−,..., −]k : V → V, such that

∑ ∑ ±[[xσ(1),..., xσ(i)], xσ(i+1),..., xσ(i+j)] = 0, i+j=k σ∈Si,j

for allx 1,..., xk ∈ V, k ∈ N. Put [−] = δ. k = 1 δ2(x) = 0 k = 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!

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 24 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry

L∞ Morphisms and L∞ Quasi-Isomorphisms

Remark

L∞ algebras build up a category: a morphism between L∞ algebras ∧k V, W, or L∞ morphism, is a family of degree k − 1 maps fk : V → W satisfying 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 betweenL ∞ morphism. The homo- topy category of L∞ algebras is then obtained by identifying homotopic L∞ morphisms. L∞ quasi-isomorphisms are isomorphisms in the homotopy category.

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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:

% p h (K, δ) o / (H, 0) , j such that [h, δ] = id − jp, and pj = id.

Homotopy Transfer Theorem

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

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

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 26 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Outline

1 Symplectic and Contact Manifolds

2 Multisymplectic and Multicontact Manifolds

3 Strong Homotopy Lie Algebras

4 L∞ Algebras from Multisymplectic and Multicontact Geometry

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 27 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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 some vector field Xσ called Hamiltonian. Remark In the1-plectic case, every0-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 the1-plectic case, it is precisely the Poisson bracket.

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The L∞ Algebra of a Pre-Multisymplectic Manifold

m−1 Let (M, ω) be a pre-m-plectic manifold and ΩHam(M, ω) Hamiltonian forms on it. Consider the truncated de Rham complex g(M, ω):

m−1 d m−2 d ∞ 0 ←− ΩHam(M, ω) ←− Ω (M) ←− · · · ←− 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

( k m−1 −(−) iX ··· iX ω if σ1,..., σk ∈ Ω (M, ω) [σ ,..., σ ] = σ1 σk Ham . 1 k 0 otherwise

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 29 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Homogeneous de Rham Complex of a Principal R×-bundle

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

Definition A differential form σ on a principal R×-bundle Me is homogeneous if L∆σ = σ, with ∆ the Euler vector field.

Let Ω◦(Me ) be homogenous forms on Me .

Remark

The de Rham differential preserves Ω◦(Me ).

Proposition

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

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 30 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Hamiltonian Fields/Forms on Multicontact Manifolds

Let (M, C) be a pre-m-contact manifold and (Me , ωe) its pre-m-plectiza- tion. In the 1-contact case, sections of the Jacobi bundleL are homogenous functions on Me . This suggests the following Definition ( − ) ( ) = An homogeneous m 1 -form σ on Me , ωe is C-Hamiltonian if iXσ ω −dσ for some projectable vector field Xσ called C-Hamiltonian. Remark In the1-contact case, all sections of L are C-Hamiltonian. Remark ( ) 7−→ − σ, τ iXσ iXτ ωe is a well-defined, skew-symmetric bracket on C-Hamiltonian forms. However, it is not a Lie bracket, in general. In the1-contact case, it is precisely the Jacobi bracket.

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The L∞ Algebra of a Pre-Multicontact Manifold

m−1 Let (M, C) be a pre-m-contact manifold and Ω◦,Ham(M, C) C-Hamiltonian forms on Me . Consider the truncated homogeneous de Rham complex g(M, C):

m−1 d m−2 d ∞ 0 ←− Ω◦,Ham(M, C) ←− Ω◦ (Me ) ←− · · · ←− C◦ (Me ) ←− 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

( k m−1 −(−) iX ··· iX ω if σ1,..., σk ∈ Ω (M, C) [σ ,..., σ ] = σ1 σk e ◦,Ham . 1 k 0 otherwise

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 32 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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) is an 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 a pre-multisymplectic manifold (Me , ωe). (Me , ωe) is an higher ana- logue of the symplectization of a contact manifold.

There is an L∞ algebra g(M, C) attached to every pre-multicontact manifold (M, C). g(M, C) is an higher analogue of the Jacobi structure of a contact manifold.

Remark A system of PDEs is geometrically a pre-multicontact manifold. One concludes that there is anL ∞ algebra attached to every system of PDEs!

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 33 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry Perspectives

Remark The multisymplectic and multicontact cases are different: g(M, ω) does not resolve infinitesimal symmetries of (M, ω) in general, and, from the homotopic 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 understood as an exterior differential system. As such, it can be infinitely pro- longed. There is an other L∞ algebra g∞ encoding both infinitesimal symmetries and formal deformations of the prolongation. It would be nice 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.

Luca Vitagliano Higher Contact Geometry and L∞ Algebras 34 / 36 Symplectic and Contact Manifolds Multisymplectic and Multicontact Manifolds Strong Homotopy Lie Algebras L∞ Algebras from Multisymplectic and Multicontact Geometry 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 and Courant 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 prequantum theory; e-print: arXiv:1304.0236.

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

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

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Thank you!

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