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
36
Embed
Higher Contact Geometry and L Algebras · Introduction: Symplectic Geometry A symplectic manifold is a manifold M equipped with a closed, non-degenerate differential2-form w. The
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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.
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ 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 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!
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ 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!
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.
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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!
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ 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 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 and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
Contact Manifolds
Let M be a smooth manifold.
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!
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ 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)-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).
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
Multicontact Manifolds
Let M be a smooth manifold.
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
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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.
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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.
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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.
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ 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 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.
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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.
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ 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) 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!
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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.