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
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Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
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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.
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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!
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Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
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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.
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Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
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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!
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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 : multisymplectic = Lie algebra : homotopy Lie algebra
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Outline
1 Symplectic and Contact Manifolds
2 Multisymplectic and Multicontact Manifolds
3 Strong Homotopy Lie Algebras
4 L∞ Algebras from Multisymplectic and Multicontact Geometry
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Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ 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
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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!
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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.
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Distributions on Manifolds
Let M be a smooth manifold.
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.
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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!
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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.
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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.
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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).
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Outline
1 Symplectic and Contact Manifolds
2 Multisymplectic and Multicontact Manifolds
3 Strong Homotopy Lie Algebras
4 L∞ Algebras from Multisymplectic and Multicontact Geometry
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Strong Homotopy Lie AlgebrasL∞ 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, 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!
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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).
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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
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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.
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(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)!
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Outline
1 Symplectic and Contact Manifolds
2 Multisymplectic and Multicontact Manifolds
3 Strong Homotopy Lie Algebras
4 L∞ Algebras from Multisymplectic and Multicontact Geometry
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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!
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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!
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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.
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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.
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Outline
1 Symplectic and Contact Manifolds
2 Multisymplectic and Multicontact Manifolds
3 Strong Homotopy Lie Algebras
4 L∞ Algebras from Multisymplectic and Multicontact Geometry
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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.
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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.
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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.
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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.
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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
.
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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!
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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.
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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.
Luca Vitagliano Higher Contact Geometry and L∞ Algebras 35 / 36
Symplectic and Contact ManifoldsMultisymplectic and Multicontact Manifolds
Strong Homotopy Lie AlgebrasL∞ Algebras from Multisymplectic and Multicontact Geometry
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Luca Vitagliano Higher Contact Geometry and L∞ Algebras 36 / 36
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