Gauge transformations of multisymplectic manifolds and L ∞ observables

Abstract

Short presentation on the development of the project introduced in the previous talk:

https://www.researchgate.net/publication/344044689_Gauge_transformations_of_multisymplectic_manifolds_and_L_infinity_observables

Full text

Gauge transformations of multisymplectic manifolds
and L∞observables
Antonio Michele Miti
Universit`a Cattolica del Sacro Cuore KU Leuven
Brescia, Italy Leuven, Belgium
Young Researchers’ Virtual Multisymplectic Geometry Workshop
Online, July 13, 2021

Outline 0
Introduction
I(Multi)-symplectic geometry: mechanical perspective
IObservability
Gauge Compatibility Problem - Symplectic Case
IPoisson algebra and Lie algebroids
ICompatibility between gauge transformations
IGeometric interpretation in pre-quantization
Gauge Compatibility Problem - MULTISymplectic Case
ILie ∞-algebra of Observables
IHomotopy comomentum maps
IVinogradov Algebroids
IRogers embedding
ICompatibility with Gauge transformations

Symplectic geometry (mechanical perspective) 1
”geometric approach” to mechanics . . .
Def: Symplectic Manifold
M, ω Smooth mfd.
non-degenerate, closed,
2-form.
Example: M=T∗Qis symplectic
with ω=dθgiven by
θ|(q,p)(v) = p(π∗v).
based on the notion of ”states”.
”algebraic approach” to mechanics .. .
Def: Classical Observables
Unital, associative, commutative alge-
bra C∞(M).
Def: Hamiltonian vector fields
vf∈X(M) such that:
ιvfω=−df (exact)
vf=Ham.v.f. pertaining to f∈C∞(M).
Def: Poisson Algebra of Observables
C∞(M) is a Poisson algebra with
{f,g}=ιvgιvfω=ω(vf,vg).
based on the notion of ”meaurable
quantities”.

Symplectic geometry (mechanical perspective) 1
”geometric approach” to mechanics . . .
Def: Symplectic Manifold
M, ω Smooth mfd.
non-degenerate, closed,
2-form.
Example: M=T∗Qis symplectic
with ω=dθgiven by
θ|(q,p)(v) = p(π∗v).
based on the notion of ”states”.
”algebraic approach” to mechanics .. .
Def: Classical Observables
Unital, associative, commutative alge-
bra C∞(M).
Def: Hamiltonian vector fields
vf∈X(M) such that:
ιvfω=−df (exact)
vf=Ham.v.f. pertaining to f∈C∞(M).
Def: Poisson Algebra of Observables
C∞(M) is a Poisson algebra with
{f,g}=ιvgιvfω=ω(vf,vg).
based on the notion of ”meaurable
quantities”.

Symplectic geometry (mechanical perspective) 1
”geometric approach” to mechanics . . .
Def: Symplectic Manifold
M, ω Smooth mfd.
non-degenerate, closed,
2-form.
Example: M=T∗Qis symplectic
with ω=dθgiven by
θ|(q,p)(v) = p(π∗v).
based on the notion of ”states”.
”algebraic approach” to mechanics .. .
Def: Classical Observables
Unital, associative, commutative alge-
bra C∞(M).
Def: Hamiltonian vector fields
vf∈X(M) such that:
ιvfω=−df (exact)
vf=Ham.v.f. pertaining to f∈C∞(M).
Def: Poisson Algebra of Observables
C∞(M) is a Poisson algebra with
{f,g}=ιvgιvfω=ω(vf,vg).
based on the notion of ”meaurable
quantities”.

Symplectic geometry (mechanical perspective) 1
”geometric approach” to mechanics . . .
Def: Symplectic Manifold
M, ω Smooth mfd.
non-degenerate, closed,
2-form.
Example: M=T∗Qis symplectic
with ω=dθgiven by
θ|(q,p)(v) = p(π∗v).
based on the notion of ”states”.
”algebraic approach” to mechanics .. .
Def: Classical Observables
Unital, associative, commutative alge-
bra C∞(M).
Def: Hamiltonian vector fields
vf∈X(M) such that:
ιvfω=−df (exact)
vf=Ham.v.f. pertaining to f∈C∞(M).
Def: Poisson Algebra of Observables
C∞(M) is a Poisson algebra with
{f,g}=ιvgιvfω=ω(vf,vg).
based on the notion of ”meaurable
quantities”.

Symplectic geometry (mechanical perspective) 1
”geometric approach” to mechanics . . .
Def: Symplectic Manifold
M, ω Smooth mfd.
non-degenerate, closed,
2-form.
Example: M=T∗Qis symplectic
with ω=dθgiven by
θ|(q,p)(v) = p(π∗v).
based on the notion of ”states”.
”algebraic approach” to mechanics .. .
Def: Classical Observables
Unital, associative, commutative alge-
bra C∞(M).
Def: Hamiltonian vector fields
vf∈X(M) such that:
ιvfω=−df (exact)
vf=Ham.v.f. pertaining to f∈C∞(M).
Def: Poisson Algebra of Observables
C∞(M) is a Poisson algebra with
{f,g}=ιvgιvfω=ω(vf,vg).
based on the notion of ”meaurable
quantities”.

Symplectic geometry (mechanical perspective) 1
”geometric approach” to mechanics . . .
Def: Symplectic Manifold
M, ω Smooth mfd.
non-degenerate, closed,
2-form.
Example: M=T∗Qis symplectic
with ω=dθgiven by
θ|(q,p)(v) = p(π∗v).
based on the notion of ”states”.
”algebraic approach” to mechanics .. .
Def: Classical Observables
Unital, associative, commutative alge-
bra C∞(M).
Def: Hamiltonian vector fields
vf∈X(M) such that:
ιvfω=−df (exact)
vf=Ham.v.f. pertaining to f∈C∞(M).
Def: Poisson Algebra of Observables
C∞(M) is a Poisson algebra with
{f,g}=ιvgιvfω=ω(vf,vg).
based on the notion of ”meaurable
quantities”.

From Symplectic to MultiSymplectic (mechanical perspective) 2
Historical motivation
Mechanics: geometrical foundations of (first-order) field theories.
Kijowski, W. Tulczyjew [KT79];
Cari˜nena, Crampin, Ibort [CCI91];
Gotay, Isenberg, Marsden, Montgomery [GIMM97];
· · ·
oThe lack of a satisfactory notion of
observables hindered the spread of this
formalism. o
?Why observables are so crucial?
Quantization! ?

From Symplectic to MultiSymplectic (mechanical perspective) 2
Historical motivation
Mechanics: geometrical foundations of (first-order) field theories.
Kijowski, W. Tulczyjew [KT79];
Cari˜nena, Crampin, Ibort [CCI91];
Gotay, Isenberg, Marsden, Montgomery [GIMM97];
· · ·
oThe lack of a satisfactory notion of
observables hindered the spread of this
formalism. o
?Why observables are so crucial?
Quantization! ?

From Symplectic to MultiSymplectic (mechanical perspective) 2
Historical motivation
Mechanics: geometrical foundations of (first-order) field theories.
Kijowski, W. Tulczyjew [KT79];
Cari˜nena, Crampin, Ibort [CCI91];
Gotay, Isenberg, Marsden, Montgomery [GIMM97];
· · ·
oThe lack of a satisfactory notion of
observables hindered the spread of this
formalism. o
?Why observables are so crucial?
Quantization! ?

Observables in n-plectic geometry 3
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Thm: Observables L∞-algebra
Ωn−1
ham (M, ω) endowed with
{σ1, σ2}=−ιv
1ιv
2ω
can be ”completed” to a
L∞−algebra.
3Skew-symmetric;
7multiplication of observables;
7Jacobi equation;
+Jacobi equation up to homotopies.
Thm: Observables Leibniz algebra
Ωn−1
ham (M, ω) endowed with
Jσ1, σ2K=Lv
1σ2.
can be ”completed” to a
DG-Leibniz algebra.
7Skew-symmetric;
7multiplication of observables;
3Jacobi equation;
+Skew-symmetric up to homotopies.

Observables in n-plectic geometry 3
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Thm: Observables L∞-algebra
Ωn−1
ham (M, ω) endowed with
{σ1, σ2}=−ιv
1ιv
2ω
can be ”completed” to a
L∞−algebra.
3Skew-symmetric;
7multiplication of observables;
7Jacobi equation;
+Jacobi equation up to homotopies.
Thm: Observables Leibniz algebra
Ωn−1
ham (M, ω) endowed with
Jσ1, σ2K=Lv
1σ2.
can be ”completed” to a
DG-Leibniz algebra.
7Skew-symmetric;
7multiplication of observables;
3Jacobi equation;
+Skew-symmetric up to homotopies.

Observables in n-plectic geometry 3
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Thm: Observables L∞-algebra
Ωn−1
ham (M, ω) endowed with
{σ1, σ2}=−ιv
1ιv
2ω
can be ”completed” to a
L∞−algebra.
3Skew-symmetric;
7multiplication of observables;
7Jacobi equation;
+Jacobi equation up to homotopies.
Thm: Observables Leibniz algebra
Ωn−1
ham (M, ω) endowed with
Jσ1, σ2K=Lv
1σ2.
can be ”completed” to a
DG-Leibniz algebra.
7Skew-symmetric;
7multiplication of observables;
3Jacobi equation;
+Skew-symmetric up to homotopies.

Observables in n-plectic geometry 3
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Thm: Observables L∞-algebra
Ωn−1
ham (M, ω) endowed with
{σ1, σ2}=−ιv
1ιv
2ω
can be ”completed” to a
L∞−algebra.
3Skew-symmetric;
7multiplication of observables;
7Jacobi equation;
+Jacobi equation up to homotopies.
Thm: Observables Leibniz algebra
Ωn−1
ham (M, ω) endowed with
Jσ1, σ2K=Lv
1σ2.
can be ”completed” to a
DG-Leibniz algebra.
7Skew-symmetric;
7multiplication of observables;
3Jacobi equation;
+Skew-symmetric up to homotopies.

Observables in n-plectic geometry 3
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Thm: Observables L∞-algebra
Ωn−1
ham (M, ω) endowed with
{σ1, σ2}=−ιv
1ιv
2ω
can be ”completed” to a
L∞−algebra.
3Skew-symmetric;
7multiplication of observables;
7Jacobi equation;
+Jacobi equation up to homotopies.
Thm: Observables Leibniz algebra
Ωn−1
ham (M, ω) endowed with
Jσ1, σ2K=Lv
1σ2.
can be ”completed” to a
DG-Leibniz algebra.
7Skew-symmetric;
7multiplication of observables;
3Jacobi equation;
+Skew-symmetric up to homotopies.

Observables in n-plectic geometry 3
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Thm: Observables L∞-algebra
Ωn−1
ham (M, ω) endowed with
{σ1, σ2}=−ιv
1ιv
2ω
can be ”completed” to a
L∞−algebra.
3Skew-symmetric;
7multiplication of observables;
7Jacobi equation;
+Jacobi equation up to homotopies.
Thm: Observables Leibniz algebra
Ωn−1
ham (M, ω) endowed with
Jσ1, σ2K=Lv
1σ2.
can be ”completed” to a
DG-Leibniz algebra.
7Skew-symmetric;
7multiplication of observables;
3Jacobi equation;
+Skew-symmetric up to homotopies.

Scope of the talk 4
Broad general question:
What is the corresponding ”correct” notion
of observables in multisymplectic geometry?
Specific technical question:
How the compatibility diagram between gauge transforma-
tion and comomentum maps extends to the n-plectic case?

Scope of the talk 4
Broad general question:
What is the corresponding ”correct” notion
of observables in multisymplectic geometry?
Specific technical question:
How the compatibility diagram between gauge transforma-
tion and comomentum maps extends to the n-plectic case?

Outline 4
Introduction
I(Multi)-symplectic geometry: mechanical perspective
IObservability
Gauge Compatibility Problem - Symplectic Case
IPoisson algebra and Lie algebroids
ICompatibility between gauge transformations
IGeometric interpretation in pre-quantization
Gauge Compatibility Problem - MULTISymplectic Case
ILie ∞-algebra of Observables
IHomotopy comomentum maps
IVinogradov Algebroids
IRogers embedding
ICompatibility with Gauge transformations

Embedding of the observables algebra in the Lie algebroid 5
Given a symplectic mfd. (M, ω) ...
(M, ω) (TM ⊕RM, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
observables
twisted standard Lie algebroid
Ψ
Lie algebras cat.
I... there is a naturally associated Poisson algebra ...
I... and also a (standard twisted) Lie Algebroid.
IA Lie algebroid is a ”controlled” ∞-dimensional Lie algebra given (in this
case) by x1
f1,x2
f2 =[x1,x2]
x1(f2)−x2(f1)−ω(x1,x2)
Thm: There exists an embedding of Lie algebras.
Ψ : C∞(M)ωΓ(TM ⊕R)ω
fv
f
f
Ψ

Embedding of the observables algebra in the Lie algebroid 5
Given a symplectic mfd. (M, ω) ...
(M, ω) (TM ⊕RM, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
observables
twisted standard Lie algebroid
Ψ
Lie algebras cat.
I... there is a naturally associated Poisson algebra ...
I... and also a (standard twisted) Lie Algebroid.
IA Lie algebroid is a ”controlled” ∞-dimensional Lie algebra given (in this
case) by x1
f1,x2
f2 =[x1,x2]
x1(f2)−x2(f1)−ω(x1,x2)
Thm: There exists an embedding of Lie algebras.
Ψ : C∞(M)ωΓ(TM ⊕R)ω
fv
f
f
Ψ

Embedding of the observables algebra in the Lie algebroid 5
Given a symplectic mfd. (M, ω) ...
(M, ω) (TM ⊕RM, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
observables
twisted standard Lie algebroid
Ψ
Lie algebras cat.
I... there is a naturally associated Poisson algebra ...
I... and also a (standard twisted) Lie Algebroid.
IA Lie algebroid is a ”controlled” ∞-dimensional Lie algebra given (in this
case) by x1
f1,x2
f2 =[x1,x2]
x1(f2)−x2(f1)−ω(x1,x2)
Thm: There exists an embedding of Lie algebras.
Ψ : C∞(M)ωΓ(TM ⊕R)ω
fv
f
f
Ψ

Embedding of the observables algebra in the Lie algebroid 5
Given a symplectic mfd. (M, ω) ...
(M, ω) (TM ⊕RM, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
observables
twisted standard Lie algebroid
Ψ
Lie algebras cat.
Thm: There exists an embedding of Lie algebras.
Ψ : C∞(M)ωΓ(TM ⊕R)ω
fv
f
f
Ψ

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.
IConsider a second gauge-related symplectic structure on M
˜ω=ω+dB with B∈Ω1(M).

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.
IThere is a natural isomorphism in the Lie Alg.oids category
(B-transformation) x
f7→ x
f−ιxB.

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.
IThere is a natural isomorphism in the Lie Alg.oids category
(B-transformation) x
f7→ x
f−ιxB.

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.
?How can we close the left-hand side? ?

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.
IConsider an infinitesimal group action gMwhich is Hamiltonian w.r.t.
both ωand ˜ω.
Ilet be f:g→C∞(M)ωand ˜
f:g→C∞(M)˜ωtwo comoment map s.t.
˜
f(ξ) = f(ξ)−ιξB

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.
IConsider an infinitesimal group action gMwhich is Hamiltonian w.r.t.
both ωand ˜ω.
Ilet be f:g→C∞(M)ωand ˜
f:g→C∞(M)˜ωtwo comoment map s.t.
˜
f(ξ) = f(ξ)−ιξB

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.
Lemma: The central pentagon commutes!

Compatibility between gauge transformations and comoment maps 6
Consider (M, ω)symplectic mfd.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
g
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ

∼
f
˜
fΨ
Lie algebras cat.
Lemma: The central pentagon commutes!
?What happens in the higher (n-plectic) case? ?

Geometric interpretation of the diagram 7
Consider (M, ω)symplectic and prequantizable.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
gQ(P, θ)
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ
Preqθ

∼

Ψ
Preqθ
Lie algebras cat.
IFix a Prequantization Bundle S1→P→Mwith connection θ,
I”infinitesimal quantomorphisms” Q(P, θ) := {Y∈X(P)|L
Yθ= 0}.
IEmbdedding through Atiah algebroid.

Geometric interpretation of the diagram 7
Consider (M, ω)symplectic and prequantizable.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
gQ(P, θ)
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ
Preqθ

∼

Ψ
Preqθ
Lie algebras cat.
IFix a Prequantization Bundle S1→P→Mwith connection θ,
I”infinitesimal quantomorphisms” Q(P, θ) := {Y∈X(P)|L
Yθ= 0}.
IEmbdedding through Atiah algebroid.

Geometric interpretation of the diagram 7
Consider (M, ω)symplectic and prequantizable.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
gQ(P, θ)
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ
Preqθ

∼

Ψ
Preqθ
Lie algebras cat.
IFix a Prequantization Bundle S1→P→Mwith connection θ,
I”infinitesimal quantomorphisms” Q(P, θ) := {Y∈X(P)|L
Yθ= 0}.
IEmbdedding through Atiah algebroid.
Lemma: The left square commutes!

Geometric interpretation of the diagram 7
Consider (M, ω)symplectic and prequantizable.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
gQ(P, θ)
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ
Preqθ

∼

Ψ
Preqθ
Lie algebras cat.
IEmbdedding through Atiah algebroid.

Geometric interpretation of the diagram 7
Consider (M, ω)symplectic and prequantizable.
(M, ω) (TM ⊕R, ρ, [·,·]ω)
C∞(M),{·,·}ω Γ(TM ⊕R),[·,·]ω
gQ(P, θ)
C∞(M),{·,·}˜ω Γ(TM ⊕R),[·,·]˜ω
(M,˜ω) (TM ⊕R, ρ, [·,·]˜ω)
observables
twisted standard Lie algebroid
gauge transf.
B-transf.
Ψ
Preqθ

∼

Ψ
Preqθ
Lie algebras cat.
IEmbdedding through Atiah algebroid.
Lemma: The left square and right triangle commute!

Outline 7
Introduction
I(Multi)-symplectic geometry: mechanical perspective
IObservability
Gauge Compatibility Problem - Symplectic Case
IPoisson algebra and Lie algebroids
ICompatibility between gauge transformations
IGeometric interpretation in pre-quantization
Gauge Compatibility Problem - MULTISymplectic Case
ILie ∞-algebra of Observables
IHomotopy comomentum maps
IVinogradov Algebroids
IRogers embedding
ICompatibility with Gauge transformations

Lie ∞-algebra of Observables (higher observables) 8
Let be (M, ω) a n-plectic manifold.
Def: L∞-algebra of observables (Rogers)
Is a cochain-complex (L,{·}1)
0L1−n. . . L2−k. . . L−1L00
Ω0(M). . . Ωn+1−k(M). . . Ωn−2(M) Ωn−1
Ham(M, ω)
:=
{·}1{·}1
:=
{·}1{·}1
:=
{·}1
:=
d d d d d
with n(skew-symmetric) multibrackets (2 ≤k≤n+ 1)
{·,...,·}k:Ωn−1
Ham(M, ω)⊗kΩn+1−k(M)
σ1⊗ · · · ⊗ σk(−)k+1 ιv
σ1· · · ιv
σkω
Higher analogue of the Poisson algebra structure associated to a symplectic mfd.
If n>1:
7we lose : multiplication of observables, Jacobi equation;
3we gain : brackets with arities different than two,
Jacobi equation up to homotopies.

Lie ∞-algebra of Observables (higher observables) 8
Let be (M, ω) a n-plectic manifold.
Def: L∞-algebra of observables (Rogers)
Is a cochain-complex (L,{·}1)
0L1−n. . . L2−k. . . L−1L00
Ω0(M). . . . . . Ωn−2(M)
:=
{·}1{·}1
:=
{·}1{·}1
:=
{·}1
:=
d d d d d Ωn−1
Ham(M, ω)
{·,...,·}k
Ωn+1−k(M)
with n(skew-symmetric) multibrackets (2 ≤k≤n+ 1)
{·,...,·}k:Ωn−1
Ham(M, ω)⊗kΩn+1−k(M)
σ1⊗ · · · ⊗ σk(−)k+1 ιv
σ1· · · ιv
σkω
Higher analogue of the Poisson algebra structure associated to a symplectic mfd.
If n>1:
7we lose : multiplication of observables, Jacobi equation;
3we gain : brackets with arities different than two,
Jacobi equation up to homotopies.

Lie ∞-algebra of Observables (higher observables) 8
Let be (M, ω) a n-plectic manifold.
Def: L∞-algebra of observables (Rogers)
Is a cochain-complex (L,{·}1)
0L1−n. . . L2−k. . . L−1L00
Ω0(M). . . . . . Ωn−2(M)
:=
{·}1{·}1
:=
{·}1{·}1
:=
{·}1
:=
d d d d d Ωn−1
Ham(M, ω)
{·,...,·}k
Ωn+1−k(M)
with n(skew-symmetric) multibrackets (2 ≤k≤n+ 1)
{·,...,·}k:Ωn−1
Ham(M, ω)⊗kΩn+1−k(M)
σ1⊗ · · · ⊗ σk(−)k+1 ιv
σ1· · · ιv
σkω
Higher analogue of the Poisson algebra structure associated to a symplectic mfd.
If n>1:
7we lose : multiplication of observables, Jacobi equation;
3we gain : brackets with arities different than two,
Jacobi equation up to homotopies.

Homotopy comomentum maps 9
Consider a Lie algebra action v:g→X(M) preserving the n-plectic form ω.
Def: Homotopy comomentum map (Callies, Fregier, Rogers, Zambon)
L∞(M, ω)
g X(M)
v
πHam
(f)
v
HCMM is an L∞-morphism (f) : g→L∞(M, ω)
lifting the infinitesimal action v:g→X(M)
(acting via Hamiltonian vector fields!)
d f1(x) = −ιvxω∀x∈g.
Lemma: HCMM unfolded [CFRZ16]
HCMM is a sequence of (graded-skew) multilinear maps:
(f) = fk: Λkg→L1−k⊆Ωn−k(M)0≤k≤n+ 1
fulfilling:
If0= 0, fn+1 = 0
Idfk(p) = fk−1(∂p)−(−1)k(k+1)
2ι(vp)ω∀p∈Λk(g),∀k=1,...n+1
Chevalley-Eilenberg boundary op.

Homotopy comomentum maps 9
Consider a Lie algebra action v:g→X(M) preserving the n-plectic form ω.
Def: Homotopy comomentum map (Callies, Fregier, Rogers, Zambon)
L∞(M, ω)
g X(M)
v
πHam
(f)
v
HCMM is an L∞-morphism (f) : g→L∞(M, ω)
lifting the infinitesimal action v:g→X(M)
(acting via Hamiltonian vector fields!)
d f1(x) = −ιvxω∀x∈g.
Lemma: HCMM unfolded [CFRZ16]
HCMM is a sequence of (graded-skew) multilinear maps:
(f) = fk: Λkg→L1−k⊆Ωn−k(M)0≤k≤n+ 1
fulfilling:
If0= 0, fn+1 = 0
Idfk(p) = fk−1(∂p)−(−1)k(k+1)
2ι(vp)ω∀p∈Λk(g),∀k=1,...n+1
Chevalley-Eilenberg boundary op.

Homotopy comomentum maps 9
Consider a Lie algebra action v:g→X(M) preserving the n-plectic form ω.
Def: Homotopy comomentum map (Callies, Fregier, Rogers, Zambon)
L∞(M, ω)
XHam(M, ω )
g X(M)
v
πHam
(f)
v
HCMM is an L∞-morphism (f) : g→L∞(M, ω)
lifting the infinitesimal action v:g→X(M)
(acting via Hamiltonian vector fields!)
d f1(x) = −ιvxω∀x∈g.
Lemma: HCMM unfolded [CFRZ16]
HCMM is a sequence of (graded-skew) multilinear maps:
(f) = fk: Λkg→L1−k⊆Ωn−k(M)0≤k≤n+ 1
fulfilling:
If0= 0, fn+1 = 0
Idfk(p) = fk−1(∂p)−(−1)k(k+1)
2ι(vp)ω∀p∈Λk(g),∀k=1,...n+1
Chevalley-Eilenberg boundary op.

Homotopy comomentum maps 9
Consider a Lie algebra action v:g→X(M) preserving the n-plectic form ω.
Def: Homotopy comomentum map (Callies, Fregier, Rogers, Zambon)
L∞(M, ω)
XHam(M, ω )
g X(M)
v
πHam
(f)
v
HCMM is an L∞-morphism (f) : g→L∞(M, ω)
lifting the infinitesimal action v:g→X(M)
(acting via Hamiltonian vector fields!)
d f1(x) = −ιvxω∀x∈g.
Lemma: HCMM unfolded [CFRZ16]
HCMM is a sequence of (graded-skew) multilinear maps:
(f) = fk: Λkg→L1−k⊆Ωn−k(M)0≤k≤n+ 1
fulfilling:
If0= 0, fn+1 = 0
Idfk(p) = fk−1(∂p)−(−1)k(k+1)
2ι(vp)ω∀p∈Λk(g),∀k=1,...n+1
Chevalley-Eilenberg boundary op.

Homotopy comomentum maps 9
Consider a Lie algebra action v:g→X(M) preserving the n-plectic form ω.
Def: Homotopy comomentum map (Callies, Fregier, Rogers, Zambon)
L∞(M, ω)
XHam(M, ω )
g X(M)
v
πHam
(f)
v
HCMM is an L∞-morphism (f) : g→L∞(M, ω)
lifting the infinitesimal action v:g→X(M)
(acting via Hamiltonian vector fields!)
d f1(x) = −ιvxω∀x∈g.
Lemma: HCMM unfolded [CFRZ16]
HCMM is a sequence of (graded-skew) multilinear maps:
(f) = fk: Λkg→L1−k⊆Ωn−k(M)0≤k≤n+ 1
fulfilling:
If0= 0, fn+1 = 0
Idfk(p) = fk−1(∂p)−(−1)k(k+1)
2ι(vp)ω∀p∈Λk(g),∀k=1,...n+1
Chevalley-Eilenberg boundary op.

Homotopy comomentum maps 9
Consider a Lie algebra action v:g→X(M) preserving the n-plectic form ω.
Def: Homotopy comomentum map (Callies, Fregier, Rogers, Zambon)
L∞(M, ω)
XHam(M, ω )
g X(M)
v
πHam
(f)
v
HCMM is an L∞-morphism (f) : g→L∞(M, ω)
lifting the infinitesimal action v:g→X(M)
(acting via Hamiltonian vector fields!)
d f1(x) = −ιvxω∀x∈g.
Lemma: HCMM unfolded [CFRZ16]
HCMM is a sequence of (graded-skew) multilinear maps:
(f) = fk: Λkg→L1−k⊆Ωn−k(M)0≤k≤n+ 1
fulfilling:
If0= 0, fn+1 = 0
Idfk(p) = fk−1(∂p)−(−1)k(k+1)
2ι(vp)ω∀p∈Λk(g),∀k=1,...n+1
Chevalley-Eilenberg boundary op.

Homotopy comomentum maps 9
Consider a Lie algebra action v:g→X(M) preserving the n-plectic form ω.
Def: Homotopy comomentum map (Callies, Fregier, Rogers, Zambon)
L∞(M, ω)
XHam(M, ω )
g X(M)
v
πHam
(f)
v
HCMM is an L∞-morphism (f) : g→L∞(M, ω)
lifting the infinitesimal action v:g→X(M)
(acting via Hamiltonian vector fields!)
d f1(x) = −ιvxω∀x∈g.
Lemma: HCMM unfolded [CFRZ16]
HCMM is a sequence of (graded-skew) multilinear maps:
(f) = fk: Λkg→L1−k⊆Ωn−k(M)0≤k≤n+ 1
fulfilling:
If0= 0, fn+1 = 0
Idfk(p) = fk−1(∂p)−(−1)k(k+1)
2ι(vp)ω∀p∈Λk(g),∀k=1,...n+1
Chevalley-Eilenberg boundary op.

Vinogradov Algebroids 10
Def: Vinogradov algebroid (higher Courant)
E,ρ,h·,·i±,[·,·]ωVector bun. E=TM ⊕(∧n−1T∗M);
anchor ρ:E TM
x
αx;
pairing h·,·i±:E⊗E∧n−2T∗M
x1
α1⊗x2
α21
2(ιx1α2±ιx2α1) ;
(higher) Courant bracket
[·,·]ω:E⊗E∧n−2T∗M
x1
α1⊗x2
α2  [x1,x2]
L
x1α2−L
x2α1−dDx1
α1,x2
α2E+ιx1ιx2ω.
I(n= 1) ⇒standard twisted Lie algebroid;
I(n= 2) ⇒standard twisted Courant algebroid;

Vinogradov L∞-algebra 11
Vin. alg.oids are NQ-manifolds (L∞-algebroids). ⇒Associated L∞-algebra.
Def: Vinogradov L∞-algebra [Zam10]
L∞(En, ω) := V
n, µ graded vector space
Vk=(X(M)⊕Ωn−1(M)k= 0,
Ωn−1+k(M)−n+ 1 ≤k<0.
-unary bracket: µ1(f) = df ;
-binary bracket: µ2(e1,e2)=[e1,e2]ω,
µ2(e1,f2) = 1
2L
X1f2;
-ternary bracket: µ3(e1,e2,e3) = −1
3h[e1,e2]ω,e3i++(cyc.) ,
µ3(f1,e2,e3) = −1
61
2(ιX1L
X2−ιX2L
X1) + ι[X1,X2]f1;
-k-ary bracket for k≥3 an odd integer: · · ·

Vinogradov L∞-algebra 11
Vin. alg.oids are NQ-manifolds (L∞-algebroids). ⇒Associated L∞-algebra.
Def: Vinogradov L∞-algebra [Zam10]
L∞(En, ω) := V
n, µ graded vector space
Vk=(X(M)⊕Ωn−1(M)k= 0,
Ωn−1+k(M)−n+ 1 ≤k<0.
-unary bracket: µ1(f) = df ;
-binary bracket: µ2(e1,e2)=[e1,e2]ω,
µ2(e1,f2) = 1
2L
X1f2;
-ternary bracket: µ3(e1,e2,e3) = −1
3h[e1,e2]ω,e3i++(cyc.) ,
µ3(f1,e2,e3) = −1
61
2(ιX1L
X2−ιX2L
X1) + ι[X1,X2]f1;
-k-ary bracket for k≥3 an odd integer: · · ·
ei=Xi
αi∈X(M)⊕Ωn−1(M) , fi∈Ln−2
k=0 Ωk(M).

Embedding observables L∞-algebra into Vinogradov L∞-algebra 12
Consider now ωn-plectic .
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
observables
twisted standard Vinogradov algebroid
Ψ
L∞-algebras cat.

Embedding observables L∞-algebra into Vinogradov L∞-algebra 12
Consider now ωn-plectic .
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
observables
twisted standard Vinogradov algebroid
Ψ
L∞-algebras cat.

Embedding observables L∞-algebra into Vinogradov L∞-algebra 12
Consider now ωn-plectic .
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
observables
twisted standard Vinogradov algebroid
Ψ
L∞-algebras cat.

Embedding observables L∞-algebra into Vinogradov L∞-algebra 12
Consider now ωn-plectic .
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
observables
twisted standard Vinogradov algebroid
Ψ
L∞-algebras cat.
Thm: Embedding of L∞-algebras Ψ : L∞(M, ω )→L∞(En, ω) [Mit21].
·consider the graded vector subspace A
Ak=(nX
α∈X(M)⊕Ωn−1(M)ιXω=−dαok= 0,
Ωn−1+k(M)−n+ 1 ≤k<0.
·restrict the two L∞-structures to πand µon A
·L∞(M, ω)∼
=(A, π)∼
=(A, µ)→L∞(En, ω)

Embedding observables L∞-algebra into Vinogradov L∞-algebra 12
Consider now ωn-plectic .
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
observables
twisted standard Vinogradov algebroid
Ψ
L∞-algebras cat.
Thm: Embedding of L∞-algebras Ψ : L∞(M, ω )→L∞(En, ω) [Mit21].
·consider the graded vector subspace A
Ak=(nX
α∈X(M)⊕Ωn−1(M)ιXω=−dαok= 0,
Ωn−1+k(M)−n+ 1 ≤k<0.
·restrict the two L∞-structures to πand µon A
·L∞(M, ω)∼
=(A, π)∼
=(A, µ)→L∞(En, ω)

Embedding observables L∞-algebra into Vinogradov L∞-algebra 12
Consider now ωn-plectic .
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
observables
twisted standard Vinogradov algebroid
Ψ
L∞-algebras cat.
Thm: Embedding of L∞-algebras Ψ : L∞(M, ω )→L∞(En, ω) [Mit21].
·consider the graded vector subspace A
Ak=(nX
α∈X(M)⊕Ωn−1(M)ιXω=−dαok= 0,
Ωn−1+k(M)−n+ 1 ≤k<0.
·restrict the two L∞-structures to πand µon A
·L∞(M, ω)∼
=(A, π)∼
=(A, µ)→L∞(En, ω)Complete proof
oup to n≥4! o

Compatibility with Gauge transformations 13
Consider now ωn-plectic and ˜ω=ω+dB:
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
g
L∞(M,˜ω)L∞En,˜ω
(M,˜ω) (En,[·,·]˜ω)
observables
twisted standard Vinogradov algebroid
gauge transf.
∼
=
B-transf.
Ψ

∼
=
Ψ
L∞-algebras cat.
IVinogradov alg.oids w.r.t cohomologous twisting closed forms are isomorphic.
IInduced isomorphism at the level of L∞-algebras
Thm:
[Mit21]
The central square commutes.
(On the nose, not ”up to homotopies”).Complete proof
oup to n≥4! o

Compatibility with Gauge transformations 13
Consider now ωn-plectic and ˜ω=ω+dB:
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
g
L∞(M,˜ω)L∞En,˜ω
(M,˜ω) (En,[·,·]˜ω)
observables
twisted standard Vinogradov algebroid
gauge transf.
∼
=
B-transf.
Ψ

∼
=
Ψ
L∞-algebras cat.
IVinogradov alg.oids w.r.t cohomologous twisting closed forms are isomorphic.
IInduced isomorphism at the level of L∞-algebras
Thm:
[Mit21]
The central square commutes.
(On the nose, not ”up to homotopies”).Complete proof
oup to n≥4! o

Compatibility with Gauge transformations 13
Consider now ωn-plectic and ˜ω=ω+dB:
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
g
L∞(M,˜ω)L∞En,˜ω
(M,˜ω) (En,[·,·]˜ω)
observables
twisted standard Vinogradov algebroid
gauge transf.
∼
=
B-transf.
Ψ

∼
=
Ψ
L∞-algebras cat.
IVinogradov alg.oids w.r.t cohomologous twisting closed forms are isomorphic.
IInduced isomorphism at the level of L∞-algebras
Thm:
[Mit21]
The central square commutes.
(On the nose, not ”up to homotopies”).Complete proof
oup to n≥4! o

Compatibility with Gauge transformations 13
Consider now ωn-plectic and ˜ω=ω+dB:
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
g
L∞(M,˜ω)L∞En,˜ω
(M,˜ω) (En,[·,·]˜ω)
observables
twisted standard Vinogradov algebroid
gauge transf.
∼
=
B-transf.
Ψ

∼
=
Ψ
L∞-algebras cat.
IConsider a Lie algebra action g→X(M) admitting HCMM w.r.t ωand ˜ω
Thm:
[Mit21]
The central square commutes.
(On the nose, not ”up to homotopies”).
Complete proof
oup to n≥4! o

Compatibility with Gauge transformations 13
Consider now ωn-plectic and ˜ω=ω+dB:
(M, ω) (En, ρ, h·,·i,[·,·]ω)
L∞(M, ω)L∞En, ω
g
L∞(M,˜ω)L∞En,˜ω
(M,˜ω) (En,[·,·]˜ω)
observables
twisted standard Vinogradov algebroid
gauge transf.
∼
=
B-transf.
Ψ

∼
=
Ψ
L∞-algebras cat.
IConsider a Lie algebra action g→X(M) admitting HCMM w.r.t ωand ˜ω
Thm:
[Mit21]
The central square commutes.
(On the nose, not ”up to homotopies”).
Complete proof
oup to n≥4! o

Thank you for your attention!

Supplementary Material

Multisymplectic geometry in a nutshell 1
Historical motivation
Mechanics: geometrical foundations of (first-order) field theories.
mechanics geometry
phase space symplectic manifold multisymplectic manifold
classical
observables Poisson algebra L∞-algebra
symmetries group actions admitting
comoment map
group actions admitting
homotopy comomentum map
| {z }
point-like particles systems | {z }
field-theoretic systems
Scope of the thesis
•Develop theory of homotopy comomentum maps
•produce new meaningful examples.

Multisymplectic geometry in a nutshell 1
Historical motivation
Mechanics: geometrical foundations of (first-order) field theories.
mechanics geometry
phase space symplectic manifold
classical
observables Poisson algebra
symmetries group actions admitting
comoment map
| {z }
point-like particles systems
mechanics geometry
phase space symplectic manifold multisymplectic manifold
classical
observables Poisson algebra L∞-algebra
symmetries group actions admitting
comoment map
group actions admitting
homotopy comomentum map
| {z }
point-like particles systems | {z }
field-theoretic systems
Scope of the thesis
•Develop theory of homotopy comomentum maps
•produce new meaningful examples.

Multisymplectic geometry in a nutshell 1
Historical motivation
Mechanics: geometrical foundations of (first-order) field theories.
mechanics geometry
phase space symplectic manifold multisymplectic manifold
classical
observables Poisson algebra L∞-algebra
symmetries group actions admitting
comoment map
group actions admitting
homotopy comomentum map
| {z }
point-like particles systems | {z }
field-theoretic systems
Scope of the thesis
•Develop theory of homotopy comomentum maps
•produce new meaningful examples.

Multisymplectic geometry in a nutshell 1
Historical motivation
Mechanics: geometrical foundations of (first-order) field theories.
mechanics geometry
phase space symplectic manifold multisymplectic manifold
classical
observables Poisson algebra L∞-algebra
symmetries group actions admitting
comoment map
group actions admitting
homotopy comomentum map
| {z }
point-like particles systems | {z }
field-theoretic systems
Scope of the thesis
•Develop theory of homotopy comomentum maps
•produce new meaningful examples.

Multisymplectic manifolds 2
Def: n-plectic manifold (Cantrijn, Ibort, De Le´on)
M, ω Smooth Mfd.
non-degenerate, closed, (n+ 1)-form.
Def: Non-degenerate (n+ 1)-form
The ω[(flat) bundle map is
injective.
ω[:TM ΛnT∗M
(x,u) (x, ιuωx)
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Examples:
•n= 1 ⇒ωis a symplectic form
•n= (dim(M)−1) ⇒ωis a volume form
•Let Qa smooth manifold, the multicotangent bundle ΛnT∗Qis naturally
n-plectic. (cfr, GIMMSY construction for classical field theories)

Multisymplectic manifolds 2
Def: n-plectic manifold (Cantrijn, Ibort, De Le´on)
M, ω Smooth Mfd.
non-degenerate, closed, (n+ 1)-form.
Def: Non-degenerate (n+ 1)-form
The ω[(flat) bundle map is
injective.
ω[:TM ΛnT∗M
(x,u) (x, ιuωx)
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Examples:
•n= 1 ⇒ωis a symplectic form
•n= (dim(M)−1) ⇒ωis a volume form
•Let Qa smooth manifold, the multicotangent bundle ΛnT∗Qis naturally
n-plectic. (cfr, GIMMSY construction for classical field theories)

Multisymplectic manifolds 2
Def: n-plectic manifold (Cantrijn, Ibort, De Le´on)
M, ω Smooth Mfd.
non-degenerate, closed, (n+ 1)-form.
Def: Non-degenerate (n+ 1)-form
The ω[(flat) bundle map is
injective.
ω[:TM ΛnT∗M
(x,u) (x, ιuωx)
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Examples:
•n= 1 ⇒ωis a symplectic form
•n= (dim(M)−1) ⇒ωis a volume form
•Let Qa smooth manifold, the multicotangent bundle ΛnT∗Qis naturally
n-plectic. (cfr, GIMMSY construction for classical field theories)

Multisymplectic manifolds 2
Def: n-plectic manifold (Cantrijn, Ibort, De Le´on)
M, ω Smooth Mfd.
non-degenerate, closed, (n+ 1)-form.
Def: Non-degenerate (n+ 1)-form
The ω[(flat) bundle map is
injective.
ω[:TM ΛnT∗M
(x,u) (x, ιuωx)
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Examples:
•n= 1 ⇒ωis a symplectic form
•n= (dim(M)−1) ⇒ωis a volume form
•Let Qa smooth manifold, the multicotangent bundle ΛnT∗Qis naturally
n-plectic. (cfr, GIMMSY construction for classical field theories)

Multisymplectic manifolds 2
Def: n-plectic manifold (Cantrijn, Ibort, De Le´on)
M, ω Smooth Mfd.
non-degenerate, closed, (n+ 1)-form.
Def: Non-degenerate (n+ 1)-form
The ω[(flat) bundle map is
injective.
ω[:TM ΛnT∗M
(x,u) (x, ιuωx)
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Examples:
•n= 1 ⇒ωis a symplectic form
•n= (dim(M)−1) ⇒ωis a volume form
•Let Qa smooth manifold, the multicotangent bundle ΛnT∗Qis naturally
n-plectic. (cfr, GIMMSY construction for classical field theories)

Multisymplectic manifolds 2
Def: n-plectic manifold (Cantrijn, Ibort, De Le´on)
M, ω Smooth Mfd.
non-degenerate, closed, (n+ 1)-form.
Def: Non-degenerate (n+ 1)-form
The ω[(flat) bundle map is
injective.
ω[:TM ΛnT∗M
(x,u) (x, ιuωx)
Def: Hamiltonian (n−1)-forms
Ωn−1
ham (M, ω) :=σ∈Ωn−1(M)∃v
σ∈X(M) : dσ=−ιv
σω
Hamilton-DeDonder-Weyl
equation
Examples:
•n= 1 ⇒ωis a symplectic form
•n= (dim(M)−1) ⇒ωis a volume form
•Let Qa smooth manifold, the multicotangent bundle ΛnT∗Qis naturally
n-plectic. (cfr, GIMMSY construction for classical field theories)

Extended Bibliography I 3
J. F. Cari˜nena, M. Crampin, and L. A. Ibort.
On the multisymplectic formalism for first order field theories.
Differential Geometry and its Applications, 1(4):345–374, 1991.
Martin Callies, Ya¨el Fr´egier, Christopher L. Rogers, and Marco Zambon.
Homotopy moment maps.
Adv. Math. (N. Y)., 303:954–1043, nov 2016.
Mark J. Gotay, James Isenberg, Jerrold E. Marsden, and Richard
Montgomery.
Momentum maps and classical relativistic fields. part i: Covariant field theory.
jan 1997.
Jerzy Kijowski and Wlodzimierz M Tulczyjew.
A Symplectic Framework for Field Theories, volume 107 of Lecture Notes in
Physics.
Springer Berlin Heidelberg, Berlin, Heidelberg, 1979.

Extended Bibliography II 4
Antonio Michele Miti.
Homotopy comomentum maps in multisymplectic geometry.
Doctoral thesis, Universit`a Cattolica del Sacro Cuore & KU Leuven, April
2021.
Marco Zambon.
L-infinity algebras and higher analogues of dirac structures and courant
algebroids.
Journal of Symplectic Geometry, 10(4):563–599, mar 2010.