2.4 Symmetries and Homotopy comomentum maps
2.4.1 Homotopy comomentum maps
When studying weakly Hamiltonian actions on symplectic manifolds, the auxiliary concept of moment map takes an exceptionally important role. The latter is in particular instrumental in many celebrated results in symplectic geometry like the Hamiltonian formulation of the Noether theorem, the classification of toric manifolds and the Kostant-Kirillov-Souriau coadjoint orbits method (see [CdS01] for a complete review). Most of these results have also a non-trivial application in the geometrical approach to mechanics (see, for instance, [AMM78]).
Reminder 2.4.5 (Moment maps in symplectic geometry). Let be (M, ω) a symplectic (i.e. 1-plectic) manifold. Consider a symplectic smooth action ϑ : G M preserving the 2-form ω. Denote by v : G → Xmsy(M, ω) the corresponding infinitesimal action by fundamental vector fields. We define the following:
• A weak moment map pertaining to ϑ is a smooth map ˆµ : M → g∗ such that dhˆµ(x), ξi = −ιvξωx ∀x ∈ M, ξ ∈ g .
• A (strong) moment map pertaining to ϑ is a Ad∗-equivariant weak moment map, i.e. the following diagram commutes in the category of smooth manifolds
M g∗
M g∗
ˆ µ
ϑg Ad∗g ˆ
µ
.
Dually, see for instance [CdS01, §22.1], one can define the following:
• A weak comoment map pertaining to ϑ is a linear map ˇµ : g → C∞(M) such that
d ˇµ(ξ) = −ιvξω ∀ξ ∈ g
• A (strong) comoment map pertaining to ϑ is a weak comoment map that is also a Lie algebra morphism, i.e. the following diagram commutes in the category of vector spaces
g⊗2 C∞(M)⊗2
g C∞(M)
ˇ µ⊗2
[·,·] [·,·]
ˇ µ
.
The duality, hence the "co" in the namings of the previous list, comes from the fact that ˇµ is the dual of ˆµ in the following sense
ˆµ(ξ)
p= hˆµp, ξi ∀ξ ∈ g, p ∈ M . (2.26) In other words
ˆµ ∈ Homsmooth M, Homvect(g, R) , ˇµ ∈ Homvect g, Homsmooth(M, R)
come respectively from the currying [na20d] of the same "evaluation" vector bundle map
µ: M × g → RM
with respect to the first or second entry.
The upshot is that a (co-)moment can exist only if the action is weakly Hamiltonian in the sense of definition 2.4.3. In that case, a weak comoment map is precisely a choice of Hamiltonian form for any fundamental vector field, hence the following diagram commutes in the category of vector spaces
C∞(M)
g X
πham
v ˇ
µ .
If the comoment map is "strong", the action is said to be (strongly) Hamiltonian and the previous diagram commutes in the category of Lie algebras.
The term "moment" comes from the following crucial examples that is prototypical in geometric mechanics:
Example 2.4.6 (Linear and angular momenta). Given an action ϑ : G Q, its lift ϑL : G M = T∗Q acts via symplectic vector fields with respect to the canonical symplectic form. In particular this action preserves the tautological 1-form θ and it can be shown to be Hamiltonian with comoment map given by
ˇµ: g C∞(M) ξ −ιvξθ .
If one takes Q = R3, Q and T∗Q ∼= R6can be interpreted as the configuration spaceand the phase space of a point-particle in the physical space. The comoment map with respect to the action of the translation or the rotation group on Q gives, respectively, the linear and angular momenta of the point-particle freely moving in space [CdS01, §22.4].
In the multisymplectic context, the generalization of the (co)momentum maps of the symplectic case leads to the more refined concept of "homotopy comomentum map ".
Definition 2.4.7(Homotopy comomentum maps [CFRZ16]). Let v : g → X(M) be a multisymplectic Lie algebra action, i.e. it preserves the symplectic form ω ∈ Ωn+1(M). We call a homotopy comomentum map pertaining to v any L∞-morphism
(f) =n
fk : Λgk →(L∞(M, ω))1−k⊆Ωn−ko
k=1,...,n
from g to L∞(M, ω) satisfying
df1(ξ) = −ιvξω ∀ξ ∈ g .
Notation 2.4.8. A group action G (M, ω) is called Hamiltonian, if the corresponding infinitesimal Lie algebra action admits a homotopy comomentum map. We will often refer to the homotopy comomentum map of a certain Lie group action understanding it as the homotopy comomentum map corresponding to the infinitesimal action for the Lie algebra of the given group.
Remark 2.4.9. More conceptually, a homotopy comomentum map is an L∞ -morphism (f) : g → L∞(M, ω) lifting the action v : g → X(M), i.e. making the following diagram commutative in the L∞-algebras category:
L∞(M, ω)
g X(M)
πham
(f )
v
where the vertical arrow πham is the trivial L∞-extension of the linear function mapping any Hamiltonian form to the unique corresponding Hamiltonian vector field given in remark 2.3.12.
Hence, comparing with remark 2.4.4, one can see that the existence of a homotopy comomentum map is a stronger condition in the sense that it requires a lift of v not only in the plain vector space category but in the L∞-algebra category
L∞(M, ω)
g Ωn−1ham(M, ω)
Xham(M, ω)
πham
v f1
(f )
πham
.
In the following we will often make use of an explicit version of definition 2.4.7 subsumed by the following lemma:
Lemma 2.4.10 ([CFRZ16]). A homotopy comomentum map (f) for the infinitesimal multisymplectic action of g on M is given explicitly by a sequence of linear maps
(f) =n
fi : Λig→Ωn−i(M) | 0 ≤ i ≤ n + 1o fulfilling a set of equations:
− fk−1(∂p) = dfk(p) + ς(k)ι(vp)ω (2.27) together with the condition
f0= fn+1= 0
for all p ∈ Λkg and k = 1, . . . n + 1. (Recall that ∂ denotes the Chevalley-Eilenberg boundary operator defined in equation (1.30) and ς(k) is the sign coefficient given in equation (2.8).)
Proof. Equation (2.27) is a simple application of remark 1.2.36 to the grounded L∞-algebra of higher observables.
Remark 2.4.11. Formula (2.27) can be read as follows: when k = 1 it tells us that v acts via Hamiltonian vector fields and f1 is a linear map choosing an Hamiltonian form f1(x) pertaining to the Hamiltonian vector field vxfor any x ∈ g. Hence, f1 is the choice of a primitive for the contraction of ω with any fundamental vector field of the action.
When k ≥ 2, equation (2.27) can be read as the condition that the auxiliary closed differential form
µk := fk−1(∂p) + ς(k)ι(vp)ω
must actually be exact, with potential −fk(p). Closure of µk is again a consequence of lemma 2.1.10 together with d ω = 0. Namely one has:
d µk = d(fk−1(∂p) + ς(k)ι(vp)ω) =
= ς(k)(−1)kι(v∂p)ω − ς(k − 1)ι(v∂p)ω =
= [−ς(k + 1) − ς(k − 1)]ι(v∂p)ω = 0 .
(2.28)
The previous remark leads to the following sufficient condition to the existence of a homotopy comomentum map:
Theorem 2.4.12 ([CFRZ16, Thm. 9.6]). Consider a Lie algebra g acting on a multisymplectic manifold (M, ω). The action v : g → X(M) admits a homotopy comomentum map if it acts by Hamiltonian vector fields, i.e. if there exists a function φ : g → Ωn−1ham(M, ω) such that d φ = −ιvω, and HdRk (M) = 0 all 1 ≤ k ≤ n − 1.
Remark 2.4.13 ([CFRZ16, Rem. 9.7]). The previous theorem applies also when the infinitesimal Lie algebra action v comes from an infinite-dimensional Lie group G that is locally exponential.
Remark 2.4.14. The reason why the term "homotopy" appears in the definition of the multisymplectic analogue of an ordinary comoment map is due to the fact that it can be interpreted as a homotopy between cochain-complexes. Observe that the components of (f) can be arranged in the following (non-commutative) diagram inside the category of vector spaces:
Vn+2
g Vn+1
g Vn
g · · · V2
g V1
g 0
0 C∞(M) Ω1(M) · · · Ωn−1(M) Ωn(M) · · ·
∂
ιn+2g ω
∂
fn+1 ιn+1g ω fn ιngω
∂
ι2gω
∂
ι1gω f1
d d d
The top line can be interpreted as the reversed Chevalley-Eilenberg chain complex, i.e \CE(g) = S•(g[1]) and the line below as a suitable shifted truncation of the de Rham complex, TrnΩ(M)[n + 1] ⊃ L, including the chain complex underlying L∞(M, ω). Hence, equation (2.27) express the condition that (f) is a chain homotopy between the chain map ιgω(see remark 2.2.24)
and the zero map, i.e.
\CE(g) TrnΩ(M)[n + 1]
ιg
0 (f )
Remark 2.4.15. The universal property of the homotopical pullback introduced in remark 2.3.15 implies that if a weakly Hamiltonian action v : g → Xham(M) satisfies the property that (ιg) := ((ιX) ◦ v) is null-homotopic, then there exists a (f) (unique modulo homotopy) such that the following diagram commutes up to homotopies
g
L∞(M, ω) 0
X1(M) B
v
∃!(f )
y
πham
(ιX)
.
The latter means that the two outer triangles commute and there exists two homotopies (ιXham) ⇒ 0 and (ιg) ⇒ 0.
Remark 2.4.16. As noted in [CFRZ16, Rem. 5.2], one can generalize the standard characterization of the image of a comoment map as a Poisson sub-algebra in symplectic geometry also in the multisymplectic case.
In the latter case, the image of (f) has to be understood as the cochain complex I ,→ Lgiven by the following components
Ik=(Im(fn) ⊂ C∞(M) if k = 1 − n Im(f1−k) ⊕ d Im(f2−k) ⊂ Ωn+k−1(M) if 2 − n ≤ k ≤ 0 . Remark 2.4.17 (Relation with other notions of multisymplectic moment maps).
It is important to remark that definition 2.4.7 is not the only notion of
"multisymplectic" moment map proposed in the literature. We mention among other the so-called covariant multimoment map[CCI91] (see also [GIM+98]), the multimoment map[MS12b] and the weak moment map[Her18b]. The relationship of these notions with the definition of homotopy comomentum map can be read in [CFRZ16, §12] and [MR20a].
A big part of chapters 5 and 3 will be devoted to give new explicit constructions of homotopy comomentum maps. Several other examples can be found in [CFRZ16][RW19].