Homotopy Moment Maps Explored

• Homotopy moment maps are the L∞-algebraic analogues of classical moment maps, acting on pre-n-plectic and multisymplectic manifolds.
• They encode higher equivariance and compatibility conditions through L∞-morphisms, linking Lie algebra cohomology, equivariant cohomology, and Noether theory.
• Their framework enables explicit constructions, obstruction analysis, and applications in field theory, general relativity, and hydrodynamics.

Homotopy moment maps are the higher, $L_\infty$-algebraic analogue of ordinary moment maps for Lie group actions preserving a closed differential form of degree $>1$. For a pre-$n$-plectic or $n$-plectic manifold $(M,\omega)$, the relevant observables no longer form an ordinary Poisson algebra of functions, but an $L_\infty$-algebra of Hamiltonian forms. A homotopy moment map is correspondingly an $L_\infty$-morphism from the symmetry Lie algebra into that observable $L_\infty$-algebra, with first component satisfying the Hamiltonian condition $d f_1(x)=-\iota_{v_x}\omega$. The higher components encode the failure of strict equivariance and organize it by coherent homotopies, linking moment-map theory to equivariant cohomology, Lie algebra cohomology, Lie kernels, transgression, and higher versions of Noether theory (Callies et al., 2013, Fregier et al., 2014).

1. Geometric setting and the $L_\infty$-algebra of observables

The basic geometric input is a manifold $>1$0 equipped with a closed $>1$1-form $>1$2. The literature distinguishes pre-$>1$3-plectic manifolds, where $>1$4, from $>1$5-plectic or multisymplectic manifolds, where $>1$6 is also nondegenerate in the sense that $>1$7. For such a form, a differential form $>1$8 is Hamiltonian if there exists a vector field $>1$9 such that

$n$0

This generalizes the symplectic relation between Hamiltonian functions and Hamiltonian vector fields (Callies et al., 2013, Shahbazi et al., 2015).

The observables of $n$1 assemble into the Rogers $n$2-algebra $n$3, concentrated in degrees $n$4. In one common convention, degree $n$5 consists of Hamiltonian $n$6-forms, negative degrees consist of ordinary lower-degree forms, the unary bracket is $n$7 on negative degrees, and for $n$8 the higher brackets are given by repeated contraction of Hamiltonian vector fields into $n$9, with sign

$n$0

For $n$1, this collapses to the ordinary Poisson Lie algebra of functions on a symplectic manifold (Callies et al., 2013, Shahbazi et al., 2015).

In the 2-plectic case, the observable algebra is a Lie 2-algebra. Its underlying graded vector space has degree $n$2 given by Hamiltonian $n$3-forms and degree $n$4 given by functions $n$5, with

$n$6

The presence of the ternary bracket is the algebraic reason that higher moment maps take values in an $n$7-algebra rather than an ordinary Lie algebra (Mammadova et al., 2019).

2. Definition and higher compatibility equations

Let $n$8 be a Lie group with Lie algebra $n$9, acting on $(M,\omega)$0 by diffeomorphisms preserving $(M,\omega)$1, with infinitesimal generators $(M,\omega)$2. A homotopy moment map—also called a homotopy comoment map in part of the literature—is an $(M,\omega)$3-morphism

$(M,\omega)$4

lifting the infinitesimal action. Its first component is a map

$(M,\omega)$5

satisfying

$(M,\omega)$6

The higher components

$(M,\omega)$7

encode the higher coherence data (Callies et al., 2013, Miti, 2021).

A standard component form of the $(M,\omega)$8-morphism identities is

$(M,\omega)$9

with $L_\infty$0 and

$L_\infty$1

Equivalently, for $L_\infty$2,

$L_\infty$3

together with the top equation at $L_\infty$4 (Callies et al., 2013, Mammadova et al., 2020).

The low-degree meaning is already characteristic. In particular,

$L_\infty$5

so the failure of the first component to preserve brackets is exact and is repaired by $L_\infty$6. This is the higher-categorical replacement of strict equivariance (Blohmann, 2021). In the symplectic case $L_\infty$7, only $L_\infty$8 survives, and the definition reduces to the ordinary moment-map equation $L_\infty$9 (Herman, 2017, Herman, 2018).

The same principle extends beyond Lie algebra domains. For a minimal Lie 2-algebra $L_\infty$0, a homotopy moment map is an $L_\infty$1-morphism

$L_\infty$2

with components

$L_\infty$3

satisfying binary and ternary compatibility equations involving the Lie 2-algebra cocycle (Mammadova et al., 2019).

3. Cohomological formulations and obstruction theory

A central development in the subject is the cohomological reformulation of homotopy moment maps as primitives of a canonical cocycle. For a $L_\infty$4-action preserving a closed $L_\infty$5-form $L_\infty$6, one considers the total complex

$L_\infty$7

where $L_\infty$8 on $L_\infty$9. If $L_\infty$0, define

$L_\infty$1

Then $L_\infty$2, so $L_\infty$3 is $L_\infty$4-closed whenever $L_\infty$5 is closed. A collection of components $L_\infty$6 is the data of a homotopy moment map exactly when the corresponding total element $L_\infty$7 satisfies

$L_\infty$8

Homotopy moment maps are therefore precisely the primitives of the canonical cocycle $L_\infty$9 (Fregier et al., 2014).

This reformulation makes the relation with equivariant cohomology explicit. In the Cartan model, suitable equivariant extensions of $d f_1(x)=-\iota_{v_x}\omega$0 produce homotopy moment maps. In particular, if $d f_1(x)=-\iota_{v_x}\omega$1 is a Cartan cocycle with

$d f_1(x)=-\iota_{v_x}\omega$2

then one obtains a homotopy moment map by explicit contraction formulas. More general Cartan cocycles also yield homotopy moment maps, and the Bott–Shulman–Stasheff model provides an alternative simplicial construction (Callies et al., 2013, Fregier et al., 2014).

The same framework yields obstruction classes. For connected $d f_1(x)=-\iota_{v_x}\omega$3, the action defines a Lie algebra cocycle

$d f_1(x)=-\iota_{v_x}\omega$4

whose class $d f_1(x)=-\iota_{v_x}\omega$5 is independent of $d f_1(x)=-\iota_{v_x}\omega$6. If a homotopy moment map exists, then $d f_1(x)=-\iota_{v_x}\omega$7. Conversely, if $d f_1(x)=-\iota_{v_x}\omega$8 and

$d f_1(x)=-\iota_{v_x}\omega$9

then a homotopy moment map exists (Callies et al., 2013). In the relative theory, a new phenomenon appears: if

$L_\infty$0

and $L_\infty$1 is connected with $L_\infty$2, then every Hamiltonian lift $L_\infty$3 extends to a relative homotopy moment map, and no separate Lie-algebra-cohomology condition is needed (Dinamo, 8 Jul 2026).

4. Weak, relative, and Lie-algebroid extensions

A prominent variant is the weak homotopy moment map, defined only on the Lie kernel

$L_\infty$4

It is a collection

$L_\infty$5

satisfying

$L_\infty$6

Any full homotopy moment map restricts to a weak one, and the $L_\infty$7-th component of a homotopy moment map is the multi-moment map of Madsen–Swann (Herman, 2018). The cohomological existence and uniqueness theory for weak maps generalizes the classical symplectic theory directly. One formulation shows that if

$L_\infty$8

then there exists a not-necessarily equivariant weak homotopy $L_\infty$9-moment map, and under the same vanishing condition equivariant weak $>1$00-moment maps are unique (Herman, 2017). Another formulation states that if

$>1$01

then a weak $>1$02-moment map exists, while

$>1$03

implies uniqueness of equivariant weak $>1$04-moment maps (Herman, 2018).

The relation between weak and full moment maps is not automatic. The extension problem is controlled by the top obstruction

$>1$05

The action admits a homotopy moment map if and only if it admits a weak moment map and $>1$06. For a fixed weak moment map, strict extension to a full homotopy moment map is governed by an additional cohomology class $>1$07 (Mammadova et al., 2020).

Two further generalizations enlarge the symmetry side. For a Lie algebroid $>1$08 over a pre-$>1$09-plectic manifold $>1$10, a homotopy momentum section is a formal sum

$>1$11

satisfying

$>1$12

For action Lie algebroids, an equivariant homotopy momentum section reproduces the standard homotopy moment map equations, while the weak version is defined on the Lie algebroid Lie kernel (Hirota et al., 2021). In the relative setting, for a smooth map $>1$13 with relative pre-$>1$14-plectic structure $>1$15, a relative homotopy moment map is an $>1$16-morphism

$>1$17

and is equivalent to an ordinary homotopy moment map on $>1$18 together with a coherent $>1$19-twisted trivialization of its pullback to $>1$20 (Dinamo, 8 Jul 2026).

5. Constructions and representative examples

Several general constructions recur throughout the literature. If $>1$21 and $>1$22 is $>1$23-invariant, then there is an explicit homotopy moment map with components

$>1$24

This covers exact pre-$>1$25-plectic forms and constant higher forms on vector spaces (Callies et al., 2013). A genuinely multisymplectic construction, with no direct symplectic analogue, is the product

$>1$26

where homotopy moment maps on the factors induce one on the product, and the observable $>1$27-algebras of the factors embed into that of the product by an explicit $>1$28-morphism (Shahbazi et al., 2015). Transgression provides another mechanism: a homotopy moment map on $>1$29 induces one on loop space $>1$30 and, more generally, on mapping spaces $>1$31 by integrating forms along $>1$32 (Fregier et al., 2014).

The theory has a substantial example base. For compact group actions on multisymplectic spheres with their standard volume form, the action admits a homotopy comoment map if and only if the sphere dimension is even or the action is not transitive (Miti, 2021). For spaces of connections on principal bundles over compact oriented $>1$33-manifolds, invariant polynomials define closed $>1$34-forms on the affine space of connections, and the gauge group admits a homotopy moment map; the reduced space of flat connections then inherits a closed $>1$35-form generalizing the Atiyah–Bott symplectic form (Callies et al., 2013).

Related higher moment-map notions appear in 2-plectic geometry. A multi-moment map for a closed $>1$36-form is an equivariant map

$>1$37

satisfying

$>1$38

where $>1$39 is the Lie kernel. Its existence and uniqueness are controlled by Lie algebra Betti numbers $>1$40 and $>1$41, and the case $>1$42 yields the $>1$43-trivial class (Madsen et al., 2010). Weak and full homotopy moment maps recover this Lie-kernel viewpoint in higher degrees (Herman, 2018).

Further explicit examples include closed $>1$44-geometry and hydrodynamics. On a closed $>1$45-manifold, a $>1$46-form $>1$47 is Hamiltonian precisely when $>1$48 lies in the $>1$49 component, and the corresponding Hamiltonian vector field is the curl,

$>1$50

with generalized Poisson bracket given by the $>1$51 cross product of curls. A diagonal $>1$52-action on $>1$53 with the standard closed $>1$54-form carries a full homotopy moment map (Herman, 2018). On $>1$55 with its standard volume form, the Lie algebra of divergence-free vector fields admits a hydrodynamical homotopy comoment map with

$>1$56

and this construction transgresses to the standard hydrodynamical comoment map (Miti, 2021).

6. Field-theoretic formulations, Noether theory, and general relativity

Homotopy moment maps enter field theory most explicitly through multisymplectic and variational-bicomplex formalisms. In the variational bicomplex of a local Lagrangian field theory, forms on the infinite jet bundle $>1$57 are bigraded by vertical and horizontal degree, with differentials $>1$58 and $>1$59 satisfying

$>1$60

For a Lagrangian $>1$61,

$>1$62

where $>1$63 is the Euler–Lagrange form and $>1$64 is a boundary form. The Lepage form

$>1$65

is a primitive of the premultisymplectic form

$>1$66

If a Lie algebra action preserves $>1$67, then

$>1$68

defines a homotopy momentum map (Blohmann, 2021).

For general relativity, this abstract mechanism is realized on the configuration bundle of Lorentzian metrics. The Hilbert–Einstein Lagrangian is $>1$69, and the Euler–Lagrange form is

$>1$70

The infinitesimal diffeomorphism action of a spacetime vector field $>1$71 prolongs to the jet bundle and splits into a strictly vertical evolutionary part $>1$72 and a strictly horizontal Cartan lift $>1$73, giving the diagonal action

$>1$74

A key point is that for general relativity this combined action is the correct one; acting only vertically is not enough (Blohmann, 2021).

The central theorem is that the Lepage form is invariant under the diagonal action,

$>1$75

and therefore the diffeomorphism action admits a homotopy momentum map

$>1$76

Its first component extends the Noether current map, and for a manifest symmetry $>1$77 one has

$>1$78

In general relativity the Noether current is

$>1$79

and the exact correction term is essential if $>1$80 is to be an $>1$81-morphism. This upgrades the usual “symmetry $>1$82 conserved current” correspondence to a homotopy-theoretic version of Noether’s first theorem (Blohmann, 2021).

Related field-theoretic structures appear in sigma models and quasi-Hamiltonian geometry. For Lie-algebroid gauged nonlinear sigma models with Wess–Zumino term, gauge invariance of the action reorganizes precisely into the homotopy momentum section equations, and the gauged sigma model has a homotopy Hamiltonian Lie algebroid structure if and only if the target pre-multisymplectic manifold carries a homotopy momentum section (Hirota et al., 2021). In the relative theory, quasi-Hamiltonian $>1$83-spaces with group-valued moment map $>1$84 fit the framework of relative 2-plectic geometry: the pair $>1$85 built from the Cartan $>1$86-form is a relative $>1$87-plectic structure, and every quasi-Hamiltonian $>1$88-space carries a canonical relative homotopy moment map, with the entire moment-map datum living on the group side and the $>1$89-component vanishing in the splitting theorem (Dinamo, 8 Jul 2026).