---
title: Homotopy Moment Maps Explored
url: https://www.emergentmind.com/topics/homotopy-moment-maps
type: topic
---

# Homotopy Moment Maps Explored

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 [1304.2051][1409.3142].

## 1. Geometric setting and the \(L_\infty\)-algebra of observables

The basic geometric input is a manifold \(M\) equipped with a closed \((n+1)\)-form \(\omega\). The literature distinguishes **pre-\(n\)-plectic** manifolds, where \(d\omega=0\), from **\(n\)-plectic** or multisymplectic manifolds, where \(\omega\) is also nondegenerate in the sense that \(\iota_v\omega=0\Rightarrow v=0\). For such a form, a differential form \(\alpha\in\Omega^{n-1}(M)\) is **Hamiltonian** if there exists a vector field \(v_\alpha\) such that
\[
d\alpha=-\,\iota_{v_\alpha}\omega.
\]
This generalizes the symplectic relation between Hamiltonian functions and Hamiltonian vector fields [1304.2051][1504.08194].

The observables of \((M,\omega)\) assemble into the Rogers \(L_\infty\)-algebra \(L_\infty(M,\omega)\), concentrated in degrees \(1-n,\dots,0\). In one common convention, degree \(0\) consists of Hamiltonian \((n-1)\)-forms, negative degrees consist of ordinary lower-degree forms, the unary bracket is \(l_1=d\) on negative degrees, and for \(k\ge 2\) the higher brackets are given by repeated contraction of Hamiltonian vector fields into \(\omega\), with sign
\[
\varsigma(k)=-(-1)^{k(k+1)/2}.
\]
For \(n=1\), this collapses to the ordinary Poisson Lie algebra of functions on a symplectic manifold [1304.2051][1504.08194].

In the 2-plectic case, the observable algebra is a **Lie 2-algebra**. Its underlying graded vector space has degree \(0\) given by Hamiltonian \(1\)-forms and degree \(-1\) given by functions \(C^\infty(M)\), with
\[
l_1(f)=df,\qquad
l_2(\alpha,\beta)=\iota_{v_\beta}\iota_{v_\alpha}\omega,\qquad
l_3(\alpha,\beta,\gamma)=\iota_{v_\gamma}\iota_{v_\beta}\iota_{v_\alpha}\omega.
\]
The presence of the ternary bracket is the algebraic reason that higher moment maps take values in an \(L_\infty\)-algebra rather than an ordinary Lie algebra [1901.10842].

## 2. Definition and higher compatibility equations

Let \(G\) be a Lie group with Lie algebra \(\mathfrak g\), acting on \(M\) by diffeomorphisms preserving \(\omega\), with infinitesimal generators \(x\mapsto v_x\). A **homotopy moment map**—also called a **homotopy comoment map** in part of the literature—is an \(L_\infty\)-morphism
\[
f:\mathfrak g\rightsquigarrow L_\infty(M,\omega)
\]
lifting the infinitesimal action. Its first component is a map
\[
f_1:\mathfrak g\to \Omega^{n-1}_{\mathrm{Ham}}(M)
\]
satisfying
\[
d f_1(x)=-\,\iota_{v_x}\omega.
\]
The higher components
\[
f_k:\Lambda^k\mathfrak g\to \Omega^{n-k}(M),\qquad 1\le k\le n,
\]
encode the higher coherence data [1304.2051][2105.05645].

A standard component form of the \(L_\infty\)-morphism identities is
\[
-\,f_{k-1}(\partial p)=d f_k(p)+\zeta(k)\,\iota_{v_p}\omega,\qquad p\in \Lambda^k\mathfrak g,
\]
with \(f_0=f_{n+1}=0\) and
\[
\zeta(k)=-(-1)^{k(k+1)/2}.
\]
Equivalently, for \(2\le k\le n\),
\[
\sum_{1\le i<j\le k}(-1)^{i+j+1} f_{k-1}\bigl([x_i,x_j],x_1,\dots,\widehat{x_i},\dots,\widehat{x_j},\dots,x_k\bigr)
= d f_k(x_1,\dots,x_k)+\varsigma(k)\,\iota(v_{x_1}\wedge\cdots\wedge v_{x_k})\omega,
\]
together with the top equation at \(k=n+1\) [1304.2051][2001.00264].

The low-degree meaning is already characteristic. In particular,
\[
\mu_1([a_1,a_2]) - d\mu_2(a_1,a_2) = \{\mu_1(a_1),\mu_1(a_2)\},
\]
so the failure of the first component to preserve brackets is exact and is repaired by \(\mu_2\). This is the higher-categorical replacement of strict equivariance [2103.07670]. In the symplectic case \(n=1\), only \(f_1\) survives, and the definition reduces to the ordinary moment-map equation \(df(\xi)=-V_\xi\lrcorner\omega\) [1711.02572][1807.01641].

The same principle extends beyond Lie algebra domains. For a minimal Lie 2-algebra \(h[1]\oplus\mathfrak g\), a homotopy moment map is an \(L_\infty\)-morphism
\[
(f_1,f_2):h[1]\oplus\mathfrak g\to L(M,\omega),
\]
with components
\[
f_1|_{\mathfrak g}:\mathfrak g\to \Omega^1_{\mathrm{Ham}}(M),\qquad
f_1|_h:h\to C^\infty(M),\qquad
f_2:\Lambda^2\mathfrak g\to C^\infty(M),
\]
satisfying binary and ternary compatibility equations involving the Lie 2-algebra cocycle [1901.10842].

## 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 \(G\)-action preserving a closed \((n+1)\)-form \(\omega\), one considers the total complex
\[
\bigl(\wedge^{\ge 1}\mathfrak g^*\otimes \Omega(M),\, d_{\mathrm{tot}}\bigr),
\]
where \(d_{\mathrm{tot}}=d_{\mathrm{CE}}+(-1)^k d\) on \(\wedge^k\mathfrak g^*\otimes\Omega(M)\). If \(\sigma\in\Omega^N(M)^G\), define
\[
\sigma_k(x_1,\dots,x_k)=\iota(v_{x_1}\wedge\cdots\wedge v_{x_k})\sigma,\qquad
\widetilde\sigma=\sum_{k=1}^N(-1)^{k-1}\sigma_k.
\]
Then \(d_{\mathrm{tot}}\widetilde\sigma=\widetilde{d\sigma}\), so \(\widetilde\omega\) is \(d_{\mathrm{tot}}\)-closed whenever \(\omega\) is closed. A collection of components \(f_k\) is the data of a homotopy moment map exactly when the corresponding total element \(f\) satisfies
\[
d_{\mathrm{tot}}f=\widetilde\omega.
\]
Homotopy moment maps are therefore precisely the primitives of the canonical cocycle \(\widetilde\omega\) [1409.3142].

This reformulation makes the relation with equivariant cohomology explicit. In the Cartan model, suitable equivariant extensions of \(\omega\) produce homotopy moment maps. In particular, if \(\omega-\mu\) is a Cartan cocycle with
\[
d\mu(x)=-\iota_{v_x}\omega,\qquad
L_{v_x}\mu(y)=\mu([x,y]),\qquad
\iota_{v_x}\mu(x)=0,
\]
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 [1304.2051][1409.3142].

The same framework yields obstruction classes. For connected \(M\), the action defines a Lie algebra cocycle
\[
c_p(x_1,\dots,x_{n+1}) = (-1)^{n+1}s(n+1)\,\omega_p(v_{x_1},\dots,v_{x_{n+1}})
\]
whose class \([c]\in H^{n+1}(\mathfrak g;\mathbb R)\) is independent of \(p\). If a homotopy moment map exists, then \([c]=0\). Conversely, if \([c]=0\) and
\[
H^k_{\mathrm{dR}}(M)=0,\qquad 1\le k\le n-1,
\]
then a homotopy moment map exists [1304.2051]. In the relative theory, a new phenomenon appears: if
\[
H^i(F)=0\quad\text{for }1\le i\le n-1,
\]
and \(N\) is connected with \(M\neq\emptyset\), then every Hamiltonian lift \(f_1\) extends to a relative homotopy moment map, and no separate Lie-algebra-cohomology condition is needed [2607.07088].

## 4. Weak, relative, and Lie-algebroid extensions

A prominent variant is the **weak homotopy moment map**, defined only on the **Lie kernel**
\[
P_{\mathfrak g,k}:=\ker(\partial:\Lambda^k\mathfrak g\to \Lambda^{k-1}\mathfrak g).
\]
It is a collection
\[
f_k:P_{\mathfrak g,k}\to \Omega^{n-k}(M)
\]
satisfying
\[
d f_k(p)= -\,\zeta(k)\, V_p\lrcorner\omega,\qquad p\in P_{\mathfrak g,k}.
\]
Any full homotopy moment map restricts to a weak one, and the \(n\)-th component of a homotopy moment map is the multi-moment map of Madsen–Swann [1807.01641]. The cohomological existence and uniqueness theory for weak maps generalizes the classical symplectic theory directly. One formulation shows that if
\[
H^0\!\left(\mathfrak g,\, P_{\mathfrak g,k}^{*}\otimes \Omega^{n-k}_{\mathrm{cl}}(M)\right)=0,
\]
then there exists a not-necessarily equivariant weak homotopy \(k\)-moment map, and under the same vanishing condition equivariant weak \(k\)-moment maps are unique [1711.02572]. Another formulation states that if
\[
H^1(\mathfrak g;P_{\mathfrak g,k})=0,
\]
then a weak \(k\)-moment map exists, while
\[
H^0(\mathfrak g;P_{\mathfrak g,k}^{\mathrm{cl}})=0
\]
implies uniqueness of equivariant weak \(k\)-moment maps [1807.01641].

The relation between weak and full moment maps is not automatic. The extension problem is controlled by the top obstruction
\[
\phi:P_{n+1,\mathfrak g}\to C^\infty(M),\qquad p\mapsto \iota_{v_p}\omega.
\]
The action admits a homotopy moment map if and only if it admits a weak moment map and \(\phi\equiv 0\). For a fixed weak moment map, strict extension to a full homotopy moment map is governed by an additional cohomology class \([\gamma]\in H^{n+1}(\widetilde C)\) [2001.00264].

Two further generalizations enlarge the symmetry side. For a Lie algebroid \((E,\rho,[\ ,\ ])\to M\) over a pre-\(n\)-plectic manifold \((M,\omega)\), a **homotopy momentum section** is a formal sum
\[
\mu=\sum_{k=0}^{n-1}\mu_k,\qquad \mu_k\in \Omega^k\!\left(M,\wedge^{n-k}E^*\right),
\]
satisfying
\[
(\nabla^E+d^E)\mu=-\,\iota_\rho\omega.
\]
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 [2110.12305]. In the **relative** setting, for a smooth map \(F:M\to N\) with relative pre-\(n\)-plectic structure \(\varpi=(\omega,\eta)\), a relative homotopy moment map is an \(L_\infty\)-morphism
\[
f:\mathfrak g\rightsquigarrow L_\infty(F,\varpi)
\]
and is equivalent to an ordinary homotopy moment map on \((N,\omega)\) together with a coherent \(\eta\)-twisted trivialization of its pullback to \(M\) [2607.07088].

## 5. Constructions and representative examples

Several general constructions recur throughout the literature. If \(\omega=d\alpha\) and \(\alpha\) is \(G\)-invariant, then there is an explicit homotopy moment map with components
\[
f_k(x_1,\dots,x_k)=(-1)^{k-1}s(k)\,\iota_{v_{x_1}}\cdots \iota_{v_{x_{k-1}}}\alpha.
\]
This covers exact pre-\(n\)-plectic forms and constant higher forms on vector spaces [1304.2051]. A genuinely multisymplectic construction, with no direct symplectic analogue, is the product
\[
(M_a\times M_b,\omega_a\wedge\omega_b),
\]
where homotopy moment maps on the factors induce one on the product, and the observable \(L_\infty\)-algebras of the factors embed into that of the product by an explicit \(L_\infty\)-morphism [1504.08194]. Transgression provides another mechanism: a homotopy moment map on \((M,\omega)\) induces one on loop space \(LM\) and, more generally, on mapping spaces \(M^\Sigma\) by integrating forms along \(\Sigma\) [1409.3142].

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 [2105.05645]. For spaces of connections on principal bundles over compact oriented \((n+1)\)-manifolds, invariant polynomials define closed \((n+1)\)-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 \((n+1)\)-form generalizing the Atiyah–Bott symplectic form [1304.2051].

Related higher moment-map notions appear in 2-plectic geometry. A **multi-moment map** for a closed \(3\)-form is an equivariant map
\[
v:M\to P_{\mathfrak g}^*
\]
satisfying
\[
d(v,p)=p\lrcorner c,\qquad p\in P_{\mathfrak g},
\]
where \(P_{\mathfrak g}=\ker(\Lambda^2\mathfrak g\to\mathfrak g)\) is the Lie kernel. Its existence and uniqueness are controlled by Lie algebra Betti numbers \(b_2(\mathfrak g)\) and \(b_3(\mathfrak g)\), and the case \(b_2(\mathfrak g)=b_3(\mathfrak g)=0\) yields the \((2,3)\)-trivial class [1012.2048]. Weak and full homotopy moment maps recover this Lie-kernel viewpoint in higher degrees [1807.01641].

Further explicit examples include closed \(G_2\)-geometry and hydrodynamics. On a closed \(G_2\)-manifold, a \(1\)-form \(\alpha\) is Hamiltonian precisely when \(d\alpha\) lies in the \(\Lambda^2_7\) component, and the corresponding Hamiltonian vector field is the curl,
\[
X_\alpha=\operatorname{curl}(\alpha),
\]
with generalized Poisson bracket given by the \(G_2\) cross product of curls. A diagonal \(T^2\)-action on \(\mathbb R^7\) with the standard closed \(G_2\)-form carries a full homotopy moment map [1807.01641]. On \(\mathbb R^3\) with its standard volume form, the Lie algebra of divergence-free vector fields admits a hydrodynamical homotopy comoment map with
\[
f_1=\flat\circ \mathrm{curl}^{-1},\qquad
f_2=\Delta^{-1}\circ \delta\circ \mu_2,
\]
and this construction transgresses to the standard hydrodynamical comoment map [2105.05645].

## 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 \(J^\infty F\) are bigraded by vertical and horizontal degree, with differentials \(\delta\) and \(d\) satisfying
\[
\delta^2=d^2=\delta d+d\delta=0.
\]
For a Lagrangian \(L\in\Omega^{0,n}(J^\infty F)\),
\[
\delta L = EL - d\gamma,
\]
where \(EL\) is the Euler–Lagrange form and \(\gamma\) is a boundary form. The Lepage form
\[
\lambda=L+\gamma
\]
is a primitive of the premultisymplectic form
\[
\omega=d\lambda=EL+\delta\gamma.
\]
If a Lie algebra action preserves \(\lambda\), then
\[
\mu_k(a_1,\dots,a_k)=\iota_{\rho(a_1)}\cdots \iota_{\rho(a_k)}\lambda
\]
defines a homotopy momentum map [2103.07670].

For general relativity, this abstract mechanism is realized on the configuration bundle of Lorentzian metrics. The Hilbert–Einstein Lagrangian is \(L=R\), and the Euler–Lagrange form is
\[
EL = -G^{ab}\,\delta g_{ab}\wedge ,
\qquad
G^{ab}=\mathrm{Ric}^{ab}-\tfrac12 R g^{ab}.
\]
The infinitesimal diffeomorphism action of a spacetime vector field \(v\in X(M)\) prolongs to the jet bundle and splits into a strictly vertical evolutionary part \(\xi_v\) and a strictly horizontal Cartan lift \(\hat v\), giving the diagonal action
\[
\rho(v)=\xi_v+\hat v.
\]
A key point is that for general relativity this combined action is the correct one; acting only vertically is not enough [2103.07670].

The central theorem is that the Lepage form is invariant under the diagonal action,
\[
L_{\rho(v)}(L+\gamma)=0\qquad\text{for all }v\in X(M),
\]
and therefore the diffeomorphism action admits a homotopy momentum map
\[
\mu:X(M)\longrightarrow L_\infty(J^\infty,EL+\delta\gamma),
\qquad
\mu_k(v_1,\dots,v_k)=\iota_{\rho(v_1)}\cdots\iota_{\rho(v_k)}(L+\gamma).
\]
Its first component extends the Noether current map, and for a manifest symmetry \(v\) one has
\[
j_v = -\iota_{\hat v}L - \iota_{\xi_v}\gamma,\qquad
\mu_1(v)= -j_v+\iota_{\hat v}\gamma.
\]
In general relativity the Noether current is
\[
j_v = 2G^{ab}v_a \wedge \iota_{\hat{\partial}_b} + d\Bigl(\tfrac12(\nabla^a v^b-\nabla^b v^a)\,\iota_{\hat{\partial}_a}\iota_{\hat{\partial}_b}\Bigr),
\]
and the exact correction term is essential if \(\mu\) is to be an \(L_\infty\)-morphism. This upgrades the usual “symmetry \(\to\) conserved current” correspondence to a homotopy-theoretic version of Noether’s first theorem [2103.07670].

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 [2110.12305]. In the relative theory, quasi-Hamiltonian \(G\)-spaces with group-valued moment map \(\mu:M\to G\) fit the framework of relative 2-plectic geometry: the pair \((\eta,\omega)\) built from the Cartan \(3\)-form is a relative \(2\)-plectic structure, and every quasi-Hamiltonian \(G\)-space carries a canonical relative homotopy moment map, with the entire moment-map datum living on the group side and the \(M\)-component vanishing in the splitting theorem [2607.07088].

Source: https://www.emergentmind.com/topics/homotopy-moment-maps