Lie 2-algebra moment maps in multisymplectic geometry

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Abstract

Consider a closed non-degenerate 3-form ω with an infinitesimal action of a Lie algebra g. Motivated by the fact that the observables associated to ω form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra g.

We formulate existence criteria and provide a construction for such homotopy moment maps, by characterizing them in terms of cohomology.

Introduction

In symplectic geometry moment maps play an important role, leading to celebrated theorems: the Marsden-Weinstein-Meyer symplectic reduction, the Atiyah-Guillemin-Sternberg convexity theorem, and the classification of toric symplectic manifolds via Delzant polytopes. Given an action of a Lie group G on a symplectic manifold M, a moment map can be equivalently described as a Lie algebra morphism g(C(M),{,}) realizing the generators of the action as Hamiltonian vector fields, where {,} is the Poisson bracket on functions that encodes the symplectic structure on M.

Here we consider 2-plectic forms, i.e. closed non-degenerate 3-forms. In that case C(M) no longer carries a Poisson bracket. However, it can be enlarged to a Lie 2-algebra (a simple kind of L-algebra) canonically attached to ω, which we denote by L(M,ω). A g moment map is then defined as an L-algebra morphism gL(M,ω) compatible with the action, see [3, Prop. 5.1]. There it is shown (in the wider setting of multisymplectic forms of arbitrary degree) that examples abound, and a link with equivariant cohomology is established.

In this article we go one step further, replacing the Lie algebra g by a Lie 2-algebra L having g as its degree 0 component. There are two main motivations for this:

  • As a g moment map is an L-algebra morphism, from an algebraic point of view it is natural to let the domain be an L-algebra rather than just a Lie algebra.

  • While an action by Hamiltonian vector fields might not admit a g moment map, it always admits a moment map for a specific Lie 2-algebra having g in degree zero, provided that H1(M)=0 ([3, Prop. 9.10], which we recall in Proposition 4.3). Furthermore, even when H1(M)0, it admits a moment map for some Lie 2-algebra (see Proposition A.1).

Main results. Let (M,ω) be a 2-plectic manifold, and let gX(M),xvx be a Lie algebra morphism taking values in Hamiltonian vector fields. Let L be a Lie 2-algebra with vanishing unary bracket, whose degree 0 component is the Lie algebra g.
  • In §3 we show that L moment maps are in bijection with the primitives of a certain 3-cocycle ω˜ (constructed out of the g action on (M,ω)) in the total complex CE(L)Ω(M). Here CE(L) denotes the Chevalley-Eilenberg complex of the Lie 2-algebra L.

  • While the (very large) complex CE(L)Ω(M) captures all the information about L moment maps, it turns out that the complex CE(L) itself captures much of this information, as we show in §4. The g action on (M,ω) defines a 3-cocycle ω3p3g in the Chevalley-Eilenberg complex of the Lie algebra g [3, §9]. Since the projection Lg is an L-morphism, ω3p can be regarded as a 3-cocycle in CE(L). We summarize Theorem 4.9 and Proposition 4.14:

    Theorem

    i) A necessary condition for the existence of L moment maps is that [ω3p]CE(L)=0.

    ii) Assume H1(M)=0 and [ω3p]CE(L)=0. Then out of every primitive ηCE(L)2 of ω3p one can construct an L moment map ϕη. This construction recovers all L moment maps, up to inner equivalence.

    Note that the above is not only an existence statement, but provides a constructive way to obtain L moment maps.

  • The condition [ω3p]CE(L)=0 is not very explicit, because it is expressed in terms of the (quite large) complex CE(L). In §5 we express this condition in terms of the familiar Lie algebra cohomology of g, see Proposition 5.3. As a consequence we can provide criteria – which are easy to check in practice – for the existence or non-existence of L moment maps. They are expressed in terms of a certain 3-cocycle cred of the Lie algebra g with values in a trivial representation, induced by the ternary bracket of the Lie 2-algebra L. We summarize these criteria as follows (Proposition 5.5 and Proposition 5.7):

    Proposition

    Assume [ω3p]g0.

    i) If [cred]g=0, then there exists no L moment map.

    ii) Assume H1(M)=0. If H3(g) is one-dimensional and [cred]g0, then there exists an L moment map.

    We also give an alternative characterization of the condition [ω3p]CE(L)=0 in Proposition 5.11

In this diagram we summarize the relation between the cohomologies (and the relevant classes) of the three complexes that appear in §3, §4, §5 respectively:H(CE(L)Ω(M))rH(CE(L))H(g)[ω˜][ω3p]CE(L)[ω3p]g Generalizations. All the results of this article also hold for (possibly degenerate) closed 3-forms, with simple modifications.

Furthermore, we expect similar results to hold when ω is an n-plectic form and L is a Lie n-algebra, for arbitrary n2. For the results of §3 we expect this because in the defining conditions for L-morphisms from a Lie n-algebra to L(M,ω), all the terms that are quadratic and higher (in the morphism) can be expressed using the prescribed Lie algebra action.1 For the results of §4, we expect this because of [3, Prop. 9.10]. However, for arbitrary n, we do not expect that existence criteria can be phrased in terms of Lie algebra cohomology as explicitly as in §5.

Conventions. Given a graded vector space V=iZVi and an integer k, we denote by V[k] the graded vector space obtained from V by shifting the degrees by k. Explicitly, its degree i component is (V[k])i=Vi+k.

Acknowledgments. M.Z. thanks Domenico Fiorenza for useful conversations. We thank the referee for her/his comments and insights. We acknowledge partial support by the long term structural funding – Methusalem grant of the Flemish Government, the FWO under EOS project G0H4518N, the FWO research project G083118N (Belgium).

Section snippets

Lie algebra actions on multisymplectic manifolds

We begin by recalling the relevant notions from multisymplectic geometry. Throughout this article, M denotes a connected manifold.

Lie 2-algebra actions on 2-plectic manifolds

Since the “observables” L(M,ω) on an n-plectic manifold form an L-algebra, it is natural to relax the definition of g moment map by allowing g to be an L-algebra rather than just a Lie algebra. In this section we do this in the simplest case, i.e., for n=2.

A cohomological characterization of Lie 2-algebra moment maps

The main observation in [6] and [12] is that there is a complex that allows to efficiently encode moment maps for Lie algebra actions, showing in particular that the latter form an affine subspace. In this section we obtain an analogous result for Lie 2-algebras.

Let h[1]g be a minimal Lie 2-algebra, let ω be a 2-plectic form on a manifold M, and let gX(M),xvx be a Lie algebra morphism taking values in Hamiltonian vector fields.

Existence results and a construction

We use the characterization of moment maps for Lie 2-algebras given in §3 to obtain existence results and construct explicit examples.

Again, let h[1]g be a minimal Lie 2-algebra, let ω be a 2-plectic form on a manifold M, and let gX(M),xvx be a Lie algebra morphism taking values in Hamiltonian vector fields.

Revisiting the existence results

As earlier, let h[1]g be a minimal Lie 2-algebra, let ω be a 2-plectic form on a manifold M, and let gX(M),xvx be a Lie algebra morphism taking values in Hamiltonian vector fields.

An answer to the existence question for h[1]g moment maps was given in Corollary 4.11 in terms of the cohomology of the Chevalley-Eilenberg complex CE(L) of the Lie 2-algebra. However, the latter complex is quite large and involved. In this section we rephrase that answer in two ways: one that is explicit and

Examples

We now present instances in which moment maps for Lie 2-algebras exist, using the explicit criteria developed in §5.1.

In this section g is always a Lie algebra, h a g-representation, and c a 3-cocycle for this representation. (We remind that this triple of data is equivalent to a minimal Lie 2-algebra structure on h[1]g, see §2.2.) Furthermore, (M,ω) is a 2-plectic manifold, which we assume to satisfyH1(M)=0, and gX(M),xvx is a Lie algebra morphism taking values in Hamiltonian vector fields.

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