Abstract
For a manifold M with an integral closed 3form ω, we construct a PU(H)bundle and a Lie groupoid over its total space, together with a curving in the sense of gerbes. If the form is nondegenerate, we furthermore give a natural Lie 2algebra quasiisomorphism from the observables of (M, ω) to the weak symmetries of the above geometric structure, generalising the prequantisation map of Kostant and Souriau.
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Acknowledgements
We thank Camille LaurentGengoux for useful discussions, notably on multiplicative tensors on Lie groupoids. We also wish to thank Jouko Mickelsson for a helpful remark concerning principal connections in infinite dimensions, and the anonymous referee for several recommendations improving the article. The research of Gabriel Sevestre was financially supported by the Région Grand Est in France.
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Appendices
Appendix A: Lie 2Algebras and Their Morphisms
In this appendix we assemble the definitions regarding Lie 2algebras and crossed modules of Lie algebras needed in this article. For a more complete treatment see [1].
Definition A.1
Let \(L_{\bullet }:=L_{1}\rightarrow L_{0}\) be a 2term complex of vector spaces. A Lie 2algebra structure on L_{∙} consists in (multi)linear graded antisymmetric maps {l_{k}1 ≤ k ≤ 3} with the degree of l_{k} being equal to 2−k, precisely

\(l_{1}:L_{1}\rightarrow L_{0}\)

\(l_{2}:{\Lambda }^{2} L_{\bullet }\rightarrow L_{\bullet }\)

\(l_{3}:{\Lambda }^{3} L_{0}\rightarrow L_{1}\)
such that the following equations hold, for x, y, z, t ∈ L_{0} and u, v ∈ L_{− 1}:
Definition A.2
A Lie algebra crossed module is given by two Lie algebras \(\mathfrak {h}\) and \(\mathfrak {g}\), a Lie algebra morphism \(\eta :\mathfrak {h}\rightarrow \mathfrak {g}\) and an action \(\vartheta :\mathfrak {g}\times \mathfrak {h}\rightarrow \mathfrak {h}\) by derivations such that

(A1)
\(\vartheta (\eta (v),w)=[v,w]_{\mathfrak {h}}\)

(A2)
\(\eta (\vartheta (X,w))=[X,\eta (w)]_{\mathfrak {g}}\)
for all \(v,w\in \mathfrak {h}\) and \(X\in \mathfrak {g}\). One denotes such a crossed module as a quadruple \((\mathfrak {h},\mathfrak {g},\eta ,\vartheta )\).
Remark A.3
A Lie algebra crossed module is naturally given the structure of a strict Lie 2algebra (i.e. a Lie 2algebra with third bracket identically 0) by setting for \(X,Y\in \mathfrak {g}\), \(v\in \mathfrak {h}\):

\(L_{0}:=\mathfrak {g}\), \(L_{1}:=\mathfrak {h}\)

l_{1} := η

\(l_{2}(X,Y):=[X,Y]_{\mathfrak {g}}\), l_{2}(X, v) := 𝜗(X, v)

l_{3} := 0
Note that one often writes [, ] for the operation l_{2} of the Lie 2algebra associated to a Lie algebra crossed module.
Example A.4
We will now describe the crossed module structure on the multiplicative vector fields on a groupoid, which is used in Section 3. We will only give the general construction without any proofs, all the details can be found in [4] (see also [23]). Let indeed \(G_{1}\rightrightarrows G_{0}\) be a Lie groupoid and \(\text {Lie}(G_{1})\rightarrow G_{0}\) the associated Lie algebroid. Recall that \(\text {Lie}(G_{1})=\ker (s_{*})_{\varepsilon (G_{0})}\). We note s, t the source respectively the target map, ε the unit map, i the inverse map, and R_{g}, L_{g} the right respectively left multiplication by an element g ∈ G_{1}. We have the right and leftinvariant vector fields on G_{1} associated to a section a ∈Γ(G_{0},Lie(G_{1})), given respectively by:
For \(\xi \in \mathfrak {X}_{mult}(G_{1})\), its action on the Lie algebroid sections is given by:
where the bracket on the right hand side is the bracket of vector fields on G_{1}. Since Γ(G_{0},Lie(G_{1})) is isomorphic as a \(C^{\infty }(G_{0})\)module to the rightinvariant vector fields on G_{1} and the resulting vector field \([\xi ,\overrightarrow {a}]\) is again rightinvariant, \([\xi ,\overrightarrow {a}]\) equals \(\overrightarrow {b}\) for a uniquely determined section b of Lie(G_{1}).
Thus the twoterm complex of Lie algebras:
where X(a) is the vector field on G_{0} associated to a via the anchor map of Lie(G_{1}), together with the action defined above, and the usual brackets on Γ(G_{0},Lie(G_{1})) and \(\mathfrak {X}_{mult}(G_{1})\), is a Lie algebra crossed module, hence a strict Lie 2algebra.
Considering a Lie 2algebra as a special case of a Lie \(\infty \)algebra one obtains immediately the following
Definition A.5
Let \((L_{\bullet }=L_{1}\rightarrow L_{0},\{l_{k}\})\) and \((L_{\bullet }^{\prime } =L_{1}^{\prime }\rightarrow L_{0}^{\prime } ,\{l_{k}^{\prime }\})\) be two Lie 2algebras. A Lie \({\infty }\)algebra morphism \(L_{\bullet }\xrightarrow {\Phi } L_{\bullet }^{\prime } \) is given by linear maps:

\({\Phi }_{1}:L_{\bullet }\rightarrow L_{\bullet }^{\prime }\)

\({\Phi }_{2}:{\Lambda }^{2} L_{\bullet }\rightarrow L_{\bullet }^{\prime } \)
with Φ_{1} of degree 0 and Φ_{2} of degree − 1, such that for every x, y, z ∈ L_{0} and u ∈ L_{− 1}:
Remark A.6
A Lie ∞algebra morphism between two Lie 2algebras is, of course, also called a Lie 2algebra morphism.
Remark A.7
Following a suggestion of Camille LaurentGengoux, we visualize a Lie 2algebra morphism \(L_{\bullet }\xrightarrow {\Phi } L^{\prime }_{\bullet }\) as below:
Note that the outer square is a commutative diagram since \({\Phi }_{1}\circ l_{1}=l^{\prime }_{1}\circ {\Phi }_{1}\). We underline that neither “subdiagrams” containing the diagonal commute, nor is Φ_{2} defined on the vector space L_{0} (nor does the form of the diagonal arrow indicate any kind of injectivity).
Since a Lie ∞algebra morphism is notably a morphism of cochain complexes, it induces a natural linear map on the level of cohomology.
Definition A.8
A cochain complex morphism is called a quasiisomorphism if the induced map in cohomology is an isomorphism. A Lie ∞algebra morphism between two Lie 2algebras (or Lie ∞algebras, in fact) is called a quasiisomorphism if viewed as a cochain morphism it is a quasiisomorphism.
Appendix B: The Case of Exact 2Plectic Manifolds
In this appendix, we expose the case of 2plectic manifolds (M, ω) with a potential χ ∈Ω^{2}(M) for the 3form ω, i.e., dχ = ω. This case is important for applications in physics, compare, e.g., [11, 26] and [27].
Recall that for \(P\xrightarrow {r}B\) and \(P^{\prime }\xrightarrow {r^{\prime }}B\) two principal S^{1}bundles over B we may form a new bundle \(P\otimes P^{\prime }\rightarrow B\) defined by
where the quotient is taken so that \((p,p^{\prime })\sim (p z,p^{\prime } z^{1})\) for p ∈ P, \(p^{\prime }\in P^{\prime }\) and z ∈ S^{1}. We may also form the dual bundle P^{∗} defined by the same fiber bundle but provided with the action p ⋅ z = pz^{− 1}.
Let \(Y\xrightarrow {\Pi }M\) be a surjective submersion and consider \(Q\xrightarrow {r} Y\) a principal S^{1}bundle, with connection 1form \(\widetilde {A}\in i{\Omega }^{1}(Q)\). We denote \(F^{\widetilde {A}}\in i{\Omega }^{2}(Y)\) the curvature of this principal connection, so that \(d\widetilde {A}=r^{*}F^{\widetilde {A}}\). Furthermore, we put \({\Pi }_{k}:Y^{[2]}\rightarrow Y\) for k = 1, 2, where π_{k} is the projection that omits the k th factor. Then
is a S^{1}central extension of the Lie groupoid \(Y^{[2]}\rightrightarrows Y\), and therefore \(\delta (Q):=(\pi _{1}^{*}Q)^{*}\otimes \pi _{2}^{*}Q\) yields a bundle gerbe. In fact (compare [22]), a bundle gerbe has vanishing DixmierDouady class precisely when it is isomorphic (in an appropriate sense) to a bundle gerbe of the form (δ(Q), Y ), and a choice of an isomorphism \(P\rightarrow \delta (Q)\) is called a trivialisation of the bundle gerbe (P, Y ).
Let us recall several observations concerning the Lie groupoid \(\delta (Q)\rightrightarrows Y\). An element of δ(Q) is written as a quadruple ([q_{1}, q_{2}], y_{1}, y_{2}) where q_{i} ∈ Q, (y_{1}, y_{2}) ∈ Y^{[2]} such that r(q_{1}) = y_{1}, r(q_{2}) = y_{2}, and [q_{1}, q_{2}] is the class stemming from the equivalence relation: \((q_{1},q_{2})\sim (q_{1}z,q_{2}z)\) for all z ∈ S^{1}. In the sequel, we denote such a quadruple by the equivalence classe [q_{1}, q_{2}]. Then the structure maps of the Lie groupoid \(\delta (Q)\rightrightarrows Y\) are given by

s([q_{1}, q_{2}]) = r(q_{2})

t([q_{1}, q_{2}]) = r(q_{1}).
Note that for [q_{1}, q_{2}], [q_{3}, q_{4}] in δ(Q), such that s([q_{1}, q_{2}]) = r(q_{2}) = r(q_{3}) = t([q_{3}, q_{4}]), we may assume without loss of generality that q_{2} = q_{3}, since the equivalence classes are defined by the orbits of the S^{1}action. Therefore the groupoid multiplication can be written as
For y ∈ Y we have furthermore
Thus denoting by \({\Delta }:Y\rightarrow Y^{[2]}\) the diagonal inclusion, the bundle Δ^{∗}(δ(Q)) is canonically trivialised. Taking for y ∈ Y any q ∈ Q_{y}, the canonical section \(\varepsilon :Y\rightarrow {\Delta }^{*}(\delta (Q))\) is given by 𝜖(y) = [q, q] and 𝜖 is taken as the unit map of the Lie groupoid \(\delta (Q)\rightrightarrows Y\). Finally the inverse map is given as
We now describe a connective structure on (δ(Q), Y ). Consider the projections \((\pi _{1}^{*}Q)^{*}\times _{Y^{[2]}}\pi _{2}^{*}Q\xrightarrow {p_{k}}Q\), where p_{k} projects to the k th factor (k = 1, 2), and the 1form \(A=p_{1}^{*}\widetilde {A}p_{2}^{*}\widetilde {A}\). Then A defines a principal connection on \((\pi _{1}^{*}Q)^{*}\otimes \pi _{2}^{*}Q\). More precisely, we have:
Lemma B.1
We keep the notations from the previous considerations. Then

(i)
the connection A on \(\delta (Q)\rightarrow Y^{[2]}\) is multiplicative,

(ii)
a curving of this connection is given by the curvature \(F^{\widetilde {A}}\) of \(\widetilde {A}\).
Therefore \((A,F^{\widetilde {A}})\) is a connective structure on the bundle gerbe (δ(Q), Y ). Moreover, the 3curvature is identically zero and thus (δ(Q), Y ) has vanishing DixmierDouady class.
Proof
For q_{1}, q_{2}, q_{3} ∈ Q and \(v_{q_{1}},v_{q_{2}},v_{q_{3}}\) tangent vectors at the respective points, we compute
where \(\text {proj}_{k}:\delta (Q){{}_{s}\times _{t}}\delta (Q)\rightarrow \delta (Q)\) is the projection onto the k th factor, for k = 1, 2. Thus A is multiplicative.
Moreover, we have
This shows that \(F^{\widetilde {A}}\) provides a curving for A, and that \((A,F^{\widetilde {A}})\) is indeed a connective structure for (δ(Q), Y ). Since \(dF^{\widetilde {A}}=0\), the 3curvature equals zero, and therefore (δ(Q), Y ) has vanishing DixmierDouady class. □
We now construct the prequantisation map for an exact 2plectic manifold. Let (M, ω) be a 2plectic manifold with ω = dχ an exact 3form. Consider the trivial bundle \({\Pi }:Y:=M\times PU(H)\rightarrow M\), equipped with the trivial connection, that we denote here by \(C\in {\Omega }^{1}(M\times PU(H))\otimes \mathfrak {pu}(H)\). Recall that C is given by
i.e., C is the pullback of the MaurerCartan form on PU(H) via the projection \(Y\rightarrow PU(H)\).
Observe that Y^{[2]} = M × PU(H) × PU(H). We then have \(\psi :M\times PU(H)\times PU(H)\rightarrow PU(H)\), \(\psi (m,g,g^{\prime })=g^{1}g^{\prime }\) (compare Lemma 2.10). We also define \(Q:=M\times U(H)\rightarrow Y\). Then \(\delta (Q):=(\pi _{1}^{*}Q)^{*}\otimes \pi _{2}^{*}Q\) is isomorphic to ψ^{∗}U(H), the pullback of the principal S^{1}bundle \(U(H)\rightarrow PU(H)\) by the map ψ. To see this, note that the elements of δ(Q) are triples \((m,[u,u^{\prime }])\), where m ∈ M, \(u,u^{\prime }\in U(H)\), and the class \([u,u^{\prime }]\) is taken with respect to \((u,u^{\prime })\sim (uz,u^{\prime }z)\) for all z ∈ S^{1}. The elements of ψ^{∗}U(H) are triples (m, g, u), where m ∈ M, g ∈ PU(H) and u ∈ U(H). Then the isomorphism \(\delta (Q)\rightarrow \psi ^{*} U(H)\) is given by
with \(U(H)\xrightarrow {q}PU(H)\) being the canonical projection. The map (∗) is welldefined and equivariant, and therefore an isomorphism of principal S^{1}bundles.
Let \(\widetilde {A}\) be the principal connection on Q defined by the MaurerCartan form on U(H), projected onto \(i\mathbb {R}=\text {Lie}(S^{1})\) via a splitting \(\mathfrak {pu}(H)\rightarrow \mathfrak {u}(H)\) (see Lemma 2.6). Let \(F^{\widetilde {A}}\) be the curvature of this connection. On the bundle \(\delta (Q)\rightarrow Y^{[2]}\), we consider as above the connection \(A=p_{1}^{*}\widetilde {A}p_{2}^{*}\widetilde {A}\) (recall that A is multiplicative by the preceding lemma). We set \(\theta ={\Pi }^{*}\chi +F^{\widetilde {A}}\). Then (A, 𝜃) is a connective structure on the bundle gerbe (δ(Q), Y ), with 3curvature (−i2π)ω. For a vector field \(X\in \mathfrak {X}(M)\), the horizontal lift X^{h} to Y with respect to the conection C is simply (X, 0), which we denote again by X. Then we have X_{2} = (X, 0, 0) and we continue to denote this vector field by X. The horizontal lift of a vector field \(Z\in \mathfrak {X}(Y^{[2]})\) with respect to the connection A will be denoted by \(\widetilde {Z}\).
We conclude with an explicit description of the components of the Lie 2algebra morphism of Theorem 4.6 in the exact case. For the sake of better readability, we omit the symbol π^{∗} for pullbacks of functions and differential forms with respect to the projection \({\Pi }:Y\rightarrow M\). Furthermore, given a vector field V on a factor of a product A × B, we denote its trivial extension to this product again by V. With these conventions, we obtain for \(f\in C^{\infty }(M)\) and \(\alpha ,\beta \in {\Omega }^{1}_{Ham}(M,d\chi )\):

Φ_{1}(f) = (0, f)

\({\Phi }_{1}(\alpha )=(\widetilde {X_{\alpha }},X_{\alpha },\iota _{X_{\alpha }}\chi +\alpha )\)

Φ_{2}(α, β) = (0, χ(X_{α}, X_{β}) + α(X_{β}) − β(X_{α})).
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Sevestre, G., Wurzbacher, T. On the Prequantisation Map for 2Plectic Manifolds. Math Phys Anal Geom 24, 20 (2021). https://doi.org/10.1007/s11040021093915
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Keywords
 Prequantisation
 Multisymplectic geometry
 Geometrisation of integral threeforms
 Multiplicative vector fields
Mathematics Subject Classification (2020)
 Primary: 58H05
 53C08
 53D50
 Secondary: 53D05