On the Prequantisation Map for 2-Plectic Manifolds

Abstract

For a manifold M with an integral closed 3-form 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 non-degenerate, we furthermore give a natural Lie 2-algebra quasi-isomorphism 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 Laurent-Gengoux 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 2-Algebras and Their Morphisms

In this appendix we assemble the definitions regarding Lie 2-algebras 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 2-term complex of vector spaces. A Lie 2-algebra structure on L consists in (multi)linear graded antisymmetric maps {lk|1 ≤ k ≤ 3} with the degree of lk 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, tL0 and u, vL− 1:

$$ \begin{array}{@{}rcl@{}} &&l_{1}(l_{2}(x,u))=l_{2}(x,l_{1}(u)) \hskip0.5cm \text{and} \hskip0.5cm l_{2}(l_{1}(u),v)=l_{2}(u,l_{1}(v))\\ &&l_{1}(l_{3}(x,y,z))+l_{2}(l_{2}(x,y),z)-l_{2}(l_{2}(x,z),y)+l_{2}(l_{2}(y,z),x)=0\\ &&l_{3}(l_{1}(u),x,y)+l_{2}(l_{2}(x,y),u)-l_{2}(l_{2}(x,u),y)+l_{2}(l_{2}(y,u),x)=0\\ && l_{3}(l_{2}(x,y),z,t)-l_{3}(l_{2}(x,z),y,t)+l_{3}(l_{2}(x,t),y,z) \\ && +l_{3}(l_{2}(y,z),x,t)-l_{3}(l_{2}(y,t),x,z)+l_{3}(l_{2}(z,t),x,y) \\ && = l_{2}(l_{3}(x,y,z),t)-l_{2}(l_{3}(x,y,t),z)+l_{2}(l_{3}(x,z,t),y)-l_{2}(l_{3}(y,z,t),x) . \end{array} $$

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

  1. (A1)

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

  2. (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 2-algebra (i.e. a Lie 2-algebra with third bracket identically 0) by setting for \(X,Y\in \mathfrak {g}\), \(v\in \mathfrak {h}\):

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

  • l1 := η

  • \(l_{2}(X,Y):=[X,Y]_{\mathfrak {g}}\), l2(X, v) := 𝜗(X, v)

  • l3 := 0

Note that one often writes [, ] for the operation l2 of the Lie 2-algebra 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 Rg, Lg the right respectively left multiplication by an element gG1. We have the right- and left-invariant vector fields on G1 associated to a section a ∈Γ(G0,Lie(G1)), given respectively by:

$$ \begin{array}{@{}rcl@{}} \overrightarrow{a}(g)=(R_{g})_{*}(a(t(g))) \text{and}\\ \overleftarrow{a}(g)=(L_{g})_{*}(i_{*})(a(t(g))) . \end{array} $$

For \(\xi \in \mathfrak {X}_{mult}(G_{1})\), its action on the Lie algebroid sections is given by:

$$ [\xi,a]:=\vartheta(\xi,a)=b \hskip0.2cm \text{if and only if} \hskip0.2cm \overrightarrow{b} = [\xi,\overrightarrow{a}], $$

where the bracket on the right hand side is the bracket of vector fields on G1. Since Γ(G0,Lie(G1)) is isomorphic as a \(C^{\infty }(G_{0})\)-module to the right-invariant vector fields on G1 and the resulting vector field \([\xi ,\overrightarrow {a}]\) is again right-invariant, \([\xi ,\overrightarrow {a}]\) equals \(\overrightarrow {b}\) for a uniquely determined section b of Lie(G1).

Thus the two-term complex of Lie algebras:

$$ {\Gamma}(G_{0}, \text{Lie}(G_{1}))\xrightarrow{\eta}\mathfrak{X}_{mult}(G_{1}), a\mapsto (\overrightarrow{a}+\overleftarrow{a},X(a)) , $$

where X(a) is the vector field on G0 associated to a via the anchor map of Lie(G1), together with the action defined above, and the usual brackets on Γ(G0,Lie(G1)) and \(\mathfrak {X}_{mult}(G_{1})\), is a Lie algebra crossed module, hence a strict Lie 2-algebra.

Considering a Lie 2-algebra 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 2-algebras. 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 Φ1 of degree 0 and Φ2 of degree − 1, such that for every x, y, zL0 and uL− 1:

$$ \begin{array}{@{}rcl@{}} &&{\Phi}_{1}(l_{1}(u))=l_{1}^{\prime}({\Phi}_{1}(u))\\ &&{\Phi}_{1}(l_{2}(x,y)) = l_{2}^{\prime}({\Phi}_{1}(x),{\Phi}_{1}(y)) + l_{1}^{\prime}({\Phi}_{2}(x,y))\\ &&{\Phi}_{1}(l_{2}(u,x)) =l_{2}^{\prime}({\Phi}_{1}(u),{\Phi}_{1}(x)) + {\Phi}_{2}(l_{1}(u),x)\\ &&{\Phi}_{2}(l_{2}(x,y),z)-{\Phi}_{2}(l_{2}(x,z),y)+{\Phi}_{2}(l_{2}(y,z),x)+{\Phi}_{1}(l_{3}(x,y,z))\\ &=&l_{2}^{\prime}({\Phi}_{1}(x),{\Phi}_{2}(y,z)) -l_{2}^{\prime}({\Phi}_{1}(y),{\Phi}_{2}(x,z))+l_{2}^{\prime}({\Phi}_{1}(z),{\Phi}_{2}(x,y)) \\ &&+l_{3}^{\prime}({\Phi}_{1}(x),{\Phi}_{1}(y),{\Phi}_{1}(z)). \end{array} $$

Remark A.6

A Lie ∞-algebra morphism between two Lie 2-algebras is, of course, also called a Lie 2-algebra morphism.

Remark A.7

Following a suggestion of Camille Laurent-Gengoux, we visualize a Lie 2-algebra morphism \(L_{\bullet }\xrightarrow {\Phi } L^{\prime }_{\bullet }\) as below:

figurea

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 Φ2 defined on the vector space L0 (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 quasi-isomorphism if the induced map in cohomology is an isomorphism. A Lie ∞-algebra morphism between two Lie 2-algebras (or Lie ∞-algebras, in fact) is called a quasi-isomorphism if viewed as a cochain morphism it is a quasi-isomorphism.

Appendix B: The Case of Exact 2-Plectic Manifolds

In this appendix, we expose the case of 2-plectic manifolds (M, ω) with a potential χ ∈Ω2(M) for the 3-form 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 S1-bundles over B we may form a new bundle \(P\otimes P^{\prime }\rightarrow B\) defined by

$$ P\otimes P^{\prime}:=(P {{}_{r}\times_{r^{\prime}}}P^{\prime})/S^{1}, $$

where the quotient is taken so that \((p,p^{\prime })\sim (p z,p^{\prime } z^{-1})\) for pP, \(p^{\prime }\in P^{\prime }\) and zS1. We may also form the dual bundle P defined by the same fiber bundle but provided with the action pz = pz− 1.

Let \(Y\xrightarrow {\Pi }M\) be a surjective submersion and consider \(Q\xrightarrow {r} Y\) a principal S1-bundle, with connection 1-form \(\widetilde {A}\in i{\Omega }^{1}(Q)\). 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 πk is the projection that omits the k th factor. Then

$$ 1\rightarrow Y\times S^{1}\rightarrow(\pi_{1}^{*}Q)^{*}\otimes \pi_{2}^{*}Q\rightarrow Y^{[2]}\rightarrow 1 $$

is a S1-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 Dixmier-Douady 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 ([q1, q2], y1, y2) where qiQ, (y1, y2) ∈ Y[2] such that r(q1) = y1, r(q2) = y2, and [q1, q2] is the class stemming from the equivalence relation: \((q_{1},q_{2})\sim (q_{1}z,q_{2}z)\) for all zS1. In the sequel, we denote such a quadruple by the equivalence classe [q1, q2]. Then the structure maps of the Lie groupoid \(\delta (Q)\rightrightarrows Y\) are given by

  • s([q1, q2]) = r(q2)

  • t([q1, q2]) = r(q1).

Note that for [q1, q2], [q3, q4] in δ(Q), such that s([q1, q2]) = r(q2) = r(q3) = t([q3, q4]), we may assume without loss of generality that q2 = q3, since the equivalence classes are defined by the orbits of the S1-action. Therefore the groupoid multiplication can be written as

$$ m([q_{1},q_{2}],[q_{2},q_{3}])=[q_{1},q_{3}]. $$

For yY we have furthermore

$$ \delta(Q)_{(y,y)}=\left( (\pi_{1}^{*}Q)^{*}\otimes \pi_{2}^{*}Q\right)_{(y,y)}=Q_{y}^{*}\otimes Q_{y}. $$

Thus denoting by \({\Delta }:Y\rightarrow Y^{[2]}\) the diagonal inclusion, the bundle Δ(δ(Q)) is canonically trivialised. Taking for yY any qQy, 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

$$ [q_{1},q_{2}]^{-1}=[q_{2},q_{1}]. $$

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 pk projects to the k th factor (k = 1, 2), and the 1-form \(A=p_{1}^{*}\widetilde {A}-p_{2}^{*}\widetilde {A}\). 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

  1. (i)

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

  2. (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 3-curvature is identically zero and thus (δ(Q), Y ) has vanishing Dixmier-Douady class.

Proof

For q1, q2, q3Q and \(v_{q_{1}},v_{q_{2}},v_{q_{3}}\) tangent vectors at the respective points, we compute

$$ \begin{array}{@{}rcl@{}} &&(m^{*}A)_{[q_{1},q_{2}],[q_{2},q_{3}]}((v_{q_{1}},v_{q_{2}}),(v_{q_{2}},v_{q_{3}}))=A_{[q_{1},q_{3}]}(v_{q_{1}},v_{q_{3}}) =\widetilde{A}_{q_{1}}(v_{q_{1}})-\widetilde{A}_{q_{3}}(v_{q_{3}})=\\ &&\widetilde{A}_{q_{1}}(v_{q_{1}})-\widetilde{A}_{q_{2}}(v_{q_{2}})+\widetilde{A}_{q_{2}}(v_{q_{2}})-\widetilde{A}_{q_{3}}(v_{q_{3}}) =A_{[q_{1},q_{2}]}(v_{q_{1}},v_{q_{2}})+A_{[q_{2},q_{3}]}(v_{q_{2}},v_{q_{3}})=\\ &&(\text{proj}_{1}^{*}A+\text{proj}_{2}^{*}A)_{[q_{1},q_{2}] ,[q_{2},q_{3}]}((v_{q_{1}},v_{q_{2}}),(v_{q_{2}},v_{q_{3}})), \end{array} $$

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

$$ dA=p_{1}^{*}d\widetilde{A}-p_{2}^{*}d\widetilde{A}=p_{1}^{*}r^{*}F^{\widetilde{A}} -p_{2}^{*}r^{*}F^{\widetilde{A}} =(t^{*}-s^{*})F^{\widetilde{A}}. $$

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 3-curvature equals zero, and therefore (δ(Q), Y ) has vanishing Dixmier-Douady class. □

We now construct the prequantisation map for an exact 2-plectic manifold. Let (M, ω) be a 2-plectic manifold with ω = dχ an exact 3-form. 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

$$ C_{(m,g)}(u_{m},v_{g})=(L_{g^{-1}})_{*g}(v_{g}), $$

i.e., C is the pullback of the Maurer-Cartan 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 ψU(H), the pullback of the principal S1-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 mM, \(u,u^{\prime }\in U(H)\), and the class \([u,u^{\prime }]\) is taken with respect to \((u,u^{\prime })\sim (uz,u^{\prime }z)\) for all zS1. The elements of ψU(H) are triples (m, g, u), where mM, gPU(H) and uU(H). Then the isomorphism \(\delta (Q)\rightarrow \psi ^{*} U(H)\) is given by

$$ (*) (m,[u,u^{\prime}])\mapsto (m,q(u),u^{-1}u^{\prime}), $$

with \(U(H)\xrightarrow {q}PU(H)\) being the canonical projection. The map (∗) is well-defined and equivariant, and therefore an isomorphism of principal S1-bundles.

Let \(\widetilde {A}\) be the principal connection on Q defined by the Maurer-Cartan 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 3-curvature (−i2π)ω. For a vector field \(X\in \mathfrak {X}(M)\), the horizontal lift Xh to Y with respect to the conection C is simply (X, 0), which we denote again by X. Then we have X2 = (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 2-algebra 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 \(\alpha ,\beta \in {\Omega }^{1}_{Ham}(M,d\chi )\):

  • Φ1(f) = (0, f)

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

  • Φ2(α, β) = (0, χ(Xα, Xβ) + α(Xβ) − β(Xα)).

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Sevestre, G., Wurzbacher, T. On the Prequantisation Map for 2-Plectic Manifolds. Math Phys Anal Geom 24, 20 (2021). https://doi.org/10.1007/s11040-021-09391-5

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Keywords

  • Prequantisation
  • Multisymplectic geometry
  • Geometrisation of integral three-forms
  • Multiplicative vector fields

Mathematics Subject Classification (2020)

  • Primary: 58H05
  • 53C08
  • 53D50
  • Secondary: 53D05