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.

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 {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 ∈ L₀ and u, v ∈ L_− 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

• (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 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 R_g, L_g the right respectively left multiplication by an element g ∈ G₁. 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, z ∈ L₀ and u ∈ L_− 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:

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 p ∈ P, $p^{\prime }\in P^{\prime }$ and z ∈ S¹. 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 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 π_k 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 q_i ∈ Q, (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 z ∈ S¹. 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 y ∈ Y 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 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

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

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 p_k 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

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

Proof

For q₁, q₂, q₃ ∈ Q 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 ψ^∗U(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 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 ∈ S¹. 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

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

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 X^h 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(X_α, X_β) + α(X_β) − β(X_α)).