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Abstract

We associate to any integrable Poisson manifold a stack, i.e., a category fibered in groupoids over a site. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. We show that two Poisson manifolds are symplectically Morita equivalent if and only if their associated stacks are isomorphic. We also discuss the non-integrable case.

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Notes

  1. Note that some references may give a slightly different definition of the Courant bracket. In this case, for convenience we are adopting the version used in [4]. When restricted to any given Dirac structure, they turn out to be equal.

  2. A more explicit construction of the groupoid structure can found in the proof of 70. Proposition in [12]. The relevant portion can be found in the paragraph beginning with ‘Now we show the converse’.

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Acknowledgements

The author would like to thank his thesis advisor Rui Loja Fernandes for his guidance and support throughout the preparation of this document. The author would also like thank Eugene Lerman and Matias del Hoyo for several comments and discussions related to this work. Finally, he is very grateful for the anonymous referee’s constructive feedback.

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Correspondence to Joel Villatoro.

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Joel Villatoro was partially supported by an AGEP-GRS fellowship under NSF Grant DMS 130847.

A Proof of Lemma 3.1

A Proof of Lemma 3.1

Proof

Suppose \(\mathcal {G}\) with \(\mathbf {s},\mathbf {t},\mathbf {u},\mathbf {m},\mathbf {i}\) is a Lie groupoid and \((\mathbf {s},\sigma ),(\mathbf {t},\tau ),(\mathbf {u},\upsilon ),(\mathbf {i},\iota )\) are morphisms in \(\mathtt {DMan}\). This data constitutes a D-Lie groupoid if and only if the gauge equation associated to each groupoid axiom holds. In the table below, we have enumerated the axioms of a groupoid and computed the corresponding equations of 2-forms.

 

Axiom

Domain

Gauge part

(G1)

\(\mathbf {s}\circ \mathbf {u}= \text{ Id }_M\)

M

\(\mathbf {u}^* \sigma + \upsilon = 0\)

(G2)

\(\mathbf {s}\circ \mathbf {m}= \mathbf {s}\circ \text{ pr }_2\)

\( \mathcal {G}^{(2)}\)

\(\mathbf {m}^* \sigma + \mu = \text{ pr }_1^* \sigma + \text{ pr }_2^* \sigma \)

(G3)

\(\mathbf {s}\circ \mathbf {i}= \mathbf {t}\)

\( \mathcal {G}\)

\(\mathbf {i}^* \sigma + \iota = \tau \)

(G4)

\(\mathbf {i}\circ \mathbf {u}= \mathbf {u}\)

M

\(\mathbf {u}^* \iota + \upsilon = \upsilon \)

(G5)

\(\mathbf {m}\circ ((\mathbf {u}\circ \mathbf {t}) \times \text{ Id }_\mathcal {G}) = \text{ Id }_\mathcal {G}\)

\( \mathcal {G}\)

\({((\mathbf {u}\circ \mathbf {t}) \times \text{ Id })}^* \mu = {(\mathbf {u}\circ \mathbf {t})}^* \sigma \)

(G6)

\(\mathbf {m}\circ (\text{ Id }_\mathcal {G}\times (\mathbf {u}\circ \mathbf {s})) = \text{ Id }_\mathcal {G}\)

\( \mathcal {G}\)

\({(\text{ Id }\times (\mathbf {u}\circ \mathbf {s}) )}^* \mu = {(\mathbf {u}\circ \mathbf {s})}^* \tau \)

(G7)

\(\mathbf {m}\circ (\mathbf {i}\times \text{ Id }_\mathcal {G}) = \mathbf {u}\circ \mathbf {s}\)

\( \mathcal {G}\)

\({(\mathbf {i}\times \text{ Id })}^* \mu - \mathbf {i}^* \sigma = \mathbf {s}^* \upsilon + \sigma \)

(G8)

\(\mathbf {m}\circ (\text{ Id }_\mathcal {G}\times \mathbf {i}) = \mathbf {u}\circ \mathbf {t}\)

\( \mathcal {G}\)

\({(\text{ Id }\times \mathbf {i})}^* \mu - \mathbf {i}^* \tau = \mathbf {t}^* \upsilon + \tau \)

(G9)

\({\begin{aligned}&\mathbf {m}\circ (\mathbf {m}(\text{ pr }_1 \times \text{ pr }_2) \times \text{ pr }_3) \\&= \mathbf {m}\circ (\text{ pr }_1 \times \mathbf {m}(\text{ pr }_2 \times \text{ pr }_3)) \end{aligned}}\)

\(\mathcal {G}^{(3)}\)

See (7.2) below

Now suppose we are supplied with 2-forms \(\sigma \) and \(\tau \) satisfying (i) and (ii) from 3.1. Take the gauge equations from (G1–G3) to be the definitions of \(\upsilon \), \(\mu \) and \(\iota \). Let \(L_\mathcal {G}:= \mathbf {s}^* L_M - \sigma \). We must show that \(\mathcal {G}\) and M together with \((\mathbf {s},\sigma ),(\mathbf {t},\tau ),(\mathbf {u},\upsilon ),(\mathbf {i},\iota )\) constitutes a well-defined D-Lie groupoid. Assumption (i) implies that \((\mathbf {s},\sigma )\) and \((\mathbf {t},\tau )\) are well-defined morphisms in \(\mathtt {DMan}\). A careful calculation shows that the remaining maps are also morphisms of Dirac structures. It remains to show that the each gauge equation in the above table holds.

The equations from (G1–G3) follow immediately by definition. The equation for (G4) holds since

$$\begin{aligned} \mathbf {u}^*(\iota ) = \mathbf {u}^*(\tau - \sigma ) = 0. \end{aligned}$$

The first equality follows from (G3) while the second follows from the fact that \(\tau -\sigma \) is multiplicative.

Next we show (G5) by computing directly.

$$\begin{aligned} {((\mathbf {u}\circ \mathbf {t}) \times \text{ Id })}^* \mu&={((\mathbf {u}\circ \mathbf {t}) \times \text{ Id })}^*( \text{ pr }_1^* \sigma + \text{ pr }_2^* \sigma - \mathbf {m}^* \sigma ) \\&= {(\mathbf {u}\circ \mathbf {t})}^* \sigma + \sigma - {(\mathbf {m}((\mathbf {u}\circ \mathbf {t}) \times \text{ Id }))}^* \sigma \\&= {(\mathbf {u}\circ \mathbf {t})}^* \sigma + \sigma - \sigma = {(\mathbf {u}\circ \mathbf {t})}^* \sigma . \end{aligned}$$

It follows from the multiplicativity of \(\tau - \sigma \) that

$$\begin{aligned} \mathbf {m}^* \tau + \mu = \text{ pr }_1^* \tau + \text{ pr }_2^* \tau . \end{aligned}$$
(7.1)

By using this expression for \(\mu \), we can show (G6) by a calculation essentially identical to (G5). Next up, we show (G7):

$$\begin{aligned} {(\mathbf {i}\times \text{ Id })}^* \mu - \mathbf {i}^* \sigma&= {(\mathbf {i}\times \text{ Id })}^* (\text{ pr }_1^* \sigma + \text{ pr }_2^* \sigma - \mathbf {m}^* \sigma ) - \mathbf {i}^* \sigma \\&= \mathbf {i}^* \sigma + \sigma - {(\mathbf {u}\circ \mathbf {s})}^* \sigma - \mathbf {i}^* \sigma \\&= -\mathbf {s}^* \mathbf {u}^* \sigma + \sigma = \mathbf {s}^* \iota + \sigma \end{aligned}$$

Since (G8) is similar, we can proceed to (G9). The gauge equation for (G9) is

$$\begin{aligned} \begin{aligned}&{(\text{ pr }_1 \times \text{ pr }_2)}^* \mu + {(\mathbf {m}\circ (\text{ pr }_1 \times \text{ pr }_2) \times \text{ pr }_3)}^* \mu \\&\quad ={(\text{ pr }_2 \times \text{ pr }_3)}^* \mu + {(\text{ pr }_1 \times \mathbf {m}\circ (\text{ pr }_2 \times \text{ pr }_3))}^* \mu \,. \end{aligned} \end{aligned}$$
(7.2)

If we apply the substitution \(\mu = \text{ pr }_1^* \sigma + \text{ pr }_2^* \sigma - \mathbf {m}^* \sigma \) throughout, we get:

$$\begin{aligned}&\text{ pr }_1^* \sigma + \text{ pr }_2^* \sigma -{(\text{ pr }_1 \times \text{ pr }_2)}^*\mathbf {m}^* \sigma + {(\text{ pr }_1 \times \text{ pr }_2)}^*\mathbf {m}^* \sigma + \text{ pr }_3^* \sigma - \mathbf {A}_L^* \sigma = \\&\quad \text{ pr }_2^* \sigma + \text{ pr }_3^* \sigma -{(\text{ pr }_2 \times \text{ pr }_3)}^*\mathbf {m}^* \sigma + {(\text{ pr }_2 \times \text{ pr }_3)}^*\mathbf {m}^* \sigma + \text{ pr }_1^* \sigma - \mathbf {A}_R^* \sigma . \end{aligned}$$

Here \(\mathbf {A}_L, \mathbf {A}_R: \mathcal {G}^{(3)} \rightarrow \mathcal {G}\) are the left and right hand associativity maps. Since \(\mathcal {G}\) is a Lie groupoid and assumed to be associative, it follows immediately that (9) holds. \(\square \)

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Villatoro, J. Poisson manifolds and their associated stacks. Lett Math Phys 108, 897–926 (2018). https://doi.org/10.1007/s11005-017-1012-5

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