vector space $S$ equipped with a positive polarization $E$ is a complex line $\delta$ together with an isomorphism $$\varphi : \delta^{\otimes 2 p} \rightarrow  \bigl( \wedge
^{\operatorname{top}} E^* \bigr)^{\otimes p}$$ We have a category whose objects are the quadruples $(S,E, \delta, \varphi)$. The morphisms from $(S_a, E_a, \delta_a,
\varphi_a)$ to $(S_b , E_b , \delta_b, \varphi_b)$ are the pairs consisting of a linear symplectomorphism $g : S_a \rightarrow S_b$ together with a morphism $\Psi : \delta_a \rightarrow \delta_b$ such that $$\varphi_b \circ
\Psi^{\otimes 2 p } = \Psi_{g, E_a, E_b}^{\otimes p}    \circ
\varphi_a$$ where the map $\Psi_{g, E_a, E_b}$ is defined as in section \[sec:sympl-line-categ\]. The composition of a morphism $(g,\Psi): a
\rightarrow b$ with a morphism $(g',\Psi
') : b \rightarrow c$ is defined as $$(g', \Psi') {\circ_{{{\mathcal{D}}}}}(g, \Psi) := \bigl( g'g, \zeta ^{1/2} ( g'gE_a,
g' E_b, E_c) \Psi' \circ \Psi \bigr)$$ where the product in the right-hand side is the usual composition of maps.

The automorphism group ${\operatorname{Mp}}_p (S,E)$ of $(S,E, \delta, \varphi)$ consists of the pair $(g, z)$ where $g$ is a linear symplectomorphism of $S$ and $z$ a complex number such that $$z^{2p}  =  \operatorname{det} ^p  ( g^{-1}  \pi_{E, gE} : E
\rightarrow E) .$$ 