${\operatorname{Sp}}(S)$ the group of linear symplectomorphism. For any $g \in {\operatorname{Sp}}(S)$, observe that the endomorphism $\Psi_{ g, E, E}$ of $\wedge ^{{\operatorname{top}}} E^*$ is the multiplication by $ \det ( g^{-1}\pi_{E, gE} : E
\rightarrow E)$.

Let $(\delta, \varphi)$ be a half-form line of $(S,E)$. Then identifying the automorphisms of $\delta$ with complex numbers, the automorphisms of $(S, E, \delta , \varphi)$ are the pairs $(g, z)$ consisting of a linear symplectomorphism $g \in {\operatorname{Sp}}(S)$ with a complex number $z$ such that $$z^2 = \det ( g^{-1}  \pi_{E, gE} : E \rightarrow E) .$$ The composition of two automorphisms is given by $$(g', z' ) \circ_{{{\mathcal{D}}}} ( g ,z ) =  \bigl( g' g , \zeta ^{1/2} ( g'gE,
g' E, E) z' z  \bigr)   .$$ Since the previous formulas don’t depend on $( \delta,\varphi)$, we denote by ${\operatorname{Mp}}(S,E)$ the group of automorphisms of $(S, E, \delta , \varphi)$. The projection $${\operatorname{Mp}}(S, E) \rightarrow {\operatorname{Sp}}(S)$$ is two to one. Recall that ${\operatorname{Sp}}( S)$ is connected with fundamental group ${{\mathbb{Z}}}$. The metaplectic group is defined as the 2-cover of the symplectic group.

The group ${\operatorname{Mp}}(S,E ) $ is connected, so it is isomorphic to the metaplectic group.

Choose a line $D$ 