A metaplectic structure ^[1] on a symplectic manifold ${\displaystyle (M,\omega )}$  is an equivariant lift of the symplectic frame bundle ${\displaystyle \pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,}$  with respect to the double covering ${\displaystyle \rho \colon {\mathrm {Mp} }(n,{\mathbb {R} })\to {\mathrm {Sp} }(n,{\mathbb {R} }).\,}$  In other words, a pair ${\displaystyle ({\mathbf {P} },F_{\mathbf {P} })}$  is a metaplectic structure on the principal bundle ${\displaystyle \pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,}$  when

a) ${\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to M\,}$  is a principal ${\displaystyle {\mathrm {Mp} }(n,{\mathbb {R} })}$ -bundle over ${\displaystyle M}$ ,

b) ${\displaystyle F_{\mathbf {P} }\colon {\mathbf {P} }\to {\mathbf {R} }\,}$  is an equivariant ${\displaystyle 2}$ -fold covering map such that

${\displaystyle \pi _{\mathbf {R} }\circ F_{\mathbf {P} }=\pi _{\mathbf {P} }}$  and ${\displaystyle F_{\mathbf {P} }({\mathbf {p} }q)=F_{\mathbf {P} }({\mathbf {p} })\rho (q)}$  for all ${\displaystyle {\mathbf {p} }\in {\mathbf {P} }}$  and ${\displaystyle q\in {\mathrm {Mp} }(n,{\mathbb {R} }).}$ 

The principal bundle ${\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to M\,}$  is also called the bundle of metaplectic frames over ${\displaystyle M}$ .

Two metaplectic structures ${\displaystyle ({\mathbf {P} _{1}},F_{\mathbf {P} _{1}})}$  and ${\displaystyle ({\mathbf {P} _{2}},F_{\mathbf {P} _{2}})}$  on the same symplectic manifold ${\displaystyle (M,\omega )}$  are called equivalent if there exists a ${\displaystyle {\mathrm {Mp} }(n,{\mathbb {R} })}$ -equivariant map ${\displaystyle f\colon {\mathbf {P} _{1}}\to {\mathbf {P} _{2}}}$  such that

${\displaystyle F_{\mathbf {P} _{2}}\circ f=F_{\mathbf {P} _{1}}}$  and ${\displaystyle f({\mathbf {p} }q)=f({\mathbf {p} })q}$  for all ${\displaystyle {\mathbf {p} }\in {\mathbf {P} _{1}}}$  and ${\displaystyle q\in {\mathrm {Mp} }(n,{\mathbb {R} }).}$ 

Of course, in this case ${\displaystyle F_{\mathbf {P} _{1}}}$  and ${\displaystyle F_{\mathbf {P} _{2}}}$  are two equivalent double coverings of the symplectic frame ${\displaystyle {\mathrm {Sp} }(n,{\mathbb {R} })}$ -bundle ${\displaystyle \pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,}$  of the given symplectic manifold ${\displaystyle (M,\omega )}$ .

Obstruction edit

Since every symplectic manifold ${\displaystyle M}$  is necessarily of even dimension and orientable, one can prove that the topological obstruction to the existence of metaplectic structures is precisely the same as in Riemannian spin geometry.^[2] In other words, a symplectic manifold ${\displaystyle (M,\omega )}$  admits a metaplectic structures if and only if the second Stiefel-Whitney class ${\displaystyle w_{2}(M)\in H^{2}(M,{\mathbb {Z} _{2}})}$  of ${\displaystyle M}$  vanishes. In fact, the modulo ${\displaystyle _{2}}$  reduction of the first Chern class ${\displaystyle c_{1}(M)\in H^{2}(M,{\mathbb {Z} })}$  is the second Stiefel-Whitney class ${\displaystyle w_{2}(M)}$ . Hence, ${\displaystyle (M,\omega )}$  admits metaplectic structures if and only if ${\displaystyle c_{1}(M)}$  is even, i.e., if and only if ${\displaystyle w_{2}(M)}$  is zero.

If this is the case, the isomorphy classes of metaplectic structures on ${\displaystyle (M,\omega )}$  are classified by the first cohomology group ${\displaystyle H^{1}(M,{\mathbb {Z} _{2}})}$  of ${\displaystyle M}$  with ${\displaystyle {\mathbb {Z} _{2}}}$ -coefficients.

As the manifold ${\displaystyle M}$  is assumed to be oriented, the first Stiefel-Whitney class ${\displaystyle w_{1}(M)\in H^{1}(M,{\mathbb {Z} _{2}})}$  of ${\displaystyle M}$  vanishes too.

Manifolds admitting a metaplectic structure edit

• Phase spaces ${\displaystyle (T^{\ast }N,\theta )\,,}$  ${\displaystyle N}$  any orientable manifold.
• Complex projective spaces ${\displaystyle {\mathbb {P} }^{2k+1}{\mathbb {C} }\,,}$  ${\displaystyle \,k\in {\mathbb {N} }_{0}\,.}$  Since ${\displaystyle {\mathbb {P} }^{2k+1}{\mathbb {C} }\,}$  is simply connected, such a structure has to be unique.
• Grassmannian ${\displaystyle Gr(2,4)\,,}$  etc.

• Metaplectic group
• Symplectic frame bundle
• Symplectic group
• Symplectic spinor bundle

• Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0