In differential geometry, given a metaplectic structure
π
P
:
P
→
M
on a
2
n
-dimensional symplectic manifold
(
M
,
ω
)
,
one defines the symplectic spinor bundle to be the Hilbert space bundle
π
Q
:
Q
→
M
associated to the metaplectic structure via the metaplectic representation. The metaplectic representation of the metaplectic group —the two-fold covering of the symplectic group— gives rise to an infinite rank vector bundle, this is the symplectic spinor construction due to Bertram Kostant.
A section of the symplectic spinor bundle
Q
is called a symplectic spinor field.
Let
(
P
,
F
P
)
be a metaplectic structure on a symplectic manifold
(
M
,
ω
)
,
that is, an equivariant lift of the symplectic frame bundle
π
R
:
R
→
M
with respect to the double covering
ρ
:
M
p
(
n
,
R
)
→
S
p
(
n
,
R
)
.
The symplectic spinor bundle
Q
is defined to be the Hilbert space bundle
Q
=
P
×
m
L
2
(
R
n
)
associated to the metaplectic structure
P
via the metaplectic representation
m
:
M
p
(
n
,
R
)
→
U
(
L
2
(
R
n
)
)
,
also called the Segal-Shale-Weil representation of
M
p
(
n
,
R
)
.
Here, the notation
U
(
W
)
denotes the group of unitary operators acting on a Hilbert space
W
.
The Segal-Shale-Weil representation is an infinite dimensional unitary representation of the metaplectic group
M
p
(
n
,
R
)
on the space of all complex valued square Lebesgue integrable square-integrable functions
L
2
(
R
n
)
.
Because of the infinite dimension, the Segal-Shale-Weil representation is not so easy to handle.
Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0
