In differential geometry, given a metaplectic structure

\pi_\colon\to M\,

on a

2n

-dimensional

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...

(M, \omega),\,

the symplectic spinor bundle is the

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

bundle

\pi_\colon\to M\,

associated to the metaplectic structure via the metaplectic representation. The metaplectic representation of the

metaplectic group In mathematics, the metaplectic group Mp2''n'' is a double cover of the symplectic group Sp2''n''. It can be defined over either real or ''p''-adic numbers. The construction covers more generally the case of an arbitrary local or finite field ...

— the two-fold covering of the

symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic g ...

— gives rise to an infinite rank

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...

; this is the symplectic spinor construction due to

Bertram Kostant Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics. Early life and education Kostant grew up in New York City, where he gradu ...

. A section of the symplectic spinor bundle

\,

is called a symplectic spinor field.

Formal definition

Let

(,F_)

be a metaplectic structure on a

(M, \omega),\,

that is, an

equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry gro ...

lift of the

symplectic frame bundle In symplectic geometry, the symplectic frame bundle of a given symplectic manifold (M, \omega)\, is the canonical principal (n,)-subbundle \pi_\colon\to M\, of the tangent frame bundle \mathrm FM\, consisting of linear frames which are symplectic ...

\pi_\colon\to M\,

with respect to the double covering

\rho\colon (n,)\to (n,).\,

The symplectic spinor bundle

\,

is defined to be the Hilbert space bundle :

=\times_L^2(^n)\,

associated to the metaplectic structure

via the metaplectic representation

\colon (n,)\to (L^2(^n)),\,

also called the Segal–Shale–Weil representation of

(n,).\,

Here, the notation

()\,

denotes the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the c ...

s acting on a

.\,

The Segal–Shale–Weil representation is an infinite dimensional

unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ' ...

of the metaplectic group

(n,)

on the space of all complex valued square Lebesgue integrable

square-integrable function In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute val ...

L^2(^n).\,

Because of the infinite dimension, the Segal–Shale–Weil representation is not so easy to handle.

Formal definition

Notes

Further reading