Contents

group theory

Contents

Definition

Double cover $Mp$ of $Sp$

For $(V,\omega)$ a symplectic vector space, the metaplectic group $Mp(V,\omega)$ is the Lie group which is the universal double cover of the symplectic group $Sp(V,\omega)$.

This has various more explicit presentations. One is by quadratic Hamiltonians: The metaplectic group is that subgroup of the quantomorphism group of the symplectic manifold $(V,\omega)$ whose elements are given by paths of Hamiltonians that are homogeneously quadratic Hamiltonians (due to Leray 81, section 1.1, see also Robbin-Salamon 93, sections 9-10). (The more general subgroup given by possibly inhomogeneous quadratic Hamiltonians this way is the extended affine symplectic group. The subgroup given by linear Hamiltonians is the Heisenberg group $Heis(V,\omega)$.)

Circle extension $Mp^c$ of $Sp$

There is also a nontrivial circle group-extension of the symplectic group, called $Mp^c$. This is the circle extension associated to the plain metaplectic group $Mp$ above, via the canonical action of $\mathbb{Z}_2$ on $U(1)$ (by complex conjugation): (Forger-Hess 79 (2.4))

\begin{aligned} Mp^c(V,\omega) & \coloneqq Mp(V,\omega) \times_{\mathbb{Z}_2} U(1) \coloneqq ( Mp(V,\omega) \times U(1) )/\mathbb{Z}_2 \end{aligned}

where the last line denotes the quotient group by the diagonal action of $\mathbb{Z}_2$.

(This is in direct analogy to the group Spin^c and its relation to Spin.)

Again, this has various more explicit presentations.

The Segal-Shale-Weil representation is the following. By the Stone-von Neumann theorem there is an essentially unique irreducible unitary representation $W$ of the Heisenberg group $Heis(V,\omega)$. This being essentially unique implies that for each element $g\in Sp(V,\omega)$ of the symplectic group, there is a unique unitary operator $U_g$ such that for all $v\in V$

$W(g(v)) = U_g W(v) U^{-1}_g \,.$

The group $Mp^c$ is the subgroup of the unitary group of all such $U_g$ for $g\in Sp(V,\omega)$. The map $U_g \mapsto g$ exhibits this as a group extension by the circle group

$U(1)\longrightarrow Mp^c(V,\omega) \longrightarrow Sp(V,\omega) \,.$

Alternatively, there is again a characterization by quadratic Hamiltonians (Robinson-Rawnsley 89, theorem (2.4)

Circle extension $MU^c$ of $U$

A symplectic vector space $(V,\omega)$ has a compatible complex structure $J$. Write

$U(V,J) \hookrightarrow Sp(V,\omega)$

for the corresponding unitary group.

Definition

The restriction (pullback) of $Mp^c$ above to this subgroup is denoted $MU^c$ in (Robinson-Rawnsley 89, p. 22)

$\array{ U(1) &=& U(1) \\ \downarrow && \downarrow \\ MU^c(V,J) &\hookrightarrow& Mp^c(V,\omega) \\ \downarrow && \downarrow \\ U(V,J) &\hookrightarrow& Sp(V,\omega) }$

(beware the notational clash with the Thom spectrum MU, which is unrelated).

Properties

Relation to the metalinear group

Inside the symplectic group $Sp(2n, \mathbb{R})$ in dimension $2n$ sits the general linear group in dimension $n$

$Gl(n,\mathbb{R}) \hookrightarrow Sp(2n,\mathbb{R})$

as the subgroup that preserves the standard Lagrangian submanifold $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$. Restriction of the metaplectic group extension along this inclusion defines the metalinear group $Ml(n)$

$\array{ Ml(n, \mathbb{R}) &\hookrightarrow& Mp(2n, \mathbb{R}) \\ \downarrow && \downarrow \\ Gl(n, \mathbb{R}) &\hookrightarrow& Sp(2n, \mathbb{R}) } \,.$

Hence a metaplectic structure on a symplectic manifold induces a metalinear structure on its Lagrangian submanifolds.

(Non-)Triviality of extensions

Proposition

The extension

$U(1)\to Mp^c(V,\omega) \to Sp(V,\omega)$

is nontrivial (does not give a split exact sequence).

Proposition

The extension

$U(1)\to MU^c(V,J) \to U(V,J)$

is trivial (does give a split exact sequence).

Corollary

Every symplectic manifold admits a metaplectic structure.

Proof

Since the unitary group $U(V,J)$ is the maximal compact subgroup of the symplectic group (see here) every $Sp(V,\omega)$-principal bundle has a reduction to a $U(V,J)$-principal bundle. By prop. this reduction in turn lifts to a $MU^c(V,J)$-structure. By def. this induces an $Mp^c$-structure under inclusion along $MU^c \hookrightarrow Mp^c$.

Original references include

• Andre Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111: 143–211. (1964).

• M. Kashiwara; Michèle Vergne, On the Segal-Shale-Weil Representations and Harmonic Polynomials, Inventiones mathematicae (1978) (EuDML, pdf)

• Michael Forger, Harald Hess, Universal metaplectic structures and geometric quantization, Comm. Math. Phys. Volume 64, Number 3 (1979), 269-278. (EUCLID)

• Jean Leray, Lagrangian analysis and quantum mechanics, MIT press 1981 pdf

Further discussion includes

• P. L. Robinson, John Rawnsley, The metaplectic representation, $Mp^c$-structures and geometric quantization, 1989

• Joel Robbin, Dietmar Salamon, Feynman path integrals on phase space and the metaplectic representation Math. Z. 221 (1996), no. 2, 307–335, (MR98f:58051, doi, pdf), also in Dietmar Salamon (ed.), Symplectic Geometry, LMS Lecture Note series 192 (1993)

• John Rawnsley, On the universal covering group of the real symplectic group, Journal of Geometry and Physics 62 (2012) 2044–2058 (pdf)

• Michel Cahen, Simone Gutt, $Spin^c$, $Mp^c$ and Symplectic Dirac Operators, Geometric Methods in Physics Trends in Mathematics 2013, pp 13-28 (pdf)

• Jan Dereziński, Christian Gérard, Mathematics of Quantization and Quantum Fields, Cambridge University Press, 2013