nLab metaplectic group



Symplectic geometry

Group Theory



Double cover MpMp of SpSp

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

Circle extension Mp cMp^c of SpSp

There is also a nontrivial circle group-extension of the symplectic group, called Mp cMp^c. This is the circle extension associated to the plain metaplectic group MpMp above, via the canonical action of 2\mathbb{Z}_2 on U(1)U(1) (by complex conjugation): (Forger-Hess 79 (2.4))

Mp c(V,ω) Mp(V,ω)× 2U(1)(Mp(V,ω)×U(1))/ 2 \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 2\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 WW of the Heisenberg group Heis(V,ω)Heis(V,\omega). This being essentially unique implies that for each element gSp(V,ω)g\in Sp(V,\omega) of the symplectic group, there is a unique unitary operator U gU_g such that for all vVv\in V

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

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

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

e.g. (Robinson-Rawnsley 89, p. 19, Dereziński-Gérard 13, def. 10.24)

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

Circle extension MU cMU^c of UU

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

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

for the corresponding unitary group.


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

U(1) = U(1) MU c(V,J) Mp c(V,ω) U(V,J) Sp(V,ω) \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).


Relation to the metalinear group

Inside the symplectic group Sp(2n,)Sp(2n, \mathbb{R}) in dimension 2n2n sits the general linear group in dimension nn

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

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

Ml(n,) Mp(2n,) Gl(n,) Sp(2n,). \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


The extension

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

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

(Robinson-Rawnsley 89, theorem (2.8))


The extension

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

is trivial (does give a split exact sequence).

(Robinson-Rawnsley 89, theorem (2.9))


Every symplectic manifold admits a metaplectic structure.

(Robinson-Rawnsley 89, theorem (6.2))


Since the unitary group U(V,J)U(V,J) is the maximal compact subgroup of the symplectic group (see here) every Sp(V,ω)Sp(V,\omega)-principal bundle has a reduction to a U(V,J)U(V,J)-principal bundle. By prop. this reduction in turn lifts to a MU c(V,J)MU^c(V,J)-structure. By def. this induces an Mp cMp^c-structure under inclusion along MU cMp cMU^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:cmp/1103904723)

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

Further discussion includes

  • P. L. Robinson, John Rawnsley, The metaplectic representation, Mp cMp^c-structures and geometric quantization, 1989 (doi:10.1090/memo/0410)

  • 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 cSpin^c, Mp cMp^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

Last revised on July 18, 2020 at 20:33:45. See the history of this page for a list of all contributions to it.