Do compact symplectic manifolds play a role in physics?

Question

In classical mechanics, the phase space is the cotangent bundle of the configuration space, and it is a symplectic manifold, but not compact.

Do compact symplectic manifolds have physical meaning? Or just of mathematical interest?

Answer 1

The phase space for a spin-$J$ system is the the two sphere $S^2$ with symplectic structure $\omega= J \sin \theta d\theta \wedge d\phi$, i.e. $J$ times the area 2-form. The unit vector ${\bf n}$ that specfies points on the sphere correponds to the direction of the spin: ${\bf S}= J{\bf n}$.

In general, any Hamiltonian system whose quantum version has finite-dimensional Hilbert space ${\mathcal H}$ will have a compact phase space whose Liouville volume $\int_M\omega^2/n!$ is proportional to ${\rm dim}({\mathcal H})$. A typical example would be the ${\rm SU}(n)$ internal symmetry dynamics described by the Wong equations in gauge theories.

Answer 2

Every Calabi-Yau manifold, being Kähler, is symplectic. Compact Calabi-Yau manifolds play an important role in string theory, though their symplectic structures did not initially seem to play an important role (for as far as I know).

However, one context in which these do play a role is in homological mirror symmetry, an attempt to formulate the concept of mirror symmetry as observed in string theory in purely mathematical terms. In it, the duality between a Calabi-Yau manifold and its mirror partner is stated in terms of the algebraic/analytic structure of one, and the symplectic structure of the other.

Answer 3

The other two answers do indeed give examples of compact symplectic spaces that can appear in physics. I would like to give some comments about how such things might arise in general, rather than an example thereof.

Starting with an arbitrary Lagrangian system, we construct the conjugate momenta and can construct from the boundary variation of the Lagrangian a 2-form which we will want to interpret as our symplectic form. However, there are no guarantees that this 2-form will actually be non-degenerate. This happens in the case of constraints (and also gauge theory), so this is not exactly an edge case insofar as application is concerned.

Vestiges of this can be seen in Dirac's approach to constrained systems, though he never went so far as working with the symplectic form itself. In the end, the resolution of these difficulties is to perform what's known as a symplectic quotient of the standard non-compact/"flat" phase space. Such quotientings may or may not result in another non-compact space, or it may be mixed in that only some directions become compact.