Projective unitary group

In mathematics, the projective unitary group PU(n) is the quotient of the unitary group U(n) by the right multiplication of its center, U(1), embedded as scalars.

Thus, elements of PU(n) correspond to equivalence classes of unitary matrices under multiplication by a constant phase θ.

This space is not SU(n) (which only requires the determinant to be one), because SU(n) still contains elements eiθI where eiθ is an n-th root of unity (since then det(eiθI) = eiθn = 1).

The projective special unitary group PSU(n) is equal to the projective unitary group, in contrast to the orthogonal case.

The connections between the U(n), SU(n), their centers, and the projective unitary groups is shown in the Figure on the right (notice that in the figure the integers are denoted

is the scalar matrices of the nth roots of unity: The natural map is an isomorphism, by the second isomorphism theorem, thus and the special unitary group SU(n) is an n-fold cover of the projective unitary group.

via: One can also define unitary groups over finite fields: given a field of order q, there is a non-degenerate Hermitian structure on vector spaces over

unique up to unitary congruence, and correspondingly a matrix group denoted

and likewise special and projective unitary groups.

is a finite simple group, (Grove 2002, Thm.

The same construction may be applied to matrices acting on an infinite-dimensional Hilbert space

When f: X → U(H) is a continuous mapping of a compact space X into the unitary group, one can use a finite dimensional approximation of its image and a simple K-theoretic trick to show that it is actually homotopic to the trivial map onto a single point.

Note that this is a purely infinite dimensional phenomenon, in contrast to the finite-dimensional cousins U(n) and their limit U(∞) under the inclusion maps which are not contractible admitting homotopically nontrivial continuous mappings onto U(1) given by the determinant of matrices.

is, as in the finite dimensional case, U(1), which again acts on the unitary group via multiplication by a phase.

As the unitary group does not contain the zero matrix, this action is free.

is defined precisely to be the space of orbits of the U(1) action on

must be of the same homotopy type as the infinite-dimensional complex projective space, which also represents

The adjoint action is defined by thinking of an element of PU(n) as an equivalence class of elements of U(n) that differ by phases.

One can then take the adjoint action with respect to any of these U(n) representatives, and the phases commute with everything and so cancel.

Thus the action is independent of the choice of representative and so it is well-defined.

These are useful in quantum mechanics, as physical states are only defined up to phase.

The projective representations of a group are classified by its second integral cohomology, which in this case is or The cohomology groups in the finite case can be derived from the long exact sequence for bundles and the above fact that SU(n) is a

The adjoint action of the infinite projective unitary group is useful in geometric definitions of twisted K-theory.

on either the Fredholm operators or the infinite unitary group is used.

is also the classifying space for the third integral cohomology group, therefore

, which is PU(n) and it acts on fields using the adjoint action described above.

In this context, the distinction between SU(n) and PU(n) has an important physical consequence.

SU(n) is simply connected, but the fundamental group of PU(n) is

, the cyclic group of order n. Therefore a PU(n) gauge theory with adjoint scalars will have nontrivial codimension 2 vortices in which the expectation values of the scalars wind around PU(n)'s nontrivial cycle as one encircles the vortex.

An example of such a vortex is the Douglas–Shenker string in SU(n) Seiberg–Witten gauge theories.