  Numerical and analytical results are reviewed, which support SO(5) symmetry
as a concept unifying superconductivity and antiferromagnetism in the
high-temperature superconductors. Exact cluster diagonalizations verify that
the low-energy states of the two-dimensional t-J and Hubbard models, widely
used microscopic models for the high-Tc cuprates, form SO(5) symmetry
multiplets. Apart from a small standard deviation ~J/10, these multiplets
become degenerate at a critical chemical potential (transition into doped
system). As a consequence, the d-wave superconducting states away from
half-filling are obtained from the higher spin states at half-filling through
SO(5) rotations. Between one and two dimensions, using weak-coupling
renormalization, a rather general ladder Hamiltonian including
next-nearest-neighbor hopping can be shown to flow to an SO(5) symmetric point.
Experimental tests and consequences such as the existence of a pi-Goldstone
mode both in the insulator and superconductor and, in particular, the
relationship between the photoemission spectra of the insulator and
superconductor, are emphasized.
