Non-integrable Phases Research Articles

Gauge invariance in Chern-Simons theory on a torus.

In the Chern-Simons gauge theory on a manifold ${T}^{2}$\ifmmode\times\else\texttimes\fi{}${R}^{1}$ (two-torus\ifmmode\times\else\texttimes\fi{}time) the unitary operators, which induce large gauge transformations shifting the nonintegrable phases of the two dinstinct Wilson-line integrals on the torus by multiples of 2\ensuremath{\pi}, do not commute with each other unless the coefficient of the Chern-Simons term is quantized. In U(1) theory this condition gives the statistics phase \ensuremath{\theta}=\ensuremath{\pi}/n (n an integer). The condition coincides with the one previously derived on a manifold ${S}^{3}$ (three-sphere) for SU(N\ensuremath{\ge}3) theory but differs by a factor 2 for SU(2) theory. The requirement of the ${Z}_{N}$ invariance in pure SU(N) gauge theory imposes a stronger constraint.

Dynamics of non-integrable phases and gauge symmetry breaking

On a multiply-connected space the non-integrable phase factor P exp(ig ∫ A μ dx μ }, a pathordered line integral along a non-contractable loop, becomes a dynamical degree of freedom in gauge theory. The dynamics of such non-integrable phases are examined in detail with the most general boundary condition for gauge fields and fermions. Sometimes the dynamics of the non-integrable phases compensate the arbitrariness in the boundary condition imposed, leading to the same physics results. In other cases the dynamics of the non-integrable phases induce spontaneous breaking of non-Abelian gauge symmetry. In other words the physically realized symmetry of the system differs from, and can be either greater or smaller than, the symmetry of the boundary condition. The effective potential for the non-integrable phases in the SU( N) gauge theory on S 1 ⊗ R 1, d−2 is computed in the one-loop approximation. It is shown that the gauge symmetry is dynamically broken in the presence of fermions in the adjoint representation, depending on the value of the boundary condition parameter.

Dynamics of Non-integrable phases and gauge symmetry breaking: Yutaka Hosotani. School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455

Dynamics of Non-integrable phases and gauge symmetry breaking: Yutaka Hosotani. School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455