Witten Index in 3D Supersymmetric Chern-Simons-Yang-Mills Theory
SU/Nordita High Energy Theory Seminar
Friday 10 August 2012
to 14:30 at
Andrei Smilga (Nantes University)
The number of vacuum states in this theory was evaluated by Witten in 1999. He obtained in particular that the index vanishes for SU(N) if the level (the CS coupling) k is less than N/2 such that supersymmetry is broken. He analyzed this theory in a LARGE spatial box in which case the low-energy dynamics is determined by only the CS term in the action, with the level being renormalized by a fermion loop.
We perform a different calculation of the index using the Born-Oppenheimer (BO) approach and putting the system into a SMALL spatial box and studying the effective Hamiltonian depending on the zero field harmonics. The latter represents a generalization of Landau Hamiltonian describing the motion in an external magnetic field. At the tree level, this magnetic field is homogeneous, whereas the gluon and fermion loops bring about singular flux lines. The magnetic flux of the lines arising due to gluon loops in integer, and they do not affect the index. The magnetic fluxes of the lines associated with the fermion loops are half integer and they affect the index.
The final result coincides with the result of Witten.