0910.0803 (A. V. Smilga)

Witten index in supersymmetric 3d theories revisited    [PDF]

A. V. Smilga

We have performed a direct calculation of Witten index in N = 1,2,3
supersymmetric Yang-Mills Chern-Simons 3d theories. We do it in the framework
of Born-Oppenheimer (BO) approach by putting the system into a small spatial
box and studying the effective Hamiltonian depending on the zero field
harmonics. At the tree level, our results coincide with the results of Witten,
but there is a difference in the way the loop effects are implemented. In
Witten's approach, one has only take into account the fermion loops, which
bring about a negative shift of the (chosen positive at the tree level)
Chern-Simons coupling k. As a result, Witten index vanishes and supersymmetry
is broken at small k. In the effective BO Hamiltonian framework, fermion, gluon
and ghost loops contribute on an equal footing. Fermion loop contribution to
the effective Hamiltonian can be evaluated exactly, and their effect amounts to
the negative shift k -> k - h/2 for N =1 and k -> k - h for N = 2,3 in the
tree-level formulae for the index. In our approach, with rather natural
assumptions on the structure of bosonic corrections, the shift k -> k + h
brought about by the gluon loops also affects the index. Since the total shift
of k is positive or zero, Witten index appears to be nonzero at nonzero k, and
supersymmetry is not broken. We discuss possible reasons for such disagreement.

View original: http://arxiv.org/abs/0910.0803