Witten index in supersymmetric 3d theories revisited

Abstract

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.

A bug in Eq.(2.20) is fixed

References in corpus (4)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Background field calculations and nonrenormalization theorems in 4d supersymmetric gauge theories and their low-dimensional descendants
• D-brane interactions in type IIB plane-wave background
• Low--dimensional sisters of Seiberg-Witten effective theory

Cited by in corpus (9)

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• Two-loop effective potential for the Wess-Zumino model in 2+1 dimensions
• Once more on the Witten index of 3d supersymmetric YM-CS theory
• Ground states of supersymmetric Yang-Mills-Chern-Simons theory
• Spin(7) duality for N=1 CS-matter theories
• Vacuum structure in 3d supersymmetric gauge theories
• Zero-mode dynamics in supersymmetric Yang-Mills-Chern-Simons theory
• Multidimensional Dirac strings and the Witten index of SYMCS theories with groups of higher rank