Once more on the Witten index of 3d supersymmetric YM-CS theory
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The problem of counting the vacuum states in the supersymmetric 3d Yang- Mills-Chern-Simons theory is reconsidered. We resolve the controversy between its original calculation in  at large volumes g 2L ≫ 1 and the calculation based on the evaluation of the effective Lagrangian in the small volume limit, g 2L ≪ 1 . We show that the latter calculation suffers from uncertainties associated with the singularities in the moduli space of classical vacua where the Born-Oppenheimer approximation breaks down. We also show that these singularities can be accurately treated in the Hamiltonian Born-Oppenheimer method, where one has to match carefully the effective wave functions on the Abelian valley and the wave functions of reduced non-Abelian QM theory near the singularities. This gives the same result as original Witten’s calculation.
KeywordsSupersymmetric gauge theory Field Theories in Lower Dimensions Chern- Simons Theories
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