Bound States at Threshold in Supersymmetric Quantum Mechanics
M. Porrati, A. Rozenberg
TL;DR
This work introduces a general method to detect normalizable ground states in supersymmetric quantum mechanics with non-compact moduli spaces by perturbing the superpotential with a function w and comparing the L2 cohomology of the perturbed and original systems. The method reduces the problem to computing the Witten index of a gapped, perturbed system and then transferring the cohomology information back to the original Hamiltonian, providing criteria for the existence of bound states at threshold. Applied to the H-monopole QM and the D0 brane QM, it proves at least one zero-energy bound state in each case (n_B − n_F = 1 for the H-monopole and, under a no-3D-bound-state assumption, for prime N D0 branes in nine dimensions). The approach avoids heavy bulk integrals and offers a framework that can extend to more complex SUSY quantum systems relevant to string theory and matrix theory, while yielding cohomology representatives of the true ground states.
Abstract
We propose a general method that allows to detect the existence of normalizable ground states in supersymmetric quantum mechanical systems with non-Fredholm spectrum. We apply our method to show the existence of bound states at threshold in two important cases: 1) the quantum mechanical system describing H-monopoles; 2) the quantum mechanics of D0 branes.