More on Homological Supersymmetric Quantum Mechanics

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In this work, we first solve complex Morse flow equations for the simplest case of a bosonic harmonic oscillator to discuss localization in the context of Picard-Lefschetz theory.

• We briefly touch on the exact non-BPS solutions of the bosonized supersymmetric quantum mechanics on algebraic geometric grounds and report that their complex phases can be accessed through the cohomology of WKB 1-form of the underlying singular spectral curve subject to necessary cohomological corrections for non-zero genus.
• Motivated by Picard-Lefschetz theory, we write down a general formula for the index of $\mathcal{N} = 4$ quantum mechanics with background $R$-symmetry gauge fields.
• We conjecture that certain symmetries of the refined Witten index and singularities of the moduli space may be used to determine the correct intersection coefficients.