More on Homological Supersymmetric Quantum Mechanics

Abstract

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. A few examples, where this conjecture holds, are shown in both linear and closed quivers with rank-one quiver gauge groups. The $R$-anomaly removal along the "Morsified" relative homology cycles also called "Lefschetz thimbles" is shown to lead to the appearance of Stokes lines. We show that the Fayet-Iliopoulos (FI) parameters appear in the intersection coefficients for the relative homology of the quiver quantum mechanics resulting from dimensional reduction of $2d$ $\mathcal{N}=(2,2)$ gauge theory on a circle and explicitly calculate integrals along the Lefschetz thimbles in $\mathcal{N}=4$ ${\mathbb{CP}}^{k-1}$ model. The Stokes jumping of coefficients and its relation to wall crossing phenomena is briefly discussed. We also find that the notion of "on-the-wall" index is related to the invariant Lefschetz thimbles under Stokes phenomena. An implication of the Lefschetz thimbles in constructing knots from quiver quantum mechanics is indicated.

29 pages, 13 figures. v3 Improved version, more references added, conclusions added, typos fixed, accepted for publication in PRD

References in corpus (8)

• Witten Index and Wall Crossing
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• Non-perturbative Contributions from Complexified Solutions in $\mathbb{C}{P}^{N-1}$ Models
• Exact Resurgent Trans-series and Multi-Bion Contributions to All Orders
• Holomorphic Anomaly and Quantum Mechanics
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Cited by in corpus (11)

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• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• $1/\epsilon$ problem in resurgence
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Complex Saddles of Charged-AdS Gravitational partition function
• Exact WKB in all sectors II: Potentials with non-degenerate saddles