More on homological supersymmetric quantum mechanics
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 nonzero genus. Motivated by Picard-Lefschetz theory, we write down a general formula for the index of 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 parameters appear in the intersection coefficients for the relative homology of the quiver quantum mechanics resulting from dimensional reduction of 2d N = (2 , 2) gauge theory on a circle and explicitly calculate integrals along the Lefschetz thimbles in 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.
- Research Organization:
- North Carolina State University, Raleigh, NC (United States)
- Sponsoring Organization:
- USDOE
- Grant/Contract Number:
- SC0013036; FG02-03ER41260
- OSTI ID:
- 1424540
- Alternate ID(s):
- OSTI ID: 1498961
- Journal Information:
- Physical Review D, Journal Name: Physical Review D Vol. 97 Journal Issue: 6; ISSN 2470-0010
- Publisher:
- American Physical SocietyCopyright Statement
- Country of Publication:
- United States
- Language:
- English
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