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Title: 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 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.

Authors:
Publication Date:
Research Org.:
North Carolina State Univ., Raleigh, NC (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1424540
Alternate Identifier(s):
OSTI ID: 1498961
Grant/Contract Number:  
SC0013036; FG02-03ER41260
Resource Type:
Published Article
Journal Name:
Physical Review D
Additional Journal Information:
Journal Name: Physical Review D Journal Volume: 97 Journal Issue: 6; Journal ID: ISSN 2470-0010
Publisher:
American Physical Society
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats

Behtash, Alireza. More on homological supersymmetric quantum mechanics. United States: N. p., 2018. Web. doi:10.1103/PhysRevD.97.065002.
Behtash, Alireza. More on homological supersymmetric quantum mechanics. United States. doi:10.1103/PhysRevD.97.065002.
Behtash, Alireza. Wed . "More on homological supersymmetric quantum mechanics". United States. doi:10.1103/PhysRevD.97.065002.
@article{osti_1424540,
title = {More on homological supersymmetric quantum mechanics},
author = {Behtash, Alireza},
abstractNote = {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.},
doi = {10.1103/PhysRevD.97.065002},
journal = {Physical Review D},
number = 6,
volume = 97,
place = {United States},
year = {2018},
month = {3}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1103/PhysRevD.97.065002

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Cited by: 2 works
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