# 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}

}

DOI: 10.1103/PhysRevD.97.065002

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Works referenced in this record:

##
Hidden Topological Angles in Path Integrals

journal, July 2015

- Behtash, Alireza; Sulejmanpasic, Tin; Schäfer, Thomas
- Physical Review Letters, Vol. 115, Issue 4

##
Phases of N = 2 theories in two dimensions

journal, August 1993

- Witten, Edward
- Nuclear Physics B, Vol. 403, Issue 1-2

##
Spectral Networks

journal, March 2013

- Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew
- Annales Henri Poincaré, Vol. 14, Issue 7

##
Supersymmetry and Morse theory

journal, January 1982

- Witten, Edward
- Journal of Differential Geometry, Vol. 17, Issue 4

##
On classification ofN=2 supersymmetric theories

journal, December 1993

- Cecotti, Sergio; Vafa, Cumrun
- Communications in Mathematical Physics, Vol. 158, Issue 3

##
Elliptic Genera of 2d $${\mathcal{N}}$$ N = 2 Gauge Theories

journal, November 2014

- Benini, Francesco; Eager, Richard; Hori, Kentaro
- Communications in Mathematical Physics, Vol. 333, Issue 3

##
Bogomol'nyi Equation for Intersecting Domain Walls

journal, August 1999

- Gibbons, G. W.; Townsend, P. K.
- Physical Review Letters, Vol. 83, Issue 9

##
BPS Quivers and Spectra of Complete $${\mathcal{N} = 2}$$ N = 2 Quantum Field Theories

journal, August 2013

- Alim, Murad; Cecotti, Sergio; Córdova, Clay
- Communications in Mathematical Physics, Vol. 323, Issue 3

##
Spontaneously broken supergauge symmetries and goldstone spinors

journal, September 1974

- Fayet, P.; Iliopoulos, J.
- Physics Letters B, Vol. 51, Issue 5

##
Wall-crossing made smooth

journal, April 2015

- Pioline, Boris
- Journal of High Energy Physics, Vol. 2015, Issue 4

##
Quiver invariants from intrinsic Higgs states

journal, July 2012

- Lee, Seung-Joo; Wang, Zhao-Long; Yi, Piljin
- Journal of High Energy Physics, Vol. 2012, Issue 7

##
Knot invariants from four-dimensional gauge theory

journal, January 2012

- Gaiotto, Davide; Witten, Edward
- Advances in Theoretical and Mathematical Physics, Vol. 16, Issue 3

##
Holomorphic disks and knot invariants

journal, August 2004

- Ozsváth, Peter; Szabó, Zoltán
- Advances in Mathematics, Vol. 186, Issue 1

##
Coulomb branch localization in quiver quantum mechanics

journal, February 2016

- Ohta, Kazutoshi; Sasai, Yuya
- Journal of High Energy Physics, Vol. 2016, Issue 2

##
2d index and surface operators

journal, March 2014

- Gadde, Abhijit; Gukov, Sergei
- Journal of High Energy Physics, Vol. 2014, Issue 3

##
${\mathbb{R}}^{3}$ Index for Four-Dimensional $\mathcal{N}=2$ Field Theories

journal, March 2015

- Alexandrov, Sergei; Moore, Gregory W.; Neitzke, Andrew
- Physical Review Letters, Vol. 114, Issue 12

##
3d analogs of Argyres-Douglas theories and knot homologies

journal, January 2013

- Fuji, Hiroyuki; Gukov, Sergei; Stošić, Marko
- Journal of High Energy Physics, Vol. 2013, Issue 1

##
On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants

journal, May 2013

- Manschot, Jan; Pioline, Boris; Sen, Ashoke
- Journal of High Energy Physics, Vol. 2013, Issue 5

##
Witten index and wall crossing

journal, January 2015

- Hori, Kentaro; Kim, Heeyeon; Yi, Piljin
- Journal of High Energy Physics, Vol. 2015, Issue 1

##
Quantum Quivers and Hall/Hole Halos

journal, October 2002

- Denef, Frederik
- Journal of High Energy Physics, Vol. 2002, Issue 10

##
Two dimensional gauge theories revisited

journal, October 1992

- Witten, Edward
- Journal of Geometry and Physics, Vol. 9, Issue 4

##
The curious incident of multi-instantons and the necessity of Lefschetz thimbles

journal, November 2015

- Behtash, Alireza; Poppitz, Erich; Sulejmanpasic, Tin
- Journal of High Energy Physics, Vol. 2015, Issue 11

##
Holomorphic anomaly and quantum mechanics

journal, December 2017

- Codesido, Santiago; Mariño, Marcos
- Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 5

##
Fivebranes and 3-manifold homology

journal, July 2017

- Gukov, Sergei; Putrov, Pavel; Vafa, Cumrun
- Journal of High Energy Physics, Vol. 2017, Issue 7

##
Analytic continuation of Liouville theory

journal, December 2011

- Harlow, Daniel; Maltz, Jonathan; Witten, Edward
- Journal of High Energy Physics, Vol. 2011, Issue 12

##
BPS graphs: from spectral networks to BPS quivers

journal, July 2017

- Gabella, Maxime; Longhi, Pietro; Park, Chan Y.
- Journal of High Energy Physics, Vol. 2017, Issue 7

##
Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling

journal, December 2014

- Tanizaki, Yuya; Koike, Takayuki
- Annals of Physics, Vol. 351

##
The Yang-Mills Equations over Riemann Surfaces

journal, March 1983

- Atiyah, M. F.; Bott, R.
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 308, Issue 1505

##
Supersymmetric localization in two dimensions

journal, October 2017

- Benini, Francesco; Le Floch, Bruno
- Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 44

##
Domain wall junctions are 1/4 BPS states

journal, February 2000

- Carroll, Sean M.; Hellerman, Simeon; Trodden, Mark
- Physical Review D, Vol. 61, Issue 6

##
Elliptic Genera of Two-Dimensional $${\mathcal{N} = 2}$$ N = 2 Gauge Theories with Rank-One Gauge Groups

journal, November 2013

- Benini, Francesco; Eager, Richard; Hori, Kentaro
- Letters in Mathematical Physics, Vol. 104, Issue 4

##
Nonperturbative contributions from complexified solutions in $\mathbb{C}{P}^{N-1}$ models

journal, November 2016

- Fujimori, Toshiaki; Kamata, Syo; Misumi, Tatsuhiro
- Physical Review D, Vol. 94, Issue 10

##
Gauge theory and knot homologies

journal, May 2007

- Gukov, S.
- Fortschritte der Physik, Vol. 55, Issue 5-7

##
BPS states, refined indices, and quiver invariants

journal, October 2012

- Lee, Seung-Joo; Wang, Zhao-Long; Yi, Piljin
- Journal of High Energy Physics, Vol. 2012, Issue 10

##
Exact resurgent trans-series and multibion contributions to all orders

journal, May 2017

- Fujimori, Toshiaki; Kamata, Syo; Misumi, Tatsuhiro
- Physical Review D, Vol. 95, Issue 10

##
Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

journal, May 2012

- Pestun, Vasily
- Communications in Mathematical Physics, Vol. 313, Issue 1

##
Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles

journal, October 2015

- Cherman, Aleksey; Dorigoni, Daniele; Ünsal, Mithat
- Journal of High Energy Physics, Vol. 2015, Issue 10

##
Dynamical breaking of supersymmetry

journal, October 1981

- Witten, Edward
- Nuclear Physics B, Vol. 188, Issue 3

##
Complexified Path Integrals, Exact Saddles, and Supersymmetry

journal, January 2016

- Behtash, Alireza; Dunne, Gerald V.; Schäfer, Thomas
- Physical Review Letters, Vol. 116, Issue 1

##
The moment map and equivariant cohomology

journal, January 1984

- Atiyah, M. F.; Bott, R.
- Topology, Vol. 23, Issue 1