skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Real homotopy theory and supersymmetric quantum mechanics

Abstract

In the context of observing string backgrounds, much work has been devoted to the question of how similar a general quantum field theory (specifically, a two-dimensional superconformal field theory) is to a sigma model. Put differently, one would like to know how well or poorly one can understand the physics of string backgrounds in terms of concepts of classical geometry. Much attention has also been given of late to the question of how geometry can be encoded in a microscopic physical description that makes no explicit reference to space and time. We revisit the first question and review both well-known and less well-known results about geometry and sigma models from the perspective of dimensional reduction to supersymmetric quantum mechanics. The consequences of arising as the dimensional reduction of a d-dimensional theory for the resulting quantum mechanics are explored. In this context, we reinterpret the minimal models of rational (more precisely, complex) homotopy theory as certain supersymmetric Fock spaces, with unusual actions of the supercharges. The data of the Massey products appear naturally as supersymmetric vacuum states that are entangled between different degrees of freedom. This connection between entanglement and geometry is, as far as we know, not well known tomore » physicists. In addition, we take note of an intriguing numerical coincidence in the context of string compactification on hyper-Kähler eight-manifolds.« less

Authors:
 [1]; ORCiD logo [2]
  1. Stony Brook Univ., NY (United States). Dept. of Physics and Astronomy
  2. Ruprecht-Karls-Univ.Heidelberg, Heidelberg (Germany). Mathematisches Inst.
Publication Date:
Research Org.:
California Inst. of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
OSTI Identifier:
1540144
Alternate Identifier(s):
OSTI ID: 1458783
Grant/Contract Number:  
SC0011632
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 59; Journal Issue: 7; Journal ID: ISSN 0022-2488
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics

Citation Formats

Kim, Hyungrok, and Saberi, Ingmar. Real homotopy theory and supersymmetric quantum mechanics. United States: N. p., 2018. Web. doi:10.1063/1.5011677.
Kim, Hyungrok, & Saberi, Ingmar. Real homotopy theory and supersymmetric quantum mechanics. United States. doi:https://doi.org/10.1063/1.5011677
Kim, Hyungrok, and Saberi, Ingmar. Tue . "Real homotopy theory and supersymmetric quantum mechanics". United States. doi:https://doi.org/10.1063/1.5011677. https://www.osti.gov/servlets/purl/1540144.
@article{osti_1540144,
title = {Real homotopy theory and supersymmetric quantum mechanics},
author = {Kim, Hyungrok and Saberi, Ingmar},
abstractNote = {In the context of observing string backgrounds, much work has been devoted to the question of how similar a general quantum field theory (specifically, a two-dimensional superconformal field theory) is to a sigma model. Put differently, one would like to know how well or poorly one can understand the physics of string backgrounds in terms of concepts of classical geometry. Much attention has also been given of late to the question of how geometry can be encoded in a microscopic physical description that makes no explicit reference to space and time. We revisit the first question and review both well-known and less well-known results about geometry and sigma models from the perspective of dimensional reduction to supersymmetric quantum mechanics. The consequences of arising as the dimensional reduction of a d-dimensional theory for the resulting quantum mechanics are explored. In this context, we reinterpret the minimal models of rational (more precisely, complex) homotopy theory as certain supersymmetric Fock spaces, with unusual actions of the supercharges. The data of the Massey products appear naturally as supersymmetric vacuum states that are entangled between different degrees of freedom. This connection between entanglement and geometry is, as far as we know, not well known to physicists. In addition, we take note of an intriguing numerical coincidence in the context of string compactification on hyper-Kähler eight-manifolds.},
doi = {10.1063/1.5011677},
journal = {Journal of Mathematical Physics},
number = 7,
volume = 59,
place = {United States},
year = {2018},
month = {7}
}

Works referenced in this record:

On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four
journal, January 2001


On the Decomposition of KÄHler Manifolds with Trivial Canonical Class
journal, April 1974


Supersymmetry and Morse theory
journal, January 1982


Action of the Lie algebra SO(5) on the cohomology of a hyperk�hler manifold
journal, January 1991

  • Verbitskii, M. S.
  • Functional Analysis and Its Applications, Vol. 24, Issue 3
  • DOI: 10.1007/BF01077967

Supersymmetry and the cohomology of (hyper) Kähler manifolds
journal, October 1997


N = 4 topological strings
journal, January 1995


Aspects of NT ⩾ 2 topological gauge theories and D-branes
journal, May 1997


Superstrings and manifolds of exceptional holonomy
journal, September 1995

  • Shatashvili, S. L.; Vafa, C.
  • Selecta Mathematica, Vol. 1, Issue 2
  • DOI: 10.1007/bf01671569

On the cohomology of Kahler and hyper-Kähler manifolds
journal, January 1996


Twisted N =2 Supersymmetry with Central Charge and Equivariant Cohomology
journal, April 1997

  • Labastida, J. M. F.; Mariño, M.
  • Communications in Mathematical Physics, Vol. 185, Issue 1
  • DOI: 10.1007/s002200050081

Cohomology of compact hyperkähler manifolds and its applications
journal, July 1996

  • Verbitsky, Mikhail
  • Geometric and Functional Analysis, Vol. 6, Issue 4
  • DOI: 10.1007/bf02247112

Structures différentiables sur les types d'homotopie rationnelle simplement connexes
journal, January 1976

  • Barge, Jean
  • Annales scientifiques de l'École normale supérieure, Vol. 9, Issue 4
  • DOI: 10.24033/asens.1315

Constraints on supersymmetry breaking
journal, July 1982


On the Algebras of BPS States
journal, October 1998

  • Harvey, Jeffrey A.; Moore, Gregory
  • Communications in Mathematical Physics, Vol. 197, Issue 3
  • DOI: 10.1007/s002200050461

Defect perturbations in Landau-Ginzburg models
journal, March 2010

  • Brunner, Ilka; Roggenkamp, Daniel; Rossi, Sebastiano
  • Journal of High Energy Physics, Vol. 2010, Issue 3
  • DOI: 10.1007/jhep03(2010)015

Sequencing BPS spectra
journal, March 2016

  • Gukov, Sergei; Nawata, Satoshi; Saberi, Ingmar
  • Journal of High Energy Physics, Vol. 2016, Issue 3
  • DOI: 10.1007/jhep03(2016)004

Infinitesimal computations in topology
journal, December 1977

  • Sullivan, Dennis
  • Publications mathématiques de l'IHÉS, Vol. 47, Issue 1
  • DOI: 10.1007/bf02684341

Chiral rings in N = 2 superconformal theories
journal, September 1989


Real homotopy theory of K�hler manifolds
journal, October 1975

  • Deligne, Pierre; Griffiths, Phillip; Morgan, John
  • Inventiones Mathematicae, Vol. 29, Issue 3
  • DOI: 10.1007/bf01389853