Octonions, Monopoles, and Knots

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Witten’s approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys–Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any G2 holonomy manifold and is a close cousin of the monopole equation of Bogomolny. Octonions play the central role in our view, in which both the seven-dimensional equations and the Haydys–Witten equations appear as reductions of the eight-dimensional Spin(7) instanton equation.

Original languageEnglish (US)
Pages (from-to)641-659
Number of pages19
JournalLetters in Mathematical Physics
Volume105
Issue number5
DOIs
StatePublished - May 1 2015

Fingerprint

Octonions
Monopole
monopoles
Knot
Khovanov Homology
Holonomy
Instantons
Systems of Partial Differential Equations
One to one correspondence
homology
System of equations
instantons
partial differential equations

Keywords

  • Gauge theory
  • knots
  • octonions
  • special holonomy

ASJC Scopus subject areas

  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

Octonions, Monopoles, and Knots. / Cherkis, Sergey.

In: Letters in Mathematical Physics, Vol. 105, No. 5, 01.05.2015, p. 641-659.

Research output: Contribution to journalArticle

@article{b76e278648694a1eb5abad39f7fdb501,
title = "Octonions, Monopoles, and Knots",
abstract = "Witten’s approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys–Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any G2 holonomy manifold and is a close cousin of the monopole equation of Bogomolny. Octonions play the central role in our view, in which both the seven-dimensional equations and the Haydys–Witten equations appear as reductions of the eight-dimensional Spin(7) instanton equation.",
keywords = "Gauge theory, knots, octonions, special holonomy",
author = "Sergey Cherkis",
year = "2015",
month = "5",
day = "1",
doi = "10.1007/s11005-015-0755-0",
language = "English (US)",
volume = "105",
pages = "641--659",
journal = "Letters in Mathematical Physics",
issn = "0377-9017",
publisher = "Springer Netherlands",
number = "5",

}

TY - JOUR

T1 - Octonions, Monopoles, and Knots

AU - Cherkis, Sergey

PY - 2015/5/1

Y1 - 2015/5/1

N2 - Witten’s approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys–Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any G2 holonomy manifold and is a close cousin of the monopole equation of Bogomolny. Octonions play the central role in our view, in which both the seven-dimensional equations and the Haydys–Witten equations appear as reductions of the eight-dimensional Spin(7) instanton equation.

AB - Witten’s approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys–Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any G2 holonomy manifold and is a close cousin of the monopole equation of Bogomolny. Octonions play the central role in our view, in which both the seven-dimensional equations and the Haydys–Witten equations appear as reductions of the eight-dimensional Spin(7) instanton equation.

KW - Gauge theory

KW - knots

KW - octonions

KW - special holonomy

UR - http://www.scopus.com/inward/record.url?scp=84937763356&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937763356&partnerID=8YFLogxK

U2 - 10.1007/s11005-015-0755-0

DO - 10.1007/s11005-015-0755-0

M3 - Article

AN - SCOPUS:84937763356

VL - 105

SP - 641

EP - 659

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 5

ER -