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.

arXiv comment21 pages, LaTeX, 1 figure

References in corpus (10)

• A Strong Coupling Test of S-Duality
• Knot Invariants and Topological Strings
• A functor-valued invariant of tangles
• Khovanov-Rozansky Homology and Topological Strings
• Knot Invariants from Four-Dimensional Gauge Theory
• Special Quantum Field Theories In Eight And Other Dimensions
• Higher Dimensional Analogues of Donaldson-Witten Theory
• Euclidean D-branes and higher-dimensional gauge theory
• Gauge theory and the division algebras
• SU(2) solutions to self-duality equations in eight dimensions

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• Holography for field theory solitons
• Integrable (2k)-Dimensional Hitchin Equations
• On an integrable deformation of Kapustin-Witten systems
• Self-dual gravity via Hitchin's equations
• Q-balls of Quasi-particles in a (2,0)-theory model of the Fractional Quantum Hall Effect