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arXiv · 1112.0030

Homological algebra of knots and BPS states

Sergei Gukov·Marko Stosic
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Abstract

It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzburg models. One important application to knot homologies is the existence of "colored differentials" that relate homological invariants of knots colored by different representations. Based on this structure, we formulate a list of properties of the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By calculating the colored HOMFLY homology for symmetric and anti-symmetric representations, we find a remarkable "mirror symmetry" between these triply-graded theories.

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Sergei Gukov, Marko Stosic. 2012-03-15. Homological algebra of knots and BPS states. https://doi.org/10.2140/gtm.2012.18.309

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