Homological algebra of knots and BPS states
arXiv:1112.0030 · doi:10.2140/gtm.2012.18.309
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.
42 pages, 5 figures, additional references and explanations concerning colored differentials and grading choices, enhanced analysis of examples
References in corpus (4)
Cited by in corpus (17)
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- Remarks on coloured triply graded link invariants
- A non-torus link from topological vertex
- Refined large N duality for knots
- Categorified sl(N) invariants of colored rational tangles
- Modules over plane curve singularities in any ranks and DAHA
- Towards formalization of the soliton counting technique for the Khovanov-Rozansky invariants in the deformed -matrix approach
- Superpolynomials of algebraic links