Homological algebra of knots and BPS states

By Sergei Gukov, Marko Stosic. 2011

136 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3be5c314ed333e6020e2a3b524f2602ffa1d4e15

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In Geometry and Topology

1. Pavel Putrov
2. Sergei Gukov
3. C. Vafa
4. I. Cherednik
5. I. Danilenko
6. Paul Wedrich
7. David E. V. Rose
8. E. Witten

In Mathematical Physics

1. I. Cherednik
2. Sergei Gukov
3. Pavel Putrov
4. C. Vafa
5. Paul Wedrich
6. David E. V. Rose
7. I. Danilenko
8. E. Witten

Citations in the corpus

• Fivebranes and 3-manifold homology (2016, background): “As in the case of knots [59], the basic building blocks of 3-manifold homologies (1.”
• Deformations of colored sl(N) link homologies via foams (2015, methodology): “Special thanks go to Marko Stošić and Daniel Tubbenhauer for numerous useful comments on a draft of this paper.”
• DAHA and iterated torus knots (2014, background): “(v) Superduality [25; 12; 22] Up to a power of q and t , (2-23) HEr;Es.”