Khovanov-Rozansky Homology and Topological Strings

By Sergei Gukov, A. Schwarz, C. Vafa. 2004

249 citations recorded by the source corpus.

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/8146d9b86f923d4fc4366a2443a9a4cda09d1946

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Sergei Gukov
2. C. Vafa
3. E. Witten
4. Tudor Dimofte
5. Pavel Putrov
6. Marcos Mariño
7. Mina Aganagic
8. I. Cherednik

In Mathematical Physics

1. Sergei Gukov
2. C. Vafa
3. E. Witten
4. Tudor Dimofte
5. Marcos Mariño
6. Mina Aganagic
7. I. Cherednik
8. Marko Stosic

In Nuclear and High Energy Physics

1. Marcos Mariño
2. C. Vafa
3. Sergei Gukov
4. E. Witten
5. Tudor Dimofte
6. D. Gaiotto
7. Mina Aganagic
8. Pavel Putrov

Citations in the corpus

• Fivebranes and 3-manifold homology (2016, methodology): “This structural property parallels what was found for knots and links [7, 10, 29].”
• Fivebranes and Knots (2011, methodology): “) When an M-theory embedding exists, it can lead to further results, as shown strikingly in [31,34,36] by analysis of geometric transitions that do follow from a string/M-theory embedding.”
• The refined topological vertex (2007, background): “In fact the results of [25] suggests that the Khovanov knot invariants are related to this refinement of the open string amplitude.”
• A new look at the path integral of quantum mechanics (2010, background): “The application involves a new way to understand the link [15] between BPS states of branes and Khovanov homology [16] of knots.”
• Knot Invariants from Four-Dimensional Gauge Theory (2011, background): “Khovanov homology has been interpreted physically [3] in terms of topological strings, building on earlier work on BPS states of open strings [4].”
• Some differentials on Khovanov–Rozansky homology (2006, background): “in [11]. Their graded Euler characteristics give the sl(N) polynomials: X I,J (−1)JqI dimHI,J N (K) = PK(q N,q). The large Nbehavior of these theories was studied first by Gukov, Schwartz, and Vafa in [5], and then later in [3], where it was conjectured that the limit of HN(K) as N→ ∞ should be a tri”
• Framed BPS states (2010, unknown): “For nonexotic BPS states our PSC is the same as the “refined index” of [8, 10, 11] up to a (convention-dependent) sign.”
• Instanton counting, Macdonald function and the moduli space of D-branes (2005, unknown): “In section two we consider Nekrasov’s formula for the SU(2) gauge theory and check that, even if ǫ1 + ǫ2 6= 0 and the self-duality is lost, it factorizes correctly as the character of SU(2)L × SU(2)R spin representation of five dimensional massive particles.”
• Matrix factorizations and link homology II (2005, methodology): “S Gukov, A Schwarz and C Vafa recently conjectured [6] that there exist integer-valued link invariants DQ;s;r depending on three integer parameters Q; s; r; that can be used to determine ranks of sl(n) link homology groups as well as the coefficients of the HOMFLYPT polynomial of a link.”
• Jones polynomials of torus knots via DAHA (2011, methodology): “The focus is on the 3-parametric superpolynomials [1, 11, 17, 18, 24, 31].”
• REFINED BPS STATE COUNTING FROM NEKRASOV'S FORMULA AND MACDONALD FUNCTIONS (2008, background): “Based on this transformation law we argue a possible relation of our refined vertex to homological invariants of the Hopf link [28, 29].”
• Quantum geometry of refined topological strings (2011, methodology): “Combining the SO(2)L subgroup of it together with the SO(2) R-symmetry, we still have an SO(2)L × SO(2)R = U(1)L ×U(1)R symmetry available [26].”
• Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry (2009, methodology): “In a similar spirit, one can establish a relation between our surface operators in N = 2 field theories, and the D-strings employed by [16] in N = 2 string theory compactifications to give a physical interpretation to the refined open topological string amplitudes[17, 18].”