Khovanov-Rozansky Homology and Topological Strings

Abstract

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.

Link Homologies and the Refined Topological Vertex

“…We will be able to compute the homology groups H g,R directly from string theory using the recent work [17], where it was shown how the topological vertex [18] (which computes topological string amplitudes in toric geometries (Table 1)) can be refined to compute Refined BPS invariants [15]. Since the topological vertex formalism is composed of open string amplitudes, this refinement together with the conjecture of [8] implies that the refined topological vertex should be computing homological link invariants, at least for the class of links which can be formulated in terms of local toric geometries. The basic example of such a link is the Hopf link.…”

“…Moreover the sign of N is correlated with its fermion number. It was proposed in [8] that there is a further charge one can consider in labeling the BPS states of M2 branes ending on M5 branes: The normal geometry to the M5 brane includes, in addition to the spacetime R 3 , and the three normal directions inside the CY, an extra R 2 plane. It was proposed there that the extra SO(2) rotation in this plane will provide an extra gradation which could be viewed as a refinement of topological strings and it was conjectured that this is related to link homologies that we will review in the next section.…”

“…Similarly, the homological sl(N ) invariant of a two-component link L can be written as a sum of connected and disconnected terms [8]:…”

“…It was proposed in [8] that the homological grading of link homologies is related to the extra charge in the extension of topological string proposed in [14]. In particular, supersymmetric states of holomorphic branes ending on Lagrangian branes, labeled by all physical charges, should reproduce homological invariants of knots and links,…”

Factorization of differential expansion for antiparallel double-braid knots

“…Far ago it was reformulated in terms of Chern-Simons theory [9]- [17], where (at least for the knots in S 3 ) it is basically reduced to a free-field calculation, and a formal answer is provided [18]- [32] in terms of representation theory of quantum groups. This, however, does not help to obtain explicit answers, except for the simplest situations, covered by the well-known databases [33][34][35].…”