Abstract
We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case.
Highlights
We consider knot invariants in the context of large N transitions of topological strings
If we have a stack of N D-branes wrapping a three manifold M 3 ⊂ T ∗M viewing T ∗M as a Calabi-Yau threefold, the large N perturbative Feynman diagrams, i.e. ‘t Hooft diagrams can be viewed as degenerate versions of holomorphic maps from Riemann surfaces with boundaries to T ∗M where the boundary of the Riemann surface is restriced to lie on M
In other words the partition function of the Chern-Simons theory is equivalent to the closed topological A-model involving Riemann surfaces without boundaries, on the resolved conifold
Summary
The conifold transition is a topology changing process relating the smooth hypersurface Xμ (2.1). By construction there is a holomorphic cylinder in Cμ,a ⊂ Xμ contained in Cμ, with one boundary component in Sμ and the second boundary component in Lμ,a This is precisely the basic set-up of large N duality in terms of lifted lagrangian cycles described in section (2.1), above equation (2.19). The second problem is whether one can construct a family of lagrangian cycles Mǫ on Y completing the geometric transition picture represented in (2.17) This will be shown to be the case for any algebraic knot in the subsection, with the caveat that the resulting Gromov-Witten theory with lagrangian boundary conditions on Mǫ is again tractable only for torus knots.
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