Lagrangians for the Gopakumar–Vafa conjecture

Abstract

This article explains how to construct immersed Lagrangian submanifolds in C 2 that are asymptotic at large distance from the origin to a given braid in the 3-sphere. The self-intersections of the Lagrangians are related to the crossings of the braid. These Lagrangians are then used to construct immersed Lagrangians in the vector bundle O(−1) ⊕ O(−1) over the Riemann sphere which are asymptotic at large distance from the zero section to braids. The verification of Gopakumar and Vafa's proposal has been slow, in part because the string theoretic arguments have not provided a geometric correspondence between a particular knot and a particular set of holomorphic curves in O(−1) ⊕ O(−1). Even so, it is a good bet, verified in part by Katz and Liu [7], Labastida, Marino and Vafa [9] and Aganagic, Klemm and Vafa [1], that such a correspondence exists and that it is mediated by a suitable Lagrangian 3-manifold sitting in O(−1) ⊕ O(−1). To be specific, the knot should determine the Lagrangian, and then a knot invariant should come as a suitable count of compact, holomorphic curves with boundary on the Lagrangian. This said, the mathematics of counting holomorphic curves with boundary on a Lagrangian submanifold dates back to Floer's original work on the Arnold conjecture

Topological Open String Amplitudes On Orientifolds

“…If one of the representations is trivial, we recover the oriented amplitude in the presence of S K , thereforeC R = W U(N) R (K). But in the general case it is not obvious how to determine C R 1 R 2 .Although there are proposals for the geometry of the Lagrangian submanifolds S K[25,41], a direct Gromov-Witten computation of the corresponding open string amplitudes seems to be very difficult. One possible way of determining C R 1 R 2 would be to translate it into a pure knot-theoretic computation in the context of Chern-Simons theory, but we haven't found a completely satisfactory solution to this problem.Although we don't know how to compute the covering amplitude for an arbitrary knot, we can still extract the f c=1 R amplitudes from the knowledge of W SO(N) R…”

Entanglement entropy and edge modes in topological string theory II: The dual gauge theory story

“…From the point of view of the original closed string theory on the resolved conifold, the Chern-Simons gauge theory can be thought of as living on the S 3 at infinity. [24,25,50] This is reminiscent of AdS/CFT.…”

“…It was suggested in [19] and later shown by [24,25,50] that the duality between the worldsheet and the Wilson loops continues to hold in topological string theory under geometric transitions. The similarity and differences of the dualities in these two cases can be seen directly by comparing Fig.…”

“…There is an S 3 at asymptotic infinity that has been omitted from the picture. [25,50] map on the branes has a direct analogue in AdS/CFT, where the anyons are replaced by heavy quarks( see figure 20 ) .…”

“…Figure 21: Duality between Wilson loop and worldsheet in topological string.There is an S 3 at asymptotic infinity that has been omitted from the picture [25,50]. …”