Chern-Simons Theory, Holography and Topological Strings

TL;DR

The paper investigates the deep connections between Chern-Simons theory, topological strings, and holography, showing how large-N dualities and geometric transitions map open string sectors with D-branes to closed string theories on dual geometries. It develops a holographic framework in which the Kahler form is dual to a 3-form flux sourced by Lagrangian branes, enabling explicit computations of all genus topological-string amplitudes for toric Calabi–Yau threefolds via the topological vertex. It further explains how A- and B-model decoupling leads to skein relations for knot invariants and demonstrates applications to gauge theory partition functions and BPS black-hole microstate counting, with the holographic dictionary grounded in string-field theory. Together, these results provide a unified lens on enumerative geometry, knot theory, and quantum gravity-inspired dualities in the topological string setting, with practical impact on computing amplitudes and invariants across noncompact Calabi–Yau geometries.

Abstract

In this note we present a brief overview of connections between Chern-Simons theory and topological strings. A prominent role in this link has been played by large N dualities and holography. We demystify this by explaining why the Kahler form should be viewed as dual to the field strength associated with a 3-form gauge potential, sourced by Lagrangian D-branes. We explain how this leads to the computation of topological string amplitudes in terms of topological vertex for toric Calabi-Yau threefolds. Furthermore, applications of topological strings to a conceptual derivation of Skein relations for link invariants as well as some of its physical applications to black hole physics are also reviewed.

Figures (4)

• Figure 1: A large number $N$ of Lagrangian D-branes wrapped around $S^3$ leads to a geometric transition where the branes disappear and are replaced by the flux on the $S^2$ which links the $S^3$. The Kahler class gets related to flux and leads to $N\lambda$ as the area of $S^2$, where $t=N\lambda$ is the string coupling.
• Figure 2: The topological Chern-Simons transitions, can be used to compute topological string amplitudes on ${\bf P}^2$ blown up at three points, which after three flops can be used to compute topological strings on ${\bf P}^2$ itself.
• Figure 3: Topolotical vertex can be defined by holographic transitions from Chern-Simons theory. It can be formulated in terms of three Lagrangian branes on ${\bf C}^3$ and the holomorphic curves ending on them can be labeled by Representations $R_i$ of $SU(N_i)$ for large $N_i$, leading to the vertex $C_{R_1,R_2,R_3}$.
• Figure 4: Worldsheet skein relations follow from recalling that changing complex structure should not change the amplitudes of the A-model. These figures represent change of complex structure from left to right. The lines in the first figure represent the end points of the worldsheet diagrams, and it shows that under that complex deformations new worldsheet maps emerge after passing through a critical point (the dashed green line) where the image of the boundary of the worldsheet intersects. The second figure shows what happens if a Lagrangian D-brane crosses a closed holomorphic curve $C^+$ as one changes the complex structure. Here it shows that new contributions emerge where the worldsheet can end on the Lagrangian brane as complex structure changes as well as a different Kahler class seen by the transformed holomorphic curve $C^-$ as its linking with $L$ has changed.