Knot Invariants and M-Theory I: Hitchin Equations, Chern-Simons Actions, and the Surface Operators

TL;DR

This paper explores advanced theoretical frameworks connecting knot invariants, M-theory, and surface operators, extending Witten's brane construction and relating it to Ooguri-Vafa models through dualities and seven-dimensional manifolds.

Contribution

It introduces a modified brane construction linking Witten's and Ooguri-Vafa's models via dualities, and constructs explicit seven-dimensional M-theory manifolds for topological theories.

Findings

Both models lead to different seven-dimensional M-theory manifolds.

Localization equations and surface operators naturally emerge from the Hamiltonian formalism.

Knot invariants are constructed using M2-brane states in these models.

Abstract

Recently Witten introduced a type IIB brane construction with certain boundary conditions to study knot invariants and Khovanov homology. The essential ingredients used in his work are the topologically twisted N = 4 Yang-Mills theory, localization equations and surface operators. In this paper we extend his construction in two possible ways. On one hand we show that a slight modification of Witten's brane construction could lead, using certain well defined duality transformations, to the model used by Ooguri-Vafa to study knot invariants using gravity duals. On the other hand, we argue that both these constructions, of Witten and of Ooguri-Vafa, lead to two different seven-dimensional manifolds in M-theory from where the topological theories may appear from certain twisting of the G-flux action. The non-abelian nature of the topological action may also be studied if we take the wrapped M2-brane states in the theory. We discuss explicit constructions of the seven-dimensional manifolds in M-theory, and show that both the localization equations and surface operators appear naturally from the Hamiltonian formalism of the theories. Knots and link invariants are then constructed using M2-brane states in both the models.