Knot invariants and M-theory: Hitchin equations, Chern-Simons actions, and surface operators

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 $\mathcal{N}=4$ 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.

Figure 1

A loop $K$, denoted by $p_2$, in the $(x_1,x_2)$ plane can be lifted up to form a knot $\mathbf{K}$, denoted by $p_1$, once we go to the Euclidean space. A nontrivial Wilson loop can now be constructed by integrating the twisted gauge field ${\mathcal{A}}_d$ along the knot $p_1$.

Figure 2

A surface operator constructed out of a M2-brane intersects the three-dimensional Euclidean boundary $\mathbf{W}$ (or ${\mathbf{W}}_3$ in the language of [11]) along a knot $\mathbf{K}$ and is stretched along the remaining $\psi$ direction, which we denote here as ${\mathbf{R}}_{+}$. As such it is a codimension two singularity both on the three-dimensional boundary ${\mathbf{W}}_3$ as well as the four-dimensional space $\mathbf{V}\equiv {\mathbf{W}}_3\times {\mathbf{R}}_{+}$.

Figure 3

An unknot configuration drawn almost parallel to the $x_1$ axis to simplify the computation of the Wilson loop. Thus away from the regions denoted by $Q_i$, we can restrict the Wilson line integral to be only along $x_1$.

Figure 4

Two Wilson lines in the three-dimensional boundary denoted by (a) is arranged so that they are parallel to the $x_1$ axis. In (b) we split them via Heegaard splitting and they are rejoined in (c) via a braid group action. This procedure allows us to introduce nontrivial structures to the Wilson lines.

Figure 5

The action of the braid group on the Wilson lines. They are distinguished by their overcrossing and undercrossing pattern. The first one has a braid group action ${\sigma }_1^{-1}$, whereas the second one has a braid group action ${\sigma }_1$.

Figure 6

Four Wilson lines are joined pairwise by identifying the respective monodromies around them.

Figure 7

Once we identify monodromies of a pair of Wilson lines, the structure of the codimension two surface operator in four-dimensional space can be formed out of two-branes. Here two such configurations are shown on a Heegaard-split three-manifold base.

Figure 8

Construction of an unknot using all the ingredients that we developed earlier. Boxes $\mathbf{A}$ represent the Wilson lines parallel to $x_1$ axis, boxes $\mathbf{B}$ denote the curving of the Wilson lines by identifying pairwise monodromies, and finally box $\mathbf{C}$ denotes the braid group action. Together they form an unknot configuration. The points $a_i$ and $b_i$ are the points where the Wilson lines end on the Heegaard-split three manifolds.

Figure 9

Construction of a trefoil knot by joining boxes $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ appropriately. The braid group action now acts twice. The points $(a_i,b_i)$ still remain the points where the Wilson lines end on the Heegaard-split manifolds.

Figure 10

A specific construction of a $(2,n)$ torus knot by joining boxes $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ appropriately. The braid group action now acts $n$ times. The points $(a_i,b_i)$ still remain the points where the Wilson lines end on the Heegaard-split manifolds. Once we extend the figure along the ${\mathbf{R}}_{+}$ (or $\psi$) direction, we will get the configuration of the surface operator.

Figure 11

The construction of a figure-8 knot using $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ boxes in a slightly different way than discussed earlier. The braid group action is now ${\sigma }_1^{-1}·{\sigma }_2·{\sigma }_1^{-1}·{\sigma }_2$ acting on the Wilson lines as shown.

Figure 12

The construction of $5_2$ knot using the $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ boxes. The braid group action is now ${\sigma }_1^3·{\sigma }_2·{\sigma }_1^{-1}·{\sigma }_2$ acting on the Wilson lines as shown.

Figure 13

The web of dualities that connect various configurations in type IIB and type IIA theories. Here we will concentrate mostly on the lower left-hand box that captures the physics of D5-branes wrapped on the two-cycle of a non-Kähler resolved conifold.

Figure 14

Knot $\mathbf{K}$ on D6-branes wrapped on ${\mathbf{S}}_{(2)}^3$ of a deformed conifold is represented by a D2-brane (or D4-brane) surface operator that intersects the D6-branes on $\mathbf{K}$. This picture is before geometric transition. After geometric transition, the D6-branes disappear and are replaced by fluxes on a non-Kähler resolved conifold, but the D2-brane (or D4-brane) state survives on the dual side retaining all information of the knot $\mathbf{K}$.

Figure 15

An example of a trefoil knot computation in the Ooguri-Vafa model. The knot invariant is now proportional to $⟨{\mathrm{\Psi }}_0|\mathrm{\Psi }⟩$, which is somewhat similar in spirit with the knot invariants computed earlier. The details however differ.