Chern-Simons theory and topological strings

A review of the relation between Chern-Simons gauge theory and topological string theory on noncompact Calabi-Yau spaces is given. This relation has made it possible to give an exact solution of topological string theory on these spaces to all orders in the string coupling constant. Here the focus is on the construction of this solution, which is encoded in the topological vertex, and the implications of the physics of string/gauge theory duality for knot theory and for the geometry of Calabi-Yau manifolds.

Figure 1

Some knots and links. In the notation $x_n^L,x$ indicates the number of crossings, $L$ the number of components (in case it is a link with $L>1$), and $n$ is a number used to enumerate knots and links in a given set characterized by $x$ and $L$. The knot $3_1$ is also known as the trefoil knot, while $4_1$ is known as the figure-eight knot. The link $2_1^2$ is called the Hopf link.

Figure 2

When computing the linking number of two knots, the crossings are assigned a sign $\pm 1$ as indicated.

Figure 3

Heegaard splitting of a three-manifold $M$ into two three-manifolds $M_1$ and $M_2$ with a common boundary $\Sigma$.

Figure 4

Performing the path integral on a solid torus with a Wilson line in representation $R$ gives the state $\mid R⟩$ in $\mathcal{H}\left({\mathbf{T}}^2\right)$.

Figure 5

A link with two disjoint unknots.

Figure 6

The index structure of the fields in the adjoint representation of $U\left(N\right)$ is represented by a double line.

Figure 7

The cubic vertex in the double-line notation.

Figure 8

A planar diagram with $h=3$.

Figure 9

A nonplanar diagram with $g=h=1$.

Figure 10

The degeneration locus graph of the ${\mathbf{T}}^2\times \mathbb{R}$ fibration of ${\mathbf{C}}^3$ in the base ${\mathbb{R}}^3$ parametrized by $(r_{\alpha },r_{\beta },r_{\gamma })$.

Figure 11

The resolved conifold $\mathcal{O}(-1)\oplus \mathcal{O}(-1)\to {\mathbb{P}}^1$ and associated graph. This manifold is made out of two ${\mathbf{C}}^3$ patches glued through a common edge.

Figure 12

The $\mathcal{O}(-3)\to {\mathbb{P}}^2$ graph. This manifold is built out of three ${\mathbf{C}}^3$ patches.

Figure 13

A partially degenerate worldsheet instanton of genus $g=0$ and with $h=3$ ending on an unknot. The instanton is made out of an honest holomorphic disk and the degenerate piece, which is a fat graph.

Figure 14

The deformed conifold $T^{*}{\mathbf{S}}^3$, regarded as a ${\mathbf{T}}^2\times \mathbb{R}$ fibration of ${\mathbb{R}}^3$. Two of the directions represent the axes of the two cylinders, and the third direction represents the real axis of the $z$ plane. The dashed line represents the ${\mathbf{S}}^3$ cycle.

Figure 15

On the left-hand side, we represent ${\mathbf{S}}^3$ as a ${\mathbf{T}}^2$ fibration over the interval. One of the circles of the torus degenerates over one end point, while the other circle degenerates over the opposite end point. Each of the degenerating circles fibers over half the interval to produce a disk $D$, and on the right-hand side the three-sphere is equivalently realized as two ${\mathbf{S}}^1\times D$ glued through an $S$ transformation.

Figure 16

A pictorial representation of the conifold transition. The deformed conifold (left) with an ${\mathbf{S}}^3$, the conifold singularity (center), and the resolved conifold (right).

Figure 17

A Calabi-Yau manifold which is a ${\mathbf{T}}^2\times \mathbb{R}$ fibration of ${\mathbb{R}}^3$. The dashed lines represent ${\mathbf{S}}^3$’s.

Figure 18

The geometric transition of the Calabi-Yau manifold depicted in Fig. 17. In the leftmost geometry there are two three-spheres, represented by dashed lines. The intermediate geometry is singular, and the right shows the planar graph associated to the smooth toric Calabi-Yau manifold after the transition. It contains three ${\mathbb{P}}^1$’s with Kähler parameters $t_1$, $t_2$, and $t$.

Figure 19

The only nondegenerate instantons contributing to the geometry of Fig. 17 come from an annulus stretching along the degeneracy locus.

Figure 20

The geometry of Fig. 17 can be cut into three pieces. The piece that contains the annulus gives by canonical quantization the state (219).

Figure 21

Two different ways of compactifying a Lagrangian submanifold located at the horizontal edge are shown. These are specified by additional lines (drawn as arrows) where the torus fibration degenerates. The dashed lines represent the compactified submanifolds, which now have the topology of ${\mathbf{S}}^3$.

Figure 22

The canonical choice of framing for the topological vertex.

Figure 23

The configuration used to derive the topological vertex amplitude. The “deformed” geometry shown on the left. The “resolved” geometry obtained by geometric transition shown on the right. It contains a resolved conifold and a ${\mathbb{P}}^1$ of size $t$. Also depicted on the left is the open strings stretched among the different branes which contribute to the amplitude.

Figure 24

Moving the Lagrangian submanifold with representation $Q_1$ to the outgoing edge.

Figure 25

The toric diagram of local ${\mathbb{P}}^1\times {\mathbb{P}}^1$.