The String Coffee Table

Recall that we were studying twisted super Yang-Mills theory with gauge group $G$ on a four-dimensional manifold which we take to be of product form

(Apparently we could more generally asume a fibration.)

In the limit where the volume of $C$ goes to 0 this is equivalent to a topological $\sigma$-model on $\Sigma$ with target the Hitchin moduli space $\mathbf{M}_\mathrm{Hit}(G,C)$ of dimension

where $g$ is the genus of $C$.

So the path integral is of the form

where we integrate over maps

For the sort of action $S$ involved here, we need a metric on target space $\mathbf{M}_\mathrm{Hit}(G,C)$. This is proportional to the canonical hyper-Kähler metric

Similarly, the $B$ field on target space is proportional to the canonical 2-form

where $I$ is the complex structure.

We had seen that a parameter $\mathbb{C}P^1 \ni t = \frac{v}{u}$ defines which topological $\sigma$ model we are dealing with, by specifyin the BRST charge to be

Now consider how the target space looks like in more detail.

We can think of it as the moduli space of pairs

Here $F = \nabla^2$ is the curvature of a connection on a $G$-bundle $E$ over $C$, and $\phi \in \Omega^{1,0}(\mathrm{ad}E)$ is called the Higgs field.

$\mathbf{M}_\mathrm{Hit}(G,C)$ is hyper Kähler (it is a hyper Kähler reduction of $\infty$-dimensional affine space of pairs $(A,\phi)$).

The hyper Kähler metric reads

Due to the target being hyper Kähler, the corresponding $\sigma$-model has $(4,4)$ supersymmetry.

In fact, $\mathbf{M}_\mathrm{Hit}$ has an entire sphere of complex structures

where $I,J,K$ with

are the basic complex structures that we shall study in the following.

More explicitly, one finds that

and

are holomorphic 1-forms on target space, for $w \in \mathbb{C} \cup \infty$.

The three basic complex structures $I$, $J$ and $K$ are obtained as the following three interesting special cases of this.

1) Let $w = \infty$. Then $\delta A_z$ and $\delta \phi_z$ are holomorphic differentials. Theis defines the complex structure $I$.

Equipped with this complex structure, the Hitchin moduli space can be thought of as the space of stable Higgs $G$-bundles

2) For $w = -i$ the holomorphic differentials are

This defines the complex structure called $J$.

In this case we can think of the Hitchin moduli space as that of stable flat Higgs bundles for the complexified gauge group

3) Finally, the third basic complex structure is the product of the first two

This corresponds to $w = -1$.

The Hitchin moduli space has an action of $S^1$ by isometries which leave $I$ invariant and rotate $J$ and $K$ via $\phi_z \mapsto e^{i\alpha} \phi_z$.

Given any complex structure $I_w$, we can define an $A$ model and a $B$ model. But actually, what we get here is not always just an $A$-model or just a $B$-model, but in general a mixture of them.

So recall how the twistig is accomplished.

We start with a 2D conformal theory with stress-energy tensor

and with R-currents $J(z)$ and $\bar J(\bar z)$.

Twisting is accomplished by performing the replacement

the point is that the precise nature of the R-current depends on the complex structure that we choose.

Hence, there is a whole sphere of R-currents. The most general twist possible is denoted $J_{w_+}$ and $\bar J_{w_-}$, depending on two parameters

In order to obtain the pure $A$-model we set

corresponding to $w_+ = - \frac{1}{\bar w_-}$.

The pure $B$-model is obtained for $w_+ = w_-$. In general, the twist yields neither of these.

We need to define the map between our twisting parameters $w_+$ and $w_-$ and the parameter $t$ from before. It turns out that the relation is

The $A$-model corresponds to $t \in \mathbb{R}$, the $B$-model to $t = \pm i$.

(Side remark: this is obtained by studying the adiabatic limit of the BPS equations.)

Next, Kapustin draw a couple of pictures depicting the sphere of complex structures, the equator of $A$-model twists and the $B$-model north and south poles. I won’t try to reproduce these here. See figure 1 on p. 21 of the Witten-Kapustin paper ($\to$).

The important point is, that, as we had seen in the previous talk, S-duality sends $t=i$ to $t=1$. This relates

The rest of the lecture was concerend with making contact to the Strominger-Yau-Zaslow picture of mirror symmetry ($\to$).

One expects, due to their work, that the target space on each side of the duality has a fibration by Lagrangian tori What is this fibration?

$\mathbf{M}_\mathrm{Hit}(G,C)$ fibers over an affine space of half the total dimension, with the generic fiber being a torus.

This fibration is complex with respect to $I$ and Lagrangian with respect to $J$ and $K$.

Let the gauge group be $G = \mathrm{GL}(n)$. Then we get

where the projection $p$ works like

So let’s run the SYZ argument. Consider a point

and regard this as a 0-brane for the $\sigma$-model,

This is a $B$-brane, since a point is a reasonable D-brane in any complex structure, hence it is in particular one with respect to $J$.

The S-dual of this $B$-brane is an $A$-brane on the mirror manifold $\mathbf{M}_\mathrm{Hit}(G,C)_K$. This is a Lagrangian submanifold, actually a fiber of the Hitchin fibration.

From the gauge theory it follows that the $A$-brane must sit over the fiber of the Hitchin fibration, so it follows that the $A$-brane must equal that fiber.

Hence fibers $p^{-1}(p(q))$ must parameterize flat connections on the fiber of the dual $\multiscripts{^L}{p}{^{-1}}(p(q))$.

So to any fat $\multiscripts{^L}{G}{_\mathbb{C}}$-connection on $C$, S-duality associates an $A$-brane on $\mathbf{M}_\mathrm{Hit}(G,C)$.

Compare this to geometric Langlands ($\to$), where one associates a $D$-module on the moduli stack of $G$-bundles $\mathrm{Bun}(G,C)$.

In fact, to any $A$-brane we can associate a $D$-module. $A$-branes are eigen-objects ($\to$) of the symplectic Hecke operator.