## June 17, 2006

### Kapustin on SYM, Mirror Symmetry and Langlands, II

#### Posted by Urs Schreiber

The second part of the lecture.

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

(1)$X=\Sigma ×C\phantom{\rule{thinmathspace}{0ex}}.$

(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 ${M}_{\mathrm{Hit}}\left(G,C\right)$ of dimension

(2)$\mathrm{dim}{M}_{\mathrm{Hit}}\left(G,C\right)=2\mathrm{dim}G\left(g-1\right)\phantom{\rule{thinmathspace}{0ex}},$

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

So the path integral is of the form

(3)$Z=\int D\varphi \mathrm{exp}\left(-S\left(\varphi \right)\right)\phantom{\rule{thinmathspace}{0ex}},$

where we integrate over maps

(4)$ph:\Sigma \to {M}_{\mathrm{Hit}}\left(G,C\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(5)$g=\frac{4\pi }{{e}^{2}}\phantom{\rule{thinmathspace}{0ex}}{g}_{\mathrm{can}}\phantom{\rule{thinmathspace}{0ex}}.$

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

(6)$B=-\frac{\theta }{2\pi }\phantom{\rule{thinmathspace}{0ex}}{\omega }_{I}\phantom{\rule{thinmathspace}{0ex}},$

where $I$ is the complex structure.

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

(7)${Q}_{\mathrm{BRST}}=u{Q}_{l}+v{Q}_{r}\phantom{\rule{thinmathspace}{0ex}}.$

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

We can think of it as the moduli space of pairs

(8)${M}_{\mathrm{Hit}}\left(G,C\right)=\left\{\left(A,\varphi \right)\mid {F}_{z\overline{z}}-\left[{\varphi }_{z},{\varphi }_{\overline{z}}\right]=0\phantom{\rule{thinmathspace}{0ex}},{D}_{\overline{z}}{\varphi }_{z}=0\right\}/\text{gauge transformations}\phantom{\rule{thinmathspace}{0ex}}.$

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

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

The hyper Kähler metric reads

(9)${\mathrm{ds}}^{2}={\int }_{C}\mathrm{tr}\left(\delta {A}_{z}{\delta }_{{A}_{\overline{z}}}+\delta {\varphi }_{z}\delta {\varphi }_{\overline{z}}\right){\mathrm{dz}}^{2}\phantom{\rule{thinmathspace}{0ex}}.$

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

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

(10)${I}_{\mathrm{general}}=aI+bJ+cK\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{a}^{2}+{b}^{2}+{c}^{2}=1\phantom{\rule{thinmathspace}{0ex}},$

where $I,J,K$ with

(11)$IJ=K$

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

More explicitly, one finds that

(12)$\delta {A}_{z}-w\delta {\varphi }_{z}$

and

(13)$\delta {A}_{\overline{z}}+{w}^{-1}\delta {\varphi }_{\overline{z}}$

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

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 $\delta {\varphi }_{z}$ 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

(14)${M}_{\mathrm{Hit}}\left(G,C{\right)}_{I}\simeq {M}_{\mathrm{stable}}\left(G,C\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(15)$\begin{array}{rl}& \delta {A}_{z}+i\delta {\varphi }_{z}\\ & \delta {A}_{\overline{z}}+i\delta {\varphi }_{\overline{z}}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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

(16)${M}_{\mathrm{Hit}}\left(G,C{\right)}_{J}\simeq {M}_{\text{stable, flat}}\left({G}_{ℂ},C\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(17)$K=IJ\phantom{\rule{thinmathspace}{0ex}}.$

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 ${\varphi }_{z}↦{e}^{i\alpha }{\varphi }_{z}$.

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

(18)$\begin{array}{rl}& T\left(z\right):={T}_{zz}\left(z\right)\\ & \overline{T}\left(\overline{z}\right):={\overline{T}}_{\overline{z}\overline{z}}\left(z\right)\end{array}$

and with R-currents $J\left(z\right)$ and $\overline{J}\left(\overline{z}\right)$.

Twisting is accomplished by performing the replacement

(19)$\begin{array}{rl}& T↦T+\frac{1}{2}{\partial }_{z}J\\ & \overline{T}↦\overline{T}+\frac{1}{2}{\overline{\partial }}_{\overline{z}}\overline{J}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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 ${\overline{J}}_{{w}_{-}}$, depending on two parameters

(20)$\left({w}_{+},{w}_{-}\right)\in ℂ{P}^{1}×ℂ{P}^{1}\phantom{\rule{thinmathspace}{0ex}}.$

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

(21)${I}_{{w}_{+}}=-{I}_{{w}_{-}}$

corresponding to ${w}_{+}=-\frac{1}{{\overline{w}}_{-}}$.

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

(22)$\begin{array}{rl}& {w}_{+}=-t\\ & {w}_{-}={t}^{-1}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

The $A$-model corresponds to $t\in ℝ$, the $B$-model to $t=±i$.

(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

(23)$A-\text{model of}{M}_{\mathrm{Hit}}\left(G,C{\right)}_{K}\phantom{\rule{thickmathspace}{0ex}}\simeq \phantom{\rule{thickmathspace}{0ex}}B-\text{model of}{M}_{\mathrm{Hit}}\left({}^{L}G,C{\right)}_{J}\phantom{\rule{thinmathspace}{0ex}}.$

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?

${M}_{\mathrm{Hit}}\left(G,C\right)$ 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 $G=\mathrm{GL}\left(n\right)$. Then we get

(24)$\begin{array}{c}{M}_{\mathrm{Hit}}\left(G,C\right)\\ p↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ V={\oplus }_{k=1}^{n}{H}^{0}\left(C,{K}_{C}^{k}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$

where the projection $p$ works like

(25)$p:\left(A,\varphi \right)↦\mathrm{tr}{\varphi }_{z}^{k}\phantom{\rule{thinmathspace}{0ex}}.$

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

(26)$q\in {M}_{\mathrm{Hit}}\left({}^{L}G,C\right)\simeq {M}_{\text{flat connections}}\left({}^{L}G_{ℂ},C\right)$

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

(27)$\varphi \left(\partial \Sigma \right)=q\phantom{\rule{thinmathspace}{0ex}}.$

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 ${M}_{\mathrm{Hit}}\left(G,C{\right)}_{K}$. 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 ${p}^{-1}\left(p\left(q\right)\right)$ must parameterize flat connections on the fiber of the dual ${}^{L}p^{-1}\left(p\left(q\right)\right)$.

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

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

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

Posted at June 17, 2006 9:30 AM UTC

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