The String Coffee Table

The goal is to understand how the S-duality conjecture ($\to$) in 4-dimensional gauge theory implies the geometric Langlands conjecture ($\to$).

So recall what Yang-Mills theory in 4-dimensions looks like.

We have some gauge group $G$ and consider functionals on $G$-bundles with connection $A$ over some spacetime $X$, which is equipped with a pseudo-Riemannian metric $g$ of signature $-1$.

The action we associate with such a configuration is given by

with the combined coupling constant

living in the upper half plane.

The partion function of the theory is given by the path integral over all bundles with connections

and the correlators are given by

as usual.

By the integrality of $F \wedge F$, the shift

does not change the partition function. So the theory is invariant under

Around 1977, several people (Olive, Montonen and others) stated the

S-duality conjecture: Yang-Mills theory is in addition invariant under the combined change

and

where $\multiscripts{^L}{G}{}$ is the Langlands dual group of $G$ ($\to$).

(Of course at that time people did not identify it as the Langlands dual group.)

So the claim would be that

But this does not really make sense, since the whole thing is only defined in the context of renormalization theory and the coupling $e^2$ is not a constant, but runs.

It was realized by Witten and Olive in 1978 that the conjecture works a little better for $N=2$ super Yang Mills. In 1997 Osborn noticed that it works really good only for $N=4$ SYM.

So we turn to that.

The field content is now

$\bullet$ the connection 1-form $A_\mu$

$\bullet$ six scalars $\phi^i$ with values in the adjoint rep of $G$

$\bullet$ fermions $\psi^p_\alpha$, $\psi_{p \dot \alpha}$.

There is an $SU(4) \simeq \mathrm{Spin}(6)$ R-symmetry acting on the fermions and scalars.

The action now looks like

It’s invariant under the supersymmetry generated by the supercharges $Q^p_\alpha$ and $\bar Q_{\dot \alpha q}$ which satisfy the super-Poincaré algebra.

In this refined context we now have

S-duality Conjecture for $N=4$ SYM:

where $q=1$ for simply laced groups, $q=2$ for doubly laced ones and $q=3$ for $G_2$. Similarly, the correlators are conjectured to satisfy

with $\multiscripts{^L}{O}{_1}$ being some dual observable.

The problem is that such pairs of dual observables are hard to come by. One of the only known examples are Wilson loop observables. These are dual to $'t Hooft$ operators.

In order to make progress, Witten suggested to twist the gauge theory in order to turn it into a purely topological theory.

This amounts to changing the spin of various fields such that one of the supercharges becomes a scalar, which may the be declared to be a BRST operator.

In order for this to work one needs to embed the holonomy group of the base space $X$ into the R-symmetry group. There are three distinct topological twists of $N=4$ SYM.

One of them, apparently called the GL twist, applies when the holonomy group is $\mathrm{Spin}(4)$. Identifying a $\mathrm{Spin}(4)$ subgroup of $\mathrm{Spin}(6)_\mathbb{R}$ sends

$\bullet$ the connection $A_\mu$ to itself,

$\bullet$ the six scalars $\phi^i$ to four 1-forms $\phi_\mu$ and two 0-forms $\phi_5$, $\phi_6$,

$\bullet$ the fermions $\psi_\alpha, \psi_{\dot\alpha}$ to a bunch of fermions denoted $\psi_\mu, \tilde \psi_\mu, \eta, \tilde \eta, \chi_{\mu\nu}$.

Doing all this we obtain two scalar supercharges called $Q_l$ and $Q_r$. Since they anticommute

the BRST charge can be any combination of the form

Hence there is now a new parameter in the game, namely the ration

The resulting theory is topological in that it does not depend on the metric anymore (though it still depends on the smooth structure).

The action for the twisted theory looks like

where a new collective parameter $\psi$ has been introduced, which now also incorporates the ratio $t$

What does S-duality mean now, for this twisted setup?

One expects

which requires that also $t$ has to be transformed

where $e^{i\phi} = -\frac{\tau}{|\tau|}$. Together with the operation

this generates a subgroup of $\mathrm{SL}(2,\mathbb{R})$.

So the twisted theory is not S-duality invariant. Rather, there is a $t$-parameterized family of theories on which $S$ duality acts. The only fixed point of this action would correspond to $t = \infty$.

Solving the equations of motion, one now finds the following. Setting the result of $Q$ applied to any of $\psi_\mu, \tilde \psi_\mu, \eta, \tilde \eta, \chi_{\mu\nu}$ to zero produces the BPS equations. These are

1)

2)

2)

Consider now two special values of the parameter $t$

i) $t = i$

In this case define a generalized connection

and its curvature

Equations 1) and 2) then imply that $\mathbf{F}= 0$, which means that in this case we are dealing with a theory of complex flat connections.

(We can drop the third equation if we complexify the gauge group.)

ii) $t=1$

For this case we get

and

Now suppose that $\theta = 0$. In this special case the S-duality operation acts as

and hence the first case $t=i$ is sent to the second case, $t=1$.

Note that in this case $\psi$ vanishes, which we can interpret as saying that perturbation theory about this case makes good sense.

In order to make further progress we factor our spacetime as

with both factors 2-dimensional. Assume there was originally a product metric on this, such that the volume of $C$ is arbitrarily small. Then the gauge theory reduces effectively to one on $\Sigma$, i.e. it becomes 2-dimensional.

As Vafa and collaboratos have figured out in 1994, the 2D field thoery here is a $\sigma$-model whose target space is the Hitchin moduli space $M_\mathrm{Hit}(G,C)$ of stable Higgs bundles, defined by

More precisely, this is true for $t=1$. It turns out that this 2D TFT is an A-model.

On the other hand, if instead we choose the dual gauge group $\multiscripts{^L}{G}{}$ and the BRST ratio parameter $t=i$, then the gauge theory on $\Sigma$ turns out to be a $\sigma$-model with target $M_\mathrm{Hit}(\multiscripts{^L}{G}{},C)$, which happens to be a B-model 2D TFT.

S-duality now amounts to the statement that these two theories are equivalent.

But, since it exchanges an A-model with a B-model, this means that S-duality now really acts like mirror symmetry ($\to$).