## June 15, 2006

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

#### Posted by Urs Schreiber

Today A. Kapustin gave the first of two ESI lecture talks on the super Yang-Mills aspect of his work with Witten on the physical realization of geometric Langlands duality ($\to$, $\to$, $\to$), following the paper

A. Kapustin & E. Witten
Electric-Magnetic Duality And the Geometric Langlands Program
hep-th/0604151.

Here is a transcript of my notes (though there is nothing here which cannot also be found in this paper).

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

(1)$S_\mathrm{YM}(A) = \int_X \left( -\frac{1}{2e^2} \;F \wedge * F + \frac{i \theta}{8\pi^2}\; F \wedge F \right) \,,$

with the combined coupling constant

(2)$\tau := \frac{\theta}{2\pi} + i \frac{4\pi}{e^2}$

living in the upper half plane.

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

(3)$Z\left( \tau,X,G \right) = \int DA \mathrm{exp} \left( S_\mathrm{YM}(A) \right)$

and the correlators are given by

(4)$\left\langle O_1,\cdots,O_n \right\rangle = \int DA \mathrm{exp} \left( S_\mathrm{YM}(A) \right) O_1,\cdots,O_n$

as usual.

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

(5)$\theta \mapsto \theta + 2\pi$

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

(6)$\tau \mapsto \tau + 1 \,.$

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

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

(7)$\tau \mapsto - \frac{1}{\tau}$

and

(8)$G \mapsto \multiscripts{^L}{G}{} \,,$

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

(9)$Z\left( \tau,X,G \right) = Z\left( -\frac{1}{\tau},X,G \right) \,.$

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

(10)$S_\mathrm{SYM} = S_\mathrm{YM} + \frac{1}{e^2} \int_X \mathrm{tr} \left( D_\mu \phi^i D^\mu \phi^i + \sum_{i \lt j} \left[ \phi^i,\phi^j \right] \right) + \text{fermionic terms} \,.$

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:

(11)$Z_\mathrm{SYM}\left( \tau,G,X \right) = Z_\mathrm{SYM}\left( -\frac{1}{q\tau},\multiscripts{^L}{G}{},X \right) \,,$

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

(12)$\left\langle O_1, \cdots, O_n \right\rangle_{\tau,G} = \left\langle \multiscripts{^L}{O}{_1}, \cdots, \multiscripts{^L}{O}{_n} \right\rangle_{-\frac{1}{q\tau},\multiscripts{^L}{G}{}} \,,$

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.

(13)$Q^2 = 0 \,.$

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

(14)$\left\lbrace Q_l,Q_r \right\rbrace = 0$

the BRST charge can be any combination of the form

(15)$Q_\mathrm{BRST} = u Q_l + v Q_r \,.$

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

(16)$t = \frac{v}{u} \,.$

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

(17)$S_\mathrm{twisted} = \frac{i\psi}{4\pi} \int_X \mathrm{tr} F \wedge F + \left\lbrace Q, \cdots \right\rbrace \,,$

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

(18)$\psi := \frac{\theta}{2\pi} + i \frac{4\pi}{e^2}\frac{t^2-1}{T^2 + 1} \,.$

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

One expects

(19)$Z_\mathrm{twisted}(\psi,G) = Z_\mathrm{twisted}(-\frac{1}{\psi},\multiscripts{^L}{G}{}) \,,$

which requires that also $t$ has to be transformed

(20)\begin{aligned} S : & \tau \mapsto -\frac{1}{\tau} \\ & t \mapsto t e^{i\phi} \end{aligned} \,,

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

(21)\begin{aligned} T : & \tau \mapsto \frac{a\tau + b}{c\tau + d} \\ & t \mapsto t \frac{c\tau + d}{|c\tau + d|} \end{aligned}

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)

(22)$(F - \phi\wedge \phi + t D\phi)^+ = 0$

2)

(23)$(F - \phi\wedge \phi - t^{-1} D\phi)^- = 0$

2)

(24)$D_\mu \phi^\mu = 0 \,.$

Consider now two special values of the parameter $t$

i) $t = i$

In this case define a generalized connection

(25)$\mathbf{A} := A + i \phi$

and its curvature

(26)$\mathbf{F} := d\mathbf{A} + \mathbf{A}\wedge\mathbf{A} \,.$

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

(27)$F - \phi\wedge \phi + D\phi = 0$

and

(28)$D \star \phi = 0 \,.$

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

(29)\begin{aligned} S : & \tau \mapsto -\frac{1}{\tau} \\ & t \mapsto -it \end{aligned}

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

(30)$X = \Sigma \times C \,,$

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

(31)\begin{aligned} & F - \phi\wedge \phi = 0 \\ & D \star \phi = 0 \end{aligned} \,.

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 ($\to$).

Posted at June 15, 2006 7:23 PM UTC

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