### S-Duality for N=2

You’ve probably noticed a lot of recent activity centered around S-duality for N=4 supersymmetric gauge theories (hint: it goes by the name of “Geometric Langlands”). There are, similarly, finite N=2 supersymmetric gauge theories, for instance $SU(N_c)$ with $N_f=2N_c$ hypermultiplets in the fundamental. One might ask whether some notion of S-duality holds for them as well. For a few, there is such a notion, but, in most cases, the construction of a dual theory has been elusive, and the answer cannot be as simple as in the N=4 case.

For N=4, if you sit at a point on the Coulomb branch (where the gauge group, $G$, is broken to the Cartan torus) and go to weak coupling, $\tau= \tfrac{\theta}{2\pi}+ \tfrac{4\pi i}{g^2}\to i\infty$, a set of BPS states, the W-bosons of the broken $G$ symmetry become massless. Conversely, if you take $\tau\to 0$, a different set of BPS states become massless, the W-bosons of the Langlands-dual gauge group, $\tilde G$. And, in fact, these are the only type of degenerations that occur, at a generic point of the Coulomb branch, as you vary the complex gauge coupling.

Not so for the finite N=2 theories. E.g., for $SU(3)$, with $N_f=6$ fundamentals, the special geometry of the Coulomb branch is governed by a family of genus-2 curves. Fixing a point on the Coulomb branch, these are parametrized by $\tau\in \mathcal{F}$, where $\mathcal{F}$ is the fundamental domain for the congruence subgroup^{1}, $\Gamma_0(2)$.
There are two cusp points in the fundamental domain of $\Gamma_0(2)$, one at $\tau=i\infty$ and the other on the real axis. The former is a traditional weak coupling singularity, where the curve denerates to a pair of $\mathbb{P}^1$s, each with 3 marked points. But the singularity at $\tau=0$ is not of that form. Instead, the curve degenerates to a torus with two marked points. This clearly can’t be of the form of some weakly-coupled gauge theory coupled to some hypermultiplets.

Argyres and Seiberg propose a different kind of dual theory, a (weakly-coupled) N=2 gauge theory coupled to an N=2 SCFT. Specifically, in the case at hand, the SCFT (first found by Minahan and Nemeschansky) has an $E_6$ global (non-R) symmetry group. Argyres and Seiberg gauge an $SU(2)$ subgroup of $E_6$. In addition to the SCFT, the $SU(2)$ gauge theory is coupled to a pair of half-hypermultiplets in the fundamental representation.

The $SU(2)$ gauge coupling goes to zero at the cusp where the original $SU(3)$ gauge coupling goes to infinity. The commutant of $SU(2)$ in $E_6$ is $SU(6)$. Rotating the two half-hypermultiplets supplies an additional $U(1)$ symmetry. Thus, the global symmetry group, $SU(6)\times U(1)$ is what one expects for 6 quark flavours.

The SCFT contributes to the $SU(2)$ $\beta$-function. After all, the dual theory is supposed to be conformally-invariant.

The current algebra of an SCFT, with global symmetry group, $\mathcal{G}$, takes the form

where the $\tensor{f}{^a^b_c}$ are the structure constants in the convention where the roots (long roots, in the non-simply-laced case) of $\mathfrak{g}$ have $\text{length}^2=2$. If we gauge a subgroup $G\subset \mathcal{G}$, the contribution of the SCFT to the $\beta$-function of the $G$ gauge theory is determined integrating the two-point function, $\langle J_\mu^a(x)J_\nu^b(0)\rangle$. In the conventions of (1), the $\beta$-function coefficient is $-2 l(\text{adj}) + 2l(R_\text{hypers}) + I_{G\subset \mathcal{G}} k_{\mathcal{G}}$ where $I_{G\subset \mathcal{G}}$ is the index of the embedding of $G$ in $\mathcal{G}$ ($I=1$ in our case) and $l(R)$ is the index of representation $R$. For $SU(N)$, $l(\text{adj})=2N$, $l(N)=1$.

So, with two half-hypers in the fundamental, obtaining vanishing $SU(2)$ $\beta$-function requires $k_{E_6}=6$. That’s a *prediction* about some correlation functions in the $E_6$ SCFT.

In particular, it implies a prediction for the 2-point function of the $SU(6)$ flavour symmetry currents. They argue that, by superconformal invariance, this 2-point function is independent of the gauge coupling. Hence, it can be evaluated in the original $SU(3)$ gauge theory at *weak coupling* and, lo and behold, the result is in perfect agreement with the predicted value, $k_{E_6}=6$.

They do some more checks, involving examining the pattern of $SU(6)$-breaking in various massive deformations of the proposed duality. They also look at N=2 $Sp(2)= Spin(5)$ gauge theory, with 12 half-hypermultiplets in the $\mathbf{4}$. This, they propose is dual to $SU(2)$, coupled to the $E_7$ SCFT. Gauging the $SU(2)$ in the maximal embedding, $SU(2)\times SO(12)\subset E_7$ leads to the correct $SO(12)$ flavour symmetry group, with $k_{E_7}=8$.

More generally, I expect, one would like to study more general conformally invariant $N=2$ gauge theory, containing both matter hypermultiplets and isolated SCFT’s. Dual pairs of such theories should provide very interesting 4D TQFTs.

^{1} We take the slightly unconventional embedding $\Gamma_0(2)=\left\{\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in PSL(2,\mathbb{Z}) \Bigr\vert {b+c=0 \pmod{2}}\right\}$. This is an index-3 subgroup of $PSL(2,\mathbb{Z})$, which acts by fractional linear transformations on $\tau' =2 \tau$. The fundamental domain, $\mathcal{F}\subset \mathcal{H}$, for $PSL(2,\mathbb{Z})$ has one cusp point, an orbifold point of order 2, and an orbifold point of order 3. The fundamental domain of $\Gamma_0(2)$ is a 3-fold cover of $\mathcal{F}$, with two cusp points and an orbifold point of order 2. Though we don’t need it, $\Gamma(2)\subset \Gamma_0(2)$ is an index-2 subgroup (hence, index-6 in $PSL(2,\mathbb{Z})$), and *its* fundamental domain has three cusp points and no orbifold points.

^{2} In $d\gt2$ dimensions, the first term in (1) is
$\frac{k_{\mathcal{G}} \delta^{a b}}{(2\pi)^d} (g_{\mu\nu}\partial^2 -\partial_\mu\partial_\nu) \frac{1}{(x^2)^{d-2}}$
In two dimensions, the $1/(x^2)^{d-2}$ is replaced by a logarithm.

## Re: S-Duality for N=2

How illuminating would it be to study these theories and their S-duals as brane realizations? I imagine it would be helpful to have a geometric description in mind. What does the S-dual N=2 gauge theory coupled to an N=2 SCFT you described look like if you do this?