Musings

In addition to the central charge, $k_G$, of the current algebra

of generators of the global symmetry group of the SCFT, there are also the conformal anomaly coefficients,

where $$\begin{array}{rl}{(\text{Weyl})}^2& =R_{\mu \nu \lambda \rho }^2-2R_{\mu \nu }^2+\frac{1}{3}{R}^2\\ (\text{Euler})& =R_{\mu \nu \lambda \rho }^2-4R_{\mu \nu }^2+R^2\end{array}$$ In a supersymmetric gauge theory, the coefficients, $a$, $c$ are one-loop exact, being related by supersymmetric Ward identities to certain R-current anomalies.

For an $𝒩=2$ supersymmetric gauge theory, with gauge group, $𝒢$, and hypermultiplets in representation $R_{\text{hyper}}$,

For an $𝒩=1$ theory, with gauge group, $𝒢$, and chiral multiplets in representation $R_{\text{chiral}}$,

Applying these formulæ (and the vanishing of the $\beta$-function) to the two dual pairs, proposed by Argyres and Seiberg,

• $\mathrm{SU}(3)$ with 6 hypermultiplets in the $3$, dual to the $E_6$ theory coupled to $\mathrm{SU}(2)$ with two half-hypers in the $2$.
• $\mathrm{Sp}(2)$ with 12 half-hypers in the $4$, dual to the $E_7$ theory coupled to $\mathrm{SU}(2)$.

allow one to extract predictions for these parameters in the $E_6$ and $E_7$ SCFTs.

Predictions from S-duality

$G$	$k_G$	$a$	$c$
$E_6$	$6$	$\frac{41}{24}$	$\frac{13}{6}$
$E_7$	$8$	$\frac{59}{24}$	$\frac{19}{6}$

These isolated SCFTs can be realized in F-Theory, by placing a D3-brane at the location of an F-theory singularity (a collection of mutually-nonlocal 7-branes, sitting on top of one another). The possible singularities of elliptically-fibered surfaces were classified by Kodaira, in terms of the singularities of the Weierstrass model $$y^2=x^3+f(z)x+g(z)$$ The order of vanishing of the discriminant, $$\Delta (z)=4f(z{)}^3+27g(z{)}^2$$ is the number of 7-branes, $n_7$. The singularities of relevance to us are the ones for which the dilaton gradient vanishes, leading to a superconformal theory on the world-volume of the D3-brane.

Kodaira singularieties with vanishing dilaton gradient

Kodaira Classification	$\mathrm{II}$	$\mathrm{III}$	$\mathrm{IV}$	$I_0^{*}$	${\mathrm{IV}}^{*}$	${\mathrm{III}}^{*}$	${\mathrm{III}}^{*}$
$G$	none	$A_1$	$A_2$	$D_4$	$E_6$	$E_7$	$E_8$
$n_7$	$2$	$3$	$4$	$6$	$8$	$9$	$10$

$G$ is the gauge group on the 7-branes, which is a global symmetry group, from the point of view of the theory on the D3-brane. When the D3-brane sits on top of the singularity, the theory is superconformal. There is also a Coulomb branch, corresponding to moving the D3-brane off the singularity, and

is the conformal dimension of the operator parametrizing the Coulomb branch. Each 7-brane contributes a deficit angle of $\pi /6$, so, overall, in the plane transverse to the 7-branes, we have the identification, $\theta \sim \theta +2\pi /\Delta$.

Instead of one D3-brane, one can put $N$ D3-branes at the singularity, and still obtain a superconformal field theory. In the large-$N$ limit, one has an AdS dual description which Ofer and Yuji exploit.

For most of the above models, the field theory description is not very useful, as the dilaton is frozen at a strongly-coupled value. But, for the $D_4$ theory, it is adjustable. At weak coupling, the configuration of 7-branes is just an ${\mathrm{O}7}^{-}$ with 4 D7 branes sitting on it, cancelling the charge. (Away from weak coupling, the orientifold plane splits into a pair of mutually-nonlocal 7-branes.) The gauge theory on the D7-branes is $\mathrm{SO}(8)$. Putting $N$ D3-branes on top of this yields an $\mathrm{Sp}(N)$ gauge theory on the D3-brane world-volume, with a hypermultiplet in the $Rank-2 Antisymmetric Tensor Representation$ (from the 3-3 strings) and half-hypermultiplets (from the 3-7 strings) in the $( Fundamental Representation \mathrm{,8})=(2N\mathrm{,8})$ of $\mathrm{Sp}(N)\times \mathrm{SO}(8)$.

One readily computes

In the limit of large-N and strong ‘t Hooft coupling, the AdS description becomes effective, and Ofer and Yuji compute

where the $O(N)$ terms are contributions from Chern-Simons couplings and the $O(1)$ terms are 1-loop corrections in the supergravity. A-priori, you might wonder whether there shouldn’t be $O(1/N)$ corrections to these formulæ. But, remarkably, they agree with (6), for all $N$. And, extrapolating down to $N=1$, they agree with the predictions from Table 1 for the $E_6$ and $E_7$ theories.

So, it seems, we now have predictions for these quantities for the other superconformal field theories in Table 2.