This is based on the recent paper by Kim, Razamat, Vafa and Zafrir.

Supersymmetric QFTs in 4d often exhibit properties that are hard to explain from first principles, for instance symmetries at IR fixed points that are not manifest in the UV description. Is there a systematic way of understanding these symmetries? One fruitful approach is to see the 4d theory as the compactification of a 6d theory on a Riemann surface. Of course, there is much freedom in the choice of the 6d theory and the Riemann surface.

The KRVZ paper considers perhaps the most minimal 6d $\mathcal{N}=(1,0)$ theory, namely the E-string model. It can be viewed as

  • theory of a small $E_8$ instanton;
  • the theory on an M5 brane probing the Horava-Witten $E_8$-wall;
  • the theory obtained by blowing up a point in the $\mathbb{C}^2$ base of F-theory;
  • the theory on an M5 brane probing a $D_4$ singularity.
This theory is then put on a general Riemann surface (with fluxes and holonomies). For certain choices of compactification parameters, the resulting 4d theory should enjoy exceptional symmetry.

One particular case is the $E_7$ surprise theory. Let us describe briefly the history of this surprise, which appeared in a 2012 paper by Dimofte and Gaiotto. We denote by $\mathcal{T}$ the 4d $\mathcal{N}=1$ theory with $SU(2)$ gauge group and $N_f=4$ flavors. This theory has a manifest $SU(8)$ symmetry, but if we ignore some details, is seems that this symmetry is in fact embedded into the larger $E_7$. To get rid of these annoying details, two options are considered:

  • Couple together two copies of $\mathcal{T}$, to give $\mathcal{T}_{II}$.
  • Couple $\mathcal{T}$ to 28 free 5d hypers to define a superconformal boundary condition $\mathcal{B}_{\mathcal{T}}$.
The superconformal index for both theories then has exact $E_7$ symmetry. This does not mean that the theory itself has an $E_7$ flavor symmetry, but this symmetry emhancement occurs at a very special point in the moduli space, as shown by Dimofte and Gaiotto.

The KRVZ paper demystifies this surprise, by providing the map to 6d theories and Riemann surfaces. More than that, the understanding of the $E_7$ surprise allows them to chart the whole correspondence between 6d and 4d in this case, and even find models with the full $E_8$ as the symmetry group of the fixed point.