Class-S Theories of Type E₈

This website is a companion to our paper arXiv:1802.09626. The physics background, and many more details can be found there. If you find this site useful in your own work, please remember to cite that paper.

There are 70 nilpotent orbits in $\mathfrak{e}_8$. Excluding the regular orbit (which corresponds to the trivial defect), that gives 69 codimension-2 defect operators (“regular punctures”) in the $E_8$ (2,0) theory. We label them by their Bala-Carter labels, where $0$ denotes the full puncture and $E_8(a_1)$ (the subregular orbit) denotes the simple puncture. Their properties are catalogued in our paper, and a quick summary can be found here.

Given a triple of defects, we can compactify the (2,0) theory on a 3-punctured sphere (fixture). There are 57,155 such fixtures, of which 7,319 are “bad” (do not lead to well-defined 4D theories). You can explore the properties of the 4D $\mathcal{N}=2$ SCFTs which arise from the remaining 49,836 here.

A 4-punctured sphere gives rise to a 4D $\mathcal{N}=2$ SCFT, presented as a gauge theory with simple gauge group $G\subset E_8$, coupled to two of the aforementioned fixtures. Different pants decompositions yield different such presentations (“S-duality frames”). You can explore the 1,028,790 4-punctured spheres (of which 3,352 are bad) and the S-duality frames of the 1,025,438 good ones here.

When viewing an S-duality frame, clicking on a fixture brings up the details for that fixture. When viewing a fixture, clicking on a puncture brings up the details for that puncture.

Have fun exploring the E₈ Theory. Or hop on over and explore the E₇ Theory instead.