Top Down Approach to 6D SCFTs

TL;DR

This paper surveys the landscape of six-dimensional superconformal field theories through a top-down lens, predominantly using F-theory to realize and classify these fixed points. It details how tensionless strings arise in decoupled setups, how anomaly polynomials are computed via Green-Schwarz mechanisms, and how RG flows organize the space of theories through tensor and Higgs branches. A central thread is the systematic classification of F-theory bases, fibers, and endpoint configurations, together with a deep link between 6D SCFTs and homomorphisms into E8, which provides consistency checks and a framework for flows. The work further discusses state counting via effective strings and indices, and concludes with broader implications for lower-dimensional compactifications and swampland considerations, highlighting both the progress and the open problems in this rapidly evolving area.

Abstract

Six-dimensional superconformal field theories (6D SCFTs) occupy a central place in the study of quantum field theories encountered in high energy theory. This article reviews the top down construction and study of this rich class of quantum field theories, in particular, how they are realized by suitable backgrounds in string / M- / F-theory. We review the recent F-theoretic classification of 6D SCFTs, explain how to calculate physical quantities of interest such as the anomaly polynomial of 6D SCFTs, and also explain recent progress in understanding renormalization group flows for deformations of such theories. Additional topics covered by this review include some discussion on the (weighted and signed) counting of states in these theories via superconformal indices. We also include several previously unpublished results as well as a new variant on the swampland conjecture for general quantum field theories decoupled from gravity. The aim of the article is to provide a point of entry into this growing literature rather than an exhaustive overview.

Figures (4)

• Figure 1: Depiction of the ADE Dynkin diagrams. Geometrically, each circle denotes a curve of self-intersection $-2$, and neighboring curves intersect once. For each graph, the subscript indicates the total number of independent curves appearing in the resolution of the corresponding ADE singularity.
• Figure 2: Depiction of the 6D SCFTs obtained from M5-branes probing an ADE singularity. On the partial tensor branch we keep the M5-branes at the orbifold singularity but separate them along a common real line. The conformal fixed point is reached by sending all the M5-branes to the same point on this interval. On the partial tensor branch, we have $\mathcal{N} = 1$ 7D super Yang-Mills theory with gauge group $G$ coupled to a collection of domain walls as given by M5-branes. The M5-branes chop up the transverse line into finite intervals, as well as two semi-infinite intervals. The left and right therefore produce flavor symmetries $G_L \simeq G$ and $G_R \simeq G$ on the left and right of the brane configuration.
• Figure 3: Depiction of the elliptic curve as a two-sheeted cover, as specified by the Weierstrass equation $y^2 = x^3 + fx + g$. Each sheet is spanned by the $x$-coordinate, which has four roots, three of which are "visible" in the cubic, with the fourth a point at infinity. Pairing these roots and joining the corresponding branch cuts, we realize a genus one curve, or $T^2$. The tube of this $T^2$ can be seen as the purple segment joining the two sheets.
• Figure 4: Geometry for a 6D F-theory vacuum. The torus fiber degenerates over a codimension one curve (red), producing vector multiplets in the 6D theory. Hypermultiplets are localized at the intersection point of two such curves (codimension two). Tensor multiplets come from dimensionally reducing the fields of type IIB string theory on these curves, and effective strings come from D3-branes wrapping the curves.