Small instanton transitions for M5 fractions

TL;DR

This work uses anomaly matching to compute the Higgs branch dimension of 6d conformal matter theories arising from M5-branes on ADE singularities, revealing that M5 fractions can recombine into full M5s while leaving frozen remnants of the singularity. The authors develop a unified formula, $d_H^{\mathrm{CFT}} = n_H - n_V + 29\,n_{GS}$, with $n_{GS}$ encoded by a Green–Schwarz-type matrix, and apply it to both unfrozen and frozen chains, including T-brane decorations. They demonstrate broad consistency with torus ($T^2$) and three-torus ($T^3$) compactifications, extending class S constructions to twisted punctures and to non-simply-laced and fully frozen cases, and propose affine-Dynkin quiver realizations for completely frozen CM on $T^3$. The results connect 6d anomaly data to 4d class S and 3d mirror descriptions, offering new checks and generalizations of existing conjectures and highlighting how freezing and recombination shape the Higgs sector of the resulting theories. Overall, the paper provides a robust framework to relate M5-brane dynamics, anomaly constraints, and lower-dimensional compactifications across a rich landscape of (frozen) conformal matter theories.

Abstract

M5-branes on an ADE singularity are described by certain six-dimensional "conformal matter" superconformal field theories. Their Higgs moduli spaces contain information about various dynamical processes for the M5s; however, they are not directly accessible due to the lack of a Lagrangian formulation. Using anomaly matching, we compute their dimensions. The result implies that M5 fractions can recombine in several different ways, where the M5s are leaving behind frozen versions of the singularity. The anomaly polynomial gives hints about the nature of the freezing. We also check the Higgs dimension formula by comparing it with various existing conjectures for the CFTs one obtains by torus compactifications down to four and three dimensions. Aided by our results, we also extend those conjectures to compactifications of theories not previously considered. These involve class S theories with twisted punctures in four dimensions, and affine-Dynkin-shaped quivers in three dimensions.

Figures (1)

• Figure 1: The central part of the picture represents fractional 2 M5-branes (dots) on a $\mathbb{R}\times \mathbb{R}^4/\Gamma_{E_6}$ singularity (red line). In this case each of the individual fractions is $1/4$ an ordinary M5. (To be precise, the M5 charges of the fractions are not the same. The fraction between $E_6$ and $\varnothing$ has charge $1/3$, while the one between $\varnothing$ and $SU(3)$ has charge $1/6$; see (\ref{['eq:M5charge']}).) We also show the gauge groups (or lack thereof) on each segment between two fractional M5s. On the top part of the picture, we show a situation where the first four fractions have recombined into a full M5; the latter can now be pulled off the singularity. On the bottom part of the picture we see a different transition, where the fractions have come together in a different way.