Proving the 6d a-theorem with the double affine Grassmannian

TL;DR

The paper proves a six-dimensional $(1,0)$ supersymmetric $a$-theorem for an infinite class of theories known as A-type orbi-instantons by embedding their Higgs-branch RG flows into the Hasse diagram of strata in the double affine Grassmannian of $E_8$. The authors establish a natural partial order on dominant affine coweights that mirrors RG-flow direction and show that the $a$-anomaly decreases along flows by expressing $a$ as a function of stratum data and proving $\Delta a>0$ for all minimal degenerations, using both analytic arguments and computer verification (via GAP). The strategy connects 6d physics to rich geometric representation theory, leveraging Kac diagrams, affine coweights, and 3d magnetic quivers to relate Higgs branches to Coulomb branches, and showing that the hierarchy of flows corresponds to transverse slices between strata. Their results complete the list of known $c$-theorems in even dimensions within this supersymmetric context and offer a blueprint for extending the analysis to other ADE types and to the full Higgs branch structure, including $k$-changing transitions. The work highlights deep interactions between RG flow irreversibility, geometric stratifications, and the moduli spaces of instantons on orbifolds, with implications for Hořava–Witten M9-wall physics and the broader interface of QFT and geometric representation theory.

Abstract

This paper contains two results of independent interest, the first being more mathematical in nature whereas the second more physical. We first show that the hierarchy of Higgs branch RG flows between the 6d $(1,0)$ SCFTs known as A-type orbi-instantons is given by the Hasse diagram of certain strata and transverse slices in the double affine Grassmannian of $E_8$. Secondly, we leverage the partial order naturally defined on this Hasse diagram to prove the $a$-theorem for orbi-instanton Higgs branch RG flows, thereby exhausting the list of $c$-theorems in the even-dimensional (supersymmetric) setting.

Figures (12)

• Figure 1: Top left: the generic point on the tensor branch of the UV orbi-instanton. Top right: the origin of the tensor branch where the orbi-instanton SCFT defined by the triple $(N,k,\rho_\infty^\text{UV})$ lives (the generic $\rho_\infty^\text{UV}$ is defined in (\ref{['eq:rhopart']})); all $N$ M5's (the full instantons) are brought on top of each other and on top of the $\mathbb{C}^2/\mathbb{Z}_k$ orbifold point, where the M9 sits (along $x^6$), fractionating into $N_\rho$ pieces (themselves acting as fractional instantons), with $N_\rho$ defined in (\ref{['eq:Nmu']}). Bottom left: an IR orbi-instanton defined by a different Kac diagram $(N,k,\rho_\infty^\text{IR})$ connected via Higgs branch RG flow to the UV theory. (The coweight corresponding to $(N,k,\rho_\infty^\text{IR})$ lies above the coweight corresponding to $(N,k,\rho_\infty^\text{UV})$ in the partial order.) To reach it, we activate a VEV for matter operators (in flavor hypermultiplets), and generically break the left flavor symmetry factor of the UV theory (a maximal subalgebra of $E_8$, including the full $E_8$ by extension). Bottom right: a product of a "lower-rank" orbi-instanton $(N-M,k,\rho_\infty^\text{UV})$ times $m$ substacks of $m_i$ M5's each, defining the partition $[m_i]$ (i.e. $\sum_{i=1}^m m_i = M$).
• Figure 2: The hierarchy of Higgs branch RG flows for $k=4$ from Fazzi:2022hal (left) and Frey:2018vpw (right), which are also shown in figure \ref{['fig:frey-rudelius']} (on the left and right respectively), and with red and blue nodes respectively in the right panel of fig:k34notvec. (Notice that in the above diagrams we are disregarding branchings between flows.) In the first the value of $n$ increases (as $-n$ decreases), whereas it remains constant in the second. The $a$ conformal anomaly decreases from top to bottom along the yellow arrow.
• Figure 3: Hierarchies of left flavor Higgsings for $k=4$: on the left $P=N+N_\rho$ is held fixed (both $N_\rho$ and $n$ do not decrease), on the right only $N$ is fixed (both $N_\rho$ and $n$ do not increase).
• Figure 4: An M-theory realization of the degenerations \ref{['bullet:i325']}, \ref{['bullet:ii325']}, \ref{['bullet:iii325']}, and \ref{['bullet:iv325']} as brane moves.
• Figure 5: A cutout from the semi-infinite periodic Hasse diagram of dominant coweights at level $k=2$, that is the hierarchy of left flavor Higgsings between orbi-instantons for $k=2$ and non-increasing numbers of full ($N$) and fractional ($N_\rho$) instantons. The labels on the transitions are the minimal degeneration types from section \ref{['sec:grass']} (and would also correspond to the result of quiver subtraction between the 3d magnetic quivers associated with the UV and IR orbi-instantons). E.g. the $\mathfrak{a}_1=A_1=\mathbb{C}^2/\mathbb{Z}_2$ flavor Higgsing is triggered by a VEV for a hypermultiplet charged under the ${\mathfrak{su}({2})}$ subalgebra of $\mathfrak{f}=E_7\oplus {\mathfrak{su}({2})}$ associated with Kac diagram $[2]$. The $a$ conformal anomaly decreases along all allowed oriented paths (RG flows).

Theorems & Definitions (29)

• Theorem 3.1: Roy
• Lemma 3.2
• proof
• Example 3.3
• Lemma 5.1
• proof
• Corollary 5.2
• proof
• Lemma 5.3
• proof