On a 5D UV completion of Argyres-Douglas theories

TL;DR

The paper presents a 5d UV completion framework for Argyres-Douglas theories by embedding their RG flows into 5d $ ext{N}=1$ SCFTs on $S^1$, analyzed through a $q$-Painlevé/gauge theory correspondence. It constructs 5d topological observables via a blowup formalism, derives NS-limit blowup factors that reproduce $q$-Painlevé $ au$-functions, and expresses these in terms of Wilson-loop data with integer $q$-polynomial coefficients. It then studies the SW limit, special symmetry points, and two four-dimensional reductions: a geometric engineering limit to pure 4d $ ext{SU}(2)$ SYM and a strongly coupled negative-coupling limit (for CS level $k=1$) yielding the AD theory $H_0$, including a detailed double-scaling construction that reproduces the PI-type equation governing the AD dynamics. The results illuminate a concrete UV completion path for AD theories and reveal a rich structure of Hurwitz-type expansions, modular properties, and phase diagrams connecting 5d, 4d, and AD regimes. Overall, the work provides a precise bridge between 5d SCFTs, Painlevé dynamics, and 4d AD points, with explicit computational tools via blowup equations and Wilson-loop expansions.

Abstract

We discuss a novel UV completion of a class of Argyres-Douglas (AD) theories in the $Ω$-background by its embedding into the renormalisation group flow from five dimensional $\mathcal{N}=1$ superconformal field theories (SCFT) on $S^1$. This is obtained via analysing these theories in the light of ($q$-)Painlevé/gauge theory correspondence, which allows to compute the five dimensional BPS partition functions as an expansion in the Wilson loop vev with integer $q$-polynomials coefficients. These are derived formulating the gauge theory on a blown-up geometry and using a five-dimensional lift of (topological) operator/state correspondence. We discuss in detail the phase diagram of the four dimensional limits, pinpointing the special AD loci. Explicit computations are reported for $\tilde E_1$ SCFT and its limit to H$_0=(A_1,A_2)$ AD theory.

Figures (5)

• Figure 1: Sakai's classification for Painlevé equations by surface type and corresponding susy theories, where we highlighted the ones we study. The red arrow corresponds to the flow from $\tilde{E}_1$ SCFT ($q$-PI equation) to 4d AD H$_0=(A_1,A_2)$ theory (PI equation). The blue arrows correspond to the 4d geometric engineering limits and the orange one to the flow to $E_0$ SCFT (local $\mathbb{P}^2$).
• Figure 2: Diagram for the blow up $X'=\hat{\mathbb{C}}^2\times S^1_\beta$. The red surface is the support $E\times S^1_\beta$ of the codimension 2 operator $I(E)^{d+1}$. The arrow illustrates the 5d topological operator/state correspondence (blow down).
• Figure 3: $(p,q)$-web for 5d SYM with $CS$ level $k=0$. For negative coupling the roles of instantons and $W$-bosons are exchanged.
• Figure 4: $(p,q)$-web for 5d SYM with $CS$ level $k=1$ for $M_{UV}>0$ (upper part) and $M_{UV}<0$ (lower part). In the limit $M_{UV}\to-\infty$ we obtain the $E_0$ theory (local $\mathbb{P}^2$ geometry).
• Figure 5: Phase diagram for the 4d scaling limits. The grey region corresponds to the trivial theory $\mathcal{T}=0$ and the dashed regions correspond to gapped theories ($\mathcal{T}\sim e^{f(s)}$) that we obtain only for generic values of $z_0$. The red point corresponds to the H$_0$ AD SCFT, the orange line to H$_0$ with no coupling deformation and the blue line and the light blue region are the SW limits of the previous two.