We propose that Argyres-Douglas theories of type $\textit{D}$ (SU($\textit{N}$)) and ($\textit{A}$ $\textit{, A}$ ) — both realizable as Type A class $$\mathcal{S}$$ theories with irregular punctures — can be obtained via a sequence of mass deformations from a common ancestor: a class $$\mathcal{S}$$ theory with only regular punctures. Building on our previous work, this result establishes that these theories ultimately originate from 6d $$\mathcal{N}=(1, 0)$$ orbi-instanton theories compactified on a torus. The requisite 4d mass deformations are realized as tractable Fayet-Iliopoulos deformations on the 3d mirror quiver. The core of our method is a constructive procedure that utilizes the Euclidean algorithm to define a chain of deformations connecting different $\textit{D}$ (SU($\textit{N}$)) theories. By reversing this chain, we recursively build a “parent” star-shaped quiver for any given ($\textit{N, p}$). This quiver is the 3d mirror theory of the required class $$\mathcal{S}$$ ancestor. We substantiate our general claims with several detailed examples that explicitly illustrate the deformation procedure.

Italy

Simone Giacomelli

William Harding

Thailand

Noppadol Mekareeya

United States

Alessandro Mininno

From regular to irregular: a unified origin for Argyres-Douglas theories

Vertex operator algebras (VOAs) arise in protected subsectors of supersymmetric quantum field theories, notably in 4d $$ \mathcal{N} $$ = 2 superconformal field theories (SCFT) via the Schur sector and in twisted 3d $$ \mathcal{N} $$ = 4 theories via boundary algebras. These constructions are connected through twisted circle compactifications, which can be best understood from the dynamics of BPS particles in the Coulomb branch of the 4d SCFT. This data is encoded in an operator $$ \hat{\Phi} $$ acting on the Hilbert space of an auxiliary quantum mechanics of BPS particles, whose trace yields the partition functions of a 3d topological field theory (TFT) bounding the VOA. We generalize this trace formula by considering higher powers of $$ \hat{\Phi} $$ , leading to a finite family of VOAs associated with a given 4d SCFT. Applying this framework to Argyres-Douglas theories labeled by ($\textit{A}$ , $\textit{G}$), where $\textit{G}$ is an ADE-type group of rank up to 8, we extract the modular data of the family of boundary VOAs via TFT partition function calculations on Seifert manifolds. Our results suggest that the modular data obtained from different powers of $$ \hat{\Phi} $$ are related by Galois transformations.

South Korea

Byeonggi Go

Qiang Jia

Heeyeon Kim

Sungjoon Kim

From BPS spectra of Argyres-Douglas theories to families of 3d TFTs

A correction to this paper has been published: https://doi.org/10.1007/JHEP11(2023)198

Armenia

Hasmik Poghosyan

Rubik Poghossian

Erratum to: A note on rank 5/2 Liouville irregular block, Painlevé I and the $$ {\mathcal{H}}_0 $$ Argyres-Douglas theory

We construct TBA equations for D-type Argyres-Douglas theories with an SU(2) flavor symmetry based on their spectral networks. We show that the solutions of these TBA equations agree with the quantum periods of the corresponding quantum Seiberg-Witten curves defined in the Nekrasov-Shatashvili limit of the Omega background, including a centrifugal correction. We study the variety of TBA systems across the Coulomb branch moduli space and find that they correspond to the Dynkin diagrams of Lie algebras in the minimal chamber, and reproduce the TBA equations for reflectionless D scattering theories at the maximally symmetric point. Numerical computations demonstrate that the quantum periods obtained from the Borel-Padé resummation and their WKB expansions are in agreement with the solutions of the TBA equations.

Japan

Katsushi Ito

Jingjing Yang

TBA equations and quantum periods for D-type Argyres-Douglas theories

We consider a family of Argyres-Douglas theories, which are 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 strongly coupled superconformal field theories (SCFTs) but share many features with 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills theories. In particular, the two central charges of these theories are the same, namely a = c. We derive a simple and illuminating formula for the Schur index of these theories, which factorizes into the product of a Casimir term and a term referred to as the Schur partition function. While the former is controlled by the anomaly, the latter is identified with the vacuum character of the corresponding chiral algebra and is expected to satisfy the modular linear differential equation. Our simple expression for the Schur partition function, which can be regarded as the generalization of MacMahon’s generalized sum-of-divisor function, allows one to numerically compute the series expansions efficiently, and furthermore find the corresponding modular linear differential equation. In a special case where the chiral algebra is known, we are able to derive the corresponding modular linear differential equation using Zhu’s recursion relation. We further study the solutions to the modular linear differential equations and discuss their modular transformations. As an application, we study the high temperature limit or the Cardy-like limit of the Schur index using its simple expression and modular properties, thus shedding light on the 1/4-BPS microstates of genuine   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs with a = c and their dual quantum gravity via the AdS/CFT correspondence.

United Kingdom

Hongliang Jiang

Modularity in Argyres-Douglas theories with a = c

The Seiberg-Witten solution to four-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 super-Yang-Mills theory with gauge group SU(N) and without hypermultiplets is used to investigate the neighborhood of the maximal Argyres-Douglas points of type   <math> <mfenced> <msub> <mi>a</mi> <mn>1</mn> </msub> <msub> <mi>a</mi> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfenced> </math>  $$ \left({\mathfrak{a}}_1,{\mathfrak{a}}_{N-1}\right) $$ . A convergent series expansion for the Seiberg-Witten periods near the Argyres-Douglas points is obtained by analytic continuation of the series expansion around the ℤ 2N  symmetric point derived in arXiv:2208.11502. Along with direct integration of the Picard-Fuchs equations for the periods, the expansion is used to determine the location of the walls of marginal stability for SU(3). The intrinsic periods and Kähler potential of the   <math> <mfenced> <msub> <mi>a</mi> <mn>1</mn> </msub> <msub> <mi>a</mi> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfenced> </math>  $$ \left({\mathfrak{a}}_1,{\mathfrak{a}}_{N-1}\right) $$  superconformal fixed point are computed by letting the strong coupling scale tend to infinity. We conjecture that the resulting intrinsic Kähler potential is positive definite and convex, with a unique minimum at the Argyres-Douglas point, provided only intrinsic Coulomb branch operators with unitary scaling dimensions ∆ &gt; 1 acquire a vacuum expectation value, and provide both analytical and numerical evidence in support of this conjecture. In all the low rank examples considered here, it is found that turning on moduli dual to ∆ ≤ 1 operators spoils the positivity and convexity of the intrinsic Kähler potential.

Sriram Bharadwaj

Eric D’Hoker

Approaching Argyres-Douglas theories

Conformal theories of the Argyres-Douglas type are notoriously hard to study given that they are isolated and strongly coupled thus lacking a lagrangian description. In flat space, an exact description is provided by the Seiberg-Witten theory. Turning on a Ω-background makes the geometry “quantum” and tractable only in the weak curvature limit. In this paper we use the AGT correspondence to derive Ω-exact formulae for the partition function, in the nearby of monopole points where the dynamics is described by irregular conformal blocks of the CFT. The results are checked against those obtained by the recursion relations coming from a conformal anomaly in the region where the two approaches overlap. The Nekrasov-Shatashvili limit is also discussed. Finally, we comment on the existence of black holes in De Sitter space whose low energy dynamics is described by an Argyres-Douglas theory.

Francesco Fucito

Jose Morales

On irregular states and Argyres-Douglas theories

We demonstrate the presence of non-invertible symmetries in an infinite family of superconformal Argyres-Douglas theories. This class of theories arises from diagonal gauging of the flavor symmetry of a collection of multiple copies of D  p (SU(N)) theories. The same set of theories that we study can also be realized from 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (1, 0) compactification on a torus. The main example in this class is the (A 2, D 4) theory. We show in detail that this specific theory bears the same structures of non-invertible duality and triality defects as those of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills with gauge algebra 𝔰𝔲(2). We extend this result to infinitely many other Argyres-Douglas theories in the same family, including those with central charges a = c whose conformal manifold is one dimensional, and those with a ≠ c whose conformal manifold has dimension larger than one. Our result is supported by examining certain special cases that can be realized in terms of theories of class 𝒮.

Federico Carta

Germany

Comments on Non-invertible Symmetries in Argyres-Douglas Theories

We present a general prescription for constructing 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 Lagrangians for the IR SCFTs that arise from the circle reduction of a large class of Argyres-Douglas theories. The resultant Lagrangian gives a realization of the Higgs branch of the 4d SCFT as a quiver variety, up to a set of decoupled interacting SCFTs with empty Higgs branches. As representative examples, we focus on the families (A  p−N−1 , A  N−1) and D  p (SU(N)). The Lagrangian in question is generically a non-ADE-type quiver gauge theory involving only unitary gauge nodes with fundamental and bifundamental hypermultiplets, as well as hypermultiplets which are only charged under the U(1) subgroups of certain gauge nodes. Our starting point is the Lagrangian 3d mirror of the circle-reduced Argyres-Douglas theory, which can be read off from the class   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  construction. Using the toolkit of the S-type operations, developed in [1], we show that the mirror of the 3d mirror for any Argyres-Douglas theory in the aforementioned families is guaranteed to be a Lagrangian theory of the above type, up to some decoupled free sectors. We comment on the extension of this procedure to other families of Argyres-Douglas theories. In addition, for the case of D  p (SU(N)) theories, we compare these 3d Lagrangians to the ones found in [2] and propose that the two are related by an IR duality. We check the proposed IR duality at the level of the three-sphere partition function for specific examples. In contrast to the 3d Lagrangians in [2], which are linear chains involving unitary-special unitary nodes, we observe that the Coulomb branch global symmetries are manifest in the 3d Lagrangians that we find.

Anindya Dey

Higgs branches of Argyres-Douglas theories as quiver varieties

We study ultraviolet (UV) incarnations of deep infrared (IR) physics on the Coulomb branch of the simplest interacting 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theory: the minimal Argyres-Douglas (MAD) theory. One of the most basic properties of the Coulomb branch is an emergent infinite-dimensional higher-spin symmetry. While the MAD theory is interacting and therefore does not have such a symmetry, we find UV operators that encode the emergent complex higher-spin symmetry on the Coulomb branch. Moreover, we show that cousins of these UV operators give rise to cousins of the IR higher-spin multiplets. In terms of superconformal representation theory, we are led to a conjecture on the exact spectrum of   <math> <msub> <mover> <mi>C</mi> <mo>¯</mo> </mover> <mrow> <mi>R</mi> <mo>,</mo> <mi>r</mi> <mfenced> <mi>j</mi> <mover> <mi>j</mi> <mo>¯</mo> </mover> </mfenced> </mrow> </msub> </math>  $$ {\overline{\mathcal{C}}}_{R,r\left(j,\overline{j}\right)} $$  multiplets in the MAD theory for all R, r, j, and   <math> <mover> <mi>j</mi> <mo>¯</mo> </mover> </math>  $$ \overline{j} $$  satisfying   <math> <mi>R</mi> <mo>+</mo> <mover> <mi>j</mi> <mo>¯</mo> </mover> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </math>  $$ R+\overline{j}-j+1=0 $$ , thereby making progress towards a full characterization of the protected spectrum. Along the way, we give a geometrical interpretation of these operators and include them in an extension of the Coulomb branch /   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 chiral operator correspondence.