Argyres-Douglas Theories Research Articles

Braided Tensor Categories Related to $${\mathcal {B}}_{p}$$ Vertex Algebras

The $${\mathcal {B}}_{p}$$ -algebras are a family of vertex operator algebras parameterized by $$p\in {\mathbb {Z}}_{\ge 2}$$ . They are important examples of logarithmic CFTs and appear as chiral algebras of type $$(A_1, A_{2p-3})$$ Argyres–Douglas theories. The first member of this series, the $${\mathcal {B}}_2$$ -algebra, are the well-known symplectic bosons also often called the $$\beta \gamma $$ vertex operator algebra. We study categories related to the $${\mathcal {B}}_{p}$$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $${\mathfrak {sl}}_{2}$$ . These categories are braided, rigid and non semi-simple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

Testing Macdonald index as a refined character of chiral algebra

We test in (An−1, Am−1) Argyres-Douglas theories with gcd(n, m) = 1 the proposal of Song’s in [1] that the Macdonald index gives a refined character of the dual chiral algebra. In particular, we extend the analysis to higher rank theories and Macdonald indices with surface operator, via the TQFT picture and Gaiotto-Rastelli-Razamat’s Higgsing method. We establish the prescription for refined characters in higher rank minimal models from the dual (An−1, Am−1) theories in the large m limit, and then provide evidence for Song’s proposal to hold (at least) in some simple modules (including the vacuum module) at finite m. We also discuss some observed mismatch in our approach for surface operators with large vortex number.

The Higgs mechanism \u2014 Hasse diagrams for symplectic singularities

We explore the geometrical structure of Higgs branches of quantum field theories with 8 supercharges in 3, 4, 5 and 6 dimensions. They are symplectic singularities, and as such admit a decomposition (or foliation ) into so-called symplectic leaves, which are related to each other by transverse slices. We identify this foliation with the pattern of partial Higgs mechanism of the theory and, using brane systems and recently introduced notions of magnetic quivers and quiver subtraction, we formalise the rules to obtain the Hasse diagram which encodes the structure of the foliation. While the unbroken gauge symmetry and the number of flat directions are obtainable by classical field theory analysis for Lagrangian theories, our approach allows us to characterise the geometry of the Higgs branch by a Hasse diagram with symplectic leaves and transverse slices, thus refining the analysis and extending it to non-Lagrangian theories. Most of the Hasse diagrams we obtain extend beyond the cases of nilpotent orbit closures known in the mathematics literature. The geometric analysis developed in this paper is applied to Higgs branches of several Lagrangian gauge theories, Argyres-Douglas theories, five dimensional SQCD theories at the conformal fixed point, and six dimensional SCFTs.

Peculiar index relations, 2D TQFT, and universality of SUSY enhancement

We study certain exactly marginal gaugings involving arbitrary numbers of Argyres-Douglas (AD) theories and show that the resulting Schur indices are related to those of certain Lagrangian theories of class mathcal{S} via simple transformations. By writing these quantities in the language of 2D topological quantum field theory (TQFT), we easily read off the S-duality action on the flavor symmetries of the AD quivers and also find expressions for the Schur indices of various classes of exotic AD theories appearing in different decoupling limits. The TQFT expressions for these latter theories are related by simple transformations to the corresponding quantities for certain well-known isolated theories with regular punctures (e.g., the Minahan-Nemeschansky E6 theory and various generalizations). We then reinterpret the TQFT expressions for the indices of our AD theories in terms of the topology of the corresponding 3D mirror quivers, and we show that our isolated AD theories generically admit renormalization group (RG) flows to interacting superconformal field theories (SCFTs) with thirty-two (Poincaré plus special) supercharges. Motivated by these examples, we argue that, in a sense we make precise, the existence of RG flows to interacting SCFTs with thirty-two supercharges is generic in a far larger class of 4D mathcal{N} = 2 SCFTs arising from compactifications of the 6D (2, 0) theory on surfaces with irregular singularities.

Argyres-Douglas theories and Liouville irregular states

We study irregular states of rank-two and three in Liouville theory, based on an ansatz proposed by D. Gaiotto and J. Teschner. Using these irregular states, we evaluate asymptotic expansions of irregular conformal blocks corresponding to the partition functions of (A1, A3) and (A1, D4) Argyres-Douglas theories for general Ω-background parameters. In the limit of vanishing Liouville charge, our result reproduces strong coupling expansions of the partition functions recently obtained via the Painlevé/gauge correspondence. This suggests that the irregular conformal block for one irregular singularity of rank 3 on sphere is also related to Painlevé II. We also find that our partition functions are invariant under the action of the Weyl group of flavor symmetries once four and two-dimensional parameters are correctly identified. We finally propose a generalization of this parameter identification to general irregular states of integer rank.

Codimension-two defects and Argyres-Douglas theories from outer-automorphism twist in 6D (2,0) theories

The 6D (2,0) theory has codimension-one symmetry defects associated to the outer-automorphism group of the underlying the simply-laced Lie algebra of ADE type. These symmetry defects give rise to twisted sectors of codimension-two defects that are either regular or irregular corresponding to simple or higher order poles of the Higgs field. In this paper, we perform a systematic study of twisted irregular codimension-two defects generalizing our earlier work in the untwisted case. In a class S setup, such twisted defects engineer 4D $\mathcal{N}=2$ superconformal field theories of the Argyres-Douglas type whose flavor symmetries are (subgroups of) nonsimply laced Lie groups. We propose formulas for the conformal and flavor central charges of these twisted theories, accompanied by nontrivial consistency checks. We also identify the 2D chiral algebra (vertex operator algebra) of a subclass of these theories and determine their Higgs branch moduli space from the associated variety of the chiral algebra.

Vanishing OPE coefficients in 4d mathcal{N}=2 SCFTs

We compute the superconformal characters of various short multiplets in 4d mathcal{N}=2 superconformal algebra, from which selection rules for operator products are obtained. Combining with the superconformal index, we show that a particular short multiplet appearing in the n-fold product of stress-tensor multiplet is absent in the (A1, A2n) Argyres-Douglas (AD) theory. This implies that certain operator product expansion (OPE) coefficients involving this multiplet vanish whenever the central charge c is identical to that of the AD theory. Similarly, by considering the n-th power of the current multiplet, we show that a particular short multiplet and OPE coefficients vanish for a class of AD theories with ADE flavor symmetry. We also consider the generalized AD theory of type (Ak−1, An−1) for coprime k, n and compute its Macdonald index using the associated W -algebra under a mild assumption. This allows us to show that a number of short multiplets and OPE coefficients vanish in this theory. We also provide a Mathematica file along with this paper, where we implement the algorithm by Cordova-Dumitrescu-Intriligator to compute the spectrum of 4d mathcal{N}=2 superconformal multiplets as well as their superconformal character.

Landscape of Simple Superconformal Field Theories in 4D.

We explore the space of renormalization group flows that originate from supersymmetric N=1 SU(2) gauge theory with one adjoint and a pair of fundamental chiral multiplets. By considering all possible relevant deformations-including coupling to gauge-singlet chiral multiplets-we find 34 fixed points in this simple setup. We observe that theories in this class exhibit many novel phenomena: emergent symmetry, decoupling of operators, and narrow distribution of central charges a/c. This set of theories includes two of the N=2 minimal Argyres-Douglas theories and their mass deformed versions. In addition, we find 36 candidate fixed point theories possessing unphysical fermionic operators-including one with central charges (a,c)≃(0.20,0.22) that are smaller than any known superconformal theory-that need further investigation.

Quantum Seiberg-Witten curve and universality in Argyres-Douglas theories

We study the quantum Seiberg-Witten (SW) curves for (A1,G)-type Argyres-Douglas (AD) theory by taking the scaling limit of the quantum SW curve of N=2 gauge theory with gauge group G. For G=Ar, the quantum SW curve of the AD theory is consistent with the scaling limit of the curve of the gauge theory. For G=Dr, we need the quantum correction to the SW curve of the AD theory, which depends on the quantization condition of the original SW curve. We also study the universality of the quantum SW curves for (A1,A3) and (A1,D4)-type AD theories.

On the correspondence between surface operators in Argyres-Douglas theories and modules of chiral algebra

We compute the Schur index of Argyres-Douglas theories of type (AN −1,AM −1) with surface operators inserted, via the Higgsing prescription proposed by D. Gaiotto, L. Rastelli and S.S. Razamat. These surface operators are obtained by turning on position-dependent vacuum expectation values of operators in a UV theory which can flow to the Argyres-Douglas theories. We focus on two series of (AN −1, AM −1) theories; one with gcd(N, M) = 1 and the other with M = N (k − 1) for an integer k ≥ 2. Our results are identified with the characters of non-vacuum modules of the associated 2d chiral algebras, which explicitly confirms a remarkable correspondence recently discovered by C. Cordova, D. Gaiotto and S.-H. Shao.

On dimensional reduction of 4d N=1 Lagrangians for Argyres-Douglas theories

Recently, it was found that certain 4d mathcal{N}=1 Lagrangians experience supersymmetry enhancement at their IR fixed point, thereby giving a Lagrangian description for a plethora of Argyres-Douglas theories. A generic feature of these Lagrangians is that a number of gauge invariant operators decouple (as free fields) along the RG-flow. These decoupled operators can be naturally taken into account from the beginning itself by introducing additional gauge singlets (sometimes called “flipping fields”) that couple to the decoupled operators via appropriate superpotential terms. It has also been checked that upon dimensionally reducing to 3d, the (A1, A2n−1) type Lagrangians only produce the expected behavior when flipping fields are included in the Lagrangian. In this paper we further investigate the role of flipping fields and find an example where the expected necessity of including the flipping fields in the dimensionally reduced Lagrangians seems to get violated. In the process we find two new dual Lagrangians for the so called 3d T [SU(2)] theory.

Argyres-Douglas theories, Painlev\xe9 II and quantum mechanics

We show in details that the all order genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the H1 Argyres-Douglas theory coupled to the Ω background. In the self-dual limit we use the Painlevé/gauge correspondence and we show that, after summing over all instanton sectors, the two-cut cubic matrix model computes the tau function of Painlevé II without taking any double scaling limit or adding any external fields. We decode such solution within the context of transseries. Finally in the Nekrasov-Shatashvili limit we connect the H1 and the H0 Argyres-Douglas theories to the quantum mechanical models with cubic and double well potentials.

Logarithmic W-algebras and Argyres-Douglas theories at higher rank

Families of vertex algebras associated to nilpotent elements of simply-laced Lie algebras are constructed. These algebras are close cousins of logarithmic W-algebras of Feigin and Tipunin and they are also obtained as modifications of semiclassical limits of vertex algebras appearing in the context of S-duality for four-dimensional gauge theories. In the case of type A and principal nilpotent element the character agrees precisely with the Schur-Index formula for corresponding Argyres-Douglas theories with irregular singularities. For other nilpotent elements they are identified with Schur-indices of type IV Argyres-Douglas theories. Further, there is a conformal embedding pattern of these vertex operator algebras that nicely matches the RG-flow of Argyres-Douglas theories as discussed by Buican and Nishinaka.

Vertex operator algebras, Higgs branches, and modular differential equations

Every four-dimensional mathcal{N}=2 superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any mathcal{N}=2 SCFT should obey a finite order modular differential equation. By way of the “high temperature” limit of the superconformal index, this allows the Weyl anomaly coefficient a to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the “Deligne-Cvitanović exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class mathcal{S} theories, and mathcal{N}=2 super Yang-Mills with mathfrak{s}mathfrak{u}(n) gauge group for small-to-moderate values of n.

Quantum periods for [formula omitted]SU(2) SQCD around the superconformal point

We study the Argyres–Douglas theories realized at the superconformal point in the Coulomb moduli space of N=2 supersymmetric SU(2) QCD with Nf=1,2,3 hypermultiplets in the Nekrasov–Shatashvili limit of the Omega-background. The Seiberg–Witten curve of the theory is quantized in this limit and the periods receive the quantum corrections. By applying the WKB method for the quantum Seiberg–Witten curve, we calculate the quantum corrections to the Seiberg–Witten periods around the superconformal point up to the fourth order in the parameter of the Omega background.

Four-dimensional lens space index from two-dimensional chiral algebra

We study the supersymmetric partition function on S1 × L(r, 1), or the lens space index of four-dimensional mathcal{N}=2 superconformal field theories and their connection to two-dimensional chiral algebras. We primarily focus on free theories as well as ArgyresDouglas theories of type (A1, Ak) and (A1, Dk). We observe that in specific limits, the lens space index is reproduced in terms of the (refined) character of an appropriately twisted module of the associated two-dimensional chiral algebra or a generalized vertex operator algebra. The particular twisted module is determined by the choice of discrete holonomies for the flavor symmetry in four-dimensions.

Notes on integral identities for 3d supersymmetric dualities

Four dimensional mathcal{N}=2 Argyres-Douglas theories have been recently conjectured to be described by mathcal{N}=1 Lagrangian theories. Such models, once reduced to 3d, should be mirror dual to Lagrangian mathcal{N}=4 theories. This has been numerically checked through the matching of the partition functions on the three sphere. In this article, we provide an analytic derivation for this result in the A2n−1 case via hyperbolic hypergeometric integrals. We study the D4 case as well, commenting on some open questions and possible resolutions. In the second part of the paper we discuss other integral identities leading to the matching of the partition functions in 3d dual pairs involving higher monopole superpotentials.

On the chiral algebra of Argyres-Douglas theories and S-duality

We study the two-dimensional chiral algebra associated with the simplest Argyres-Douglas type theory with an exactly marginal coupling, i.e., the (A3, A3) theory. Near a cusp in the space of the exactly marginal deformations (i.e., the conformal manifold), the theory is well-described by the SU(2) gauge theory coupled to isolated Argyres-Douglas theories and a fundamental hypermultiplet. In this sense, the (A3, A3) theory is an Argyres-Douglas version of the mathcal{N} = 2 SU(2) conformal QCD. By studying its Higgs branch and Schur index, we identify the minimal possible set of chiral algebra generators for the (A3, A3) theory, and show that there is a unique set of closed OPEs among these generators. The resulting OPEs are consistent with the Schur index, Higgs branch chiral ring relations, and the BRST cohomology conjecture. We then show that the automorphism group of the chiral algebra we constructed contains a discrete group G with an S3 subgroup and a homomorphism G → S4 × Z2. This result is consistent with the S-duality of the (A3, A3) theory.

Bootstrapping the (A1, A2) Argyres-Douglas theory

We apply bootstrap techniques in order to constrain the CFT data of the (A1, A2) Argyres-Douglas theory, which is arguably the simplest of the Argyres-Douglas models. We study the four-point function of its single Coulomb branch chiral ring generator and put numerical bounds on the low-lying spectrum of the theory. Of particular interest is an infinite family of semi-short multiplets labeled by the spin ℓ. Although the conformal dimensions of these multiplets are protected, their three-point functions are not. Using the numerical bootstrap we impose rigorous upper and lower bounds on their values for spins up to ℓ = 20. Through a recently obtained inversion formula, we also estimate them for sufficiently large ℓ, and the comparison of both approaches shows consistent results. We also give a rigorous numerical range for the OPE coefficient of the next operator in the chiral ring, and estimates for the dimension of the first R-symmetry neutral non-protected multiplet for small spin.

Argyres-Douglas matter and S-duality. Part II

We study S-duality of Argyres-Douglas theories obtained by compactification of 6d (2,0) theories of ADE type on a sphere with irregular punctures. The weakly coupled descriptions are given by the degeneration limit of auxiliary Riemann sphere with marked points, among which three punctured sphere represents isolated superconformal theories. We also discuss twisted irregular punctures and their S-duality.