In this first of two papers, we explain in detail the simplest example of a broader set of relations between apparently very different theories. Our example relates $\mathfrak{su}(2)$ $\mathcal{N}=4$ super Yang-Mills (SYM) to a theory we call "$(3,2)$." This latter theory is an exactly marginal diagonal $SU(2)$ gauging of three $D_3(SU(2))$ Argyres-Douglas (AD) theories. We begin by observing that the Schur indices of these two theories are related by an algebraic transformation that is surprisingly reminiscent of index transformations describing spontaneous symmetry breaking on the Higgs branch. However, this transformation breaks half the supersymmetry of the SYM theory as well as its full $\mathcal{N}=2$ $SU(2)_F$ flavor symmetry. Moreover, it does so in an interesting way when viewed through the lens of the corresponding 2D vertex operator algebras (VOAs): affine currents of the small $\mathcal{N}=4$ super-Virasoro algebra at $c=-9$ get mapped to the $\mathcal{A}(6)$ stress tensor and some of its conformal descendants, while the extra supersymmetry currents on the $\mathcal{N}=4$ side get mapped to higher-dimensional fermionic currents and their descendants on the $\mathcal{A}(6)$ side. We prove these relations are facets of an exact graded vector space isomorphism (GVSI) between these two VOAs. This GVSI respects the $U(1)_r$ charge of the parent 4D theories. We briefly sketch how more general $\mathfrak{su}(N)$ $\mathcal{N}=4$ SYM theories are related to an infinite class of AD theories via generalizations of our example. We conclude by showing that, in this class of theories, the $\mathcal{A}(6)$ VOA saturates a new inequality on the number of strong generators.
We study chiral algebras associated with Argyres-Douglas theories engineered from M5 brane. For the theory engineered using 6d $(2,0)$ type $J$ theory on a sphere with a single irregular singularity (without mass parameter), its chiral algebra is the minimal model of W algebra of $J$ type. For the theory engineered using an irregular singularity and a regular full singularity, its chiral algebra is the affine Kac-Moody algebra of $J$ type. We can obtain the Schur index of these theories by computing the vacua character of the corresponding chiral algebra.
The Schur-index of the $(A_1, X_n)$-Argyres-Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the $A_{\text{odd}}$ and $D_{\text{even}}$-type Argyres-Douglas theories. The vertex operator algebra corresponding to $A_{2p-3}$-Argyres-Douglas theory is the logarithmic $\mathcal B_p$-algebra of [1], while the one corresponding to $D_{2p}$, denoted by $\mathcal W_p$, is realized as a non-regular Quantum Hamiltonian reduction of $L_{k}(\mathfrak{sl}_{p+1})$ at level $k=-(p^2-1)/p$. For all $n$ one observes that the quantum Hamiltonian r…
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…
We study the Schur index, the Zhu's $C_2$ algebra, and the Macdonald index of a four dimensional $\mathcal{N}=2$ Argyres-Douglas (AD) theories from the structure of the associated two dimensional $W$-algebra. The Schur index is derived from the vacuum character of the corresponding $W$-algebra and can be rewritten in a very simple form, which can be easily used to verify properties like level-rank dualities, collapsing levels, and S-duality conjectures. The Zhu's $C_2$ algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu's $C_2$ al…
The behaviour of the chiral condensates in the SU(2) gauge theory with broken N=2 supersymmetry is reviewed. The calculation of monopole, dyon, and charge condensates is described. It is shown that the monopole and charge condensates vanish at the Argyres-Douglas point where the monopole and charge vacua collide. This phenomenon is interpreted as a deconfinement of electric and magnetic charges at the Argyres-Douglas point.
We find $\mathcal{N}=1$ Lagrangian gauge theories that flow to generalized Argyres-Douglas theories with $\mathcal{N}=2$ supersymmetry. We find that certain SU quiver gauge theories flow to generalized Argyres-Douglas theories of type $(A_{k-1}, A_{mk-1})$ and $(I_{m, k m}, S)$. We also find quiver gauge theories of SO/Sp gauge groups flowing to the $(A_{2m-1}, D_{2mk+1})$, $(A_{2m}, D_{2m(k-1)+k})$ and $D_{m(2k+2)}^{m(2k+2)}[m]$ theories.
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.
In this paper, by using the factorization equation of the ${\cal N}=2$ theory we study ${\cal N}=1$ supersymmetric SO(N) gauge theory in Argyres-Douglas points . We suppose that maximal confinement occurs in the system and give a tree level superpotential. Then, we obtain general Picard-fuchs equations for glueball superfield which are hypergeometric equations having regular singular points corresponding to Argyres-Douglas points. Furthermore, we study the solution of these differential equations and calculate the effective superpotential. Finally, we study scaling behavior of the chiral opera…
Strebel differentials are a special class of quadratic differentials with several applications in string theory. In this note we show that finding Strebel differentials with integral lengths is equivalent to finding generalized Argyres-Douglas singularities in the Coulomb moduli space of a U(N) $\N=2$ gauge theory with massive flavours. Using this relation, we find an efficient technique to solve the problem of factorizing the Seiberg-Witten curve at the Argyres-Douglas singularity. We also comment upon a relation between more general Seiberg-Witten curves and Belyi maps.
The Coulomb branch indices of Argyres-Douglas theories on $L(k,1)\times S^{1}$ are recently identified with matrix elements of modular transforms of certain $2d$ vertex operator algebras in a particular limit. A one parameter generalization of the modular transformation matrices of $(2N+3,2)$ minimal models are proposed to compute the full Coulomb branch index of $(A_{1},A_{2N})$ Argyres-Douglas theories on the same space. Morever, M-theory construction of these theories suggests direct connection to the refined Chern-Simons theory. The connection is made precise by showing how the modular tra…
We obtain a large class of new 4d Argyres-Douglas theories by classifying irregular punctures for the 6d (2,0) superconformal theory of ADE type on a sphere. Along the way, we identify the connection between the Hitchin system and three-fold singularity descriptions of the same Argyres-Douglas theory. Other constructions such as taking degeneration limits of the irregular puncture, adding an extra regular puncture, and introducing outer-automorphism twists are also discussed. Later we investigate various features of these theories including their Coulomb branch spectrum and central charges.
We make a preliminary investigation into twisted $A_{2n}$ theories of class S. Contrary to a common piece of folklore, we establish that theories of this type realise a variety of models of Argyres-Douglas type while utilising only regular punctures. We present an in-depth analysis of all twisted $A_2$ trinion theories, analyse their interrelations via partial Higgsing, and discuss some of their generalised S-dualities.
As regular conformal blocks describe the N=2 superconformal gauge theories in four dimensions, irregular conformal blocks are expected to reproduce the instanton partition functions of the Argyres-Douglas theories. In this paper, we construct matrix models which reproduce the irregular conformal conformal blocks of the Liouville theory on sphere, by taking a colliding limit of the Penner-type matrix models. The resulting matrix models have not only logarithmic terms but also rational terms in the potential. We also discuss their relation to the Argyres-Douglas type theories.
Using the $u$-plane integral as a tool, we derive a formula for the partition function of the simplest nontrivial (topologically twisted) Argyres-Douglas theory on compact, oriented, simply connected, four-manifolds without boundary and with $b_2^+>0$. The result can be expressed in terms of classical cohomological invariants and Seiberg-Witten invariants. Our results hint at the existence of standard four-manifolds that are not of Seiberg-Witten simple type.
We construct an N=1 theory with gauge group U(nN) and degree n+1 tree level superpotential whose matrix model spectral curve develops an A_{n+1} Argyres-Douglas singularity. We evaluate the coupling constants of the low-energy U(1)^n theory and show that the large N expansion is singular at the Argyres-Douglas points. Nevertheless, it is possible to define appropriate double scaling limits which are conjectured to yield four dimensional non-critical string theories as proposed by Ferrari. In the Argyres-Douglas limit the n-cut spectral curve degenerates into a solution with n/2 cuts for even n…
3d mirrors for all 4d $\mathcal{N}=2$ Argyres-Douglas (AD) theories engineered using 6d $(2,0)$ theory are found. The basic steps are: 1): Find a punctured sphere representation for the AD theories (this is achieved in our previous studies of S duality); 2): Attach a 3d theory for each puncture; 3): Glue together the 3d theory for each puncture. We found the 3d mirror quiver gauge theory for the AD theories engineered using 6d $A$ and $D$ type theories. These 3d mirrors are useful for studying the properties of original 4d theory such as Higgs branch, S-duality, etc; We also construct many new…
We continue the study of Lagrangian descriptions of $\mathcal{N}=2$ Argyres-Douglas theories. We use our recent interpretation in terms of sequential confinement to guess the Lagrangians of all the Argyres-Douglas models with Abelian three dimensional mirror. We find classes of four dimensional $\mathcal{N}=1$ quivers that flow in the infrared to generalized Argyres-Douglas theories, such as the $(A_k,A_{kN+N-1})$ models. We study in detail how the $\mathcal{N}=1$ chiral rings map to the Coulomb and Higgs Branches of the $\mathcal{N}=2$ CFT's. The three dimensional mirror RG flows are shown to…
We conjecture a precise relationship between the Schur limit of the superconformal index of four-dimensional $\mathcal{N}=2$ field theories, which counts local operators, and the spectrum of BPS particles on the Coulomb branch. We verify this conjecture for the special case of free field theories, $\mathcal{N}=2$ QED, and $SU(2)$ gauge theory coupled to fundamental matter. Assuming the validity of our proposal, we compute the Schur index of all Argyres-Douglas theories. Our answers match expectations from the connection of Schur operators with two-dimensional chiral algebras. Based on our resu…
We study the BPS spectrum of four-dimensional $\mathcal{N}=2$ superconformal field theory of Argyres-Douglas type, obtained via twisted compactification of six-dimensional $A_{N-1}$ $(2,0)$ theory on a sphere with an irregular puncture, by using spectral networks. We give strong evidence of the equivalence of $\mathcal{N}=2$ superconformal field theories from six-dimensional theories of different ranks by systematically comparing the chamber structure and wall-crossing phenomena.
We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N=(2, 0) theory of type $J$ on a punctured sphere. We denote the AD theories as $(J^b[k],Y)$, where $J^b[k]$ and $Y$ represent an irregular and a regular singularity respectively. We restrict to the `minimal' case where $J^b[k]$ has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra $\mathcal{W}^{k_{2d}}(J,Y)$, where $k_{2d}=-h+ \frac{b}{b+k}$ with $h$ b…
We study S duality of four dimensional N=2 Argyres-Douglas (AD) theory engineered from 6d A_{N-1} (2,0) theory. We find a (p,q) sequence of SCFTs, here (p,q) is co-prime and class S theory defined on sphere corresponds to class (0,1) theory. We represent these theories by a sphere with marked points, and S duality is interpreted as different pants decompositions of the same punctured sphere. The weakly coupled gauge theory description involves gauging AD matter which is represented by three punctured sphere.
The calculation of physical quantities in certain quantum field theories such as those of the Argyres-Douglas type is notoriously hard, due to the lack of a Lagrangian description. Here we tackle this problem following two alternative approaches. On the one hand, we use localization on the four-sphere to compute two-correlators and OPE coefficients in Argyres-Douglas superconformal theories. On the other hand, we use the conformal bootstrap machinery to put stringent bounds on such coefficients, only relying on the knowledge of central charge and conformal dimension of the operators. We compar…
We construct a large class of Argyres-Douglas type theories by compactifying six dimensional (2,0) A_N theory on a Riemann surface with irregular singularities. We give a complete classification for the choices of Riemann surface and the singularities. The Seiberg-Witten curve and scaling dimensions of the operator spectrum are worked out. Three dimensional mirror theory and the central charges a and c are also calculated for some subsets, etc. Our results greatly enlarge the landscape of N=2 superconformal field theory and in fact also include previous theories constructed using regular singu…
The lowest critical point of one unitary matrix model with cosine plus logarithmic potential is known to correspond with the $(A_1, A_3)$ Argyres-Douglas (AD) theory and its double scaling limit derives the Painlevé II equation with parameter. Here, we consider the critical points associated with all cosine potentials and determine the scaling operators, their vevs and their scaling dimensions from perturbed string equations at planar level. These dimensions agree with those of $(A_1,A_{4k-1})$ AD theory.
We determine the structure of 1-form symmetries for all 4d $\mathcal{N} = 2$ theories that have a geometric engineering in terms of type IIB string theory on isolated hypersurface singularities. This is a large class of models, that includes Argyres-Douglas theories and many others. Despite the lack of known gauge theory descriptions for most such theories, we find that the spectrum of 1-form symmetries can be obtained via a careful analysis of the non-commutative behaviour of RR fluxes at infinity in the IIB setup. The final result admits a very compact field theoretical reformulation in term…
We present new $AdS_5$ solutions in 11d supergravity and identify them as the gravity duals of a class of 4d $\mathcal N = 2$ SCFTs of Argyres-Douglas type, engineered by a stack of M5-branes wrapped on a sphere with one irregular puncture and one regular puncture. The gravity solutions feature an internal M5-brane source, which is dual to the irregular puncture. Thanks to a novel Stückelberg mechanism involving an axion originating from the expansion of the M-theory 3-form, one of the isometry generators on the gravity side is not mapped to a continuous symmetry on the SCFT side.
General formulas for the scalar modes a_i and a^i_D in the Seiberg-Witten SU(N) setting are derived in the cases with and without massive hypermultiplets. Subsequently these formulas are applied in a study of the SU(3) Argyres-Douglas point. We use this example to study the question, whether the scalar modes admit an interpretation in terms of BPS mass states everywhere in moduli space. The paper collects, in an appendix, various facts on the function Lauricella F_D^(n), which naturally appears in the derived formulas.
We study geometric engineering of Argyres-Douglas superconformal theories realized by type IIB strings propagating in singular Calabi-Yau threefolds. We use this construction to count the degeneracy of light BPS states under small perturbations away from the conformal point, by computing the degeneracy of D3-branes wrapped around supersymmetric 3-cycles in the Calabi-Yau. We find finitely many BPS states, the number of which depends on how this deformation is done, similarly to the degeneracy of kink solutions for the deformation of N=2 Landau-Ginzburg superconformal theories in two dimensions…
We compute the Schur index of Argyres-Douglas theories of type $(A_{N-1},A_{M-1})$ 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 $(A_{N-1},A_{M-1})$ theories; one with ${\rm gcd}(N,M)=1$ and the other with $M=N(k-1)$ for an integer $k\geq 2$. For these two series of Argyres-Douglas theories, our results are identical to the char…
We study the quantum Seiberg-Witten (SW) curves for $(A_1, G)$-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 $G=A_r$, the quantum SW curve of the AD theory is consistent with the scaling limit of the curve of the gauge theory. For $G=D_r$, 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 $(A_1, A_3)$ and $(A_1, D_4)$-type AD theories.
We show in details that the all-orders genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the $H_1$ 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 trans-series. Finally in the Nekrasov-Shatashvili limit we connect the $H_1$ and th…
N=2 Seiberg-Witten theories allow an interesting interplay between the Argyres-Douglas loci, singularity structures and wall-crossing formulae. In this paper we investigate this connection by first studying the singularity structures of hyper-elliptic Seiberg-Witten curves for pure N=2 gauge theories with SU(r+1) and Sp(2r) gauge groups, and propose new methods to locate the Argyres-Douglas loci in the moduli space, where multiple mutually non-local BPS states become massless. In a region of the moduli space, we compute dyon charges for all 2r+2 and 2r+1 massless dyons for SU(r+1) and Sp(2r) g…
We consider chiral condensates in SU(2) gauge theory with broken N=2 supersymmetry. The matter sector contains an adjoint multiplet and one fundamental flavor. Matter and gaugino condensates are determined by integrating out the adjoint field. The only nonperturbative input is the Affleck-Dine-Seiberg (ADS) superpotential generated by one instanton plus the Konishi anomaly. These results are consistent with those obtained by the `integrating in' procedure, including a reproduction of the Seiberg-Witten curve from the ADS superpotential. We then calculate monopole, dyon, and charge condensates …
We study the two-dimensional chiral algebra associated with the simplest Argyres-Douglas type theory with an exactly marginal coupling, i.e., the $(A_3,A_3)$ 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 $(A_3,A_3)$ 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 chir…
We use Coulomb branch indices of Argyres-Douglas theories on $S^1 \times L(k,1)$ to quantize moduli spaces ${\cal M}_H$ of wild/irregular Hitchin systems. In particular, we obtain formulae for the "wild Hitchin characters" -- the graded dimensions of the Hilbert spaces from quantization -- for four infinite families of ${\cal M}_H$, giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in ${\cal M}_H$ under the $U(1)$ Hitchin action, and a limit of them can be identif…
AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with interwiners we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state, with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conform…
We apply bootstrap techniques in order to constrain the CFT data of the $(A_1,A_2)$ 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 $\ell$. 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 t…
We consider N = 2 super Yang-Mills theory with SU(2) gauge group and a single quark hypermultiplet in the fundamental representation. For a specific value of the quark bare mass and at a certain point in the moduli space of vacua, the central charges corresponding to two mutually non-local electro-magnetic charges vanish simultaneously, indicating the possibility of massless such states in the spectrum. By realizing the theory as an M-theory configuration, we show that these states indeed exist in the spectrum near the critical point.
We construct a new class of three-dimensional topological quantum field theories (3d TQFTs) by considering generalized Argyres-Douglas theories on $S^1 \times M_3$ with a non-trivial holonomy of a discrete global symmetry along the $S^1$. For the minimal choice of the holonomy, the resulting 3d TQFTs are non-unitary and semisimple, thus distinguishing themselves from theories of Chern-Simons and Rozansky-Witten types respectively. Changing the holonomy performs a Galois transformation on the TQFT, which can sometimes give rise to more familiar unitary theories such as the $(G_2)_1$ and $(F_4)_…
We initiate a study of the holographic duals of a class of four-dimensional $\mathcal{N}=2$ superconformal field theories that are engineered by wrapping M5-branes on a sphere with an irregular puncture. These notably include the strongly-coupled field theories of Argyres-Douglas type. Our solutions are obtained in 7d gauged supergravity, where they take the form of a warped product of $AdS_5$ and a "half-spindle." The irregular puncture is modeled by a localized M5-brane source in the internal space of the gravity duals. Our solutions feature a realization of supersymmetry that is distinct fr…
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 $(A_1,A_3)$ and $(A_1,D_4)$ 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 irregula…
We generalize S-duality to N=2 superconformal field theories (SCFTs) with Coulomb branch operators of non-integer scaling dimension. As simple examples, we find minimal generalizations of the S-dualities discovered in SU(2) gauge theory with four fundamental flavors by Seiberg and Witten and in SU(3) gauge theory with six fundamental flavors by Argyres and Seiberg. Our constructions start by weakly gauging diagonal SU(2) and SU(3) flavor symmetry subgroups of two copies of a particular rank-one Argyres-Douglas theory (along with sufficient numbers of hypermultiplets to guarantee conformality o…
We propose a Nekrasov-type formula for the instanton partition functions of four-dimensional N=2 U(2) gauge theories coupled to (A_1,D_{2n}) Argyres-Douglas theories. This is carried out by extending the generalized AGT correspondence to the case of U(2) gauge group, which requires us to define irregular states of the direct sum of Virasoro and Heisenberg algebras. Using our formula, one can evaluate the contribution of the (A_1,D_{2n}) theory at each fixed point on the U(2) instanton moduli space. As an application, we evaluate the instanton partition function of the (A_3,A_3) theory to find …
We calculate the superconformal indices of the $\mathcal N=2$ superconformal field theories realized on $N$ coincident D3-branes in 7-brane backgrounds with constant axiodilaton via the AdS/CFT correspondence. We include the finite-$N$ corrections as the contribution of D3-branes wrapped around 3-cycles in the internal space. We take only single-wrapping contributions into account for simplicity. We also determine the orders of the next-to-leading corrections which we do not calculate. The orders are relatively high, and we obtain many trustable terms. We give the results for $N=1,2,3$ explici…
We provide evidence for the existence of a new strongly-coupled four dimensional $\mathcal{N}=2$ superconformal field theory arising as a non-trivial IR fixed point on the Coulomb branch of the mass-deformed superconformal Lagrangian theory with gauge group $G_2$ and four fundamental hypermultiplets. Notably, our analysis proceeds by using various geometric constraints to bootstrap the data of the theory, and makes no explicit reference to the Seiberg-Witten curve. We conjecture a corresponding VOA and check that the vacuum character satisfies a linear modular differential equation of fourth o…
We revisit the study of singular points in the Coulomb branch of $\mathcal{N}=2$ SQCD in four dimensions with gauge group $SU(N)$ and odd number of flavors. For certain choices of the mass parameters these vacua are not lifted by a mass term for the chiral multiplet in the adjoint representation. By using recent results about the M5 brane description of $\mathcal{N}=1$ theories we study the resulting vacua and argue that the low-energy effective theory has a simple Lagrangian description involving a free chiral multiplet in the adjoint representation of the flavor symmetry group, a system some…
We study superconformal indices of 4d N=2 class S theories with certain irregular punctures called type $I_{k, N}$. This class of theories include generalized Argyres-Douglas theories of type $(A_{k-1}, A_{N-1})$ and more. We conjecture the superconformal indices in certain simplified limits based on the TQFT structure of the class S theories by writing an expression for the wave function corresponding to the puncture $I_{k, N}$. We write the Schur limit of the wave function when $k$ and $N$ are coprime. When $k=2$, we also conjecture a closed-form expression for the Hall-Littlewood index and …
We present a general prescription for constructing 3d $\mathcal{N}=4$ 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 hyperm…
We study the quantum spectral curve of the Argyres-Douglas theories in the Nekrasov-Sahashvili limit of the Omega-background. Using the ODE/IM correspondence we investigate the quantum integrable model corresponding to the quantum spectral curve. We show that the models for the $A_{2N}$-type theories are non-unitary coset models $(A_1)_1\times (A_1)_{L}/(A_1)_{L+1}$ at the fractional level $L=\frac{2}{2N+1}-2$, which appear in the study of the 4d/2d correspondence of ${\cal N}=2$ superconformal field theories. Based on the WKB analysis, we clarify the relation between the Y-functions and the q…
The 6d $(2,0)$ theory has codimension-one symmetry defects associated to the outer-automorphism group of the underlying ADE Lie algebra. 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 flav…
In [arXiv:1805.05057 [hep-th]],[arXiv:1812.00811 [hep-th]], the partition function of the Gross-Witten-Wadia unitary matrix model with the logarithmic term has been identified with the $τ$ function of a certain Painlevé system, and the double scaling limit of the associated discrete Painlevé equation to the critical point provides us with the Painlevé II equation. This limit captures the critical behavior of the $su(2)$, $N_f =2$ $\mathcal{N}=2$ supersymmetric gauge theory around its Argyres-Douglas $4D$ superconformal point. Here, we consider further extension of the model that contains the $…
We study $SU(2)$ gauge theories coupled to $(A_1,D_N)$ theories with or without a fundamental hypermultiplet. For even $N$, a formula for the contribution of $(A_1,D_N)$ to the Nekrasov partition function was recently obtained by us with Y.~Sugawara and T.~Uetoko. In this paper, we generalize it to the case of odd $N$ in the classical limit, under the condition that the relevant couplings and vacuum expectation values of Coulomb branch operators of $(A_1,D_N)$ are all turned off. We apply our formula to the $(A_2,A_5)$ theory to find that its prepotential is related to that of the $SU(2)$ gaug…
We study singularities of algebraic curves associated with 3d N=2 theories that have at least one global flavor symmetry. Of particular interest is a class of theories T_K labeled by knots, whose partition functions package Poincare polynomials of the S^r-colored HOMFLY homologies. We derive the defining equation, called the super-A-polynomial, for algebraic curves associated with many new examples of 3d N=2 theories T_K and study its singularity structure. In particular, we catalog general types of singularities that presumably exist for all knots and propose their physical interpretation. A …
