Argyres-Douglas Theories Research Articles

Argyres-Douglas theories, chiral algebras and wild Hitchin characters

We use Coulomb branch indices of Argyres-Douglas theories on S1 × L(k, 1) to quantize moduli spaces {mathrm{mathcal{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 {mathrm{mathcal{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 {mathrm{mathcal{M}}}_H under the U(1) Hitchin action, and a limit of them can be identified with matrix elements of the modular transform STkS in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.

Supersymmetric Gauge Theories with Decoupled Operators and Chiral Ring Stability.

We propose a general way to complete supersymmetric theories with operators below the unitarity bound, adding gauge-singlet fields that enforce the decoupling of such operators. This makes it possible to perform all usual computations, and to compactify on a circle. We concentrate on a duality between an N=1 SU(2) gauge theory and the N=2 A_{3} Argyres-Douglas theory, mapping the moduli space and chiral ring of the completed N=1 theory to those of the A_{3} model. We reduce the completed gauge theory to 3D, finding a 3D duality with N=4 supersymmetric QED (SQED) with two flavors. The naive dimensional reduction is instead N=2 SQED. Crucial is a concept of chiral ring stability, which modifies the superpotential and allows for a 3D emergent global symmetry.

Vertex operator algebras of Argyres-Douglas theories from M5-branes

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d mathcal{N}=left(2, 0right) theory of type J on a punctured sphere. We denote the AD theories as (Jb[k], Y), where Jb[k] and Y represent an irregular and a regular singularity respectively. We restrict to the ‘minimal’ case where Jb[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}}left(J, Y right) , where {k}_{2d}=-h+frac{b}{b+k} with h being the dual Coxeter number of J. We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for b = h. We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.

Line defect Schur indices, Verlinde algebras and U(1)r fixed points

Given an mathcal{N}=2 superconformal field theory, we reconsider the Schur index ℐL(q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that ℐL(q) admits an expansion in terms of characters of the chiral algebra mathcal{A} introduced by Beem et al., with simple coefficients υL,β(q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients υL,β(q) is linearly related to the vacuum expectation values 〈L〉 in U(1)r -invariant vacua of the theory compactified on S1. This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on U(1)r -invariant vacua, and a Verlindelike algebra associated to mathcal{A} . Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type (A1, A2), (A1, A4), (A1, A6), (A1, D3) and (A1, D5). In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.

Lagrangians for generalized Argyres-Douglas theories

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 (Ak , AkN +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 land on the mathcal{N}=4 complete graph quivers. We also compactify to three dimensions the gauge theory dual to (A1, D4), and find the expected Abelianization duality with mathcal{N}=4 SQED with 3 flavors.

Mathcal{N} =1 Lagrangians for generalized Argyres-Douglas theories

We find mathcal{N} = 1 Lagrangian gauge theories that flow to generalized ArgyresDouglas theories with mathcal{N} = 2 supersymmetry. We find that certain SU quiver gauge theories flow to generalized Argyres-Douglas theories of type (Ak−1, Amk−1) and (Im,km, S). We also find quiver gauge theories of SO/Sp gauge groups flowing to the (A2m−1, D2mk+1), (A2m, D2m(k−1)+k) and Dm(2k + 2)m(2k + 2)[m] theories.

Abelianization and sequential confinement in 2 + 1 dimensions

We consider the lagrangian description of Argyres-Douglas theories of type A2N −1, which is a SU(N) gauge theory with an adjoint and one fundamental flavor. An appropriate reformulation allows us to map the moduli space of vacua across the duality, and to dimensionally reduce. Going down to three dimensions, we find that the adjoint SQCD “abelianizes”: in the infrared it is equivalent to a mathcal{N}=4 linear quiver theory. Moreover, we study the mirror dual: using a monopole duality to “sequentially confine” quivers tails with balanced nodes, we show that the mirror RG flow lands on mathcal{N}=4 SQED with N flavors. These results make the supersymmetry enhancement explicit and provide a physical derivation of previous proposals for the three dimensional mirror of AD theories.

On Painlevé/gauge theory correspondence

We elucidate the relation between Painlev\'e equations and four-dimensional rank one ${\cal N= 2}$ theories by identifying the connection associated to Painlev\'e isomonodromic problems with the oper limit of the flat connection of the Hitchin system associated to gauge theories and by studying the corresponding renormalisation group flow. Based on this correspondence we provide long-distance expansions at various canonical rays for all Painlev\'e functions in terms of magnetic and dyonic Nekrasov partition functions for ${\cal N= 2}$ SQCD and Argyres-Douglas theories at self-dual Omega background $\epsilon_1+\epsilon_2= 0$, or equivalently in terms of $c= 1$ irregular conformal blocks.

Dualities of deformed mathcal{N}=2 SCFTs from link monodromy on D3-brane states

We study D3-brane theories that are dually described as deformations of two different mathcal{N}=2 superconformal theories with massless monopoles and dyons. These arise at the self-intersection of a seven-brane in F-theory, which cuts out a link on a small three-sphere surrounding the self-intersection. The spectrum is studied by taking small loops in the three-sphere, yielding a link-induced monodromy action on string junction D3-brane states, and subsequently quotienting by the monodromy. This reduces the differing flavor algebras of the mathcal{N}=2 theories to the same flavor algebra, as required by duality, and projects out charged states, yielding an mathcal{N}=1 superconformal theory on the D3-brane. In one, a deformation of a rank one Argyres-Douglas theory retains its SU(2) flavor symmetry and exhibits a charge neutral flavor triplet that is comprised of electron, dyon, and monopole string junctions. From duality we argue that the monodromy projection should also be imposed away from the conformal point, in which case the D3-brane field theory appears to exhibit confinement of electrons, dyons, and monopoles. We will address the mathematical counterparts in a companion paper.

Macdonald index and chiral algebra

For any 4d mathcal{N} = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type (A1, A2n) and (A1, D2n+1) where the chiral algebras are given by Virasoro and widehat{mathfrak{su}}(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.

ODE/IM correspondence and the Argyres-Douglas theory

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 A2N -type theories are non-unitary coset models (A1)1 × (A1)L/(A1)L+1 at the fractional level L=frac{2}{2N+1}-2 , which appear in the study of the 4d/2d correspondence of mathcal{N} = 2 superconformal field theories. Based on the WKB analysis, we clarify the relation between the Y-functions and the quantum periods and study the exact Bohr-Sommerfeld quantization condition for the quantum periods. We also discuss the quantum spectral curves for the D and E type theories.

W-algebras for Argyres–Douglas theories

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 -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014), while the one corresponding to $$D_{2p}$$ , denoted by , is realized as a non-regular quantum Hamiltonian reduction of $$L_{k}(\mathfrak {sl}_{p+1})$$ at level . For all n one observes that the quantum Hamiltonian reduction of the vertex operator algebra of $$D_n$$ -Argyres–Douglas theory is the vertex operator algebra of $$A_{n-3}$$ -Argyres–Douglas theory. As a corollary, one realizes the singlet and triplet algebras (the vertex algebras associated to the best understood logarithmic conformal field theories) as quantum Hamiltonian reductions as well. Finally, characters of certain modules of these vertex operator algebras and the modular properties of their meromorphic continuations are given.

Surface defects and chiral algebras

We investigate superconformal surface defects in four-dimensional mathcal{N}=2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfeld-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. In each case we find perfect agreement with the predicted characters.

Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index.

We find a four-dimensional N=1 gauge theory which flows to the minimal interacting N=2 superconformal field theory, the Argyres-Douglas theory, in the infrared up to the extra free chiral multiplets. The gauge theory is obtained from a certain N=1 preserving deformation of the N=2 SU(2) gauge theory with four fundamental hypermultiplets. From this description, we compute the full superconformal index and find agreements with the known results in special limits.

Nekrasov and Argyres–Douglas theories in spherical Hecke algebra representation

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 intertwiners 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 conformal states corresponding to Argyres–Douglas theory, which involves summation of functions over Young diagrams.

Strong coupling in F-theory and geometrically non-Higgsable seven-branes

Geometrically non-Higgsable seven-branes carry gauge sectors that cannot be broken by complex structure deformation, and there is growing evidence that such configurations are typical in F-theory. We study strongly coupled physics associated with these branes. Axiodilaton profiles are computed using Ramanujan's theories of elliptic functions to alternative bases, showing explicitly that the string coupling is O(1) in the vicinity of the brane; that it sources nilpotent SL(2,Z) monodromy and therefore the associated brane charges are modular; and that essentially all F-theory compactifications have regions with order one string coupling. It is shown that non-perturbative SU(3) and SU(2) seven-branes are related to weakly coupled counterparts with D7-branes via deformation-induced Hanany–Witten moves on (p,q) string junctions that turn them into fundamental open strings; only the former may exist for generic complex structure. D3-brane near these and the Kodaira type II seven-branes probe Argyres–Douglas theories. The BPS states of slightly deformed theories are shown to be dyonic string junctions.

Mathcal{N}=1 deformations and RG flows of mathcal{N}=2 SCFTs

We study certain mathcal{N}=1 preserving deformations of four-dimensional mathcal{N}=2 superconformal field theories (SCFTs) with non-abelian flavor symmetry. The deformation is described by adding an mathcal{N}=1 chiral multiplet transforming in the adjoint representation of the flavor symmetry with a superpotential coupling, and giving a nilpotent vacuum expectation value to the chiral multiplet which breaks the flavor symmetry. This triggers a renormalization group flow to an infrared SCFT. Remarkably, we find classes of theories flow to enhanced mathcal{N}=2 supersymmetric fixed points in the infrared under the deformation. They include generalized Argyres-Douglas theories and rank-one SCFTs with non-abelian flavor symmetries. Most notably, we find renormalization group flows from the deformed conformal SQCDs to the (A1, An) Argyres-Douglas theories. From these “Lagrangian descriptions,” we compute the full superconformal indices of the (A1, An) theories and find agreements with the previous results. Furthermore, we study the cases, including the TN and R0,N theories of class mathcal{S} and some of rank-one SCFTs, where the deformation gives genuine mathcal{N}=1 fixed points.

N $$ \mathcal{N} $$ =1 Deformations and RG flows of N $$ \mathcal{N} $$ =2 SCFTs, part II: non-principal deformations

We continue to investigate the $\mathcal{N}=1$ deformations of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) labeled by a nilpotent element of the flavor symmetry. This triggers a renormalization group (RG) flow to an $\mathcal{N}=1$ SCFT. We systematically analyze all possible deformations of this type for certain classes of $\mathcal{N}=2$ SCFTs: conformal SQCDs, generalized Argyres-Douglas theories and the $E_6$ SCFT. We find a number of examples where the amount of supersymmetry gets enhanced to $\mathcal{N}=2$ at the end point of the RG flow. Most notably, we find that the $SU(N)$ and $Sp(N)$ conformal SQCDs can be deformed to flow to the Argyres-Douglas (AD) theories of type $(A_1, D_{2N-1})$ and $(A_1, D_{2N})$ respectively. This RG flow therefore allows us to compute the full superconformal index of the $(A_1,D_N)$ class of AD theories. Moreover, we find an infrared duality between $\mathcal{N}=1$ theories where the fixed point is described by an $\mathcal{N}=2$ AD theory. We observe that the classes of examples that exhibit supersymmetry enhancement saturate certain bounds for the central charges implied by the associated two-dimensional chiral algebra.

Small deformation of a simple N=2 superconformal theory

We study an interesting relevant deformation of the simplest interacting N=2 SCFT---the original Argyres-Douglas (AD) theory. We argue that, although this deformation is not strictly speaking Banks-Zaks like (certain operator dimensions change macroscopically), there are senses in which it constitutes a mild deformation of the parent AD theory: the exact change in the "a" anomaly is small and is essentially saturated at one loop. Moreover, contributions from IR operators that have a simple description in the UV theory reproduce a particular limit of the IR index to a remarkably high order. These results lead us to conclude that the IR theory is an interacting N=1 SCFT with particularly small "a" and "c" central charges and that this theory sheds some interesting light on the spectrum of its AD parent.

Infrared computations of defect Schur indices

We conjecture a formula for the Schur index of N=2 four-dimensional theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS excitations. We test our proposal in a variety of examples for SU(2) gauge theories, either conformal or asymptotically free. We use the conjecture to compute these defect-enriched Schur indices for theories which lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate in various examples that line defect indices can be expressed as sums of characters of the associated two-dimensional chiral algebra and that for Argyres-Douglas theories the line defect OPE reduces in the index to the Verlinde algebra.