Two-dimensional Chiral Algebra Research Articles

Twisted holography without conformal symmetry

We discuss the notion of translation-invariant vacua for 2d chiral algebras and relate it to the notion of the associated variety. The two-dimensional chiral algebra associated to four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 U(N) SYM has a conjectural holographic dual involving the B-model topological string theory. We study the effect of non-zero vacuum expectation values on the chiral algebra correlation functions and derive a holographic dual Calabi-Yau geometry. We test our proposal by a large N analysis of correlation functions of determinant operators, whose saddles can be matched with semi-classical configurations of “Giant Graviton” D-branes in the bulk.

Superconformal Algebras and Holomorphic Field Theories

We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find is closely related to algebras studied by Faonte–Hennion–Kapranov, Hennion–Kapranov, and the second author with Gwilliam in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; further deforming by a superconformal element induces the Koszul resolution of a plane in mathbb {C}^2 cong {mathbb {R}}^4. The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras.

Comments on chiral algebras and \u03a9-deformations

Every six-dimensional mathcal{N} = (2, 0) SCFT on R6 contains a set of protected operators whose correlation functions are controlled by a two-dimensional chiral algebra. We provide an alternative construction of this chiral algebra by performing an Ω-deformation of a topological-holomorphic twist of the mathcal{N} = (2, 0) theory on R6 and restricting to the cohomology of a specific supercharge. In addition, we show that the central charge of the chiral algebra can be obtained by performing equivariant integration of the anomaly polynomial of the six-dimensional theory. Furthermore, we generalize this construction to include orbifolds of the R4 transverse to the chiral algebra plane.

Supersymmetric localization in AdS5 and the protected chiral algebra

mathcal{N}=4 super Yang-Mills theory admits [1] a protected subsector isomorphic to a two-dimensional chiral algebra, obtained by passing to the cohomology of a certain supercharge. In the large N limit, we expect this chiral algebra to have a dual description as a subsector of IIB supergravity on AdS5 × S5. This subsector can be carved out by a version of supersymmetric localization, using the bulk analog of the boundary supercharge. We illustrate this procedure in a simple model, the theory of an mathcal{N}=4 vector multiplet in AdS5, for which a convenient off-shell description is available. This model can be viewed as the simplest truncation of the full AdS5 × S5 supergravity, in which case the vector multiplet should be taken in the adjoint representation of {mathfrak{g}}_F=mathfrak{s}mathfrak{u}{(2)}_F . Localization yields Chern-Simons theory on AdS3 with gauge algebra {mathfrak{g}}_F , whose boundary dual is the affine Lie algebra {widehat{mathfrak{g}}}_F . We comment on the generalization to the full bulk theory. We propose that the large N limit of the chiral algebra of mathcal{N}=4 SYM is again dual to Chern-Simons theory, with gauge algebra a suitable higher-spin superalgebra.

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.

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.

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.

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.

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.

Conformal manifolds in four dimensions and chiral algebras

Any superconformal field theory (SCFT) in four dimensions has a sector of operators related to a two-dimensional chiral algebra containing a Virasoro sub-algebra. Moreover, there are well-known examples of isolated SCFTs whose chiral algebra is a Virasoro algebra. In this note, we consider the chiral algebras associated with interacting SCFTs possessing an exactly marginal deformation that can be interpreted as a gauge coupling (i.e., at special points on the resulting conformal manifolds, free gauge fields appear that decouple from isolated SCFT building blocks). At any point on these conformal manifolds, we argue that the associated chiral algebras possess at least three generators. In addition, we show that there are examples of SCFTs realizing such a minimal chiral algebra: they are certain points on the conformal manifold obtained by considering the low-energy limit of type IIB string theory on the three complex-dimensional hypersurface singularity . The associated chiral algebra is the theory of Feigin, Feigin, and Tipunin. As byproducts of our work, we argue that (i) a collection of isolated theories can be conformally gauged only if there is a SUSY moduli space associated with the corresponding symmetry current moment maps in each sector, and (ii) SCFTs with have hidden fermionic symmetries (in the sense of fermionic chiral algebra generators).

N = 2 $$ \mathcal{N}=2 $$ central charge bounds from 2d chiral algebras

We study protected correlation functions in $\mathcal{N} = 2$ SCFT whose description is captured by a two-dimensional chiral algebra. Our analysis implies a new analytic bound for the $c$-anomaly as a function of the flavor central charge $k$, valid for any theory with a flavor symmetry $G$. Combining our result with older bounds in the literature puts strong constraints on the parameter space of $\mathcal{N}=2$ theories. In particular, it singles out a special set of models whose value of $c$ is uniquely fixed once $k$ is given. This set includes the canonical rank one $\mathcal{N}=2$ SCFTs given by Kodaira's classification.

Stress-tensor OPE in N = 2 $$ \mathcal{N}=2 $$ superconformal theories

We carry out a detailed superspace analysis of the OPE of two N=2 stress-tensor multiplets. Knowledge of the multiplets appearing in the expansion, together with the two-dimensional chiral algebra description of N=2 SCFTs, imply an analytic bound on the central charge c. This bound is valid for any N=2 SCFT regardless of its matter content and flavor symmetries, and is saturated by the simplest Argyres-Douglas fixed point. We also present a partial conformal block analysis for the scalar superconformal primary of the multiplet.

Schur indices, BPS particles, and Argyres-Douglas theories

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 results we propose that the chiral algebra of the generalized Argyres-Douglas theory $(A_{k-1},A_{N-1})$ with $k$ and $N$ coprime, is the vacuum sector of the $(k,k+N)$ $W_{k}$ minimal model, and that the Schur index is the associated vacuum character.

On the superconformal index of Argyres–Douglas theories

We conjecture a closed-form expression for the Schur limit of the superconformal index of two infinite series of Argyres–Douglas (AD) superconformal field theories (SCFTs): the and the theories. While these SCFTs can be realized at special points on the Coulomb branch of certain gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the and indices by using a relation to two-dimensional q-deformed Yang–Mills theory and data from the class construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the and SCFTs. As checks of our conjectures, we study the consistency of our results with an S-duality recently discussed by us in collaboration with Giacomelli and Papageorgakis, we reproduce known Higgs branch relations, we check consistency with a series of renormalization group flows, and we verify that the small S1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S1 reduction of our indices in a separate paper.

W $$ \mathcal{W} $$ symmetry in six dimensions

Six-dimensional conformal field theories with $(2,0)$ supersymmetry are shown to possess a protected sector of operators and observables that are isomorphic to a two-dimensional chiral algebra. We argue that the chiral algebra associated to a $(2,0)$ theory labelled by the simply-laced Lie algebra $\mathfrak{g}$ is precisely the W algebra of type $\mathfrak{g}$, for a specific value of the central charge. Simple examples of observables that are made accessible by this correspondence are the three-point functions of half-BPS operators. For the $A_n$ series, we compare our results at large $n$ to those obtained using the holographic dual description and find perfect agreement. We further find protected chiral algebras that appear on the worldvolumes of codimension two defects in $(2,0)$ SCFTs. This construction has likely implications for understanding the microscopic origin of the AGT correspondence.

Chiral algebras of class S $$ \mathcal{S} $$

Four-dimensional N=2 superconformal field theories have families of protected correlation functions that possess the structure of two-dimensional chiral algebras. In this paper, we explore the chiral algebras that arise in this manner in the context of theories of class S. The class S duality web implies nontrivial associativity properties for the corresponding chiral algebras, the structure of which is best summarized in the language of generalized topological quantum field theory. We make a number of conjectures regarding the chiral algebras associated to various strongly coupled fixed points.

Infinite Chiral Symmetry in Four Dimensions

We describe a new correspondence between four-dimensional conformal field theories with extended supersymmetry and two-dimensional chiral algebras. The meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four-dimensional theory. Infinite chiral symmetry has far-reaching consequences for the spectral data, correlation functions, and central charges of any four-dimensional theory with \({\mathcal{N}=2}\) superconformal symmetry.