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Motivated by the fundamental role that bosonic and fermionic symmetries play in physics, we study (non-invertible) one-form symmetries in $2 + 1$d consisting of topological lines with bosonic and fermionic self-statistics. We refer to these lines as Bose-Fermi-Braided (BFB) symmetries and argue that they can be classified. Unlike the case of generic anyonic lines, BFB symmetries are closely related to groups. In particular, when BFB lines are non-invertible, they are non-intrinsically non-invertible. Moreover, BFB symmetries are, in a categorical sense, weakly group theoretical. Using this understanding, we study invariants of renormalization group flows involving non-topological QFTs with BFB symmetry., Comment: 56 pages and 10 figures

Coulomb branches of vacua are the most universal moduli spaces that arise in local unitary interacting 4d $\mathcal{N}=2$ superconformal field theories (SCFTs). In these theories, $1/2$-BPS primaries parameterize the Coulomb branches and form (anti-)chiral rings. We define the notion of a Coulomb branch operator algebra, $\mathcal{A}_{\mathcal{C}}$, that contains these chiral and anti-chiral rings along with infinitely many more operators and products that are less protected by supersymmetry. Using a universal symmetry, $\mathcal{I}\cong\mathbb{Z}_2$, that arises from studying the superconformal group, we give $\mathcal{I}$ selection rules for $\mathcal{A}_{\mathcal{C}}$ and, more generally, for arbitrary products in the local operator algebra of any 4d $\mathcal{N}=2$ SCFT. Defining the notion of a "Coulombic" SCFT, we propose explanations for certain phenomena in a 4d/2d correspondence involving 4d $\mathcal{N}=2$ theories and 2d vertex operator algebras. Finally, by considering deformations of $\mathcal{I}$, we explore the case of $\mathcal{N}>2$ SCFTs., Comment: 29 pages; v2: references added, minor typos fixed, results unchanged

We study general maps from the space of rational CFTs with a fixed chiral algebra and associated Chern-Simons (CS) theories to the space of qudit stabilizer codes with a fixed generalized Pauli group. We consider certain natural constraints on such a map and show that the map can be described as a graph homomorphism from an orbifold graph, which captures the orbifold structure of CFTs, to a code graph, which captures the structure of self-dual stabilizer codes. By studying explicit examples, we show that this graph homomorphism cannot always be a graph embedding. However, we construct a physically motivated map from universal orbifold subgraphs of CFTs to operators in a generalized Pauli group. We show that this map results in a self-dual stabilizer code if and only if the surface operators in the bulk CS theories corresponding to the CFTs in question are self-dual. For CFTs admitting a stabilizer code description, we show that the full abelianized generalized Pauli group can be obtained from twisted sectors of certain 0-form symmetries of the CFT. Finally, we connect our construction with SymTFTs, and we argue that many equivalences between codes that arise in our setup correspond to equivalence classes of bulk topological surfaces under fusion with invertible surfaces., Comment: 57 pages, 12 figures

We characterize discrete (anti-)unitary symmetries and their non-invertible generalizations in $2+1$d topological quantum field theories (TQFTs) through their actions on line operators and fusion spaces. We explain all possible sources of non-invertibility that can arise in this context. Our approach gives a simple $2+1$d proof that non-invertible generalizations of unitary symmetries exist if and only if a bosonic TQFT contains condensable bosonic line operators (i.e., these non-invertible symmetries are necessarily "non-intrinsic"). Moving beyond unitary symmetries and their non-invertible cousins, we define a non-invertible generalization of time-reversal symmetries and derive various properties of TQFTs with such symmetries. Finally, using recent results on 2-categories, we extend our results to corresponding statements in $2+1$d quantum field theories that are not necessarily topological., Comment: 79 pages, 33 figures

We ask when it is possible to construct arbitrary unitary multiplets of the superconformal algebra with eight Poincar\'e supercharges that are compatible with locality from (continuous deformations of) representations in free field theories. We answer this question in two, three, and five dimensions. In four dimensions, we resort to an intricate but self-consistent web of conjectures. If correct, these conjectures imply various new non-perturbative constraints on short multiplets in any local unitary 4d $\mathcal{N}=2$ superconformal field theory and on an unusual set of related vertex algebras. Throughout, we connect our results with properties of deformations in the space of theories., Comment: 30 pages

We study the conditions under which 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) have multiplets housing operators that are chiral with respect to an $\mathcal{N}=1$ subalgebra. Our main focus is on the set of often-ignored and relatively poorly understood $\overline{\mathcal{B}}$ representations. These multiplets typically evade direct detection by the most popular non-perturbative 4d $\mathcal{N}=2$ tools and correspondences. In spite of this fact, we demonstrate the ubiquity of $\overline{\mathcal{B}}$ multiplets and show they are associated with interesting phenomena. For example, we give a purely algebraic proof that they are present in all local unitary $\mathcal{N}>2$ SCFTs. We also show that $\overline{\mathcal{B}}$ multiplets exist in $\mathcal{N}=2$ theories with rank greater than one and a conformal manifold or a freely generated Coulomb branch. Using recent topological quantum field theory results, we argue that certain $\overline{\mathcal{B}}$ multiplets exist in broad classes of theories with the $\mathbb{Z}_2$-valued 't Hooft anomaly for $Sp(N)$ global symmetry. Motivated by these statements, we then study the question of whether $\overline{\mathcal{B}}$ multiplets exist in rank-one SCFTs with exactly $\mathcal{N}=2$ SUSY. We conclude with various open questions., Comment: 44 pages; v2: minor typos corrected, additional explanations, results unchanged

In quantum field theory (QFT) above two spacetime dimensions, one is usually only able to construct exact operator maps from the ultraviolet (UV) to the infrared (IR) of strongly coupled renormalization group (RG) flows for the most symmetry-protected observables. Famous examples include maps of chiral rings in 4d $\mathcal{N}=2$ supersymmetry. In this letter, we construct the first non-perturbative UV/IR map for less protected operators: starting from a particularly "simple" UV strongly coupled non-Lagrangian 4d $\mathcal{N}=2$ QFT, we show that a universal non-chiral quarter-BPS ring can be mapped exactly and bijectively to the IR. In particular, strongly coupled UV dynamics governing infinitely many null states manifest in the IR via Fermi statistics of free gauginos. Using the concept of arc space, this bijection allows us to compute the exact UV Macdonald index in the IR., Comment: v1: 10 pages and 1 figure; v2: accepted PRL version (title changed but results unchanged)

We study ultraviolet (UV) incarnations of deep infrared (IR) physics on the Coulomb branch of the simplest interacting 4D $\mathcal{N}=2$ 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 $\bar{\mathcal{C}}_{R,r(j,\bar j)}$ multiplets in the MAD theory for all $R$, $r$, $j$, and $\bar j$ satisfying $R+\bar j -j+1=0$, 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 / $\mathcal{N}=2$ chiral operator correspondence., Comment: 36 pages; 3 figures; 2 tables

Despite the power of supersymmetry, finding exact closed-form expressions for the protected operator spectra of interacting superconformal field theories (SCFTs) is difficult. In this paper, we take a step towards a solution for the "simplest" interacting 4D $\mathcal{N}=2$ SCFT: the minimal Argyres-Douglas (MAD) theory. We present two results that go beyond the well-understood Coulomb branch and Schur sectors. First, we find the exact closed-form spectrum of multiplets containing operators that are chiral with respect to any $\mathcal{N}=1\subset\mathcal{N}=2$ superconformal subalgebra. We argue that this "full" chiral sector (FCS) is as simple as allowed by unitarity for a theory with a Coulomb branch and that, up to a rescaling of $U(1)_r$ quantum numbers and the vanishing of a finite number of states, the MAD FCS is isospectral to the FCS of the free $\mathcal{N}=2$ Abelian gauge theory. In the language of superconformal representation theory, this leaves only the spectrum of the poorly understood $\bar{\mathcal{C}}_{R,r(j,\bar j)}$ multiplets to be determined. Our second result sheds light on these observables: we find an exact closed-form answer for the number of $\bar{\mathcal{C}}_{0,r(j,0)}$ multiplets, for any $r$ and $j$, in the MAD theory. We argue that this sub-sector is also as simple as allowed by unitarity for a theory with a Coulomb branch and that there is a natural map to the corresponding sector of the free $\mathcal{N}=2$ Abelian gauge theory. These results motivate a conjecture on the full local operator algebra of the MAD theory., Comment: 34 pages and 2 appendices; v2: references added and typos corrected

We give a general construction relating Narain rational conformal field theories (RCFTs) and associated 3d Chern-Simons (CS) theories to quantum stabilizer codes. Starting from an abelian CS theory with a fusion group consisting of $n$ even-order factors, we map a boundary RCFT to an $n$-qubit quantum code. When the relevant 't Hooft anomalies vanish, we can orbifold our RCFTs and describe this gauging at the level of the code. Along the way, we give CFT interpretations of the code subspace and the Hilbert space of qubits while mapping error operations to CFT defect fields., Comment: 20 pages; 3 figures; 4 appendices

Long ago, Argyres and Douglas discovered a particularly simple interacting 4D $\mathcal{N}=2$ superconformal field theory (SCFT) on the Coulomb branch of $SU(3)$ $\mathcal{N}=2$ super Yang-Mills. Further hints of the theory's simplicity arise due to the fact that it has the smallest possible value of the $c$ central charge among unitary interacting $\mathcal{N}=2$ SCFTs. A main purpose of this note is to uncover additional aspects of this minimal Argyres-Douglas (MAD) theory's simplicity. In particular, we argue that: (1) the MAD theory shares an infinite set of large spin thresholds in part of its operator spectrum with the free $\mathcal{N}=2$ Maxwell theory (this data is therefore invariant under generic $\mathcal{N}=2$-preserving renormalization group flows to the IR) and (2) the MAD theory has, at every order in the natural grading, the smallest number of "Schur" operators of any unitary $\mathcal{N}=2$ theory (interacting or free). We then show that property (1) has a suitable generalization for all $(A_1, A_{2k})$ cousins of the MAD theory. In particular, the corresponding large spin thresholds encode generic renormalization group flows within this class. This construction therefore gives a different handle on these flows from the one provided by the Seiberg-Witten description. To emphasize the importance of these spin thresholds, we abstractly study theories with "enough matter" to form Higgs branches and argue that infinitely many spin thresholds are small or vanishing., Comment: 37 pages; v2: references added

We study Galois actions on $2+1$D topological quantum field theories (TQFTs), characterizing their interplay with theory factorization, gauging, the structure of gapped boundaries and dualities, 0-form symmetries, 1-form symmetries, and 2-groups. In order to gain a better physical understanding of Galois actions, we prove sufficient conditions for the preservation of unitarity. We then map out the Galois orbits of various classes of unitary TQFTs. The simplest such orbits are trivial (e.g., as in various theories of physical interest like the Toric Code, Double Semion, and 3-Fermion Model), and we refer to such theories as unitary "Galois fixed point TQFTs." Starting from these fixed point theories, we study conditions for preservation of Galois invariance under gauging 0-form and 1-form symmetries (as well as under more general anyon condensation). Assuming a conjecture in the literature, we prove that all unitary Galois fixed point TQFTs can be engineered by gauging 0-form symmetries of theories built from Deligne products of certain abelian TQFTs., Comment: 88 pages; 10 figures; 2 appendices; v2: typos fixed; reference added

We systematically study 4D $\mathcal{N}=2$ superconformal field theories (SCFTs) that can be constructed via type IIB string theory on isolated hypersurface singularities (IHSs) embedded in $\mathbb{C}^4$. We show that if a theory in this class has no $\mathcal{N}=2$-preserving exactly marginal deformation (i.e., the theory is isolated as an $\mathcal{N}=2$ SCFT), then it has no 1-form symmetry. This situation is somewhat reminiscent of 1-form symmetry and decomposition in 2D quantum field theory. Moreover, our result suggests that, for theories arising from IHSs, 1-form symmetries originate from gauge groups (with vanishing beta functions). One corollary of our discussion is that there is no 1-form symmetry in IHS theories that have all Coulomb branch chiral ring generators of scaling dimension less than two. In terms of the $a$ and $c$ central charges, this condition implies that IHS theories satisfying $a<{1\over24}(15r+2f)$ and $c<{1\over6}(3r+f)$ (where $r$ is the complex dimension of the Coulomb branch, and $f$ is the rank of the continuous 0-form flavor symmetry) have no 1-form symmetry. After reviewing the 1-form symmetries of other classes of theories, we are motivated to conjecture that general interacting 4D $\mathcal{N}=2$ SCFTs with all Coulomb branch chiral ring generators of dimension less than two have no 1-form symmetry., Comment: 84 pages; v2: references added, minor typos fixed, minor additional result (direct proof of corollary 1)

We study the implications of the anyon fusion equation $a\times b=c$ on global properties of $2+1$D topological quantum field theories (TQFTs). Here $a$ and $b$ are anyons that fuse together to give a unique anyon, $c$. As is well known, when at least one of $a$ and $b$ is abelian, such equations describe aspects of the one-form symmetry of the theory. When $a$ and $b$ are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and $a$ and $b$ lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian $a$ and $b$ can also be an indication of zero-form symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the case of discrete gauge theories based on the largest Mathieu group, $M_{24}$), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems., Comment: 65 pages; 3 figures; 3 appendices; v2: some clarifications in section 3; reference added; v3: brief additional discussion in introduction and references added; v4: summary of results added, typos fixed, additional discussion--to appear in Quantum

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., Comment: 11 pages; 5 figures

Long ago, Arad and Herzog (AH) conjectured that, in finite simple groups, the product of two conjugacy classes of length greater than one is never a single conjugacy class. We discuss implications of this conjecture for non-abelian anyons in 2+1-dimensional discrete gauge theories. Thinking in this way also suggests closely related statements about finite simple groups and their associated discrete gauge theories. We prove these statements and provide some physical intuition for their validity. Finally, we explain that the lack of certain dualities in theories with non-abelian finite simple gauge groups provides a non-trivial check of the AH conjecture., Comment: 10 pages; 3 figures; 2 Appendices; v2: references updated and additional results added

We revisit certain natural algebraic transformations on the space of 3D topological quantum field theories (TQFTs) called "Galois conjugations." Using a notion of multiboundary entanglement entropy (MEE) defined for TQFTs on compact 3-manifolds with disjoint boundaries, we give these abstract transformations additional physical meaning. In the process, we prove a theorem on the invariance of MEE along orbits of the Galois action in the case of arbitrary Abelian theories defined on any link complement in $S^3$. We then give a generalization to non-Abelian TQFTs living on certain infinite classes of torus link complements. Along the way, we find an interplay between the modular data of non-Abelian TQFTs, the topology of the ambient spacetime, and the Galois action. These results are suggestive of a deeper connection between entanglement and fusion., Comment: 11 figures; 3 appendices

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 $E_6$ 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 (Poincar\'e 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., Comment: 52 pages and 21 figures

We continue our program of mapping data of 4D $\mathcal{N}=2$ superconformal field theories (SCFTs) onto observables of 2D chiral rational conformal field theories (RCFTs) by revisiting an infinite set of strongly coupled Argyres-Douglas (AD) SCFTs and their associated logarithmic 2D chiral algebras. First, we turn on discrete flavor fugacities (for continuous flavor symmetries) in a known correspondence between certain unrefined characters of these logarithmic theories and unrefined characters of a set of unitary 2D chiral RCFTs. Motivated by this discussion, we then study 4D Higgs branch renormalization group flows (i.e., flows activated by vevs for which only $su(2)_R\subset su(2)_R\times u(1)_R$ is spontaneously broken) emanating from our AD theories from the point of view of the unitary 2D theories and find some surprises. In particular, we argue that certain universal pieces of the topological data underlying the 2D chiral algebra representations associated with the 4D infrared (IR) theory can be computed, via Galois conjugation, in the topological quantum field theory (TQFT) underlying the unitary ultraviolet (UV) chiral RCFT. The mapping of this topological data from UV to IR agrees with the fact that, in our theories, the moduli spaces we study consist of free hypermultiplets at generic points if and only if the UV TQFT is a theory of abelian anyons., Comment: 33 pages and 1 figure

Proceedings of the workshop "Boundary and Defect Conformal Field Theory: Open Problems and Applications," Chicheley Hall, Buckinghamshire, UK, 7-8 Sept. 2017., Comment: Conference proceedings of "Boundary and Defect Conformal Field Theory: Open Problems and Applications." Invited submission to Proceedings of the Royal Society A. Section 17 is a more condensed version of arXiv:1711.05049

We initiate a study of an infinite set of renormalization group flows with accidental supersymmetry enhancement. The ultraviolet fixed points are strongly interacting four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) with no known Lagrangian descriptions, and the infrared fixed points are SCFTs with thirty-two (Poincar\'e plus special) supercharges., Comment: 10 figures, 21 pages, v2: references added, minor typos corrected, results unchanged

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this note, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) by performing simple calculations involving sets of non-unitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full $\mathcal{N}=2$ superconformal algebra. As a byproduct, we find relations between characters in unitary and non-unitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction., Comment: 7 pages; 1 appendix; v2: version to appear in PRL; minor typo corrected; references added; title changed to conform to PRL guidelines (old title: "A Non-Unitary Surprise")

Using the chiral algebra bootstrap, we revisit the simplest Argyres-Douglas (AD) generalization of Argyres-Seiberg S-duality. We argue that the exotic AD superconformal field theory (SCFT), $T_{3,{3\over2}}$, emerging in this duality splits into a free piece and an interacting piece, T_X, even though this factorization seems invisible in the Seiberg-Witten (SW) curve derived from the corresponding M5-brane construction. Without a Lagrangian, an associated topological field theory, a BPS spectrum, or even an SW curve, we nonetheless obtain exact information about T_X by bootstrapping its chiral algebra, chi(T_X), and finding the corresponding vacuum character in terms of Affine Kac-Moody characters. By a standard 4D/2D correspondence, this result gives us the Schur index for T_X and, by studying this quantity in the limit of small S^1, we make contact with a proposed S^1 reduction. Along the way, we discuss various properties of T_X: as an N=1 theory, it has flavor symmetry SU(3)XSU(2)XU(1), the central charge of chi(T_X) matches the central charge of the bc ghosts in bosonic string theory, and its global SU(2) symmetry has a Witten anomaly. This anomaly does not prevent us from building conformal manifolds out of arbitrary numbers of T_X theories (giving us a surprisingly close AD relative of Gaiotto's T_N theories), but it does lead to some open questions in the context of the chiral algebra / 4D N=2 SCFT correspondence., Comment: 30+10 pages; 4 appendices; 6 figures

We generalize, in a manifestly Weyl-invariant way, our previous expressions for irregular singularity wave functions in two-dimensional SU(2) q-deformed Yang-Mills theory to SU(N). As an application, we give closed-form expressions for the Schur indices of all (A_{N-1}, A_{N(n-1)-1}) Argyres-Douglas (AD) superconformal field theories (SCFTs), thus completing the computation of these quantities for the (A_N, A_M) SCFTs. With minimal effort, our wave functions also give new Schur indices of various infinite sets of "Type IV" AD theories. We explore the discrete symmetries of these indices and also show how highly intricate renormalization group (RG) flows from isolated theories and conformal manifolds in the ultraviolet to isolated theories and (products of) conformal manifolds in the infrared are encoded in these indices. We compare our flows with dimensionally reduced flows via a simple "monopole vev RG" formalism. Finally, since our expressions are given in terms of concise Lie algebra data, we speculate on extensions of our results that might be useful for probing the existence of hypothetical SCFTs based on other Lie algebras. We conclude with a discussion of some open problems., Comment: 34 pages, 2 figures

Motivated by the three-dimensional topological field theory / two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, that can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic "topological symmetries" that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry / topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones., Comment: 54 pages and 17 figures; v2: reference added; preprint number added; appendix B of v1 has evolved into a separate publication

We explicitly construct and list all unitary superconformal multiplets, along with their index contributions, in five and six dimensions. From this data, we uncover various unifying themes in the representation theory of five- and six-dimensional superconformal field theories. At the same time, we provide a detailed argument for the complete classification of unitary irreducible representations in five dimensions using a combination of physical and mathematical techniques., Comment: 118 pages; v2 typos corrected and references added ; v3 classification of 5D unitary multiplets corrected and C-type multiplets removed ; v4 discussion of the procedure for obtaining irreps of the SCA improved and typos corrected, published version

Any N=2 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 N=2 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 x_1^3+x_2^3+x_3^3+A x_1x_2x_3+w^2=0. The associated chiral algebra is the A(6) 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) N=2 SCFTs with a>=c have hidden fermionic symmetries (in the sense of fermionic chiral algebra generators)., Comment: 25 pages and 1 figure; v2: minor improvements in discussion; results unchanged

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., Comment: 8 pages; v2: reference added, typos corrected, minor additional comments, results unchanged; v3: minor improvements, results unchanged

We conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2)_R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S^1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general N=2 superconformal field theories., Comment: 42 pages, 2 figures, and 2 tables

In a recent paper, we proposed closed-form expressions for the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) superconformal field theories (SCFTs) in the Schur limit. Following up on our results, we turn our attention to the small S^1 regime of these indices. As expected on general grounds, our study reproduces the S^3 partition functions of the resulting dimensionally reduced theories. However, we show that in all cases---with the exception of the reduction of the (A_1, D_4) SCFT---certain imaginary partners of real mass terms are turned on in the corresponding mirror theories. We interpret these deformations as R symmetry mixing with the topological symmetries of the direct S^1 reductions. Moreover, we argue that these shifts occur in any of our theories whose four-dimensional N=2 superconformal U(1)_R symmetry does not obey an SU(2) quantization condition. We then use our R symmetry map to find the four-dimensional ancestors of certain three-dimensional operators. Somewhat surprisingly, this picture turns out to imply that the scaling dimensions of many of the chiral operators of the four-dimensional theory are encoded in accidental symmetries of the three-dimensional theory. We also comment on the implications of our work on the space of general N=2 SCFTs., Comment: 31 pages; version 2: small typos corrected; results unchanged; published version

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 (A_1,A_{2n-3}) and the (A_1,D_{2n}) theories. While these SCFTs can be realized at special points on the Coulomb branch of certain N=2 gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the (A_1, A_{2n-3}) and (A_1, D_{2n}) indices by using a relation to two-dimensional q-deformed Yang-Mills theory and data from the class S construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the (A_1, A_3) and (A_1, D_4) 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 S^1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S^1 reduction of our indices in a separate paper., Comment: 46 pages, 1 figure; v2: typos corrected, arXiv identifier of reference [41] added

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 of the gauging). As we explore the resulting conformal manifold of the SU(2) SCFT, we find an action of S-duality on the parameters of the theory that is reminiscent of Spin(8) triality. On the other hand, as we explore the conformal manifold of the SU(3) theory, we find that an exotic rank-two SCFT emerges in a dual SU(2) description., Comment: 49 pages, 4 figures; v2: typos corrected, reference added, brief additional comments in section 4; results unchanged

In this note we begin a systematic study of compact conformal manifolds of SCFTs in four dimensions (our notion of compactness is with respect to the topology induced by the Zamolodchikov metric). Supersymmetry guarantees that such manifolds are Kahler, and so the simplest possible non-trivial compact conformal manifold in this set of geometries is a complex one-dimensional projective space. We show that such a manifold is indeed realized and give a general prescription for constructing complex N-dimensional projective space conformal manifolds as certain small N=2->N=1 breaking deformations of strongly interacting N=2 SCFTs. In many cases, our prescription reduces the construction of such spaces to a study of the N=2 chiral ring. We also give an algorithm for constructing more general compact spaces of SCFTs., Comment: 19 pages; typos corrected; reference added

We study certain higher-spin chiral operators in N=2 superconformal field theories (SCFTs). In Lagrangian theories, or in theories related to Lagrangian theories by generalized Argyres-Seiberg-Gaiotto duality ("type A" theories in our classification), we give a simple superconformal representation theory proof that such operators do not exist. This argument is independent of the details of the superconformal index. We then use the index to show that if a theory is not of type A but has an N=2-preserving deformation by a relevant operator that takes it to a theory of this type in the infrared, the ultraviolet theory cannot have these higher-spin operators either. As an application of this discussion, we give a simple prescription to extract the 2a-c conformal anomaly directly from the superconformal index. We also comment on how this procedure works in the holographic limit., Comment: 1+40 pages; version 3: reference added; footnote with additional details regarding OPE of exotic operators; to appear in JHEP; results unchanged

We study lower bounds on the minimal distance in theory space between four-dimensional superconformal field theories (SCFTs) connected via broad classes of renormalization group (RG) flows preserving various amounts of supersymmetry (SUSY). For N=1 RG flows, the ultraviolet (UV) and infrared (IR) endpoints of the flow can be parametrically close. On the other hand, for RG flows emanating from a maximally supersymmetric SCFT, the distance to the IR theory cannot be arbitrarily small regardless of the amount of (non-trivial) SUSY preserved along the flow. The case of RG flows from N=2 UV SCFTs is more subtle. We argue that for RG flows preserving the full N=2 SUSY, there are various obstructions to finding examples with parametrically close UV and IR endpoints. Under reasonable assumptions, these obstructions include: unitarity, known bounds on the c central charge derived from associativity of the operator product expansion, and the central charge bounds of Hofman and Maldacena. On the other hand, for RG flows that break N=2 to N=1, it is possible to find IR fixed points that are parametrically close to the UV ones. In this case, we argue that if the UV SCFT possesses a single stress tensor, then such RG flows excite of order all the degrees of freedom of the UV theory. Furthermore, if the UV theory has some flavor symmetry, we argue that the UV central charges should not be too large relative to certain parameters in the theory., Comment: 41 pages and 2 figures; references updated; minor improvements in examples section

We study certain small supersymmetry-breaking perturbations of a large class of strongly coupled four-dimensional R-symmetric renormalization group (RG) flows between superconformal field theories in the ultraviolet (UV) and the infrared (IR). We analyze the conditions under which these perturbations scale to zero at leading order in the deep IR, resulting in accidental supersymmetry. Furthermore, we connect the emergence of IR supersymmetry in this context with a quantity that was recently conjectured to be larger at the UV starting points of the underlying supersymmetric flows than at the corresponding IR endpoints, and we propose a bound on emergent supersymmetry. Along the way, we prove a simple and useful non-perturbative theorem regarding the IR behavior of global flavor currents. Our results suggest general ways in which light stop particles can emerge and potentially influence physics at the Large Hadron Collider., Comment: Published JHEP version. Introduction slightly streamlined; one additional comment; equations and results unchanged

In this note, we study a large class of four-dimensional R-symmetric theories, and we describe a new quantity, \tau_U, which is well-defined in these theories. Furthermore, we conjecture that this quantity is larger in the ultraviolet (UV) than in the infrared (IR), i.e. that \tau_U^{UV}>\tau_U^{IR}. While we do not prove this inequality in full generality, it is straightforward to show that our conjecture holds in the subset of theories that do not have accidental symmetries. In addition, we subject our inequality to an array of non-trivial tests in theories with accidental symmetries and dramatically different dynamics both in N=1 and N=2 supersymmetry and find that our inequality is obeyed. One interesting consequence of this conjecture is that the mixing of accidental symmetries with the IR superconformal R current is bounded by the UV quantity, \tau_U^{UV}. To demonstrate the potential utility of this bound, we apply it to the somewhat mysterious SU(2) gauge theory of Intriligator, Seiberg, and Shenker and show that our conjecture, if correct, implies that this theory flows in the IR to an interacting superconformal field theory., Comment: 50 pages; additional discussion; minor changes; results unchanged

In many strongly-coupled systems, the infrared dynamics is described by different degrees of freedom from the ultraviolet. It is then natural to ask how operators written in terms of the microscopic variables are mapped to operators composed of the macroscopic ones. Certain types of operators, like conserved currents, are simple to map, and in supersymmetric theories one can also follow the chiral ring. In this note, we consider supersymmetric theories and extend the mapping to anomalous currents (and gaugino bilinears). Our technique is completely independent of subtleties associated with the renormalization group, thereby shedding new light on previous approaches to the problem. We demonstrate the UV/IR mapping in several examples with different types of dynamics, emphasizing the uniformity and simplicity of the approach. Natural applications of these ideas include the effects of soft breaking on the dynamics of various theories and new models of electroweak symmetry breaking., Comment: 23 pages; a few additional comments; results unchanged

An important unanswered question in quantum field theory is to understand precisely under which conditions scale invariance implies invariance under the full conformal group. While the general answer in two dimensions has been known for over twenty years, a precise non-perturbative relation between scale and conformal invariance in higher dimensions has been lacking. In this note, we specialize to four dimensions and give a full quantum mechanical proof that certain unitary R-symmetric fixed points are necessarily superconformal. Among other consequences, this result implies that the infrared fixed points of N=1 supersymmetric quantum chromodynamics are superconformal., Comment: 21 pages; references added; brief additional comments; results unchanged

We construct an N=2 supersymmetric generalization of the N=1 supercurrent formalism of Komargodski and Seiberg (KS) and use it to show that N=2 theories with linear superconformal anomalies cannot break SUSY under certain broad assumptions. This result suggests that there are no metastable SUSY breaking vacua in a large class of theories that includes N=2 Super Yang-Mills (SYM)., Comment: 19 pages; minor revisions; JHEP version

In theories of phyiscs beyond the Standard Model (SM), visible sector fields often carry quantum numbers under additional gauge symmetries. One could then imagine a scenario in which these extra gauge symmetries play a role in transmitting supersymmetry breaking from a hidden sector to the Supersymmetric Standard Model (SSM). In this paper we present a general formalism for studying the resulting hidden sectors and calculating the corresponding gauge mediated soft parameters. We find that a large class of generic models features a leading universal contribution to the soft scalar masses that only depends on the scale of Higgsing, even if the model is strongly coupled. As a by-product of our analysis, we elucidate some IR aspects of the correlation functions in General Gauge Mediation. We also discuss possible phenomenological applications., Comment: 22 pages, 2 figures; references added

We explore various aspects of General Gauge Mediation(GGM). We present a reformulation of the correlation functions used in GGM, and further elucidate their IR and UV properties. Additionally we clarify the issue of UV sensitivity in the calculation of the soft masses in the MSSM, highlighting the role of the supertrace over the messenger spectrum. Finally, we present weakly coupled messenger models which fully cover the parameter space of GGM. These examples demonstrate that the full parameter space of GGM is physical and realizable. Thus it should be considered a valid basis for future phenomenological explorations of gauge mediation., Comment: 27 pages, minor changes, typos fixed in appendix B

It is well known that D-brane instantons can generate contributions to the effective superpotential of gauge theories living on D-branes which are perturbatively forbidden by global U(1) symmetries. We extend this idea to theories with supersymmetry breaking, studying the effect of D-brane instantons stretched between the SUSY-breaking and visible sectors. Analogously to what happens in the SUSY case, this mechanism can give rise to perturbatively forbidden soft terms (among other effects). We introduce and discuss general properties of instanton mediation. We illustrate our ideas in simple Type IIB toroidal orientifolds. As a bi-product, we present a string theory realization of a Polonyi hidden sector., Comment: 42 pages, 7 figures. V2. typos fixed and references added