5d ray index is the 5d superconformal index in the presence of a ray-like defect. An elementary ray defect flows to a fundamental Wilson ray in IR gauge theory. In 5d $\mathcal{N}=1$ superconformal field theory(SCFT) with $E_n$ global symmetry, we compute the ray index associated with an adjoint Wilson ray. We show that the operators that appear in the index are non-trivially charged under the center of the global symmetry $E_n$. The charge under the center being twice compared with the center charge carried by the elementary ray operator indicates a non-trivial OPE between the ray-like defects in the UV SCFT.
We use the F-theory realization of 6D superconformal field theories (SCFTs) to study the corresponding spectrum of stringlike, i.e. surface defects. On the tensor branch, all of the stringlike excitations pick up a finite tension, and there is a corresponding lattice of string charges, as well as a dual lattice of charges for the surface defects. The defect group is data intrinsic to the SCFT and measures the surface defect charges which are not screened by dynamical strings. When non-trivial, it indicates that the associated theory has a partition vector rather than a partition function. We c…
We use holography to study codimension-2 surface defects in 5d SCFTs engineered by $(p,q)$ 5-brane webs. The three-dimensional defects are realized by D3-branes ending on the brane web. We identify the holographic representation of the defects in Type IIB $AdS_6$ solutions as probe D3-branes, and study conformal and non-conformal defects which, respectively, preserve one half and one quarter of the supersymmetry. For a sample of 5d SCFTs, including the $T_N$ theories, we provide explicit solutions for conformal and non-conformal defects. For the conformal defects we obtain their contribution t…
We study the $(1,0)$ six-dimensional SCFTs living on defects of non-geometric heterotic backgrounds (T-fects) preserving a $E_7\times E_8$ subgroup of $E_8\times E_8$. These configurations can be dualized explicitly to F-theory on elliptic K3-fibered non-compact Calabi-Yau threefolds. We find that the majority of the resulting dual threefolds contain non-resolvable singularities. In those cases in which we can resolve the singularities we explicitly determine the SCFTs living on the defect. We find a form of duality in which distinct defects are described by the same IR fixed point. For instan…
We discuss supersymmetric surface defects in compactifications of six dimensional minimal conformal matter of type SU(3) and SO(8) to four dimensions. The relevant field theories in four dimensions are N=1 quiver gauge theories with SU(3) and SU(4) gauge groups respectively. The defects are engineered by giving space-time dependent vacuum expectation values to baryonic operators. We find evidence that in the case of SU(3) minimal conformal matter the defects carry SU(2) flavor symmetry which is not a symmetry of the four dimensional model. The simplest case of a model in this class is SU(3) SQ…
We study ${\mathcal N}=(1,0)$ six-dimensional theories living on defects of non-geometric backgrounds of the $E_8\times E_8$ and the $\text{Spin}(32)/{\mathbb Z}_2$ heterotic strings. Such configurations can be analyzed by dualizing to F-theory on elliptic K3-fibered non-compact Calabi-Yau threefolds. The majority of the resulting dual threefolds turn out to contain singularities which do not admit a crepant resolution. When the singularities can be resolved crepantly, the theories living on the defect are explicitly determined and reveal a form of duality in which distinct defects are describ…
Tokai University（东海大学）；Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Southeast University（东南大学）
本文提出三维超对称规范理论中线缺陷 Witten 指标的一般公式，并发现 M2-膜 SCFT 中 Wilson 线缺陷指标的递推关系，实现高秩指标由低秩数据递归确定。
In conformal field theories (CFTs) of dimension $d>3$, two-dimensional (2d) conformal defects are characterised in part by central charges defined via the defect's contribution to the trace anomaly. However, in general for interacting CFTs these central charges are difficult to calculate. For superconformal 2d defects in supersymmetric (SUSY) CFTs (SCFTs), we show how to compute these defect central charges from the SUSY partition function either on $S^d$ with defect along $S^2$, or on $S^1 \times S^{d-1}$ with defect along $S^1 \times S^1$. In the latter case we propose that defect central ch…
We construct new families of $\text{AdS}_2 \times S^2 \times S^2$ solutions with 4 supercharges in Type II supergravities. We show that subclasses of these solutions can be interpreted in terms of defect branes embedded in 4d $\mathcal{N} = 4 $ SYM, or orbifolds thereof. This is explicitly realised by showing that the solutions asymptote locally to $\text{AdS}_5 \times S^5/\mathbb{Z}_n$, in Type IIB, or its T-dual background, in Type IIA. The latter is a Gaiotto-Maldacena geometry realised on an intersection of D4 and NS5 branes. We extend the Type IIA solutions to include D6 branes, and inter…
We initiate the classification of unitary superconformal defects in unitary superconformal field theories (SCFT) of diverse spacetime dimensions $3\leq d \leq 6$. Our method explores general constraints from the defect superconformal symmetry, unitarity, as well as consistency conditions of local bulk-defect couplings. Such features are common to all superconformal defects regardless of any Lagrangian description. In particular, modified Ward identities of conserved currents in the presence of the defect induce a distinguished set of conformal primary operators on the defect worldvolume, which…
We study the $d=11$ gravity dual $AdS_4\times N(1,1)$ of the $d=3$ $\mathcal{N}=3$ flavored Chern-Simons-matter (CSM) theory. In the dual gravity side, we analyze the M5-brane making the edge on the boundary of $AdS_4$ and derive the quantized Hall conductivity of the dual gauge theory. In the gauge theory side, this M5-brane intersects the gauge theory at the codimension-one defect.
We summarize the results and the ideas in 1707.06152, 1707.06154, 1807.07768 where new warped $\mathrm{AdS}_3$ backgrounds are derived in massive IIA string theory by uplifting exact solutions in $\mathcal N=1$, $d=7$ and $\mathcal N=(1,1)$, $d=6$ gauged supergravities. These solutions are respectively asymptotically $\mathrm{AdS}_7$ and $\mathrm{AdS}_6$ and they are related to the D2-D4-NS5-D6-D8 brane intersection. We provide a particular supergravity solution in 10d describing this bound state and we discuss the relations between its near-horizon geometry and the uplifts from 6d and 7d. The…
We construct a new family of $\text{AdS}_2\times S^2\times S^2$ solutions to Type IIA supergravity with 4 supercharges acting with non-Abelian T-duality on the recent class constructed in arXiv:2107.12277. We focus on a particular solution in this class asymptoting locally to an $\text{AdS}_5$ Gaiotto-Maldacena geometry. This solution allows for a line defect interpretation within the 4d $\mathcal{N}=2$ SCFT dual to this geometry, that we study in detail. We show that the defect branes, consisting on a non-trivial intersection of D2-D4-NS5-F1 branes, can be interpreted as baryon vertices withi…
The operator product expansion (OPE) in 4d (super)conformal field theory is of broad interest, for both formal and phenomenological applications. In this paper, we use conformal perturbation theory to study the OPE of nearly-free fields coupled to SCFTs. Under fairly general assumptions, we show that the OPE of a chiral operator of dimension $Δ= 1+ε$ with its complex conjugate always contains an operator of dimension less than $2 Δ$. Our bounds apply to Banks-Zaks fixed points and their generalizations, as we illustrate using several examples.
Non-invertible defects from the Conway SCFT to K3 sigma models II: duality and Fibonacci defects
II. Institut für Theoretische Physik, Universität Hamburg（汉堡大学第二理论物理研究所）；Istituto Nazionale di Ricerca Metrologica（意大利国家计量研究所）；Northeastern University（东北大学）；Dipartimento di Fisica e Astronomia, Università di Padova（帕多瓦大学物理与天文系）；INFN, sez. di Padova（意大利国家核物理研究所帕多瓦分部）
The numerical conformal bootstrap is used to study mixed correlators in $\mathcal{N}=1$ superconformal field theories (SCFTs) in $d=4$ spacetime dimensions. Systems of four-point functions involving scalar chiral and real operators are analyzed, including the case where the scalar real operator is the zero component of a global conserved current multiplet. New results on superconformal blocks as well as universal constraints on the space of 4D $\mathcal{N}=1$ SCFTs with chiral operators are presented. At the level of precision used, the conditions under which the putative "minimal" 4D $\mathca…
We study a rich set of four-dimensional superconformal field theories (SCFTs) with both central charges identical: $a = c$. These are constructed via the diagonal $\mathcal{N}=2$ or $\mathcal{N}=1$ gauging of the flavor symmetry $G$ of a collection of $\mathcal{N}=2$ Argyres-Douglas theories of type $\mathcal{D}_p(G)$, with or without adjoint chiral multiplets, in arXiv:2106.12579 and arXiv:2111.12092. We compute superconformal indices of some theories where the rank of $G$ is low, performing a refined test for unitarity, and further determine the relevant and marginal operator content in deta…
The Coulomb and Higgs indices of the 3d $\mathcal{N}=4$ $U(N)$ ADHM theories can be decorated by line defect operators as the line defect correlators. We obtain exact closed-form expressions and various non-trivial algebraic relations for the correlators of the Wilson lines in the fundamental and (anti)symmetric representations by means of the Hall-Littlewood expansion, the Fermi-gas method and the residue calculation. From the large $N$ limit of the correlators we obtain the single particle gravity indices which are expected to encode the spectra of fluctuation modes on the gravity duals of l…
In this paper we compute the vacuum expectation value of the Wilson loop and its correlators with chiral primary operators in $\mathcal{N}=2, 4$ superconformal $U(N)$ gauge theories at large $N$. After localization these quantities can be computed in terms of a deformed $U(N)$ matrix model. The Wilson loops we deal with are in the fundamental and symmetric representations.
The study of exactly marginal deformations of superconformal field theories is a topic that has received considerable attention due to their rich properties. We investigate the $\mathcal{N}=2$ preserving exactly marginal operators of 3d $S$-fold SCFTs. Two families of such theories are considered: one is constructed by gauging the diagonal flavour symmetry of the $T(U(2))$ and $T(U(3))$ theories, and the other by gauging the diagonal flavour symmetry of the $T^{[2,1^2]}_{[2,1^2]}(SU(4))$ theory. In both families, it is possible to turn on a Chern--Simons level for each gauge group and to coupl…
One can derive a large class of new $\mathcal{N}=1$ SCFTs by turning on $\mathcal{N}=1$ preserving deformations for $\mathcal{N}=2$ Argyres-Dougals theories. In this work, we use $\mathcal{N}=2$ superconformal indices to get indices of $\mathcal{N}=1$ SCFTs, then use these indices to derive chiral rings of $\mathcal{N}=1$ SCFTs. For a large class of $\mathcal{N}=2$ theories, we find that the IR theory contains only free chirals if we deform the parent $\mathcal{N}=2$ theory using the Coulomb branch operator with smallest scaling dimension. Our results provide interesting lessons on studies of …
The content of the OPE of two 1/2 BPS operators in N=4 SCFT$_4$ is given by their superspace three-point functions with a third, a priori long operator. For certain 1/2 BPS short superfields these three-point functions are uniquely determined by superconformal invariance. We focus on the cases where the leading ($θ=0$) components lie in the tensor products $[0,m,0]\otimes[0,n,0]$ and $[m,0,0]\otimes[0,0,n]$ of SU(4). We show that the shortness conditions at the first two points imply selection rules for the supermultiplet at the third point. Our main result is the identification of all possibl…
We study the conformal bootstrap constraints for 4D $\mathcal{N}=1$ superconformal field theories containing a chiral operator $ϕ$ and the chiral ring relation $ϕ^2=0$. Hints for a minimal interacting SCFT in this class have appeared in previous numerical bootstrap studies. We perform a detailed study of the properties of this conjectured theory, establishing that the corresponding solution to the bootstrap constraints contains a $\text{U}(1)_R$ current multiplet and estimating the central charge and low-lying operator spectrum of this theory.
In this work we construct the crossing symmetry equations for mixed correlators of two long and two BPS operators in 4D $\mathcal{N}=1$ SCFTs. The analysis presented here illustrates how our general group theoretic approach to long superblocks and tensor structures of superconformal algebras can be applied to give explicit ready-to-use expressions. In the case at hand, we obtain a system of four crossing symmetry equations for the relevant OPE coefficients. One of these four equations coincides with the equation found and analysed by Li, Meltzer and Stergiou by restricting to the superprimary …
Applying the Casimir operator to four-point functions in CFTs allows us to find the conformal blocks for any external operators. In this work, we initiate the program to find the superconformal blocks, using the super Casimir operator, for $4D$ ${\mathcal N}=1$ SCFTs. We begin by finding the most general four-point function with zero $U(1)_R$-charge, including all the possible nilpotent structures allowed by the superconformal algebra. We then study particular cases where some of the operators satisfy shortening conditions. Finally, we obtain the super Casimir equations for four point-function…
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, …
We show how to treat the superconformal algebras with eight Poincaré supercharges in a unified manner for spacetime dimension $2 < d\leq 6$. This formalism is ideally suited for analyzing the quadratic Casimir operator of the superconformal algebra and its use in deriving superconformal blocks. We illustrate this by an explicit construction of the superconformal blocks, for any value of the spacetime dimension, for external protected scalar operators which are the lowest component of flavor current multiplets.
We consider four-dimensional N=2 superconformal field theories based on ADE quiver diagrams. We use the procedure of hep-th/0206079 and compute the exact anomalous dimensions of operators with large U(1)_R charge to all orders in perturbation in the planar limit. The results are in agreement with the string computation in the dual pp-wave backgrounds.
We discuss a candidate for a minimal interacting 4-dimensional N=1 superconformal field theory (SCFT). The model contains a chiral primary operator u satisfying the chiral ring relation u^2=0, and its scaling dimension is Δ(u)=1.5. The model is derived by turning on a N=1 preserving deformation of N=2 A2 Argyres-Douglas theory. The central charges are given by (a,c)=(263/768, 271/768) ~ (0.342,0.353). There is no moduli space of vacua, no flavor symmetry, and the chiral ring is finite.
We analyze the relation between the large N limit of 6 dimensional superconformal field theories with eight supercharges and M theory on orbifolds of AdS_7xS^4. We use the known spectrum of Kaluza-Klein harmonics of supergravity on AdS_7xS^4 and we take their orbifold projection to get information about the chiral primary operators of 6 dimensional SCFT which is realized on the worldvolume of M5 brane sitting at the orbifold singularities.
We investigate the one-dimensional defect SCFT defined on the $1/2$ BPS Wilson line/loop in ABJ(M) theory. We show that the supermatrix structure of the defect imposes a covariant supermatrix representation of the supercharges. Exploiting this covariant formulation, we prove the existence of a long multiplet whose highest weight state is a constant supermatrix operator. At weak coupling, we study this operator in perturbation theory and confirm that it acquires a non-trivial anomalous dimension. At strong coupling, we conjecture that this operator is dual to the lowest bound state of fluctuati…
We study the short multiplet spectrum in 4d $\mathcal{N}=2$ superconformal theories of low rank using the full superconformal indices and the selection rules from the superconformal representation theory. We find a universal expression for the leading terms for the superconformal index of rank one $H_0, H_1, H_2, D_4, E_6, E_7$ theories. From this result, we argue that certain short multiplets appear in the operator product expansions involving stress-tensor, conserved current, and Coulomb branch operator vanish. We also apply the same procedure to 5d superconformal theories and find that $E_1…
We compute the superconformal partial waves of the four-point correlator $\langle JJJJ\rangle$, in which the external operator $J$ is the superconformal primary of the $4D$ $\mathcal{N}=2$ stress-tensor multiplet $\mathcal{J}$. We develop the superembedding formalism for the superconformal field theories (SCFTs) with extended supersymmetry. In $\mathcal{N}=2$ SCFTs, the three-point functions $\langle \mathcal{J}\mathcal{J}\mathcal{O}\rangle$ with general multiplet $\mathcal{O}$ contain two independent nilpotent superconformal invariants and new superconformal tensor structures, which can be ni…
Turning on N=2 supersymmetry-preserving relevant operators in a 4-dimensional N=2 superconformal field theory (SCFT) corresponds to a complex deformation compatible with the rigid special Kahler geometry encoded in the low energy effective action. Field theoretic consistency arguments indicate that there should be many distinct such relevant deformations of each SCFT fixed point. Some new supersymmetry-preserving complex deformations are constructed of isolated rank 1 SCFTs. We also make predictions for the dimensions of certain Higgs branches for some rank 1 SCFTs.
We extend SO(4,2) covariant lightcone embedding methods of four-dimensional CFTs to N=1 superconformal field theory (SCFT). Manifest superconformal SU(2,2|1) invariance is achieved by realizing 4D superconformal space as a surface embedded in the projective superspace spanned by certain complex chiral supermatrices. Because SU(2,2|1) acts linearly on the ambient space, the constraints on correlators implied by superconformal Ward identities are automatically solved in this formalism. Applications include new, compact expressions for correlation functions containing one anti-chiral superfield a…
We construct an index for BPS operators supported on a ray in five dimensional superconformal field theories with exceptional global symmetries. We compute the $E_n$ representations (for $n=2,\dots,7$) of operators of low spin, thus verifying that while the expression for the index is only SO$(2n-2)\times$U(1) invariant, the index itself exhibits the full $E_n$ symmetry (at least up to the order we expanded). The ray operators we studied in 5d can be viewed as generalizations of operators constructed in a Yang-Mills theory with fundamental matter by attaching an open Wilson line to a quark. Fo…
We use the superconformal bootstrap to derive exact relations between OPE coefficients in three-dimensional superconformal field theories with ${\cal N} \geq 4$ supersymmetry. These relations follow from a consistent truncation of the crossing symmetry equations that is associated with the cohomology of a certain supercharge. In ${\cal N} = 4$ SCFTs, the non-trivial cohomology classes are in one-to-one correspondence with certain half-BPS operators, provided that these operators are restricted to lie on a line. The relations we find are powerful enough to allow us to determine an infinite numb…
We study $3d$ SCFTs in the superspace formalism and discuss superfields and on-shell higher spin current multiplets in free $3d$ SCFTs with $\mathcal{N}= 1,2,3,4$ and $6$ superconformal symmetry. For $\mathcal{N}=1$ 3d SCFTs we determine the superconformal invariants in superspace needed for constructing 3-point functions of higher spin operators, find the non-linear relations between the invariants and consequently write down all the independent invariant structures, both parity even and odd, for various 3-point functions of higher spin operators.
We compute the superconformal characters of various short multiplets in 4d N=2 superconformal algebra, from which selection rules for operator products are obtained. Combining with the superconformal index, we show that a particular short multiplet appearing in the n-fold product of stress-tensor multiplet is absent in the $(A_1, A_{2n})$ Argyres-Douglas (AD) theory. This implies that the operator product expansion (OPE) coefficients involving this multiplet vanish whenever the central charge $c$ is identical to that of the AD theory. Similarly, by considering the n-th power of the current mul…
Defect entanglement entropy for superconformal RG Interfaces
University of Oviedo（奥维耶多大学）；Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA)（阿斯图里亚斯空间科学与技术大学研究所）；University of Crete（克里特大学）
