We present three infinite families of supersymmetric Type IIB backgrounds with AdS$_4$, AdS$_3$ and AdS$_2$ factors, dual to SCFTs in $3$, $2$ and $1$ space-time dimensions respectively. These field theories emerge at low energies, after a twisted compactification of a family of five dimensional ${\cal N}=2$ SCFTs on hyperbolic spaces. The holographic flows across dimensions are explicitly computed. We also discuss a family of SUSY breaking backgrounds, dual to a QCD-like quiver with massive (bi)fundamental matter. Some field theoretical observables are computed for these theories at the fixed points and along the flow.
We study the compactification of 5d SCFTs to 4d on a circle with a twist in a discrete global symmetry element of these SCFTs. We present evidence that this leads to various 4d N=2 isolated SCFTs. These include many known theories as well as seemingly new ones. The known theories include the recently discovered rank 1 SU(4) SCFT and its mass deformations. One application of the new SCFTs is in the dual descriptions of the 4d gauge theory SU(N)+1S+(N-2)F. Also interesting is the appearance of a theory with rank 1 and $F_4$ global symmetry.
We study the $3d$ $\mathcal{N}=2$ theories resulting from the compactification of a family of $5d$ SCFTs on a torus with flux in the global symmetry. The family of $5d$ SCFTs used in the analysis is the one that UV completes the $5d$ $SU(N+1)$ gauge theories with Chern--Simons level $k$ and $N_f$ fundamental hypermultiplets, generalizing the previous investigation of the torus compactifications of the rank 1 Seiberg $E_{N_f+1}$ SCFT (which is the $N=1$ member of the family). This construction systematically yields three-dimensional theories presenting highly non-trivial non-perturbative phenom…
We consider compactifications of very Higgsable 6d $N =(1,0)$ SCFTs on $T^2$ with non-trivial Stiefel-Whitney classes (or equivalently 't Hooft magnetic fluxes) introduced for their flavor symmetry groups. These systems can also be studied as twisted $S^1$ compactifications of the corresponding 5d theories. We deduce various properties of the resulting 4d $N=2$ SCFTs by combining these two viewpoints. In particular, we find that all 4d rank-1 $N =2$ SCFTs with a dimension-6 Coulomb branch operator with flavor symmetry $\mathfrak{e}_8$, $\mathfrak{usp}(10)$, $\mathfrak{su}(4) $ and $\mathfrak{s…
We study 6d superconformal field theories (SCFTs) compactified on a circle with arbitrary twists. The theories obtained after compactification, often referred to as 5d Kaluza-Klein (KK) theories, can be viewed as starting points for RG flows to 5d SCFTs. According to a conjecture, all 5d SCFTs can be obtained in this fashion. We compute the Coulomb branch prepotential for all 5d KK theories obtainable in this manner and associate to these theories a smooth local genus one fibered Calabi-Yau threefold in which is encoded information about all possible RG flows to 5d SCFTs. These Calabi-Yau thre…
A broad class of holographic duals for 4d $\mathcal N=2$ SCFTs is based on the general half-BPS $AdS_5$ solutions to M-theory constructed by Lin, Lunin and Maldacena, and their Kaluza-Klein reductions to Type IIA. We derive the equations governing spin 2 fluctuations around these solutions. The resulting partial differential equations admit families of universal solutions which are constructed entirely out of the generic data characterizing the background. This implies the existence of families of spin 2 operators in the dual SCFTs, in short superconformal multiplets which we identify.
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…
The center-flavor symmetry of a gauge theory specifies the global form of consistent gauge and flavor bundle background field configurations. For 6d gauge theories which arise from a tensor branch deformation of a superconformal field theory (SCFT), we determine the global structure of such background field configurations, including possible continuous Abelian symmetry and R-symmetry bundles. Proceeding to the conformal fixed point, this provides a prescription for reading off the global form of the continuous factors of the zero-form symmetry, including possible non-trivial mixing between fla…
Compactifying 6d superconformal field theories (SCFTs) to 4d $\mathcal{N}=1$ theories on two-punctured spheres (tubes) and tori with flux is realized using duality domain walls in 5d $\mathcal{N}=1$ Kaluza-Klein (KK) theories, which are usually denoted by $flux$ $domain$ $walls$. We revisit this construction and study it in detail from the 5d perspective, specifically rephrasing it using the box graph description of the extended Coulomb branch phases of 5d theories. This perspective could be helpful in understanding how to equivalently realize the 4d $\mathcal{N}=1$ models from geometric engin…
We study holographic renormalization group flows from four-dimensional $\mathcal{N}=2$ SCFTs to either $\mathcal{N}=2$ or $\mathcal{N}=1$ SCFTs. Our approach is based on the framework of five-dimensional half-maximal supergravity with general gauging, which we use to study domain wall solutions interpolating between different supersymmetric AdS$_5$ vacua. We show that a holographic RG flow connecting two $\mathcal{N}=2$ SCFTs is only possible if the flavor symmetry of the UV theory admits an ${\rm SO}(3)$ subgroup. In this case the ratio of the IR and UV central charges satisfies a universal r…
We study entanglement entropy and the free energy in recently constructed holographic duals for 5d SCFTs in type IIB supergravity. The solutions exhibit mild singularities, which could potentially complicate holographic applications. We use the relation of the entanglement entropy for a spherical entangling surface to the free energy of the field theory on the five sphere as a well-motivated benchmark to assess how problematic the singularities are. The holographic supergravity computations give well-defined results for both quantities and they satisfy the expected relations. This supports the…
We study general properties of the mapping between 5$d$ and 4$d$ superconformal field theories (SCFTs) under both twisted circle compactification and tuning of local relevant deformation and CB moduli. After elucidating in generality when a 5$d$ SCFT reduces to a 4$d$ one, we identify nearly all $\mathcal{N}=1$ 5$d$ SCFT parents of rank-2 4$d$ $\mathcal{N}=2$ SCFTs. We then use this result to map out the mass deformation trajectories among the rank-2 theories in 4$d$. This can be done by first understanding the mass deformations of the 5$d$ $\mathcal{N}=1$ SCFTs and then map them to 4$d$. The …
We study supersymmetric $AdS_4\times Σ_2$ and $AdS_3\times Σ_3$ solutions in half-maximal gauged supergravity in six dimensions with $SU(2)_R\times SU(2)$ gauge group. The gauged supergravity is obtained by coupling three vector multiplets to the pure $F(4)$ gauged supergravity. The $SU(2)_R$ R-symmetry together with the $SO(3)\sim SU(2)$ symmetry of the vector multiplets are gauged. The resulting gauged supergravity admits supersymmetric $AdS_6$ critical points with $SO(4)\sim SU(2)\times SU(2)$ and $SO(3)\sim SU(2)_{\textrm{diag}}$ symmetries. The former corresponds to five-dimensional $N=2$…
We employ a non-compact gauging of four-dimensional maximal supergravity to construct a two-parameter family of AdS$_4$ J-fold solutions preserving $\mathcal{N}=2$ supersymmetry. All solutions preserve $\mathfrak{u}(1)\times\mathfrak{u}(1)$ global symmetry and in special limits we recover the previously known $\mathfrak{su}(2)\times\mathfrak{u}(1)$ invariant $\mathcal{N}=2$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$ invariant $\mathcal{N}=4$ J-fold solutions. This family of AdS$_4$ backgrounds can be uplifted to type IIB string theory and is holographically dual to the conformal manifold of …
We show that the string worldsheet theory of Gaiotto-Maldacena holographic duals to N=2 superconformal field theories generically fails to be classically integrable. We demonstrate numerically that the dynamics of a winding string configuration possesses a non-vanishing Lyapunov exponent. Furthermore an analytic study of the Normal Variational Equation fails to yield a Liouvillian solution. An exception to the generic non-integrability of such backgrounds is provided by the non-Abelian T-dual of $AdS_5 \times S^5$. Here by virtue of the canonical transformation nature of the T-duality classica…
Building on recent progress in the study of compactifications of $6d$ $(1,0)$ superconformal field theories (SCFTs) on Riemann surfaces to $4d$ $\mathcal{N}=1$ theories, we initiate a systematic study of compactifications of $5d$ $\mathcal{N}=1$ SCFTs on Riemann surfaces to $3d$ $\mathcal{N}=2$ theories. Specifically, we consider the compactification of the so-called rank 1 Seiberg $E_{N_f+1}$ SCFTs on tori and tubes with flux in their global symmetry, and put the resulting $3d$ theories to various consistency checks. These include matching the (usually enhanced) IR symmetry of the $3d$ theori…
We study the holographic description of ${\cal N}=2$ Super Conformal Field Theories in four dimensions first given by Gaiotto and Maldacena. We present new expressions that holographically calculate characteristic numbers of the CFT and associated Hanany-Witten set-ups, or more dynamical observables, like the central charge. A number of examples of varying complexity are studied and some proofs for these new expressions are presented. We repeat this treatment for the case of the marginally deformed Gaiotto-Maldacena theories, presenting an infinite family of new solutions and compute some of i…
Compactifications of 6D superconformal field theories (SCFTs) on four-manifolds generate a large class of novel 2d quantum field theories. We consider in detail the case of the rank one simple non-Higgsable cluster 6D SCFTs. On the tensor branch of these theories, the gauge group is simple and there are no matter fields. For compactifications on suitably chosen Kahler surfaces, we present evidence that this provides a method to realize 2d SCFTs with N = (0,2) supersymmetry. In particular, we find that reduction on the tensor branch of the 6D SCFT yields a description of the same 2d fixed point…
We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi--Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d $\mathcal{N}=(1,0)$ SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi--Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, thes…
We study the twisted compactifications of five-dimensional Seiberg SCFTs, with $SU_\mathcal{M}(2)\times E_{N_f+1}$ flavor symmetry, on a generic Riemann surface that preserves four supercharges. The five-dimensional SCFTs are obtained from the decoupling limit of $N$ D4-branes probing a geometry of $N_f<8$ D8-branes and an O8-plane. In addition to the R-symmetry, we can also twist the flavor symmetry by turning on background flux on the Riemann surface. In particular, in the string theory construction of the five-dimensional SCFTs, the background flux for the $SU_\mathcal{M}(2)$ has a geometri…
Using six-dimensional Euclidean $F(4)$ gauged supergravity we construct a holographic renormalization group flow for a CFT on $S^5$. Numerical solutions to the BPS equations are obtained and the free energy of the theory on $S^5$ is determined holographically by calculation of the renormalized on-shell supergravity action. In the process, we deal with subtle issues such as holographic renormalization and addition of finite counterterms. We then propose a candidate field theory dual to these solutions. This tentative dual is a supersymmetry-preserving deformation of the strongly-coupled non-Lag…
We study compactifications of an infinite family of four-dimensional $\mathcal{N}=1$ SCFTs on a Riemann surface in the presence of arbitrary background fluxes for global symmetries. The four-dimensional parent theories have holographic Sasaki-Einstein duals in type IIB string theory. We compute central charges and R-charges of baryonic operators in the resulting two-dimensional $\mathcal{N}=(0,2)$ theories in three distinct ways: from the field theory side utilizing c-extremization, its recently discovered geometric dual formulation, and holographically using new AdS$_3$ duals of two-dimension…
We investigate the holographic Rényi entropy for two-dimensional $\mathcal N=(2,2)$ superconformal field theory (SCFT), which is dual to $\mathcal N=2$ supergravity in AdS$_3$ background. In SCFT we have the stress tensor, current, and their supersymmetric partners, and in supergravity we have the graviton, vector field, and two gravitinos. We get the Rényi mutual information of two short intervals on complex plane in expansion by the cross ratio $x$ to order $x^4$, and Rényi entropy of one interval on torus in expansion by $q=\exp(-2πβ/L)$, with $β$ being the inverse temperature and $L$ being…
According to a conjecture, all 5d SCFTs should be obtainable by rank-preserving RG flows of 6d SCFTs compactified on a circle possibly twisted by a background for the discrete global symmetries around the circle. For a 6d SCFT admitting an F-theory construction, its untwisted compactification admits a dual M-theory description in terms of a "parent" Calabi-Yau threefold which captures the Coulomb branch of the compactified 6d SCFT. The RG flows to 5d SCFTs can then be identified with a sequence of flop transitions and blowdowns of the parent Calabi-Yau leading to "descendant" Calabi-Yau threef…
We probe type IIA geometries with \emph{folded} semiclassical strings those encode the strong coupling dynamics associated with long operators in a class of SCFTs pertaining to spacetimes whose dimension vary from $6d$ down to $2d$. We particularly focus on folded string configurations those are extended through stack of flavor $ Dp $ ($ p=6,8 $) branes localized along the internal manifold. Considering some specific examples within the realm of $ \mathcal{N}=2 $ linear quivers, we show that the associated string spectrum exhibits a pole as the string approaches flavor D6 branes. We identify t…
We study four-dimensional $N=4$ gauged supergravity coupled to six vector multiplets with semisimple gauge groups $SO(4)\times SO(4)$, $SO(3,1)\times SO(3,1)$ and $SO(4)\times SO(3,1)$. All of these gauge groups are dyonically embedded in the global symmetry group $SO(6,6)$ via its maximal subgroup $SO(3,3)\times SO(3,3)$. For $SO(4)\times SO(4)$ gauge group, there are four $N=4$ supersymmetric $AdS_4$ vacua with $SO(4)\times SO(4)$, $SO(4)\times SO(3)$, $SO(3)\times SO(4)$ and $SO(3)\times SO(3)$ symmetries, respectively. These $AdS_4$ vacua correspond to $N=4$ SCFTs in three dimensions with …
Following a recent proposal, we delineate a general procedure to classify 5d SCFTs via compactifications of 6d SCFTs on a circle (possibly with a twist by a discrete global symmetry). The path from 6d SCFTs to 5d SCFTs can be divided into two steps. The first step involves computing the Coulomb branch data of the 5d KK theory obtained by compactifying a 6d SCFT on a circle of finite radius. The second step involves computing the limit of the KK theory when the inverse radius along with some other mass parameters is sent to infinity. Under this RG flow, the KK theory reduces to a 5d SCFT. We il…
We study $5d$ gauge theories that go in the UV to $6d$ $\mathcal{N}$$=(1,0)$ SCFT. We focus on these theories that can be engineered in string theory by brane webs. Given a theory in this class, we propose a method to determine the $6d$ SCFT it goes to. We also discuss the implication of this to the compactification of the resulting $6d$ SCFT on a torus to $4d$. We test and demonstrate this method with a variety of examples.
We study compactifications on Riemann surfaces with punctures of N=(1,0) 6d SCFTs with a one dimensional tensor branch and no continuous global symmetries. The effective description of such models on the tensor branch is in terms of pure gauge theories with decoupled tensor. For generic Riemann surfaces, the resulting theories in four dimensions are expected to have N=1 supersymmetry. We compute the anomalies expected from the resulting 4d theories by integrating the anomaly polynomial of the 6d theory on the Riemann surface. For the cases with 6d gauge models with gauge groups SU(3) and SO(8)…
We study marginal deformations of a class of three-dimensional $\mathcal{N}=2$ SCFTs that admit a holographic dual description in massive type IIA supergravity. We compute the dimension of the conformal manifold in these SCFTs and identify a special submanifold with enhanced flavor symmetry. We show how to apply the TsT transformation of Lunin-Maldacena to construct explicit supersymmetric AdS$_4$ solutions of massive type IIA supergravity with non-trivial internal fluxes that are the holographic dual of these marginal deformations. Finally, we also briefly comment on a class of RG flows betwe…
Using the F-theory realization, we identify a subclass of 6d (1,0) SCFTs whose compactification on a Riemann surface leads to N = 1 4d SCFTs where the moduli space of the Riemann surface is part of the moduli space of the theory. In particular we argue that for a special case of these theories (dual to M5 branes probing ADE singularities), we obtain 4d N = 1 theories whose space of marginal deformations is given by the moduli space of flat ADE connections on a Riemann surface.
We consider the torus compactifications with flux of a class of $6d$ $(1,0)$ SCFTs that can be engineered as the low-energy theories on M$5$-branes near an M$9$-plane on a $C^2/Z_2$ singularity. Specifically, we concentrate on the two SCFTs where the $Z_2$ orbifold action acts non-trivially on the $E_8$ global symmetry. We analyze this problem by compactifying to $5d$, where we can exploit the relation to $5d$ duality domain walls. By a suitable guess of the domain wall theories, the resulting $4d$ theories can be conjectured. These can then be tested by comparing their properties, notably ano…
We find an explicit supergravity background dual to the $Ω$-deformation of a four-dimensional $\mathcal{N}=2$ SCFT on $\mathbb{R}^4$. The solution can be constructed in the five-dimensional ${\cal N}=4^+$ gauged supergravity and has a nontrivial self-dual 2-form. When uplifted to type IIB supergravity the background is a deformation of AdS$_5\times S^5$ which preserves 16 supercharges. We also discuss generalizations of this solution corresponding to turning on a vacuum expectation value for a scalar operator in the dual SCFT.
We formulate geometric conditions necessary for engineering 5d superconformal field theories (SCFTs) via M-theory compactification on a local Calabi-Yau 3-fold. Extending the classification of the rank 1 cases, which are realized geometrically as shrinking del Pezzo surfaces embedded in a 3-fold, we propose an exhaustive classification of local 3-folds engineering rank 2 SCFTs in 5d. This systematic classification confirms that all rank 2 SCFTs predicted using gauge theoretic arguments can be realized as consistent theories, with the exception of one family which is shown to be non-perturbativ…
We present new $AdS_5$ solutions in 11d supergravity and identify them as the gravity duals of a class of 4d $\mathcal N = 2$ SCFTs of Argyres-Douglas type, engineered by a stack of M5-branes wrapped on a sphere with one irregular puncture and one regular puncture. The gravity solutions feature an internal M5-brane source, which is dual to the irregular puncture. Thanks to a novel Stückelberg mechanism involving an axion originating from the expansion of the M-theory 3-form, one of the isometry generators on the gravity side is not mapped to a continuous symmetry on the SCFT side.
All known interacting 6D superconformal field theories (SCFTs) have a tensor branch which includes anti-chiral two-forms and a corresponding lattice of string charges. Automorphisms of this lattice preserve the Dirac pairing and specify discrete global and gauge symmetries of the 6D theory. In this paper we compute this automorphism group for 6D SCFTs. This discrete data determines the geometric structure of the moduli space of vacua. Upon compactification, these automorphisms generate Seiberg-like dualities, as well as additional theories in discrete quotients by the 6D global symmetries. Whe…
We study holographic RG flow solutions within four-dimensional $N=4$ gauged supergravity obtained from type IIA and IIB string theories compactified on $T^6/\mathbb{Z}_2\times \mathbb{Z}_2$ orbifold with gauge, geometric and non-geometric fluxes. In type IIB non-geometric compactifications, the resulting gauged supergravity has $ISO(3)\times ISO(3)$ gauge group and admits an $N=4$ $AdS_4$ vacuum dual to an $N=4$ superconformal field theory (SCFT) in three dimensions. We study various supersymmetric RG flows from this $N=4$ SCFT to $N=4$ and $N=1$ non-conformal field theories in the IR. The flo…
In this paper we investigate the holographic Rényi entropy in $\mathcal N=1$ supergravity (SUGRA) in AdS$_3$ spacetime, which is dual to the two-dimensional $\mathcal N=(1,1)$ superconformal field theory (SCFT). We consider both cases of two short intervals on a line with zero temperature and one interval on a circle with low temperature. In SUGRA side we consider contributions of both graviton and gravitino, and in SCFT side we consider contributions of both stress tensor $T$, $\bar T$ and their superpartners $G$, $\bar G$. We find matches between SUGRA and SCFT results.
We develop a technique for counting the number of stress tensor multiplets in a 4D $\mathcal{N}=2$ SCFT. This provides a simple diagnostic for when an isolated (non-Lagrangian) SCFT is a product of two (or more) such theories. In class-S, the basic building blocks are the isolated SCFTs arising from the compactification of a 6D (2,0) theory on a 3-punctured sphere ("fixture"). We apply our technique to determine when a fixture is a product SCFT. The answer is that this phenomenon is surprisingly rare. In the low-rank $A_{N-1}$, $D_N$ theories and the $E_6$ theory studied by the first author an…
We find an infinite family of $4D$ $\mathcal{N}=2$ interacting superconformal field theories which enter the description of the strong-coupling limit of $SU(2N+1)$ gauge theories with hypermultiplets in the $\wedge^2(\square)+\text{Sym}^2(\square)$ . These theories arise from the compactification of the $6D$ $(2,0)$ theory of type $A_{2N}$ on a sphere with two full twisted punctures and one minimal untwisted puncture. For $N=1$, this theory is the "new" rank-1 SCFT with $Δ(u)=3$ of Argyres and Wittig. Using the superconformal index, we finally pin down the properties of this theory.
We prove that there does not exist any weak coupling limit in the space of superconformal field theories in five and six dimensions, based on an analysis of the representation theory of the corresponding superconformal algebras. Holographically, this implies that superstring theories on AdS$_6$ and AdS$_7$ do not admit tensionless limits. Finally, we discuss the implications of our result on the existence of an action for coincident M5-branes.
We determine the supersymmetric AdS_7 backgrounds of seven-dimensional half-maximal gauged supergravities and show that they do not admit any deformations that preserve all 16 supercharges. We compare this result to the conformal manifold of the holographically dual (1,0) superconformal field theories and show that accordingly its representation theory implies that no supersymmetric marginal operators exist.
