We apply c-extremization, whose proof we review in full detail, to study two-dimensional N=(0,2) superconformal field theories arising from the low-energy dynamics of D3-branes wrapped on Riemann surfaces and M5-branes wrapped on four-manifolds. We compute the exact central charges of these theories using anomalies and c-extremization. In all cases we also construct AdS_3 supergravity solutions of type IIB and eleven-dimensional supergravity, which are holographic duals to the field theories at large N, and exactly reproduce the central charges computed via c-extremization.
We study the holographic dual to $c$-extremization for 2d $(0,2)$ superconformal field theories (SCFTs) that have an AdS$_3$ dual realized in Type IIB with varying axio-dilaton, i.e. F-theory. M/F-duality implies that such AdS$_3$ solutions can be mapped to AdS$_2$ solutions in M-theory, which are holographically dual to superconformal quantum mechanics (SCQM), obtained by dimensional reduction of the 2d SCFTs. We analyze the corresponding map between holographic $c$-extremization in F-theory and $\mathcal{I}$-extremization in M-theory, where in general the latter receives corrections relative…
Recently a general principle, called c-extremization, which determines the exact R-symmetry of two-dimensional SCFTs with N = (0,2) supersymmetry was identified. In this note we show that the supergravity dual corresponds to the extremization of the T-tensor of N = 2 gauged supergravity in three-dimensions. To support this claim, we demonstrate that the expected central charge of SCFTs arising from twisted compactifications of four-dimensional N = 4 SYM on Riemann surfaces, whose gravity dual is a reduction of five-dimensional U(1)^3 gauged supergravity, is recovered in the three-dimensional f…
We consider 4d N=1 SCFTs, topologically twisted on compact constant curvature Riemann surfaces, giving rise to 2d N=(0,2) SCFTs. The exact R-current of these 2d SCFT extremizes the central charge c_{2d}, similarly to the 4d picture, where the exact R-current maximizes the central charge a_{4d}. There are global currents that do not mix with the R-current in 4d but their mixing becomes non trivial in 2d. In this paper we study the holographic dual of this process by analyzing a 5d N=2 truncation of T^{1,1} with one Betti vector multiplet, dual to the baryonic current on the CFT side. The hologr…
We analyze topological mass terms of BF type arising in supersymmetric M-theory compactifications to $AdS_5$. These describe spontaneously broken higher-form gauge symmetries in the bulk. Different choices of boundary conditions for the BF terms yield dual field theories with distinct global discrete symmetries. We discuss in detail these symmetries and their 't Hooft anomalies for 4d $\mathcal N = 1$ SCFTs arising from M5-branes wrapped on a Riemann surface without punctures, including theories from M5-branes at a $\mathbb Z_2$ orbifold singularity. The anomaly polynomial is computed via infl…
We study CFT data of 3-dimensional superconformal field theories (SCFTs) arising from wrapped two M5-branes on closed hyperbolic 3-manifolds. Via so-called 3d/3d correspondence, central charges of these SCFTs are related to a $SL(2)$ Chern-Simons (CS) invariant on the 3-manifolds. We give a rigorous definition of the invariant in terms of resurgence theory and a state-integral model for the complex CS theory. We numerically evaluate the central charges for several closed 3-manifolds with small hyperbolic volume. The computation suggests that the wrapped M5-brane systems give infinitely many di…
We show that there is a family of two-dimensional $(0,2)$ SCFTs associated with twisted compactifications of the four-dimensional $\mathcal{N}=1$ Leigh-Strassler fixed point on a closed hyperbolic Riemann surface. We calculate the central charges for this class of theories using anomalies and $c$-extremization. In a suitable truncation of the five-dimensional maximal supergravity, we construct supersymmetric $AdS_3$ solutions that are holographic duals of those two-dimensional $(0,2)$ SCFTs. We also exhibit supersymmetric domain wall solutions that are holographically dual to the RG flows betw…
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 find a large class of two-dimensional $\mathcal{N}=(0,2)$ SCFTs obtained by compactifying four-dimensional $\mathcal{N}=1$ quiver gauge theories on a Riemann surface. We study these theories using anomalies and $c$-extremization. The gravitational duals to these fixed points are new AdS$_3$ solutions of IIB supergravity which we exhibit explicitly. Along the way we uncover a universal relation between the conformal anomaly coefficients of four-dimensional and two-dimensional SCFTs connected by an RG flow across dimensions. We also observe an interesting novel phenomenon in which the superco…
The three dimensional exact R symmetry of N=2 SCFTs extremizes the partition function localized on a three sphere. Here we verify this statement at weak coupling. We give a detailed analysis for two classes of models. The first one is an SU(N)_k gauge theory at large k with both fundamental and adjoint matter fields, while the second is a flavored version of the ABJ theory, where the CS levels are large but they do not necessarily sum up to zero. We study in both cases superpotential deformations and compute the R charges at different fixed points. When these fixed points are connected by an R…
We consider supersymmetric AdS$_3 \times Y_7$ and AdS$_2 \times Y_9$ solutions of type IIB and $D=11$ supergravity, respectively, that are holographically dual to SCFTs with $(0,2)$ supersymmetry in two dimensions and $\mathcal{N}=2$ supersymmetry in one dimension. The geometry of $Y_{2n+1}$, which can be defined for $n\ge 3$, shares many similarities with Sasaki-Einstein geometry, including the existence of a canonical R-symmetry Killing vector, but there are also some crucial differences. We show that the R-symmetry Killing vector may be determined by extremizing a function that depends only…
Within the framework of four dimensional conformal supergravity we consider $\mathcal{N}=1,2,3,4$ supersymmetric theories generally twisted along the abelian subgroups of the R-symmetry and possibly other global symmetry groups. Upon compactification on constant curvature Riemann surfaces with arbitrary genus we provide an extensive classification of the resulting two dimensional theories according to the amount of supersymmetry that is preserved. Exploiting the c-extremization prescription introduced in arXiv:1211.4030 we develop a general procedure to obtain the central charge for 2d $\mathc…
We construct supersymmetric $AdS_3$ solutions of $D=11$ supergravity, dual to $d=2$, $\mathcal{N}=(0,2)$ SCFTs, that are associated with M5-branes wrapping two different four-dimensional orbifolds. In one case the orbifold is a spindle fibred over another spindle, while in the other it is a spindle fibred over a Riemann surface with genus $g>1$. We show that the central charges of the $d=2$ SCFTs calculated from the gravity solutions agree with field theory computations using anomaly polynomials. The new $D=11$ solutions are obtained after constructing a new consistent Kaluza-Klein truncation …
We provide a string theory embedding for N = 1 superconformal field theories defined by bipartite graphs inscribed on a disk. We realize these theories by exploiting the close connection with related N = 2 generalized (A_(k-1), A_(n-1)) Argyres-Douglas theories. The N = 1 theory is obtained from spacetime filling D5-branes wrapped on an algebraic curve and NS5-branes wrapped on special Lagrangians of C^2 which intersect along the BPS flow lines of the corresponding N = 2 Argyres-Douglas theory. Dualities of the N = 1 field theory follow from geometric deformations of the brane configuration wh…
Coincident M2 branes at a conical singularity are related to M theory on AdS_4 x X_7 for an appropriate 7 dimensional Sasaki-Einstein manifold X_7. For X_7=Q^{1,1,1}=(SU(2) x SU(2) x SU(2))/(U(1) x U(1)) which was found sometime ago, the infrared limit of the theory on N M2 branes was constructed recently. It is the SU(N) x SU(N) x SU(N) gauge theories with three series of chiral fields A_i, i=1,2 transforming in the (N, \bar{N},1) representation, B_j, j=1,2 transforming in the (1,N, \bar{N}) representation and C_k, k=1,2 transforming in the (\bar{N},1,N) representation. From the scalar Laplac…
AdS_3 solutions dual to N = (0,2) SCFTs arise when D3-branes wrap Kahler two-cycles in manifolds with SU(4) holonomy. Here we review known AdS_3 solutions and identify the corresponding three-dimensional gauged supergravities, solutions of which uplift to type IIB supergravity. In particular, we discuss gauged supergravities dual to twisted compactifications on Riemann surfaces of both N=4 SYM and N =1 SCFTs with Sasaki-Einstein duals. We check in each case that c-extremization gives the exact central charge and R symmetry. For completeness, we also study AdS_3 solutions from intersecting D3-b…
We consider D3-brane gauge theories at an arbitrary toric Calabi-Yau 3-fold cone singularity that are then further compactified on a Riemann surface $Σ_g$, with an arbitrary partial topological twist for the global $U(1)$ symmetries. This constitutes a rich, infinite class of two-dimensional $(0,2)$ theories. Under the assumption that such a theory flows to a SCFT, we show that the supergravity formulas for the central charge and $R$-charges of BPS baryonic operators of the dual AdS$_3$ solution may be computed using only the toric data of the Calabi-Yau 3-fold and the topological twist parame…
Quantum field theories (QFT) in the presence of defects exhibit new types of anomalies which play an important role in constraining the defect dynamics and defect renormalization group (RG) flows. Here we study surface defects and their anomalies in conformal field theories (CFT) of general spacetime dimensions. When the defect is conformal, it is characterized by a conformal $b$-anomaly analogous to the $c$-anomaly of 2d CFTs. The $b$-theorem states that $b$ must monotonically decrease under defect RG flows and was proven by coupling to a spurious defect dilaton. We revisit the proof by deriv…
We construct supersymmetric $AdS_3\times Σ$ solutions of minimal gauged supergravity in $D=5$, where $Σ$ is a two-dimensional orbifold known as a spindle. Remarkably, these uplift on $S^5$, or more generally on any regular Sasaki-Einstein manifold, to smooth solutions of type IIB supergravity. The solutions are dual to $d=2$, $\mathcal{N}=(0,2)$ SCFTs and we show that the central charge for the gravity solution agrees with a field theory calculation associated with D3-branes wrapped on $Σ$. Unlike for smooth $Σ$ the superconformal R-symmetry mixes with the $U(1)$ isometry of the spindle.
Large classes of AdS$_p$ supergravity backgrounds describing the IR dynamics of $p$-branes wrapped on a Riemann surface are determined by a solution to the Liouville equation. The regular solutions of this equation lead to the well-known wrapped brane supergravity solutions associated with the constant curvature metric on a compact Riemann surface. We show that some singular solutions of the Liouville equation have a physical interpretation as explicit point-like brane sources on the Riemann surface. We uncover the details of this picture by focusing on $\mathcal{N}=1$ theories of class $\math…
We construct supersymmetric $AdS_5\times Σ$ solutions of $D=7$ gauged supergravity, where $Σ$ is a two-dimensional orbifold known as a spindle. These uplift on $S^4$ to solutions of $D=11$ supergravity which have orbifold singularites. We argue that the solutions are dual to $d=4$, $\mathcal{N}=1$ SCFTs that arise from $N$ M5-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. In contrast to the usual topological twist solutions, the superconformal R-symmetry mixes with the isometry of the spindle in the IR, and we verify this via a field theory calcula…
We study M theory on $AdS_4 \times \RP^7$ corresponding to 3 dimensional ${\cal N}=8$ superconformal field theory which is the strong coupling limit of 3 dimensional super Yang-Mills theory. For SU(N) theory, a wrapped M5 brane on $\RP^5$ can be interpreted as baryon vertex. For $SO(N)/Sp(2N)$ theory, by using the property of (co-)homology of $\RP^7$, we classify various wrapping branes and consider domain walls and the baryon vertex.
We construct supersymmetric AdS$_4\timesΣ$ solutions of $D=6$ gauged supergravity, where $Σ$ is a two-dimensional orbifold known as a spindle. These uplift to solutions of massive type IIA supergravity using a general prescription, that we describe. We argue that these solutions correspond to the near-horizon limit of a system of $N_f$ D8-branes, together with $N$ D4-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. The dual field theories are $d=3$, ${\cal N }= 2$ SCFTs that arise from a twisted compactification of the $d=5$, ${\cal N}=1$ $USp(2N)$ g…
We study M theory on $AdS_7 \times \RP^4$ corresponding to 6 dimensional SO(2N) $(0, 2)$ superconformal field theory on a circle which becomes 5 dimensional super Yang-Mills theory at low energies. For SU(N) (0,2) theory, a wrapped D4 brane on $§^4$ which is connected to a D4 brane on the boundary of $AdS_7$ by N fundamental strings can be interpreted as baryon vertex. For SO(2N) (0,2) theory, by using the property of homology of $\RP^4$, we classify various wrapping branes. Then we consider particles, strings, twobranes, domain walls and the baryon vertex in Type IIA string theory.
We study the effects of the introduction of a $\vartheta$ term in minimal gauged supergravity in four dimensions. We show why this term is not present in supergravity duals of field theories arising on wrapped $M2$-branes, but is there in the case of $M5$-branes wrapping hyperbolic manifolds $Σ_3$, and compute the higher-derivative corrections. Having proved that the on-shell supergravity action of any supersymmetric solution can be expressed in terms of data from the fixed points of a Killing vector, we show that it is proportional to a complex topological invariant of $Σ_3$. This is consiste…
At the level of the bosonic fields, we construct consistent Kaluza--Klein reductions of D=11 supergravity on $Σ_3\times S^4$, where $Σ_3=H^3/Γ,S^3/Γ$ or $R^3/Γ$ where $Γ$ is a discrete group of isometries. The result is the bosonic content of an N=2 D=4 gauged supergravity with a single vector multiplet and two hypermultiplets, whose scalar fields parametrise $SU(1,1)/U(1)\times G_{2(2)}/SO(4)$. %, and the gauging is only in the hypermultiplet When $Σ_3=H^3/Γ$ the D=4 theory has an $AdS_4$ vacuum which uplifts to the known supersymmetric $AdS_4\times H^3/Γ\times S^4$ solution of D=11 supergrav…
Through consistent Kaluza-Klein reduction, we construct 3D $\mathcal{N} =2$ gauged supergravities corresponding to twisted compactifications of M5-branes on a product of constant curvature Riemann surfaces, including Kähler-Einstein four-manifolds. We extend the reduction to fermionic supersymmetry variations in order to determine the 3D Killing spinor equations and classify all timelike supersymmetric solutions. As a by-product, we identify an infinite class of new supersymmetric warped $AdS_3$ (Gödel) and warped $d S_3$ solutions. Moreover, we show that the superpotential $T$ encodes the cen…
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…
We use F-theory to classify possibly all six-dimensional superconformal field theories (SCFTs). This involves a two step process: We first classify all possible tensor branches allowed in F-theory (which correspond to allowed collections of contractible spheres) and then classify all possible configurations of seven-branes wrapped over them. We describe the first step in terms of "atoms" joined into "radicals" and "molecules," using an analogy from chemistry. The second step has an interpretation via quiver-type gauge theories constrained by anomaly cancellation. A very surprising outcome of o…
We study the 't Hooft anomalies of four-dimensional superconformal field theories that arise from M5-branes wrapped on a punctured Riemann surface. In general there are two independent contributions to the anomalies. There is a bulk term obtained by integrating the anomaly polynomial of the world-volume theory on the M5-branes over the Riemann surface; this contribution knows about the punctures only through its dependence on the Euler characteristic of the surface. The second set of contributions comes from local data at the punctures; these terms are independent from the bulk data of the sur…
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.
We study a supersymmetric, rotating, electrically charged black hole in AdS$_{4}$ which is a solution of four-dimensional minimal gauged supergravity. Using holography we show that the free energy on $S^3$ and the superconformal index of the dual three-dimensional $\mathcal N=2$ SCFT, in the planar limit, are related in a simple universal way. This result applies to large classes of SCFTs constructed from branes in string and M-theory which we discuss in some detail. For theories of class $\mathcal R$, which arise from $N$ M5-branes wrapped on hyperbolic three-manifolds, we show that the super…
We find a class of new supersymmetric dyonic black holes in four-dimensional maximal gauged supergravity which are asymptotic to the ${\rm SU(3)}\times {\rm U(1)}$ invariant AdS$_4$ Warner vacuum. These black holes can be embedded in eleven-dimensional supergravity where they describe the backreaction of M2-branes wrapped on a Riemann surface. The holographic dual description of these supergravity backgrounds is given by a partial topological twist on a Riemann surface of a three-dimensional $\mathcal{N}=2$ SCFT that is obtained by a mass-deformation of the ABJM theory. We compute explicitly t…
We study the interplay between four-derivative 4d gauged supergravity, holography, wrapped M5-branes, and theories of class $\mathcal{R}$. Using results from Chern-Simons theory on hyperbolic three-manifolds and the 3d-3d correspondence we are able to constrain the two independent coefficients in the four-derivative supergravity Lagrangian. This in turn allows us to calculate the subleading terms in the large-$N$ expansion of supersymmetric partition functions for an infinite class of three-dimensional $\mathcal{N}=2$ SCFTs of class $\mathcal{R}$. We also determine the leading correction to th…
In the context of the $AdS_4\times Q^{111}$ solution of D=11 supergravity we construct supersymmetric zero temperature black brane solutions that interpolate between $AdS_4$ in the UV and $AdS_2\times \mathbb{R}^2$ in the IR. The dual N=2 SCFT has a $U(1)^2$ baryonic symmetry and the solutions carry electric charge with respect to one of the U(1) factors and magnetic charge with respect to the other. The solutions describe stable zero temperature ground states of the deformed SCFT which have finite entropy density. We also construct analogous supersymmetric solutions that flow to $AdS_2\times …
We construct families of supersymmetric AdS$_3\times Y_7$ and AdS$_3\times Y_8$ solutions to type IIB string theory and M-theory, respectively. Here $Y_7$ is an $S^5$ fibration over $Σ$, while $Y_8$ is an $S^4$ fibration over $Σ_g\times Σ$, where $Σ_g$ is a Riemann surface of genus $g>1$ and $Σ$ is a two-dimensional orbifold known as a spindle. We interpret the solutions as near-horizon limits of $N$ D3-branes wrapped on $Σ$ and $N$ M5-branes wrapped on $Σ_g\times Σ$, respectively. These are holographically dual to $d=2$, $(0,2)$ SCFTs, and we show that the central charge and superconformal R-…
We construct a consistent Kaluza-Klein reduction of $D=11$ supergravity on $Σ_2\times S^4$, where $Σ_2=S^2,\mathbb{R}^2$ or $H^2$, or a quotient thereof, at the level of the bosonic fields. The result is a gauged $N=4$, $D=5$ supergravity theory coupled to three vector multiplets, with the gauging lying in an $SO(2)\times SE(3)\subset SO(5,3)$ subgroup of the $SO(1,1)\times SO(5,3)$ global symmetry group of the ungauged theory. For $Σ_2=H^2$, the $D=5$ theory has a maximally supersymmetric $AdS_5$ vacuum which uplifts to the known solution of $D=11$ supergravity corresponding to M5-branes wrap…
We consider M-theory on (T^2\times R^2)/Z_n with M5 branes wrapped on R^2. One can probe this background with M5 branes wrapped on T^2. The theories on the probes provide many new examples of N=2 field theories without Lagrangian description. All these theories have Coulomb branches, and we find the corresponding Seiberg-Witten curves. The exact solution is encoded in a Hitchin system on an orbifolded torus with punctures. The theories we consider also arise from D3 probes in F-theory on K3\times K3 orbifolds. Interestingly, the relevant F-theory background has frozen Z_n singularities which a…
We study the D-brane spectrum on a two-parameter Calabi-Yau model. The analysis is based on different tools in distinct regions of the moduli space: wrapped brane configurations on elliptic fibrations near the large radius limit, and SCFT boundary states at the Gepner point. We develop an explicit correspondence between these two classes of objects, suggesting that boundary states are natural quantum generalizations of bundles. We also find interesting D-brane dynamics in deep stringy regimes. The most striking example is, perhaps, that nonsupersymmetric D6-D0 and D4-D2 large radius configurat…
We consider general aspects of the realization of R and non-R flavor symmetries in the AdS_5 x H_5 dual of 4d N=1 superconformal field theories. We find a general prescription for computing the charges under these symmetries for baryonic operators, which uses only topological information (intersection numbers) on H_5. We find and discuss a new correspondence between the nodes of the SCFT quiver diagrams and certain divisors in the associated geometry. We also discuss connections between the non-R flavor symmetries and the enhanced gauge symmetries in non-conformal theories obtained by adding w…
We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a general…
One of the hallmarks of 6D superconformal field theories (SCFTs) is that on a partial tensor branch, all known theories resemble quiver gauge theories with links comprised of 6D conformal matter, a generalization of weakly coupled hypermultiplets. In this paper we construct 4D quiverlike gauge theories in which the links are obtained from compactifications of 6D conformal matter on Riemann surfaces with flavor symmetry fluxes. This includes generalizations of super QCD with exceptional gauge groups and quarks replaced by 4D conformal matter. Just as in super QCD, we find evidence for a conform…
