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Abstract We study the anomalies of two-dimensional BPS defects in six-dimensional N $$ \mathcal{N} $$ = (1, 0) superconformal field theories. Using a holographic description of these defects furnished by probe D4-branes in AdS7 solutions of ten-dimensional type IIA supergravity, we compute the two independent defect Weyl anomalies from the on-shell action for a spherical defect and defect sphere entanglement entropy. We find agreement between the holographic prediction for the defect A-type anomaly coming from the defect sphere free energy and the leading large N contribution to the defect ’t Hooft anomaly found using anomaly inflow. We also find agreement between the holographic computation of the expectation value of a surface operator wrapping a torus and the supersymmetric localization computation for a circular Wilson loop in N $$ \mathcal{N} $$ = 1 super Yang-Mills theory on . Lastly, we holographically compute the defect gravitational anomaly from the Wess-Zumino action of the probe D4-brane, which provides a subleading large N correction to the defect A-type anomaly.

Abstract Finsler geometry is a natural generalization of (pseudo-)Riemannian geometry, where the line element is not the square root of a quadratic form but a more general homogeneous function. Parameterizing this in terms of symmetric tensors suggests a possible interpretation in terms of higher-spin fields. We will see here that, at linear level in these fields, the Finsler version of the Ricci tensor leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling. Nonlinear terms can also be systematically analyzed, suggesting a possible interacting structure. No particular choice of spacetime dimension is needed. The Stueckelberg mechanism breaks gauge transformations to a redundancy that does not change the geometry. This creates a serious issue: non-transverse modes are not eliminated, at least for the versions of Finsler dynamics examined in this paper.

Abstract For the Laplacian of an n-Riemannian manifold X, the Weyl law states that the k-th eigenvalue is asymptotically proportional to (k/V)2/n , where V is the volume of X. We show that this result can be derived via physical considerations by demanding that the gravitational potential for a compactification on X behaves in the expected (4+n)-dimensional way at short distances. In simple product compactifications, when particle motion on X is ergodic, for large k the eigenfunctions oscillate around a constant, and the argument is relatively straightforward. The Weyl law thus allows to reconstruct the four-dimensional Planck mass from the asymptotics of the masses of the spin 2 Kaluza-Klein modes. For warped compactifications, a puzzle appears: the Weyl law still depends on the ordinary volume V, while the Planck mass famously depends on a weighted volume obtained as an integral of the warping function. We resolve this tension by arguing that in the ergodic case the eigenfunctions oscillate now around a power of the warping function rather than around a constant, a property that we call weighted quantum ergodicity. This has implications for the problem of gravity localization, which we discuss. We show that for spaces with Dp-brane singularities the spectrum is discrete only for p = 6, 7, 8, and for these cases we rigorously prove the Weyl law by applying modern techniques from RCD theory.

Abstract In models with extra dimensions, matter particles can be easily localized to a ‘brane world’, but gravitational attraction tends to spread out in the extra dimensions unless they are small. Strong warping gradients can help localize gravity closer to the brane. In this note we give a mathematically rigorous proof that the internal wave-function of the massless graviton is constant as an eigenfunction of the weighted Laplacian, and hence is a power of the warping as a bound state in an analogue Schrödinger potential. This holds even in presence of singularities induced by thin branes. We also reassess the status of AdS vacuum solutions where the graviton is massive. We prove a bound on scale separation for such models, as an application of our recent results on KK masses. We also use them to estimate the scale at which gravity is localized, without having to compute the spectrum explicitly. For example, we point out that localization can be obtained at least up to the cosmological scale in string/M-theory solutions with infinite-volume Riemann surfaces; and in a known class of N $$ \mathcal{N} $$ = 4 models, when the number of NS5- and D5-branes is roughly equal.

Abstract We introduce a class of 3d theories consisting of strongly-coupled N $$ \mathcal{N} $$ = 4 systems coupled to N $$ \mathcal{N} $$ = 3 Chern-Simons gauge multiplets, which exhibit N $$ \mathcal{N} $$ = 4 enhancements when a peculiar condition on the Chern-Simons levels is met. An example is the SU(N)3 Chern-Simons theory coupled to the 3d T N theory, which enhances to N $$ \mathcal{N} $$ = 4 when 1/k 1 + 1/k 2 + 1/k 3 = 0. We also show that some but not all of these N $$ \mathcal{N} $$ = 4 enhancements can be understood by considering M5-branes on a special class of Seifert manifolds. Our construction provides a large class of N $$ \mathcal{N} $$ = 4 theories which have not been studied previously.

Abstract Vacuum compactifications may suffer from instabilities under small perturbations or tunnel effects; both are difficult to analyze. In this paper we consider the issue from a higher-dimensional perspective. We first look at how stability works for supersymmetric vacua, where it is widely expected to hold. We first show that the nucleation of brane bubbles in type II AdS compactifications is forbidden in the probe approximation by a simple argument involving pure spinors and calibrations. We then adapt familiar positive-energy theorems directly to M-theory and type II supergravity, rather than to their effective lower-dimensional reductions, also showing how to consistently include localized sources. We finally initiate an analysis of how these arguments might be extended to non-supersymmetric vacua. In M-theory, at the lower-derivative level, we find that the most natural modifications fail to stabilize the skew-whiffed and Englert vacua.

Abstract Warped AdS3 solutions in 10 dimensional supergravity that preserve N $$ \mathcal{N} $$ = (1, 1) supersymmetry are considered. Sufficient geometric conditions for their existence, and to stop the AdS3 factor experiencing an enhancement to AdS4, are presented. The internal space of such solutions decomposes as a foliation of M6 over an interval where M6 supports either an SU(3)- or SU(2)-structure. The former case is classified in terms of torsion classes and new solutions are found.

Abstract We consider gravity compactifications whose internal space consists of small bridges connecting larger manifolds, possibly noncompact. We prove that, under rather general assumptions, this leads to a massive spin-two field with very small mass. The argument involves a recently-noticed relation to Bakry-Émery geometry, a version of the so-called Cheeger constant, and the theory of synthetic Ricci lower bounds. The latter technique allows generalizations to non-smooth spaces such as those with D-brane singularities. For AdS d vacua with a bridge admitting an AdS d+1 interpretation, the holographic dual is a CFT d with two CFT d−1 boundaries. The ratio of their degrees of freedom gives the graviton mass, generalizing results obtained by Bachas and Lavdas for d = 4. We also prove new bounds on the higher eigenvalues. These are in agreement with the spin-two swampland conjecture in the regime where the background is scale-separated; in the opposite regime we provide examples where they are in naive tension with it.

Abstract In a broad class of gravity theories, the equations of motion for vacuum compactifications give a curvature bound on the Ricci tensor minus a multiple of the Hessian of the warping function. Using results in so-called Bakry-Émery geometry, we put rigorous general bounds on the KK scale in gravity compactifications in terms of the reduced Planck mass or the internal diameter. We reexamine in this light the local behavior in type IIA for the class of supersymmetric solutions most promising for scale separation. We find that the local O6-plane behavior cannot be smoothed out as in other local examples; it generically turns into a formal partially smeared O4.

Abstract For several classes of BPS vacua, we find a procedure to modify the PDEs that imply preserved supersymmetry and the equations of motion so that they still imply the latter but not the former. In each case we trace back this supersymmetry-breaking deformation to a distinct modification of the pure spinor equations that provide a geometrical interpretation of supersymmetry. We give some concrete examples: first we generalize the Imamura class of Mink6 solutions by removing a symmetry requirement, and then derive some local and global solutions both before and after breaking supersymmetry.

Abstract In previous work, we found ten-dimensional solutions to the supergravity equations of motion with a dS4 factor and O8-planes. We generalize this analysis and obtain other solutions in the same spirit, with an O8+ and an O6 − . We examine our original solutions in more detail, focusing in particular on the O8 − singularities and on the issues created by their boundary conditions. We also point out some previously known super- symmetric AdS solutions with the same local behavior at their O8 − singularity.

Abstract In this work we study ten-dimensional solutions to type IIA string theory of the form AdS4 × X 6 which contain orientifold planes and preserve N $$ \mathcal{N} $$ = 1 supersymmetry. In particular, we consider solutions which exhibit some key features of the four-dimensional DGKT proposal for compactifications on Calabi-Yau manifolds with fluxes, and in this sense may be considered their ten-dimensional uplifts. We focus on the supersymmetry equations and Bianchi identities, and find solutions to these that are valid at the two-derivative level and at first order in an expansion parameter which is related to the AdS cosmological constant. This family of solutions is such that the background metric is deformed from the Ricci-flat one to one exhibiting SU(3) × SU(3)-structure, and dilaton gradients and warp factors are induced.

Abstract AdS7 supersymmetric solutions in type IIA have been classified, and they are infinitely many. Moreover, every such solution has a non-supersymmetric sister. In this pa- per, we study the perturbative and non-perturbative stability of these non-supersymmetric solutions, focusing on cases without orientifolds. Perturbatively, we first look at the KK spectrum of spin-2 excitations. This does not exhibit instabilities, but it does show that there is no separation of scales for either the BPS and the non-BPS case, thus proving for supersymmetric AdS7 a well-known recent conjecture. We then use 7d gauged supergravity and a brane polarization computation to access part of the spectrum of KK scalars. The result signals an instability for all non-supersymmetric solutions except those that have a single D8 on each side. We finally look at non-perturbative instabilities, and find that NS5 bubbles make these remaining solutions decay.

Abstract We study the recently proposed AdS 7/CFT6 dualities for a class of 6d N $$ \mathcal{N} $$ = (1, 0) theories that flow on the tensor branch to long linear quiver gauge theories. We find a precise agreement in the symmetries and in the spectrum of charged states between the 6d SCFTs and their conjectured AdS 7 duals. We also confirm a recent conjecture that a discrete S N symmetry relating the baryons in the quiver theories is in fact gauged.

Abstract Placing D3-branes at conical Calabi-Yau threefold singularities produces many AdS5/CFT4 duals. Recent progress in differential geometry has produced a technique (called K-stability) to recognize which singularities admit conical Calabi-Yau metrics. On the other hand, the algebraic technique of non-commutative crepant resolutions, involving matrix factorizations, has been developed to associate a quiver to a singularity. In this paper, we put together these ideas to produce new AdS5/CFT4 duals, with special emphasis on non-toric singularities.

Abstract We find non-supersymmetric AdS8 solutions of type IIA supergravity. The internal space is topologically an S 2 with a U(1) isometry. The only non-zero flux is F 0; an O8 sourcing it is present at the equator of the S 2. The warping function and dilaton are non-constant. It is also possible to add D8-branes on top of the O8. Possible destabilizing brane bubbles (whose presence would be suggested by the weak-gravity conjecture) are either absent or collapsing. Our solutions are candidate holographic duals to unitary interacting CFTs in seven dimensions with exceptional global symmetry. We also present analogous non-supersymmetric AdS d solutions for general d which are supported only by F 0.

Abstract In several contexts, supersymmetry can be reformulated in terms of calibrations, namely forms whose integrals measure minimal energies. It has been conjectured that this should be possible in general. For type II supergravity, we present a new system of equations which realizes this expectation. Besides the customary D-brane calibrations, it also includes NS5-brane and KK5-monopole calibrations. It is equivalent to supersymmetry under the assumption that the Killing vector associated to supersymmetry is timelike. No assumption is made on a factorization of spacetime. We also obtain a version of the system which is manifestly S-invariant and we present an application to near-horizon backgrounds. Using calibration, a definition of central charges in purely gravitational terms is given.

Abstract A notable class of superconformal theories (SCFTs) in six dimensions is parameterized by an integer N , an ADE group G, and two nilpotent elements μ L,R in G. Nilpotent elements have a natural partial ordering, which has been conjectured to coincide with the hierarchy of renormalization-group flows among the SCFTs. In this paper we test this conjecture for G = SU(k), where AdS7 duals exist in IIA. We work with a seven-dimensional gauged supergravity, consisting of the gravity multiplet and two SU(k) non-Abelian vector multiplets. We show that this theory has many supersymmetric AdS7 vacua, determined by two nilpotent elements, which are naturally interpreted as IIA AdS7 solutions. The BPS equations for domain walls connecting two such vacua can be solved analytically, up to a Nahm equation with certain boundary conditions. The latter admit a solution connecting two vacua if and only if the corresponding nilpotent elements are related by the natural partial ordering, in agreement with the field theory conjecture.

Abstract We show that all known 6D SCFTs can be obtained iteratively from an underlying set of UV progenitor theories through the processes of “fission” and “fusion”. Fission consists of a tensor branch deformation followed by a special class of Higgs branch deformations characterized by discrete and continuous homomorphisms into flavor symmetry algebras. Almost all 6D SCFTs can be realized as fission products. The remainder can be constructed via one step of fusion involving these fission products, whereby a single common flavor symmetry of decoupled 6D SCFTs is gauged and paired with a new tensor multiplet at the origin of moduli space, producing an RG flow “in reverse” to the UV. This leads to a streamlined labeling scheme for all known 6D SCFTs in terms of a few pieces of group theoretic data. The partial ordering of continuous homomorphisms s u 2 → g flav $$ \mathfrak{s}\mathfrak{u}(2)\to {\mathfrak{g}}_{\mathrm{flav}} $$ for g flav $$ {\mathfrak{g}}_{\mathrm{flav}} $$ a flavor symmetry also points the way to a classification of 6D RG flows.

Abstract We initiate a classification of N = 2 $$ \mathcal{N}=2 $$ supersymmetric AdS4 solutions of (massive) type IIA supergravity. The internal space is locally equipped with either an SU(2) or an identity structure. We focus on the SU(2) structure and determine the conditions it satisfies, dictated by supersymmetry. Imposing as an Ansatz that the internal space is complex, we reduce the problem of finding solutions to a Riccati ODE, which we solve analytically. We obtain in this fashion a large number of new families of solutions, both regular as well as with localized O8-planes and conical Calabi-Yau singularities. We also recover many solutions already discussed in the literature.

Abstract We study the interpretation of O7+-planes in F-theory, mainly in the context of the six-dimensional models. In particular, we study how to assign gauge algebras and matter content to seven-branes and their intersections, and the implication of anomaly cancellation in our construction, generalizing earlier analyses without any O7+-planes. By including O7+-planes we can realize 6d superconformal field theories hitherto unobtainable in F-theory, such as those with hypermultiplets in the symmetric representation of su. We also examine a couple of compact models. These reproduce some famous perturbative models, and in some cases enhance their gauge symmetries non-perturbatively.

Abstract The geometry of the N=3 $$ \mathcal{N}=3 $$, SO(4)-invariant, AdS4 solution of massive type IIA supergravity that uplifts from the N=3 $$ \mathcal{N}=3 $$ vacuum of D = 4 N=8 $$ \mathcal{N}=8 $$ dyonic ISO(7) supergravity is investigated. Firstly, a D = 4, SO(4)-invariant restricted duality hierarchy is constructed and used to uplift the entire, dynamical SO(4)-invariant sector to massive type IIA. The resulting consistent uplift formulae are used to obtain a new local expression for the N=3 $$ \mathcal{N}=3 $$ AdS4 solution in massive IIA and analyse its geometry. Locally, the internal S 6 geometry corresponds to a warped fibration of S 2 and a hemisphere of S 4. This can be regarded as a warped generalisation of the usual twistor fibration geometry. Finally, the triplet of Killing spinors corresponding to the N=3 $$ \mathcal{N}=3 $$ solution are constructed and shown to obey the massive type IIA Killing spinor equations.

Abstract We construct non-geometric AdS4 solutions of IIB string theory where the fields in overlapping patches are glued by elements of the S-duality group. We obtain them by suitable quotients of compact and non-compact geometric solutions. The quotient procedure suggests CFT duals as quiver theories with links involving the so-called T [U(N)] theory. We test the validity of the non-geometric solutions (and of our proposed holographic duality) by computing the three-sphere partition function Z of the CFTs. A first class of solutions is obtained by an S-duality quotient of Janus-type non-compact solutions and is dual to 3d N=4 $$ \mathcal{N}=4 $$ SCFTs; for these we manage to compute Z of the dual CFT at finite N, and it agrees perfectly with the supergravity result in the large N limit. A second class has five-branes, it is obtained by a Möbius-like S-quotient of ordinary compact solutions and is dual to 3d N=3 $$ \mathcal{N}=3 $$ SCFTs. For these, Z agrees with the supergravity result if one chooses the limit carefully so that the effect of the fivebranes does not backreact on the entire geometry. Other limits suggest the existence of IIA duals.

Abstract In supergravity compactifications, there is in general no clear prescription on how to select a finite-dimensional family of metrics on the internal space, and a family of forms on which to expand the various potentials, such that the lower-dimensional effective theory is supersymmetric. We propose a finite-dimensional family of deformations for regular Sasaki-Einstein seven-manifolds M 7, relevant for M-theory compactifications down to four dimensions. It consists of integrable Cauchy-Riemann structures, corresponding to complex deformations of the Calabi-Yau cone M 8 over M 7. The non-harmonic forms we propose are the ones contained in one of the Kohn-Rossi cohomology groups, which is finite-dimensional and naturally controls the deformations of Cauchy-Riemann structures. The same family of deformations can be also described in terms of twisted cohomology of the base M 6, or in terms of Milnor cycles arising in deformations of M 8. Using existing results on SU(3) structure compactifications, we briefly discuss the reduction of M-theory on our class of deformed Sasaki-Einstein manifolds to four-dimensional gauged supergravity.

Abstract We analyze general N=2 $$ \mathcal{N}=2 $$ supersymmetric AdS4 solutions of type IIB supergravity. Utilizing a set of pure spinor equations directly adapted to N=2 $$ \mathcal{N}=2 $$, the necessary and sufficient conditions for supersymmetry are reduced to a concise system of partial differential equations for two functions which determine the solutions. We show that using this system analytic solutions can be generated, thus potentially expanding the rather limited set of known AdS4 solutions in type IIB supergravity.

Abstract We find AdS5 solutions holographically dual to compactifications of six- dimensional N $$ \mathcal{N} $$ = (1, 0) supersymmetric field theories on Riemann surfaces with punctures. We simplify a previous analysis of supersymmetric AdS5 IIA solutions, and with a suitable Ansatz we find explicit solutions organized in three classes, where an O8-D8 stack, D6-and D4-branes are simultaneously present, localized and partially localized. The D4-branes are smeared over the Riemann surface and this is interpreted as the presence of a uniform distribution of punctures. For the first class we identify the corresponding six-dimensional theory as an E-string theory coupled to a quiver gauge theory. The second class of solutions lacks D6-branes and its central charge scales as n 5/2, suggesting a five-dimensional origin for the dual field theory. The last class has elements of the previous two.

Abstract The worldvolume theory of M5-branes on an ADE singularity ℝ 5/Γ G can be Higgsed in various ways, corresponding to the possible nilpotent orbits of G. In the F-theory dual picture, this corresponds to activating T-brane data along two stacks of 7-branes and yields a tensor branch realization for a large class of 6D SCFTs. In this paper, we show that the moduli spaces and anomalies of these T-brane theories are related in a simple, universal way to data of the nilpotent orbits. This often works in surprising ways and gives a nontrivial confirmation of the conjectured properties of T-branes in F-theory. We use this result to formally engineer a class of theories where the IIA picture naïvely breaks down. We also give a proof of the a-theorem for all RG flows within this class of T-brane theories.

Abstract M5-branes on an ADE singularity are described by certain six-dimensional “conformal matter” superconformal field theories. Their Higgs moduli spaces contain information about various dynamical processes for the M5s; however, they are not directly accessible due to the lack of a Lagrangian formulation. Using anomaly matching, we compute their dimensions. The result implies that M5 fractions can recombine in several different ways, where the M5s are leaving behind frozen versions of the singularity. The anomaly polynomial gives hints about the nature of the freezing. We also check the Higgs dimension formula by comparing it with various existing conjectures for the CFTs one obtains by torus compactifications down to four and three dimensions. Aided by our results, we also extend those conjectures to compactifications of theories not previously considered. These involve class S $$ \mathcal{S} $$ theories with twisted punctures in four dimensions, and affine-Dynkin-shaped quivers in three dimensions.

Abstract We classify Minkowski4 solutions in type IIA supergravity, with N $$ \mathcal{N} $$ = 2 supersymmetry and an SU(2) R-symmetry of a certain type. Many subcases can be reduced to relatively simple PDEs, among which we recover various intersecting brane systems, and AdS d solutions, d = 5, 6, 7, and in particular the recently found general massive AdS7 solutions. Imposing compactness of the internal six-manifold we obtain promising solutions with localized D-branes and O-planes.