Marginal deformations of 3d N=2\mathcal{N}=2 CFTs from AdS4_4 backgrounds in generalised geometry

We study exactly marginal deformations of 3d $\mathcal{N}=2$ CFTs dual to AdS$_4$ solutions in eleven-dimensional supergravity using generalised geometry. Focussing on Sasaki-Einstein backgrounds, we find that marginal deformations correspond to turning on a four-form flux on the internal space at first order. Viewing this as the deformation of a generalised structure, we derive a general expression for the four-form flux in terms of a holomorphic function. We discuss the explicit examples of S$^7$, Q$^{1,1,1}$ and M$^{1,1,1}$ and, using an obstruction analysis, find the conditions for the first-order deformations to extend all orders, thus identifying which marginal deformations are exactly marginal. We also show how the all-orders $\gamma$-deformation of Lunin and Maldacena can be encoded as a tri-vector deformation in generalised geometry and outline how to recover the supergravity solution from the generalised metric.Comment: 37 pages; references added in v

The exceptional generalised geometry of supersymmetric AdS flux backgrounds

We analyse generic AdS flux backgrounds preserving eight supercharges in $D=4$ and $D=5$ dimensions using exceptional generalised geometry. We show that they are described by a pair of globally defined, generalised structures, identical to those that appear for flat flux backgrounds but with different integrability conditions. We give a number of explicit examples of such "exceptional Sasaki-Einstein" backgrounds in type IIB supergravity and M-theory. In particular, we give the complete analysis of the generic AdS$_5$ M-theory backgrounds. We also briefly discuss the structure of the moduli space of solutions. In all cases, one structure defines a "generalised Reeb vector" that generates a Killing symmetry of the background corresponding to the R-symmetry of the dual field theory, and in addition encodes the generic contact structures that appear in the $D=4$ M-theory and $D=5$ type IIB cases. Finally, we investigate the relation between generalised structures and quantities in the dual field theory, showing that the central charge and R-charge of BPS wrapped-brane states are both encoded by the generalised Reeb vector, as well as discussing how volume minimisation (the dual of $a$- and $\mathcal{F}$-maximisation) is encoded.Comment: 40 page

Eigenvalues and eigenforms on Calabi-Yau threefolds

We present a numerical algorithm for computing the spectrum of the Laplace-de Rham operator on Calabi-Yau manifolds, extending previous work on the scalar Laplace operator. Using an approximate Calabi-Yau metric as input, we compute the eigenvalues and eigenforms of the Laplace operator acting on $(p,q)$-forms for the example of the Fermat quintic threefold. We provide a check of our algorithm by computing the spectrum of $(p,q)$-eigenforms on $\mathbb{P}^{3}$.Comment: 36 pages, 18 figures, 4 tables; v2: increased number of points in numerical integratio

N=(2,0)N=(2,0) AdS3_3 Solutions of M-theory

We consider the most general solutions of eleven-dimensional supergravity preserving $N=2$ supersymmetry whose metrics are warped products of three-dimensional anti-de Sitter space with an eight-dimensional manifold, focusing on those that are dual to two-dimensional (2,0) superconformal field theories. We give a set of necessary and sufficient conditions for a solution to be supersymmetric, which can be phrased, in the general case, in terms of a local SU(2) structure and its intrinsic torsion. We show that these supergravity backgrounds always admit a nowhere-vanishing Killing vector field that preserves the solution and encodes the U(1) R-symmetry of the dual field theory. We illustrate our results with examples which have appeared in the literature, including those with SU(4), G$_2$ and SU(3) structures, and discuss new classes of solutions.Comment: 40 pages plus appendices; submission includes two Mathematica notebook

Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold

We use numerical methods to obtain moduli-dependent Calabi-Yau metrics and from them the moduli-dependent massive tower of Kaluza-Klein states for the one-parameter family of quintic Calabi-Yau manifolds. We then compute geodesic distances in their K\"ahler and complex structure moduli space using exact expressions from mirror symmetry, approximate expressions, and numerical methods and compare the results. Finally, we fit the moduli-dependence of the massive spectrum to the geodesic distance to obtain the rate at which states become exponentially light. The result is indeed of order one, as suggested by the swampland distance conjecture. We also observe level-crossing in the eigenvalue spectrum and find that states in small irreducible representations of the symmetry group tend to become lighter than states in larger irreducible representations.Comment: references added and minor changes, 10 pages, 5 figure

Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold

We use numerical methods to obtain moduli-dependent Calabi-Yau metrics, and from them, the moduli-dependent massive tower of Kaluza-Klein states for the one-parameter family of quintic Calabi-Yau manifolds. We then compute geodesic distances in their Kähler and complex structure moduli space using exact expressions from mirror symmetry, approximate expressions, and numerical methods, and we compare the results. Finally, we fit the moduli dependence of the massive spectrum to the geodesic distance to obtain the rate at which states become exponentially light. The result is indeed of order 1, as suggested by the swampland distance conjecture. We also observe level crossing in the eigenvalue spectrum and find that states in small irreducible representations of the symmetry group tend to become lighter than states in larger irreducible representations

Exactly marginal deformations from exceptional generalised geometry

We apply exceptional generalised geometry to the study of exactly marginal deformations of $\mathcal{N}=1$ SCFTs that are dual to generic AdS$_5$ flux backgrounds in type IIB or eleven-dimensional supergravity. In the gauge theory, marginal deformations are parametrised by the space of chiral primary operators of conformal dimension three, while exactly marginal deformations correspond to quotienting this space by the complexified global symmetry group. We show how the supergravity analysis gives a geometric interpretation of the gauge theory results. The marginal deformations arise from deformations of generalised structures that solve moment maps for the generalised diffeomorphism group and have the correct charge under the generalised Reeb vector, generating the R-symmetry. If this is the only symmetry of the background, all marginal deformations are exactly marginal. If the background possesses extra isometries, there are obstructions that come from fixed points of the moment maps. The exactly marginal deformations are then given by a further quotient by these extra isometries. Our analysis holds for any $\mathcal{N}=2$ AdS$_5$ flux background. Focussing on the particular case of type IIB Sasaki-Einstein backgrounds we recover the result that marginal deformations correspond to perturbing the solution by three-form flux at first order. In various explicit examples, we show that our expression for the three-form flux matches those in the literature and the obstruction conditions match the one-loop beta functions of the dual SCFT.Comment: 52 page

Calabi-Yau Three-folds: Poincare Polynomials and Fractals

We study the Poincare polynomials of all known Calabi-Yau three-folds as constrained polynomials of Littlewood type, thus generalising the well-known investigation into the distribution of the Euler characteristic and Hodge numbers. We find interesting fractal behaviour in the roots of these polynomials in relation to the existence of isometries, distribution versus typicality, and mirror symmetry.Comment: 14 pages, 6 figures, invited contribution to the Max Kreuzer Memorial Volume, based on MPhys project of the first author under the supervision of the second, at the University of Oxfor

Generalised geometry for supersymmetric flux backgrounds

We present a geometric description of flux backgrounds in supergravity that preserve eight supercharges using the language of (exceptional) generalised geometry. These “exceptional Calabi–Yau” geometries generalise complex, symplectic and hyper-Kähler geometries, where integrability is equivalent to supersymmetry for the background. The integrability conditions take the form of vanishing moment maps for the “generalised diffeomorphism group”, and the moduli spaces of structures appear as hyper-Kähler and symplectic quotients. Our formalism applies to generic D=4,5,6 backgrounds preserving eight supercharges in both type II and eleven-dimensional supergravity. We include a number of examples of flux backgrounds that can be reformulated as exceptional Calabi–Yau geometries. We extend this analysis and show that generic AdS flux backgrounds in D=4,5 are also described by exceptional generalised geometry, giving what one might call “exceptional Sasaki–Einstein” geometry. These backgrounds always admit a “generalised Reeb vector” that generates a Killing symmetry of the background, corresponding to the R-symmetry of the dual field theory. We also discuss the relation between generalised structures and quantities in the dual field theory. We then consider deformations of these generalised structures. For AdS5 backgrounds in type IIB, a first-order deformation amounts to turning on three-form fluxes that preserve supersymmetry. We find the general form of these fluxes for any Sasaki–Einstein space and show that higher-order deformations are obstructed by the moment map for the symmetry group of the undeformed background. In the dual field theory, this corresponds to finding those marginal deformations that are exactly marginal. We give a number of examples and match to known expressions in the literature. We also apply our formalism to AdS5 backgrounds in M-theory, where the first-order deformation amounts to turning on a four-form flux that preserves supersymmetry.Open Acces

Geometric flows and supersymmetry

We study the relation between supersymmetry and geometric flows driven by the Bianchi identity for the three-form flux $H$ in heterotic supergravity. We describe how the flow equations can be derived from a functional that appears in a rewriting of the bosonic action in terms of squares of supersymmetry operators. On a complex threefold, the resulting equations match what is known in the mathematics literature as "anomaly flow". We generalise this to seven- and eight-manifolds with G$_2$ or Spin(7) structures and discuss examples where the manifold is a torus fibration over a K3 surface. In the latter cases, the flow simplifies to a single scalar equation, with the existence of the supergravity solution implied by the long-time existence and convergence of the flow. We also comment on the $\alpha'$ expansion and highlight the importance of using the proper connection in the Bianchi identity to ensure that the flow's fixed points satisfy the supergravity equations of motion.Comment: 47 pages plus appendice