## Publications

113 results found

Hulík O, Malek E, Valach F,
et al., 2024, Y-algebroids and E<inf>7(7)</inf>× ℝ<sup>+</sup>-generalised geometry, *Journal of High Energy Physics*, Vol: 2024

We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to D ≥ 5 dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted Y, that provides exactly the necessary properties to also describe compactifications to D = 4 dimensions. We classify “M-exact” E7-algebroids and show that this precisely matches the form of the generalised tangent space of E7(7) × ℝ+-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids.

Smith GR, Waldram D, 2023, M-theory moduli from exceptional complex structures, *The Journal of High Energy Physics*, Vol: 2023, ISSN: 1029-8479

We continue the analysis of the geometry of generic Minkowski N = 1, D = 4flux compactifications in M-theory using exceptional generalised geometry, including thecalculation of the infinitesimal moduli spaces. The backgrounds can be classified into twoclasses: type-0 and type-3. For type-0, we review how the moduli arise from standard deRham cohomology classes. We also argue that, under reasonable assumptions, there areno appropriate sources to support compact flux backgrounds for this class and so the onlysolutions are in fact G2 geometries. For type-3 backgrounds, given a suitable ∂0∂¯0-lemma,we show that the moduli can be calculated from a cohomology based on an involutive subbundle of the complexified tangent space. Using a simple spectral sequence we prove quitegenerally that the presence of flux can only reduce the number of moduli compared withthe fluxless case. We then use the formalism to calculate the moduli of heterotic M-theoryand show they match those of the dual Hull-Strominger system as expected.

Josse G, Malek E, Petrini M,
et al., 2022, The higher-dimensional origin of five-dimensional N=2 gauged supergravities, *The Journal of High Energy Physics*, Vol: 2022, ISSN: 1029-8479

Using exceptional generalised geometry, we classify which five-dimensional N = 2 gauged supergravities can arise as a consistent truncation of 10-/11-dimensional supergravity. Exceptional generalised geometry turns the classification into an algebraic problem of finding subgroups GS ⊂ USp(8) ⊂ E6(6) that preserve exactly two spinors. Moreover, the intrinsic torsion of the GS structure must contain only constant singlets under GS, and these, in turn, determine the gauging of the five-dimensional theory. The resulting five-dimensional theories are strongly constrained: their scalar manifolds are necessarily symmetric spaces and only a small number of matter multiplets can be kept, which we completely enumerate. We also determine the largest reductive and compact gaugings that can arise from consistent truncations.

Ashmore A, Petrini M, Tasker EL,
et al., 2022, Exactly marginal deformations and their supergravity duals, *Physical Review Letters*, Vol: 128, Pages: 1-6, ISSN: 0031-9007

We study the space of supersymmetric AdS5 solutions of type IIB supergravity corresponding to the conformal manifold of the dual N=1 conformal field theory. We show that the background geometry naturally encodes a generalized holomorphic structure, dual to the superpotential of the field theory, with the existence of the full solution following from a continuity argument. In particular, this work allows us to address the long-standing problem of finding the gravity dual of the generic N=1 deformations of N=4 conformal field theory: even if we are not able to give it in a fully explicit form, we provide a proof-of-existence of the supergravity solution. Using this formalism, we derive a new result for the Hilbert series of the deformed field theories.

Bugden M, Hulik O, Valach F,
et al., 2022, Exceptional algebroids and Type IIB superstrings, *Fortschritte Der Physik/Progress of Physics*, Vol: 70, Pages: 1-8, ISSN: 0015-8208

In this note we study exceptional algebroids, focusing on their relation to type IIB superstring theory. We show that a IIB-exact exceptional algebroid (corresponding to the group urn:x-wiley:00158208:media:prop202100104:prop202100104-math-0001, for urn:x-wiley:00158208:media:prop202100104:prop202100104-math-0002) locally has a standard form given by the exceptional tangent bundle. We derive possible twists, given by a flat urn:x-wiley:00158208:media:prop202100104:prop202100104-math-0003-connection, a covariantly closed pair of 3-forms, and a 5-form, and comment on their physical interpretation. Using this analysis we reduce the search for Leibniz parallelisable spaces, and hence maximally supersymmetric consistent truncations, to a simple algebraic problem. We show that the exceptional algebroid perspective also gives a simple description of Poisson–Lie U-duality without spectators and hence of generalised Yang–Baxter deformations.

Tennyson D, Waldram D, 2021, Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory, *The Journal of High Energy Physics*, Vol: 201, Pages: 1-64, ISSN: 1029-8479

We present a detailed study of a new mathematical object in E6(6)ℝ+ generalised geometry called an ‘exceptional complex structure’ (ECS). It is the extension of a conventional complex structure to one that includes all the degrees of freedom of M-theory or type IIB supergravity in six or five dimensions, and as such characterises, in part, the geometry of generic supersymmetric compactifications to five-dimensional Minkowkski space. We define an ECS as an integrable U*(6) × ℝ+ structure and show it is equivalent to a particular form of involutive subbundle of the complexified generalised tangent bundle L1 ⊂ Eℂ. We also define a refinement, an SU*(6) structure, and show that its integrability requires in addition a vanishing moment map on the space of structures. We are able to classify all possible ECSs, showing that they are characterised by two numbers denoted ‘type’ and ‘class’. We then use the deformation theory of ECS to find the moduli of any SU*(6) structure. We relate these structures to the geometry of generic minimally supersymmetric flux backgrounds of M-theory of the form ℝ4,1 × M, where the SU*(6) moduli correspond to the hypermultiplet moduli in the lower-dimensional theory. Such geometries are of class zero or one. The former are equivalent to a choice of (non-metric-compatible) conventional SL(3, ℂ) structure and strikingly have the same space of hypermultiplet moduli as the fluxless Calabi-Yau case.

Bugden M, Hulik O, Valach F,
et al., 2021, G-Algebroids: a unified framework for exceptional and generalised geometry, and poisson-lie duality, *Fortschritte Der Physik/Progress of Physics*, Vol: 69, Pages: 1-11, ISSN: 0015-8208

We introduce the notion of urn:x-wiley:00158208:media:prop202100028:prop202100028-math-0001-algebroid, generalising both Lie and Courant algebroids, as well as the algebroids used in urn:x-wiley:00158208:media:prop202100028:prop202100028-math-0002 exceptional generalised geometry for urn:x-wiley:00158208:media:prop202100028:prop202100028-math-0003. Focusing on the exceptional case, we prove a classification of “exact” algebroids and translate the related classification of Leibniz parallelisable spaces into a tractable algebraic problem. After discussing the general notion of Poisson–Lie duality, we show that the Poisson–Lie U-duality is compatible with the equations of motion of supergravity.

Cassani D, Josse G, Petrini M,
et al., 2021, N=2 consistent truncations from wrapped M5-branes, *The Journal of High Energy Physics*, Vol: 2021, Pages: 1-69, ISSN: 1029-8479

We discuss consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold M, preserving minimal N = 2 supersymmetry in five dimensions. These are based on GS ⊆ USp(6) structures for the generalised E6(6) tangent bundle on M, such that the intrinsic torsion is a constant GS singlet. We spell out the algorithm defining the full bosonic truncation ansatz and then apply this formalism to consistent truncations that contain warped AdS5 ×w M solutions arising from M5-branes wrapped on a Riemann surface. The generalised U(1) structure associated with the N = 2 solution of Maldacena-Nuñez leads to five-dimensional supergravity with four vector multiplets, one hypermultiplet and SO(3) × U(1) × ℝ gauge group. The generalised structure associated with “BBBW” solutions yields two vector multiplets, one hypermultiplet and an abelian gauging. We argue that these are the most general consistent truncations on such backgrounds.

Ashmore A, Strickland-Constable C, Tennyson D,
et al., 2021, Generalising G₂ geometry: involutivity, moment maps and moduli, *The Journal of High Energy Physics*, Vol: 2021, Pages: 1-66, ISSN: 1029-8479

We analyse the geometry of generic Minkowski N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E7(7) × ℝ+ generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

Ashmore A, Strickland-Constable C, Tennyson D,
et al., 2020, Heterotic backgrounds via generalised geometry: moment maps and moduli, *The Journal of High Energy Physics*, Vol: 2020, Pages: 1-46, ISSN: 1029-8479

We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an SU(3) × Spin(6 + n) structure within O(6, 6 + n) × ℝ+ generalised geometry. Supersymmetry of the background is encoded in the existence of an involutive subbundle of the generalised tangent bundle and the vanishing of a moment map for the action of diffeomorphisms and gauge symmetries. We give both the superpotential and the Kähler potential for a generic background, showing that the latter defines a natural Hitchin functional for heterotic geometries. Intriguingly, this formulation suggests new connections to geometric invariant theory and an extended notion of stability. Finally we show that the analysis of infinitesimal deformations of these geometric structures naturally reproduces the known cohomologies that count the massless moduli of supersymmetric heterotic backgrounds.

Cassani D, Josse G, Petrini M,
et al., 2019, Systematics of consistent truncations from generalised geometry, *The Journal of High Energy Physics*, Vol: 2019, Pages: 1-60, ISSN: 1029-8479

We present a generalised geometry framework for systematically constructing consistent truncations of ten- and eleven-dimensional supergravity preserving varying fractions of supersymmetry. Truncations arise when there is a reduced structure group GS of the exceptional generalised geometry, such that the intrinsic torsion is a GS -singlet. The matter content of the truncated theory follows from group-theoretical arguments, while the gauging is determined by the sub-algebra of generalised diffeomorphisms generated by the GS -singlet vectors. After discussing the general ideas across different spacetime dimensions and amounts of supersymmetry, we provide detailed formulae for truncations to gauged half-maximal supergravity in five dimensions. In particular, we establish an expression for the generalised metric on the exceptional tangent bundle, which determines the scalar truncation ansatz. As applications, we show that this formalism gives a simple derivation of a new consistent truncation of type IIB supergravity on β-deformed Lunin-Maldacena geometries, yielding half-maximal supergravity coupled to two vector multiplets, and of the truncation of eleven-dimensional supergravity on Maldacena-Núñez geometries, given by S4 twisted over a Riemann surface, which leads to half-maximal supergravity coupled to three vector multiplets.

Lee K, Strickland-Constable C, Waldram D, 2017, Spheres, generalised parallelisability and consistent truncations, *Fortschritte der Physik / Progress of Physics*, Vol: 65, ISSN: 0015-8208

We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten- and eleven-dimensional supergravity. In all cases the reduction manifold admits a “generalised parallelisation” with a frame algebra with constant coefficients. The consistent truncation then arises as a generalised version of a conventional Scherk–Schwarz reduction with the frame algebra encoding the embedding tensor of the reduced theory. The key new result is that all round-sphere math formula geometries admit such generalised parallelisations with an math formula frame algebra. Thus we show that the remarkable consistent truncations on S3, S4, S5 and S7 are in fact simply generalised Scherk–Schwarz reductions. This description leads directly to the standard non-linear scalar-field ansatze and as an application we give the full scalar-field ansatz for the type IIB truncation on S5.

Lee K, Strickland-Constable C, Waldram D, 2017, New Gaugings and Non-Geometry, *Fortschritte der Physik / Progress of Physics*, Vol: 65, ISSN: 0015-8208

We discuss the possible realisation in string/M theory of the recently discovered family of four-dimensional maximal math formula gauged supergravities, and of an analogous family of seven-dimensional half-maximal math formula gauged supergravities. We first prove a no-go theorem that neither class of gaugings can be realised via a compactification that is locally described by ten- or eleven-dimensional supergravity. In the language of Double Field Theory and its M theory analogue, this implies that the section condition must be violated. Introducing the minimal number of additional coordinates possible, we then show that the standard S3 and S7 compactifications of ten- and eleven-dimensional supergravity admit a new class of section-violating generalised frames with a generalised Lie derivative algebra that reproduces the embedding tensor of the math formula and math formula gaugings respectively. The physical meaning, if any, of these constructions is unclear. They highlight a number of the issues that arise when attempting to apply the formalism of Double Field Theory to non-toroidal backgrounds. Using a naive brane charge quantisation to determine the periodicities of the additional coordinates restricts the math formula gaugings to an infinite discrete set and excludes all the math formula gaugings other than the standard one.

Ashmore A, Gabella M, Grana M,
et al., 2017, Exactly marginal deformations from exceptional generalised geometry, *Journal of High Energy Physics*, Vol: 2017, ISSN: 1029-8479

We apply exceptional generalised geometry to the study of exactly marginal deformations of NN = 1 SCFTs that are dual to generic AdS5 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 NN = 2 AdS5 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.

Ashmore A, Waldram D, 2017, Exceptional Calabi-Yau spaces: the geometry of N=2 backgrounds with flux, *FORTSCHRITTE DER PHYSIK-PROGRESS OF PHYSICS*, Vol: 65, ISSN: 0015-8208

Ashmore A, Petrini M, Waldram D, 2016, The exceptional generalised geometry of supersymmetric AdS flux backgrounds, *Journal of High Energy Physics*, Vol: 2016, ISSN: 1029-8479

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 AdS5 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 FF-maximisation) is encoded.

Cassani D, de Felice O, Petrini M,
et al., 2016, Exceptional generalised geometry for massive IIA and consistent reductions, *Journal of High Energy Physics*, Vol: 2016, ISSN: 1126-6708

We develop an exceptional generalised geometry formalism for massive typeIIA supergravity. In particular, we construct a deformation of the generalised Lie derivative,which generates the type IIA gauge transformations as modified by the Romans mass.We apply this new framework to consistent Kaluza-Klein reductions preserving maximalsupersymmetry. We find a generalised parallelisation of the exceptional tangent bundleon S6, and from this reproduce the consistent truncation ansatz and embedding tensorleading to dyonically gauged ISO(7) supergravity in four dimensions. We also discussclosely related hyperboloid reductions, yielding a dyonic ISO(p, 7 − p) gauging. Finally,while for vanishing Romans mass we find a generalised parallelisation on Sd, d = 4, 3, 2,leading to a maximally supersymmetric reduction with gauge group SO(d + 1) (or larger),we provide evidence that an analogous reduction does not exist in the massive theory.

Coimbra A, Strickland-Constable C, Waldram D, 2016, Supersymmetric backgrounds and generalised special holonomy, *Classical and Quantum Gravity*, Vol: 33, ISSN: 1361-6382

We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form ${{\mathbb{R}}}^{D-\mathrm{1,1}}\times M$. Using the language of ${E}_{d(d)}\times {{\mathbb{R}}}^{+}$ generalised geometry, we show that, for $D\geqslant 4$, preserving minimal supersymmetry is equivalent to the manifold M having generalised special holonomy and list the relevant holonomy groups. We conjecture that this result extends to backgrounds preserving any number of supersymmetries. As a prime example, we consider ${ \mathcal N }=1$ in D = 4. The corresponding generalised special holonomy group is ${SU}(7)$, giving the natural M theory extension to the notion of a G 2 manifold, and, for Type II backgrounds, reformulating the pure spinor ${SU}(3)\times {SU}(3)$ conditions as an integrable structure.

Grana M, Louis J, Theis U,
et al., 2015, Quantum corrections in string compactifications on SU(3) structure geometries, *Journal of High Energy Physics*, Vol: 2015, ISSN: 1029-8479

We investigate quantum corrections to the classical four-dimensional lowenergyeffective action of type II string theory compactified on SU(3) structure geometries.Various methods previously developed for Calabi-Yau compactifications are adopted toconstrain – under some simple assumptions about the low-energy degrees of freedom – theleading perturbative corrections to the moduli space metrics in both α0 and the stringcoupling constant. We find that they can be parametrized by a moduli dependent functionin the hypermultiplet sector and a constant in the vector multiplet sector. We argue thatunder specific additional assumption they take – in complete analogy to the Calabi-Yaucase – a universal form which depends only on the Euler characteristic of the six-dimensionalcompact space.

Coimbra A, Minasian R, Triendl H,
et al., 2014, Generalised geometry for string corrections, *Journal of High Energy Physics*, Vol: 2014, ISSN: 1029-8479

We present a general formalism for incorporating the string corrections in generalised geometry, which necessitates the extension of the generalised tangent bundle. Not only are such extensions obstructed, string symmetries and the existence of a welldefined effective action require a precise choice of the (generalised) connection. The action takes a universal form given by a generalised Lichnerowitz-Bismut theorem. As examples of this construction we discuss the corrections linear in α′ in heterotic strings and the absence of such corrections for type II theories.

Coimbra A, Strickland-Constable C, Waldram D, 2014, Supergravity as generalised geometry II: Ed(d) x R+ and M theory, *Journal of High Energy Physics*, Vol: 2014, ISSN: 1029-8479

We reformulate eleven-dimensional supergravity, including fermions, in termsof generalised geometry, for spacetimes that are warped products of Minkowski space witha d-dimensional manifold M with d ≤ 7. The reformulation has an Ed(d) × R+ structuregroup and it has a local H˜d symmetry, where H˜d is the double cover of the maximallycompact subgroup of Ed(d). The bosonic degrees for freedom unify into a generalisedmetric, and, defining the generalised analogue D of the Levi-Civita connection, one findsthat the corresponding equations of motion are the vanishing of the generalised Riccitensor. To leading order, we show that the fermionic equations of motion, action andsupersymmetry variations can all be written in terms of D. Although we will not givethe detailed decompositions, this reformulation is equally applicable to type IIA or IIBsupergravity restricted to a (d−1)-dimensional manifold. For completeness we give explicitexpressions in terms of H˜4 = Spin(5) and H˜7 = SU(8) representations for d = 4 and d = 7.

Coimbra A, Strickland-Constable C, Waldram D, 2014, Ed(d) × R+ generalised geometry, connections and M theory, *Journal of High Energy Physics*, Vol: 2014, ISSN: 1029-8479

We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a d-dimensional manifold for all d ≤ 7. The theory is based on an extended tangent space which admits a natural Ed(d)×R+Ed(d)×R+ action. The bosonic degrees of freedom are unified as a “generalised metric”, as are the diffeomorphism and gauge symmetries, while the local O(d) symmetry is promoted to Hd, the maximally compact subgroup of Ed(d). We introduce the analogue of the Levi-Civita connection and the Ricci tensor and show that the bosonic action and equations of motion are simply given by the generalised Ricci scalar and the vanishing of the generalised Ricci tensor respectively. The formalism also gives a unified description of the bosonic NSNS and RR sectors of type II supergravity in d − 1 dimensions. Locally the formulation also describes M-theory variants of double field theory and we derive the corresponding section condition in general dimension. We comment on the relation to other approaches to M theory with Ed(d) symmetry, as well as the connections to flux compactifications and the embedding tensor formalism.

Coimbra A, Strickland-Constable C, Waldram D, 2012, Generalised Geometry and type II Supergravity, *Fortschritte der Physik / Progress of Physics*, Vol: 60, Pages: 982-986, ISSN: 0015-8208

Ten-dimensional type II supergravity can be reformulated as a generalised geometrical analogue of Einstein gravity, defined by an O(9,1) × O(1,9) ⊂ O(10,10) × ℝ+ structure on the generalised tangent space. To leading order in the fermion fields, this allow one to rewrite the action, equations of motion and supersymmetry variations in a simple, manifestly Spin(9,1) × Spin(1,9)-covariant form.

Gauntlett JP, Sonner J, Waldram D, 2011, Universal fermionic spectral functions from string theory, *Phys.Rev.Lett.*, Vol: 107, Pages: 241601-241601

Gauntlett JP, Sonner J, Waldram D, 2011, Spectral function of the supersymmetry current, *JHEP*, Vol: 1111, Pages: 153-153

Coimbra A, Strickland-Constable C, Waldram D, 2011, Supergravity as Generalised Geometry I: Type II Theories, *JHEP*, Vol: 1111, Pages: 091-091

Gabella M, Gauntlett JP, Palti E,
et al., 2010, AdS(5) Solutions of Type IIB Supergravity and Generalized Complex Geometry, *Commun.Math.Phys.*, Vol: 299, Pages: 365-408-365-408

Gabella M, Gauntlett JP, Palti E,
et al., 2009, The Central charge of supersymmetric AdS(5) solutions of type IIB supergravity, *Phys.Rev.Lett.*, Vol: 103, Pages: 051601-051601

Grana M, Minasian R, Petrini M,
et al., 2009, T-duality, Generalized Geometry and Non-Geometric Backgrounds, *JHEP*, Vol: 0904, Pages: 075-075

Gauntlett JP, Kim S, Varela O,
et al., 2009, Consistent supersymmetric Kaluza-Klein truncations with massive modes, *JHEP*, Vol: 0904, Pages: 102-102

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