arXiv subjects

Daniel Waldram

Publications and source records attributed to Daniel Waldram.

At least 37 records · Page 2Linked to original sources

arXiv2015

Exceptional Calabi--Yau spaces: the geometry of $\mathcal{N}=2$ backgrounds with flux

In this paper we define the analogue of Calabi--Yau geometry for generic $D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory. We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in $E_{7(7)}\times\mathbb{R}^+$ generalised geometry. Such "exceptional Calabi--Yau" geometries are determined by two generalised objects that parametrise hyper- and vector-multiplet degrees of freedom and generalise conventional complex, symplectic and hyper-Kahler geometries. The integrability conditions for both hyper- and vector-multiplet structures are given by the vanishing of moment maps for the "generalised diffeomorphism group" of diffeomorphisms combined with gauge transformations. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to $D=5$ and $D=6$ flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in $O(d,d)\times\mathbb{R}^+$ generalised geometry.

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arXiv2015

New gaugings and non-geometry

We discuss the possible realisation in string/M theory of the recently discovered family of four-dimensional maximal $SO(8)$ gauged supergravities, and of an analogous family of seven-dimensional half-maximal $SO(4)$ 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 $S^3$ and $S^7$ 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 $SO(4)$ and $SO(8)$ 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 $SO(4)$ gaugings to an infinite discrete set and excludes all the $SO(8)$ gaugings other than the standard one.

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arXiv2014

Quantum Corrections in String Compactifications on SU(3) Structure Geometries

We investigate quantum corrections to the classical four-dimensional low-energy effective action of type II string theory compactified on SU(3) structure geometries. Various methods previously developed for Calabi-Yau compactifications are adopted to determine - under some simple assumptions about the low-energy degrees of freedom - the leading perturbative corrections to the moduli space metrics in both alpha' and the string coupling constant. We find - in complete analogy to the Calabi-Yau case - that the corrections take a universal form dependent only on the Euler characteristic of the six-dimensional compact space.

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arXiv2014

Supersymmetric Backgrounds and Generalised Special Holonomy

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 $\mathbb{R}^{D-1,1}\times M$. Using the language of $E_{d(d)}\times\mathbb{R}^+$ generalised geometry, we show that, for $D\geq 4$, 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.

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arXiv2014

Generalised geometry for string corrections

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 well-defined 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.

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arXiv2014

Spheres, generalised parallelisability and consistent truncations

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 $S^d$ geometries admit such generalised parallelisations with an $SO(d+1)$ frame algebra. Thus we show that the remarkable consistent truncations on $S^3$, $S^4$, $S^5$ and $S^7$ 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 $S^5$.

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arXiv2013

$E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory

We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a $d$-dimensional manifold for all $d\leq7$. The theory is based on an extended tangent space which admits a natural $E_{d(d)} \times \mathbb{R}^+$ 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 $H_d$, the maximally compact subgroup of $E_{d(d)}$. 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 $E_{d(d)}$ symmetry, as well as the connections to flux compactifications and the embedding tensor formalism.

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arXiv2013

Supergravity as Generalised Geometry II: $E_{d(d)} \times \mathbb{R}^+$ and M theory

We reformulate eleven-dimensional supergravity, including fermions, in terms of generalised geometry, for spacetimes that are warped products of Minkowski space with a $d$-dimensional manifold $M$ with $d\leq7$. The reformation has a $E_{d(d)} \times \mathbb{R}^+$ structure group and is has a local $\tilde{H}_d$ symmetry, where $\tilde{H}_d$ is the double cover of the maximally compact subgroup of $E_{d(d)}$. The bosonic degrees for freedom unify into a generalised metric, and, defining the generalised analogue $D$ of the Levi-Civita connection, one finds that the corresponding equations of motion are the vanishing of the generalised Ricci tensor. To leading order, we show that the fermionic equations of motion, action and supersymmetry variations can all be written in terms of $D$. Although we will not give the detailed decompositions, this reformulation is equally applicable to type IIA or IIB supergravity restricted to a $(d-1)$-dimensional manifold. For completeness we give explicit expressions in terms of $\tilde{H}_4=\mathit{Spin}(5)$ and $\tilde{H}_7=\mathit{SU}(8)$ representations for $d=4$ and $d=7$.

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arXiv2013

Supergravity as Generalised Geometry I: Type II Theories

We reformulate ten-dimensional type II supergravity as a generalised geometrical analogue of Einstein gravity, defined by an $O(9,1)\times O(1,9)\subset O(10,10)\times\mathbb{R}^+$ structure on the generalised tangent space. Using the notion of generalised connection and torsion, we introduce the analogue of the Levi-Civita connection, and derive the corresponding tensorial measures of generalised curvature. We show how, to leading order in the fermion fields, these structures allow one to rewrite the action, equations of motion and supersymmetry variations in a simple, manifestly $\mathit{Spin}(9,1)\times\mathit{Spin}(1,9)$-covariant form.

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arXiv2012

Generalised Geometry and type II Supergravity

Ten-dimensional type II supergravity can be reformulated as a generalised geometrical analogue of Einstein gravity, defined by an $O(9,1)\times O(1,9)\subset O(10,10)\times\mathbb{R}^+$ 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 $Spin(9,1)\times Spin(1,9)$-covariant form.

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arXiv2011

Spectral function of the supersymmetry current

We continue our study of the retarded Green's function of the universal fermionic supersymmetry current ("supercurrent") for the most general class of d=3 N=2 SCFTs with D=10 or D=11 supergravity duals by studying the propagation of the Dirac gravitino in the electrically charged AdS-Reissner-Nordström black-brane background of N=2 minimal gauged supergravity in D=4. We expand upon results presented in a companion paper, including the absence of a Fermi surface and the appearance of a soft power-law gap at zero temperature. We also present the analytic solution of the gravitino equation in the AdS_2 X R^2 background which arises as the near-horizon limit at zero temperature. In addition we determine the quasinormal mode spectrum.

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arXiv2011

Universal fermionic spectral functions from string theory

We carry out the first holographic calculation of a fermionic response function for a strongly coupled $d=3$ system with an explicit D=10 or D=11 supergravity dual. By considering the supersymmetry current, we obtain a universal result applicable to all d=3 N=2 SCFTs with such duals. Surprisingly, the spectral function does not exhibit a Fermi surface, despite the fact that the system is at finite charge density. We show that it has a phonino pole and at low frequencies there is a depletion of spectral weight with a power-law scaling which is governed by a locally quantum critical point.

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arXiv2010

AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry

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 generalized complex geometry, which is an extension of Kahler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.

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arXiv2009

E7(7) formulation of N=2 backgrounds

In this paper we reformulate N=2 supergravity backgrounds arising in type II string theory in terms of quantities transforming under the U-duality group E7(7). In particular we combine the Ramond--Ramond scalar degrees of freedom together with the O(6,6) pure spinors which govern the Neveu-Schwarz sector by considering an extended version of generalised geometry. We give E7(7)-invariant expressions for the Kahler and hyperkahler potentials describing the moduli space of vector and hypermultiplets, demonstrating that both correspond to standard E7(7) coset spaces. We also find E7(7) expressions for the Killing prepotentials defining the scalar potential, and discuss the equations governing N=1 vacua in this formalism.

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arXiv2009

The central charge of supersymmetric AdS_5 solutions of type IIB supergravity

We show that generic supersymmetric AdS_5 solutions of type IIB supergravity admit a canonical contact structure. This structure determines the central charge of the dual field theory and the conformal dimension of operators dual to supersymmetric wrapped D3-branes. Hence both quantities can be calculated using incomplete information about the solutions, allowing us to prove that they are rational numbers for solutions with a U(1) R-symmetry, in agreement with field theory expectations. We also discuss related Duistermaat-Heckman integrals and localization formulae.

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arXiv2009

SU(3) x SU(3) compactification and mirror duals of magnetic fluxes

This paper analyses type II string theories in backgrounds which admit an SU(3) x SU(3) structure. Such backgrounds are designed to linearly realize eight out of the original 32 supercharges and as a consequence the low-energy effective action can be written in terms of couplings which are closely related to the couplings of four-dimensional N=2 theories. This generalizes the previously studied case of SU(3) backgrounds in that the left- and right-moving sector each have a different globally defined spinor. Given a truncation to a finite number of modes, these backgrounds lead to a conventional four-dimensional low-energy effective theory. The results are manifestly mirror symmetric and give terms corresponding to the mirror dual couplings of Calabi-Yau compactifications with magnetic fluxes. It is argued, however, that generically such backgrounds are non-geometric and hence the supergravity analysis is not strictly valid. Remarkably, the naive generalization of the geometrical expressions nonetheless appears to give the correct low-energy effective theory.

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arXiv2009

Consistent supersymmetric Kaluza--Klein truncations with massive modes

We construct consistent Kaluza--Klein reductions of D=11 supergravity to four dimensions using an arbitrary seven-dimensional Sasaki--Einstein manifold. At the level of bosonic fields, we extend the known reduction, which leads to minimal N=2 gauged supergravity, to also include a multiplet of massive fields, containing the breathing mode of the Sasaki--Einstein space, and still consistent with N=2 supersymmetry. In the context of flux compactifications, the Sasaki--Einstein reductions are generalizations of type IIA SU(3)-structure reductions which include both metric and form-field flux and lead to a massive universal tensor multiplet. We carry out a similar analysis for an arbitrary weak G_2 manifold leading to an N=1 supergravity with massive fields. The straightforward extension of our results to the case of the seven-sphere would imply that there is a four-dimensional Lagrangian with N=8 supersymmetry containing both massless and massive spin two fields. We use our results to construct solutions of M-theory with non-relativistic conformal symmetry.

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arXiv2008

M-theory, exceptional generalised geometry and superpotentials

We discuss the structure of "exceptional generalised geometry" (EGG), an extension of Hitchin's generalised geometry that provides a unified geometrical description of backgrounds in eleven-dimensional supergravity. On a d-dimensional background, as first described by Hull, the action of the generalised geometrical O(d,d) symmetry group is replaced in EGG by the exceptional U-duality group E_d(d). The metric and form-field degrees of freedom combine into a single geometrical object, so that EGG naturally describes generic backgrounds with flux, and there is an EGG analogue of the Courant bracket which encodes the differential geometry. Our focus is on the case of seven-dimensional backgrounds with N=1 four-dimensional supersymmetry. The corresponding EGG is the generalisation of a G_2-structure manifold. We show it is characterised by an element ϕin a particular orbit of the 912 representation of E_7(7), which defines an SU(7) (subset of E_7(7)) structure. As an application, we derive the generic form of the four-dimensional effective superpotential, and show that it can be written in a universal form, as a homogeneous E_7(7)-invariant functional of ϕ.

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