Beyond E11

We study the non-linear realisation of E11 originally proposed by West with particular emphasis on the issue of linearised gauge invariance. Our analysis shows even at low levels that the conjectured equations can only be invariant under local gauge transformations if a certain section condition that has appeared in a different context in the E11 literature is satisfied. This section condition also generalises the one known from exceptional field theory. Even with the section condition, the E11 duality equation for gravity is known to miss the trace component of the spin connection. We propose an extended scheme based on an infinite-dimensional Lie superalgebra, called the tensor hierarchy algebra, that incorporates the section condition and resolves the above issue. The tensor hierarchy algebra defines a generalised differential complex, which provides a systematic description of gauge invariance and Bianchi identities. It furthermore provides an E11 representation for the field strengths, for which we define a twisted first order self-duality equation underlying the dynamics.Comment: 97 pages. v2: Minor changes, references added. Published versio

Separability in Cohomogeneity-2 Kerr-NUT-AdS Metrics

The remarkable and unexpected separability of the Hamilton-Jacobi and Klein-Gordon equations in the background of a rotating four-dimensional black hole played an important role in the construction of generalisations of the Kerr metric, and in the uncovering of hidden symmetries associated with the existence of Killing tensors. In this paper, we show that the Hamilton-Jacobi and Klein-Gordon equations are separable in Kerr-AdS backgrounds in all dimensions, if one specialises the rotation parameters so that the metrics have cohomogeneity 2. Furthermore, we show that this property of separability extends to the NUT generalisations of these cohomogeneity-2 black holes that we obtained in a recent paper. In all these cases, we also construct the associated irreducible rank-2 Killing tensor whose existence reflects the hidden symmetry that leads to the separability. We also consider some cohomogeneity-1 specialisations of the new Kerr-NUT-AdS metrics, showing how they relate to previous results in the literature.Comment: Latex, 15 pages, minor typos correcte

Universal Area Product Formulas for Rotating and Charged Black Holes in Four and Higher Dimensions

We present explicit results for the product of all horizon areas for general rotating multicharge black holes, both in asymptotically flat and asymptotically anti–de Sitter spacetimes in four and higher dimensions. The expressions are universal, and depend only on the quantized charges, quantized angular momenta and the cosmological constant. If the latter is also quantized these universal results may provide a ‘‘looking glass’’ for probing the microscopics of general black holes

Mass And Force Relations For Einstein-Maxwell-Dilaton Black Holes

We investigate various properties of extremal dyonic static black holes in Einstein-Maxwell-Dilaton theory. Using the fact that the long-range force between two identical extremal black holes always vanishes, we obtain a simple first-order ordinary differential equation for the black hole mass in terms of its electric and magnetic charges. Although this equation appears not to be solvable explicitly for general values of the strength a of the dilatonic coupling to the Maxwell field, it nevertheless provides a powerful way of characterising the black hole mass and the scalar charge. We make use of these expressions to derive general results about the long-range force between two non-identical extremal black holes. In particular, we argue that the force is repulsive whenever a>1 and attractive whenever a<1 (it vanishes in the intermediate BPS case a=1). The sign of the force is also correlated with the sign of the binding energy between extremal black holes, as well as with the convexity or concavity of the surface characterizing the extremal mass as a function of the charges. Our work is motivated in part by the Repulsive Force Conjecture and the question of whether long range forces between non-identical states can shed new light on the Swampland.Comment: 34 pages, 3 figure

Generalized Dualities and Supergroups

Using a recently developed formulation of double field theory in superspace, the graviton, $B$-field, gravitini, dilatini, and Ramond-Ramond bispinor are encoded in a single generalized supervielbein. Duality transformations are encoded as orthosymplectic transformations, extending the bosonic $O(D,D)$ duality group, and these act on all constituents of the supervielbein in an easily computable way. We first review conventional non-abelian T-duality in the Green-Schwarz superstring and describe the dual geometries in the language of double superspace. Since dualities are related to super-Killing vectors, this includes as special cases both abelian and non-abelian fermionic T-duality. We then extend this approach to include Poisson-Lie T-duality and its generalizations, including the generalized coset construction recently discussed in arXiv:1912.11036. As an application, we construct the supergeometries associated with the integrable $\lambda$ and $\eta$ deformations of the $AdS_5 \times S^5$ superstring. The deformation parameters $\lambda$ and $\eta$ are identified with the possible one-parameter embeddings of the supergravity frame within the doubled supergeometry. In this framework, the Ramond-Ramond bispinors are directly computable purely from the algebraic data of the supergroup.Comment: 85 pages; v2: references added, additional comments in introduction and conclusio

Consistent Truncations and Dualities

Recent progress in generalised geometry and extended field theories suggests a deep connection between consistent truncations and dualities, which is not immediately obvious. A prime example is generalised Scherk-Schwarz reductions in double field theory, which have been shown to be in one-to-one correspondence with Poisson-Lie T-duality. Here we demonstrate that this relation is only the tip of the iceberg. Currently, the most general known classes of T-dualities (excluding mirror symmetry) are based on dressing cosets. But as we discuss, they can be further extended to the even larger class of generalised cosets. We prove that the latter give rise to consistent truncations for which the ansatz can be constructed systematically. Hence, we pave the way for many new examples of T-dualities and consistent truncations. The arising structures result in covariant tensors with more than two derivatives and we argue how they might be key to understand generalised T-dualities and consistent truncations beyond the leading two derivative level.Comment: 43 page

M-theory on special holonomy spaces

We review the construction of regular p-brane solutions of M-theory and string theory with less than maximal supersymmetry whose transverse spaces have metrics with special holonomy, and where additional fluxes allow brane resolutions via transgression terms. We summarize the properties of resolved M2-branes and fractional D2-branes, whose transverse spaces are Ricci flat eight-dimensional and seven-dimensional spaces of special holonomy. © 2002 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87637/2/53_1.pd

Ricci-Flat Metrics, Harmonic Forms and Brane Resolutions

 We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel metric, on the tangent bundle of S n+1 . We obtain explicit results for all the metrics, and show how they can be obtained from first-order equations derivable from a superpotential. We then provide an explicit construction for the harmonic self-dual (p, q) -forms in the middle dimension p+q=(n+1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p, p) -forms are L 2 -normalisable, while for (p, q) -forms the degree of divergence grows with . We also construct a set of Ricci-flat metrics whose level surfaces are U(1) bundles over a product of N Einstein-Kähler manifolds, and we construct examples of harmonic forms there. As an application, we construct new examples of deformed supersymmetric non-singular M2-branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2,1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1,2) and (2,1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42006/1/32320457.pd