Duality covariant non-BPS first order systems

We study extremal black hole solutions to four dimensional N=2 supergravity based on a cubic symmetric scalar manifold. Using the coset construction available for these models, we define the first order flow equations implied by the corresponding nilpotency conditions on the three-dimensional scalar momenta for the composite non-BPS class of multi-centre black holes. As an application, we directly solve these equations for the single-centre subclass, and write the general solution in a manifestly duality covariant form. This includes all single-centre under-rotating non-BPS solutions, as well as their non-interacting multi-centre generalisations.Comment: 31 pages, v2: Discussion of the quadratic constraint clarified, references added, typos corrected, published versio

Subtracted Geometry From Harrison Transformations

We consider the rotating non-extremal black hole of N=2 D=4 STU supergravity carrying three magnetic charges and one electric charge. We show that its subtracted geometry is obtained by applying a specific SO(4,4) Harrison transformation on the black hole. As previously noted, the resulting subtracted geometry is a solution of the N=2 S=T=U supergravity.Comment: 11 pages main text; total 24 pages; Latex file; v2 typos corrected + ref added; v3 results significantly strengthened, changes in section 3.1 and appendix C, version to appear in JHE

Multi-black holes from nilpotent Lie algebra orbits

For N \ge 2 supergravities, BPS black hole solutions preserving four supersymmetries can be superposed linearly, leading to well defined solutions containing an arbitrary number of such BPS black holes at arbitrary positions. Being stationary, these solutions can be understood via associated non-linear sigma models over pseudo-Riemaniann spaces coupled to Euclidean gravity in three spatial dimensions. As the main result of this paper, we show that whenever this pseudo-Riemanniann space is an irreducible symmetric space G/H*, the most general solutions of this type can be entirely characterised and derived from the nilpotent orbits of the associated Lie algebra Lie(G). This technique also permits the explicit computation of non-supersymmetric extremal solutions which cannot be obtained by truncation to N=2 supergravity theories. For maximal supergravity, we not only recover the known BPS solutions depending on 32 independent harmonic functions, but in addition find a set of non-BPS solutions depending on 29 harmonic functions. While the BPS solutions can be understood within the appropriate N=2 truncation of N=8 supergravity, the general non-BPS solutions require the whole field content of the theory.Comment: Corrected version for publication, references adde

An R^4 non-renormalisation theorem in N=4 supergravity

We consider the four-graviton amplitudes in CHL constructions providing four-dimensional N=4 models with various numbers of vector multiplets. We show that in these models the two-loop amplitude has a prefactor of d^2R^4. This implies a non-renormalisation theorem for the R^4 term, which forbids the appearance of a three-loop ultraviolet divergence in four dimensions in the four-graviton amplitude. We connect the special nature of the R^4 term to the U(1) anomaly of pure N=4 supergravity.Comment: v2: added comments about one-loop UV divergences. Assorted stylistic corrections. Added references. v3: Eq. III.21 corrected and assorted minor corrections and clarifications. Version to be published. v4: minor corrections. 18 pages. one figur

Non-extremal black holes from the generalised r-map

We review the timelike dimensional reduction of a class of five-dimensional theories that generalises 5D, N = 2 supergravity coupled to vector multiplets. As an application we construct instanton solutions to the four-dimensional Euclidean theory, and investigate the criteria for solutions to lift to static non-extremal black holes in five dimensions. We focus specifically on two classes of models: STU-like models, and models with a block diagonal target space metric. For STU-like models the second order equations of motion of the four-dimensional theory can be solved explicitly, and we obtain the general solution. For block diagonal models we find a restricted class of solutions, where the number of independent scalar fields depends on the number of blocks. When lifting these solutions to five dimensions we show, by explicit calculation, that one obtains static non-extremal black holes with scalar fields that take finite values on the horizon only if the number of integration constants reduces by exactly half.Comment: 22 pages. Based on talk by OV at "Black Objects in Supergravity School" (BOSS2011), INFN, Frascati, Italy, 9-13 May, 201

Extremal Multicenter Black Holes: Nilpotent Orbits and Tits Satake Universality Classes

Four dimensional supergravity theories whose scalar manifold is a symmetric coset manifold U[D=4]/Hc are arranged into a finite list of Tits Satake universality classes. Stationary solutions of these theories, spherically symmetric or not, are identified with those of an euclidian three-dimensional sigma-model, whose target manifold is a Lorentzian coset U[D=3]/H* and the extremal ones are associated with H* nilpotent orbits in the K* representation emerging from the orthogonal decomposition of the algebra U[D=3] with respect to H*. It is shown that the classification of such orbits can always be reduced to the Tits-Satake projection and it is a class property of the Tits Satake universality classes. The construction procedure of Bossard et al of extremal multicenter solutions by means of a triangular hierarchy of integrable equations is completed and converted into a closed algorithm by means of a general formula that provides the transition from the symmetric to the solvable gauge. The question of the relation between H* orbits and charge orbits W of the corresponding black holes is addressed and also reduced to the corresponding question within the Tits Satake projection. It is conjectured that on the vanishing locus of the Taub-NUT current the relation between H*-orbit and W-orbit is rigid and one-to-one. All black holes emerging from multicenter solutions associated with a given H* orbit have the same W-type. For the S^3 model we provide a complete survey of its multicenter solutions associated with all of the previously classified nilpotent orbits of sl(2) x sl(2) within g[2,2]. We find a new intrinsic classification of the W-orbits of this model that might provide a paradigm for the analogous classification in all the other Tits Satake universality classes.Comment: 83 pages, LaTeX; v2: few misprints corrected and references adde

All the timelike supersymmetric solutions of all ungauged d=4 supergravities

We determine the form of all timelike supersymmetric solutions of all N greater or equal than 2, d=4 ungauged supergravities, for N less or equal than 4 coupled to vector supermultiplets, using the $Usp(n+1,n+1)-symmetric formulation of Andrianopoli, D'Auria and Ferrara and the spinor-bilinears method, while preserving the global symmetries of the theories all the way. As previously conjectured in the literature, the supersymmetric solutions are always associated to a truncation to an N=2 theory that may include hypermultiplets, although fields which are eliminated in the truncations can have non-trivial values, as is required by the preservation of the global symmetry of the theories. The solutions are determined by a number of independent functions, harmonic in transverse space, which is twice the number of vector fields of the theory (n+1). The transverse space is flat if an only if the would-be hyperscalars of the associated N=2 truncation are trivial.Comment: v3: Some changes in the introduction. Version to be published in JHE

First-order flows and stabilisation equations for non-BPS extremal black holes

We derive a generalised form of flow equations for extremal static and rotating non-BPS black holes in four-dimensional ungauged N = 2 supergravity coupled to vector multiplets. For particular charge vectors, we give stabilisation equations for the scalars, analogous to the BPS case, describing full known solutions. Based on this, we propose a generic ansatz for the stabilisation equations, which surprisingly includes ratios of harmonic functions.Comment: 27 pages; v2: presentation improved and references added as in the published versio

Octonionic black holes

Using algebraic tools inspired by the study of nilpotent orbits in simple Lie algebras, we obtain a large class of solutions describing interacting non-BPS black holes in N=8 supergravity, which depend on 44 harmonic functions. For this purpose, we consider a truncation E6(6) / Sp(8,R) of the non-linear sigma model describing stationary solutions of the theory, which permits a reduction of algebraic computations to the multiplication of 27 by 27 matrices. The lift to N=8 supergravity is then carried out without loss of information by using a pertinent representation of the moduli parametrizing E7(7) / SU(8) in terms of complex valued Hermitian matrices over the split octonions, which generalise the projective coordinates of exceptional special Kaehler manifolds. We extract the electromagnetic charges, mass and angular momenta of the solutions, and exhibit the duality invariance of the black holes distance separations. We discuss in particular a new type of interaction which appears when interacting non-BPS black holes are not aligned. Finally we will explain the possible generalisations toward the description of the most general stationary black hole solutions of N=8 supergravity.Comment: 90 pages, Corrected version for publication in JHE

String cosmology from Poisson-Lie T-dual sigma models on supermanifolds

We generalize the formulation of Poisson-Lie T-dual sigma models on manifolds to supermanifolds. In this respect, we formulate 1+1 dimensional string cosmological models on the Lie supergroup C^3 and its dual (A_1,1 + 2A)^0_(1,0,0), which are coupled to two fermionic fields. Then, we solve the equations of motion of the models and show that there is a essential singularity for the metric of the original model and its dual.Comment: 17 pages, Appendix and three references have adde