Extremal black holes and nilpotent orbits

The stationary solutions of a large variety of (super)gravity theories can be described within a non-linear sigma model G / H* coupled to Euclidean gravity in three-dimensions, for which G is a simple group and H* a non-compact real form of its maximal compact subgroup. The absence of naked singularities in four dimensions requires the G Noether charge in 3D to satisfy a characteristic equation that determines it in function of the mass, the NUT charge and the electro-magnetic charges of the solution. It follows that the Noether charge associated to extremal black holes must lie in a certain Lagrangian submanifold of a nilpotent orbit of G. Constructing a suitable parameterisation of this Lagrangian, we are able to determine the so-called `fake superpotential' that governs the radial dependency of the scalar fields.Comment: Invited talk at the XVI International Congress on Mathematical Physics, Prague, 3-8 August 200

1/8 BPS black hole composites

We show that the 1/8 BPS condition for composite stationary black holes can be rewritten as a first order system of differential equations associated to the nilpotent orbit in which lie the Noether charges of the black holes. Solving these equations, we prove that the most general 1/8 BPS black hole composites are solutions of the N=2 truncation of the theory associated to the quaternions. This system of first order differential equations generalises to the non-BPS solutions with a vanishing central charge at the horizon in N=2, 4 supergravity theories with a symmetric moduli space. We solve these equations for the exceptional N=2 supergravity associated to the octonions.Comment: 19 page

Shadows and Twisted Variables

We explain how a new type of fields called shadows and the use of twisted variables allow for a better description of Yang-Mills supersymmetric theories. (Based on lectures given in Cargese, June 2006.)Comment: Cargese Jun 200

Loops in exceptional field theory

We study certain four-graviton amplitudes in exceptional field theory in dimensions $D\geq 4$ up to two loops. As the formulation is manifestly invariant under the U-duality group $E_{11-D}(\mathbb{Z})$, our resulting expressions can be expressed in terms of automorphic forms. In the low energy expansion, we find terms in the M-theory effective action of type $R^4$, $\nabla^4R^4$ and $\nabla^6 R^4$ with automorphic coefficient functions in agreement with independent derivations from string theory. This provides in particular an explicit integral formula for the exact string theory $\nabla^6 R^4$ threshold function. We exhibit moreover that the usual supergravity logarithmic divergences cancel out in the full exceptional field theory amplitude, within an appropriately defined dimensional regularisation scheme. We also comment on terms of higher derivative order and the role of the section constraint for possible counterterms.Comment: 1+78 pages: v2: Added references and typos corrected. Version accepted by JHEP. v3: corrected discussion of associativity in EFT and other typo

The two D6R4 type invariants and their higher order generalisation

We show that there are two distinct classes of D6R4 type supersymmetry invariants in maximal supergravity. The second class includes a coupling in F2D4R4 that generalises to 1/8 BPS protected F2kD4R4 couplings. We work out the supersymmetry constraints on the corresponding threshold functions, and argue that the functions in the second class satisfy to homogeneous differential equations for arbitrary k>0, such that the corresponding exact threshold functions in type II string theory should be proportional to Eisenstein series, which we identify. This analysis explains in particular that the exact D6R4 threshold function is the sum of an Eisenstein function and a solution to an inhomogeneous Poisson equation in string theory.Comment: 53 page

non-BPS walls of marginal stability

We explore the properties of non-BPS multi-centre extremal black holes in ungauged N=2 supergravity coupled to n_v vector multiplets, as described by solutions to the composite non-BPS linear system. After setting up an explicit description that allows for arbitrary non-BPS charges to be realised at each centre, we study the structure of the resulting solutions. Using these results, we prove that the binding energy of the composite is always positive and we show explicitly the existence of walls of marginal stability for generic choices of charges. The two-centre solutions only exist on a hypersurface of dimension n_v+1 in moduli space, with an n_v-dimensional boundary, where the distance between the centres diverges and the binding energy vanishes.Comment: 54 pages, 1 figur

Minimal unitary representations from supersymmetry

We compute the supersymmetry constraints on the R^4 type corrections in maximal supergravity in dimension 8, 6, 4 and 3, and determine the tensorial differential equations satisfied by the function of the scalar fields multiplying the R^4 term in the corresponding invariants. The second order derivative of this function restricted to the Joseph ideal vanishes in dimension lower than six. These results are extended to the d^4 R^4 and the d^6 R^4 corrections, based on the harmonic superspace construction of these invariants in the linearised approximation. We discuss the solutions of these differential equations and analysis the consequences on the non-perturbative type II low energy string theory effective action.Comment: 84 pages, Corrected version for publication in JHEP, additional comment on d^6 R^4 in four dimension

A bubbling bolt

We present a new solvable system, solving the equations of five-dimensional ungauged N=1 supergravity coupled to vector multiplets, that allows for non-extremal solutions and reduces to a known system when restricted to the floating brane Ansatz. A two-centre globally hyperbolic smooth geometry is obtained as a solution to this system, describing a bubble linking a Gibbons--Hawking centre to a charged bolt. However this solution turns out to violate the BPS bound, and we show that its generalisation to an arbitrary number of Gibbons--Hawking centres never admits a spin structure.Comment: 36 page

Interacting non-BPS black holes

We explain how to exploit systematically the structure of nilpotent orbits to obtain a solvable system of equations describing extremal solutions of (super-)gravity theories, i.e. systems that can be solved in a linear way. We present the procedure in the case of the STU model, where we show that all extremal solutions with a flat three-dimensional base are fully described with the help of three different nilpotent orbits: the BPS, the almost-BPS and the composite non-BPS. The latter describes a new class of solutions for which the orientation of half of the constituent branes have been inverted with respect to the BPS one, such that all the centres are intrinsically non-BPS, and interact with each others. We finally recover explicitly the ensemble of the almost-BPS solutions in our formalism and present an explicit two-centre solution of the new class.Comment: 49 page