Multi-black holes from nilpotent Lie algebra orbits

For N 3 2{\mathcal{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-Riemannian spaces coupled to Euclidean gravity in three spatial dimensions. As the main result of this paper, we show that whenever this pseudo-Riemannian space is an irreducible symmetric space \mathfrakG/\mathfrakH^*{\mathfrak{G}/\mathfrak{H}^*}, the most general solutions of this type can be entirely characterised and derived from the nilpotent orbits of the associated Lie algebra \mathfrakg{\mathfrak{g}}. This technique also permits the explicit computation of non-supersymmetric extremal solutions which cannot be obtained by truncation to N=2{\mathcal{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{\mathcal{N}=2} truncation of N=8{\mathcal{N}=8} supergravity, the general non-BPS solutions require the whole field content of the theory.     

Stationary D = 4 black holes in supergravity: The issue of real nilpotent orbits

The complete classification of the nilpotent orbits of SO(2,2)² in the representation (2,2,2,2) , achieved in 14 , is applied to the study of multi‐center, asymptotically flat, extremal black hole solutions to the STU model. These real orbits provide an intrinsic characterization of regular single‐center solutions, which is invariant with respect to the action of the global symmetry group SO(4,4), underlying the stationary solutions of the model, and provide stringent regularity constraints on multi‐centered solutions. The known almost‐BPS and composite non‐BPS solutions are revisited in this setting. We systematically provide, for the relevant SO(2,2)²‐nilpotent orbits of the global Noether charge matrix, regular representatives thereof. This analysis unveils a composition law of the orbits according to which those containing regular multi‐centered solutions can be obtained as combinations of specific single‐center orbits defining the constituent black holes. Some of the SO(2,2)²‐orbits of the total Noether charge matrix are characterized as “intrinsically singular” in that they cannot contain any regular solution.     

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.     

Ann monopole solution with 4n — 1 degrees of freedom

An exact static monopole solution, possessingn units of magnetic charge and (4n-1) degrees of freedom, is constructed, generalising the recent work of Ward on two monopole solutions. The equations solved are those of anSU(2) gauge theory with adjoint representation Higgs field in the (BPS) limit of vanishing Higgs potential. The number of degrees of freedom is maximal for self-dual solutions. The construction is described in a deductive way, within the framework of the Atiyah-Ward formalism for self-dual gauge fields.     

Existence of Static BPS Monopoles and Dyons in Arbitrary (4p – 1)-Dimensional Spaces

We prove the existence and uniqueness of a static and radially symmetric BPS monopole of unit topological charge in an arbitrary (4p – 1)-dimensional space descended from the generalized Yang–Mills theory in 4p dimensions and formulated and presented in a recent study of Radu and Tchrakian. This monopole solution also gives rise to an electrically and magnetically charged soliton, called dyon, in the same spacetime setting through the well-known Julia–Zee correspondence. Our method is based on a dynamical shooting approach depending on two shooting parameters which provides an effective framework for constructing the BPS monopole and dyon solutions in general dimensions.     

Moduli Space of BPS Walls in Supersymmetric Gauge Theories

Existence and uniqueness of the solution are proved for the ‘master equation’ derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with N _F charged matter hypermultiplets with eight supercharges. This proof establishes that the solutions of the BPS equations are completely characterized by the moduli matrices divided by the V-equivalence relation for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(N _C) gauge theory with N _F hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(N _C) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences.     

Some BPS configurations of the BLG theory

We obtain BPS configurations of the BLG theory and its variant including mass terms for scalars and fermions in addition to a background field with different world-volume and R-symmetries. Three cases are considered, with world-volume symmetries SO(1,1) and SO(2) and preserving different amounts of supersymmetry. In the former case we obtain a singular configuration preserving N=(3,3) supersymmetry and an one-quarter BPS configuration corresponding to intersecting M2-M5-M5-branes. In the latter instance the BPS equations are reduced to those in the self-dual Chern-Simons theory with two complex scalars. In want of an exact solution, we find a topological vortex solution numerically in this case. Other solutions are given by combinations of domain walls.     

BPS solutions to a generalized Maxwell–Higgs model

We look for topological BPS solutions of an Abelian Maxwell–Higgs theory endowed by non-standard kinetic terms to both gauge and scalar fields. Here, the non-usual dynamics are controlled by two positive functions, G(|ϕ|) and w(|ϕ|), which are related to the self-dual scalar potential V(|ϕ|) of the model by a fundamental constraint. The numerical results we found present interesting new features, and contribute to the development of the recent issue concerning the study of generalized models and their applications.     

The self-dual string soliton in AdS<Subscript>4</Subscript>×S<Superscript>7</Superscript> spacetime

We construct self-dual string soliton solutions in AdS₄×S⁷ spacetime, starting from the covariant equations of motion of the M5-brane. We study the properties of the solutions and find that their actions are linearized, indicating the BPS nature of the solutions, and we find that they have the same electric and magnetic charges. The straight string soliton solution represents the configuration of the membranes ending on a M5-brane with a straight string intersection, and it behaves like the spiky solution in flat spacetime. The spherical string soliton solution, which could be related to the straight one by a conformal transformation, represents the membranes ending on a M5-brane with a spherical intersection.     

Abelian and non‐abelian D‐brane effective actions

In this Ph.D. thesis, accepted at the Vrije Universiteit Brussel, we review and elaborate on a method to find the D‐brane effective action, based on BPS equations. Firstly, both for the Yang‐Mills action and the Born‐Infeld action it is shown that these configurations are indeed BPS, i.e. solutions to these equations saturate a Bogomolny bound and leave some supersymmetry unbroken. Next, we use the BPS equations as a tool to construct the D‐brane effective action and require that (a deformation of) these equations should still imply the equations of motion in more general cases. In the abelian case we managed to calculate all order in α′ four‐derivative corrections to the effective action and the BPS equations while in the non‐abelian case we obtained the effective action up to order α′⁴. Furthermore, we discuss a check based on the spectrum of strings stretching between intersecting branes. Finally, this Ph.D. thesis also discusses the construction of a boundary superspace which would be the first step to use the method of Weyl invariance in N = 2 superspace in order to again construct the D‐brane effective action. A more detailed summary of each section can be found in the introduction.     

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 ${\mathcal{N}=1}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{\mathcal{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 K?hler-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.     

Magnetic monopoles

These are notes of the first part of the lectures given at the JINR-ISU Baikal Summer School on Physics of Elementary Particles and Astrophysics (July 2010). I review classical monopole solutions of the SU(2) Yang-Mills-Higgs theory providing a pedagogical introduction into to the theory of non-Abelian monopoles both in the BPS limit and beyond of it. I briefly discuss monopole dynamics, the idea of the moduli space and some of the basic properties which are connected with the field theoretical aspects of these classical solutions.     

On the Tetrahedrally Symmetric Monopole

We study SU(2) BPS monopoles with spectral curves of the form η ³+χ(ζ ⁶+b ζ ³−1) = 0. Previous work has established a countable family of solutions to Hitchin’s constraint that L ² was trivial on such a curve. Here we establish that the only curves of this family that yield BPS monopoles correspond to tetrahedrally symmetric monopoles. We introduce several new techniques making use of a factorization theorem of Fay and Accola for theta functions, together with properties of the Humbert variety. The geometry leads us to a formulation purely in terms of elliptic functions. A more general conjecture than needed for the stated result is given.     

Massive nonlinear sigma models and BPS domain walls in harmonic superspace

Four-dimensional massive nonlinear sigma models and BPS wall solutions are studied in the off-shell harmonic superspace approach in which supersymmetry is manifest. The general nonlinear sigma model can be described by an analytic harmonic potential which is the hyper-Kähler analog of the Kähler potential in theory. We examine the massive nonlinear sigma model with multi-center four-dimensional target hyper-Kähler metrics and derive the corresponding BPS equation. We study in some detail two particular cases with the Taub-NUT and double Taub-NUT metrics. The latter embodies, as its two separate limits, both Taub-NUT and Eguchi–Hanson metrics. We find that domain wall solutions exist only in the double Taub-NUT case including its Eguchi–Hanson limit.     

Quaternionic and hyper-Kähler metrics from generalized sigma models

The problem of finding new metrics of interest, in the context of SUGRA, is reduced to two stages: first, solving a generalized BPS sigma model with full quaternionic structure proposed by the authors and, second, constructing the hyper-Kähler metric, or suitable deformations of this condition, taking advantage of the correspondence between the quaternionic left-regular potential and the hyper-Kähler metric of the target space. As illustration, new solutions are obtained using generalized Q-sigma model for Wess–Zumino type superpotentials. Explicit solutions analog to the Berger?s sphere and Abraham–Townsend type are given and generalizations of 4-dimensional quaternionic metrics, product of complex ones, are shown and discussed.     

Hořava–Witten stability: eppur si muove

We construct exact time-dependent solutions of the supergravity equations of motion in which two initially non-singular branes, one with positive and the other with negative tension, move together and annihilate each other in an all-enveloping spacetime singularity. Among our solutions are the Hořava–Witten solution of heterotic M-theory and a Randall–Sundrum I type solution, both of which are supersymmetric, i.e. BPS, in the time-independent case. In the absence of branes our solutions are of Kasner type, and the source of instability may ascribed to a failure to stabilise some of the modulus fields of the compactification. It also raises questions about the viability of models based on some sorts of negative tension brane.     

M‐theory solutions invariant under D(2,1; γ) ⊕ D(2,1;γ)

We simplify and extend the construction of half‐BPS solutions to 11‐dimensional supergravity, with isometry superalgebra D(2,1;γ) ⊕ D(2,1;γ). Their space‐time has the form AdS₃× S³× S³ warped over a Riemann surface Σ. It describes near‐horizon geometries of M2 branes ending on, or intersecting with, M5 branes along a common string. The general solution to the BPS equations is specified by a reduced set of data (γ, h, G), where γ is the real parameter of the isometry superalgebra, and h and G are functions on Σ whose differential equations and regularity conditions depend only on the sign of γ. The magnitude of γ enters only through the map of h,G onto the supergravity fields, thereby promoting all solutions into families parametrized by |γ|. By analyzing the regularity conditions for the supergravity fields, we prove two general theorems: (i) that the only solution with a 2‐dimensional CFT dual is AdS₃× S³× S³× ℝ², modulo discrete identifications of the flat ℝ², and (ii) that solutions with γ < 0 cannot have more than one asymptotic higher‐dimensional AdS region. We classify the allowed singularities of h and G near the boundary of Σ, and identify four local solutions: asymptotic AdS₄/Z₂ or AdS₇^′ regions; highly‐curved M5‐branes; and a coordinate singularity called the “cap”. By putting these “Lego” pieces together we recover all known global regular solutions with the above symmetry, including the self‐dual strings on M5 for γ <0, and the Janus solution for γ > 0, but now promoted to families parametrized by |γ|. We also construct exactly new regular solutions which are asymptotic to AdS₄/Z₂ for γ < 0, and conjecture that they are a different superconformal limit of the self‐dual string. Finally, we construct exactly γ > 0 solutions with highly curved M5‐brane regions, which are the formal continuation of the self‐dual string solutions across the decompactification point at γ = 0.     

Monopole and dyon bound states in N = 2 supersymmetric Yang-Mills theories

We study the existence of monopole bound states saturating the BPS bound in N = 2 supersymmetric Yang-Mills theories. We describe how the existence of such bound states relates to the topology of index bundles over the moduli space of BPS solutions. Using an L² index theorem, we prove the existence of certain BPS states predicted by Seiberg and Witten based on their study of the vacuum structure of N = 2 Yang-Mills theories.     

The strong-coupling spectrum of the Seiberg-Witten theory

We carefully study the global structure of the solution of the N = 2 supersymmetric pure Yang-Mills theory with gauge group SU(2) obtained by Seiberg and Witten. We exploit itsZsymmetry and describe the curve in moduli space where BPS states can become unstable, separating the strong-coupling from the weak-coupling region. This allows us to obtain the spectrum of stable BPS states in the strong-coupling region: we prove that only the two particles responsible for the singularities of the solution (the magnetic monopole and the dyon of unit electric charge) are present in this region. Our method also permits us to very easily obtain the weak-coupling spectrum, without using semiclassical methods. We discuss how the BPS states disintegrate when crossing the border from the weak- to the strong-coupling region.