We find the geometry of all supersymmetric type I backgrounds by solving the gravitino and dilati... more We find the geometry of all supersymmetric type I backgrounds by solving the gravitino and dilatino Killing spinor equations, using the spinorial geometry technique, in all cases. The solutions of the gravitino Killing spinor equation are characterized by their isotropy group in Spin(9, 1), while the solutions of the dilatino Killing spinor equation are characterized by their isotropy group in the subgroup Σ(P) of Spin(9, 1) which preserves the space of parallel spinors P. Given a solution of the gravitino Killing spinor equation with L parallel spinors, L = 1, 2, 3, 4, 5, 6, 8, the dilatino Killing spinor equation allows for solutions with N supersymmetries for any 0 < N ≤ L. Moreover for L = 16, we confirm that N = 8, 10, 12, 14, 16. We find that in most cases the Bianchi identities and the field equations of type I backgrounds imply a further reduction of the holonomy of the supercovariant connection. In addition, we show that in some cases if the holonomy group of the supercovariant connection is precisely the isotropy group of the parallel spinors, then all parallel spinors are Killing and so there are no backgrounds with N < L supersymmetries.
Abstract We will start by reviewing some aspects of bosonic string theory and then move on to the... more Abstract We will start by reviewing some aspects of bosonic string theory and then move on to the fermionic supersymmetric string, ie the so called superstring. The focus will here be on the veri cation of the supersymmetry of the action. In chapter 3 we will consider eleven-dimensional supergravity and in particular solve the Bianchi identities. We will in chapter 4 nally examine the p= 2 brane in eleven-dimensional supergravity and the emphasis will here be on the {-symmetry of the action. i
Abstract In recent years dramatic progress has been made in the understanding of the nonperturbat... more Abstract In recent years dramatic progress has been made in the understanding of the nonperturbative structure of superstring theory and M-theory. Central to this progress are non-perturbative, solitonic objects collectively referred to as p-branes. In this thesis, comprising an introductory text and three appended research papers, we are going to briefly review superstring theory and M-theory. Emphasis will be given to the dynamics of p-branes, which is the subject of Papers I-III.
We utilize the classification of IIB horizons with 5-form flux to present a unified description f... more We utilize the classification of IIB horizons with 5-form flux to present a unified description for the geometry of AdS_n, n=3,5,7 solutions. In particular, we show that all such backgrounds can be constructed from 8-dimensional 2-strong Calabi-Yau geometries with torsion which admit some additional isometries. We explore the geometry of AdS_3 and AdS_5 solutions but we do not find AdS_7 solutions.
In ten dimensions, there exist five consistent string theories and in eleven dimensions there is ... more In ten dimensions, there exist five consistent string theories and in eleven dimensions there is a unique supergravity theory. When trying to find the fundamental theory of nature this is clearly an ``embarrassment of riches''. In the mid 90s, however, it was discovered that these theories are all related via a kind of transformation called duality. The six theories are therefore only facets of a (largely unknown) underlying theory referred to as M- theory. This theory is intrinsically non-perturbative and therefore very hard to study. Witten has suggested that until we know more about M-theory, M can stand for `magic', `mystery' or `membrane', according to taste. In this thesis, comprising an introductory text and eight appended research papers, we are going to describe some of the methods used to study M-theory. Central to this analysis are non-perturbative, solitonic objects collectively referred to as p-branes, whose properties are studied in Papers I-IV and VI. In Paper II we generalize the Goldstone mechanism to the case of tensor fields of arbitrary rank, providing an understanding of the emergence of vector and tensor fields on branes in terms of broken symmetries. In Papers III and IV we find brane solutions with finite field strengths on the brane. Lately, noncommutative theories decoupled from closed strings have been discovered. These theories are defined on branes with critical field strengths and are studied and extended in Papers VI and VII. In Paper V we generalize eleven dimensional supergravity to obtain the most general geometrical structure in eleven dimensional superspace. The motivation for this is to examine what constraints supersymmetry imposes on possible correction terms arising from M-theory. To facilitate this study the Mathematica package GAMMA, which is capable of performing Γ-matrix algebra and Fierz transformations, were developed and is presented in Paper VIII.
]. It is shown that all supersymmetric solutions of IIB supergravity with more than 28 Killing sp... more ]. It is shown that all supersymmetric solutions of IIB supergravity with more than 28 Killing spinors are locally maximally supersymmetric.
We give a superspace description of D=3, N=8 supergravity. The formulation is off-shell in the se... more We give a superspace description of D=3, N=8 supergravity. The formulation is off-shell in the sense that the equations of motion are not implied by the superspace constraints (but an action principle is not given). The multiplet structure is unconventional, which we connect to the existence of a "Dragon window", that is modules occurring in the supercurvature but not in the supertorsion. According to Dragon's theorem this cannot happen above three dimensions. We clarify the relevance of this window for going on the conformal shell, and discuss some aspects of coupling to conformal matter.
We classify under some assumptions the IIB black hole horizons with 5-form flux preserving more t... more We classify under some assumptions the IIB black hole horizons with 5-form flux preserving more than 2 supersymmetries. We find that the spatial horizon sections with non-vanishing flux preserving 4 supersymmetries are locally isometric either to S 1 × S 3 × T 4 or to S 1 × S 3 × K 3 and the associated near horizon geometries are locally isometric to AdS 3 × S 3 × T 4 and AdS 3 × S 3 × K 3, respectively. The near horizon geometries preserving more than 4 supersymmetries are locally isometric to {{R}^{1,1}} × {T^8}.
We investigate the near horizon geometry of IIB supergravity black holes with non-vanishing 5-for... more We investigate the near horizon geometry of IIB supergravity black holes with non-vanishing 5-form flux preserving at least two supersymmetries. We demonstrate that there are three classes of solutions distinguished by the choice of Killing spinors. We find that the spatial horizon sections of the class of solutions with an SU(4) invariant pure Killing spinor are hermitian manifolds and admit a hidden Kähler with torsion (KT) geometry compatible with the SU(4) structure. Moreover the Bianchi identity of the 5-form, which also implies the field equations, can be expressed in terms of the torsion $ d\left( {\omega \wedge H} \right) = \partial \bar{\partial }{\omega^2} = 0 $ , where ω is a Hermitian form. We give several examples of near horizon geometries which include group manifolds, group fibrations over KT manifolds and uplifted geometries of lower dimensional black holes. Furthermore, we show that the class of solutions associated with a Spin(7) invariant spinor is locally a product $ {\mathbb{R}^{1,1}} \times \mathcal{S} $ , where $ \mathcal{S} $ is a holonomy Spin(7) manifold.
We show that eleven-dimensional supergravity backgrounds with thirty one supersymmetries, N=31, a... more We show that eleven-dimensional supergravity backgrounds with thirty one supersymmetries, N=31, admit an additional Killing spinor and so they are locally isometric to maximally supersymmetric ones. This rules out the existence of simply connected eleven-dimensional supergravity preons. We also show that N=15 solutions of type I supergravities are locally isometric to Minkowski spacetime.
We adapt the spinorial geometry method to investigate supergravity backgrounds with near maximal ... more We adapt the spinorial geometry method to investigate supergravity backgrounds with near maximal number of supersymmetries. We then apply the formalism to show that the IIB supergravity backgrounds with 31 supersymmetries preserve an additional supersymmetry and so they are maximally supersymmetric. This rules out the existence of IIB supergravity preons.
We construct maximal D = 8 gauged supergravities by the reduction of D = 11 supergravity over thr... more We construct maximal D = 8 gauged supergravities by the reduction of D = 11 supergravity over three-dimensional group manifolds. Such manifolds are classified into two classes, A and B, and eleven types. This Bianchi classification carries over to the gauged supergravities. The class A theories have 1/2 BPS domain wall solutions that uplift to purely gravitational solutions consisting of 7D Minkowski and a 4D Euclidean geometry. These geometries are generically singular. The two regular exceptions correspond to the nearhorizon limit of the single-or double-centre Kaluza-Klein monopole. In contrast, the class B supergravities are defined by a set of equations of motion that cannot be integrated to an action and allow for no 1/2 BPS domain walls.
We review how the classification of all supersymmetric backgrounds of IIB supergravity can be red... more We review how the classification of all supersymmetric backgrounds of IIB supergravity can be reduced to the evaluation of the Killing spinor equations and their integrability conditions, which contain the field equations, on five types of spinors. This is an extension of the work [hep-th/0503046] to IIB supergravity. By using the explicit expressions for the Killing spinor equations evaluated on the five types of spinors the Killing spinor equations become a linear system in terms of the fluxes, the geometry and the spacetime derivatives of the functions that determine the Killing spinors. This system can be solved to express the fluxes in terms of the geometry and to determine the conditions on the geometry of any supersymmetric background. Similarly, the integrability conditions of the Killing spinor equations are turned into a linear system. This can be used to determine the field equations that are implied by the Killing spinor equations for any supersymmetric background. These linear systems simplify for generic backgrounds with maximal and half-maximal number of H-invariant Killing spinors, H ⊂ Spin(9,1). In the maximal case, the Killing spinor equations factorise, whereas in the half-maximal case they do not.
We present all isotropy groups and associated $\Sigma$ groups, up to discrete identifications of ... more We present all isotropy groups and associated $\Sigma$ groups, up to discrete identifications of the component connected to the identity, of spinors of eleven-dimensional and type II supergravities. The $\Sigma$ groups are products of a Spin group and an R-symmetry group of a suitable lower dimensional supergravity theory. Using the case of SU(4)-invariant spinors as a paradigm, we demonstrate that the $\Sigma$ groups, and so the R-symmetry groups of lower-dimensional supergravity theories arising from compactifications, have disconnected components. These lead to discrete symmetry groups reminiscent of R-parity. We examine the role of disconnected components of the $\Sigma$ groups in the choice of Killing spinor representatives and in the context of compactifications.
We review the recent progress made towards the classification of supersymmetric solutions in ten ... more We review the recent progress made towards the classification of supersymmetric solutions in ten and eleven dimensions with emphasis on those of IIB supergravity. In particular, the spinorial geometry method is outlined and adapted to nearly maximally supersymmetric backgrounds. We then demonstrate its effectiveness by classifying the maximally supersymmetric IIB G-backgrounds and by showing that N = 31 IIB solutions do not exist.
We outline the solution of the Killing spinor equations of the heterotic supergravity. In additio... more We outline the solution of the Killing spinor equations of the heterotic supergravity. In addition, we describe the classification of all half supersymmetric solutions.
This paper is concerned with the problem of coupling the N=8 superconformal Bagger-Lambert-Gustav... more This paper is concerned with the problem of coupling the N=8 superconformal Bagger-Lambert-Gustavsson (BLG) theory to N=8 conformal supergravity in three dimensions. We start by constructing the on-shell N=8 conformal supergravity in three dimensions consisting of a Chern-Simons type term for each of the gauge fields: the spin connection, the SO(8) R-symmetry gauge field and the spin 3/2 Rarita-Schwinger (gravitino) field. We then proceed to couple this theory to the BLG theory. The final theory should have the same physical content, i.e., degrees of freedom, as the ordinary BLG theory. We discuss briefly the properties of this "topologically gauged" BLG theory and why this theory may be useful.
We find the geometry of all supersymmetric type I backgrounds by solving the gravitino and dilati... more We find the geometry of all supersymmetric type I backgrounds by solving the gravitino and dilatino Killing spinor equations, using the spinorial geometry technique, in all cases. The solutions of the gravitino Killing spinor equation are characterized by their isotropy group in Spin(9, 1), while the solutions of the dilatino Killing spinor equation are characterized by their isotropy group in the subgroup Σ(P) of Spin(9, 1) which preserves the space of parallel spinors P. Given a solution of the gravitino Killing spinor equation with L parallel spinors, L = 1, 2, 3, 4, 5, 6, 8, the dilatino Killing spinor equation allows for solutions with N supersymmetries for any 0 < N ≤ L. Moreover for L = 16, we confirm that N = 8, 10, 12, 14, 16. We find that in most cases the Bianchi identities and the field equations of type I backgrounds imply a further reduction of the holonomy of the supercovariant connection. In addition, we show that in some cases if the holonomy group of the supercovariant connection is precisely the isotropy group of the parallel spinors, then all parallel spinors are Killing and so there are no backgrounds with N < L supersymmetries.
Abstract We will start by reviewing some aspects of bosonic string theory and then move on to the... more Abstract We will start by reviewing some aspects of bosonic string theory and then move on to the fermionic supersymmetric string, ie the so called superstring. The focus will here be on the veri cation of the supersymmetry of the action. In chapter 3 we will consider eleven-dimensional supergravity and in particular solve the Bianchi identities. We will in chapter 4 nally examine the p= 2 brane in eleven-dimensional supergravity and the emphasis will here be on the {-symmetry of the action. i
Abstract In recent years dramatic progress has been made in the understanding of the nonperturbat... more Abstract In recent years dramatic progress has been made in the understanding of the nonperturbative structure of superstring theory and M-theory. Central to this progress are non-perturbative, solitonic objects collectively referred to as p-branes. In this thesis, comprising an introductory text and three appended research papers, we are going to briefly review superstring theory and M-theory. Emphasis will be given to the dynamics of p-branes, which is the subject of Papers I-III.
We utilize the classification of IIB horizons with 5-form flux to present a unified description f... more We utilize the classification of IIB horizons with 5-form flux to present a unified description for the geometry of AdS_n, n=3,5,7 solutions. In particular, we show that all such backgrounds can be constructed from 8-dimensional 2-strong Calabi-Yau geometries with torsion which admit some additional isometries. We explore the geometry of AdS_3 and AdS_5 solutions but we do not find AdS_7 solutions.
In ten dimensions, there exist five consistent string theories and in eleven dimensions there is ... more In ten dimensions, there exist five consistent string theories and in eleven dimensions there is a unique supergravity theory. When trying to find the fundamental theory of nature this is clearly an ``embarrassment of riches''. In the mid 90s, however, it was discovered that these theories are all related via a kind of transformation called duality. The six theories are therefore only facets of a (largely unknown) underlying theory referred to as M- theory. This theory is intrinsically non-perturbative and therefore very hard to study. Witten has suggested that until we know more about M-theory, M can stand for `magic', `mystery' or `membrane', according to taste. In this thesis, comprising an introductory text and eight appended research papers, we are going to describe some of the methods used to study M-theory. Central to this analysis are non-perturbative, solitonic objects collectively referred to as p-branes, whose properties are studied in Papers I-IV and VI. In Paper II we generalize the Goldstone mechanism to the case of tensor fields of arbitrary rank, providing an understanding of the emergence of vector and tensor fields on branes in terms of broken symmetries. In Papers III and IV we find brane solutions with finite field strengths on the brane. Lately, noncommutative theories decoupled from closed strings have been discovered. These theories are defined on branes with critical field strengths and are studied and extended in Papers VI and VII. In Paper V we generalize eleven dimensional supergravity to obtain the most general geometrical structure in eleven dimensional superspace. The motivation for this is to examine what constraints supersymmetry imposes on possible correction terms arising from M-theory. To facilitate this study the Mathematica package GAMMA, which is capable of performing Γ-matrix algebra and Fierz transformations, were developed and is presented in Paper VIII.
]. It is shown that all supersymmetric solutions of IIB supergravity with more than 28 Killing sp... more ]. It is shown that all supersymmetric solutions of IIB supergravity with more than 28 Killing spinors are locally maximally supersymmetric.
We give a superspace description of D=3, N=8 supergravity. The formulation is off-shell in the se... more We give a superspace description of D=3, N=8 supergravity. The formulation is off-shell in the sense that the equations of motion are not implied by the superspace constraints (but an action principle is not given). The multiplet structure is unconventional, which we connect to the existence of a "Dragon window", that is modules occurring in the supercurvature but not in the supertorsion. According to Dragon's theorem this cannot happen above three dimensions. We clarify the relevance of this window for going on the conformal shell, and discuss some aspects of coupling to conformal matter.
We classify under some assumptions the IIB black hole horizons with 5-form flux preserving more t... more We classify under some assumptions the IIB black hole horizons with 5-form flux preserving more than 2 supersymmetries. We find that the spatial horizon sections with non-vanishing flux preserving 4 supersymmetries are locally isometric either to S 1 × S 3 × T 4 or to S 1 × S 3 × K 3 and the associated near horizon geometries are locally isometric to AdS 3 × S 3 × T 4 and AdS 3 × S 3 × K 3, respectively. The near horizon geometries preserving more than 4 supersymmetries are locally isometric to {{R}^{1,1}} × {T^8}.
We investigate the near horizon geometry of IIB supergravity black holes with non-vanishing 5-for... more We investigate the near horizon geometry of IIB supergravity black holes with non-vanishing 5-form flux preserving at least two supersymmetries. We demonstrate that there are three classes of solutions distinguished by the choice of Killing spinors. We find that the spatial horizon sections of the class of solutions with an SU(4) invariant pure Killing spinor are hermitian manifolds and admit a hidden Kähler with torsion (KT) geometry compatible with the SU(4) structure. Moreover the Bianchi identity of the 5-form, which also implies the field equations, can be expressed in terms of the torsion $ d\left( {\omega \wedge H} \right) = \partial \bar{\partial }{\omega^2} = 0 $ , where ω is a Hermitian form. We give several examples of near horizon geometries which include group manifolds, group fibrations over KT manifolds and uplifted geometries of lower dimensional black holes. Furthermore, we show that the class of solutions associated with a Spin(7) invariant spinor is locally a product $ {\mathbb{R}^{1,1}} \times \mathcal{S} $ , where $ \mathcal{S} $ is a holonomy Spin(7) manifold.
We show that eleven-dimensional supergravity backgrounds with thirty one supersymmetries, N=31, a... more We show that eleven-dimensional supergravity backgrounds with thirty one supersymmetries, N=31, admit an additional Killing spinor and so they are locally isometric to maximally supersymmetric ones. This rules out the existence of simply connected eleven-dimensional supergravity preons. We also show that N=15 solutions of type I supergravities are locally isometric to Minkowski spacetime.
We adapt the spinorial geometry method to investigate supergravity backgrounds with near maximal ... more We adapt the spinorial geometry method to investigate supergravity backgrounds with near maximal number of supersymmetries. We then apply the formalism to show that the IIB supergravity backgrounds with 31 supersymmetries preserve an additional supersymmetry and so they are maximally supersymmetric. This rules out the existence of IIB supergravity preons.
We construct maximal D = 8 gauged supergravities by the reduction of D = 11 supergravity over thr... more We construct maximal D = 8 gauged supergravities by the reduction of D = 11 supergravity over three-dimensional group manifolds. Such manifolds are classified into two classes, A and B, and eleven types. This Bianchi classification carries over to the gauged supergravities. The class A theories have 1/2 BPS domain wall solutions that uplift to purely gravitational solutions consisting of 7D Minkowski and a 4D Euclidean geometry. These geometries are generically singular. The two regular exceptions correspond to the nearhorizon limit of the single-or double-centre Kaluza-Klein monopole. In contrast, the class B supergravities are defined by a set of equations of motion that cannot be integrated to an action and allow for no 1/2 BPS domain walls.
We review how the classification of all supersymmetric backgrounds of IIB supergravity can be red... more We review how the classification of all supersymmetric backgrounds of IIB supergravity can be reduced to the evaluation of the Killing spinor equations and their integrability conditions, which contain the field equations, on five types of spinors. This is an extension of the work [hep-th/0503046] to IIB supergravity. By using the explicit expressions for the Killing spinor equations evaluated on the five types of spinors the Killing spinor equations become a linear system in terms of the fluxes, the geometry and the spacetime derivatives of the functions that determine the Killing spinors. This system can be solved to express the fluxes in terms of the geometry and to determine the conditions on the geometry of any supersymmetric background. Similarly, the integrability conditions of the Killing spinor equations are turned into a linear system. This can be used to determine the field equations that are implied by the Killing spinor equations for any supersymmetric background. These linear systems simplify for generic backgrounds with maximal and half-maximal number of H-invariant Killing spinors, H ⊂ Spin(9,1). In the maximal case, the Killing spinor equations factorise, whereas in the half-maximal case they do not.
We present all isotropy groups and associated $\Sigma$ groups, up to discrete identifications of ... more We present all isotropy groups and associated $\Sigma$ groups, up to discrete identifications of the component connected to the identity, of spinors of eleven-dimensional and type II supergravities. The $\Sigma$ groups are products of a Spin group and an R-symmetry group of a suitable lower dimensional supergravity theory. Using the case of SU(4)-invariant spinors as a paradigm, we demonstrate that the $\Sigma$ groups, and so the R-symmetry groups of lower-dimensional supergravity theories arising from compactifications, have disconnected components. These lead to discrete symmetry groups reminiscent of R-parity. We examine the role of disconnected components of the $\Sigma$ groups in the choice of Killing spinor representatives and in the context of compactifications.
We review the recent progress made towards the classification of supersymmetric solutions in ten ... more We review the recent progress made towards the classification of supersymmetric solutions in ten and eleven dimensions with emphasis on those of IIB supergravity. In particular, the spinorial geometry method is outlined and adapted to nearly maximally supersymmetric backgrounds. We then demonstrate its effectiveness by classifying the maximally supersymmetric IIB G-backgrounds and by showing that N = 31 IIB solutions do not exist.
We outline the solution of the Killing spinor equations of the heterotic supergravity. In additio... more We outline the solution of the Killing spinor equations of the heterotic supergravity. In addition, we describe the classification of all half supersymmetric solutions.
This paper is concerned with the problem of coupling the N=8 superconformal Bagger-Lambert-Gustav... more This paper is concerned with the problem of coupling the N=8 superconformal Bagger-Lambert-Gustavsson (BLG) theory to N=8 conformal supergravity in three dimensions. We start by constructing the on-shell N=8 conformal supergravity in three dimensions consisting of a Chern-Simons type term for each of the gauge fields: the spin connection, the SO(8) R-symmetry gauge field and the spin 3/2 Rarita-Schwinger (gravitino) field. We then proceed to couple this theory to the BLG theory. The final theory should have the same physical content, i.e., degrees of freedom, as the ordinary BLG theory. We discuss briefly the properties of this "topologically gauged" BLG theory and why this theory may be useful.
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Papers by Ulf Gran