Spinorial geometry techniques have recently been used to classify all half supersymmetric solutions in gauged five dimensional supergravity with vector multiplets. In this paper we consider solutions for which at least one of the Killing spinors generates a timelike Killing vector. We obtain coordinate transformations which considerably simplify the solutions, and in a number of cases, we obtain explicitly some additional Killing vectors which were hidden in the original analysis.
The isometries of $AdS_5$ space and supersymmetric $AdS_5\otimes S_1$ space are nonlinearly realized on four dimensional Minkowski space. The resultant effective actions in terms of the Nambu-Goldstone modes are constructed. The dilatonic mode governing the motion of the Minkowski space probe brane into the covolume of supersymmetric $AdS_5$ space is found to be unstable and the bulk of the $AdS_5$ space is unable to sustain the brane. No such instablility appears in the non-supersymmetric case.
This talk summarizes the study of superfield gaugings of isometries of extended supersymmetric mechanics in hep-th/0605211, hep-th/0611247 and arXiv:0706.0706. The gauging procedure provides a manifestly supersymmetric realization of d=1 automorphic dualities which interrelate various irreducible off-shell multiplets of d=1 extended supersymmetry featuring the same number of physical fermions but different divisions of bosonic fields into the physical and auxiliary subsets. We concentrate on the most interesting N=4 case and demonstrate that, with a suitable choice of the symmetry to be gauged…
The near-horizon geometry of an asymptotically AdS_5 supersymmetric black hole discovered by Gutowski and Reall is analysed. After lifting the solution to 10 dimensions, we explicitly solve the Killing spinor equations in both Poincare and global coordinates. It is found that exactly four supersymmetries are preserved which is twice the number for the full black hole. The full set of isometries is constructed and the isometry supergroup is shown to be SU(1,1|1) X SU(2) X U(3). We further study half-BPS configurations of D3-branes in the near-horizon geometry in Poincare and global coordinates.…
We examine T-duality transformations for supersymmetric strings with target space geometry with compact abelian isometries. We consider the partition function of these models and we show that although T-duality is not a symmetry, due to an anomaly, it relates type IIA to type IIB strings. In this way we extend the corresponding result for toroidal compactification to the general case of non-trivial backgrounds with abelian isometries and for world sheets of any genera.
The geometry of (2,1) supersymmetric sigma-models is reviewed and the conditions under which they have isometry symmetries are analysed. Certain potentials are constructed that play an important role in the gauging of such symmetries. The gauged action is found for a special class of models.
We analyse the geometry of four-dimensional bosonic manifolds arising within the context of $N=4, D=1$ supersymmetry. We demonstrate that both cases of general hyper-Kähler manifolds, i.e. those with translation or rotational isometries, may be supersymmetrized in the same way. We start from a generic N=4 supersymmetric three-dimensional action and perform dualization of the coupling constant, initially present in the action. As a result, we end up with explicit component actions for $N=4, D=1$ nonlinear sigma-models with hyper-Kähler geometry (with both types of isometries) in the target spac…
We build up a class of N=2 supersymmetric non-linear $σ$-models in an N=1 superspace based on the Atiyah-Ward space-time of (2+2)-signature metric. We also discuss the gauging of isometries of the associated hyper-Kählerian target spaces and present the resulting gauge-covariant supersymmetric action functional.
We present a supersymmetric non-linear $\s$-model built up in the $N=1$ superspace of Atiyah-Ward space-time. A manifold of the Kähler type comes out that is restricted by a particular decomposition of the Kähler potential. The gauging of the $\s$-model isometries is also accomplished in superspace.
We use the embedding tensor formalism to analyse maximally symmetric backgrounds of N=2 gauged supergravities which have the full N=2 supersymmetry. We state the condition for N=2 vacua and discuss some of their general properties. We show that if the gauged isometries leave the SU(2) R-symmetry invariant, then the N=2 vacuum must be Minkowski. This implies that there are no AdS backgrounds with eight unbroken supercharges in the effective N=2 supergravity of six-dimensional SU(3) x SU(3) structure compactifications of type II string theory and M-theory. Combined with previous results on N=1 v…
In this paper, we study the half-supersymmetric time-dependent configurations in M-theory and their matrix models. We find a large class of 11D supergravity solutions, which keeps sixteen supersymmetries. Furthermore, we investigate the isometries of these configurations and show that in general these configurations have no supernumerary supersymmetries. And also we define the Matrix models in these backgrounds following Discrete Light-Cone Quantization (DLCQ) prescription.
We present a systematic classification of half-supersymmetric solutions of gauged N=2, D=5 supergravity coupled to an arbitrary number of abelian vector multiplets for which at least one of the Killing spinors generate a time-like Killing vector.
We determine the supersymmetric AdS_7 backgrounds of seven-dimensional half-maximal gauged supergravities and show that they do not admit any deformations that preserve all 16 supercharges. We compare this result to the conformal manifold of the holographically dual (1,0) superconformal field theories and show that accordingly its representation theory implies that no supersymmetric marginal operators exist.
Using the boundary Yang-Baxter equations and exact results on the bulk $S$-matrices, we compute exact boundary scattering amplitudes of the supersymmetric sine-Gordon model with integrable boundary potentials.
We study classical integrability of the supersymmetric U(N) $σ$ model with the Wess-Zumino-Witten term on full and half plane. We demonstrate the existence of nonlocal conserved currents of the model and derive general recursion relations for the infinite number of the corresponding charges in a superfield framework. The explicit form of the first few supersymmetric charges are constructed. We show that the considered model is integrable on full plane as a concequence of the conservation of the supersymmetric charges. Also, we study the model on half plane with free boundary, and examine the c…
By using O(7,7) transformations, to deform D6-branes, we obtain half-supersymmetric bound state solutions of type IIA supergravity, containing D6, D4, D2, D0, F1-branes and waves. We lift the solutions to M-theory which gives half-supersymmetric M-theory bound states, e.g. KK6-M5-M5-M5-M2-M2-M2-MW. We also take near horizon limits for the type IIA solutions, which gives supergravity duals of 7-dimensional non-commutative open string theory (with space-time and space-space non-commutativity), non-commutative Yang-Mills theory (with space-space and light-like non-commutativity) and an open D4-br…
Partial Isometries are important constructs that help give nontrivial solutions once a simple solution is known. We generalize this notion to Extended Partial Isometries and include operators which have right inverses but no left inverses (or vice versa). We find a large class of such operators and conjecture the moduli space to be the Hilbert Scheme of Points. Further, we apply this technique to find instanton solutions of a noncommutative N=1 supersymmetric gauge theory in six dimensions and show that this construction yields nontrivial solutions for other noncommutative gauge theories. The …
We classify half-supersymmetric solutions of gauged N=2, D=5 supergravity coupled to an arbitrary number of abelian vector multiplets for which all of the Killing spinors generate null Killing vectors. We show that there are four classes of solutions, and in each class we find the metric, scalars and gauge field strengths. When the scalar manifold is symmetric, the solutions correspond to a class of local near horizon geometries recently found by Kunduri and Lucietti.
We present a maximally supersymmetric IIB string background. The geometry is that of a conformally flat lorentzian symmetric space G/K with solvable G, with a homogeneous five-form flux. We give the explicit supergravity solution, compute the isometries, the 32 Killing spinors, and the symmetry superalgebra, and then discuss T-duality and the relation to M-theory.
The supersymmetric sinh-Gordon model on a half-line with integrable boundary conditions is considered perturbatively to verify conjectured exact reflection factors to one loop order. Propagators for the boson and fermion fields restricted to a half-line contain several novel features and are developed as prerequisites for the calculations.
We classify the half-supersymmetric "domain walls", i.e. branes of codimension one, in toroidally compactified IIA/IIB string theory and show to which gauged supergravity theory each of these domain walls belong. We use as input the requirement of supersymmetric Wess-Zumino terms, the properties of the E11 Kac-Moody algebra and the embedding tensor formalism. We show that the number of half-supersymmetric domain walls is a multiple of the number of corresponding central charges in the supersymmetry algebra, where the multiplicity is related to the degeneracy of the BPS conditions.
The entropy of a half-BPS black hole in N=4 supersymmetric heterotic string compactification is independent of the details of the charge vector and is a function only of the norm of the charge vector calculated using the appropriate Lorenzian metric. Thus in order for this to agree with the degeneracy of the elementary string states, the latter must also be a function of the same invariant norm. We show that this is true for generic CHL compactifications to all orders in a power series expansion in the inverse charges, but there are exponentially suppressed corrections which do depend on the d…
We study consistent truncations of M-theory to gauged N=2 supergravity in four dimensions, based on a large class of SU(3)-structures in seven dimensions. We show that the gauging involves isometries of the vector multiplet scalar manifold as well as the Heisenberg algebra and a special isometry of the hyperscalar manifold. As a result, non-abelian gauge groups and new non-trivial scalar potentials are generated. Then we specialize to all homogeneous SU(3)-structures supporting supersymmetric AdS4 vacua. These are the Stiefel manifold V52, the Aloff-Wallach spaces N(k,l), the seven-sphere (see…
We analyze four- and six-derivative couplings in the low energy effective action of $D=3$ string vacua with half-maximal supersymmetry. In analogy with an earlier proposal for the $(\nablaΦ)^4$ coupling, we propose that the $\nabla^2(\nablaΦ)^4$ coupling is given exactly by a manifestly U-duality invariant genus-two modular integral. In the limit where a circle in the internal torus decompactifies, the $\nabla^2(\nablaΦ)^4$ coupling reduces to the $D^2 F^4$ and $R^2 F^2$ couplings in $D=4$, along with an infinite series of corrections of order $e^{-R}$, from four-dimensional 1/4-BPS dyons whos…
In these notes we attempt to give a pedagogical introduction to the work of Seiberg and Witten on S-duality and the exact results of N=2 supersymmetric gauge theories with and without matter. The first half is devoted to a review of monopoles in gauge theories and the construction of supersymmetric gauge theories. In the second half, we describe the work of Seiberg and Witten.
The complete Type IIB supergravity solutions with 16 supersymmetries are obtained on the manifold $AdS_4 \times S^2 \times S^2 \times Σ$ with $SO(2,3) \times SO(3) \times SO(3)$ symmetry in terms of two holomorphic functions on a Riemann surface $Σ$, which generally has a boundary. This is achieved by reducing the BPS equations using the above symmetry requirements, proving that all solutions of the BPS equations solve the full Type IIB supergravity field equations, mapping the BPS equations onto a new integrable system akin to the Liouville and Sine-Gordon theories, and mapping this integrabl…
We investigate half-supersymmetric domain wall solutions of four maximally supersymmetric D=9 massive supergravity theories obtained by Scherk-Schwarz reduction of D=10 IIA and IIB supergravity. One of the theories does not have a superpotential and does not allow domain wall solutions preserving any supersymmetry. The other three theories have superpotentials leading to half-supersymmetric domain wall solutions, one of which has zero potential but non-zero superpotential. The uplifting of these domain wall solutions to ten dimensions leads to three classes of half-supersymmetric Type IIB 7-br…
Using the duality between color and kinematics, we construct two-loop four-point scattering amplitudes in $\mathcal{N}=2$ super-Yang-Mills (SYM) theory coupled to $N_f$ fundamental hypermultiplets. Our results are valid in $D\le 6$ dimensions, where the upper bound corresponds to six-dimensional chiral $\mathcal{N}=(1,0)$ SYM theory. By exploiting a close connection with $\mathcal{N}=4$ SYM theory - and, equivalently, six-dimensional $\mathcal{N}=(1,1)$ SYM theory - we find compact integrands with four-dimensional external vectors in both the maximally-helicity-violating (MHV) and all-chiral-v…
Subject to some relatively mild assumptions, we derive the complete form of all timelike half-supersymmetric solutions to N=2, D=4 gauged supergravity coupled to an arbitrary number of abelian vector multiplets. This is done using spinorial geometry techniques. Explicit examples are given for a simple prepotential. Among the solutions, there are near-horizon geometries of extremal rotating BPS black holes still to be discovered, with a nontrivial dependence of the scalar fields on one of the horizon coordinates.
We provide the explicit gauging of all the SU(2,1) isometries of one N=2 supergravity hypermultiplet, which spans SU(2,1)/U(2) coset space parameterized in terms of two complex projective coordinate fields z_1 and z_2. We derive the full, explicit Killing prepotential that specifies the most general superpotential. As an application we consider the supersymmetric flow (renormalization group) equations for: (i) the flow from a null singularity to the flat, supersymmetric space-time and (ii) the flow that violates c-theorem with the superpotential crossing zero.
We construct an action for the N=2 supersymmetric sine-Gordon model on the half-line, which we argue is both supersymmetric and integrable. The boundary interaction depends on three continuous boundary parameters, as well as the bulk mass parameter.
The gauging of isometries in general sigma-models which include fermionic terms which represent the interaction of strings with background Yang-Mills fields is considered. Gauging is possible only if certain obstructions are absent. The quantum gauge anomaly is discussed, and the (1,0) supersymmetric generalisation of the gauged action given.
In a recent paper Dargis and Mathieu introduced integrodifferential odd flows for the supersymmetric KdV equation. These flows are obtained from the nonlocal conservation laws associated with the fourth root of its Lax operator. In this note I show that only half of these flows are of the standard Lax form, while the remaining half provide us with hamiltonians for an SKdV-type reduction of a new supersymmetric hierarchy. This new hierarchy is shown to be closely related to the Jacobian supersymmetric KP-hierarchy of Mulase and Rabin. A detailed study of the algebra of additional symmetries of …
In this work a thorough study of a number of specific supersymmetric sigma-models with extended supersymmetry is performed within the context of generalised complex geometry. More specifically the supersymmetric Wess-Zumino-Witten model on a variety of group manifolds was considered. By explicitly calculating the admissible complex structures and the associated pure spinors on the target manifold a full characterisation of the different possible geometries is provided. By using this approach the various aspects of generalised Kähler geometry can be studied in detail. Also considered are the va…
We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence w…
We study BPS domain wall solutions of 5-dimensional N=2 supergravity where isometries of the hypermultiplet geometry have been gauged. We derive the corresponding supersymmetric flow equations and define an appropriate c-function. As an example we discuss a domain wall solution of Freedman, Gubser, Pilch and Warner which is related to a RG-flow in a dual superconformal field theory.
The half-BPS boundary conditions preserving $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,4)$ supersymmetry in 3d $\mathcal{N}=4$ supersymmetric gauge theories are examined. The BPS equations admit decomposition of the bulk supermultiplets into specific boundary supermultiplets of preserved supersymmetry. Nahm-like equations arise in the vector multiplet BPS boundary condition preserving $\mathcal{N}=(0,4)$ supersymmetry and Robin-type boundary conditions appear for the hypermultiplet coupled to vector multiplet when $\mathcal{N}=(2,2)$ supersymmetry is preserved. The half-BPS boundary conditions ar…
We first classify all supersymmetric solutions of the 3-dimensional half-maximal ungauged supergravity that possess a timelike Killing vector coming from the Killing spinor bilinear by considering their identification under the complexification of the local symmetry of the theory. It is found that only solutions that preserve $16/2^n, 1 \leq n \leq 3$ real supersymmetries are allowed. We then classify supersymmetric solutions under the real local symmetry of the theory and we are able to solve the equations of motion for all of them. It is shown that all such solutions can be expressed as a di…
General half-BPS A-type boundary conditions are formulated for N=2 supersymmetric field theories on compact 3-manifolds with boundary. We observe that under suitable conditions manifolds of the real A-type admitting two complex supersymmetries (related by charge conjugation) possess, besides a contact structure, a natural integrable toric foliation. A boundary, or a general co-dimension-1 defect, can be inserted along any leaf of this preferred foliation to produce manifolds with boundary that have the topology of a solid torus. We show that supersymmetric field theories on such manifolds can …
We find half supersymmetric AdS-embeddings in AdS_5 x S^5 corresponding to all quarter BPS orthogonal intersections of D3-branes with Dp-branes. A particular case is the Karch-Randall embedding AdS_4 x S^2. We explicitly prove that these embeddings are supersymmetric by showing that the kappa symmetry projections are compatible with half of the target space Killing spinors and argue that all these cases lead to AdS/dCFT dualities involving a CFT with a defect. We also find an asymptotically AdS_4 x S^2 embedding that corresponds to a holographic RG-flow on the defect. We then consider the pp-w…
Super coset spaces play an important role in the formulation of supersymmetric theories. The aim of this paper is to review and discuss the geometry of super coset spaces with particular focus on the way the geometrical structures of the super coset space G/H are inherited from the super Lie group G. The isometries of the super coset space are discussed and a definition of Killing supervectors - the supervectors associated with infinitesimal isometries - is given that can be easily extended to spaces other than coset spaces.
We describe all supersymmetric solutions of the heterotic string which preserve 8 supersymmetries and show that are distinguished by the holonomy, ${\rm hol}(\hat\nabla)$, of the connection, $\hat\nabla$, with skew-symmetric torsion. The ${\rm hol}(\hat\nabla) \subseteq SU(2)$ solutions are principal bundles over a 4-dimensional hyper-Kähler manifold equipped with a anti-self-dual connection and fibre group $G$ which has Lie algebra, ${\mathfrak Lie} (G)=\bR^{5,1}$, $\mathfrak{sl}(2,\bR)\oplus \mathfrak{su}(2)$ or $\mathfrak{cw}_6$. Some of the solutions have the interpretation as 5-branes wra…
We consider the supergravity backgrounds that correspond to supersymmetric Wilson line operators in the context of AdS/CFT correspondence. We study the gravitino and dilatino conditions of the IIB supergravity under the appropriate ansatz, and obtain some necessary conditions for a supergravity background that preserves the same symmetry as the supersymmetric Wilson lines. The supergravity solutions are characterized by continuous version of maya diagrams. This diagram is related to the eigenvalue distribution of the Gaussian matrix model. We also consider the similar backgrounds of the 11-dim…
We compute supersymmetric indices which count local operators at certain half-BPS interfaces and quarter-BPS junctions of interfaces in four-dimensional $\mathcal{N}=4$ Super Yang-Mills theory. We use the indices as very stringent tests of a variety of string theory-inspired conjectures about the action of S-duality on such defects.
We study $N=2$ seven-dimensional gauged supergravity coupled to three vector multiplets with $SO(4)$ gauge group. The resulting gauged supergravity contains 10 scalars consisting of the dilaton and 9 vector multiplet scalars parametrized by $SO(3,3)/SO(3)\times SO(3)$ coset manifold. The maximally supersymmetric $AdS_7$ vacuum with unbroken $SO(4)$ symmetry is identified with a $(1,0)$ SCFT in six dimensions. We find one new supersymmetric $AdS_7$ critical point preserving $SO(3)_{\textrm{diag}}\subset SO(3)\times SO(3)\sim SO(4)$ and study a holographic RG flow interpolating between the $SO(4…
The problem of finding supersymmetric brane configurations in the near-horizon attractor geometry of a Calabi-Yau black hole with magnetic-electric charges (p^I,q_I) is considered. Half-BPS configurations, which are static for some choice of global AdS2 coordinate, are found for wrapped brane configurations with essentially any four-dimensional charges (u^I,v_I). Half-BPS multibrane configurations can also be found for any collection of wrapped branes provided they all have the same sign for the symplectic inner product p^Iv_I-u^Iq_I of their charges with the black hole charges. This contrasts…
We study supersymmetric AdS$_D$ backgrounds of eleven-dimensional or type II supergravity preserving $\mathcal{N}$ supersymmetries using generalised geometry. We show that a large class correspond precisely to spaces admitting a generalised $G_{D,\mathcal{N}}$ structure with a weak integrability condition, which we call "weak generalised holonomy". This class contains all such backgrounds with odd $D$, but for even $D$ we must impose a technical assumption concerning the inner products of Killing spinors. It is known that all compact smooth solutions are included, though our analysis is purely…
We construct the cubic interaction vertex and dynamically generated supercharges in light-cone superstring field theory for a large class half-supersymmetric D-branes in the plane-wave background. We show that these satisfy the plane-wave superalgebra at first order in string coupling. The cubic interaction vertex and dynamical supercharges presented here are given explicitly in terms of oscilators and can be used to compute three-point functions of open strings with endpoints on half-supersymmetric D-branes.
We compute correlators of chiral operators in half-twisted (0,2) supersymmetric gauged linear sigma models. Our results give simple algebraic formulas for a (0,2) generalization of genus zero Gromov-Witten invariants of compact toric varieties. We derive compact expressions for deformed quantum cohomology relations and apply our general method to several examples.
We study theories of 4D, N=1 supersymmetric massless, arbitrary higher half odd-integer superspins. A new series of such theories is found to exist for arbitrary superspin Y (Y=s+1/2 for any integer s). The lowest member (s=1) of the series is the original off-shell formulation of 4D, N=1 supergravity first presented by Breitenlohner in 1977.
All purely bosonic supersymmetric solutions of minimal supergravity in five dimensions are classified. The solutions preserve either one half or all of the supersymmetry. Explicit examples of new solutions are given, including a large family of plane-fronted waves and a maximally supersymmetric analogue of the Gödel universe which lifts to a solution of eleven dimensional supergravity that preserves 20 supersymmetries.
We show that supersymmetric flux vacua with intermediate SU(2) structure is closely related to some special classes of half-flat structures. More concretely, solutions of the SUSY equations IIA possess a symplectic half-flat structure, whereas solutions of the SUSY equations IIB admit a half-flat structure which is in certain sense near to the balanced condition. Using this result we show that compact simply connected manifolds do not admit type IIB solutions. New solutions of the SUSY equations IIA and IIB are constructed from hyperkähler 4-manifolds, special hypo 5-manifolds and 6-dimensiona…
On The Consistent Supersymmetric Reduction Of Heterotic Supergravity With Geometrically Arising Yang-Mills Symmetries
George P. & Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University（乔治·P和辛西娅·伍兹基础物理与天文学研究所，德克萨斯农工大学）；DAMTP, Centre for Mathematical Sciences, Cambridge University（剑桥大学数学科学中心DAMTP）
The four dimensional Godel spacetime is known to have the structure M_3 x R. It is also known that the three-dimensional factor M_3 is an exact solution of three-dimensional gravity coupled to a Maxwell-Chern-Simons theory. We build in this paper a N=2 supergravity extension for this action and prove that the Godel background preserves half of all supersymmetries.
Half-supersymmetric geometries of N=2 five-dimensional gauged supergravity have recently been fully classified using spinorial geometry techniques. We use this classification to determine all possible regular half-supersymmetric near-horizon geometries, assuming that all of the gauge-invariant spinor bilinears are regular at the horizon. Four geometries are found, two of which have been found previously in hep-th/0611351 and arXiv:0708.3695.
