Self-dual Tensor Research Articles

Quaternionic extension of the double Taub-NUT metric

Starting from the generic harmonic superspace action of the quaternion-Kähler sigma models and using the quotient approach we present, in an explicit form, a quaternion-Kähler extension of the double Taub-NUT metric. It possesses U(1)× U(1) isometry and supplies a new example of non-homogeneous Einstein metric with self-dual Weyl tensor.

Uniqueness properties of the Kerr metric

We obtain a geometrical condition on vacuum, stationary,asymptotically flat spacetimes which is necessary and sufficientfor the spacetime to be locally isometric to Kerr. Namely, weprove a theorem stating that an asymptotically flat, stationary,vacuum spacetime such that the so-called Killing form is aneigenvector of the self-dual Weyl tensor must be locallyisometric to Kerr. Asymptotic flatness is a fundamentalhypothesis of the theorem, as we demonstrate by writing down thefamily of metrics obtained when this requirement is dropped.This result indicates why the Kerr metric plays such animportant role in general relativity. It may also be ofinterest in order to extend the uniqueness theorems of blackholes to the non-connected and to the non-analytic case.

Consistency of the AdS7×S4 reduction and the origin of self-duality in odd dimensions

We discuss the full nonlinear Kaluza–Klein (KK) reduction of the original formulation of d=11 supergravity on AdS7×S4 to gauged maximal (N=4) supergravity in 7 dimensions. We derive the full nonlinear embedding of the d=7 fields in the d=11 fields (“the ansatz”) and check the consistency of the ansatz by deriving the d=7 supersymmetry laws from the d=11 transformation laws in the various sectors. The ansatz itself is nonpolynomial but the final d=7 results are polynomial. The correct d=7 scalar potential is obtained. For most of our results the explicit form of the matrix U connecting the d=7 gravitino to the Killing spinor is not needed, but we derive the equation which U has to satisfy and present the general solution. Requiring that the expressionδF=dδA in d=11 can be written as δd(fields in d=7), we find the ansatz for the 4-form F. It satisfies the Bianchi identities. The corresponding ansatz for the 3-form A modifies the geometrical proposal by Freed et al. by including d=7 scalar fields. A first order formulation for A in d=11 is needed to obtain the d=7 supersymmetry laws and the action for the nonabelian selfdual antisymmetric tensor field Sαβγ,A. Therefore selfduality in odd dimensions originates from a first order formalism in higher dimensions.

An actionfor the (2,0) self-dual tensormultiplet in a conformal supergravity background

We present the action for a self-dual tensor in six dimensions, coupledto a (2,0) conformal supergravity background. This action gives riseto the expected equations of motion. An alternative look upon one of thegauge symmetries clarifies its role in the supersymmetry transformationrules and the realization of the algebra.

Abelian vectors and self-dual tensors in six-dimensional supergravity

In this note we describe the most general coupling of abelian vector and tensor multiplets to six-dimensional (1,0) supergravity. As was recently pointed out, it is of interest to consider more general Chern-Simons couplings to abelian vectors of the type H r=dB r−1/2 c rabA adA b , with c r matrices that may not be simultaneously diagonalized. We show that these couplings can be related to Green-Schwarz terms of the form B r c r ab F a F b , and how the complete local Lagrangian, that embodies factorized gauge and supersymmetry anomalies (to be disposed of by fermion loops) is uniquely determined by Wess-Zumino consistency conditions, aside from an arbitrary quartic coupling for the gauginos.

S-duality for linearized gravity

We develope the analogue of S-duality for linearized gravity in (3+1)-dimensions. Our basic idea is to consider the self-dual (anti-self-dual) curvature tensor for linearized gravity in the context of the Macdowell–Mansouri formalism. We find that the strong-weak coupling duality for linearized gravity is an exact symmetry and implies small-large duality for the cosmological constant.

Anomalies and inflow on D-branes and O-planes

We derive the general form of the anomaly for chiral spinors and self-dual antisymmetric tensors living on D-brane and O-plane intersections, using both path-integral and index theorem methods. We then show that the anomalous couplings to RR forms of D-branes and O-planes in a general background are precisely those required to cancel these anomalies through the inflow mechanism. This allows, for instance, for local anomaly cancellation in generic orientifold models, the relevant Green-Schwarz term being given by the sum of the anomalous couplings of all the D-branes and O-planes in the model.

Finite tensor deformations of supergravity solitons

We consider brane solutions where the tensor degrees of freedom are excited. Explicit solutions to the full non-linear supergravity equations of motion are given for the M5 and D3 branes, corresponding to finite selfdual tensor or Born-Infeld field strengths. The solutions are BPS-saturated and half-supersymmetric. The resulting metric space-times are analysed.

Self-dual tensors and gravitational anomalies in 4 n + 2 dimensions

Starting from a manifestly Lorentz- and diffeomorphism-invariant classical action we perform a perturbative derivation of the gravitational anomalies for chiral bosons in 4 n + 2 dimensions. The manifest classical invariance is achieved using a newly developed method based on a scalar auxiliary field and two new bosonic local symmetries. The resulting anomalies coincide with the ones predicted by the index theorem. In the two-dimensional case, moreover, we perform an exact covariant computation of the effective action for a chiral boson (a scalar) which is seen to coincide with the effective action for a two-dimensional complex Weyl fermion. All these results support the quantum reliability of the new, at the classical level manifestly invariant, method.

Anomaly-free tensor-Yang–Mills system and its dual formulation

We consider the (1,0) supersymmetric Yang–Mills multiplet coupled to a self-dual tensor multiplet in six dimensions. It is shown that the counterterm required to cancel the one-loop gauge anomaly modifies the classical equations of motion previously obtained by Bergshoeff, Sezgin and Sokatchev (BSS). We discuss the supermultiplet structure of the anomalies exhibited in the resulting equations of motion. The anomaly corrected field equations agree with the global limit, recently obtained by Duff, Liu, Lu and Pope, of a matter coupled supergravity theory in six dimensions. We also obtain the dual formulation of the BSS model in which the tensor multiplet is free while the field equations of the Yang–Mills multiplet contain the fields of the tensor multiplet at the classical level.

New couplings of six-dimensional supergravity

We describe the couplings of six-dimensional supergravity, which contain a self-dual tensor multiplet, to n T anti-self-dual tensor matter multiplets, nv vector multiplets and n H hypermultiplets. The scalar fields of the tensor multiplets form a coset SO( n T, 1)/ SO( n T), while the scalars in the hypermultiplets form quarernionic Kähler-symmetric spaces, the generic example being Sp( n H, 1)/ Sp( n H) ⊗ Sp (1). The gauging of the compact subgroup Sp( n H) ⊗ Sp(1) is also described. These results generalize previous ones in the literature on matter couplings of N = 1 supergravity in six dimensions.

The six-dimensional self-dual tensor

The equations of motion for a self-interacting self-dual tensor in six dimensions are extracted from the equations describing the M-theory five-brane. These equations are presented in a self-contained, six-dimensional Lorentz-covariant form. In particular, it is shown that the field-sttrength tensor satisfies a non-linear generalised self-duality constraint. The self-duality equation is rewritten in five-dimensional notation and shown to be identical to the corresponding equation in the non-covariant formalism.

Coupling a self-dual tensor to gravity in six dimensions

A recent result concerning interacting theories of self-dual tensor gauge fields in six dimensions is generalized to include coupling to gravity. The formalism makes five of the six general coordinate invariances manifest, whereas the sixth one requires a non-trivial analysis. The result should be helpful in formulating the world-volume action of the M theory five-brane.

Interacting chiral gauge fields in six dimensions and Born-Infeld theory

Dimensional reduction of a self-dual tensor gauge field in 6D gives an Abelian vector gauge field in 5D. We derive the conditions under which an interacting 5D theory of an Abelian vector gauge field is the dimensional reduction of a 6D Lorentz invariant interacting theory of a self-dual tensor. Then we specialize to the particular 6D theory that gives 5D Born-Infeld theory. The field equation and Lagrangian of this 6D theory are formulated with manifest 5D Lorentz invariance, while the remaining Lorentz symmetries are realized non-trivially. A string soliton with finite tension and self-dual charge is constructed.

Couplings of self-dual tensor multiplet in six dimensions

The (1,0) supersymmetry in six dimensions admits a tensor multiplet which contains a second-rank antisymmetric tensor field with a self-dual field strength and a dilaton. We describe the fully supersymmetric coupling of this multiplet to a Yang - Mills multiplet, in the absence of supergravity. The self-duality equation for the tensor field involves a Chern - Simons modified field strength, the gauge fermions and an arbitrary dimensionful parameter.

Gauge theory of supergravity based only on a self-dual spin connection.

A gauge theory of supergravity is constructed based only on the supersymmetric self-dual spin connection associated to the supergroup OSp $(1|4)$. We show that Jacobson's supergravity action arises naturally from our proposed action. It is formulated by taking the self-dual part of the MacDowell-Mansouri gauge theory of supergravity. In this sense, our quadratic action in the supersymmetric self-dual curvature tensor provides a relation between these two important previous extensions of supergravity.

On the relation of four-dimensional N = 2, 4 supersymmetric string backgrounds to integrable models

In this letter we discuss the relation of four-dimensional, N = 2 supersymmetric string backgrounds to integrable models. In particular we show that non-Kählerian gravitational backgrounds with one U(1) isometry plus non-trivial antisymmetric tensor and dilaton fields arise as the solutions of the Liouville equation or, for the case of vanishing central charge deficit, as the solutions of the continual Toda equation. When performing an Abelian duality transformation, a particular class of solutions of the continual Toda equation leads to the well-known gravitational Eguchi-Hanson instanton background with self-dual curvature tensor.

On self-dual gravity.

We study the Ashtekar-Jacobson-Smolin equations that characterize four-dimensional complex metrics with self-dual Riemann tensor. We find that we can characterize any self-dual metric by a function that satisfies a nonlinear evolution equation, to which the general solution can be found iteratively. This formal solution depends on two arbitrary functions of three coordinates. We study the symmetry algebra of these equations and find that they admit a generalized ${W}_{\ensuremath{\infty}}\ensuremath{\bigoplus}{W}_{\ensuremath{\infty}}$ algebra. We then find the associated conserved quantities which are found to have vanishing Poisson brackets (up to surface terms). We construct explicitly some families of solutions that depend on two free functions of two coordinates, included in which are the multi-center metrics of Gibbons and Hawking. Finally, in an appendix, we show how our formulation of self-dual gravity is equivalent to that of Pleba\~nski.

Neighbours of Einstein's equations-some new results

There exists an infinite-parameter set of 'neighbours' of Einstein's equations (at least if complex metrics are admitted). New results reported include exact solutions with self-dual Riemann tensor for a large class of these models, partial results concerning matter couplings, and a discussion of metrical signatures in the real case.

On extended taub-NUT metrics

Much attention has been paid to the (Euclidean) Taub-NUT metric because the geodesic on this space describes approximately the motion of two well-separated interacting monopoles. It is also well known that the Taub-NUT metric admits a Kepler-type symmetry. In this paper, the Taub-NUT metric is extended so that it still admits a Kepler-type symmetry. The geodesics of this metric will be investigated. In particular, regularization of singular geodesics is studied by use of a method from dynamical systems. Further, some geometrical properties of the extended Taub-NUT metric are cleared up. In order that the extended Taub-NUT metric either has a self-dual Riemann curvature tensor or is an Einstein metric, it is necessary and sufficient that it coincides with the original Taub-NUT metric up to a constant factor. Furthermore, a class of extended Taub-NUT metrics have a self-dual Weyl curvature tensor is found. This class of metrics, of course, includes the Taub-NUT metric.