Self-dual Two-forms Research Articles

Lorentzian Cayley form

Cayley four-form Φ on an eight-dimensional manifold M is a real differential form of a special algebraic type, which determines a Riemannian metric on M as well as a unit real Weyl spinor. It defines a Spin(7) structure on M, and this Spin(7) structure is integrable if and only if Φ is closed. We introduce the notion of a complex Cayley form. This is a one-parameter family of complex four-forms Φτ on M of a special algebraic type. Each Φτ determines a real Riemannian metric on M, as well as a complex unit Weyl spinor ψτ. The subgroup of GL(8,R) that stabilizes Φτ,τ≠0 is SU(4), and Φτ defines on M an SU(4) structure. We show that this SU(4) structure is integrable if and only if Φτ is closed.We carry out a similar construction for the split signature case. There are now two one-parameter families of complex Cayley forms. A complex Cayley form of one type defines an SU(2,2) structure, a form of the other type defines an SL(4,R) structure on M. As in the Riemannian case, these structures are integrable if and only if the corresponding complex Cayley forms are closed. Our central observation is that there exists a special member of the second one-parameter family of complex Cayley forms, which we call the Lorentzian Cayley form. This four-form has the property that it calibrates two four-dimensional subspaces H,H⊥ that have the property that the induced metric on H,H⊥ is Lorentzian. In particular, in a basis adapted to such a calibration, the Lorentzian Cayley form is built from the complex self-dual two-forms for H,H⊥. We explain how these observations solve a certain puzzle that existed in the context of four-dimensional Lorentzian geometry.

Quantum cosmology of pure connection general relativity

We study homogeneous cosmological models in formulations of general relativity with cosmological constant based on a (complexified) connection rather than a spacetime metric, in particular in a first order theory obtained by integrating out the self-dual two-forms in the chiral Plebański formulation. Classical dynamics for the Bianchi IX model are studied in the Lagrangian and Hamiltonian formalism, where we emphasise the reality conditions needed to obtain real Lorentzian solutions. The solutions to these reality conditions fall into different branches, which in turn lead to different real Hamiltonian theories, only one of which is the usual Lorentzian Bianchi IX model. We also show the simpler case of the flat Bianchi I model, for which both the reality conditions and dynamical equations simplify considerably. We discuss the relation of a real Euclidean version of the same theory to this complex theory. Finally, we study the quantum theory of homogeneous and isotropic models, for which the pure connection action for general relativity reduces to a pure boundary term and the path integral is evaluated immediately, reproducing known results in quantum cosmology. An intriguing aspect of these theories is that the signature of the effective spacetime metric, and hence the interpretation of the cosmological constant, are intrinsically ambiguous.

Harmonic forms on asymptotically ADS metrics

In this paper we study the rotationally invariant harmonic cohomology of a two-parameter family of Einstein metrics g which admits a cohomogeneity one action of SU(2) × U(1) and has AdS asymptotics. Depending on the values of the parameters, g is either of NUT type, if the fixed-point locus of the U(1) action is zero-dimensional, or of bolt type, if it is two-dimensional. We find that if g is of NUT type then the space of SU(2)-invariant harmonic two-forms is three-dimensional and consists entirely of self-dual forms; if g is of bolt type it is four-dimensional. In both cases we explicitly determine a basis. The pair (g, F) for F a self-dual harmonic two-form is also a solution of the bosonic sector of 4D supergravity. We determine for which choices it is a supersymmetric solution and the amount of preserved supersymmetry.

New locally (super)conformal gauge models in Bach-flat backgrounds

For every conformal gauge field {h}_{alpha (n)overset{cdot }{alpha }(m)} in four dimensions, with n ≥ m > 0, a gauge-invariant action is known to exist in arbitrary conformally flat backgrounds. If the Weyl tensor is non-vanishing, however, gauge invariance holds for a pure conformal field in the following cases: (i) n = m = 1 (Maxwell’s field) on arbitrary gravitational backgrounds; and (ii) n = m + 1 = 2 (conformal gravitino) and n = m = 2 (conformal graviton) on Bach-flat backgrounds. It is believed that in other cases certain lower-spin fields must be introduced to ensure gauge invariance in Bach-flat backgrounds, although no closed-form model has yet been constructed (except for conformal maximal depth fields with spin s = 5/2 and s = 3). In this paper we derive such a gauge-invariant model describing the dynamics of a conformal gauge field {h}_{alpha (3)overset{cdot }{alpha }} coupled to a self-dual two-form. Similar to other conformal higher-spin theories, it can be embedded in an off-shell superconformal gauge-invariant action. To this end, we introduce a new family of mathcal{N} = 1 superconformal gauge multiplets described by unconstrained prepotentials ϒα(n), with n > 0, and propose the corresponding gauge-invariant actions on conformally-flat backgrounds. We demonstrate that the n = 2 model, which contains {h}_{alpha (3)overset{cdot }{alpha }} at the component level, can be lifted to a Bach-flat background provided ϒα(2) is coupled to a chiral spinor Ωα. We also propose families of (super)conformal higher-derivative non-gauge actions and new superconformal operators in any curved space. Finally, through considerations based on supersymmetry, we argue that the conformal spin-3 field should always be accompanied by a conformal spin-2 field in order to ensure gauge invariance in a Bach-flat background.

Hermitian-Einstein metrics from noncommutative U(1) instantons

We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean ${\mathbb R}^4$ to symmetric two-tensors introduced in "Gravitational instantons from gauge theory," H. S. Yang and M. Salizzoni, Phys. Rev. Lett. (2006) 201602, [hep-th/0512215]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose $U(1)$ instantons on a noncommutative ${\mathbb C}^2$ as the self-dual two-form, from which we derive a family of Hermitian-Einstein metrics. We also discuss the condition when the metric becomes K\"ahler.

Abelian M5-brane on Sq6

We compute the conformal anomaly of the Abelian M5 brane on a conical deformation Sq6 of the round six-sphere. Our results agree with the corresponding results on S1×H5 that were obtained in J. Nian and Y. Zhou, “Renyi entropy of a free (2, 0) tensor multiplet and its supersymmetric counterpart,” Phys. Rev. D 93(12), 125010 (2016). For the free energies, we obtain missing Casimir energy contributions, inconsequential for the Renyi entropies, and we obtain the proposed constant shift for the Renyi entropy of the self-dual two-form.

Spacetime characterizations of Λ-vacuum metrics with a null Killing 2-form

An exhaustive list of four-dimensional Λ-vacuum spacetimes admitting a Killing vector ξ whose self-dual Killing two-form is null is obtained assuming that the self-dual Weyl tensor is proportional to the tensor product of by itself. Our analysis complements previous results (Mars 1999 Class Quantum Grav. 16 2507–23; Mars and Senovilla 2015 Ann. H. Poincaré 16 1509–50) concerning the case with non-null . We analyze both cases with Λ vanishing or not. In the latter case we prove that must hold necessarily, and we find a characterization of the Einstein spacetimes conformal to pp-waves. In the former case we obtain spacetime characterizations of vacuum plane waves and of the stationary vacuum Brinkmann spacetimes. At the light of the full set of results, old and new, we reformulate the case with non-null and . We finally present a table collecting the results for both null, and non-null, .

M5-branes on S 2 × M 4: Nahm’s equations and 4d topological sigma-models

We study the 6d N=(0,2) superconformal field theory, which describes multiple M5-branes, on the product space S^2 x M_4, and suggest a correspondence between a 2d N=(0,2) half-twisted gauge theory on S^2 and a topological sigma-model on the four-manifold M_4. To set up this correspondence, we determine in this paper the dimensional reduction of the 6d N=(0,2) theory on a two-sphere and derive that the four-dimensional theory is a sigma-model into the moduli space of solutions to Nahm's equations, or equivalently the moduli space of k-centered SU(2) monopoles, where k is the number of M5-branes. We proceed in three steps: we reduce the 6d abelian theory to a 5d Super-Yang-Mills theory on I x M_4, with I an interval, then non-abelianize the 5d theory and finally reduce this to 4d. In the special case, when M_4 is a Hyper-Kahler manifold, we show that the dimensional reduction gives rise to a topological sigma-model based on tri-holomorphic maps. Deriving the theory on a general M_4 requires knowledge of the metric of the target space. For k=2 the target space is the Atiyah-Hitchin manifold and we twist the theory to obtain a topological sigma-model, which has both scalar fields and self-dual two-forms.

Plebański formulation of general relativity: a practical introduction

We give a pedagogical introduction into an old, but unfortunately not commonly known formulation of GR in terms of self-dual two-forms due to in particular Jerzy Plebanski. Our presentation is rather explicit in that we show how the familiar textbook solutions: Schwarzschild, Volkoff-Oppenheimer, as well as those describing the Newtonian limit, a gravitational wave and the homogeneous isotropic Universe can be obtained within this formalism. Our description shows how Plebanski formulation gives quite an economical alternative to the usual metric and frame-based schemes for deriving Einstein equations.

Field representations of vector supersymmetry

We study some field representations of vector supersymmetry with superspin Y = 0 and Y = 1/2 and nonvanishing central charges. For Y = 0, we present two multiplets composed of four spinor fields, two even and two odd, and we provide a free action for them. The main differences between these two multiplets are the way the central charge operators act and the compatibility with the Majorana reality condition on the spinors. One of the two is related to a previously studied spinning particle model. For Y = 1/2, we present a multiplet composed of one even scalar, one odd vector and one even selfdual two-form, which is a truncation of a known representation of the tensor supersymmetry algebra in Euclidean spacetime. We discuss its rotation to Minkowski spacetime and provide a set of dynamical equations for it, which are however not derived from a Lagrangian. We develop a superspace formalism for vector supersymmetry with central charges and we derive our multiplets by superspace techniques. Finally, we discuss some representations with vanishing central charges.

Non-semisimple Gaugings of D = 5 N = 8 Supergravity

The recent construction of the non-semisimple gaugings of maximal D=5 supergravity is reported here. This construction is worked out in the so-called rheonomic approach, based on Free Differential Algebras and the solution of their Bianchi identities. In this approach the dualization mechanism between one-forms and two-forms is more transparent. The lagrangian is unnecessary since the field equations follow from closure of the supersymmetry algebra. These theories contain 12-r self-dual two-forms and 15+r gauge vectors, r of which are abelian and neutral. Such theories, whose existence has been proved and their supersymmetry algebra constructed, have potentially interesting properties in relation with domain wall solutions and the trapping of gravity.

The geometry of self-dual two-forms

We show that self-dual two-forms in 2n-dimensional spaces determine a n2−n+1-dimensional manifold 𝒮2n and the dimension of the maximal linear subspaces of 𝒮2n is equal to the (Radon–Hurwitz) number of linearly independent vector fields on the sphere S2n−1. We provide a direct proof that for n odd 𝒮2n has only one-dimensional linear submanifolds. We exhibit 2c−1-dimensional subspaces in dimensions which are multiples of 2c, for c=1,2,3. In particular, we demonstrate that the seven-dimensional linear subspaces of 𝒮8 also include among many other interesting classes of self-dual two-forms, the self-dual two-forms of Corrigan, Devchand, Fairlie, and Nuyts [Nucl. Phys. B 214, 452 (1983)] and a representation of 𝒞l7 given by octonionic multiplication. We discuss the relation of the linear subspaces with the representations of Clifford algebras.

TOPOLOGICAL HALF-FLAT GRAVITY

A topological field theory modeling the moduli space of half-flat gravitational instantons is presented, in which the fundamental gravitional fields are a trio of self-dual two-forms and an SU(2) connection. The theory can be understand as a topological version of the (Euclidean) Einstein gravity.