Twisted Self-duality Research Articles

Twisted Self-Duality

Twisted Self-Duality

Manifest electric-magnetic duality in linearized conformal gravity

We derive a manifestly duality-symmetric formulation of the action principle for conformal gravity linearized around Minkowski space-time. The analysis is performed in the Hamiltonian formulation, the fourth-order character of the equations of motion requiring the formal treatment of the three-dimensional metric perturbation and the extrinsic curvature as independent dynamical variables. The constraints are solved in terms of two symmetric potentials that are interpreted as a dual three-dimensional metric and a dual extrinsic curvature. The action principle can be written in terms of these four dynamical variables, duality acting as simultaneous rotations in the respective spaces spanned by the three-dimensional metrics and the extrinsic curvatures. A twisted self-duality formulation of the equations of motion is also provided.

E 9 exceptional field theory. Part II. The complete dynamics

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E9 exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends {mathfrak{e}}_9 by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity.

Prepotentials for linearized supergravity

Linearized supergravity in arbitrary dimension is reformulated into a first order formalism which treats the graviton and its dual on the same footing at the level of the action. This generalizes previous work by other authors in two directions: (1) we work in arbitrary space-time dimension, and (2) the gravitino field and supersymmetry are also considered. This requires the construction of conformally invariant curvatures (the Cotton fields) for a family of mixed symmetry tensors and tensor-spinors, whose properties we prove (invariance; completeness; conformal Poincaré lemma). We use these geometric tools to solve the Hamiltonian constraints of the graviton and gravitino through the introduction of prepotentials (two tensor fields for the graviton, one tensor-spinor for the gravitino) enjoying (linearized) conformal invariance. The action and equations of motion take a geometrically simple form in terms of the Cotton tensor(-spinors) of the prepotentials. In particular, the equations of motion of the graviton are equivalent to twisted self-duality conditions. We express the supersymmetric transformations of the graviton and gravitino into each other in terms of the prepotentials. We also reproduce the dimensional reduction of supergravity within the prepotential formalism. Finally, our formulas in dimension five are recovered from the dimensional reduction of the already known prepotential formulation of the 6D maximally supersymmetric theory.

Erratum: Twisted self-duality for higher spin gauge fields and prepotentials [Phys. Rev. D 94 , 105027 (2016)

Erratum: Twisted self-duality for higher spin gauge fields and prepotentials [Phys. Rev. D <b>94</b> , 105027 (2016)

Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds

We give the global mathematical formulation of a class of generalized four-dimensional theories of gravity coupled to scalar matter and to Abelian gauge fields. In such theories, the scalar fields are described by a section of a surjective pseudo-Riemannian submersion π over space–time, whose total space carries a Lorentzian metric making the fibers into totally-geodesic connected Riemannian submanifolds. In particular, π is a fiber bundle endowed with a complete Ehresmann connection whose transport acts through isometries between the fibers. In turn, the Abelian gauge fields are “twisted” by a flat symplectic vector bundle defined over the total space of π. This vector bundle is endowed with a vertical taming which locally encodes the gauge couplings and theta angles of the theory and gives rise to the notion of twisted self-duality, of crucial importance to construct the theory. When the Ehresmann connection of π is integrable, we show that our theories are locally equivalent to ordinary Einstein-Scalar-Maxwell theories and hence provide a global non-trivial extension of the universal bosonic sector of four-dimensional supergravity. In this case, we show using a special trivializing atlas of π that global solutions of such models can be interpreted as classical “locally-geometric” U-folds. In the non-integrable case, our theories differ locally from ordinary Einstein-Scalar-Maxwell theories and may provide a geometric description of classical U-folds which are “locally non-geometric”.

Dual formulation of covariant nonlinear duality-symmetric action of kappa-symmetric D3-brane

We study the construction of covariant nonlinear duality-symmetric actions in dual formulation. Essentially, the construction is the PST-covariantisation and nonlinearisation of Zwanziger action. The covariantisation made use of three auxiliary scalar fields. Apart from these, the construction proceed in a similar way to that of the standard formulation. For example, the theories can be extended to include interactions with external fields, and that the theories possess two local PST symmetries. We then explicitly demonstrate the construction of covariant nonlinear duality-symmetric actions in dual formulation of DBI theory, and D3-brane. For each of these theories, the twisted selfduality condition obtained from duality-symmetric actions are explicitly shown to match with the duality relation between field strength and its dual from the one-potential actions. Their on-shell actions between the duality-symmetric and the one-potential versions are also shown to match. We also explicitly prove kappa-symmetry of the covariant nonlinear duality-symmetric D3-brane action in dual formulation.

Twisted self-duality for higher spin gauge fields and prepotentials

We show that the equations of motion for (free) integer higher spin gauge fields can be formulated as twisted self-duality conditions on the higher spin curvatures of the spin-$s$ field and its dual. We focus on the case of four spacetime dimensions, but formulate our results in a manner applicable to higher spacetime dimensions. The twisted self-duality conditions are redundant and we exhibit a non-redundant subset of conditions, which have the remarkable property to involve only first-order derivatives with respect to time. This non-redundant subset equates the electric field of the spin-$s$ field (which we define) to the magnetic field of its dual (which we also define), and vice versa. The non-redundant subset of twisted self-duality conditions involve the purely spatial components of the spin-$s$ field and its dual, and also the components of the fields with one zero index. One can get rid of these gauge components by taking the curl of the equations, which does not change their physical content. In this form, the twisted self-duality conditions can be derived from a variational principle that involves prepotentials, which are the higher spin generalizations of the prepotentials previously found in the spins 2 and 3 cases. The prepotentials have again the intriguing feature of possessing both higher spin diffeomorphism invariance and higher spin conformal geometry. The tools introduced in an earlier paper for handling higher spin conformal geometry turn out to be crucial for streamlining the analysis. In four spacetime dimensions where the electric and magnetic fields are tensor fields of the same type, the twisted self-duality conditions enjoy an $SO(2)$ electric-magnetic invariance. We explicitly show that this symmetry is an "off-shell symmetry" (i.e., a symmetry of the action and not just of the equations of motion). Remarks on the extension to higher dimensions are given.

Strings, branes and the self-dual solutions of Exceptional Field Theory

It has been shown that membranes and fivebranes are wave-like or monopole-like solutions in some higher dimensional theory. Here the picture is completed by combining the wave and monopole solutions into a single solution of Exceptional Field Theory. This solution solves the twisted self-duality constraint. The 1/2 BPS brane spectrum, consisting of fundamental, solitonic and Dirichlet branes and their bound states, in ten- and eleven-dimensional supergravity may all be extracted from this single solution of Exceptional Field Theory. The solution’s properties such as its asymptotic behavior at the core and at infinity are investigated.

Exceptional field theory. II.E7(7)

We introduce exceptional field theory for the group E_{7(7)}, based on a (4+56)-dimensional spacetime subject to a covariant section condition. The `internal' generalized diffeomorphisms of the coordinates in the fundamental representation of E_{7(7)} are governed by a covariant `E-bracket', which is gauged by 56 vector fields. We construct the complete and unique set of field equations that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. Among them feature the non-abelian twisted self-duality equations for the 56 gauge vectors. We discuss the explicit solutions of the section condition describing the embedding of the full, untruncated 11-dimensional and type IIB supergravity, respectively. As a new feature compared to the previously constructed E_{6(6)} formulation, some components among the 56 gauge vectors descend from the 11-dimensional dual graviton but nevertheless allow for a consistent coupling by virtue of a covariantly constrained compensating 2-form gauge field.

Twisted self-duality for linearized gravity inDdimensions

The linearized Einstein equations in D spacetime dimensions can be written as twisted self-duality equations expressing that the linearized curvature tensor of the graviton described by a rank-two symmetric tensor, is dual to the linearized curvature tensor of the "dual graviton" described by a tensor of (D-3,1) Young symmetry type. In the case of 4 dimensions, both the graviton and its dual are rank-two symmetric tensors (Young symmetry type (1,1)), while in the case of 11 space-time dimensions relevant to M-theory, the dual graviton is described by a tensor of (8,1) Young symmetry type. We provide in this paper an action principle that yields the twisted self-duality conditions as equations of motion, keeping the graviton and its dual on equal footing. In order to construct a local, quadratic, variational principle for the twisted linear self-duality equations, it is necessary to introduce two "prepotentials". These are also tensors of mixed Young symmetry types and are obtained by solving the Hamiltonian constraints of the Hamiltonian formulation either of the Pauli-Fierz action for the graviton or of the Curtright action for its dual, the resulting actions being identical. The prepotentials enjoy interesting gauge invariance symmetries, which are exhibited and generalize the gauge symmetries found in D=4. A variational principle where the basic variables are the original Pauli-Fierz field and its dual can also be given but contrary to the prepotential action, the corresponding action is non-local in space -- while remaining local in time. We also analyze in detail the Hamiltonian structure of the theory and show that the graviton and its dual are canonically conjugate in a sense made precise in the text.

Constitutive relations and Schrödinger’s formulation of nonlinear electromagnetic theories

Abstract We present a systematic study of nonlinear and higher derivatives extensions of electromagnetism. We clarify when action functionals S[F] can be explicitly obtained from arbitrary (not necessarily self-dual) nonlinear equations of motion. We show that the “Deformed twisted self-duality condition” proposal originated in the context of supergravity counterterms is actually the general framework needed to discuss self-dual theories starting from a variational principle. We generalize to nonlinear and higher derivatives theories Schrödinger formulation of Born-Infeld theory, and for the latter, and more in general for nonlinear theories, we derive a closed form expression of the corresponding deformed twisted self-duality conditions. This implies that the hypergeometric expression entering these duality conditions and leading to Born-Infeld theory satisfies a hidden quartic equation.

Bispinor auxiliary fields in duality-invariant electrodynamics revisited

Motivated by recent progress in studying the duality-symmetric models of nonlinear electrodynamics, we revert to the auxiliary tensorial (bispinor) field formulation of the $O(2)$ duality proposed by us in Ivanov and Zupnik [Nucl. Phys. B618, 3 (2001); Yad. Fiz. 67, 2212 (2004)]. In this approach, the entire information about the given duality-symmetric system is encoded in the $O(2)$-invariant interaction Lagrangian which is a function of the auxiliary fields ${V}_{\ensuremath{\alpha}\ensuremath{\beta}}$, ${\overline{V}}_{\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\alpha}}\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\beta}}}$. We extend this setting to duality-symmetric systems with higher derivatives and show that the recently employed ``nonlinear twisted self-duality constraints'' amount to the equations of motion for the auxiliary tensorial fields in our approach. Some other related issues are briefly discussed and a few instructive examples are explicitly worked out.

Covariant actions for models with nonlinear twisted self-duality

We describe a systematic way of the generalization, to models with non-linear duality, of the space-time covariant and duality-invariant formulation of duality-symmetric theories in which the covariance of the action is ensured by the presence of a single auxiliary scalar field. It is shown that the duality-symmetric action should be invariant under the two local symmetries characteristic of this approach, which impose constraints on the form of the action similar to those of Gaillard and Zumino and in the non-covariant formalism. We show that the (twisted) self-duality condition obtained from this action upon integrating its equations of motion can always be recast in a manifestly covariant form which is independent of the auxiliary scalar and thus corresponds to the conventional on-shell duality-symmetric covariant description of the same model. Supersymmetrization of this construction is briefly discussed.

Action for twisted self-duality

One may write the Maxwell equations in terms of two gauge potentials, one electric and one magnetic, by demanding that their field strengths should be dual to each other. This requirement is the condition of twisted self-duality. It can be extended to $p$-forms in spacetime of $D$ dimensions, and it survives the introduction of a variety of couplings among forms of different rank, and also to spinor and scalar fields, which emerge naturally from supergravity. In this paper we provide a systematic derivation of the action principle, whose equations of motion are the condition of twisted self-duality. The derivation starts from the standard Maxwell action, extended to include the aforementioned couplings, and proceeds via the Hamiltonian formalism through the resolution of Gauss's law. In the pure Maxwell case we recover in this way an action that had been postulated by other authors, through an ansatz based on an action given earlier by us for untwisted self-duality. When Chern-Simons couplings are included, our action is, however, new. The derivation from the standard extended Maxwell action implies of course that the theory is Lorentz invariant and can be locally coupled to gravity. Nevertheless we include a direct compact Hamiltonian proof of these properties, which is based on the surface-deformation algebra. The symmetry in the dependence of the action on the electric and magnetic variables is manifest, since they appear as canonical conjugates. Spacetime covariance, although present, is not manifest.

Actions for non-abelian twisted self-duality

The dynamics of abelian vector and antisymmetric tensor gauge fields can be described in terms of twisted self-duality equations. These first-order equations relate the p-form fields to their dual forms by demanding that their respective field strengths are dual to each other. It is well known that such equations can be integrated to a local action that carries on equal footing the p-forms together with their duals and is manifestly duality invariant. Space–time covariance is no longer manifest but still present with a non-standard realization of space–time diffeomorphisms on the gauge fields. In this paper, we give a non-abelian generalization of this first-order action by gauging part of its global symmetries. The resulting field equations are non-abelian versions of the twisted self-duality equations. A key element in the construction is the introduction of proper couplings to higher-rank tensor fields. We discuss possible applications (to Yang–Mills and supergravity theories) and comment on the relation to previous no-go theorems.

Twisted self-duality of dimensionally reduced gravity and vertex operators

The Geroch group, isomorphic to the SL(2, R) affine Kac-Moody group, is an infinite-dimensional solution generating group of Einstein's equations with two surface orthogonal commuting Killing vectors. We introduce another solution generating group for these equations, the dressing group, and discuss its connection with the Geroch group. We show that it acts transitively on a dense subset of moduli space. We use a new Lax pair expressing a twisted self-duality of this system and we study the dressing problem associated to it. We also describe how to use vertex operators to solve the reduced Einstein's equations.

Conformal internal symmetry of 2d σ-models coupled to gravity and a dilaton

General relativity reduced to two dimensions possesses a large group of symmetries that exchange classical solutions. The associated Lie algebra is known to contain the affine Kac-Moody algebra A 1 (1) and half of a real Witt algebra. In this paper we exhibit the full symmetry under the semi-direct product of Lie A( 1 (1)) by the Witt algebra Lie ( W ), Furthermore we exhibit the corresponding hidden gauge symmetries. We show that the theory can be understood in terms of an infinite dimensional potential space involving all degrees of freedom: the dilaton as well as matter and gravitation. In the dilaton sector the linear system that extends the previously known Lax pair has the form of a twisted self-duality constraint that is the analog of the self-duality constraint arising in extended supergravities in higher space-time dimensions. Our results furnish a group theoretical explanation for the simultaneous occurrence of two spectral parameters, a constant one (= y) and a variable one (= t). They hodl for all 2d non-linear σ-models that are obtained by dimensional reduction of G/ H models in three dimensions coupled to pure gravity. In that case the Lie algebra is Lie ( W ⋉ G (1)) ; this symmetry acts on a set of off-shell fields (in a fixed gauge) and preserves the equations of motion.