Self-duality Constraint Research Articles

Deformation of d = 4, mathcal{N} ≥ 5 supergravities breaks nonlinear local supersymmetry

We study d = 4, mathcal{N} ≥ 5 supergravities and their deformation via candidate counterterms, with the purpose to absorb UV divergences. We generalize the earlier studies of deformation and twisted self-duality constraint to the case with unbroken local mathcal{H} -symmetry in presence of fermions. We find that the deformed action breaks nonlinear local supersymmetry. We show that all known cases of enhanced UV divergence cancellations are explained by nonlinear local supersymmetry.This result implies, in particular, that if mathcal{N} = 5 supergravity at five loop will turn out to be UV divergent, the deformed theory will be BRST inconsistent. If it will be finite, it will be a consequence of nonlinear local supersymmetry and E7-type duality.

Twisted Self-Duality

Twisted Self-Duality

M5-brane in the superspace approach

Motivated by Sen's spacetime prescription for the construction of theories with self-dual field strengths, we present a rigid superspace Lagrangian describing noninteracting tensor multiplets living on a stack of M5-branes and containing all the physical constraints on the fields, yielding the on-shell matching of the degrees of freedom. The geometric superspace approach adopted here offers a natural realization of superdiffeomorphisms and is particularly well suited for the coupling to supergravity. However, within this formulation the (anti-)self-duality property of the 3-form field strengths is lost when the superspace Lagrangian is trivially restricted to spacetime. We propose two main paths to address this issue: a first-order superspace extension of Sen's spacetime results, which, once trivially restricted to spacetime, yields all the dynamical equations including the (anti-)self-duality constraint on the 3-form field strengths, and a possible way to obtain a full superspace description of the theory, based on integral forms.

A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry

We consider a non-relativistic limit of the bosonic sector of eleven-dimensional supergravity, leading to a theory based on a covariant ‘membrane Newton-Cartan’ (MNC) geometry. The local tangent space is split into three ‘longitudinal’ and eight ‘transverse’ directions, related only by Galilean rather than Lorentzian symmetries. This generalises the ten-dimensional stringy Newton-Cartan (SNC) theory. In order to obtain a finite limit, the field strength of the eleven-dimensional four-form is required to obey a transverse self-duality constraint, ultimately due to the presence of the Chern-Simons term in eleven dimensions. The finite action then gives a set of equations that is invariant under longitudinal and transverse rotations, Galilean boosts and local dilatations. We supplement these equations with an extra Poisson equation, coming from the subleading action. Reduction along a longitudinal direction gives the known SNC theory with the addition of RR gauge fields, while reducing along a transverse direction yields a new non-relativistic theory associated to D2 branes. We further show that the MNC theory can be embedded in the U-duality symmetric formulation of exceptional field theory, demonstrating that it shares the same exceptional Lie algebraic symmetries as the relativistic supergravity, and providing an alternative derivation of the extra Poisson equation.

Born sigma-models for para-Hermitian manifolds and generalized T-duality

We give a covariant realization of the doubled sigma-model formulation of duality-symmetric string theory within the general framework of para-Hermitian geometry. We define a notion of generalized metric on a para-Hermitian manifold and discuss its relation to Born geometry. We show that a Born geometry uniquely defines a worldsheet sigma-model with a para-Hermitian target space, and we describe its Lie algebroid gauging as a means of recovering the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold. Applying the Kotov–Strobl gauging leads to a generalized notion of T-duality when combined with transformations that act on Born geometries. We obtain a geometric interpretation of the self-duality constraint that halves the degrees of freedom in doubled sigma-models, and we give geometric characterizations of non-geometric string backgrounds in this setting. We illustrate our formalism with detailed worldsheet descriptions of closed string phase spaces, of doubled groups where our notion of generalized T-duality includes non-abelian T-duality, and of doubled nilmanifolds.

T-dual superstring Lagrangian with double zweibeins

We present superstring Lagrangians with manifest T-duality. The Lagrangian version of the section conditions are necessary to make Lagrangians to be general coordinate invariant. We show the general solution of section conditions. The D-dimensional left and right moving currents are the 2D-dimensional chiral current which causes the chiral boson problem. We solve the problem by adding the unphysical 2D-dimensional anti­selfdual current with the selfduality constraints. The Lagrange multipliers of the selfduality constraints play the role of the worldsheet zweibein allowing the Weyl invariant and Lorentz symmetric worldsheet. Doubling the zweibein makes the type II κ-symmetry splitting into two sets of the type It κ-symmetries.

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.

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.

Unification of Type-II Strings andTDuality

We present a unified description of the low-energy limits of type-II string theories. This is achieved by a formulation that doubles the space-time coordinates in order to realize the T-duality group O(10,10) geometrically. The Ramond-Ramond fields are described by a spinor of O(10,10), which couples to the gravitational fields via the Spin(10,10) representative of the so-called generalized metric. This theory, which is supplemented by a T-duality covariant self-duality constraint, unifies the type-II theories in that each of them is obtained for a particular subspace of the doubled space.

Selfduality of [formula omitted] reduction of gravity coupled to a σ-model

Dimensional reduction in two dimensions of gravity in higher dimension, or more generally of d=3 gravity coupled to a σ-model on a symmetric space, is known to possess an infinite number of symmetries. We show that such a bidimensional model can be embedded in a covariant way into a σ-model on an infinite symmetric space, built on the semidirect product of an affine group by the Witt group. The finite theory is the solution of a covariant selfduality constraint on the infinite model. It has therefore the symmetries of the infinite symmetric space. (We give explicit transformations of the gauge algebra.) The usual physical fields are recovered in a triangular gauge, in which the equations take the form of the usual linear systems which exhibit the integrable structure of the models. Moreover, we derive the constraint equation for the conformal factor, which is associated to the central term of the affine group involved.

Superconformal Selfdual Sigma-Models

A range of bosonic models can be expressed as (sometimes generalized) $\sigma$-models, with equations of motion coming from a selfduality constraint. We show that in D=2, this is easily extended to supersymmetric cases, in a superspace approach. In particular, we find that the configurations of fields of a superconformal $\mathfrak{G}/\mathfrak{H}$ coset models which satisfy some selfduality constraint are automatically solutions to the equations of motion of the model. Finally, we show that symmetric space $\sigma$-models can be seen as infinite-dimensional $\tfG/\tfH$ models constrained by a selfduality equation, with $\tfG$ the loop extension of $\mathfrak{G}$ and $\tfH$ a maximal subgroup. It ensures that these models have a hidden global $\tfG$ symmetry together with a local $\tfH$ gauge symmetry.

Bosonization and duality in arbitrary dimensions: New results

A generic massive Thirring Model in three space-time dimensions exhibits a correspondence with a topologically massive bosonized gauge action associated to a self-duality constraint, and we write down a general expression for this relationship. We also generalize this structure to $d$ dimensions, by adopting the so-called doublet approach, recently introduced. In particular, a non- conventional formulation of the bosonization technique in higher dimensions (in the spirit of $d=3$), is proposed and, as an application, we show how fermionic (Thirring-like) representations for bosonic topologically massive models in four dimensions may be built up.

Kaluza-Klein bundles and manifolds of exceptional holonomy

We show how in the presence of RR two-form field strength the conditions for preserving supersymmetry on six- and seven-dimensional manifolds lead to certain generalizations of monopole equations. For six dimensions the string frame metric is Kaehler with the complex structure that descends from the octonions if in addition we assume F^{(1,1)}=0. The susy generator is a gauge covariantly constant spinor. For seven dimensions the string frame metric is conformal to a G_2 metric if in addition we assume the field strength to obey a selfduality constraint. Solutions to these equations lift to geometries of G_2 and Spin(7) holonomy respectively.

Strongly coupled gravity and duality

A strong coupling limit of theories whose low-energy effective field theory is 5-dimensional N=8 supergravity is proposed in which the gravitational coupling becomes large. It is argued that, if this limit exists, it should be a 6-dimensional theory with (4,0) supersymmetry compactified on a circle whose radius gives the 5-dimensional Planck length. The sector corresponding to the D=5 supergravity multiplet is a (4,0) D=6 superconformal field theory based on the (4,0) multiplet with 27 self-dual 2-forms, 42 scalars and, instead of a graviton, a fourth-rank tensor gauge field satisfying a self-duality constraint. The superconformal field theory has 32 supersymmetries and 32 conformal supersymmetries and its dimensional reduction gives the maximal supergravity in five dimensions. Electromagnetic duality generalises to a gravitational triality.

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.

Self-Duality Condition and Critical Potentials

Abstract It is shown that the self-duality constraint on the scalar field (combined with the equations of motion) by itself leads to the critical forms for the potential that minimizes the energy functional in the Chern-Simons-Higgs (CSH) system. If we have only the Chern-Simons (CS) term in the SL (2, R) gauge group one obtains a formalism that yields the equations of motion of a variety on non-linear models in two dimensions when the curvature is set equal to zero.

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.

An action principle for self-dual Yang-Mills and Einstein equations

Abstract We propose actions from which the self-dual Yang-Mills and Einstein equations in four dimensions follow by straightforward variation. This off-shell formulation essentially involves harmonic variables. After reformulation in harmonic space, the self-duality constraints can be derived from actions with the help of Lagrange multipliers. The latter turn out not to describe additional degrees of freedom. The off-shell theories obtained can be quantized in a standard way.

On a gauge theory of the self-dual field and its quantization

A gauge theory of self-dual fields is constructed by adding a Wess-Zumino term to the recently studied formulation based on a second-order scalar field lagrangian carrying with it an auxiliary vector field to take care of the self-duality constraint in a linear fashion. The two versions are quantized using the BRST formulation following the BFV procedure. No violation of microcausality occurs and the action of the ordinary scalar field may not be written as the sum of the actions of the self- and anti-self-dual fields.

Quantization of self-dual field revisited.

A self-dual field is described by the Lagrangian for an ordinary scalar field with a term added to it to take care of the self-duality constraint. A self-consistent Hamiltonian formulation is obtained using Dirac's method. The constraints are second class, the auxiliary field drops out of the Hamiltonian, and the quantized theory does not show any violation of causality.