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Exploring the geometry of supersymmetric double field theory
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Abstract
The geometry of N=1 supersymmetric double field theory is revisited in superspace. In order to maintain the constraints on the torsion tensor, the local tangent space group of O(D) x O(D) must be expanded to include a tower of higher dimension generators. These include a generator in the irreducible hook representation of the Lorentz group, which gauges the shift symmetry (or ambiguity) of the spin connection. This gauging is possible even in the purely bosonic theory, where it leads to a Lorentz curvature whose only non-vanishing pieces are the physical ones: the generalized Einstein tensor and the generalized scalar curvature. A relation to the super-Maxwell$_\infty$ algebra is proposed. The superspace Bianchi identities are solved up through dimension two, and the component supersymmetry transformations and equations of motion are explicitly (re)derived.
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A bstract
We construct an infinite system of non-linear duality equations, including fermions, that are invariant under global E 11 and gauge invariant under generalised diffeomorphisms upon the imposition of a suitable section constraint. We use finite-dimensional fermionic representations of the R-symmetry K ( E 11 ) to describe the fermionic contributions to the duality equations. These duality equations reduce to the known equations of E 8 exceptional field theory or eleven-dimensional supergravity for appropriate (partial) solutions of the section constraint. Of key importance in the construction is an indecomposable representation of E 11 that entails extra non-dynamical fields beyond those predicted by E 11 alone, generalising the known constrained p -forms of exceptional field theories. The construction hinges on the tensor hierarchy algebra extension of $$ {\mathfrak{e}}_{11} $$ e 11 , both for the bosonic theory and its supersymmetric extension.
A bstract
We study systematically various extensions of the Poincaré superalgebra. The most general structure starting from a set of spinorial supercharges Q α is a free Lie superalgebra that we discuss in detail. We explain how this universal extension of the Poincaré superalgebra gives rise to many other algebras as quotients, some of which have appeared previously in various places in the literature. In particular, we show how some quotients can be very neatly related to Borcherds superalgebras. The ideas put forward also offer some new angles on exotic branes and extended symmetry structures in M-theory.
A bstract
We formulate the locally supersymmetric E 7(7) exceptional field theory in a (4 + 56|32) dimensional superspace, corresponding to a 4D N = 8 “external” superspace augmented with an “internal” 56-dimensional space. This entails the unification of external diffeomorphisms and local supersymmetry transformations into superdiffeomorphisms. The solutions to the superspace Bianchi identities lead to on-shell duality equations for the p -form field strengths for p ≤ 4. The reduction to component fields provides a complete description of the on-shell supersymmetric theory. As an application of our results, we perform a generalized Scherk-Schwarz reduction and obtain the superspace formulation of maximal gauged supergravity in four dimensions parametrized by an embedding tensor.
We construct a supersymmetric extension of double field theory that
realizes the ten-dimensional Majorana-Weyl local supersymmetry. In terms
of a stringy differential geometry we proposed earlier, our action
consists of five simple terms -- two bosonic plus three fermionic -- and
manifests not only diffeomorphism and one-form gauge symmetry of
B-field, but also O(10,10) T-duality as well as a direct product of two
local Lorentz symmetries, SO(1,9) \times SO(9,1). A gauge fixing that
identifies the double local Lorentz groups reduces our action to the
minimal supergravity in ten dimensions.
While the fundamental object in Riemannian geometry is a metric, closed string theories call for us to put a two-form gauge field and a scalar dilaton on an equal footing with the metric. Here we propose a novel differential geometry which treats the three objects in a unified manner, manifests not only diffeomorphism and one-form gauge symmetry but also O(D, D) T-duality, and enables us to rewrite the known low energy effective action of them as a single term. We comment that the notion of cosmological constant naturally changes.
Field theory is an area in physics with a deceptively compact notation. Although general purpose computer algebra systems, built around generic list-based data structures, can be used to represent and manipulate field-theory expressions, this often leads to cumbersome input formats, unexpected side-effects, or the need for a lot of special-purpose code. This makes a direct translation of problems from paper to computer and back needlessly time-consuming and error-prone. A prototype computer algebra system is presented which features -like input, graph data structures, lists with Young-tableaux symmetries and a multiple-inheritance property system. The usefulness of this approach is illustrated with a number of explicit field-theory problems.
The covariant and kappa-symmetric action for superstring in direct product of two flat D=10 N=1 superspaces is presented. It is given by the sum of supersymmetric generalization of two copies of chiral boson actions constructed with the use of the Pasti–Sorokin–Tonin (PST) technique. The chirality of 8 ‘left’ bosons and 8 ‘left’ fermions and the anti-chirality of their ‘right’ counterparts are obtained as gauge fixed version of the equations of motion, so that the physical degrees of freedom are essentially those of the II Green–Schwarz superstring. Our action is manifestly T-duality invariant as the fields describing oscillating and winding modes enter it on equal footing.
We construct the supersymmetric completion of E6(6)-covariant exceptional field theory. The theory is based on a (5 + 27)-dimensional generalized space-time subject to a covariant section constraint. The fermions are tensors under the local Lorentz group SO(1, 4) × USp(8) and transform as weighted scalars under the E6(6) (internal) generalized diffeomorphisms. We present the complete Lagrangian and prove its invariance under supersymmetry. Upon explicit solution of the section constraint the theory embeds full D = 11 supergravity and IIB supergravity, respectively.
We give the supersymmetric extension of exceptional field theory for E7(7), which is based on a (4 + 56)-dimensional generalized spacetime subject to a covariant constraint. The fermions are tensors under the local Lorentz group SO(1, 3) × SU(8) and transform as scalar densities under the E7(7) (internal) generalized diffeomorphisms. The supersymmetry transformations are manifestly covariant under these symmetries and close, in particular, into the generalized diffeomorphisms of the 56-dimensional space. We give the fermionic field equations and prove supersymmetric invariance. We establish the consistency of these results with the recently constructed generalized geometric formulation of D = 11 supergravity.
We reformulate eleven-dimensional supergravity, including fermions, in terms of generalised geometry, for spacetimes that are warped products of Minkowski space with a d-dimensional manifold M with d ≤ 7. The reformulation has an E d( d) × + structure group and it has a local symmetry, where is the double cover of the maximally compact subgroup of E d( d). The bosonic degrees for freedom unify into a generalised metric, and, defining the generalisked analogue D of the Levi-Civita connection, one finds that the corresponding equations of motion are the vanishing of the generalised Ricci tensor. To leading order, we show that the fermionic equations of motion, action and supersymmetry variations can all be written in terms of D. Although we will not give the detailed decompositions, this reformulation is equally applicable to type IIA or IIB supergravity restricted to a ( d - 1)-dimensional manifold. For completeness we give explicit expressions in terms of = Spin(5) and = SU(8) representations for d = 4 and d = 7.
We construct a supersymmetric extension of the Lorentz and Yang-Mills Chern-Simons terms in ten dimensions. In terms of dimensionful parameters α(Lorentz) and β (Yang-Mills), we obtain the complete O(α) supersymmetrization. Furthermore, we present the leading O(α2) and O(αβ) corrections required by supersymmetry, and conjecture that the result is complete to this order. Finally, we briefly discuss the O(α3) and O(αβ2) modifications. A crucial ingredient in these constructions is the existence of an intimate relationship between the ten-dimensional supergravity and Yang-Mills multiplets.
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