Symplectic Frame Research Articles

On multifield Born and Born-Infeld theories and their non-Abelian generalizations

Starting from a recently proposed linear formulation in terms of auxiliary fields, we study $n$-field generalizations of Born and Born-Infeld theories. In this description the Lagrangian is quadratic in the vector field strengths and the symmetry properties (including the characteristic self-duality) of the corresponding non-linear theory are manifest as on-shell duality symmetries and depend on the choice of the (homogeneous) manifold spanned by the auxiliary scalar fields and the symplectic frame. By suitably choosing these defining properties of the quadratic Lagrangian, we are able to reproduce some known multi-field Born-Infeld theories and to derive new non-linear models, such as the $n$-field Born theory. We also discuss non-Abelian generalizations of these theories obtained by choosing the vector fields in the adjoint representation of an off-shell compact global symmetry group $K$ and replacing them by non-Abelian, $K$-covariant field strengths, thus promoting $K$ to a gauge group.

Observations on BI from N = 2 $$ \mathcal{N}=2 $$ supergravity and the general Ward identity

The multi-vector generalization of a rigid, partially-broken $$ \mathcal{N}=2 $$ supersymmetric theory is presented as a rigid limit of a suitable gauged $$ \mathcal{N}=2 $$ supergravity with electric, magnetic charges and antisymmetric tensor fields. This on the one hand generalizes a known result by Ferrara, Girardello and Porrati while on the other hand allows to recover the multi-vector BI models of [4] from $$ \mathcal{N}=2 $$ supergravity as the end-point of a hierarchical limit in which the Planck mass first and then the supersymmetry breaking scale are sent to infinity. We define, in the parent supergravity model, a new symplectic frame in which, in the rigid limit, manifest symplectic invariance is preserved and the electric and magnetic Fayet-Iliopoulos terms are fully originated from the dyonic components of the embedding tensor. The supergravity origin of several features of the resulting rigid supersymmetric theory are then elucidated, such as the presence of a traceless SU(2)- Lie algebra term in the Ward identity and the existence of a central charge in the supersymmetry algebra which manifests itself as a harmless gauge transformation on the gauge vectors of the rigid theory; we show that this effect can be interpreted as a kind of “superspace non-locality” which does not affect the rigid theory on space-time. To set the stage of our analysis we take the opportunity in this paper to provide and prove the relevant identities of the most general dyonic gauging of Special-Kaehler and Quaternionic-Kaehler isometries in a generic $$ \mathcal{N}=2 $$ model, which include the supersymmetry Ward identity, in a fully symplectic-covariant formalism.

Scattering amplitudes in N = 2 $$ \mathcal{N}=2 $$ Maxwell-Einstein and Yang-Mills/Einstein supergravity

We expose a double-copy structure in the scattering amplitudes of the generic Jordan family of N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity theories in four and five dimensions. The Maxwell-Einstein supergravity amplitudes are obtained through the color/kinematics duality as a product of two gauge-theory factors; one originating from pure N=2 super-Yang-Mills theory and the other from the dimensional reduction of a bosonic higher-dimensional pure Yang-Mills theory. We identify a specific symplectic frame in four dimensions for which the on-shell fields and amplitudes from the double-copy construction can be identified with the ones obtained from the supergravity Lagrangian and Feynman-rule computations. The Yang-Mills/Einstein supergravity theories are obtained by gauging a compact subgroup of the isometry group of their Maxwell-Einstein counterparts. For the generic Jordan family this process is identified with the introduction of cubic scalar couplings on the bosonic gauge-theory side, which through the double copy are responsible for the non-abelian vector interactions in the supergravity theory. As a demonstration of the power of this structure, we present explicit computations at tree-level and one loop. The double-copy construction allows us to obtain compact expressions for the supergravity superamplitudes which are naturally organized as polynomials in the gauge coupling constant.

Lagrangian Curves in a 4-Dimensional Affine Symplectic Space

Lagrangian curves in \(\mathbb {R}^{4}\) entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in \(\mathbb {R}^{4}\) and determine Lagrangian geodesics.

On invariant structures of black hole charges

Abstract We study “minimal degree” complete bases of duality- and “horizontal”- invariant homogeneous polynomials in the flux representation of two-centered black hole solutions in two classes of D = 4 Einstein supergravity models with symmetric vector multiplets’ scalar manifolds. Both classes exhibit an SL(2, $ \mathbb{R} $ ) “horizontal” symmetry which mixes the two centers. The first class encompasses $ \mathcal{N} = {2} $ and $ \mathcal{N} = {4} $ matter-coupled theories, with semisimple U-duality given by SL(2, $ \mathbb{R} $ ) × SO(m,n); the analysis is carried out in the so-called Calabi-Vesentini symplectic frame (exhibiting maximal manifest covariance) and until order six in the fluxes included. The second class, exhibiting a non-trivial “horizontal” stabilizer SO(2), includes $ \mathcal{N} = {2} $ minimally coupled and $ \mathcal{N} = 3 $ matter coupled theories, with U-duality given by the pseudounitary group U(r,s) (related to complex flux representations). Finally, we comment on the formulation of special Kähler geometry in terms of “generalized” groups of type E 7.

Topics in cubic special geometry

We reconsider the sub-leading quantum perturbative corrections to \documentclass[12pt]{minimal}\begin{document}$\mathcal {N }=2$\end{document}N=2 cubic special Kähler geometries. Imposing the invariance under axion-shifts, all such corrections (but the imaginary constant one) can be introduced or removed through suitable, lower unitriangular symplectic transformations and dubbed Peccei-Quinn (PQ) transformations. Since PQ transformations do not belong to the d = 4 U-duality group G4, in symmetric cases they generally have a non-trivial action on the unique quartic invariant polynomial \documentclass[12pt]{minimal}\begin{document}$\mathcal {I}_{4}$\end{document}I4 of the charge representation \documentclass[12pt]{minimal}\begin{document}$\mathbf {R}$\end{document}R of G4. This leads to interesting phenomena in relation to theory of extremal black hole attractors; namely, the possibility to make transitions between different charge orbits of \documentclass[12pt]{minimal}\begin{document}$\mathbf {R}$\end{document}R, with corresponding change of the supersymmetry properties of the supported attractor solutions. Furthermore, a suitable action of PQ transformations can also set \documentclass[12pt]{minimal}\begin{document}$\mathcal {I}_{4}$\end{document}I4 to zero, or vice versa it can generate a non-vanishing \documentclass[12pt]{minimal}\begin{document}$\mathcal {I}_{4}$\end{document}I4: this corresponds to transitions between “large” and “small” charge orbits, which we classify in some detail within the “special coordinates” symplectic frame. Finally, after a brief account of the action of PQ transformations on the recently established correspondence between Cayley's hyperdeterminant and elliptic curves, we derive an equivalent, alternative expression of \documentclass[12pt]{minimal}\begin{document}$\mathcal {I}_{4}$\end{document}I4, with relevant application to black hole entropy.

Iwasawa N=8 attractors

Starting from the symplectic construction of the Lie algebra e7(7) due to Adams, we consider an Iwasawa parametrization of the coset E7(7)SU(8), which is the scalar manifold of N=8, d=4 supergravity. Our approach, and the manifest off-shell symmetry of the resulting symplectic frame, is determined by a noncompact Cartan subalgebra of the maximal subgroup SL(8,R) of E7(7). In the absence of gauging, we utilize the explicit expression of the Lie algebra to study the origin of E7(7)SU(8) as scalar configuration of a 18-BPS extremal black hole attractor. In such a framework, we highlight the action of a U(1) symmetry spanning the dyonic 18-BPS attractors. Within a suitable supersymmetry truncation allowing for the embedding of the Reissner–Nördstrom black hole, this U(1) action is interpreted as nothing but the global R-symmetry of pure N=2 supergravity. Moreover, we find that the above mentioned U(1) symmetry is broken down to a discrete subgroup Z4, implying that all 18-BPS Iwasawa attractors are nondyonic near the origin of the scalar manifold. We can trace this phenomenon back to the fact that the Cartan subalgebra of SL(8,R) used in our construction endows the symplectic frame with a manifest off-shell covariance which is smaller than SL(8,R) itself. Thus, the consistence of the Adams–Iwasawa symplectic basis with the action of the U(1) symmetry gives rise to the observed Z4 residual nondyonic symmetry.

More onN=8attractors

We examine a few simple extremal black hole configurations of $\mathcal{N}=8$, $d=4$ supergravity. We first elucidate the relation between the BPS Reissner-N\ordstrom black hole and the non-BPS Kaluza-Klein dyonic black hole. Their classical entropy, given by the Bekenstein-Hawking formula, can be reproduced via the attractor mechanism by suitable choices of symplectic frame. Then, we display the embedding of the axion-dilaton black hole into $\mathcal{N}=8$ supergravity.

On numerical calculation in symplectic approach for elasticity problems

The symplectic approach proposed and developed by Zhong et al. in 1990s for elasticity problems is a rational analytical method, in which ample experience is not needed as in the conventional semi-inverse method. In the symplectic space, elasticity problems can be solved using the method of separation of variables along with the eigenfunction expansion technique, as in traditional Fourier analysis. The eigensolutions include those corresponding to zero and nonzero eigenvalues. The latter group can be further divided into α- and β-sets. This paper reformulates the form of β-set eigensolutions to achieve the stability of numerical calculation, which is very important to obtain accurate results within the symplectic frame. An example is finally given and numerical results are compared and discussed.