Topics in cubic special geometry

Abstract

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.