Topics in Cubic Special Geometry

TL;DR

This paper explores how quantum corrections and Peccei-Quinn transformations affect the structure of cubic special Kähler geometries and black hole charge orbits, revealing new phenomena in extremal black hole solutions.

Contribution

It demonstrates how PQ transformations alter the quartic invariant and charge orbits, impacting black hole entropy and supersymmetry properties in N=2 theories.

Findings

PQ transformations can change charge orbits and supersymmetry properties.

Transitions between large and small charge orbits are possible via PQ actions.

An alternative expression for the quartic invariant I4 relates to black hole entropy.

Abstract

We reconsider the sub-leading quantum perturbative corrections to N=2 cubic special Kaehler 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, 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 I4 of the charge representation 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 R, with corresponding change of the supersymmetry properties of the supported attractor solutions. Furthermore, a suitable action of PQ transformations can also set I4 to zero, or vice versa it can generate a non-vanishing 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 I4, with relevant application to black hole entropy.