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Register a new userSeparation of Attractors in 1-modulus Quantum Corrected Special Geometry
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We study the attractor equations for a quantum corrected prepotential F=t^3+ilambda, with lambda in R,which is the only correction which preserves the axion shift symmetry and modifies the geometry. By performing computations in the ``magnetic charge configuration, we find evidence for interesting phenomena (absent in the classical limit of vanishing lambda). For a certain range of the quantum parameter lambda we find a ``separation of attractors, i.e. the existence of multiple solutions to the Attractor Equations for fixed supporting charge configuration. Furthermore, we find that, away from the classical limit, a ``transmutation of the supersymmetry-preserving features of the attractors takes place when lambda reaches a particular critical value.
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We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of $mathcal{N}=2$, $D=4$ ungauged supergravity obtained as reduction of minimal, matter-coupled $D=5$ supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of $L(q,P,dot{P})$ models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is an homogeneous polynomial of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with $dot{P}=0$, such as $L(q,P)$, $1leqslant qleqslant 3$ and $Pgeqslant 1$,or $L(q,1)$, $4leqslant qleqslant 9$, and one model with $dot{P}=1$, namely $L(4,1,1)$. We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of $L(1,P)$ $Pgeqslant 2$ non-symmetric models of $mathcal{N}=2$, $D=4$ supergravity.
The scalars in vector multiplets of N=2 supersymmetric theories in 4 dimensions exhibit `special Kaehler geometry, related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the definition of special geometry. We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kaehler metric to one that occurs in N=2 supersymmetry. We treat the rigid as well as the local supersymmetry case. The connection is made to moduli spaces of Riemann surfaces and Calabi-Yau 3-folds. The conditions for the existence of a prepotential translate to a condition on the choice of canonical basis of cycles.
The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative couplings is not defined in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is defined in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the effective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus $gleq3$ contributions are constructed explicitly for a general class of effective actions associated with a special-Kahler target space and are shown to satisfy the holomorphic anomaly equation of perturbative type-II topological string theory. This identification of a topological string free energy from an effective action is primarily based on conceptual arguments and does not involve any of its more specific properties. It is fully consistent with known results. A general theorem is presented that captures some characteristic features of the equivalence, which demonstrates at the same time that non-holomorphic deformations of special geometry can be dealt with consistently.
A symplectically invariant definition of special Kahler geometry is discussed. Certain aspects hereof are illustrated by means of Calabi-Yau moduli spaces.
Recently a boundary energy-momentum tensor $T_{zz}$ has been constructed from the soft graviton operator for any 4D quantum theory of gravity in asymptotically flat space. Up to an anomaly which is one-loop exact, $T_{zz}$ generates a Virasoro action on the 2D celestial sphere at null infinity. Here we show by explicit construction that the effects of the IR divergent part of the anomaly can be eliminated by a one-loop renormalization that shifts $T_{zz}$.
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