Seminars

The superconformal indeces of 4d SCFTs count their BPS states and, holographically, give information on the microstates of the dual black holes. A possible approach to the exact computation of the index requires solving the so-called Bethe Ansatz equations (BAE). In this talk, I will introduce a rewriting of the BAE in terms of elliptic functions, focusing on the case with 16 supercharges. Such a rewriting permits 1) to quantitatively capture the conjectural relation of BAE with the Calogero-Moser integrable system (and the related continuous solutions) and 2) to rephrase the BAE in terms of configurations of lines on a plane intersecting an elliptic curve.

It is established that black holes may
be viewed as thermodynamic systems.
Extremal black holes, which are at zero
temperature,
still show interesting (thermodynamical)
features and have attracted a lot of
attention: some uniqueness theorems on
near horizon geometry
of them have been proved, indicating
presence of an AdS_2 factor in these
geometries. There is
a class of extremal black holes for
which entropy vanishes (have vanishing
horizon area).
Here we present three theorems about the
near horizon geometry of such Extremal
Vanishing Horizon (EVH) black holes:
I) Near horizon EVH geometry contains a
three dimensional maximally symmetric
subspace.
II) For theories which satisfy strong
energy condition, this 3d subspace
generically is a locally AdS_3 space.
III) In near horizon, near-EVH black
hole this AdS_3 geometry turns into a
BTZ black hole.

Motivated by the fact that any near
horizon geometry is a solution to the
theory to
which its parent black hole is a
solution, we construct solutions in 4 and 5
dimensional Einstein-Maxwell-Scalar
theories and gauged supergravities and
also in d dimensional pure Einstein
theory. We prove a special rigidity
theorem: all solutions to five
dimensional Einstein-Maxwell- theories
which are SO(2,2) symmetric, necessary
have an extra axial U(1) symmetry.
At the end, we relate the near horizon
geometries of known EVH black holes to
the SO(2,2) solutions we
have classified earlier.

Freudenthal duality can be defined as an anti-involutive, non-linear map
acting on symplectic spaces. It was originally introduced in
four-dimensional Maxwell-Einstein theories coupled to non-linear sigma
models of scalar fields.
After a general introduction on some aspects of extended (super)gravity
theories and the structure of their U-orbits, I will consider their
U-duality Lie groups "of type E7", and the corresponding notion of
Freudenthal duality.
I will elucidate and comment on the relation between the Hessian of the
black hole entropy and the pseudo-Riemannian, rigid, para-special Kaehler
metric of the pre-homogeneous vector spaces associated to the U-orbits.
I will conclude with some hints for further future developments.

I describe the recent progress in understanding which set of boundary data yield exact 4-dimensional Einstein spaces that are algebraically special. The corresponding boundary data can be viewed, in most cases, as 3-dimensional relativistic fluids, not necessarily in the hydrodynamic regime.