My work is concerned with various aspects of quantum gravity and unified
theories, especially supergravity, superstring and supermembrane theories. More
specifically, I am presently interested in the following topics:
Dimensionally reduced gravity and supergravity: (i) Infinite
dimensional symmetries generalizing the Geroch group, especially the ones
based on exceptional Lie groups which arise in the dimensional reduction
of maximal supergravity.
(ii) Exact quantization of such models as "midi-superspace"
models of quantum gravity in the framework of the theory of integrable
systems, e.g. by means of the so-called isomonodromic ansatz. Identification
of observables (non-local conserved charges) and implementation of the
Geroch group at the quantum level.
(iii) Search for a realization of hyperbolic Kac Moody algebras, and
especially E10, in gravity and supergravity (and possibly M Theory) by
generalizing and extending the BKL-type description of the chaotic
oscillations of the metric near a spacelike singularity. In this context,
I have recently become interested in string (or M Theory) cosmology, as I
believe that the chaos which occurs in all "supertheories" may invalidate
many of the approximations made in standard (e.g. inflationary) cosmology.
String and superstring theory: (i) Mathematical aspects,
in particular the relation between indefinite Kac Moody algebras and Borcherds
algebras; affine vertex operator constructions (at arbitrary level); explicit
construction and investigation of root spaces of hyperbolic Kac Moody algebras
(such as E10). (ii) Higher-dimensional supersymmetric extended objects;
supermembranes and supersymmetric matrix models; construction of vertex operators
for the supermembrane; analysis of BPS multiplets; understanding the role
of d=11 supergravity in string theory.
Maximal gauged supergravities in three dimensions: our recent
construction of maximal (N=16) gauged supergravities in three dimensions
has revealed a surprising wealth of structures, which we are currently
trying to explore: these theories accommodate all the exceptional groups (in
various real forms), with certain supergroups (again including exceptional
ones) as background isometries. These theories are not only of interest
in their own right, but should play an important role in the so-called
AdS/CFT correspondence, and might also become relevant for understanding
certain issues in three-dimensional differential geometry.
Canonical gravity: One of my long term interests is to find a
generalization of Ashtekars variables for higher-dimensional gravity and
supergravity by fusion of gravitational and non-gravitational degrees of
freedom. I would also like to understand better some of the conceptual issues
of quantum gravity in the framework of supergravity and string theory.
A key issue in my opinion is the apparent incompatibility of a
deSitter-type universe and all the (maximally supersymmetric) "supertheories"
currently on the market (which strongly prefer Anti-deSitter) and under
study as candidates for a unified theory.