Research statement (short version)

we develop a geometric description of KO-homology, and an analytic one, based on the KK-theory
of real C ∗ - algebras. One of the main mathematical achievement of this work is an explicit construction
of an isomorphism between the two descriptions, based on the Cl(n)-index theorem for
real vector bundles. Via topological invariants, we construct a homological Chern character, and
describe torsion effects in D-brane charges. Moreover, we elucidate the importance of the grading
in the KO-homology groups, and give a better interpretation of its cycles in terms of wrapping
D-branes. Finally, for the first time we apply K-homology to the study of Ramond-Ramond fluxes.
The hydrogen atom and reduction. This is a project I was involved in soon after graduating,
and before beginning my PH.D. studies.
Reduction procedures have been widely used as a powerful tool in the study of classical systems,
and symplectic manifolds with symmetries. It is known, in particular, that different classes of
completely integrable systems can be obtained as reductions of free, or simpler systems in higher
dimensions, allowing for a complete solution of the system’s dynamics. Although reduction procedures
have had a lot of consideration in the classical setting, the same has not happened in the
quantum setting, at least not in a systematic way. For this reason, through the study of the well
know hydrogen atom, in [3] we try to make steps forward the development of a quantum reduction
procedure. Our main point is based on the algebraic description of the classical reduction procedure,
which translates as an homomorphism of Lie algebras of vector fields, associated to suitable
maps between the configuration spaces. In the same spirit, we propose a quantum reduction procedure
based on homomorphisms of algebras of differential operators on manifolds. In particular, we
introduce a general notion of reduction for differential operators on an arbitrary manifold, which
can be used in various physical context, as not necessarily linked to quantum mechanics. Then we
apply this technique to show that the hydrogen atom in four dimension can be obtained as a quantum
reduction of a family of harmonic oscillators, allowing an easy computation of its spectrum,
and a straightforward analysis of its algebra of symmetries.
Future Research
In my future reasearch I would continue to investigate the mathematical aspects of gauge and string
theories in non trivial geometrical and topological settings. An interesting case is given by the homological
analysis of D-branes when a B-field is present, where twisted K-homology is the right
framework to be in. In particular, twisted K-homology comes in different geometric realisations,
but it’s not clear which one encodes the correct description of D-branes. Indeed, investigating the
mathematical relation among them can shed light on new physical phenomenons not yet seen in
the current formalism.
Another aspect I would concentrate on is the further development of the mathematical methods
used in [7], in order to have a better understanding of the global properties of Ramond-Ramond
fields on orbifolds, in particular their dynamics, and of nonperturbative aspects of orbifolded string
theory, e.g. the presence of possible orbifold anomalies when D-branes are considered.
As my interests span various fields of mathematics and physics, I will also be very willing to increase
my knowledge of mathematical methods not only in the area of geometry, topology and algebra,
but in the mathematics of (quantum) field theory in general.
References
[1] P. Baum and R.G. Douglas. K-homology and index theory. Proc. Symp. Pure Math. 38, 117–
173 (1982)