Workshop Geometry and Physics
22 Novembre 2007
Dipartimento di Matematica, Aula C
10:30- 11:20
String Theory and non compact Toric Varieties
Sergio Cacciatori, Università dell’Insubria
String theories predict for space-time to a Lorentzian manifold with (at least) ten dimensions.
Usually one thinks that six of the spatial dimensions are compactified on a very thin compact
variety, a Calabi-Yau manifold, to justify physical observations. However, recently some
alternatives to compactification have been developed, which make use of non compact complex
varieties in place of Calabi-Yau manifolds. After a short introduction to physical motivations, we
will show how some results for the compact cases (related to mirror symmetry) seem to extend for
the non compact ones, at least for the case of non compact toric varieties. However many of such
results are either partial or expressed in form of conjectures, and deserve a deeper mathematical
understanding.
11:30-12:20
Black Holes in 3D Gravity and Earthquakes on Hyperbolic
Surfaces with Boundary
Francesco Bonsante, Università degli Studi di Pavia
An interesting feature of 3D-gravity is that many global aspects of the realistic gravity have some
realization in this simple model. In particular examples of constant curvature (2+1)-spacetimes
containing black holes are well known and studied. Moduli spaces of such spacetimes have been
completely described by Barbot in terms of Teichmüller space of the spatial slice. In a recent work
with Schlenker, we use this description to get an earthquake theorem for hyperbolic surfaces with
geodesic boundary. In the talk I will describe 3d-spacetimes containing black holes and the relation
with Teichmüller theory.
14:00-14:50
Vertex Algebras avant Borcherds
Domenico Fiorenza, Università “La Sapienza” di Roma
We present a geometric setting for basic CFT/string theory constructions. It is not a rigorous
mathematical treatment of string theory, rather it is the “rules of the game”: once one believes (or
imposes) that these rules apply to path integrals for the Polyakov action, one is immediately led to
the whole machinery of vertex algebras: algebras of currents, OPEs, Ward identities, Virasoro
operators, the state-field correspondence.
15:00-16:00
Enumerative Invariants of Calabi-Yau Spaces
Balazs Szendroi, University of Oxford
I’ll discuss two seemingly very different enumerative problems for Calabi-Yau threefolds, that of
counting curves and sheaves, and their conjectural relation. I will also show how to compute some
invariants in examples. I then discuss how at least one of the enumerative problems also makes
sense in a non-commutative setting, and make further computations, leading to some pretty infinite
products.