Geometric transitions and integrable systems
Tony Pantev, University of Pennsylvania
This is a report on a joint project in progress with
E.Diaconescu, R.Donagi, B.Florea and A.Grassi. We study geometric
transitions of Calabi-Yau manifolds from the point of view of the
derived category of coherent sheaves. A geometric transition is a
familiar process in algebraic geometry in which two components of the
moduli of Calabi-Yau manifolds intersect along a common boundary. The
typical situation involves a space $M$ of CY manifolds all containing
contractible rational curves and another space $L$ parameterizing all
smoothings of the singular CYs that one obtains after contracting the
curves. We develop a non-linear version of a quantization argument
used by Dijkgraaf-Vafa in the toric setup, to quantize the spaces of
sheaves supported on the exceptional curves contained in the CYs in
$M$. This allows us to reconstruct the space $L$ directly from $M$
without a reference to the geometric transition process. To achive
this we specialize to a specific sublocus $S \subset M$ parameterizing
Calabi-Yau manifolds containing a whole curve of singularities. Using
the local geometry of $M$ near $S$ we show that when
singularity type is constant along the curve and is of type {\sf
A-D-E}, the nested sequence of moduli spaces $S \subset M \subset L$
can be linearized by a rather subtle version of the deformation to the
normal cone construction. Furthermore, the linearized moduli spaces
admit interpretations as moduli spaces of (non-compact) linearizations
of the original Calabi-Yau manifolds. Remarkably enough this process
algebraizes the Hodge theory of the family $L$. Specifically we show
that the analytic integrable system of intermediate Jacobians over $L$
osculates to first order an algebraic integrable system over the
linearization of $L$ whose fibers are products of the intermediate
jacobians of resolutions of CYs in $S$ and Prym varieties which are
fibers of a Hitchin system for the corresponding curve of
singularities. Moreover, the structure group of the Hitchin system is
precisely the {\sf A-D-E} group labeling the type of singularities.
This gives us a geometric way of describing the Dijkgraaf-Vafa
quantization procedure in a non-linear setup. It also allows us to
compute the quantum superpotential asymptotically in a large
variety of examples. The results have immediate applications to
computing open Gromov-Witten invariants, non-commutative geometry,
quantization of $D$-branes and matrix quantum mechanics.