$L^2$ harmonic forms in geometry and string theory

March 17 to March 21, 2004

at the

American Institute of Mathematics, San Jose, California

organized by

Tamas Hausel, Eugenie Hunsicker, and Rafe Mazzeo

Original Announcement

This workshop will be devoted to bringing together mathematicians and physicists who are working on the circle of ideas concerning:

The asymptotic geometry of manifolds with special holonomy;
New analytic techniques to study elliptic theory on manifolds with asymptotically regular geometry;
The role of compactifications in this analysis;
$L^2$ Hodge theory on such spaces;
Other techniques from topology and elsewhere related to these questions, including intersection cohomology, etc.;
The construction and nature of various Yang-Mills moduli spaces over such spaces.

Hodge theory plays a central role in geometric analysis and algebraic geometry, and many sophisticated mathematical tools have been developed to understand the relationship of L² harmonic differential forms on singular or noncompact spaces to the underlying topology of these spaces. Important milestones in this development include the signature theorem for manifolds with boundary due to Atiyah-Patodi-Singer, the relationship between Hodge theory on spaces with conic singularities and intersection cohomology theory, as developed originally by Cheeger, the work of Zucker, Stern and Saper on L² cohomology of (Hermitian) locally symmetric spaces, as well as other more recent works by Carron and Hitchin, to mention just a few.

Recently and quite independently, ideas related to duality in string theory have led physicists to a series of questions and conjectures concerning L² harmonic forms on various classes of noncompact manifolds. Perhaps the most famous of these is the `Sen conjecture' which makes predictions about the L2 cohomology of the monopole moduli spaces over R³ (with respect to their natural hyperKahler metrics); several other related conjectures appear in papers of Vafa and Witten, Brandhuber, Gomis, Gubser and Gukov, and other work of Sen.

The motivation for this workshop is that physicists are generating many important new ideas and directions in this field, concerning Hodge theory on various classes of complete manifolds with special holonomy (e.g. gravitational instantons, G₂ manifolds, etc.). As is often the case when mathematics and physics connect in new ways, the mathematical community has yet to absorb many of these ideas, and likewise, the physicists are not always aware of some of the newer and more powerful mathematical techniques which have been developed in this area. This makes it a very propitious time to try to bridge this gap.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.