(0,2) Mirror Symmetry and Heterotic Gromov-Witten Invariants (10w5047)

Arriving in Banff, Alberta Sunday, March 7 and departing Friday March 12, 2010

Organizers

(University of Texas)

(University of Pennsylvania)

Ilarion Melnikov (Max Planck Institute for Gravitational Physics (Albert Einstein Institute))

(University of Chicago)

Eric Sharpe (Virginia Tech)

Objectives

The developments described above lead to a number of important and timely questions:

- What is the mathematical structure of the (0,2) quantum cohomology relations?
The quantum cohomology relations in a (2,2) SCFT are relations in the chiral ring. These relations are deformed in a natural and computable fashion when bundle deformations are turned on and {cal E} is no longer TM. While in the (2,2) these relations are quantum modifications to the usual intersection form on cohomology, in the (0,2) case the relations are modifying the intersection product on certain sheaf cohomology groups. Clarifying this structure will be important for further developments.

- Is there a quantum restriction formula for (0,2) theories? The correlators of chiral operators continue to make sense off the (2,2) locus. At genus zero, these correlators give rise to deformed generating functions of Gromov-Witten invariants (and the Yukawa couplings). So far, we only have
techniques to explicitly compute these for toric varieties and Landau-Ginzburg models, but there are good reasons to believe that
the (2,2) quantum restriction formula will have a (0,2) generalization.


- Is there a modification of the special geometry structure that incorporates the bundle moduli with the more familiar Kahler and complex
structure moduli? The structure of the SCFT moduli space is certainly modified off the (2,2) locus. A sufficient understanding of
the moduli space geometry, including the singular loci and associated monodromies will be very helpful in formulating a (0,2) mirror map
and will have important physical implications.

- Can the mirror map be extended to include bundle moduli?
Given a model with {cal E} = TM, we know there exists a mirror theory with target space W and bundle {cal F} = TW. The mirror isomorphism
exchanges the complex structure and complexified Kahler moduli of the two spaces. We expect that the mirror isomorphism survives
off the (2,2) locus and involves a map between moduli of {cal E}to M to moduli of {cal F} to W.

- What is the general structure of (0,2) mirror symmetry?
There exists evidence for mirror pairs of (0,2) theories with {cal E} not a deformation of TM; however,
the general structure behind these pairs remains mysterious. It is likely that ideas developed and tested
for theories with a (2,2) locus will be useful to shed light on the more general structure.


To investigate these questions we propose to bring together a number of experts on (2,2) mirror symmetry together with
investigators who have been recently studying the (0,2) theories. A small intensive workshop seems like a perfect way to
familiarize the various groups with the general state of current research and to give impetus to further study. The recent developments
in (0,2) theories, combined with the growing interest in heterotic strings, both from the mathematical and physical perspectives make
this a perfect time to schedule such a meeting. This topic presents a great opportunity for a fruitful collaboration between mathematicians and
physicists, and we anticipate the meeting to lead to progress on the issues outlined above, both through discussions at the workshop as well
as new collaborations fostered by the meeting.