# Oberwolfach Reports

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**Volume 2, Issue 4, 2005, pp. 3019–3060**

**DOI: 10.4171/OWR/2005/53**

Mini-Workshop: Heterotic Strings, Derived Categories, and Stacks

Organized by: Bjorn Andreas (1), Harald A. Posch (2), Eric Sharpe (3) and Ping Xu (4)

(1) Institut für Mathematik, Humboldt-Universität zu Berlin, Burgstr. 26, D-10099, BERLIN, GERMANY(2) Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090, WIEN, AUSTRIA

(3) Department of Physics, University of Utah, 115 South 1400 East, Suite 201, UT 84112-0830, SALT LAKE CITY, UNITED STATES

(4) Department of Mathematics, The Pennsylvania State University, 210 McAllister Building, PA 16802, UNIVERSITY PARK, UNITED STATES

The miniworkshop \emph{Heterotic strings, derived categories, and stacks},
organised by Bj\"orn Andreas (Berlin), Emanuel Scheidegger (Vienna) and
Eric Sharpe (Utah)
was held November 13th--November 19th, 2005. This
meeting was well attended with 14 participants with
broad geographic representation.
This workshop was a nice blend of researchers with various
backgrounds in both mathematics and physics.

The three topics represent areas of mathematics and physics with
significant technical overlap. Heterotic strings are types of string theories
whose compactifications involve complex K\"ahler manifolds with holomorphic
vector bundles, and most of the complications revolve around those vector
bundles. Derived categories (of coherent sheaves) have an obvious mathematical
link with holomorphic vector bundles, and appear physically in studies of
D-brane/antibrane systems. Details of the physical model in which
derived categories enter physics are also closely related to the details
of the physical model in which stacks enter physics: in each case, only
a distinguished subclass of presentations can be realized physically,
and the nonuniqueness of presentations in that subclass is conjectured
to be washed out
by a physical process called renormalization group flow.

These topics also form elements of generalizations of a conjectured
generalization of ``mirror symmetry.'' Mirror symmetry is a symmetry
exchanging pairs of complex K\"ahler manifolds with trivial
canonical bundle. It has been of interest to algebraic geometers because
it provides a new approach to enumerative geometry: (usually difficult)
curve-counting questions were mapped to comparatively trivial questions
about the mirror manifold. Mirror symmetry was originally developed
for spaces, but recently has been extended to stacks.
One of the conjectured generalizations of mirror symmetry, known
as ``(0,2) mirror symmetry,'' exchanges pairs consisting of
complex K\"ahler manifolds with holomorphic vector bundles,
and is an analogue of ordinary mirror symmetry for heterotic strings.
Another generalization, known as ``homological mirror symmetry,''
exchanges derived categories of coherent sheaves on one of the
mirrors with a derived Fukaya category of the other.
As the topics of this miniworkshop show up in these new
areas of mirror symmetry, this miniworkshop could have
instead been titled ``New developments in mirror symmetry.''

Since understanding these topics involves an interplay between
mathematics and physics, for this miniworkshop we brought together
a collection of both mathematicians and physicists.

B. Andreas, V. Braun, and E. Scheidegger spoke specifically on
mathematical aspects of heterotic strings, and E. Sharpe gave an
overview of a few current problems in heterotic strings.
A. Tomasiello spoke on mirror symmetry in flux backgrounds, using
ideas recently developed by Hitchin to extend mirror symmetry for type II
strings. (The same ideas can also, it is thought, be used to solve
certain technical problems in understanding heterotic strings in flux
backgrounds, as discussed in E. Sharpe's talk.)
D. Ploog spoke on general aspects of derived categories
and Fourier-Mukai transforms, then
U. Bruzzo and D. Hernandez Ruiperez gave a collection
of talks on Fourier-Mukai transforms, relevant to both derived
categories (encoding automorphisms thereof) and heterotic strings
(encoding T-dualities).
E. Macri spoke on pi-stability,
a physical aspect of derived categories. K.-G. Schlesinger and
C. Lazaroiu spoke on $A_{\infty}$ and $L_{\infty}$ algebras,
as relevant to open and closed string field theory, and which play
a role in the physical understanding of derived categories.
Finally, E. Sharpe and P. Horja gave a collection of talks on physical aspects
of stacks.

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