# Workshop: Algebra, Geometry and Physics of BPS States

**Date: **November 12 - 14, 2012**Venue:** HIM lecture hall, Poppelsdorfer Allee 45**Organizers: **Nils Carqueville, Sven Meinhardt, Jacopo Stoppa

The rise of BPS invariants in mathematics is connected with the pioneering work of Donaldson and Thomas around 2000, which in particular provided a new way of counting curves in Calabi-Yau 3-folds. Some years later the theory was greatly extended by Behrend, Joyce, Kontsevich and Soibelman (and others) and it became clear that the original goal of counting curves was just the tip of an iceberg. Nowadays Donaldson-Thomas theory is regarded as a new way of thinking about moduli spaces of objects in Calabi-Yau 3-categories, as for instance the derived category of coherent sheaves or the Fukaya category of special Lagrangians in a Calabi-Yau 3-fold or the derived category of representations of quivers with potential.

Historically BPS states first appeared in theoretical physics, where they describe certain ground states in supersymmetric field theories. In the context of string and M-theory BPS states have been successfully used to count black hole microstates as well as for detailed proposals to construct (homological) link invariants.

In recent years another aspect of the theory has been discovered. As suggested by physicists, numerical BPS invariants can be used to construct new hyperkähler metrics and there should be a close relationship with integrable systems. A deep connection between Calabi-Yau 3-categories and integrable systems is conjectured and some progress has been made involving Bridgeland stability conditions and quadratic differentials on Riemann surfaces.

The aim of the workshop was to provide a platform for young scientists, both from mathematics and physics, to share their ideas and experiences with BPS invariants. The workshop was part of the Junior Hausdorff Program on Mathematical Physics and organized by the group D working on „BPS states“.

Speakers: Chris Brav, Tom Bridgeland, Vittoria Bussi, Emanuel Diaconescu, Lothar Goettsche, Martin Kool, Andrew Morrison, Daniel Persson, Markus Reineke, Daniel Roggenkamp, Roberto Volpato