3-Calabi-Yau triangulated categories T are a major area of research in Geometry and String Theory. Examples include derived categories Dbcoh X of coherent sheaves and the derived Fukaya category DbF(X) on a Calabi-Yau 3-fold X, where Kontsevich’s Homological Mirror Symmetry Conjecture explains mirror symmetry between Calabi-Yau 3-folds X, X* as equivalences of categories Dbcoh X = DbF(X*), DbF(X) = Dbcoh X*.
For such a 3-Calabi-Yau category T one can consider the moduli space Stab(T) of Bridgeland stability conditions on T, and for each σ in Stab(T) one can (under good conditions) define Donaldson-Thomas (D-T) invariants DTα(σ), which are rational numbers ‘counting’ σ-semistable objects in class α in Knum(T).
A nice example, which will be covered at the conference, is a class of categories T of derived representations of a quiver with superpotential (Q,W), in which Stab(T) and DTα(σ) can be described by work of Bridgeland and Smith in terms of quadratic differentials on a Riemann surface.
Recent work of Tom Bridgeland and coauthors explains how to encode D-T invariants into interesting geometric structures on Stab(T), involving Stokes phenomena and Riemann-Hilbert problems for singular flat connections, and connected (via Tom’s paper with Ian Strachan) to complex hyperkahler manifolds and twistor theory. All this is related to work of Kontsevich and Soibelman on analytic stability data and resurgence, and to a circle of ideas in String Theory, including work of Gaiotto, Moore and Neitzke, and topics such as resurgence, WKB analysis, and line operators.
The conference will explain these ideas with introductory talks, and aim to promote communication between Geometers and String Theorists, to better understand this new and fast-moving area.
This Simons workshop will be streamed live to the YouTube channel “SCSHGAP – Robert Bryant, Director”. That should be sufficient for most readers/spectators, though anyone expecting to participate in the discussion sessions can apply to the organizers for a private Zoom link.