Spherical DG-functors

Seidel-Thomas twists are autoequivalences of the derived category D(X)
of an algebraic variety X. They are the mirror symmetry analogues of Dehn
twists along Lagrangian spheres on a symplectic manifold. Given an object
E in D(X) with numerical properties of such a sphere, Seidel and Thomas
defined the spherical twist of D(E) along E and proved it to be an
autoequivalence. It has been understood for a while that this should
generalise to the notion of the twist along a spherical functor into
D(X). In full generality this was obstructed by some well-known
imperfections of working with triangulated categories. In this talk, I
present joint work with Rina Anno, where we fix this by working with the
standard DG-enhancement of D(X). We define the notion of a spherical
DG-functor and give the braiding criteria for twists along such functors.