Seminar (1/2): Quadratic differentials as stability conditions

Speaker: Tom Sutherland, University of Oxford

2012.11.22 | Christine Dilling

Date Tue 04 Dec
Time 16:15 17:15
Location Aud. D3

Abstract

In these two talks I will outline how a stratum of the space of quadratic differentials on a marked surface with poles at the marked points of fixed orders at least two and simple zeros can be interpreted as a space of stability conditions of an associated triangulated category. This idea has its origin in the work of the physicists Gaiotto, Moore and Neitzke and comes to fruition in upcoming work of Bridgeland and Smith.

In the first talk I will describe how the horizontal foliation of a generic such quadratic differential decomposes the surface into a finite number of horizontal strips and half-planes. The dual graph to the horizontal strips underlies a quiver with potential such that the quivers associated to two quadratic differentials in the same stratum differ by a sequence of so-called mutations.

There is an interesting triangulated category whose objects are finite-dimensional modules over the Ginzburg algebra of a quiver with potential. By a result of Keller and Yang, mutation of the quiver with potential induces an equivalence of the associated triangulated categories. Thus to each stratum of the space of quadratic differentials there is an associated triangulated category.

The second talk will be devoted to the natural question: does the stratum of quadratic differentials define an interesting invariant of the triangulated category? The answer is that it can be interpreted as the space of stability conditions of the triangulated category in the sense of Bridgeland. I will give an introduction to stability conditions on triangulated categories and outline how quadratic differentials define stability conditions.

The first talk is intended to be accessible to all and will not assume any familiarity with homological algebra. For the second talk it will be very helpful to have met a triangulated category before, the prime example being the derived category of an abelian category.

 

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