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Braid and Phantom - I

Braid and Phantom - I

Let C be a smooth projective curve of genus at least 2, and let N be the moduli space of stable rank 2 vector bundles on C with a fixed odd determinant. We construct a semi-orthogonal decomposition of the bounded derived category of N conjectured by M.S, Narasimhan. It consists of two blocks for each i-th symmetric power of C for i = 0, ..., g−2, and one block for the (g−1)-st symmetric power. The proof consists of two parts. Semi-orthogonality, which is proved jointly with Sebastian Torres, relies on hard vanishing theorems for vector bundles on the moduli space of stable pairs. The second part, elimination of the phantom, requires an analysis of weaving patterns in derived categories. In this talk we will focus on the first pattern, the Farey Twill, which is based on the theory of windows into derived categories of quotient stacks introduced by Teleman and Halpern-Leistner.

## Department of Mathematics and Statistics