Semi-orthogonal decompositions of moduli spaces

Expositora: Jenia Tevelev de la University of Massachusetts Amherst. 
 
Resumen: 

From an algebraic geometry perspective on physics, space is modeled by an algebraic variety X and fields by sections of vector bundles. This leads to various invariants of X such as K-theory, moduli spaces of vector bundles on X, and the derived category D(X) of complexes of vector bundles. In fact, D(X) has such a rich structure that X can often be reconstructed from it, for example if X is a negatively curved space like a compact Riemann surface of genus >1 or a positively curved space (Fano variety) such as the moduli space N of rank 2 stable vector bundles on C with fixed odd determinant. Derived categories of Fano varieties are especially interesting because they can always be broken into semi-orthogonal blocks, which could be either derived categories of other algebraic varieties or entirely new categories, which are then viewed as non-commutative spaces. I will explain some ideas that go into our recent proof, joint with Sebastian Torres, of the Narasimhan conjecture: D(N) admits a semi-orthogonal decomposition into derived categories of symmetric powers of C.