From the Donaldson-Uhlenbeck-Yau theorem to stability in mirror symmetry
Typ
Examensarbete för masterexamen
Program
Physics and astronomy (MPPAS), MSc
Publicerad
2019
Författare
Eurenius, Björn
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
We give an introduction to the mathematical formulation of Yang-Mills theory. In particular we
derive the Hermitian-Yang-Mills equation and show that Hermitian-Yang-Mills connections can be
described as the zeroes of the corresponding moment map. We then introduce deformed Hermitian-
Yang-Mills equations by considering arbitrary moment maps.
Furthermore we introduce slope stability of holomorphic vector bundles and show that holomorphic
vector bundles have a unique Harder-Narasimhan ltration. We then give a proof of the
Donaldson-Uhlenbeck-Yau theorem in the case of algebraic surfaces, which states that there is a
one-to-one correspondence between stable bundles and bundles that admit an irreducible Hermitian-
Yang-Mills connection.
Finally we look at stability in the context of homological mirror symmetry. We discuss Bridgeland
stability conditions on the bounded derived category of coherent sheafs Db Coh(M) over a
K ahler manifold M and discuss how it connects to the deformed Hermitian-Yang-Mills equation.
Beskrivning
Ämne/nyckelord
Donaldson-Uhlenbeck-Yau theorem , deformed Hermitian-Yang-Mills , Kobayashi-Hitchin correspondence , Mirror symmetry , Bridgeland stability