In 2011, in the context of Toen-Vezzosi’s theory of Derived
Algebraic Geometry, Pantev-Toen-Vaquie-Vezzosi https://arxiv.org/abs/1111.3209
introduced the notion of ‘k-shifted symplectic structure’
on a derived scheme or derived stack. They showed that derived
moduli stacks of coherent sheaves on a Calabi-Yau m-fold
have (2–m)-shifted symplectic structures. They also defined
‘k-shifted Lagrangians’ in k-shifted symplectic
stacks.
The theory has been applied in Donaldson-Thomas theory and its
generalizations, which study enumerative invariants of coherent
sheaves on Calabi-Yau 3- or 4-folds. It has been developed in
several important ways since the original paper.
This class will build on the reading group run by Chenjing Bu and
Andres Ibanez Nunez in TT24, though you do not need to have gone
to this. As the subject is very technical and tends to long
papers, we will not aim to present results in detail, but instead
to provide a survey or broad overview. The idea is that each week,
a volunteer will explain an important paper in the field.
Provisional
programme:
Week 1:
Dominic,
introduction.
Week 3:
Chenjing, on Calaque-Haugseng-Scheimbauer,
‘The AKSZ Construction in Derived Algebraic Geometry as an
Extended Topological
Field Theory’.
Week 4:
Another
volunteer (preferable), or Dominic on CY4 virtual classes.
Week 5:
Nick Kuhn.
Week 6:
? Sam Moore, on
Blanc-Katzarkov-Pandit, ‘Generators
in
formal deformations of categories’.
Week 7:
Lukas, on
something exciting.
Week 8:
Open problem
session.