# 2021-04-22

## Gromov-Witten Invariants in Derived AG

• My main question: does introducing derived stacks somehow make some computation easier? #todo/questions

• I haven’t found any explicit computations of these, but sources alluded to actual counts (numbers) conjecturally coming from physics, where a few have been verified.
• Integrating over a fundamental class :

• moduli space of stable maps

• Appearance of Calabi-Yau in Physics

• Mirror symmetry of CYs:

• The major types of “moduli” style invariants

• See quantum invariants
• Why care about coherent sheaves? #todo/questions

• Donaldson-Thomas invariants are supposed to relate to Gromov-Witten invariants :

• Niceness of spaces:

## Derived Stacks

• We can’t prove the Tate conjecture? I guess this is an arithmetic analog of the Hodge conjecture. Serre’s book calls some isomorphism the Tate conjecture and says it’s proved though.

• Pithy explanation of a derived scheme : a space which can be covered by Zariski opens $$Y\cong \operatorname{Spec}A^*$$ where $$A\in { \mathsf{c}{\mathsf{dg \mathsf{Alg} } }}_{k}$$.

• schemes and stacks can be very singular.

• Derived schemes and derived stacks act a bit like smooth, nonsingular objects.

• Morphisms behave like they are transverse?
• Derived modular stacks of quasicoherent sheaves over $$X$$ remember the entire deformation theory of sheaves on $$X$$.

• The homology of its “tangent space” at a point $$[E]$$ is $$\operatorname{Ext} ^*(E, E)$$, which only holds in restricted degrees if you only use a non-derived moduli scheme or stack.
#web/quick-notes #todo/questions