2021-04-22

Gromov-Witten Invariants in Derived AG

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

• Operads :

• Appearance of Calabi-Yau in Physics

  • Related: String theory, Mirror Symmetry
  attachments/image_2021-04-22-12-12-17.png❗

• Mirror symmetry of CYs:

  attachments/image_2021-04-22-12-12-51.png❗

• The major types of “moduli” style invariants

  attachments/image_2021-04-22-12-13-46.png❗
  • See quantum invariants

• Why care about coherent sheaves? #unanswered_questions

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

  attachments/image_2021-04-22-12-17-02.png❗

• Niceness of spaces:

  attachments/image_2021-04-22-12-17-44.png❗

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 \(A\in {\mathsf{cdga} }_{k}\).

  • [[scheme|schemes]] and [[stack|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 sheaf|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.