---
date: 2021-04-22
tags: [ web/quick-notes ]
---


# 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](fundamental%20class.md) :
  
 ![](attachments/image_2021-04-22-11-57-59.png)
 
![](attachments/image_2021-04-22-11-59-01.png)

- [moduli space](moduli%20space.md) of [stable maps](stable%20maps)

![](attachments/image_2021-04-22-12-00-48.png)

![](attachments/image_2021-04-22-12-02-20.png)

- [Operads](Operads) :

![](attachments/image_2021-04-22-12-05-38.png)

- Appearance of [Calabi-Yau](Calabi-Yau.md) in [Physics](Physics.md)
	- Related: [Unsorted/string theory](Unsorted/string%20theory.md), [Unsorted/mirror symmetry](Unsorted/mirror%20symmetry.md)

  ![](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](quantum%20invariants)

- Why care about [coherent sheaves](coherent%20sheaves.md)?
	#todo/questions 

![](attachments/image_2021-04-22-12-14-47.png)

- [Donaldson-Thomas invariants](Donaldson-Thomas%20invariants.md) are supposed to relate to [Gromov-Witten invariants](Gromov-Witten%20invariants.md) :

  ![](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](Tate%20conjecture.md)?
  I guess this is an arithmetic analog of the [Hodge conjecture](Hodge%20conjecture). 
  Serre's book calls some isomorphism the Tate conjecture and says it's proved though.

- Pithy explanation of a [derived scheme](derived%20scheme.md) : a space which can be covered by Zariski opens $Y\cong \spec A^*$ where $A\in \cdga_{k}$.

  - [schemes](scheme.md) and [stacks](Unsorted/stacks%20MOC.md) can be very singular. 
  
  - [Derived schemes](Derived%20schemes) and [derived stacks](derived%20stacks.md) act a bit like smooth, nonsingular objects.
	  - Morphisms behave like they are transverse?

- Derived modular stacks of [quasicoherent sheaves](quasicoherent%20sheaf.md) over $X$ remember the entire [deformation theory](deformation%20theory.md) of sheaves on $X$.
  - The homology of its "tangent space" at a point $[E]$ is $\Ext^*(E, E)$, which only holds in restricted degrees if you only use a non-derived moduli scheme or stack.