---
title: Gromov-Witten invariants
aliases:
- Gromov-Witten invariants
- GW
- Gromov-Witten
- J-holomorphic curve
- J-holomorphic
- Gromov-Witten theory
created: 2022-03-18T15:17
updated: 2024-01-10T12:17
---
---
- Tags
- #geomtop/symplectic-topology #higher-algebra/stacks #AG/moduli-spaces
- Refs:
- [ ] [Tony Yue's talk on GW with derived stacks](attachments/2022.03.18_Tony_Yue_YU_Stanford.pdf) #resources/talks
- [ ] A long blog post: #resources/notes
- [ ] Sample computations:
- [ ] What is a pseudoholomorphic curve? [what-is pseudoholomorphic](attachments/what-is%20pseudoholomorphic.pdf)
- [ ] Relation to [DT](Donaldson-Thomas%20invariants.md):
- Links:
- [derived scheme](Unsorted/derived%20scheme.md)
- [derived stack](Unsorted/derived%20algebraic%20geometry.md)
- [derived mapping stack](derived%20mapping%20stack)
- [intersection theory](intersection%20theory.md) due to Behrend-Fantechi
- [virtual fundamental class](virtual%20fundamental%20class)
- [moduli stack](moduli%20space.md)
- [cotangent complex](Unsorted/cotangent%20complex.md)
- [virtual fundamental class](Unsorted/virtual%20fundamental%20class.md)
- [infty-category](Unsorted/infinity%20categories.md)
- [Day convolution](Unsorted/Day%20convolution.md)
- [cone category](cone%20category.md)
- [homotopy coherence](homotopy%20coherence)
- [Kuranishi charts](Kuranishi%20charts)
- [Unsorted/pseudoholomorphic curve](Unsorted/pseudoholomorphic%20curve.md)
- Standard ingredients:
- [simple curves](simple%20curves) and [somewhere injectivity](somewhere%20injectivity)
- [Hessian](Hessian.md) in [Morse homology](Unsorted/Morse%20homology.md)
- [Gromov compactness](Gromov%20compactness.md)
- [Unsorted/Sard-Smale](Unsorted/Sard-Smale.md)
- [Elliptic bootstrapping](Elliptic%20bootstrapping)
- [bubbling](bubbling)
- [isoperimetric inequality](isoperimetric%20inequality)
- [evaluation maps](evaluation%20maps.md)
- [adjunction formula](Unsorted/adjunction%20formula.md)
- [[psi class]]
- [Donaldson-Thomas](Unsorted/Donaldson-Thomas%20invariants.md)
---
# Gromov-Witten invariants
The computation of the Gromov-Witten theory of compact Calabi-Yau varieties (in all genera) has been a central problem in mathematics and physics for the last twenty years. The genus-zero theory has been known since the middle 90's by the celebrated mirror theorems of Givental and Lian-Liu-Yau. On the other hand, the computation of the higher genus theory for compact Calabi-Yau manifolds is unfortunately a hard problem for both mathematicians and physicists. The genus-one theory was computed by Zinger almost ten years ago after a great deal of hard work. So far, the computations in the genus $g \geq 2$ are out of mathematicians' reach.
---
Relation to [mirror symmetry](mirror%20symmetry.md): 
See [period](period.md).
# Motivations
Relation to [Seiberg-Witten](Unsorted/Seiberg-Witten%20theory.md):
The idea of the proof is to deform the Seiberg-Witten equations so they get "close" to a Cauchy-Riemann operator on the line bundle of the Chern class $\varepsilon$. Solutions to the Seiberg-Witten equations then correspond to an almost-holomorphic section of the line bundle, and the zero set of the section is a $J$-holomorphic curve representing $\varepsilon$.
Classical approach: perfect obstruction theory due to Behrend-Fantechi, Li-Tian.
In the [nonarchimedean](nonarchimedean.md) setting: use [derived algebraic geometry](Unsorted/derived%20algebraic%20geometry.md), count curves with [K-Theory (quantum K-invariants)](Unsorted/K-theory.md).
The stack of [stable maps](Unsorted/stable%20map.md) is a substack of a [derived mapping stack](derived%20mapping%20stack).
The derived module stack $\RR\bar{M}_{g, n}(X)$ of $n\dash$pointed genus $g$ stable maps into $X$ is representable by a derived $k\dash$analytic stack [locally of finite presentation](locally%20of%20finite%20presentation) and [derived lci](derived%20lci.md).
Relation to [G-theory](G-theory) and motivation for the definition of [derived stacks](Unsorted/derived%20stacks.md) and [DAG](Unsorted/derived%20algebraic%20geometry.md):
# The virtual fundamental class
See the [Novikov ring](Novikov%20ring).
# Potential and quantum products
# J-holomorphic curve
## Gromov compactness
## Elliptic regularity
# Relation to FJRW theory
The [[FJRW theory]] of $X$ should be equivalent to the Gromov-Witten theory of $X_W$ (the [Landau-Ginzberg](Unsorted/Landau%20Ginzberg.md)/[CY](Unsorted/Calabi-Yau.md) coorespondence).
# Cohomological field theory
