- Tags
- Refs:
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Links:
- derived scheme
- derived stack
- derived mapping stack
- intersection theory due to Behrend-Fantechi
- virtual fundamental class
- moduli stack
- cotangent complex
- virtual fundamental class
- infty-category
- Day convolution
- cone category
- homotopy coherence
- Kuranishi charts
- Unsorted/pseudoholomorphic curve
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Standard ingredients:
- simple curves and somewhere injectivity
- Hessian in Morse homology
- Gromov compactness
- Unsorted/Sard-Smale
- Elliptic bootstrapping
- bubbling
- isoperimetric inequality
- evaluation maps
- adjunction formula
- psi class
- Donaldson-Thomas
Gromov-Witten invariants
Motivations
Relation to Seiberg-Witten: 
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 setting: use derived algebraic geometry, count curves with K-Theory (quantum K-invariants).
The stack of stable maps is a substack of a derived mapping stack.
The derived module stack \({\mathbf{R}}\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{g, n}(X)\) of \(n{\hbox{-}}\)pointed genus \(g\) stable maps into \(X\) is representable by a derived \(k{\hbox{-}}\)analytic stack locally of finite presentation and derived lci.
Relation to G-theory and motivation for the definition of derived stacks and DAG: 
The virtual fundamental class
See the Novikov ring.
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/CY coorespondence).
