Wall-crossing in genus zero quasimap theory and mirror maps
By Ionut Ciocan-Fontanine, Bumsig Kim. 2013
110 citations recorded by the source corpus.
For each positive rational number epsilon, the theory of epsilon-stable quasimaps to certain GIT quotients W//G developed in arXiv:1106.3724[math.AG] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to epsilon --> 0. For epsilon >1 one obtains the usual Gromov-Witten theory of W//G, while the other theories are new. However, they are all expected to contain the same information and in particular the numerical invariants should be related by wall-crossing formulas. In this paper we analyze the genus zero picture and find that the wall-crossing in this case significantly generalizes toric mirror symmetry (the toric cases correspond to abelian groups G). In particular, we give a geometric interpretation of the mirror map as a generating series of quasimap invariants. We prove our wall-crossing formulas for all targets W//G which admit a torus action with isolated fixed points, as well as for zero loci of sections of homogeneous vector bundles on such W//G.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/c82bfccc693aed9ea76a1565e6993187f88a8d5a
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In Geometry and Topology
- Bumsig Kim
- Ionut Ciocan-Fontanine
- Christopher Woodward
- Daewoong Cheong
- Dhruv Ranganathan
- K. Rietsch
- I. Smith
- Tom Coates
In Mathematical Physics
- Bumsig Kim
- Ionut Ciocan-Fontanine
- Christopher Woodward
- Daewoong Cheong
- Dhruv Ranganathan
- I. Smith
- H. Iritani
- D. Abramovich
Citations in the corpus
- QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, background): “Also, X can be a projective variety rather than a vector space.”
- Higher genus quasimap wall-crossing for semi-positive targets (2013, background): “bility conditions. The asymptotic stability condition ε= 0+ corresponds to being in the first chamber (0, 1 β(Lθ) ] for all β. For a long list of examples to which quasimap theory applies, see §2.8 of [CK2]. 2.3. ε-quasimap invariants, ε-CohFT. Let Q[Eff(W,G,θ)] be the semigroup ring. We write qβ for ”