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The Witten equation and its virtual fundamental cycle

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arxiv 0712.4025 v3 pith:KJRV6YRV submitted 2007-12-24 math.AG hep-thmath-phmath.APmath.MPmath.SG

classification math.AGhep-thmath-phmath.APmath.MPmath.SG
keywords cyclevirtualequationtheoryextendedperturbationwittenassociated
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We study a system of nonlinear elliptic PDEs associated with a quasi-homogeneous polynomial. These equations were proposed by Witten as the replacement for the Cauchy-Riemann equation in the singularity (Landau-Ginzburg) setting. We introduce a perturbation to the equation and construct a virtual cycle for the moduli space of its solutions. Then, we study the wall-crossing of the deformation of the virtual cycle under perturbation and match it to classical Picard-Lefschetz theory. An extended virtual cycle is obtained for the original equation. Finally, we prove that the extended virtual cycle satisfies a set of axioms similar to those of Gromov-Witten theory and r-spin theory.

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