The Witten equation and its virtual fundamental cycle
The Witten equation and its virtual fundamental cycle
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发表时间：
2007-12
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通讯作者：
Huijun Fan;Tyler Jarvis;Y. Ruan
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作者：
Huijun Fan;Tyler Jarvis;Y. Ruan
研究了一类与拟齐次多项式相关的非线性椭圆型偏微分方程组。这些方程是由维滕提出的，作为在奇异性（朗道-金斯堡）设置下柯西-黎曼方程的替代。我们引入一个扰动方程，并构造一个虚拟的循环的模空间的解决方案。然后，我们研究了扰动下虚环变形的跨壁性，并将其与经典的Picard-Lefschetz理论相匹配。得到了原方程的一个扩展的虚环。最后，我们证明了扩展的虚圈满足一组类似于Gromov-Witten理论和r-旋理论的公理.
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.