二维扭曲西格玛模型、镜像手征de Rham复合物和扭曲广义镜像对称
Two-Dimensional Twisted Sigma Models, the Mirror Chiral de Rham Complex, and Twisted Generalised Mirror Symmetry
摘要：
本文研究复厄米流形$X$上全纯规范丛$\mathcal E$上的二维“B-扭曲”$(0,2)$杂化西格玛模型的微扰性质.我们表明，该模型可以很自然地描述在“手征微分算子”的数学理论。特别是，物理异常的西格玛模型可以被重新解释为一个障碍，以全球定义的相关层的顶点超代数来自自由共形场论描述的模型局部$X$。此外，人们也可以获得一个新的理解的西格玛模型单圈β函数单独的全纯数据。在$(2,2)$基因座处，可以根据Malikov等人在\cite{GMS1}中定义的$\it{mirror}$“Chiral de Rham复合物”（或CDR）来描述拓扑B模型的所得半扭曲变体。通过镜像对称，我们还可以推导出卡-丘镜像流形对上镜像CDR层上同调与原始CDR层上同调之间的各种关系式。分析的半扭曲模型的非凯勒群流形的扭转也允许一个得出结论，相应的层CDR（及其镜像）是一致的数学建立的结果由本Bassat在\cite{ben}的镜像对称性的广义复杂的流形。因此，这些结论表明一个有趣的相关性的层CDR在最近的研究广义镜像对称。
英文摘要：
In this paper, we study the perturbative aspects of a "B-twisted" two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted as an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on $X$. In addition, one can also obtain a novel understanding of the sigma model one-loop beta function solely in terms of holomorphic data. At the $(2,2)$ locus, one can describe the resulting half-twisted variant of the topological B-model in terms of a $\it{mirror}$ "Chiral de Rham complex" (or CDR) defined by Malikov et al. in \cite{GMS1}. Via mirror symmetry, one can also derive various conjectural expressions relating the sheaf cohomology of the mirror CDR to that of the original CDR on pairs of Calabi-Yau mirror manifolds. An analysis of the half-twisted model on a non-Kähler group manifold with torsion also allows one to draw conclusions about the corresponding sheaves of CDR (and its mirror) that are consistent with mathematically established results by Ben-Bassat in \cite{ben} on the mirror symmetry of generalised complex manifolds. These conclusions therefore suggest an interesting relevance of the sheaf of CDR in the recent study of generalised mirror symmetry.
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