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arXiv 1306.0373math-phhep-thmath.AGmath.MP
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拓扑场论中几何结构的BRST不变形变

BRST-Invariant Deformations of Geometric Structures in Topological Field Theories

A. A. Bytsenko, M. Chaichian, A. Tureanu, F. L. Williams

备注：
44 pages. arXiv admin note: text overlap with arXiv:0909.3643, arXiv:hep-th/0210057, arXiv:math/0308080, arXiv:hep-th/0405232 by other authors
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摘要：

研究了形式向量场的李代数$W_n$及其在A模型中扰动变形全纯辛结构和B模型中带边界的Calabi-Yau流形中的应用.顶点算子代数和BRST上同调中的一个相关概念是椭圆亏格（单圈弦配分函数）。我们证明了椭圆亏格可以用双曲三几何的谱函数来表示（它继承了BRST类算子的上同调结构）。我们证明了变形的等价类由DG-代数${\mathfrak A} = (A, Q)$的Hochschild上同调理论描述，$Q =\bar{\partial}+\partial_{\rm deform}$，它被定义为$(-1)^n Q +d_{\rm Hoch}$的上同调。这里$\bar{\partial}$是初始的未变形BRST算子，而$\partial_{\rm deform}$是其代数是线性向量场的李代数${\rm gl}_n$的变形部分.本文讨论了仿射空间$X$和群${\rm Ext}_{X}^n({\cal O}_{\triangle}, {\cal O}_{\triangle})$的调和结构$(HT^\bullet(X); HΩ_\bullet(X))$（HKR同构）的识别，以及体-边界变形配对。

英文摘要：

We study a Lie algebra of formal vector fields $W_n$ with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. A relevant concept in the vertex operator algebra and the BRST cohomology is that of the elliptic genera (the one-loop string partition function). We show that the elliptic genera can be written in terms of spectral functions of the hyperbolic three-geometry (which inherits the cohomology structure of BRST-like operator). We show that equivalence classes of deformations are described by a Hochschild cohomology theory of the DG-algebra ${\mathfrak A} = (A, Q)$, $Q =\bar{\partial}+\partial_{\rm deform}$, which is defined to be the cohomology of $(-1)^n Q +d_{\rm Hoch}$. Here $\bar{\partial}$ is the initial non-deformed BRST operator while $\partial_{\rm deform}$ is the deformed part whose algebra is a Lie algebra of linear vector fields ${\rm gl}_n$. We discuss the identification of the harmonic structure $(HT^\bullet(X); HΩ_\bullet(X))$ of affine space $X$ and the group ${\rm Ext}_{X}^n({\cal O}_{\triangle}, {\cal O}_{\triangle})$ (the HKR isomorphism), and bulk-boundary deformation pairing.

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