Smoothness theorem for differential BV algebras
Smoothness theorem for differential BV algebras
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DOI：
10.1112/jtopol/jtn019
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发表时间：
2007-07
影响因子：
1.1
通讯作者：
John Terilla
中科院分区：
文献类型：
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作者：
John Terilla
给定一个微分 Batalin–Vilkovisky 代数 (V, Q, Δ.) 相关的奇微分分级李代数 (V, Q, + Δ, [,].) 始终是平滑形式。量子微分分级李代数 Lℏ:=(V[[ℏ]],Q+ℏΔ,[,]) 并不总是光滑的形式，但当它是时——例如，当 ∂‐∂ 引理的 Q-Δ 版本成立时——L 的同调性上存在一个弱 Frobenius 流形结构，这在应用中很重要，并且与量子相关函数相关。在本文中，我们证明 Lħ 是光滑形式当且仅当与复数 (V[[ℏ]],Q+ℏΔ) 上的过滤 Fp:=ℏpV[[ℏ]] 相关的谱序列在 E1 处退化。先验地，这种退化意味着一阶障碍物的集合消失，我们证明所有障碍物都会消失。对于那些源自 Calabi-Yau 范畴的 Hochschild 复形的微分 BV 代数，该谱序列的简并是非交换 Hodge 到 deRham 简并的表达式，由 Kontsevich 和 Soibelman 猜想，并由 Kaledin 证明在某些情况下成立。本文的结果表明，非交换霍奇到德拉姆简并猜想等价于量子主方程存在通用解。在本文的最后，提到了一些物理方面的考虑。
Given a differential Batalin–Vilkovisky algebra (V, Q, Δ.) the associated odd differential graded Lie algebra (V, Q, + Δ, [,].) is always smooth formal. The quantum differential graded Lie algebra Lℏ:=(V[[ℏ]],Q+ℏΔ,[,]) is not always smooth formal, but when it is — for example, when a Q‐Δ version of the ∂‐∂ Lemma holds — there is a weak Frobenius manifold structure on the homology of L that is important in applications and relevant to quantum correlation functions. In this paper, we prove that Lħ is smooth formal if and only if the spectral sequence associated to the filtration Fp:=ℏpV[[ℏ]] on the complex (V[[ℏ]],Q+ℏΔ) degenerates at E1. A priori, this degeneration means that a collection of first‐order obstructions vanish and we prove that it follows that all obstructions vanish. For those differential BV algebras that arise from the Hochschild complex of a Calabi–Yau category, the degeneration of this spectral sequence is an expression of the noncommutative Hodge to deRham degeneration, conjectured by Kontsevich and Soibelman and proved to hold in certain cases by Kaledin. The results in this paper imply that the noncommutative Hodge to de Rham degeneration conjecture is equivalent to the existence of a versal solution to the quantum master equation. At the end of the paper, some physical considerations are mentioned.