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关于一类共形 $\mathcal{E}$-模型及其手征泊松代数

On a class of conformal $\mathcal{E}$-models and their chiral Poisson algebras

Sylvain Lacroix

备注：
v3: References added, 68+10 pages
发表信息：
JHEP 06 (2023) 045
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AI总结：

本文研究一类共形 $\mathcal{E}$-模型，提出确保单圈共形不变性的代数条件，实现可观测量解耦为两个手征泊松代数，并为简并情形构建手征场与经典 $\mathcal{W}$-代数，为量子化奠定基础。

摘要：

本文研究了一类 $\mathcal{E}$-模型中的共形点。$\mathcal{E}$-模型是以当前泊松代数表述的 $\sigma$-模型，其李理论定义允许对其动力学和单圈重整化群流进行纯代数描述。我们利用这些结果，对此类模型的定义数据提出了一个简单的代数条件，该条件确保其单圈共形不变性，并使其可观测量解耦为两个手征泊松代数，分别描述理论中的经典左行场和右行场。对于所谓的非简并 $\mathcal{E}$-模型，这些手征扇区形成两个当前代数，且该模型在实现为 $\sigma$-模型时呈现 WZW 理论的形式。简并 $\mathcal{E}$-模型（其中当前代数的子代数被规范）的情况更为复杂：共形条件产生更广泛的理论类别，包括规范 WZW 模型，但也包含其他看似不同的例子，然而这些例子有时却被发现与其他李代数上的规范 WZW 模型相关。对于此类模型，我们构建了 parafermionic 型的非局域手征场以及更高自旋的局域场，形成经典 $\mathcal{W}$-代数。特别地，我们找到了构建这些局域手征场的显式且高效的算法。这些结果（以及论文末尾讨论的潜在推广）为使用二维共形场论的标准算子形式体系对一大类共形 $\mathcal{E}$-模型进行量子化开辟了道路。

英文摘要：

In this paper, we study conformal points among the class of $\mathcal{E}$-models. The latter are $σ$-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate $\mathcal{E}$-models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a $σ$-model. The case of degenerate $\mathcal{E}$-models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical $\mathcal{W}$-algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal $\mathcal{E}$-models using the standard operator formalism of two-dimensional CFT.

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