在二次基上探索$\mathcal{W}_{\infty}$
Exploring $\mathcal{W}_{\infty}$ in the quadratic basis
摘要：
我们研究了手征代数$\mathcal{W}_{\infty}$中的算子乘积展开式，首先利用了原生成场基中的结合性条件，其次利用了来自自由场表示的不同基，其中OPE取一个简单的二次型.在二次基的结果可以用某些双局部组合的生成字段complexly书面和我们推测一个封闭形式的公式在此基础上的完整的OPE。接下来，我们表明，交换关系以及关联函数可以很容易地计算使用这些双局域字段的属性。在本文的最后一部分中，我们验证了与以前得出的结果的一致性，通过研究极小模型的$\mathcal{W}_{\infty}$和比较他们的$\mathcal{W}_{\infty}$到$\mathcal{W}_N$已知的减少。我们得到的结果很好地说明了三重对称性在$\mathcal{W}_{\infty}$表示理论中的作用。
英文摘要：
We study the operator product expansions in the chiral algebra $\mathcal{W}_{\infty}$, first using the associativity conditions in the basis of primary generating fields and second using a different basis coming from the free field representation in which the OPE takes a simpler quadratic form. The results in the quadratic basis can be compactly written using certain bilocal combinations of the generating fields and we conjecture a closed-form formula for the complete OPE in this basis. Next we show that the commutation relations as well as correlation functions can be easily computed using properties of these bilocal fields. In the last part of this paper we verify the consistency with results derived previously by studying minimal models of $\mathcal{W}_{\infty}$ and comparing them to known reductions of $\mathcal{W}_{\infty}$ to $\mathcal{W}_N$. The results we obtain illustrate nicely the role of triality symmetry in the representation theory of $\mathcal{W}_{\infty}$.
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