水平无穷大的Fock空间与顶点算子的特征
Fock Space of Level infinity and Characters of Vertex Operators
摘要：
我们提出了一个顶点算子的迹的扩展，并解释了迹的表示论解释。具体地说，我们考虑了一个扭曲的顶点操作无限多Casimir操作和计算其迹作为一个字符公式。为此，我们定义了无限能级的Fock空间$\mathfrak{F}^{\infty}$。然后，证明了$\mathfrak{gl}_{\infty}$和$\mathfrak{a}_{\infty}=\widehat{\mathfrak{gl}}_{\infty}$之间的对偶性，给出了$\mathfrak{F}^{\infty}$分解为$\mathfrak{gl}_{\infty}$和$\mathfrak{a}_{\infty}$具有联合最高权向量的不可约表示.将Fock空间$\mathfrak{F}^{\infty}$分解为最高权表示提供了计算和解释扩展迹的方法。
英文摘要：
We present an extension of the trace of a vertex operator and explain a representation-theoretic interpretation of the trace. Specifically, we consider a twist of the vertex operator with infinitely many Casimir operators and compute its trace as a character formula. To do this, we define the Fock space of infinite level $\mathfrak{F}^{\infty}$. Then, we prove a duality between $\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}=\widehat{\mathfrak{gl}}_{\infty}$ of Howe type, which provides a decomposition of $\mathfrak{F}^{\infty}$ into irreducible representations with joint highest weight vector for $\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}$. The decomposition of the Fock space $\mathfrak{F}^{\infty}$ into highest weight representations provides a method to calculate and interpret the extended trace.
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