范畴量子体积算符
Categorical Quantum Volume Operator
AI总结：
本文将量子体积算符从SU(2)不可约表示推广到带状融合范畴和球面融合范畴，证明在酉范畴下算符厄米，并以SU(2)_k为例验证了k→∞时恢复标准结果。
摘要：
我们提出了量子体积算符的一种推广，用于量化弯曲三维离散几何中的体积。在其标准形式中，量子体积算符由四面体构造而成，这些四面体的面被赋予 $\mathrm{SU}(2)$ 的不可约表示。在这里，我们展示了两种等价构造，允许融合范畴中的一般对象作为自由度。首先，我们计算了带状融合范畴的体积算符。这包括重要的一类模张量范畴（例如量子双），它们是任意子模型的构建块。其次，我们通过放宽闭合约束的范畴类比（称为四面体对称性），进一步将体积算符推广到球面融合范畴。在这两种情况下，只要输入范畴是酉的，我们就得到一个厄米的体积算符。作为一个说明性例子，我们考虑 $\mathrm{SU}(2)_k$ 的情形，并证明在极限 $k\rightarrow\infty$ 下可以恢复标准的 $\mathrm{SU}(2)$ 体积算符。
英文摘要：
We present a generalization of the quantum volume operator quantifying the volume in curved three-dimensional discrete geometries. In its standard form, the quantum volume operator is constructed from tetrahedra whose faces are endowed with irreducible representations of $\mathrm{SU}(2)$. Here, we show two equivalent constructions that allow general objects in fusion categories as degrees of freedom. First, we compute the volume operator for ribbon fusion categories. This includes the important class of modular tensor categories (such as quantum doubles), which are the building blocks of anyon models. Second, we further generalize the volume operator to spherical fusion categories by relaxing the categorical analog of the closure constraint (known as tetrahedral symmetry). In both cases, we obtain a volume operator that is Hermitian, provided that the input category is unitary. As an illustrative example, we consider the case of $\mathrm{SU}(2)_k$ and show that the standard $\mathrm{SU}(2)$ volume operator is recovered in the limit $k\rightarrow\infty$.
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