We interpret aspects of the Schur indices, that were identified with characters of highest weight modules in Virasoro ( p, p ′ ) = (2 , 2 k + 3) minimal models for k = 1 , 2 , . . . , in terms of paths that first appeared in exact solutions in statistical mechanics. From that, we propose closed-form fermionic sum expressions, that is, q, t -series with manifestly non-negative coefficients, for two infinite-series of Macdonald indices of ( A 1 , A 2 k ) Argyres- Douglas theories that correspond to t -refinements of Virasoro ( p, p ′ ) = (2 , 2 k + 3) minimal model characters, and two rank-2 Macdonald indices that correspond to t -refinements of W 3 $$ {\mathcal{W}}_3 $$ non-unitary minimal model characters. Our proposals match with computations from 4d N $$ \mathcal{N} $$ = 2 gauge theories via the TQFT picture, based on the work of J Song [ 75 ].

中文翻译：

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麦克唐纳指数的闭式费米子表达式

我们解释了 Schur 指数的各个方面，在 Virasoro ( p, p ′ ) = (2 , 2 k + 3) k = 1 , 2 , ... 的最小模型中用最高权重模块的特征来识别。. . ，就首次出现在统计力学精确解中的路径而言。由此，我们针对 ( A 1 , A 2 k ) Argyres-Douglas 理论的两个无限级数麦克唐纳指数提出了闭式费米子和表达式，即具有明显非负系数的 q, t 级数到 Virasoro ( p, p ′ ) = (2 , 2 k + 3) 个最小模型字符的 t 细化，以及对应于 W 3 $$ {\mathcal{W}} 的 t 细化的两个排名 2 的 Macdonald 指数_3 $$ 非幺正最小模型字符。基于 J Song [75] 的工作，我们的建议通过 TQFT 图片与 4d N $$ \mathcal{N} $$ = 2 规范理论的计算相匹配。