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arXiv 2403.03439hep-thmath.COmath.NT
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$SU(N)$ Chern-Simons理论的线缺陷半指数

Line defect half-indices of $SU(N)$ Chern-Simons theories

Tadashi Okazaki, Douglas J. Smith

备注：
51 pages, v3: published version in JHEP
发表信息：
JHEP06(2024)006
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摘要：

本文研究了3d $\mathcal{N}=2$超对称$SU(N)$的Wilson线缺陷半指数，在规范场的Neumann边界条件下，研究了$k\le -N$能级的Chern-Simons理论，以及2d Fermi多重态和基本3d手征多重态对规范异常的抵消作用.我们利用$q$ -级数的展开式，得到了一些精确的结果，并作了一些说明.我们发现了几个有趣的连接与文献中已知的特殊功能，包括罗杰斯-拉马努金功能，我们猜想积分表示，和外观的Appell-Lerch和某些威尔逊线半指数巨正则系综揭示了一个意想不到的外观模拟模块化功能。我们还发现有趣的$q$差分方程有关的半指数威尔逊线半指数。其中一些结果也有一个描述的对偶理论与Dirichlet边界条件的向量多重态的对偶理论。

英文摘要：

We study the Wilson line defect half-indices of 3d $\mathcal{N}=2$ supersymmetric $SU(N)$ Chern-Simons theories of level $k\le -N$ with Neumann boundary conditions for the gauge fields, together with 2d Fermi multiplets and fundamental 3d chiral multiplets to cancel the gauge anomaly. We derive some exact results and also make some conjectures based on expansions of the $q$-series. We find several interesting connections with special functions known in the literature, including Rogers-Ramanujan functions for which we conjecture integral representations, and the appearance of Appell-Lerch sums for certain Wilson line half-index grand canonical ensembles which reveal an unexpected appearance of mock modular functions. We also find intriguing $q$-difference equations relating half-indices to Wilson line half-indices. Some of these results also have a description in terms of a dual theory with Dirichlet boundary conditions for the vector multiplet in the dual theory.

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