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arXiv 1902.07538hep-th
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塞弗特流形上的陈-西蒙斯理论与矩阵模型

Chern-Simons Theory on Seifert Manifold and Matrix Model

Arghya Chattopadhyay, Suvankar Dutta, Neetu

备注：
1+37 pages, nine figures, v2: typos and grammatical corrections, minor text modification matching the published version
发表信息：
Phys. Rev. D 100, 126009 (2019)
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摘要：

讨论了Seifert流形上具有秩$N$和水平$k$的Chern-Simons（CS）理论.这些理论的配分函数可以表示为模变换矩阵的函数，这些模变换矩阵在与边界Wess-Zumino-Witten（WZW）模型相关的仿射李代数$u(N)_k$的不同可积表示上求和。利用模变换矩阵的性质，我们将这些理论的配分函数表示为酉矩阵模型。证明了酉矩阵的特征值是离散的，且与相应的可积Young图的钩长成正比。结果，在大的$N$极限中，本征值密度形成上限。我们考虑CS理论在$S^2\times S^1$上与基本物质耦合，并用模变换矩阵表示配分函数。解决这个模型在大$N$，我们发现占主导地位的可积表示，并显示如何大$N$表示是相互相关的换位的杨图作为一个结果的水平秩对偶。接下来，我们考虑$U(N)$ CS理论上的$S^3$和观察到，在塞弗特框架的主导表示不再是一个可积的表示后，临界值的't Hooft耦合。我们还表明，CS上$S^3$承认多个（两个间隙阶段）大$N$阶段具有相同的自由能。

英文摘要：

Chern-Simons (CS) theories with rank $N$ and level $k$ on Seifert manifold are discussed. The partition functions of such theories can be written as a function of modular transformation matrices summed over different integrable representations of affine Lie algebra $u(N)_k$ associated with boundary Wess-Zumino-Witten (WZW) model. Using properties of modular transform matrices we express the partition functions of these theories as a unitary matrix model. We show that, the eigenvalues of unitary matrices are discrete and proportional to hook lengths of the corresponding integrable Young diagram. As a result, in the large $N$ limit, the eigenvalue density develops an upper cap. We consider CS theory on $S^2\times S^1$ coupled with fundamental matters and express the partition functions in terms of modular transformation matrices. Solving this model at large $N$ we find the dominant integrable representations and show how large $N$ representations are related to each other by transposition of Young diagrams as a result of level rank duality. Next we consider $U(N)$ CS theory on $S^3$ and observed that in Seifert framing the dominant representation is no longer an integrable representation after a critical value of 't Hooft coupling. We also show that CS on $S^3$ admits multiple (two-gap phase) large $N$ phases with the same free energy.

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