Chern-Simons theory with the exceptional gauge group as a refined topological string

• Published in: arXiv.org 2020-07
• Main Author: Mkrtchyan, R L
• Format: Article
• Language: English
• Subjects: Field theory (physics); Parameters; Partitions (mathematics); Quantum theory; String theory; Topology
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2007.09346

We present the partition function of Chern-Simons theory with the exceptional gauge group on three-sphere in the form of a partition function of the refined closed topological string with relation \(2\tau=g_s(1-b) \) between single K\"ahler parameter \(\tau\), string coupling constant \(g_s\) and refinement parameter \(b\), where \(b=\frac{5}{3},\frac{5}{2},3,4,6\) for \(G_2, F_4, E_6, E_7, E_8\), respectively. The non-zero BPS invariants \(N^d_{J_L,J_R}\) (\(d\) - degree) are \(N^2_{0,\frac{1}{2}}=1, N^{11}_{0,1}=1\). Besides these terms, partition function of Chern-Simons theory contains term corresponding to the refined constant maps of string theory. Derivation is based on the universal (in Vogel's sense) form of a Chern-Simons partition function on three-sphere, restricted to exceptional line \(Exc\) with Vogel's parameters satisfying \(\gamma=2(\alpha+\beta)\). This line contains points, corresponding to the all exceptional groups. The same results are obtained for \(F\) line \(\gamma=\alpha+\beta\) (containing \(SU(4), SO(10)\) and \(E_6\) groups), with the non-zero \(N^2_{0,\frac{1}{2}}=1, N^{7}_{0,1}=1\). In both cases refinement parameter \(b\) (\(=-\epsilon_2/\epsilon_1\) in terms of Nekrasov's parameters) is given in terms of universal parameters, restricted to the line, by \(b=-\beta/\alpha\).