$U(1)$-Gauge Field Theories on $G_2$-Manifolds

In this paper, we investigate two types of $U(1)$-gauge field theories on $G_2$-manifolds. One is the $U(1)$-Yang-Mills theory which admits the classical instanton solutions, we show that $G_2$-manifolds emerge from the anti-self-dual $U(1)$ instantons, which is an analogy of Yang's result for Calabi-Yau manifolds. The other one is the higher-order $U(1)$-Chern-Simons theory as a generalization of K\"{a}hler-Chern-Simons theory, by suitable choice of gauge and regularization technique, we calculate the partition function under semiclassical approximation.