One-loop vacuum energy in 10D super-Yang-Mills theory on magnetized tori with/without 4D N=1 supersymmetric completion

Abstract: We compute the one-loop vacuum energy in 10 dimensional (10D) super Yang-Mills theory compactified on $\mathbb{R}^{1,3}\times (\mathbb{T}^2)^3$ in the presence of the Abelian magnetic fluxes. The regularization of infinite Kaluza-Klein (KK) sum is achieved by the use of Barnes $\zeta$-functions, which can be applied to the case with Landau level structure of KK mass spectrum. We apply the technique to two different models of the 10D super Yang-Mills theory: The one is to introduce the magnetic flux background to the 10D super Yang-Mills action. The other is to first embed 10D super Yang-Mills action into 4D $\mathcal{N}=1$ superspace and to introduce magnetic fluxes. The two model buildings result in tree level potential. The KK mass spectrum of two models look different and we compute the one-loop vacuum energy of each case. Nevertheless, both of the KK mass spectra have the Landau level structure and we can apply the regularization method in the same way. We show that despite the differences the resultant one-loop vacuum energy of both models coincide with each other.