Gaugino Condensation in M-theory on S^1/Z_2

TL;DR

The paper derives the four-dimensional gaugino condensate potential in M-theory on $S^1/Z_2$ using Horava–Witten backgrounds, showing the potential is delta-free and mirrors the weakly coupled heterotic form, with a flux-quantization rule fixing an otherwise arbitrary zero-mode parameter. A detailed reduction yields a moduli-dependent exponential superpotential $W^{(χ)} \\sim h \, e^{- rac{6\\pi}{b_0 \\alpha_{ m GUT}}(S-eta T)}$, where the parameter $\beta$ can be sizable in strong coupling. The generalized flux quantization over open four-cycles fixes the zero mode via an integer $n$, and consistency with phenomenology forces $n=0$ so $\lambda=0$, aligning the strong-coupling result with the weakly coupled limit up to $\beta$-driven corrections. Finally, soft SUSY-breaking terms are computed, showing that $eta$-dependent corrections to the Kähler potential and gauge kinetic function can raise gaugino masses to order $m_{3/2}$ for $F^T$-direction breaking, while scalar masses remain small even at strong coupling.

Abstract

In the low energy limit of for M-theory on S^1/Z_2, we calculate the gaugino condensate potential in four dimensions using the background solutions due to Horava. We show that this potential is free of delta-function singularities and has the same form as the potential in the weakly coupled heterotic string. A general flux quantization rule for the three-form field of M-theory on S^1/Z_2 is given and checked in certain limiting cases. This rule is used to fix the free parameter in the potential originating from a zero mode of the form field. Finally, we calculate soft supersymmetry breaking terms. We find that corrections to the Kahler potential and the gauge kinetic function, which can be large in the strongly coupled region, contribute significantly to certain soft terms. In particular, for supersymmetry breaking in the T-modulus direction, the small values of gaugino masses and trilinear couplings that occur in the weakly coupled, large radius regime are enhanced to order m_3/2 in M-theory. The scalar soft masses remain small even, in the strong coupling M-theory limit.