N=1 Supersymmetric Vacua in Heterotic M-Theory

Abstract

In the first lecture, we derive the five-dimensional effective action of strongly coupled heterotic string theory for the complete (1,1) sector of the theory by performing a reduction, on a Calabi-Yau three-fold, of M-theory on S1/Z2. The supersymmetric ground state of the theory is a multi-charged BPS three-brane domain wall, which we construct in general. In this first lecture, we assume the ``standard'' embedding of the spin connection into the E8 gauge connection on one orbifiold fixed plane. In the second lecture, we generalize these results to ``non-standard'' embeddings. That is, we allow for general E8 X E8 gauge bundles and for the presence of five-branes. Properties of these ``non-perturbative''vacua, as well as of the resulting low-energy theories, are discussed. In the last lecture, we review the spectral cover formalism for constructing both U(n) and SU(n) holomorphic vector bundles on elliptically fibered Calabi-Yau three-folds which admit a section. Restricting the structure group to SU(n), we derive, in detail, a set of rules for the construction of three-family particle physics theories with phenomenologically relevant gauge groups. We illustrate these ideas by constructing several explicit three-family non-perturbative vacua.

Figures (3)

• Figure 1: Orbifold interval with boundaries at $0$, $\pi\rho$ and $N$ five-branes at $x_1,\dots ,x_N$. The mirror interval from $0$ to $-\pi\rho$ is suppressed in this diagram.
• Figure 2: Intersection of a five-brane wrapped on the holomorphic cycle ${\cal C}_2^{(n)}$ and a four-cycle ${\cal C}_4$. In this example the five-brane contributes two units of magnetic charge on ${\cal C}_4$.
• Figure 3: Orbifold dependence of a massless mode $(\sqrt{2}/\pi\epsilon_S)b$ for four five-branes at $(z_1,z_2,z_3,z_4)=(0.2,0.6,0.8,0.8)$ with charges $(\beta^{(1)},\beta^{(2)},\beta^{(3)},\beta^{(4)})=(1,1,1,1)$ and instanton numbers $(\beta^{(0)},\beta^{(4)})=(-1,-3)$.