Lectures on Heterotic M-Theory

We present three lectures on heterotic M -theory and a fourth lecture extending this theory to more general orbifolds. In Lecture 1, Hořava-Witten theory is briefly discussed. We then compactify this theory on Calabi-Yau threefolds, choosing the “standard” embedding of the spin connection in the gauge connection. We derive, in detail, both the five-dimensional effective action and the associated actions of the four-dimensional “end-of-the-world” branes. Lecture 2 is devoted to showing that this theory naturally admits static, N = 1 supersymmetry preserving BPS threebranes, the minimal vacuum having two such branes. One of these, the “visible” brane, is shown to support a three-generation E6 grand unified theory, whereas the other emerges as the “hidden” brane with unbroken E8 gauge group. Thus heterotic M -theory emerges as a fundamental paradigm for so-called “brane world” scenarios of particle physics. In Lecture 3 , we introduce the concept of “non-standard” embeddings. These are shown to permit a vast generalization of allowed vacua, leading on the visible brane to new grand unified theories, such as SO(10) and SU(5), and to the standard model SU(3)C × SU(2)L × U(1)Y . It is demonstrated that non-standard embeddings generically imply the existence of five-branes in the bulk space. The physical properties of these bulk branes is discussed in detail. Finally, in Lecture 4 we move beyond HořavaWitten theory and consider orbifolds larger than S/Z2. For explicitness, we consider 1 M -theory orbifolds on S/Z2×T /Z2, discussing their anomaly structure in detail and completely determining both the untwisted and twisted sector spectra. 1 Lecture 1: The Five-Dimensional Effective Theory In this first lecture, we introduce our notation and briefly discuss the theory of the strongly coupled heterotic sting introduced by Hořava and Witten. In this theory, there is an elevendimensional bulk space bounded on either end of the x-direction by two “end-of-the-world” ten-dimensional nine-branes, each supporting an N = 1, E8 supergauge theory. We then begin our construction of heterotic M-theory by compactifying the Hořava-Witten theory on a Calabi-Yau threefold. This leads to a five-dimesional bulk space bounded at the ends of the fifth dimesion by two end-of-the-world four-dimensional three-branes. Assuming, in this lecture, the “standard” embedding of the spin connection into one of the E8 gauge connections we derive, in detail, both the five-dimensional bulk space effective action and the associated actions of the four-dimensional boundary branes. We end this lecture by discussing some of the properties of this effective theory and explicitly giving the N = 2 supersymmetry transformations of the bulk space quantum fields. We begin by briefly reviewing the description of strongly coupled heterotic string theory as 11-dimensional supergravity with boundaries, as given by Hořava and Witten [1, 2]. Our conventions are as follows. We will consider eleven-dimensional spacetime compactified on a Calabi-Yau space X, with the subsequent reduction down to four dimensions effectively provided by a double-domain-wall background, corresponding to an S/Z2 orbifold. We use coordinates x with indices I, J,K, · · · = 0, · · · , 9, 11 to parameterize the full 11–dimensional space M11. Throughout these lectures, when we refer to orbifolds, we will work in the “upstairs” picture with the orbifold S/Z2 in the x –direction. We choose the range x ∈ [−πρ, πρ] with the endpoints being identified. The Z2 orbifold symmetry acts as x → −x11. Then there exist two ten–dimensional hyperplanes fixed under the Z2 symmetry which we denote by M (i) 10 , i = 1, 2. Locally, they are specified by the conditions x 11 = 0, πρ. Barred indices Ī , J̄ , K̄, · · · = 0, · · · , 9 are used for the ten–dimensional space orthogonal to the orbifold. We use indices A,B,C, · · · = 4, · · ·9 for the Calabi–Yau space. All fields will be required to have a definite behaviour under the Z2 orbifold symmetry in D = 11. We demand a bosonic field Φ to be even or odd; that is, Φ(x) = ±Φ(−x11). For a spinor Ψ the condition is Γ11Ψ(−x) = Ψ(x) so that the projection to one of the orbifold planes leads to a ten–dimensional Majorana–Weyl spinor with positive chirality. Spinors in eleven dimensions will be Majorana spinors with 32 real components throughout the paper. The bosonic part of the action is of the form S = SSG + SYM (1.1) 1 where SSG is the familiar 11–dimensional supergravity SSG = − 1 2κ2 ∫