This is the second of the series of articles on the geometry of String Theory compactifications. Before reading this note, the interested reader may want to read the first note, where the concept of *compactification background* is introduced in the context of String Theory and M-Theory compactifications.

As it is well known, to be well-defined , String Theory and M-Theory require respectively a ten and a eleven dimensional space-time. In the simplest Kaluza-Klein compactification scheme, the extra dimensions are described by a compact six or seven dimensional manifold *M*^{i}*,* whereas the four extended uncompactified dimensions are described by standard four-dimensional flat space *E*. The compact manifold that describes the extra-dimensions is sometimes called in the literature the *internal manifold,* and the manifold describing the non-compact dimensions is sometimes referred as the *external manifold.*

Given this set-up, the first thing we need to do in order to compactify the theory is to find which are the internal manifolds *allowed *by String Theory or M-Theory. Once we know the allowed manifolds, we can pick one of them and then perform an explicit compactification on that particular compactification background. Notice that there is in principle an infinite number of six or seven dimensional manifolds, and that String Theory will not be well-defined in most of them. Therefore, we need to find out which are the admissible ones, that is, which are the internal manifolds on which String Theory is well-defined. This is a very hard problem to be solved using the complete String Theory, so it is customary to resort to some kind of approximation: oft, string theorists use the Supergravity approximation to String Theory, as it is explained here. Since the ultimate goal of compactifying String Theory is to describe low-energy physics, and Supergravity is the low-energy approximation of String Theory, this procedure looks like a sensible approximation. Later, if one is interested on a finer result, one can consider String Theory corrections to the Supergravity approximation, but this usually makes the problem very difficult again, since it many times implies, although not always, the inclusion in the theory of higher derivative terms.

Therefore, we will consider from now on ten-dimensional Supergravity, the low-energy limit of Superstring Theory, and eleven-dimensional Supergravity, the low-energy limit of M-Theory as effective theories of Superstring Theory and M-theory, respectively. Therefore, under this approximation, the task of finding admissible internal manifolds for compactifications of String Theory and M-theory translates into finding the internal manifolds allowed by these Supergravity theories. This is equivalent to finding the internal manifolds that solve the equations of motion of these Supergravity theories.

Before continuing, we need to introduce some extra structure on our internal manifold *M*, since it is necessary for the correct formulation of Supergravity (or String Theory) on it. The first object that we need to introduce, and the only one that we will explicitly mention, is the metric. In particular, *M *must be equipped with a Riemannian metric *g*. Loosely speaking, a metric structure on *M* gives a canonical way to measure the distance between two points in the manifold, which thus depends on the metric used. A metric can be used to measure areas and volumes inside the manifold too. Intuitively speaking, it is very reasonable to pick up a way to measure distances and volumes if we want to formulate any sensible physical theory on *M.* But aside from this intuitive perspective, the importance of the metric *g* steams from the fact that it describes the gravitational interaction, in the very same way as Einstein himself proposed a hundred years ago in his theory of General Relativity. It is important to point out that the external space *E* must be equipped with a metric not of Riemannian type but of Lorentzian type, as it has been experimentally proven in many occasions. The Lorentzian metric on *E *together with the Riemannian metric on *M* combine to give a Lorentzian metric in the complete ten or eleven dimensional space, as it is required to have an admissible theory of gravity.

Associated to the metric *g *on *M* there is a canonical, metric-compatible, torsion-free connection on the tangent bundle of *M, * which lifts to a canonical connection on its spin bundle if *M *is oriented and spin, something that we will assume henceforth. This is the Levi-Civita (spin) connection, and its curvature is a tensor field of extreme importance, since it appears in the equations of motion of Supergravity coupled to the other fields of the theory.

Aside from the metric, Supergravity theories contain several other fields coupled among each other in a very non-linear way, a phenomena that becomes apparent when writing the equations of motion of the theory, which consist of a very non-linear system of partial differential equations, involving various sorts of tensors and spinors. Solving this system of equations give us the admissible internal manifolds *M *in which Supergravity is well-defined, which are supposed to be a *good approximation* to the internal manifolds in which String Theory is well defined, although in principle that is not completely guaranteed.

Solving the Supergravity system of differential equations is an extremely difficult task. Fortunately, the elegant mathematical structure of Supergravity comes to rescue. In particular, the invariance of Supergravity under supersymmetry transformations becomes of paramount importance. This is because supersymmetry transformations define a special class of solutions, namely those solutions invariant under supersymmetry transformations, which are then called *supersymmetric solutions. *Supersymmetric solutions are very interesting mainly for two reasons ^{1}:

- It can be shown that they satisfy simpler equations than the Supergravity equations of motion. These are the so called
*Killing spinor equations*, which are first-order differential equations instead of being second-order differential equations, as it happens with the Supergravity equations of motion. Solving the Killing spinor equations usually implies almost all the equations of motion of Supergravity, and thus they provide a huge simplification in the task of finding Supergravity solutions. The Killing spinor equations are spinorial equations which beautifully connect supersymmetric Supergravity solutions to the existence of spinors constant respect to some particular generalized connection, and in particular to the rich branch of mathematics called spin geometry. The Killing spinor equations associated to a given Supergravity theory correspond simply the vanishing of the supersymmetry transformations of the fermions of the theory. - Supergravity supersymmetric solutions are protected against String Theory corrections, and therefore they are good approximated backgrounds for String Theory and not only for Supergravity. In fact, some supersymmetric Supergravity solutions are exact String Theory backgrounds and do not receive any corrections at all.

For these reasons, the pioneers in the search of Supergravity solutions focused their attention to the class of supersymmetric solutions among all Supergravity solutions. It turns out that supersymmetric solutions can be mathematically completely classified using elegant spinorial geometric methods. In particular, the search of Supergravity solutions has been an important motivation for the mathematical study of manifolds equipped with different sorts of Killing spinors.

Given that we are interested in compactification backgrounds that solve the equations of motion, we may wonder about the existence of supersymmetric compactification backgrounds, since they will be easier to find and classify than other compactification backgrounds. Fortunately, it turns out that there exist supersymmetric compatification backgrounds and that they are indeed a particularly interesting class of supersymmetric solutions.

Let us summarize the situation so far:

- We were interested in finding the compactification backgrounds where String/M-Theory is well defined, to then compactify String/M-Theory in such backgrounds.
- In order to simplify the problem, we considered the Supergravity approximation of String Theory, and focused our attention on compactification backgrounds where Supergravity is well defined. This is the same as the compactification backgrounds that solve the Supergravity equations of motion.
- Since solving the Supergravity equations of motion is still a very difficult problem, we introduced a particular class of solutions that enjoys nice properties and are easier to obtain and classify: these are the supersymmetric solutions, which solve first order partial differential equations (plus some or none second order equations), instead of only second order differential equations.
- We finally focus our attention in those compactification backgrounds that are supersymmetric, namely we focus on supersymmetric compactification backgrounds.

Therefore, the problem has been reduced to finding supersymmetric compactification backgrounds and thus it is much more tractable, although it is still very non-trivial. Obtaining explicit supersymmetric compactification backgrounds is equivalent to solving the Killing spinor equations of the Supergravity theory plus, if any, the Supergravity equations of motion not implied by the Killing spinor equations. The simplest physically interesting scenario is to take in the Killing spinor equations as many identically vanishing Supergravity fields as possible, without completely trivializing the problem. This was indeed the procedure pursued by the string theorists P. Candelas, Gary T. Horowitz, Andrew Strominger and Edward Witten for Heterotic Supergravity (slightly corrected by String Theory effects) in the seminal paper ^{2} of 1985, where Calabi-Yau manifolds were introduced for the first time in the context of String Theory by using its effective theory, namely Heterotic Supergravity^{1}. What they indeed did is to take as zero all Supergravity fields but the metric and the bosonic gauge sector, which is given by the curvature of a connection on a particular principal bundle over the internal manifold. At the supersymmetric level, the bosonic gauge sector does not modify the type of geometry of the internal manifold *M*, it merely implies the existence of a particular holomorphic vector bundle with base *M* and whose curvature is subject to the Supergravity equations of motion. This sector can be consistently truncated (set to zero) and therefore we will forget about it in the following. The Killing spinor equations reduce then to a single equation on the internal six-dimensional compact manifold *M. *This equation simply imposes that the supersymmetry spinor is constant respect to the Levi-Civita spin connection of the manifold *M*. Hence, we have arrived to the following result (Candelasa et al (1985)):

There is a class of admissible Heterotic internal manifolds characterized as being six-dimensional, Riemannian compact manifolds admitting a constant spinor respect to the Levi-Civita spin connection.

And here is where the magic *commence*. It turns out that such manifolds had been studied in great deep by mathematicians long before they appeared in String Theory. For example, they are particular instances of manifolds of *special holonomy, * which were classified in the fifties by Berger using the classification of simple Lie groups by Wilhelm Killing and Élie Cartan, in a beautiful program consisting of finding all the possible Lie groups appearing as the holonomy groups of non-symmetric Riemannian manifolds Marcel Berger (1955) “Sur les groupes d’holonomie homogènes de variétés à connexion affine et des variétés riemanniennes” *Bulletin de la Société Mathématique de France* Volume: 83, page 279-330. We are now just one step away from obtaining that *M *is indeed a Calabi-Yau manifold, but proving this will require to deal a little bit with reducible Riemannian manifolds, symmetric spaces, holonomy theory, spin geometry and Kähler manifolds. For the interested reader, this will be the subject of the next note.

Notes:

1 From now on, and until said otherwise, we will focus exclusively on Heterotic String Theory and its effective theory, namely Heterotic Supergravity.

i For simplicity we will drop the subscript, understanding that *M *may refer to a six-dimensional or seven-dimensional manifold.

I am waiting for the third note, hoping for that you haven’t forgotten of us, yours lectors.