In the previous articles (I, II), we have characterized the simplest class of supersymmetric heterotic compactification backgrounds. In particular, we have finished the second article with the following result:

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

The purpose of this note is to explore the properties of this class of admissible internal manifolds, showing in particular that they are Calabi-Yau manifolds in a precise sense that we will specify later.

What the Supergravity analysis of the compactification tells us is that any oriented, spin, six-dimensional compact manifold admitting a parallel spinor is a valid compactification background. In order to get a grasp on the kind of manifolds which satisfy these conditions, let us first consider the following possibility:

- The admissible internal space
*M*_{6 }is a Riemannian symmetric manifold.

The technical definition of a Riemannian symmetric manifold goes as follows. A Riemannian symmetric manifold is a Riemannian manifold (*M,g) *with Riemannian metric *g* such that for every point *p *in *M** *there exists an isometry of *g* reversing every geodesic passing through *p* and such that *p *is an isolated fixed point of the isometry^{1}. On a first look, this definition does not seem to be very intuitive, but if we assume *M *to be complete, it can be reformulated in an equivalent, perhaps more understandable way. Let us thus assume that (*M,g) *is a complete symmetric space, and let us denote by *G* the connected component to the identity of its isometry group. Then *M *is of the form *G/H*, where *H* is a closed subgroup of *G *satisfying some further conditions, which we will not explicitly need here, see for example^{1} for more details and further references. The symmetric Riemannian metric on *G/H* is the invariant metric under *G.* Manifolds of the form *G/H*, where *G *is a Lie group and H is a closed subgroup of G, are called *coset spaces* or *homogeneous *manifolds. Intuitively, homogeneous manifolds locally *look *the same, as Riemannian manifolds, around every point. Symmetric spaces are then a particular type of homogeneous manifolds satisfying some extra conditions, and they are nowadays relatively well understood. For example, they were completely classified by E. Cartan already in 1925, and they have been the subject of intensive study since then. Given that symmetric spaces are relatively well-known and have been already classified, it is interesting to see if *our M*_{6} could be a symmetric space since in that case we would have at our disposal all of the existing results in the mathematical literature about this class of manifolds.

As we have said, *M*_{6} is equipped with a parallel spinor. Remarkably enough, this implies that *M*_{6} is Ricci-flat, as can be seen from the integrability condition of the parallel spinor, which roughly speaking simply consists on acting two times on the spinor with the spin connection. Therefore, if *M*_{6 }is a symmetric space, it must be a Ricci-flat symmetric space. It turns out that a Ricci-flat symmetric space must be necessarily flat and thus in the compact case it is finitely covered by a torus of the appropriate dimension. This leaves us with a very restricted and relatively simple class of compactification backgrounds, which, however, has played a very relevant role in String Theory. For instance, toroidal compactifications preserve all supersymmetries and therefore by compactifying eleven-dimensional Supergravity on a *d-*torus one can obtain the maximally supersymmetric ungauged Supergravities in *(11-d)*-dimensions. In fact, this was the method used in reference ^{2} in order to construct the maximally supersymmetric Supergravity theory in four dimensions: a very beautiful theory based on the symmetric Riemannian manifold *E7(7)/SU(8)*, where *E7(7)* stands for the maximally non-compact real form of the complex exceptional Lie group *E7* and *SU(8)* is its maximal compact subgroup.

Since assuming that *M*_{6} is a symmetric Riemannian manifold only gives a very limited class of compactification backgrounds, we will assume in the following that *M*_{6 }is not symmetric. As we have stated in (*), *M*_{6 }is equipped with a parallel spinor. This implies that the holonomy of the spin connection sits inside the stabilizer of the spinor field, namely the subgroup *G* of *Spin(6)* that leaves the spinor invariant. The group *Spin(6)* sits inside the real Clifford algebra *Cl(6),* and its irreducible real representation is obtained from the irreducible real representation of *Cl(6)* by the restriction to *Spin(6). *Now, *Cl(6)* has a unique irreducible real representation of dimension eight, which turns out to be irreducible also as a real representation for and *Spin(6)*. Vectors belonging to this representation are the so-called *Majorana spinors *in the physics community. Such real representation is nonetheless equipped with an invariant complex structure, given by the volume form of *Cl(6)*, so upon complexification we obtain two inequivalent complex representations of *Spin(6)*, namely the positive and negative chirality complex representations of *Spin(6)*. These representations correspond respectively to the fundamental representation and the complex-conjugate fundamental representation of *SU(4)* by using the isomorphism *Spin(6) = SU(4)*. String theorists prefer to work in this case with complex representations, so let us do so too: we will take a spinor of *Spin(6)* to be an object in the fundamental representation of *SU(4). *It can be shown then that any non-zero spinor is stabilized by *SU(3)* inside *SU(4).* In order to intuitively understand this, the reader may consider the case in which the spinor is just (1,0,0,0): its stabilizer is given by the matrices in *SU(4)* that can be written in two blocks, a 1×1 block given by the number one and a 3×3 block which is an element of *SU(3)*.

Therefore, from the discussion above, we deduce that the holonomy of the Levi-Civita spin connection of *M*_{6} is equal or contained in *SU(3), *which in turn implies that the holonomy *Hol(g) *of the Levi-Civita connection is also equal or contained in *SU(3)*^{2}.* *Remarkably enough, when P. Candelas, Gary T. Horowitz, Andrew Strominger and Edward Witten obtained this result in 1985 ^{3}, mathematicians had already been concerned with this class of manifolds for more than thirty years. Since we are assuming that *M*_{6} is not a symmetric space, we know from Berger’s list that only the following possibilities can happen, namely^{3}:

*Hol(g) = Identity.*In this case,*M*_{6}is flat and hence it is finitely covered by the six-dimensional torus. We obtain thus the same result as when we considered*M*_{6}to be a symmetric manifold.*Hol(g) = SU(2).*In this case*M*_{6}locally a Riemannian product of an irreducible*K3-*surface and a flat two-dimensional manifold.*Hol(g) = SU(3).*In this case, we finally conclude that*M*_{6}a Calabi-Yau: according to the definition usually used in String Theory, a Calabi-Yau manifold is indeed a six-dimensional manifold admitting a metric of*SU(3)*-holonomy.

Some comments are in order. In case 2) what we really find is that *M*_{6} is locally the Riemannian product of a four-dimensional Calabi-Yau manifold and a flat manifold. In the same way that a six-dimensional manifold is Calabi-Yau if it admits a metric of *SU(3)*-holonomy, a four-dimensional manifold is Calabi-Yau if it admits a metric of *SU(2)*-holonomy, and the definition is similar for every even dimension: a Calabi-Yau *k-*fold is a *2k*-dimensional manifold admitting a metric of *SU(k) *holonomy. In the String-Theory literature, this definition is sometimes relaxed and manifolds admitting a metric of holonomy properly contained in *SU(k)* are also considered as Calabi-Yau manifolds. Notice however that the topological properties of a manifold admitting a metric of holonomy *SU(k)* may be very different from the topological properties of a manifold admitting a metric of holonomy properly contained in *SU(k).*

Every four-dimensional Calabi-Yau manifold is a *K3* surface and conversely, every *K3* surface admits metrics of *SU(2)* holonomy. For more information about *K3* surfaces in relation to String Theory the reader can consult reference ^{4}. Let us just say here that all *K3* surfaces are diffeomorphic, or in other words, they share the same underlying smooth four-dimensional manifold, which is simply connected. We have just used the arguably most common definition of Calabi-Yau manifold in the String Theory community, but the reader should be warned that there are several inequivalent definitions of Calabi-Yau manifolds, some of them not even implying the existence of metrics of *SU(k)-*holonomy.

In contrast to what happens in four-dimensions, in six dimensions there are plenty of topologically inequivalent Calabi-Yau manifolds. Let us then focus now on case 3). We have shown that the following result holds:

There is a class of admissible Heterotic internal manifolds characterized as being six-dimensional Calabi-Yau manifolds.

Let us now explore some of the properties of Calabi-Yau manifolds. We have obtained that *M*_{6} is a Calabi-Yau manifold after imposing the existence of a parallel spinor on *M*_{6}. The converse is also true: every Calabi-Yau manifold admits a parallel spinor. Hence, from the integrability condition of such spinor we obtain that every Calabi-Yau manifold admits a Ricci-flat metric, so every Calabi-Yau manifold satisfies the vacuum Einstein equations in a Riemannian signature. Moreover, every Calabi-Yau manifold is a Kähler manifold, so in particular it is complex: this can be seen by using the General Holonomy Principle. Roughly speaking, the General Holonomy Principle states that a Riemannian manifold *(M,g)* with a metric of holonomy *Hol(g)* admits globally defined and nowhere vanishing parallel tensors stabilized by *Hol(g). *In our case *Hol(g) = SU(3)* and it can be seen that in six dimensions *SU(3)* leaves invariant a complex structure, which, in addition, must be parallel. This implies that it is integrable and that in fact *M*_{6} is Kähler. Kähler manifolds form a very important class of complex manifolds, characterized as having a Riemannian metric, an integrable complex structure and a symplectic form coexisting in a compatible way. Therefore, they lie at the common intersection of the manifolds studied in Riemannian geometry, Complex geometry, and Symplectic geometry.

Aside from the Kähler structure, in six dimensions *SU(3)* leaves invariant a (3,0)-form, which hence must be parallel. It can be shown that on a compact Ricci-flat Kähler manifold a (3,0)-form is parallel if and only if it is holomorphic. Then, *M*_{6} is equipped with an everywhere non-vanishing holomorphic three-form, which trivializes the canonical bundle of *M*_{6}. All of this together gives us another definition of Calabi-Yau manifold: a compact Kähler manifold with holomorphically trivial canonical bundle. Thanks to the Calabi-Yau theorem, every Calabi-Yau manifold in this sense admits a Ricci-flat Kähler metric of holonomy equal or contained in *SU(3).*

Compact Calabi-Yau manifolds enjoy a very special property (in dimension six or higher^{4}): they are projective algebraic. This means that such a Calabi-Yau manifold is isomorphic as a complex manifold to a complex submanifold of complex projective space of the appropriate dimension. This result follows from the fact that every complex line bundle on a Calabi-Yau manifold is holomorphic and therefore Calabi-Yau manifolds admit plenty of holomorphic line bundles. Some among these many holomorphic line bundles admit enough holomorphic sections to construct an embedding of the Calabi-Yau manifold in complex projective space as a complex submanifold, see reference ^{5} for more information and further references. Therefore Calabi-Yau manifolds can be studied using complex algebraic geometry, and they are indeed subject of intense research in algebraic geometry.

In the next article, I will explain the basics of the moduli space of Calabi-Yau structures on a Calabi-Yau manifold and the basics of the moduli space of polystable holomorphic vector bundles over them. Both spaces are of utmost relevance in order to obtain the effective theory describing a Heterotic compactification on a Calabi-Yau manifold.

Notes:

1This is the definition of a globally Riemannian symmetric space.

2Notice that the opposite is also true. This follows using that *SU(3)* is connected and simply connected.

3Strictly speaking, we refer to the restricted Holonomy group, since we are not assuming *M*_{6 }to be simply connected.

4The result does not hold in real dimension four, as there exist *non-projective* *K3*-surfaces.

## References

- “Compact manifolds with special holonomy” Dominic Joyce, Oxford University Press, 2000. ↩
- “The SO(8) supergravity” E. Cremmer and B. Julia (1979). Nucl. Phys. B 159: 141–212. ↩
- “Vacuum configurations for superstrings” P. Candelas, Gary T. Horowitz, Andrew Strominger and Edward Witten. Nuclear Physics B Volume 258, 1985, Pages 46–74. ↩
- “K3 Surfaces and String Duality” Paul S. Aspinwall, TASI Lectures 1996. ↩
- “Differential Analysis on Complex Manifolds” R. O. Wells, Springer; 3rd edition (October 31, 2007). ↩