Non-perturbative Calabi-Yau Compactifications

M-theory Vacua In this section we review some Calabi-Yau compactifications of M and F-Theories. In the first part of these lecture we described the perturbative CY compactifications, the purpose of the present section is see how these compactifications behaves in the light of duality and D-brane theory (for excellent reviews see [26,27]). The presence of D-branes or M-branes, in the case of M theory, modifies the perturbative CY compactifications, here we briefly describe these modifications. Assume that the eleven-dimensional spacetime is $Y = M \times {\bf S}^1/{\bf Z}_2 \times {\cal K}$, with ${\cal K}$ being a Calabi-Yau three-fold. Here we consider that ${\cal K}$ is a elliptic fibration, since they are favored by CY compactifications of M and F theories. These spacetime corresponds of having two copies (planes) of $X=M \times {\cal K}$ at the two boundaries of the orbifold. According to the Horava-Witten theory, anomalies cancellation involves that one ${\cal N}=1$ vector supermultiplet of the $E_8$ super Yang-Mills theory has to be captured in each orbifold fixed plane $X_i$, $i=1,2$. According to the perturbative description it is necessary to specify now two stable or semi-stable holomorphic vector bundles $V_i$ on ${\cal K}$ with arbitrary group structure. For the heterotic-M theory compactifications the structure group has to be a subgroup of $E_8$. For simplicity we restrict ourselves to SU$(n_i)$ vector bundles $V_i$ over ${\cal K}$. The presence of fivebranes is of extreme importance here, since it allows more flexibility to construct such vector bundles $V_i$ which leads to more realistic particle physics models. From the modified Bianchi identity and the anomaly cancellation condition of the orbifold system and the fivebranes wrapped on holomorphic two-cycles of ${\cal K}$, leads that these bundles are subject to the cohomological constraint of the second Chern classes $c_2(V_1) + c_2(V_2) + [W] = c_2(T{\cal K})$, where $[W]$ is the topological class associated to the fivebranes. The description of the low-energy physics requires of the computation of the first three Chern classes of the holomorphic bundles $V_i$ over ${\cal K}$ and thus determine completely a non-perturbative vacuum. M and F theories compactifications require that ${\cal K}$ must be a holomorphic elliptic fibration. Thus the construction of these bundles are nontrivial. truecm Construction of the Gauge Bundles over Elliptic Fibrations An holomorphic elliptically fibered Calabi-Yau three-fold is a fibration

$${\cal K} \buildrel{\pi}\over{\to} B$$

where $B$ is an auxiliary complex two-dimensional manifold, $\pi$ is an holomorphic mapping, and for each $b \in B$, $\pi^{-1}(\{b\})$ is isomorphic to an elliptic curve $E_b$. In addition we require the existence of a global section $\sigma: B \to {\cal K}$ of this fibration. The elliptic fibration can be characterized by a single line bundle ${\cal L}$ over $B$, ${\cal L} \to B$, whose fiber is the cotangent space to the elliptic curve, $T^*E_b$. This bundle satisfies the condition: ${\cal L} = K_B^{-1}$ with $K_B$ being the canonical bundle of $B$, under the usual condition that the canonical bundle $K_{\cal K}$ has vanishing first Chern class $c_1(K_{\cal K})=0$. While the global section is specified giving the bundles $K_B^{- \otimes 4}$ and $K_B^{- \otimes 6}$. These conditions are known to be satisfied by base spaces $B$ corresponding to del Pezzo, Hirzebruch and Enriques surfaces. For elliptic fibrations, Friedman, Morgan and Witten [28] found that the second Chern class of the holomorphic tangent bundle $T{\cal K}$ can be written in terms of the Chern classes of $B$ as follows

$$c_2(T{\cal K}) = c_2(B) + 11 c_1^2(B) + 12 \sigma c_1(B),$$ (59)

where $c_1(B)$ and $c_2(B)$ are the first and the second class of $B$ and $\sigma$ is a two-form and represents the Poincaré dual of mentioned global section of the elliptic fibration. One can construct the semi-stable SU$(n_i)$ holomorphic bundles $V_i$ on ${\cal K}$ through the specification of two line bundles $\widehat{\cal L}$ with first Chern class $\eta \equiv c_1(\widehat{\cal L})$ and $\widehat{\cal W}$ with corresponding first Chern class $c_1(\widehat{\cal W})$ depending on some parameters $n, \sigma, c_1(B), \eta$ and $\lambda$. Thus the bundle $\widehat{\cal W}$ is completely specified by the elliptic fibration and the line bundle $\widehat{\cal L}$. The condition that $c_1(\widehat{\cal W}) \in {\bf Z}$ leads to the relation $\lambda = m +{1 \over 2}$ for $n$ odd and $\lambda = m$ and $\eta = c_1(B)$ mod 2, for $n$ even, $m \in {\bf Z}$. Thus the Chern classes of the SU$(n)$ gauge bundle $V$ are

• $c_1(V)=0$
• $c_2(V) = \eta \sigma - {1 \over 24}c_1^2(B)(n^3 - n) + {1 \over 2} \big(\lambda^2 - {1 \over 4} \big) n \eta \big(\eta -n c_1(B)\big)$
• $c_3(V) = 2 \lambda \sigma \eta \big(\eta - n c_1(B)\big).$

In order to construct realistic particle physics models we take a given base space $B$ and compute its corresponding Chern classes $c_1(B)$ and $c_2(B)$. Compute the relevant Chern classes of the SU$(n)$ gauge bundles $V$. The constraints above reduce the number of consistent physical non-perturbative vacua. Given appropriate $\eta$ and $\lambda$ one can determine completely the physical Chern classes. truecm Counting the Number of Families The number of families of leptons and quarks of the four-dimensional theory is related to the number of zero modes of the Dirac operator coupled to gauge field, which is a connection on the SU$(n)$ bundle $V$ on the elliptic fibered Calabi-Yau three-fold ${\cal K}$. In order to count the number of the chiral fermionic zero modes, one can consider the following cases:

• $SU(3) \times E_6 \subset E_8:$ ${\bf 248} = ({\bf 8},{\bf 1}) \oplus ({\bf 1},{\bf 78}) \oplus ({\bf 3},{\bf 27}) \oplus({\bf 3}^*,{\bf 27}^*).$
• $SU(4) \times SO(10) \subset E_8:$ ${\bf 248} = ({\bf 15},{\bf 1}) \oplus ({\bf 1},{\bf 45}) \oplus ({\bf 4},{\bf 16}) \oplus({\bf 4}^*,{\bf 16}^*).$
• $SU(5) \times SU(5) \subset E_8:$ ${\bf 248} = ({\bf 24},{\bf 1}) \oplus ({\bf 1},{\bf 24}) \oplus({\bf 10},{\bf 5... ...({\bf 10}^*,{\bf 5}^*) \oplus ({\bf 5},{\bf 10}^*) \oplus ({\bf 5}^*,{\bf 10}).$

The matter representations appear in the fundamental representation of the gauge group SU$(n)$. The index of the Dirac operator gives

$$\delta = index(\not \! \! D_{\cal K}) = \int_{\cal K} ch(V) td({\cal K}) = {1\over 2} \int_{\cal K} c_3(V)$$ (60)

where $td({\cal K})$ is the Todd class of ${\cal K}$. From explitit formula for $c_3(V)$ we get that the number of generations is given by $\delta = \lambda \eta \big( \eta - n c_1(B)\big)$. truecm

Acknowledgements

I am very grateful to F. Quevedo, E. Cifuentes and W. Aldana for the opportunity to give these lectures at the Universidad de San Carlos, Guatemala. It is a pleasure to thank them and the Universidad de San Carlos for their hospitality.