Calabi-Yau Compactifications in Perturbative String Theory

In order to connect superstring theories to the observed 4-dimensional spacetime physics, we have to reduce the critical dimension $D=10$ to four dimensions. To preserve certain supersymmetry consistent with chirality in four dimensions it is necessary to require some properties to the ten dimensional spacetime $X$. Perhaps the simplest ansatz is to assume that the four-dimensional Minkowski spacetime $M$ and a six-dimensional internal space ${\cal K}$ factorizes as $X \cong M \times {\cal K}$, where ${\cal K}$ has tiny dimensions and unobservable in our present experiments. It is worth to say that this factorization ansatz is not unique and other possibility is the warped compactification of the celebrated Randall-Sundrum scenarios, which are nicely reviewed in Ref. [9]. It is useful to classify the compactifications according to how much supersymmetries is broken, because this number is related with the quantum corrections that we shall consider. We choose ${\cal K}$ to be a manifold with the property that a certain number of supersymmetries are preserved³. We are now looking for conditions in the background which leave some supersymmetry unbroken. These conditions are given by null variations of the Fermi fields. Consider the diagonal metric for ten-dimensional spacetime $X$ given by $G_{IJ}= f(y) \eta_{\mu \nu} + G_{mn}(y)$ where $y$ denotes the compactified coordinates and $I,J=0,...,9$, $\mu ,\nu =0,...,3$, $m,n=4,...,9$. For $D=10$, ${\cal N}=1$ heterotic string theory the Fermi fields variations are:

• gravitino: $\nabla \psi_{\mu}=\Delta_{\mu}\varepsilon$, $\delta \psi_m=(\partial_m+\frac{1}{4}\Omega^-_{mnp}\Gamma^{np})\varepsilon,$
• dilatino: $\delta \xi =(\Gamma^m\partial_m\phi -\frac{1}{12}\Gamma^{mnp}H_{mnp})\varepsilon,$
• gaugino: $\delta \lambda = F_{mn}\Gamma^{mn}\varepsilon,$

where $\varepsilon$ is a Weyl-Majorana spinor in ten dimensions, $\Omega^-_{mnp}$ is the internal component of $\Omega^-_{MNP} = \omega_{MNP} - {1 \over 2} H_{MNP}$, $\Gamma$ are the Dirac matrices. The compactification ansatz $X=M \times {\cal K}$ breaks the Lorentz group SO(9,1) into SO $(3,1) \times {\rm SO}(6)$. In the spinor representation 16 the Weyl-Majorana supersymmetry parameter $\varepsilon_{\alpha \beta}$ decomposes as $\varepsilon(y) \to \varepsilon_{\alpha \beta}(y) + \varepsilon^*_{\alpha \beta}(y)$ under ${\bf 16} \to ({\bf 2},{\bf 4}) \oplus ({\bf 2}^*,{\bf 4}^*)$. The general form of $\varepsilon_{\alpha \beta}$ is $\varepsilon_{\alpha \beta} = u_{\alpha} \zeta_{\beta}(y)$ with $u_{\alpha}$ an arbitrary Weyl spinor. When we put the condition that Fermi fields variations vanish, then each internal spinor $\zeta_{\beta}(y)$ gives the minimal (${\cal N}=1$) $D=4$ supersymmetric algebra. Now, by the null fermi fields variations we can find conditions in the background fields assuming that $H_{mnp} =0$. These are:

• $\delta \zeta =0 \Rightarrow \partial_m\Phi =0,$
• $\delta \psi =0 \Longrightarrow G_{\mu \nu}=\eta_{\mu \nu},$
• $\delta \psi_m =0 \Rightarrow \nabla_m\zeta =0.$

The last equation tell us that $\zeta_{\beta}$ is covariantly constant on the internal space ${\cal K}$, and implies that ${\cal K}$ is Ricci-flat. This is because $[\nabla_m,\nabla_n]\zeta =\frac{1}{4}R_{mnpq}\Gamma^{pq}\zeta=0$. For this reason, in general $\Gamma^{pq}$ do not belong to SO(6) but to SU(3), which is a subgroup that leaves one component of the spinor $\zeta$ invariant. Thus the compact manifold ${\cal K}$ must have SU(3) holonomy. The second unbroken susy condition implies that the warped factor $f(y)$ in metric is 1 and the metric $G_{IJ}$ is unwarped. Finally, the first condition implies that the dilation is constant. This is a Calabi-Yau three-fold. A Calabi-Yau three-fold is also a Kähler manifold in which the first Chern class zero i.e. $c_1(T{\cal K})=0.$ Any Calabi-Yau manifold possesses a unique Ricci-flat metric. When we consider ${\cal N}=1$ heterotic string theory on Calabi-Yau three-fold we obtain a four-dimensional chiral theory with spacetime supersymmetry ${\cal N}=1$. In fact, compactification on manifolds of SU(3) holonomy preserves $1/4$ of supersymmetry. If we consider ${\cal N} =2$ theories (for example, type II superstrings) in $D=10$ dimensions, after compactification on a Calabi-Yau three-fold we obtain ${\cal N} =2$ theories in $D=4$. In addition to the CY-threefold structure for ${\cal K}$ the unbroken susy condition $\delta \lambda^a = 0 = F^{a}_{mn} \Gamma^{mn} \varepsilon,$ leads to the equations in complex coordinates

$$F_{IJ} = F_{\bar{I},\bar{J}}= 0, \ \ \ \ \ \ G^{I \bar{J}} F_{I \bar{J}} = 0.$$ (28)

These equations require to specify a gauge subbundle $V$ of a $E_8 \times E_8$ gauge bundle over ${\cal K}$ and a gauge connection $A$ on $V$ with curvature $F$. The condition $F_{IJ} = F_{\bar{I},\bar{J}}= 0$ tell us that the subbundle $V$ as well as the corresponding connection should be holomorphic. The second condition $G^{I\bar{J}} F_{I \bar{J}}=0$ is the celebrated Donaldson-Uhlenbeck-Yau equation for $A$. This equation has a unique solution if the bundle $V$ is stable and if it is satisfied the integrability condition $\int_{\cal K} \Omega^{n-1} \land c_1(V) =0,$ where $\Omega$ is the Kähler form of ${\cal K}$. There is a further condition to be satisfied by the connection $A$, the Bianchi identity for $H$ and $F$, it is given by

$$dH = tr R\land R - {1 \over 30} tr F\land F.$$ (29)

The only solution is $tr R\land R \propto tr F\land F$ which implies that $c_2(T{\cal K}) = c_2(V)$. This situation is usually known as the standard embedding of the spin connection in the gauge connection ant it is a method to determine the connection $A$ on $V$. Thus in the compactification of phenomenological interest of the heterotic theory with the ansatz $X=M \times {\cal K}$, the internal space has to be a Calabi-Yau three-fold and one has to specify a stable, holomorphic vector bundle $V$ over $X$ (or ${\cal K}$) satisfying $c_1(V)=0$ and $c_2(V) = c_2(TX)$. If $V$ is a SU$(n)$ vector bundle over $X$ the subgroups of $E_8 \times E_8$ that commutes are $E_6$, SO(10) and SU(5) for $n=3,4,5$ respectively. This leads to GUTs in four dimensions justly with the gauge groups $E_6$, SO(10) or SU(5). truecm Massless Spectrum In order to describe the impact of the characteristics of ${\cal K}$ and $V$ on the properties of the spectrum of the four dimensional theory we start by decomposing the ten-dimensional Dirac operator under $M \times {\cal K}$ into

$$\not \! \! D^{(10)} = \sum_{I=0}^{9} \Gamma^I D_{I} = \not \! \! D^{(4)} + \not \! \! D_{\cal K},$$ (30)

where $\not \! \! D^{(4)} = \sum_{I=0}^3 \Gamma^I D_I$ and $\not \! \! D_{\cal K} = \sum_{J=4}^{9} \Gamma^J D_J$. Dirac equation in ten dimensions is

$$\not \! \! D^{(10)} \Psi(x^I,y^J) = \big(\not \! \! D^{(4)} + \not \! \! D_{\cal K} \big) \Psi(x^I,y^J).$$ (31)

Thus the spectrum of the Dirac operator $\not \! \! D_{\cal K}$ on ${\cal K}$ determines the massive spectrum of fermions in four dimensions. In ten dimensions the Lorentz group only has real spinor representations and the Clifford modules decomposes as: $S^{(10)} = S^{(10)}_+ \oplus S^{(10)}_-$. Positive and negative chirality are distinguised by $\Gamma^{(10)} = \Gamma^0 \Gamma^1 \dots \Gamma^{9}$. $CPT$ theorem implies that we must take only one chirality

$$\Gamma^{(10)} \Psi = + \Psi .$$ (32)

Decompose the spinor representation of SO(1,9) under ${\rm SO}(1,3) \times {\rm SO}(6)$ with $\Gamma^{(10)} = \Gamma^{(4)} \cdot \Gamma^{(6)}$ where $\Gamma^{(4)}= i \Gamma^0 \Gamma^1 \Gamma^2 \Gamma^3$ and $\Gamma^{(6)} = -i \Gamma^4 \Gamma^5 \dots \Gamma^{9}$. One solution with $\Gamma^{(10)} = +1$ is given by $\Gamma^{(4)} = \Gamma^{(6)}$ and then the spin bundle decomposes under $M \times {\cal K}$ as

$$\widehat{S}^{(10)} = \bigg(\widehat{S}^{(4)}_+ \otimes \wid... ...igg(\widehat{S}^{(4)}_- \otimes \widehat{S}^{\cal K}_-\bigg).$$ (33)

Now solve the Dirac equation with the ansatz $\Psi(x^I,y^J) = \sum_m \phi_m(x^I) \otimes \chi_m(y^J) = \sum_m \psi_m$ and $\not \! \! D'_{\cal K} \chi_m = \lambda_m \chi_m$. It leads to

$$\big(\not \! \! D'^{(4)} + \lambda_m \big) \psi_m = 0$$ (34)

where $\not \! \! D'^{(4)} = \Gamma^{(4)} \not \! \! D^{(4)}$ and $\not \! \! D'_{\cal K} = \Gamma^{(4)} \not \! \! D_{\cal K}$. $\not \! D_{\cal K}$ is an elliptic operator on the compact manifold ${\cal K}$, this implies that that operator has a finite number of fermion zero modes. Massless fermions in four dimensions originate as zero modes of the Dirac operators $\not \! D^{\cal K}$ of the internal manifold ${\cal K}$. By the Atiyah-Singer theorem, a topological invariant of ${\cal K}$ containing the information of the chiral fermions on ${\cal K}$ is given by the index of the Dirac operator

$$Index(\not \! \! D_{\cal K}) = N^{\lambda =0}_+ - N^{\lambda =0}_-,$$ (35)

for chiral fermions on ${\cal K}$ with $\Gamma^{(6)} = \pm 1.$ Here $N^{\lambda =0}_{\pm}$ are the number of positive and negative chiral zero modes. In $2k +2$ dimensions this index is vanishing. We need to couple gauge fields coming from the heterotic string theory. Recall that they are $E_8 \times E_8$ valued gauge fields. The standard embedding of the spin connection in the gauge connection leads to the chain of maximal subgroups: $SO(6) \times SO(10) \subset SO(16) \subset E_8$. This breaks SO(16) to SO(10). The computation of the $\Delta$ for this case yields $\Delta = \oplus_i L_i \otimes {\cal R}_i$ where $L_i$ are irreps of SO(6) and ${\cal R}_i$ are complex irreps of SO(10). These latter determine the irreps where are distributed the massless fermions of the four-dimensional theory. The former irreps $L_i$ determine the number of fermionic chiral zero modes described by the topological index $\delta = Index(D_{\cal K})$. This is given by

$$\delta = N^{\lambda=0}_{\Gamma^{(6)}= +1} - N^{\lambda=0}_{... ...{\cal K} ch(V) td({\cal K}) = {1\over 2} \int_{\cal K} c_3(V)$$

and from the solution $\Gamma^{(6)} = \Gamma^{(4)}$ it determines the chiral fermion families in four dimensions

$$\delta= N^{\lambda=0}_{\Gamma^{(4)}= +1} - N^{\lambda=0}_{\Gamma^{(4)}= -1}.$$ (36)

Thus the theory in four dimensions has $\Delta = \oplus_i \delta_i {\cal R}_i$ where

$$\Delta = \delta \bigg( {\bf 16} \ominus {\bf 16}^* \bigg),$$ (37)

where $\delta = \chi({\cal K})/2$ with $\chi({\cal K})$ is the Euler number of ${\cal K}.$ truecm