Notes on Calabi-Yau compactification I

02-12

Critical superstring theory requires the spacetime dimensions to be 10, or maybe 11 in strong coupling limit. in order to get 10 down to 4, the most straightforward possibility is that six or sevenof the dimensions are compactified on an internal manifold, whose size issufficiently small to have escaped detection. In the mid 1980s Calabi–Yau manifolds were firstconsidered for compactifying the six extra dimensions, and they were shownto be phenomenologically rather promising. In contrast to the circle, theydo not have isometries, and part of their role is to break symmetries ratherthan to make them. The second way to deal with the extra dimension is the brane world scenario where the gauge fields as the ends of the open strings are confined on the D3 brane while gravity as the massless modes of closed strings could escape to higher dimensional space. In this note, we would focus on the former case.

1. Complex manifold

The definition of complex manifold is similar to the definition of real manifold except that there are some additional restrictions for complex manifolds. Briefly speaking, an n-dimensional complex manifold is locally $mathbb{C}^{n}$ and the transition function of different coordinate charts is holomorphic.

One can also start from an almost complex manifold which is a real 2n-dimensional manifold M with an almost complex structure $J:TM o TM$ satisfying $J^{2}=-I$

A necessary and sufficient condition for J to be a complex structure is that the almost complex manifold is integrable, which is equivalent to the condition that the Nijenhuis tensor vanishes. Nijenhuis tensor $N:TMotimes TM o TM$ is given by

$N(X,Y)=[JX, JY]-J[X,JY]-J[JX,Y]-[X,Y]$

In local coordinates it reads

$N^{a}_{bc}=J^{d}_{b}partial_{[d}J^{a}_{c]}-J^{d}_{c}partial_{[d}J^{a}_{b]}$

An n-form in real manifold with complex structure could be decomposed as $Omega^{n}(M)=igoplus_{p+q=m}Omega^{p,q}(M)$

where a (p,q) form $omega^{p,q}$ in $Omega^{p,q}$ is a complex differential form with p holomorphic pieces and q anti-holomorphic pieces. In local coordinates, we can write

$omega^{p,q}=omega^{p,q}_{a_{1}cdots a_{p}ar{b}_{1}cdotsar{b}_{q}}dz^{a_{1}}wedge cdotswedge dz^{a_{p}}wedge dar{z}^{ar{b}_{1}}wedge cdotswedge dar{z}^{ar{b}_{q}}$

The exterior derivative $d:Omega^{n} oOmega^{n+1}$ has a decomposition as well:

$d=partial+ar{partial}$

where

$partial:Omega^{p,q}(M) oOmega^{p+1,q}(M)$ , $ar{partial}:Omega^{p,q}(M) oOmega^{p,q+1}(M)$

Then $d^{2}=0$ gives $partial^{2}=ar{partial}^{2}=ar{partial}partial+partialar{partial}=0$ . The integrability condition is equivalent to $ar{partial}^{2}=0$ . Since $ar{partial}^{2}=0$ , we can define Dolbeault cohomology by

$H^{p,q}_{ar{partial}}(M)=frac{ ext{Ker}(ar{partial}:Omega^{p,q} oOmega^{p,q+1})}{ ext{Im}(ar{partial}:Omega^{p,q-1} oOmega^{p,q})}$

which is isomorphic to $H^{q}(Omega^{p}(M))$ (?ech-Dolbeault).

2. K?hler manifold

A Hermitian metric of a complex manifold is a positive definite inner product $g:TMotimes ar{T}M omathbb{C}$ (Here we are using the notation $T_{mathbb{R}}Motimes mathbb{C}=TMoplus ar{T}M$ ). In local coordinates we can write $g=g_{aar{b}}dz^{a}otimes dar{z}^{ar{b}}$ .

The Hermitian condition for a real manifold with complex structure is

$g(X,Y)=g(JX,JY)$

In terms of the components $J^{a}_{b}$ , we have $J_{ab}=-J_{ba}$ . Therefore we can define a two form $omega=frac{1}{2}J_{ab}dx^{a}wedge dx^{b}$

In complex coordinates we can build a (1,1) form:

$omega=frac{i}{2}g_{aar{b}}dz^{a}wedge dar{z}^{ar{b}}$

If $omega$ is closed, say, $domega =0$ , $omega$ is called a K?hler form and the corresponding manifold M is a K?hler manifold. Locally one can find a K?hler potential K such that $omega=ipartialar{partial}K$ .

One important consequence of K?hlerian condition is found by calculating Laplacians. The adjoint of an operator is defined via inner products, i.e., $langle alpha, partialeta angle=langle partial^{dag}alpha,eta angle$ , where $alpha$ is a (p,q) form and $eta$ is (n+1-p, n-q) form. The Laplacians are given by

$Delta_{partial}=partialpartial^{dag}+partial^{dag}partial$ , $Delta_{ar{partial}}=ar{partial}ar{partial}^{dag}+ar{partial}^{dag}ar{partial}$ .

For K?hler manifold, we have

$Delta_{d}=2Delta_{partial}=2Delta_{ar{partial}}$

Therefore we have a decomposition of de Rham cohomology into $Delta_{ar{partial}}$ - cohomology, which could be represented by Dolbeault cohomology:

$H^{r}(M)=igoplus_{p+q=r}H^{p,q}(M)$

The Hodge numbers and Betti numbers are given by

$h^{p,q}(M)= ext{dim}H^{p,q}(M)$ , $b_{r}(M)= ext{dim}H^{r}(M)=sum_{p+q=r}h^{p,q}(M)$

Further, we have $h^{p,q}=h^{n-p,n-q}$ given by Hodge dual, and $h^{p,q}=h^{q,p}$ by complex conjugation.

3. Calabi-Yau manifold

The Calabi-Yau theorem was first conjectured by Calabi and then proven by Yau. The theorem is powerful in that its quite hard to determine whether M admits a Ricci flat metric, while the first Chern class is relatively simpler to compute.

A Calabi-Yau n-fold M is a compact (sometimes compactness is not a requirement), complex, K?hler manifold with one of the following equivalent conditions:

The canonical bundle of M trivial (M has vanishing first Chern class)
M has nowhere-vanishing holomorphic n-form
M is Ricci flat
M has K?hler metric with global holonomy contained in SU(n)

In the next note, we will see how the physical requirement of 4 dimensional external spacetime would lead us to the Calabi-Yau condition of the internal space.

Reference

[1] Becker K, Becker M, Schwarz J H. String theory and M-theory: A modern introduction[M]. Cambridge University Press, 2006.

[2] Hori K, Vafa C. Mirror symmetry[J]. arXiv preprint hep-th/0002222, 2000.

[3] Greene B. String theory on Calabi-Yau manifolds[J]. arXiv preprint hep-th/9702155, 1997.