When one does string compactification on a Calabi-Yau 3-fold. The parameters in Kähler moduli and complex moduli gives the scalar fields in 4-dimensions. It is claimed that the Kähler potentials of the CY moduli space gives the kinetic terms of the scalar fields in 4d. Could anyone let me know why?

I know that a consistent coupling of a SUSY multiplet containing scalars with supergravity requires the scalar kinetic term comes from a Kähler potential. But I am not sure why precisely this Kähler potential coincides with the one for CY moduli spaces in the case of string compactification. Could anyone explain it to me? Thanks!

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    $\begingroup$ Unfortunately I don't know the answer, but I've tried to share this question around and hopefully someone else who does will stumble upon it. Welcome to Physics Stack Exchange! $\endgroup$ – David Z May 12 '12 at 23:44

In order to describe physics of 4 dimensional space-time starting from 10 dimensional space, we consider that the extra 6 dimensional space has very tiny size (about the Planck length). This is called the comopactification in string theory. Then the symmetry of theory requires this internal space to be Ricci flat: \begin{equation} R_{IJ}=0 \end{equation} where $I$ and $J$ run from 0 to 6.

Since C-Y manifolds have the structure of the complex manifolds, it is appropriate to use the indices of the complex manifolds, i.e. $i,j=1,2,3$. The metric on C-Y manifolds is of type (1,1) $G_{i,\bar j}$ and we obtain the closed (1,1)-form from the metric \begin{equation} \omega =\sqrt{-1}G_{i\bar j} dz^i\wedge d\bar z^{\bar j} . \end{equation} And there exists nowhere vanishing holomorphic 3-form $\Omega_{klm}$ since the Ricci tensor vanishes.

In what follows we consider the moduli of C-Y manifolds. If the geometric object can be continuously deformed with its geometric properties preserved, we call the parameter of this deformation moduli. Suppose that the metric $G_{IJ}(y)$ is given on a C-Y manifold $M$ where $y$ is the local coordinate of $M$. And assume that the metric change $G_{IJ}+g_{IJ}$ and this new metric also gives RIcci flat. Then taking the first order of the deformation of metric, we have \begin{equation} \Delta_6 g_{IJ}(y)=0 \end{equation} where $\Delta_6$ is a 6 dimensional Laplacian. Thus, the deformation of the metric which preserves the definition of C-Y manifolds is given by the eigenfuntion of the Laplacian with its eigenvalue zero. In general, the eigenvalues of $-\Delta_6$ take zero and positive values: \begin{equation} -\Delta_6 f_{IJ}^\alpha(y)=m_\alpha^2 f_{IJ}^\alpha(y), \,\,\,\alpha=1,2,\cdots. \end{equation} Let us consider the case that the deformation of metric $g_{IJ}$ is of type (1,1), $\delta g_{i\bar j}$, and of type (2,0) , $\delta g_{i j}$, respectively. Generally, the metric on a K\"ahler manifold is of type (1,1) and $g_{i\bar j}$ gives the deformation preserving this type. This is called the deformation of K\"ahler structure. The deformation of K\"ahler structure is described by a solution of the equation \begin{equation} \Delta_6 \omega_{i j}(y)=0, \end{equation} i.e. it is given by a harmonic (1,1)-form. The number of a harmonic (1,1)-form is provided by the Hodge number $h_{1,1}$ of the manifold $M$. On the other hand, the type (2,0) deformation of metric implies the deformation of complex structure on a complex manifold. Using the complex conjugate $\bar \Omega$ of $\Omega$ , we obtain \begin{equation} \chi_{i\bar j \bar k} \equiv \delta g_{ij} G^{j\bar k} \bar \Omega_{\bar k\bar l\bar m}. \end{equation} Therefore the deformation of complex structure is described by harmonic (1,2)-form and the degree of freedom of the deformation becomes the Hodge number $h_{1,2}=h_{2,1}$. As we see, C-Y manifolds have two kinds of the deformation parameters, the K\"ahler parameters and the parameters of complex structure, which are called moduli, and the K\"ahler parameters correspond to the degree of the change of the size and the parameters of complex structure correspond to that of the deformation of the shape. The metric of the complex structure moduli space is \begin{equation} G_{\alpha\bar\beta}^{\rm mod}=-\frac{i\int\chi_\alpha\wedge\bar\chi_{\bar \beta}}{i\int\Omega\wedge\bar\Omega} \end{equation} Recalling that the metric $G^{\rm mod}_{\alpha\bar\beta}$ of the complex structure moduli space can be obtained from the K\"ahler potential $\cal K$ \begin{equation} G_{\alpha\bar\beta}^{\rm mod}=\partial_{\alpha}\partial_{\bar\beta}{\cal K} \ , \end{equation} one finds that the K\"ahler potential can be written as \begin{equation} {\cal K}=-\log\int\left(i\int \Omega\wedge\bar\Omega \right) \ . \end{equation}

Let us choose the basis $C_a$ $(a=1,\cdots, h_{1,1})$ of the 4-cycle as the duals of the harmonic (1,1)-form $\omega^a\equiv\omega^a_{i\bar j}d\!z^i\wedge d\! z^{\bar j}$ $(a=1,\cdots, h_{1,1})$. Then we can expand \begin{equation} \ast C+\sqrt{-1}\omega=\sum t_a \omega^a \end{equation} where $\omega$ is the K\"ahler form and $C$ is 4-th antisymmetric tensor field which is the partner of the gravitational field. Then the coefficients of the expansion is given by \begin{equation} t_a =\int _{C_a} (C+\sqrt{-1}\ast\omega). \end{equation} These are the parameters of the complexified K\"ahler moduli.

Similarly, let us choose the basis $A_a$ and $B_a$ $(a=0,1,\cdots, h_{1,2})$ of 3-cycle so that the intersection numbers satisfy $A_a\cap B_b=\delta_{ab}, A_a\cap A_b=B_a\cap B_b=0$. In this case, it is known that the parameters are taken as the moduli of the deformation of complex structure \begin{equation} z_a=\int_{A_a} \Omega, \,\,\, a=a,\cdots, h_{1,2}. \end{equation} And also it is know that the integral of $\Omega$ over the cycle $B_a$ can be written \begin{equation} \frac{\partial F}{\partial z_a}=\int _{B_a} \Omega,\,\,\, a=a,\cdots, h_{1,2} \end{equation} where $F$ is a holomorphic function of $z_a$ and is called the prepotential.

Under the compactification of a C-Y manifold fields of 10 dimensional theory can be expanded by the eigenfunctions of the 6 dimensional Laplacian \begin{equation} f_{IJ}(x,y)=\sum_\alpha \phi^\alpha(x)f_{IJ}^\alpha(y). \end{equation} Then the wave function of 10 dimensional theory reduces to the 4 dimensional field equation: \begin{eqnarray} &&\Delta_{10}f_{IJ}(x,y)=(\Delta_{4}+\Delta_{6})\sum_\alpha\phi^\alpha(x)f_{IJ}^\alpha(y)=0 \ && \rightarrow (\Delta_{4}-m_\alpha^2) \phi^\alpha(x)=0. \end{eqnarray} Hence the scalar field $\phi_\alpha$ with mass $m_\alpha$ shows up in 4 dimensional space corresponding to the eigenvalue $m_\alpha^2$ of the six dimensional Laplacian. Especially, the mass of the scalar field corresponding to the moduli of the manifold becomes zero and the massless scalar particle, moduli particle shows up in the four dimensional effective theory. Then the expectation value of the vacuum corresponding to the parameter $\{t_a, z_a\}$. Since the nonzero eigenvalue $m_\alpha$ of the Laplacian is inversely proportional to the square of the size of the space $M$ and acquire very heavy mass, we can neglect them.

${\rm \bf Note\ Added}$: Once Type II string theory is compactified on a Calabi-Yau manifold $M$, the 4d low-energy effective theory is described by ${\cal N}=2$ supergravity. The field contents of ${\cal N}=2$ supergravity consist of the Weyl (gravity) multiplet, vector multiplets and hypermultiplets. The effective actions for vector multiplets and hypermultiplets are described by non-linear sigma model with target spaces, the vector multiplet moduli space and the hypermultiplet moduli space respectively. In particular, the kinetic terms of the scalars $\phi^i$ in the vector multiplets and the hypermultiplets can be written as \begin{equation} \int_{M_4}d^4x\sqrt{g}G_{ij}\partial_\mu \phi^i \partial^\mu \phi^j +\cdots \end{equation} where $G_{ij}$ is the metric of the moduli space. In type IIA compactications, the vector multiplet moduli space coincide with the complex K\"ahler moduli and the hypermultiplet moduli space is the complex structure moduli space. In type IIB, vice-versa. In type IIB compactifications, the low energy effective action of the vector multiplets are dictated by the prepotential $F$ due to supersymmetry. Due to mirror symmetry we can restrict our attention to one of the two type II theories.

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  • $\begingroup$ This is very nice +1, but it doesn't answer the question, which is about the Kahler potential of the scalars, the field dependence of the kinetic terms, or the derivative terms in the equation of motion. It's not asking why the moduli correspond to massless fields, nor is it asking what gives rise to the superpotential for the massive case. The question is entirely restricted to the massless scalars parametrizing the Kahler moduli--- it is asking why the kinetic term of these fields coincides with the Kahler potential of the Moduli space. Maybe its only true in SUGRA approximation. $\endgroup$ – Ron Maimon May 14 '12 at 1:35
  • $\begingroup$ Thanks for the answering Satoshi. But indeed my question is why the kinetic term of these fields coincides with the Kahler metric of the CY moduli space. $\endgroup$ – color May 15 '12 at 10:56
  • $\begingroup$ I added the comments. Hope it answers your question. $\endgroup$ – Satoshi Nawata May 16 '12 at 21:37
  • $\begingroup$ @SatoshiNawata: Your comment is the statement of the thing, but he wants to know why the metric on the 4d-fields is the same as the metric on the moduli space. It's intuitively reasonable that they should be the same, since constant values of the fields correspond to different manifolds, and the local spring-constant "should be" the metric distance between the manifolds. But this requires an argument, why isn't it G plus the Ricci curvature of the moduli space? That the two notions of metric coincide is probably easy to derive in SUGRA, by derivative counting, but I didn't work it out. $\endgroup$ – Ron Maimon May 17 '12 at 7:28
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    $\begingroup$ @Ron Maimon It's because the low energy effective theory is non-linear sigma model with a target manifold the moduli space of CY. $\endgroup$ – Satoshi Nawata May 18 '12 at 6:45

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