Tree-level Couplings

Let us consider first the couplings generated at string tree-level and also tree-level in the sigma-model expansion [34,35]. Besides the 4D Poincaré symmetry, supersymmetry and gauge symmetries which determine the Cremmer et al Lagrangian, we also can use the `axionic' symmetry: $B_{MN}\rightarrow B_{MN}+{\rm closed form}$ This is a symmetry which for the 4D fields would imply that $S,T,U$ can be shifted by an arbitrary imaginary constant. There are also two scaling properties of the 4D Lagrangian $S\rightarrow \lambda S$, $G_{\mu\nu}\rightarrow \lambda G_{\mu\nu}$ for which the Lagrangian scales as ${\cal L}\rightarrow \lambda {\cal L}$. Also, given a scale $\Lambda$, define $\tau=\kappa\Lambda^4$ where $\kappa$ gives Newton's constant in 10D. The transformations $S\rightarrow \tau^{-1/2} S$, $T\rightarrow \tau^{1/2} T$ with similar transformations for the other fields, imply that the Lagrangian should scale as ${\cal L}(\kappa)\rightarrow \tau^{-1/2}{\cal L} (\Lambda^{-4})$. These scaling properties are not symmetries of the Lagrangian but of the classical field equations and so they can be used to restrict the form of the tree-level effective action only.

Using these symmetries we can extract the full dependence of the effective action on the dilaton field $S$, which is the most generic field in all compactifications. We conclude that at tree-level in both expansions [35]:

$$K(S,T,U,Q^I) = -\log(S+S^*)+\hat K(T,U,Q)$$

$$W(S,T,U,Q^I) = y_{IJK}Q^{I}Q^{J}Q^{K}$$

$$f_{ab}(S,T,U,Q^I ) = S\, \delta_{ab}$$ (18)

With $\hat K$ still undetermined. This is however a very crude approximation. What we really want is to know these functions at tree-level in the string expansion but exact in the sigma model expansion. This should be achievable because many of the 4D models are exact 2D CFTs as we saw in the previous chapter. We can still extract very useful information from equation (18). As we said above, the axionic symmetries imply that to all orders in sigma-model expansion the superpotential does not depend on $T,U$ and it is just a cubic function of the matter fields $Q^I$. This is important for several reasons: First, we know the field $T$ comes from the internal components of the metric and controls the loop expansion of the worldsheet action. If $W$ does not depend on $T$ it means that it cannot get any corrections in sigma model perturbation theory! [23]. Therefore the only $T,U$ dependence of the (exact) tree-level superpotential is due to nonperturbative effects in the worldsheet, in particular all nonrenormalizable couplings in the superpotential are exponentially suppressed ($\sim e^{-T}$)[36]. A way to see that there are nonperturbative worldsheet corrections to the string tree-level superpotential is to realize that the axionic symmetry shifting $T$ by an imaginary constant, is broken by nonperturbative worldsheet effects to $T\rightarrow T+i\,n,\, n\in{\bf Z}$. This is nothing but one of the $SL(2,{\bf Z})_{T,U}$ transformation for toroidal orbifold compactifications ($a=b=d=1, c=0$ in eq. (10)). Therefore the only conditions these symmetries impose on $W$ is that it should transform as a modular form of a given weight ( $W\rightarrow (cT+d)^{-3}\, W$ for the simplest toroidal orbifolds with $T$ the overall size of the compactification space)[37]. In fact, explicit calculations for specific orbifold models show that

$$W_{tree}(T,Q^I)=\chi_{IJK}(T)\, Q^{I}Q^{J}Q^{K}+\cdots$$ (19)

with $\chi(T)$ a particular modular form of $SL(2,{\bf Z})$ or any other duality group and the ellipsis represent higher powers of $Q$, exponentially suppressed. The identification of $\chi(T)$ with modular forms was a highly nontrivial check of the explicit orbifold calculations which were preformed in refs. [38] without any relation (nor knowledge) of the underlying duality symmetry $SL(2,{\bf Z})$. This kind of symmetry puts also strong constraints to the higher order, nonrenormalizable, corrections to $W$, since each matter field $Q$ transforms in a particular way under that symmetry ( $Q\rightarrow(cT+d)^n\, Q$ with $n$ the modular weight of $Q$). There are also other discrete symmetries, as those defined by the point group ${\cal P}$ and space group ${\cal S}$ of an orbifold which have to be respected by the superpotential $W$. These `selection rules' are very important to find vanishing couplings and uncover flat directions which can be used to break the original gauge symmetries and construct quasi-realistic models.

Second, and more important, the superpotential above does not depend on $S$ which is the string loop-counting parameter, and therefore $W_{tree}$ does not get renormalized in string perturbation theory! [39]. This means that we only need to compute $W$ at the tree level and it will not be changed by radiative corrections. This is the string version of the standard non-renormalization theorems of supersymmetric theories. Also for $Q=0$ the superpotential vanishes, independent of the values of $S,T,U$ ( $W(S,T,U,Q=0)=0$)! There are not self couplings among the `moduli' fields and therefore they represent flat directions in field space (see for instance [40] ). Notice that due to the non-renormalization theorems, this result is exact in string perturbation theory!. The only possibilty we have to lift this vacuum degeneracy is by nonperturbative string effects.

The quantity we have less information on, even at tree-level, is the Kahler potential $\hat K(T,U,Q)$. It has been computed only for several simple cases. For instance in the simplest possible Calabi Yau compactification ( $h_{1,1}=1, h_{2,1}=0$) a consistent truncation from the 10D action gives [34]:

$$K=-\log(S+S^*)-3\log(T+\overline T+Q \overline Q)$$ (20)

Curiously enough, the second term appeared in the so-called `no-scale models' studied before string theory [41]. This form holds also for the untwisted fields of orbifold compactifications, but the dependence on the twisted fields is not known. It also gives the appropriate result in the large radius limit of Calabi-Yau compactifications although it gets non-perturbative worldsheet corrections relevant at small radii. In order to find the exact tree-level Kähler potential, the best that has been done so far is to write the Kähler potential as an expansion in the matter fields [83]:

$$K = - \log(S+\overline S)+K^M(T,\overline T, U,\overline U)+K^Q(T,\overline T, U,\overline U)Q\overline Q$$

$$+ Z(T,\overline T, U,\overline U)(QQ+\overline Q\overline Q)+{\cal O}(Q^3),$$ (21)

and compute the moduli dependent quantities $K^M,K^Q,K^E$. This has been done explicitly for some $(2,2)$ orbifold compactifications. For instance for factorized orbifolds, that is orbifolds of a 6D torus which is the product of three 2D tori $T^2$, the dependence on the corresponding moduli fields is given by

$$K^M = -\sum_a\log(T_a+\overline T_a)-\sum_m\log(U_m+\overline U_m),$$

$$K^Q = \prod_{a,m}\left(T_a+\overline T_a\right)^{n_m^I}\, \left(U_m+\overline U_m\right)^{n_a^I},$$ (22)

and $Z(T,\overline T, U,\overline U)=0$. Giving rise to the Kahler potential:

$$K = -\log(S+\overline S)-\sum_a\log(T_a+\overline T_a)-\sum_m\log(U_m+\overline U_m)$$

$$+\sum_I \left\vert Q_I\right\vert^2\prod_{a,m}\left(T_a+\overline T_a\right)^{n_m^I}\, \left(U_m+\overline U_m\right)^{n_a^I}$$ (23)

Where the fractional numbers $n_m^I, n_a^I$ are the `modular weights' of the fields $Q_I$ with respect to the duality symmetries related to the moduli $T_a$ or $U_m$. For instance, under $T$ duality, the fields $Q^I$ transform as:

$$Q^I\rightarrow (ic_m T_m+d_m)^{n_m^I}\, Q^I$$ (24)

Furthermore, for $(2,2)$ (Calabi-Yau) models, there is a very interesting observation [43]. Since these compactifications are also compactifications of type II strings, and in that case the corresponding 4D theory has $N=2$ supersymmetry, the moduli dependent part of the Kähler potential, has to have the same dependence as for the $N=2$ models which are much more restrictive. This underlying $N=2$ structure has been very fruitful to extract information on $(2,2)$ models and comes with the name of `special geometry'. This restricts the function $K_M$ which gives the metric in the moduli space. First, the moduli space of the $h_{1,1}$ forms $T_a$ and the $h_{2,1}$ forms $U_a$ factorizes and so:

$$K^M(T,\overline T, U,\overline U)=K^T(T,\overline T)+K^U(U,\overline U)$$ (25)

For which the eq. (22) is a particular case. Second, in $N=2$ supergravity, the full Lagrangian is completely determined by a single holomorphic function, the prepotential. The Kähler potential for the $T_a$ fields is a function of the prepotential $F(T_a)$, given by [44]:

$$K^T(T,\overline T)=-\log\left(F+\overline F-\frac{1}{2}\left(F_T-\overline {F_T}\right) \left(T-\overline T\right)\right)$$ (26)

Where on the right hand side, the subindices mean differentiation. A similar expression holds for $K_U$ in terms of a second prepotential $G(U_m)$. Furthermore, the moduli dependence of the cubic terms in the superpotential

$$W_{tree}=\frac{1}{3}W_{abc}(T_a) Q^aQ^bQ^c+\frac{1}{3}W_{mnp}(U_m)\hat Q^m\hat Q^n\hat Q^p+\cdots ,$$ (27)

is also given by the functions $F(T_a)$ and $G(U_m)$ since the Yukawa couplings are given by

$$W_{abc}(T_a) = \partial_a\partial_b\partial_c F(T_a)$$

$$W_{mnp}(U_m) = \partial_m\partial_n\partial_p G(U_m).$$ (28)

Since $F$ and $G$ are holomorphic, they may get similar constraints as the superpotential above. In particular, since $T$ counts sigma model loops, then $G(U)$ is not renormalized and so the $U$ dependent part of the Kähler potential $K_U(U,\overline U))$ is given exactly by the tree-level result!. Similarly, the Yukawa couplings $W_{mnp}(U_m)$ are exact at tree-level. Here is where the mirror symmetry explained in the previous section plays an important role. Since by mirror symmetry we understand that the compactification on the original manifold ${\cal M}$ and its mirror ${\cal W}$ represent the same CFT, and so the same string model, in the version with the manifold ${\cal W}$, the roles of $T$ and $U$ are interchanged, therefore, computing the $U$ dependent part of the Kähler potential in ${\cal W}$ (which is exact at tree level) gives the $T$ dependent part of the Kähler potential in ${\cal M}$! This fact has been used to compute explicitly the moduli dependent part of the Kähler potential in some examples. This overcomes the Calabi-Yau problem of not knowing the exact CFT behind the compactification, at least for these couplings.