Low Energy String Effective Action

For consistent quantization of string theory it is important for the theory to be invariant under conformal rescaling of the metric. The example of the string moving in a generalised background (3.4) does not have this Weyl invariance at the classical level. Weyl invariance can be realized at the quantum level if the background fields satisfy certain differential equations. These equation can be calculated in a series expansion in the parameter $\alpha'$ and in the case of the non-linear sigma model (3.4) they become [14][15]:

$$0 = D-26 + 3 \alpha'(R+ 4\nabla_\mu \Phi \nabla^\mu \Phi - 4\nabla^2 \Phi + {\textstyle{1\over 12}} H^2) + {\cal O}(\alpha'^2),$$

$$0 = R_{\mu\nu} + {\textstyle{1\over 4}}H_{\mu\lambda \rho} H_{\nu}{}^{\lambda \rho} - 2\nabla_\mu \nabla_\nu \Phi + {\cal O}(\alpha'),$$ (3.5)

$$0 = \nabla_\lambda H^\lambda {}_{\mu\nu} - 2 (\nabla_l \Phi) H^\lambda {}_{\mu\nu} + {\cal O}(\alpha'),$$

where $R$ is the D-dimensional space-time curvature scalar and the field strength tensor $H_{\mu\nu\rho}$ of the antisymmetric field tensor $B_{\mu\nu}$ is defined as

$$H_{\mu\nu\rho} = \partial _{[\mu} B_{\nu\rho]}$$

These equations can be interpreted as the equations of motion of a so called Sting Effective Action. In the limit $\alpha' \to 0$ (zero-slope limit) this becomes the Low Energy String Effective Action:

$$S_{eff} = \int d^Dx \sqrt{\vert g\vert} e^{-\Phi} \left\{-\... ...{\textstyle{1\over 12}} H_{\mu\nu\rho} H^{\mu\nu\rho} \right\}$$ (3.6)

Note that $B_{\mu\nu}$ is a gauge field

$$\delta B_{\mu\nu} = \partial _{[\mu} \Lambda_{\nu]}$$

Because this parameter $\alpha'$ is a typical measure of length of the string, in the zero-slope limit the string reduces to a point and we get back a theory of point-particles. This action thus tells us something about the low energy behaviour of string theory. Weyl invariance requiers the first term in the action to vanish, so quantization of a bosonic string can only be realised in 26-dimensions. For Superstrings this critical dimension is $D=10$.

If we rescale the metric in the following way

$$g_{\mu\nu} \to e^{2\Phi/(D-2)} g_{\mu\nu},$$

the low energy string effective action becomes a $D$-dimensional modified Einstein action (this means a $D$-dimensional gravity coupled to some (exotic) matter fields). So in the case of Heterotic strings (a certain type of superstrings) the bosonic part of the low energy string effective action becomes

$$S = \int d^{10}x \sqrt{-g} \left\{ -R - \partial _\mu \Phi\... ...{1\over 12}} e^{-2\Phi} H_{\mu\nu\rho} H^{\mu\nu\rho} \right\}$$ (3.7)

But we want to have a theory in $4$-dimensions. We have to compactify the extra dimensions so they become invisible at low energies. This compactification or dimensional reduction can be done using the Kaluza-Klein method.