String Compactifications
An alternative to describing compactifications via a solvable conformal field theory is the perturbative approach around a geometric supergravity background at large radius. For this purpose one analyzes the string equations of motion at leading order in
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String Compactifications
Abstract An alternative to describing compactifications via a solvable conformal field theory is the perturbative approach around a geometric supergravity background at large radius. For this purpose one analyzes the string equations of p motion at leading order in a typical length scale L= ˛ 0 . We describe this approach in detail for a class of backgrounds which preserve some amount of space-time supersymmetry in four-dimensions: compactification on Calabi-Yau manifolds. But we start with a brief discussion of the string equations of motion as the requirement of vanishing beta-functions of the non-linear sigma model for a string moving in a curved background. We then derive a generalization of T-duality to manifolds with isometries. This leads to the so-called Buscher rules. We then introduce some of the mathematical tools which are required for an adequate treatment of Calabi-Yau compactifications. With them at hand we consider compactifications of the type II and heterotic superstring on Calabi-Yau manifolds and discuss the structure of their moduli spaces. In an appendix we fix our notation and review some concepts of Riemannian geometry. The derivations of some results which are used in the main text are also relegated to the appendix.
14.1 Conformal Invariance and Space-Time Geometry In the first part of this book we have constructed string theories in Minkowski spacetime. Quantum consistency required the number of space-time dimensions to be 10 for the fermionic and 26 for the bosonic string. Both are higher than the four large dimensions which we observe around us. The main consistency condition, the absence of a conformal anomaly of the combined matter and ghost system, has other solutions besides the one which leads to Minkowski space in the critical dimension. Other solutions are more complicated CFTs with or without a geometric interpretation as a compactification from dcrit to four dimensions. We have already discussed simple examples in Chap. 10: tori and toroidal orbifolds. In the present chapter we study more general compactifications R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 14, © Springer-Verlag Berlin Heidelberg 2013
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14 String Compactifications
and some of their properties, using geometrical tools. In the next chapter we will discuss compactifications using CFT methods. We start with a discussion of the implications which follow from the requirement of conformal invariance of the world sheet theory. This can be illustrated by considering the bosonic string propagating through a space-time M with metric G .X /, rather than . In this case the world-sheet action is SP D
1 4 ˛ 0
Z d 2
p h h˛ˇ G .X / @˛ X @ˇ X :
(14.1)
˙
The -model metric is also called the string frame metric. In conformal gauge (14.1) no longer reduces to a free scalar field theory. One has to deal with an interacting theory, a non-linear sigma model with target space metric G .X /: 1 S
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