Book contents
- Frontmatter
- Contents
- Preface
- A note on choice of metric
- Text website
- Part 1 Effective field theory: the Standard Model, supersymmetry, unification
- Part 2 Supersymmetry
- Part 3 String theory
- 20 Introduction
- 21 The bosonic string
- 22 The superstring
- 23 The heterotic string
- 24 Effective actions in ten dimensions
- 25 Compactification of string theory I. Tori and orbifolds
- 26 Compactification of string theory II. Calabi–Yau compactifications
- 27 Dynamics of string theory at weak coupling
- 28 Beyond weak coupling: non-perturbative string theory
- 29 Large and warped extra dimensions
- 30 Coda: where are we headed?
- Part 4 The appendices
- References
- Index
25 - Compactification of string theory I. Tori and orbifolds
from Part 3 - String theory
Published online by Cambridge University Press: 17 May 2010
- Frontmatter
- Contents
- Preface
- A note on choice of metric
- Text website
- Part 1 Effective field theory: the Standard Model, supersymmetry, unification
- Part 2 Supersymmetry
- Part 3 String theory
- 20 Introduction
- 21 The bosonic string
- 22 The superstring
- 23 The heterotic string
- 24 Effective actions in ten dimensions
- 25 Compactification of string theory I. Tori and orbifolds
- 26 Compactification of string theory II. Calabi–Yau compactifications
- 27 Dynamics of string theory at weak coupling
- 28 Beyond weak coupling: non-perturbative string theory
- 29 Large and warped extra dimensions
- 30 Coda: where are we headed?
- Part 4 The appendices
- References
- Index
Summary
We don't live in a ten-dimensional world, and certainly not in a twenty-sixdimensional world without fermions. But if we don't insist on Lorentz invariance in all directions, there are other possible ways to construct consistent string theories. In this chapter we will uncover many consistent string theories in four dimensions (and in others). If anything, our problem will shortly be an embarrassment of riches: we will see that there are vast numbers of possible string constructions. The connection of these various constructions to one another is not always clear. Many of these can be obtained from one another by varying expectation values of light fields (moduli). One might imagine that others could be obtained by exciting massive fields as well. In general, though, this is not known, and, in any case, the meaning of such connections in a theory of gravity is obscure. But before exploring these deep and difficult questions, we need to acquire some experience with constructing strings in different dimensions.
Compactification in field theory: the Kaluza–Klein program
The idea that space-time might be more than four-dimensional was first put forward by Kaluza and Klein shortly after Einstein published his general theory of relativity. They argued that, in this case, five-dimensional general coordinate invariance would give rise to both four-dimensional general coordinate invariance and a U(1) gauge invariance, unifying electromagnetism and gravity. In modern language, they considered the possibility that space-time is five-dimensional, with the structure M4 × S1. This is, on first exposure, a bizarre concept, but its implications are readily understood by considering a toy model. Take a single scalar field, Φ, in five dimensions.
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- Supersymmetry and String TheoryBeyond the Standard Model, pp. 373 - 400Publisher: Cambridge University PressPrint publication year: 2007