Abstract
In the previous chapter we have learned that in toroidal compactifications of the bosonic string there are, in addition to the Kaluza-Klein gauge bosons familiar from field theory, further massless vectors of purely stringy origin. However, we did not show that these massless vectors are gauge bosons of a non-Abelian gauge group G, transforming in the adjoint representation. The necessary mathematical tool to do this is the theory of infinite dimensional (current) algebras, the so-called affine Kač-Moody algebras. They are the subject of this chapter for which we assume some familiarity with the structure of finite dimensional Lie algebras.
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Notes
- 1.
Since we consider only compact G’s, we can take the Cartan-Killing metric to be \({\delta }^{ab}\)in which case \({f}^{abc} = {f{}^{ab}}_{c}\)is antisymmetric in all indices. The generators T aare Hermitian and the structure constants are real.
- 2.
Semi-simple finite dimensional Lie algebras do not possess non-trivial central extensions.
- 3.
The Cartan matrix is often denoted \({A}_{ij} = 2{\alpha }_{i} \cdot {\alpha }_{j}/{\alpha }_{j}^{2}\). For simply laced Lie algebras it reduces to our g ij .
- 4.
We also include the case n = 1 with \({D}_{1} \sim SO(2) \sim U(1)\).
- 5.
A highest weight state \(\vert {m}_{0},D\rangle\)satisfies \({E}^{\alpha }\vert {m}_{0},D\rangle = 0\quad \forall \text{ positive }\alpha \). It means that \(\alpha +{ m}_{0}\)is not a weight vector for any positive root \(\alpha \). The other states in the same representation are obtained by acting with lowering operators on the highest weight state. Any irreducible representation of ghas a unique highest weight state—the other weights \(m\)have the property that \({m}_{0} -m\)is a sum of positive roots. The highest weight of the adjoint representation is called highest root \(\psi \)with \({\psi }^{2} = 2\).
- 6.
Spin(2n), neven, is simply connected and has center \({\mathbb{Z}}_{2} \times {\mathbb{Z}}_{2}\). If we divide by the diagonal \({\mathbb{Z}}_{2}\)we get SO(2n) with only (0) and (V) conjugacy classes. If we divide by one of the \({\mathbb{Z}}_{2}\)we are left with the (0) and one of the spinor conjugacy classes.
- 7.
We already encountered Lorentzian lattices when we discussed the torus compactification of the bosonic and the heterotic strings.
- 8.
This is the dimensionless chiral field X R , introduced in (10.59), mapped to the complex z-plane.
- 9.
Use the relation \({e}^{iq\cdot \beta }f(p){e}^{-iq\cdot \beta } = f(p-\beta )\)for any function \(f(p)\).
- 10.
The chiral world-sheet fermions mapped to the complex plane are of this kind. We will discuss the conformal field theory of fermions in detail in Chap. 12.
- 11.
The corresponding highest weight representations are also called integrable representations.
- 12.
These identities remain also true for higher genus partition functions.
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© 2012 Springer-Verlag Berlin Heidelberg
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). Conformal Field Theory II: Lattices and Kač-Moody Algebras. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_11
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DOI: https://doi.org/10.1007/978-3-642-29497-6_11
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