Abstract
In Chap. 4 we have demonstrated the usefulness of conformal field theory as a tool for the bosonic string. In the same way as conformal symmetry was a remnant of the reparametrization invariance of the bosonic string in conformal gauge, superconformal invariance is a remnant of local supersymmetry of the fermionic string in super-conformal gauge. This leads us to consider superconformal field theory. In many aspects our discussion of superconformal field theory parallels that of conformal field theory, and we will treat those rather briefly. Of special interest are N = 2 superconformal field theories, as they are related to space-time supersymmetry. These theories show some new features which we will present in more detail.
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Notes
- 1.
As in Chap. 4we will mainly only consider the holomorphic part of the theory. Note that whereas both sectors of the theory are conformally invariant, it is possible that only one of them, say the holomorphic one, exhibits superconformal invariance. This is for instance the case in the heterotic string theory. We should also mention that superconformal invariance can also appear in the internal sector of the bosonic string.
- 2.
We are considering N = 1 superspace. If one introduces several Grassmann odd coordinates \({\theta }_{i},\bar{{\theta }}_{i}\)one arrives at extended superspaces.
- 3.
In later chapters we will repeatedly encounter the situation where T F is the sum of several terms, for instance a space-time part and an internal part. In all cases all parts of T F must satisfy the same boundary conditions. This is because it is the total T F which couples to the world-sheet gravitino.
- 4.
The classical orthosymplectic super-Lie-algebras osp(m | n) with n = 2pare matrix algebras defined by MG + GM st = 0 where \(G\,=\,\begin{array}{cc} {Vdash }_{m}& 0 \\ 0 & {\mathbb{J}}_{n} \end{array}\)with \({\mathbb{J}}_{n}\,=\,\begin{array}{cc} 0 & Vdash \\ - Vdash & 0\end{array}\)is the invariant metric. A general element Xhas the form \(M\,=\,\begin{array}{cc} A& B\\ C &D \end{array}\)of which the m×mmatrix Aand the n×nmatrix Dare associated with the bosonic generators while Band Care associated with the fermionic generators. The supertranspose of Xis defined as \({X}^{st}\,=\,\begin{array}{cc} {A}^{\mathrm{T}} & {C}^{\mathrm{T}} \\ - {B}^{\mathrm{T}} & {D}^{\mathrm{T}} \end{array}\). The maximal bosonic subalgebra is o(m) ×sp(n). osp(m | n) has \({ 1 \over 2} [{(m + n)}^{2} + n - m]\)generators of which mnare fermionic. An explicit matrix representation of osp(1 | 2) is \({L}_{0}\,=\,{ 1 \over 2} \begin{array}{ccc} 0 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} ,\,{G}_{-1/2} = \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0\end{array}\), \({G}_{+1/2}\,=\,{G}_{-1/2}^{st},\,{L}_{\pm 1}\,=\,{({G}_{\pm 1/2})}^{2}\).
- 5.
Note that in the N = 1 discrete series R supersymmetry is unbroken for even monly.
- 6.
This is the action of Chap.7, after going to superconformal gauge and performing a Wick rotation to an Euclidean world-sheet, i.e. replacing τ → − iτ and mapping from the Euclidean cylinder to the plane. For the fermions we replace \({\rho }^{0} \rightarrow -i{\rho }^{0}\)and \(\bar{\psi } \rightarrow i{\psi }^{T}{\rho }^{0}\). In contrast to previous chapters we use dimensionless scalar fields X μ. This amounts to setting α′ = 2. To restore α′one replaces \({X}^{\mu } \rightarrow \sqrt{{ 2 \over \alpha \prime } }{X}^{\mu }\)and \({k}^{\mu } \rightarrow \sqrt{{ \alpha \prime \over 2} } {k}^{\mu }\). The integration over the Grassmann coordinates is \(\int olimits olimits {d}^{2}\theta \bar{\theta }\theta = 1\).
- 7.
More accurately, for the heterotic string one needs N = (0, 2) two-dimensional world-sheet supersymmetry. Similarly, for orientifolds of type II superstring theory, one needs N = (2, 2) supersymmetry on the world-sheet.
- 8.
Concretely, this realization is given by the coset
$$\frac{\widehat{su}{(2)}_{k} \times \widehat{ u}{(1)}_{2}} {\widehat{u}{(1)}_{k+2}} ,$$(12.114)where \(\widehat{u}{(1)}_{k}\)is the CFT of a free boson compactified on a circle with radius \(\sqrt{\alpha \prime k}\)introduced in Sect. 10.3.
- 9.
The spectral flow operators \({U}_{\pm { 1 \over 2} }\)and the supercurrents \({T}_{F}^{\pm }\)are simple currents of any N = 2 SCFT.
- 10.
Since L 0and J 0commute, they can be simultaneously diagonalized and we can restrict the discussion to eigenstates of L 0and J 0.
- 11.
It is straightforward to work out the algebra in terms of the modes. All mode numbers are integers and there is a finite-dimensional subalgebra generated by L 0, ± 1and G 0, ± 1.
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). Conformal Field Theory III: Superconformal Field Theory. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_12
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DOI: https://doi.org/10.1007/978-3-642-29497-6_12
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