Conformal Field Theory III: Superconformal Field Theory
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 inv
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Conformal Field Theory III: Superconformal Field Theory
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 D 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.
12.1 N D 1 Superconformal Symmetry In Chaps. 7 and 8 we have met the superconformal algebra when we studied the fermionic string in superconformal gauge. As in the bosonic string, this is a particular realization of a much more general structure, that of superconformal field theories (SCFT). Recall that the generators of superconformal transformations are the conserved energy-momentum tensor T .z/ and the conserved world-sheet supercurrent TF .z/.1 The basic objects of superconformal field theory are chiral conformal (or primary) superfields. Their transformation properties under superconformal transformations are most convincingly motivated in superspace.2 The
1 As in Chap. 4 we 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 D 1 superspace. If one introduces several Grassmann odd coordinates i ; Ni one arrives at extended superspaces.
R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 12, © Springer-Verlag Berlin Heidelberg 2013
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12 Conformal Field Theory III: Superconformal Field Theory
coordinates of superspace are z D .z; / and zN D .Nz; N /. The Grassmann parity of z; zN is even and that of and N is odd and 2 D N 2 D 0. Due to this property any N superfield, i.e. function on superspace, has a finite Taylor expansion in and : ˚.zz; zN/ D 0 .z; zN/ C 1 .z; zN/ C N N1 .z; zN/ C N 2 .z; zN/ ;
(12.1)
where the Grassmann parity of 0 and 2 is opposite to that of 1 and N1 . The Grassmann parity of a superfield ˚ is that of its lowest component 0 . The (holomorphic) super-interval between two points z 1 and z 2 in superspace is defined as z 12 D z1 z2 1 2
(12.2)
and similarly for the anti-holomorphic super interval. It is invariant under translations in superspace whose generators are the super derivatives D D @ C @z ;
N zN : DN
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