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Erschienen in:
2019 | OriginalPaper | Buchkapitel
12. The Superconformal Action Functional
verfasst von : Enno Keßler
Erschienen in: Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional
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Abstract
This chapter treats the superconformal action functional. For any map Φ: M → N from a super Riemann surface to an arbitrary Riemannian supermanifold N and any \( \operatorname {\mathrm {U}}(1)\)-metric m on M, the action functional A(Φ, m) is a function in \(\mathcal {O}_B\) that depends on the super Riemann surface structure on M and the map Φ. We will see that the action functional A(Φ, m) is a natural generalization of the action functional of harmonic maps on Riemann surfaces in several aspects.
The first analogy between the harmonic action functional and the superconformal action functional are the conformal properties. The harmonic action functional on Riemann surfaces is conformally invariant, that is depends only on the conformal class or complex structure on the surface. The superconformal action functional A(Φ, m) is superconformally invariant. That is, A(Φ, m) does not depend on the given \( \operatorname {\mathrm {U}}(1)\)-structure but rather only on the \( \operatorname {\mathrm {SCL}}\)-structure or super Riemann surface structure. Furthermore, the harmonic action functional is diffeomorphism invariant, whereas the superconformal action functional is superdiffeomorphism invariant.