Abstract
We study the symmetries and invariances of a version of the action functional of the nonlinear sigma model with gravitino, as considered in Jost et al. (2017). The action is invariant under rescaled conformal transformations, super Weyl transformations, and diffeomorphisms. In particular cases the functional possesses a degenerate supersymmetry. The corresponding conservation laws lead to a geometric interpretation of the energy–momentum tensor and supercurrent as holomorphic sections of appropriate bundles.
Highlights
The main motivation for the introduction of the two-dimensional nonlinear supersymmetric sigma model in quantum field theory, or super gravity and super string theory, are its symmetries, see for instance [3, 9]
The conservation laws corresponding to those four symmetries are expressed in terms of the energy-momentum tensor and the supercurrent, which is the variation of the action with respect to the gravitino
We summarize our discussion about the Dirac action in the following theorem
Summary
The main motivation for the introduction of the two-dimensional nonlinear supersymmetric sigma model in quantum field theory, or super gravity and super string theory, are its symmetries, see for instance [3, 9]. Nonlinear sigma model, gravitino, Noether’s theorem, symmetry, energy-momentum tensor, supercurrent, Dirac-harmonic map. As a peculiarity in the case that the domain is two dimensional, the conformal invariance E(φ; e2ug) = E(φ; g), ∀u ∈ C∞(M ) Those two symmetries link the theory of harmonic maps on a Riemann surface to the Teichmüller theory, see [16]. The conservation laws corresponding to diffeomorphism invariance and conformal invariance are expressed in terms of the energy-momentum tensor. The conservation law corresponding to the diffeomorphism invariance of the Dirichlet energy is that the energy-momentum tensor is divergence-free on shell 2. The conservation laws corresponding to those four symmetries are expressed in terms of the energy-momentum tensor and the supercurrent, which is the variation of the action with respect to the gravitino. We make a supplement about the divergence operators, both on symmetric 2-tensors and on spinors, which are frequently used in the context
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