Abstract
We study some analytical and geometric properties of a two-dimensional nonlinear sigma model with gravitino which comes from supersymmetric string theory. When the action is critical w.r.t. variations of the various fields including the gravitino, there is a symmetric, traceless, and divergence-free energy-momentum tensor, which gives rise to a holomorphic quadratic differential. Using it we obtain a Pohozaev type identity and finally we can establish the energy identities along a weakly convergent sequence of fields with uniformly bounded energies.
Highlights
The two-dimensional nonlinear sigma models constitute important models in quantum field theory. They have physical applications and geometric implications, and their properties have been the focus of important lines of research. They arise as two-dimensional harmonic maps and pseudoholomorphic curves
In modern physics the basic matter fields are described by vector fields as well as spinor fields, which are coupled by supersymmetries
In the 1970s a supersymmetric two-dimensional nonlinear sigma model was proposed in [6, 14]; the name “supersymmetric” comes from the fact that the action functional is invariant under certain transformations of the matter fields; see for instance [13, 18]
Summary
The two-dimensional nonlinear sigma models constitute important models in quantum field theory. One can rewrite the extrinsic equations without labeling indices, but we want to derive estimates and see how the second fundamental form A affects the system; we adopt this formulation This action functional is closely related to Dirac-harmonic maps with curvature term. They constitute a simplified version of the model considered in this paper and describe the behavior of the nonlinear sigma models in degenerate cases The symmetries of this action functional always play an important role in the study of the solution spaces and here especially the rescaled conformal invariance. Known as the blow-up procedure, we can get some solutions with vanishing gravitinos, i.e., Dirac-harmonic maps with curvature term, defined on the standard sphere S2 with target manifold (N, h), known as “bubbles”. These conclusions are similar to those for harmonic maps and Dirac-harmonic maps and some of its variants in e.g. [7, 20, 24, 29, 35], one has to pay special attention to the critical gravitinos
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