Abstract
Dirac-harmonic maps are critical points of an action functional that is motivated from the nonlinear σ-model of quantum field theory. They couple a harmonic map like field with a nonlinear spinor field. In this article, we shall discuss the latest progress on heat flow approaches for the existence of Dirac-harmonic maps under appropriate boundary conditions. Also, we discuss the refined blow-up analysis for two types of approximating Dirac-harmonic maps arising from those heat flow approaches.
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Notes
Here and in the sequel, for simplicity of notations, when talking about a sequence of (ϕα,ψα) for α ↘ 1, we mean the sequence of \((\phi _{\alpha _{k}},\psi _{\alpha _{k}})\) for a given sequence of αk ↘ 1.
Compared to the usual rescaling, i.e. \(\left (\phi _{\alpha }\left (x_{\alpha }+\lambda _{\alpha } x\right ),\sqrt {\lambda _{\alpha }}\psi _{\alpha }\left (x_{\alpha }+\lambda _{\alpha } x\right )\right )\), for a blow-up sequence of Dirac-harmonic maps given in [5], here the additional factor \(\lambda _{\alpha }^{\alpha -1}\) comes from the fact that α-Dirac-harmonic maps are not conformally invariant.
Here we have used the fact that the unique spin structure on \(\mathbb {S}^{2} \setminus \{p\}\) extends to the unique spin structure on \(\mathbb {S}^{2}\) and so does the associated spinor bundle.
It is easy to check that a rescaled α-Dirac-harmonic map, e.g. \(\left (\phi _{\alpha }(\lambda _{\alpha } x), \lambda _{\alpha }^{\alpha -1}\sqrt {\lambda _{\alpha }}\psi _{\alpha }(r_{\alpha } x)\right )\) is locally a critical point of this functional, we refer to Section 5 in [19] for details. We refer to the beginning of Section 2 in [29] for the analogous case of α-harmonic maps.
Let us explain the transformation of the spinor part. In fact, it can be seen as a linear transformation (i.e. \(\lambda _{\alpha }^{\alpha -1}\psi _{\alpha }\)) composed with a conformal transformation (i.e. \(\sqrt {\lambda _{\alpha }}\psi _{\alpha }(x_{\alpha }+\lambda _{\alpha } x)\)). Since α-Dirac-harmonic maps are not conformally invariant, to get unified bubble equations, we need an additional factor \(\lambda _{\alpha }^{\alpha -1}\) in the scaling.
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Liu, L., Zhu, M. Boundary Value Problems for Dirac-Harmonic Maps and Their Heat Flows. Vietnam J. Math. 49, 577–596 (2021). https://doi.org/10.1007/s10013-021-00484-w
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DOI: https://doi.org/10.1007/s10013-021-00484-w
Keywords
- Dirac-harmonic map
- Dirac-harmonic map flow
- α-Dirac-harmonic map
- α-Dirac-harmonic map flow
- Dirichlet boundary
- Chiral boundary