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Boundary Value Problems for Dirac-Harmonic Maps and Their Heat Flows

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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

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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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Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, 430079, Wuhan, People’s Republic of China

    Lei Liu

  2. School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240, Shanghai, People’s Republic of China

    Miaomiao Zhu

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  1. Lei Liu
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  2. Miaomiao Zhu
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Correspondence to Miaomiao Zhu.

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Dedicated to Professor Jürgen Jost’s 65th birthday.

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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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