Abstract We construct a global geometric model for the bosonic sector and Killing spinor equations of four-dimensional $$\mathcal {N}=1$$N=1 supergravity coupled to a chiral non-linear sigma model and a $$\mathrm {Spin}^{c}_0$$Spin0c structure. The model involves a Lorentzian metric g on a four-manifold M, a complex chiral spinor and a map $$\varphi :M\rightarrow \mathcal {M}$$φ:M→M from M to a complex manifold $$\mathcal {M}$$M endowed with a novel geometric structure which we call a chiral triple. Using this geometric model, we show that if M is spin then the Kähler-Hodge condition on $$\mathcal {M}$$M suffices to guarantee the existence of an associated $$\mathcal {N}=1$$N=1 chiral geometric supergravity. This positively answers a conjecture proposed by D. Z. Freedman and A. V. Proeyen. We dimensionally reduce the Killing spinor equations to a Riemann surface X, obtaining a novel system of partial differential equations for a harmonic map with potential $$\varphi :X\rightarrow \mathcal {M}$$φ:X→M. We characterize all Riemann surfaces X admitting supersymmetric solutions with vanishing superpotential, showing that such solutions $$\varphi $$φ are holomorphic maps satisfying a certain condition involving the canonical bundle of X and the chiral triple of the theory. Furthermore, we determine the biholomorphism type of all Riemann surfaces carrying supersymmetric solutions with complete Riemannian metric and finite-energy scalar map $$\varphi $$φ.

中文翻译：

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$${\mathcal {N}}=1$$ N = 1 黎曼曲面上的几何超重力和手征三重

摘要 我们构建了一个全局几何模型的玻色扇区和四维 $$\mathcal {N}=1$$N=1 超重力耦合到手征非线性 sigma 模型和 $$\mathrm {自旋}^{c}_0$$Spin0c 结构。该模型涉及四流形 M 上的洛伦兹度量 g、复手征旋量和映射 $$\varphi :M\rightarrow \mathcal {M}$$φ:M→M 从 M 到复流形 $$\ mathcal {M}$$M 具有一种新颖的几何结构，我们称之为手性三元组。使用这个几何模型，我们证明如果 M 是自旋的，那么 $$\mathcal {M}$$M 上的 Kähler-Hodge 条件足以保证存在关联的 $$\mathcal {N}=1$$N= 1 手征几何超重力。这肯定地回答了 DZ Freedman 和 AV Proeyen 提出的猜想。我们将Killing Spinor方程从维度上简化为黎曼曲面X，从而获得了具有潜在$$\varphi :X\rightarrow \mathcal {M}$$φ:X→M的调和映射的偏微分方程的新系统。我们刻画了所有允许超势能消失的超对称解的黎曼曲面 X，表明这些解 $$\varphi $$φ 是满足特定条件的全纯映射，涉及 X 的规范丛和理论的手征三元组。此外，我们使用完全黎曼度量和有限能量标量映射 $$\varphi $$φ 确定所有携带超对称解的黎曼曲面的双全同胚类型。我们刻画了所有允许超势能消失的超对称解的黎曼曲面 X，表明这些解 $$\varphi $$φ 是满足特定条件的全纯映射，涉及 X 的规范丛和理论的手征三元组。此外，我们使用完全黎曼度量和有限能量标量映射 $$\varphi $$φ 确定所有携带超对称解的黎曼曲面的双全同胚类型。我们刻画了所有允许超势能消失的超对称解的黎曼曲面 X，表明这些解 $$\varphi $$φ 是满足特定条件的全纯映射，涉及 X 的规范丛和理论的手征三元组。此外，我们使用完全黎曼度量和有限能量标量映射 $$\varphi $$φ 确定所有携带超对称解的黎曼曲面的双全同胚类型。