$\mathcal{N}=1$ Geometric Supergravity and chiral triples on Riemann surfaces

Abstract

We construct a global geometric model for the bosonic sector and Killing spinor equations of four-dimensional $\mathcal{N}=1$ supergravity coupled to a chiral non-linear sigma model and a Spin${}_0^c$ structure. The model involves a Lorentzian metric $g$ on a four-manifold $M$, a complex chiral spinor and a map $\phi :M\to \mathcal{M}$ from $M$ to a complex manifold $\mathcal{M}$ endowed with a novel geometric structure which we call chiral triple. Using this geometric model, we show that if $M$ is spin the Kähler-Hodge condition on a complex manifold $\mathcal{M}$ is enough to guarantee the existence of an associated $\mathcal{N}=1$ chiral geometric supergravity, positively answering 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 $\phi :X\to \mathcal{M}$ from $X$ into the Kähler moduli space $\mathcal{M}$ of the theory. We characterize all Riemann surfaces admitting supersymmetric solutions with vanishing superpotential, proving that they consist on holomorphic maps of Riemann surfaces into $\mathcal{M}$ satisfying certain compatibility condition with respect to the canonical bundle of $X$ and the chiral triple of the theory. Furthermore, we classify the biholomorphism type of all Riemann surfaces carrying supersymmetric solutions with complete Riemannian metric and finite-energy scalar map.

arXiv comment41 pages. Minor revision, typos fixed

References in corpus (20)

• N=2 Supergravity and N=2 Super Yang-Mills Theory on General Scalar Manifolds: Symplectic Covariance, Gaugings and the Momentum Map
• On smooth Cauchy hypersurfaces and Geroch's splitting theorem
• Duality Twists, Orbifolds, and Fluxes
• Geometric Constructions of Nongeometric String Theories
• Special Kahler Manifolds
• Generalized Cylinders in Semi-Riemannian and Spin Geometry
• U-Duality and Central Charges in Various Dimensions Revisited
• Special complex manifolds
• Polyvector Super-Poincare Algebras
• Geometry of all supersymmetric four-dimensional $\mathcal{N}=1$ supergravity backgrounds
• The geometric algebra of Fierz identities in arbitrary dimensions and signatures
• The supersymmetric solutions and extensions of ungauged matter-coupled N=1,d=4 supergravity
• Domain walls and instantons in N=1, d=4 supergravity
• Supersymmetry and spin structures
• Parallal Spinors on Pseudo-Riemannian SpinC Manifolds
• Topology of supersymmetric N=1, D=4 supergravity horizons
• Generalized Einstein-Scalar-Maxwell theories and locally geometric U-folds
• Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds
• A Geometric Formulation of Supersymmetry
• Ultracold horizons in gauged N=1 d=4 supergravity

Cited by in corpus (4)

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