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**Supersymmetric \(\mathrm{AdS}_{3} \times S^{2}\) M-theory geometries with fluxes.**
*(English)*
Zbl 1290.81109

Summary: Motivated by a recent observation that the LLM geometries admit 1/4-BPS M5-brane probes with worldvolume \(\mathrm{AdS}_{3} \times \Sigma_{2} \times S^{1}\) preserving the R-symmetry, \(\mathrm{SU}(2)\times\mathrm U(1)\), we initiate a classification of the most general \(\mathrm{AdS}_{3} \times S^{2}\) geometries in M-theory dual to two-dimensional chiral \(\mathcal{N} = \left( {4,0} \right)\) SCFTs. We retain all field strengths consistent with symmetry and derive the torsion conditions for the internal six-manifold, \(M_{6}\), in terms of two linearly independent spinors. Surprisingly, we identify three Killing directions for \(M_{6}\), but only two of these generate isometries of the overall ansatz. We show that the existence of this third direction depends on the norm of the spinors. With the torsion conditions derived, we establish the MSW solution as the only solution in the class where \(M_6\) is an SU(3)-structure manifold. Then, specialising to the case where the spinors define an SU(2)-structure, we note that supersymmetry dictates that all magnetic fluxes necessarily thread the \(S^{2}\). Finally, by assuming that the two remaining Killing directions are parallel and aligned with one of the two vectors defining the SU(2)-structure, we derive a general relationship for the two spinors before extracting a known class of solutions from the torsion conditions.

### MSC:

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

83E50 | Supergravity |

81R60 | Noncommutative geometry in quantum theory |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

20C35 | Applications of group representations to physics and other areas of science |

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\textit{E. Ó. Colgáin} et al., J. High Energy Phys. 2010, No. 8, Paper No. 114, 29 p. (2010; Zbl 1290.81109)

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