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M-theory solutions invariant under \(D(2,1; \gamma) \oplus D(2,1;\gamma)\). (English) Zbl 1338.81310

Summary: We simplify and extend the construction of half-BPS solutions to 11-dimensional supergravity, with isometry superalgebra \(D(2,1; \gamma) \oplus D(2,1;\gamma)\). Their space-time has the form \(\mathrm{AdS}_{3} \times \mathrm{S}^{3} \times \mathrm{S}^{3}\) warped over a Riemann surface \(\Sigma\). It describes near-horizon geometries of M2 branes ending on, or intersecting with, M5 branes along a common string. The general solution to the BPS equations is specified by a reduced set of data \((\gamma, h, G)\), where \(\gamma\) is the real parameter of the isometry superalgebra, and \(h\) and \(G\) are functions on \(\Sigma\) whose differential equations and regularity conditions depend only on the sign of \(\gamma\). The magnitude of \(\gamma\) enters only through the map of h,G onto the supergravity fields, thereby promoting all solutions into families parametrized by \(|\gamma|\). By analyzing the regularity conditions for the supergravity fields, we prove two general theorems: (i) that the only solution with a 2-dimensional CFT dual is \(\mathrm{AdS}_{3} \times \mathrm{S}^{3} \times \mathrm{S}^{3} \times \mathbb{R}^{2}\), modulo discrete identifications of the flat \(\mathbb{R}^{2}\), and (ii) that solutions with \(\gamma < 0\) cannot have more than one asymptotic higher-dimensional AdS region. We classify the allowed singularities of \(h\) and \(G\) near the boundary of \(\Sigma\), and identify four local solutions: asymptotic \(\mathrm{AdS}_{4}/Z_{2}\) or \(\mathrm{AdS}_{7}^{\prime}\) regions; highly-curved M5-branes; and a coordinate singularity called the “cap”. By putting these “Lego” pieces together we recover all known global regular solutions with the above symmetry, including the self-dual strings on M5 for \(\gamma <0\), and the Janus solution for \(\gamma > 0\), but now promoted to families parametrized by \(|\gamma|\). We also construct exactly new regular solutions which are asymptotic to \(\mathrm{AdS}_{4}/Z_{2}\) for \(\gamma < 0\), and conjecture that they are a different superconformal limit of the self-dual string. Finally, we construct exactly \(\gamma > 0\) solutions with highly curved M5-brane regions, which are the formal continuation of the self-dual string solutions across the decompactification point at {\(\gamma\)} = 0.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83E50 Supergravity
83E15 Kaluza-Klein and other higher-dimensional theories
17A70 Superalgebras
81T20 Quantum field theory on curved space or space-time backgrounds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14H55 Riemann surfaces; Weierstrass points; gap sequences
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