Superconformal geometry in three dimensions

Journal Article · Sat Feb 01 04:00:00 EST 1992 · Journal of Mathematical Physics (New York); (United States)

DOI:https://doi.org/10.1063/1.529754· OSTI ID:5508637

Merkulov, S A ^[1]

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1. Theoretical Problems Department, USSR Academy of Sciences, ul. Vesnina 12, Moscow SU-121002 (USSR)

A category of superconformal (3{vert bar}2)-dimensional supermanifolds is introduced and studied. It is shown that each superconformal structure has associated with it a distinguished vector bundle (or {ital local} {ital supertwistor} {ital bundle}) equipped with a superconnection that has to be regarded as a supersymmetric extension of three-dimensional conformal Cartan's connection. The curvature of the superconnection provides a superconformal invariant that vanishes if and only if the (3{vert bar}2) supermanifold is locally isomorphic to isotropic super-Grassmanian GI(0{vert bar}2;C{sup 1{vert bar}4};{ital b}) with {ital b}{element of}0Sp(1{vert bar}4). A choice of scale of superconformal structure on a 3{vert bar}2-supermanifold M determines canonically a volume form on M that induces on the underlying manifold M{sub rd} the action of {ital N}=1, {ital D}=3 supergravity with cosmological constant. It is proved that a superconformal structure on M can be encoded into the holomorphic structure of the associated with M (3{vert bar}1)-dimensional complex superspace of null supergeodesics, N. A twistor correspondence between free holomorphic data on N and solutions of various field equations (including supersymmetric Einstein equations with cosmological constant) on M is established.

• OSTI ID: 5508637
• Journal Information: Journal of Mathematical Physics (New York); (United States), Journal Name: Journal of Mathematical Physics (New York); (United States) Vol. 33:2; ISSN JMAPA; ISSN 0022-2488
• Country of Publication: United States
• Language: English

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