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Hidden \(Q\)-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions. (English) Zbl 1305.81124

Summary: We disclose the mathematical structure underlying the gauge field sector of the recently constructed non-abelian superconformal models in six space-time dimensions. This is a coupled system of 1-form, 2-form, and 3-form gauge fields. We show that the algebraic consistency constraints governing this system permit to define a Lie 3-algebra, generalizing the structural Lie algebra of a standard Yang-Mills theory to the setting of a higher bundle. Reformulating the Lie 3-algebra in terms of a nilpotent degree 1 BRST-type operator \(Q\), this higher bundle can be compactly described by means of a \(Q\)-bundle; its fiber is the shifted tangent of the \(Q\)-manifold corresponding to the Lie 3-algebra and its base the odd tangent bundle of space-time equipped with the de Rham differential. The generalized Bianchi identities can then be retrieved concisely from \(Q^2 = 0\), which encode all the essence of the structural identities. Gauge transformations are identified as vertical inner automorphisms of such a bundle, their algebra being determined from a \(Q\)-derived bracket.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
83E15 Kaluza-Klein and other higher-dimensional theories
81T60 Supersymmetric field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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