Yang-Mills theories in dimensions 3,4,6,10 and Bar-duality
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Yang-Mills theories in dimensions 3,4,6,10 and Bar-duality

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https://arxiv.org/pdf/hep-th/0503165.pdf
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Abstract

In this note we give a homological explanation of "pure spinors" in YM theories with minimal amount of supersymmetries. We construct A_{\infty} algebras A for every dimension D=3,4,6,10, which for D=10 coincides with homogeneous coordinate ring of pure spinors with coordinate lambda^{alpha}. These algebras are Bar-dual to Lie algebras generated by supersymmetries, written in components. The algebras have a finite number of higher multiplications. The main result of the present note is that in dimension D=3,6,10 the algebra A\otimes \Lambda[\theta^{\alpha}]\otimes Mat_n with a differential D is equivalent to Batalin-Vilkovisky algebra of minimally supersymmetric YM theory in dimension D reduced to a point. This statement can be extended to nonreduced theories.

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