\(L_{\infty}\) algebras for extended geometry from Borcherds superalgebras. (English) Zbl 1478.17030
In this paper, some relevant aspects of the exceptional Lie algebras in connection with gauge field theory are explored. The approach is based on a variation to the ghost of the Batalin-Vilkovisky formalism, As the Borcherds superalgebra \(\mathcal{B}(\mathfrak{g}_{l+1})\) can be seen as a double extension of \(\mathfrak{g}_r\) and the \(L_\infty\)-brackets are described in terms of the underlying Borcherds algebra, the authors show that all even brackets above 2-brackets vanish, and that the coefficients appearing in the brackets are given by Bernoulli numbers. It is shown that the conclusion is valid in the absence of ancillary transformations at ghost number 1. Possible extensions of the construction to the case with ancillary transformations are discussed, and strong evidence that such a generalization should be based on the tensor hierarchy algebra as the suitable underlying algebra is presented.
Reviewer: Rutwig Campoamor Stursberg (Madrid)
MSC:
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
| 81T99 | Quantum field theory; related classical field theories |
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