Multisymplectic actions of compact Lie groups on spheres (Q2658228)

This paper is devoted to the study of multisymplectic structures and in particular to the existence of (homotopy) comoment maps for high-dimensional spheres seen as multisymplectic manifolds. Mainly, the authors prove that for any compact Lie group \(G\) acting multisymplectically and effectively on the \(n\)-sphere \(S^n\) endowed with the standard volume form, the action admits a comoment map if and only if \(n\) is even or the action is transitive. This result has interesting application to the case of \(\operatorname{SO}(n)\) acting on \(S^m\) for \(m,n>1\) with either \(n=m\) or \(n=m+1\). In particular, for the action of \(\operatorname{SO}(4)\) on \(S^3\), one can show that there is no such comoment map but one can overcome this problem by extending the Lie algebra \(\mathfrak{so}(n)\) centrally to some suitable \(L_\infty\)-algebra. The main theorem uses another important result which states that a compact Lie group preserving the pre-multisymplectic form \(\omega\colon G\times M\to M\). If \([\omega]\in H^\bullet(M)\) lies in the image of \(H_G^\bullet(M)\to H^\bullet(M)\), then the action \(\vartheta\colon G\times M\to M\), representing a compact Lie group acting on a pre-multisymplectic manifold preserving the pre-multisymplcetic form \(\omega\), admits a comoment map. We have denoted by \(H_G^\bullet(M):=H^\bullet((M\times EG)/G)\) the equivariant cohomology. The authors prove this statement without choosing a model for equivariant cohomology.