2-plectic manifolds and its relation with Courant algebroids (Q2911884)

An \(n\)-plectic structure on a manifold is nothing but an \((n+1)\)-form that is closed and nondegenerate (i.e., with a vanishing kernel). As a generalization of a symplectic structure, it clearly is of use in physics, but some caution is advised. Given the extreme generality of the notion, one should carefully select only the really relevant features from the numerous new ones that arise when passing from \(2\)-tensors to higher order ones. Thus, it is certainly reasonable to build the topic of \(n\)-plectic manifolds, not on a merely speculative basis, but with reference to relevant \(n\)-plectic structures that have appeared in the established literature from time-to-time. A relevant \(2\)-plectic structure arises in connection with Courant algebroids. This notion was defined in [\textit{Z.-J. Liu}, \textit{A.~Weinstein} and \textit{P.~Xu}, J.~Differential Geom. 45, No.~3, 547--574 (1997; Zbl 0885.58030)], as a generalization (and consequent axiomatization) of a bracket which was first introduced on \(TM\oplus T^\ast M\) by \textit{T.~J. Courant} in [Trans. Am. Math. Soc. 319, No.~2, 631--661 (1990; Zbl 0850.70212)]. Later, P.~Ševera noted that exact Courant algebroids are classified by the third de Rham cohomology of the base manifold, and \(3\)-forms in a given class correspond to a suitable notion of connection on the corresponding exact Courant algebroid. This is how \(2\)-plectic structures come into play.NEWLINENEWLINEAn extensive discussion about the interplay between \(2\)-plectic structures and Courant algebroids is the main theme of [\textit{C.~L.~Rogers}, ``Courant algebroids from categorified symplectic geometry'', \url{arXiv:1001.0040}]. The article under review deals with two issues concerning this relationship. The first relates \(2\)-Lagrangian submanifolds with isomorphisms of certain Courant algebroids. The other aims to characterize, in terms of \(2\)-plectic structures, Lie group actions that can be inherited by the corresponding Courant algebroids.NEWLINENEWLINEReviewer's remark: I have several concerns, a few of which I mention here, about the results and their presentation. For instance, in my view, the statements of both theorems about \(2\)-Lagrangian submanifolds in Section~4 express quite inaccurately what is really proved (apart from the issue of incorrect wording, such as `\(2\)-plectic submanifold' in place of `\(2\)-Lagrangian submanifold', in the first statement). And even if the Courant algebroids involved were adequately specified in the statements, I would still not gain any new insight, beyond those provided by the two earlier results that are patched together in both proofs. Nor can I imagine how the statements could be useful as a technical reference. Similar objections could be raised with respect to the result on Lie groups actions. Moreover, that statement seems to be incorrect, if one considers the trivial action of a Lie group not contained in \(\text{Ham}(M,\omega)\). These results are preceded by a quite longer introductory and preparatory discussion, which again contains some inaccuracies. These aside, the English and the mathematical style are plain, and the reader is comfortably introduced to the subject of the work.