Implosion, contraction and Moore-Tachikawa (Q6573200)

In this survey paper, the authors start out by discussing the implosion construction in symplectic geometry. This construction produces, out of a symplectic \(K\)-manifold (\(K\) is a compact Lie group), a new symplectic manifold \(M_\text{impl}\) which carries an action of the maximal torus \(T\subset K\). For a fixed group \(K\), there is a ``universal'' symplectic implosion, which results from applying the construction to the cotangent bundle \(T^*K\cong K\times \mathfrak{k}^*\), equipped with the action induced by the standard right-action of \(K\) on itself. The resulting manifold \(T^*K_\text{impl}\) is universal in the sense that it can be used to recover the implosion of any other symplectic \(K\)-manifold. The authors also consider its hyper-Kähler and complex symplectic analogs in detail, extending certain aspects of the theory to a more general setting than found in previous work.\N\NThe next step for the authors is to show how the implosion construction naturally fits in with the concept of a Moore-Tachikawa category. In this category, the objects are groups, morphisms are symplectic \(K_1\times K_2\)-spaces, and composition of morphisms is carried out by a reduction of the diagonal action on the product of two such spaces. Since \(T^*K_\text{impl}\) carries a \(K\times T\)-action, it can be viewed as an element of \(\mathrm{Hom}(K,T)\), and can therefore be composed with any symplectic \(K\)-manifold \(M\in \mathrm{Hom}(1,K)\). The resulting element of \(\mathrm{Hom}(1,T)\) is exactly \(M_\text{impl}\). In other words: \(M_\text{impl} \cong (M\times T^*K_\text{impl})/\!\!/ \Delta K\).\N\NThe authors observe that the composition of morphisms \(M_1\in \mathrm{Hom}(K_1,K_2)\), \(M_2\in \mathrm{Hom}(K_2,K_3)\) in the Moore-Tachikawa category inherits an anti-diagonal \(K_2\)-action if \(K_2\) is abelian. Taking \(K_1 = K_3 = K\) and \(K_2 = T\), we see that composing a symplectic \(K\)-manifold with the universal implosion twice, first regarded as an element of \(\mathrm{Hom}(K,T)\) and then as an element of \(\mathrm{Hom}(T,K)\), produces a new \(K\times T\)-manifold, which happens to have the same dimension as \(M\). This is called the contraction of \(M\). The authors show that this contraction construction can be treated in much the same way as implosion, including hyper-Kähler and complex symplectic versions, and end with some examples. In particular, for \(\mathrm{SU}(2)\) and \(\mathrm{SU}(3)\) the universal contractions are identified as Swann bundles over positive Wolf spaces, i.e., symmetric quaternionic Kähler manifolds with positive scalar curvature.