论文标题
QuaternionicKähler歧管的对称性与$ S^1 $ -smmetry
Symmetries of quaternionic Kähler manifolds with $S^1$-symmetry
论文作者
论文摘要
我们研究了由HK/QK对应获得的QuaternionicKähler歧管的对称性。对于任何谎言代数$ \ mathfrak {g} $的无穷小型自动形态，我们将$ \ mathfrak {g} $的核心估计，由无限型自动形态代表，由此产生的quaternionicquaternionickähler歧管。更具体地说，我们研究了通过$ C $地图结构的单循环变形获得的指标，证明了最初的投影特殊特殊kähler歧管的无穷小型自动形态的代数产生了相应的单循环造成的单循环$ C $ C $ C $ -MAP $ C $ -MAP的杀人领域的代数。作为应用程序，我们表明这种构建最多将自动形态群体的同一均匀性提高。特别是，如果初始歧管是均匀的，则一环变形度量最多是同构性。例如，我们考虑$ su（n，2）/s（u（n）\ times u（2））$上的对称QuaternionicKählerMetric的单循环变形，我们证明这是同一性的。这个家庭概括了所谓的通用超人（$ n = 1 $），为此我们确定整个等轴测组。
We study symmetry properties of quaternionic Kähler manifolds obtained by the HK/QK correspondence. To any Lie algebra $\mathfrak{g}$ of infinitesimal automorphisms of the initial hyper-Kähler data we associate a central extension of $\mathfrak{g}$, acting by infinitesimal automorphisms of the resulting quaternionic Kähler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the $c$-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special Kähler manifold gives rise to a Lie algebra of Killing fields of the corresponding one-loop deformed $c$-map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous then the one-loop deformed metric is of cohomogeneity at most one. As an example, we consider the one-loop deformation of the symmetric quaternionic Kähler metric on $SU(n,2)/S(U(n)\times U(2))$, which we prove is of cohomogeneity exactly one. This family generalizes the so-called universal hypermultiplet ($n=1$), for which we determine the full isometry group.
