Hyper-Kähler metrics arising from tri-Hamiltonian actions on solvable Lie groups

M.L. BARBERIS, I. DOTTI, A FINO

Cancún, México

Congreso; Segundo Congreso Latinoameriano de Matemáticos; 2004

Umalca

Given a hyper-Kähler metric g on a manifold M and an action of a Lie group K on M preserving the hyper-Kähler structure, under certain topological assumptions there exists a K-equivariant moment map m: M --> ( k*)3  .  The quotient construction, due to N. Hitchin et al. [Comm. Math. Phys. 108 (1987), 535--589],  gives a procedure to obtain a new hyper-Kähler structure as follows: let x = (x1,x2,x3 ) in  ( k*)3    be a regular value of m such that xi   lies in the center of  k*  for  i=1,2,3. If the action of  K on m-1(x) is free and proper, then the quotient  m-1(x) / K  is hyper-Kähler. Although this construction is a powerful method for showing existence of hyper-Kähler metrics, finding the metric explicitely and studying completeness may be difficult. In this work we consider the case M=G, a Lie group with a left invariant hyper-Kähler  metric g which is necessarily flat. Any such Lie group is two step solvable: its universal cover  is a semidirect product Hp x Hq   where the action of the first  factor on the second is determined by fixing a torus in Sp(q). We apply the quotient construction with  K = Rs   contained in Hp   acting on G by left  translations, for s less than p and q. We can show that in this case m-1(x) / K is closed in the  complete riemannian manifold  G/K , thus the hyper-Kähler metric on the quotient is complete. Moreover, we show that  m-1(x) / K is diffeomorphic to 4(p+q-s) dimensional Euclidean space and we obtain the explicit description of the hyper-Kähler metric on the quotient in local coordinates. We relate some of our results with those in [G. W.Gibbons, P. Rychenkova, R. Goto, Commun. Math. Phys. 186 (1997), 581--599].