We present an asymptotically flat and stationary multi-black lens solution with biaxisymmetry of U (1) × U (1) as a supersymmetric solution in the five-dimensional minimal ungauged supergravity. We show that the spatial cross section of each degenerate Killing horizon admits different lens space topologies of L(n, 1) = S 3 /Z n as well as a sphere S 3 .Moreover, we show that in contrast to the higher dimensional Majumdar-Papapertrou multiblack hole and multi-BMPV black hole spacetimes, the metric is smooth on each horizon even if the horizon topology is spherical.
We give a complex two-dimensional noncommutative locally symmetric Kähler manifold via a deformation quantization with separation of variables. We present an explicit formula of its star product by solving the system of recurrence relations given by Hara–Sako. In the two-dimensional case, this system of recurrence relations gives two types of equations corresponding to the two coordinates. From the two types of recurrence relations, symmetrized and antisymmetrized recurrence relations are obtained. The symmetrized one gives the solution of the recurrence relation. From the antisymmetrized one, the identities satisfied by the solution are obtained. The star products for [Formula: see text] and [Formula: see text] are constructed by the method obtained in this study, and we verify that these star products satisfy the identities.
A construction methods of noncommutative locally symmetric K\"ahler manifolds via a deformation quantization with separation of variables was proposed by Sako-Suzuki-Umetsu and Hara-Sako. This construction gives the recurrence relations to determine the star product. These recurrence relations were solved for the case of the arbitrary one-dimensional ones, $N$-dimensional complex space, complex projective space and complex hyperbolic space. For any two-dimensional case, authors found the solution of the recurrence relations. In this paper, we review the solution and make the star product for two-dimensional complex projective space as a concrete example of this solution.
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