Bianchi type A hyper-symplectic and hyper-K\"ahler metrics in 4D

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de Andrés, Luis C.; Fernández, Marisa; Ivanov, Stefan; Santisteban, José A.; Ugarte, Luis; Vassilev, Dimiter;
  • Subject: High Energy Physics - Theory | Mathematics - Differential Geometry
    arxiv: Mathematics::Differential Geometry | Mathematics::Complex Variables | Mathematics::Symplectic Geometry

We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper... View more
  • References (45)
    45 references, page 1 of 5

    2.3. The Heisenberg group H3, Bianchi type II, Gibbons-Hawking class. Consider the two-step nilpotent Heisenberg group H3 defined by the structure equations 3.1. Bianchi type IX hyper-symplectic metrics and hyper-K¨ahler Bianchi type V III hyperK¨ahler metrics. Let G = SU (2) = S3 be described by the structure equations (2.10). We evolve the SU (2) structure according to (2.8). Using (2.10), we reduce the evolution equations (3.4) to the already considered system (2.20). This establishes a correspondence between triaxial Bianchi type IX hyper-symplectic metrics and triaxial hyperK¨ahler Bianchi type V III hyper-K¨ahler metrics. The general solution is given by (2.21). Taking f = f1f2f3 in (2.21) and all fi different we obtain explicit expression of a triaxial hyper-symplectic metric (3.6), where the forms e1, e2, e3 and the functions f1, f2, f3, f are given by (2.11) and (2.21), respectively. A particular solution is obtained by letting a1 = a2 = 0, a3 = 1a6 in (2.21) which gives

    [1] Andrada, A., & Dotti, I., Double Products and Hypersymplectic Structures on R4n, Commun. Math. Phys. 262 (2006) 1-16.

    [2] Atiyah, M.F., & Hitchin, N., The geometry and dinamics of magnetic monopoles, M.B. Porter Lectures, Rice University, Princeton University Press, Princeton, New-York, 1988.

    [3] Barberis, M.L., Hyper-K¨ahler metrics conformal to left invariant metrics on four-dimensional Lie groups, Math. Phys. Anal. Geom. 6 (2003), no. 1, 1-8.

    [4] Bartocci, C., & Mencattini, I., Hyper-symplectic structures on integrable systems, J. Geom. Phys. 50 (2004), 339-344.

    [5] Belinskii, V.A., Gibbons, G.W., Page, D. N., & Pope, C. N., Asymptotically Euclidean Bianchi IX metrics in quantum gravity, Phys. Lett. B 76 (1978), 433-435.

    [6] Blaˇziˆc, N., & Vukmiroviˆc, S., Para-hypercomplex structures on a four-dimensional Lie group, Contemporary geometry and related topics, River Edge, N.J. 2004, World Sci. Publishing, pp. 41-56.

    [7] Bourliot, F., Estes, J., Petropoulos, P.M., & Spindel, Ph., Gravitational instantons, self-duality, and geometric flows, Phys. Rev. D 81 (2010), 104001 (5 pp).

    [8] Bourliot, F., Estes, J., Petropoulos, P.M., & Spindel, Ph., G3-homogeneous gravitational instantons, Class. Quantum Grav. 27 (2010), 102001 (9 pp).

    [9] Calabi, E., M´etriques k¨ahl´eriennes et fibr´es holomorphes. Ann. Sci. E´cole Norm. Sup. (4) 12 (1979), no.2, 269-294.

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