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A natural Lie super-algebra bundle on rank 3 WSD manifolds. (English) Zbl 1158.81023
Summary: We prove the existence of a natural \(*\)-Lie super-algebra bundle on any orientable WSD manifold of rank 3. We describe in detail the associated Lie super-algebra \(\mathcal L_{3,\mathbb C}\) of global sections. We show that \(\mathcal L_{3,\mathbb C}\) is a product of \(\mathbf{sl}(4,\mathbb C)\) with the full special linear super-algebras of some graded vector spaces isotypical with respect to a natural action of \(\mathbf{so}(3,\mathbb R)\). We give an explicit description of a geometrically natural real form of \(\mathcal L_{3,\mathbb C}\). This real form is made up of \(\mathbf{so}(3,\mathbb R)\)-invariant operators which preserve the Poincaré pairing on the bundle of forms.

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
58C50 Analysis on supermanifolds or graded manifolds
17A70 Superalgebras
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
81T60 Supersymmetric field theories in quantum mechanics
Full Text: DOI arXiv
[1] M. Grassi, Mirror symmetry and self-dual manifolds, math.DG/0202016. 2002
[2] Grassi, M., Self-dual manifolds and mirror symmetry for the quintic threefold, Asian J. math, 9, 79-102, (2005) · Zbl 1085.14035
[3] Strominger, A.; Yau, S.T.; Zaslow, E., Mirror symmetry is T-duality, Nuclear phys., B479, 243-259, (1996), hep-th/9606040 · Zbl 0896.14024
[4] M. Grassi, Polysymplectic spaces, \(s\)-Kähler manifolds and lagrangian fibrations, math.DG/0006154. 2000
[5] G. Gaiffi, M. Grassi, A geometric realization of \(\mathbf{sl}(6, \mathbb{C})\), Rend. Sem. Mat. Padova (in press). http://rendiconti.math.unipd.it/forthcoming.php?lan=english
[6] Carmeli, C.; Cassinelli, G.; Toigo, A.; Varadarajan, V.S., Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles, Commun. math. phys., 263, 217-258, (2006) · Zbl 1124.22007
[7] Varadarajan, V.S., Supersymmetry for mathematicians: an introduction, (2004), AMS, Courant L.N. 11 · Zbl 1142.58009
[8] Batyrev, V., Dual polyhedra and mirror symmetry for calabi – yau hypersurfaces in toric varieties, J. algebraic geom., 3, 493-535, (1994) · Zbl 0829.14023
[9] Bruzzo, U.; Marelli, G.; Pioli, F., A Fourier transform for sheaves on real tori part II. relative theory, J. geom. phys., 41, 312-329, (2002) · Zbl 1071.14515
[10] Candelas, P.; De la Ossa, X.C.; Green, P.S.; Parkes, L., A pair of calabi – yau manifolds as an exactly soluble superconformal theory, Nuclear phys., B359, 21-74, (1991) · Zbl 1098.32506
[11] Greene, B.R.; Plesser, M.R., Duality in calabi – yau moduli space, Nuclear phys., B338, 15-37, (1990)
[12] Greene, B.R.; Vafa, C.; Warner, N.P., Calabi – yau manifolds and renormalization group flows, Nuclear phys., B324, 371-390, (1989) · Zbl 0744.53044
[13] Gromov, M., Metric structures for riemannian and non-Riemannian spaces, vol. 152, (1999), Birkhäuser P.M. Boston
[14] Gross, M.; Wilson, P.M.H., Large complex structure limits of \(K 3\) surfaces, J. differential. geom., 55, 475-546, (2000) · Zbl 1027.32021
[15] Guillemin, V., Moment maps and combinatorial invariants of Hamiltonian \(\mathbb{T}^n\)-spaces, vol. 122, (1994), Birkhäuser P.M. · Zbl 0828.58001
[16] A. McInroy, Orbifold mirror symmetry for complex tori, preprint
[17] Kontsevich, M.; Soibelman, Y., Homological mirror symmetry and torus fibrations, (), 203-263 · Zbl 1072.14046
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