Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 105, 14 pages      arXiv:1105.3935      http://dx.doi.org/10.3842/SIGMA.2011.105

Dolbeault Complex on S4\{·} and S6\{·} through Supersymmetric Glasses

Andrei V. Smilga
SUBATECH, Université de Nantes, 4 rue Alfred Kastler, BP 20722, Nantes 44307, France

Received June 22, 2011, in final form November 09, 2011; Published online November 15, 2011

Abstract
S4 is not a complex manifold, but it is sufficient to remove one point to make it complex. Using supersymmetry methods, we show that the Dolbeault complex (involving the holomorphic exterior derivative ∂ and its Hermitian conjugate) can be perfectly well defined in this case. We calculate the spectrum of the Dolbeault Laplacian. It involves 3 bosonic zero modes such that the Dolbeault index on S4\{·} is equal to 3.

Key words: Dolbeault; supersymmetry.

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References

  1. Eguchi T., Gilkey P.B., Hanson A.J., Gravitation, gauge theories and differemtial geometry, Phys. Rep. 66 (1980), 213-393.
  2. Ivanov E.A., Smilga A.V., Dirac operator on complex manifolds and supersymmetric quantum mechanics, arXiv:1012.2069.
  3. Hull C.M., The geometry of supersymmetric quantum mechanics, hep-th/9910028.
  4. Smilga A.V., How to quantize supersymmetric theories, Nuclear Phys. B 292 (1987), 363-380.
  5. Smilga A.V., Supersymmetric proof of the Hirzebruch-Riemann-Roch theorem for non-Kähler manifolds, arXiv:1109.2867.
  6. Bismut J.-M., A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), 681-699.
  7. Smilga A.V., Non-integer flux: why it does not work, arXiv:1104.3986.
  8. Shifman M.A., Smilga A.V., Vainshtein A.I., On the Hilbert space of supersymmetric quantum systems, Nuclear Phys. B 299 (1988), 79-90.
  9. Wu T.T., Yang C.N., Dirac monopole without strings: monopole harmonics, Nuclear Phys. B 107 (1976), 365-380.
  10. Konyushikhin M.A., Smilga A.V., Self-duality and supersymmetry, Phys. Lett. B 689 (2010), 95-100, arXiv:0910.5162.
    Ivanov E.A., Konyushikhin M.A., Smilga A.V., SQM with nonabelian self-dual fields: harmonic superspace description, J. High Energy Phys. 2010 (2010), no. 5, 033, 13 pages, arXiv:0912.3289.
  11. Howe P.S., Papadopoulos G., Ultra-violet behaviour of two-dimensional supersymmetric non-linear σ-models, Nuclear Phys. B 289 (1987), 264-276.
    Howe P.S., Papadopoulos G., Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B 379 (1996), 80-86, hep-th/9602108.
    Verbitsky M., Hyperkähler manifolds with torsion, supersymmetry and Hodge theory, Asian J. Math. 6 (2002), 679-712, math.AG/0112215.
  12. Delduc F., Ivanov E.A., N=4 mechanics of general (4,4,0) multiplets, arXiv:1107.1429.
  13. Hirzebruch F., Arithmetic genera and the theorem of Riemann-Roch for algebraic varietes, Proc. Nat. Acad. Sci. USA 40 (1954), 110-114.
    Hirzebruch F., Topological methods in algebraic geometry, Springer-Verlag, Berlin, 1978.
    Atiyah M.F., Singer I.M., The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484-530.
    Atiyah M.F., Singer I.M., The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546-604.
    Atiyah M.F., Singer I.M., The index of elliptic operators. IV, Ann. of Math. (2) 93 (1971), 119-138.
    Atiyah M.F., Singer I.M., The index of elliptic operators. V, Ann. of Math. (2) 93 (1971), 139-149.
  14. Alvarez-Gaumé L., Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. 90 (1983), 161-173.
    Friedan D., Windey P., Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly, Nuclear Phys. B 235 (1984), 395-416.
    Windey P., Supersymmetric quantum mechanics and the Atiyah-Singer index theorem, Acta Phys. Polon. B 15 (1984), 435-452.
  15. Cecotti S., Girardello L., Functional measure, topology and dynamical supersymmetry breaking, Phys. Lett. B 110 (1982), 39-43.
    Girardello L., Imbimbo C., Mukhi S., On constant configurations and evaluation of the Witten index, Phys. Lett. B 132 (1983), 69-74.
  16. Obukhov Y.N., Spectral geometry of the Riemann-Cartan space-time, Nuclear Phys. B 212 (1983), 237-254.
    Peeters K., Waldron A., Spinors on manifolds with boundary: APS index theorem with torsion, J. High Energy Phys. 1999 (1999), no. 2, 024, 42 pages, hep-th/9901016.
  17. Blok B.Yu., Smilga A.V., Effective zero-mode Hamiltonian in supersymmetric chiral nonabelian gauge theories, Nuclear Phys. B 287 (1987), 589-600.

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