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


SIGMA 13 (2017), 027, 28 pages      arXiv:1609.01509      https://doi.org/10.3842/SIGMA.2017.027

Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds

Ana Lucia Garcia-Pulido and Rafael Herrera
Centro de Investigación en Matemáticas, A. P. 402, Guanajuato, Gto., C.P. 36000, México

Received October 10, 2016, in final form April 19, 2017; Published online April 23, 2017

Abstract
We prove the rigidity and vanishing of several indices of ''geometrically natural'' twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.

Key words: almost even-Clifford Hermitian manifolds; index of elliptic operator; twisted Dirac operators; circle action by automorphisms.

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References

  1. Arizmendi G., Garcia-Pulido A.L., Herrera R., A note on the geometry and topology of almost even-Clifford Hermitian manifolds, arXiv:1606.00774.
  2. Arizmendi G., Herrera R., Centralizers of spin subalgebras, J. Geom. Phys. 97 (2015), 77-92, arXiv:1503.06168.
  3. Atiyah M.F., Hirzebruch F., Spin-manifolds and group actions, in Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, 18-28.
  4. Atiyah M.F., Singer I.M., The index of elliptic operators. III, Ann. of Math. 87 (1968), 546-604.
  5. Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004.
  6. Bott R., Taubes C., On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989), 137-186.
  7. Bröcker T., tom Dieck T., Representations of compact Lie groups, Graduate Texts in Mathematics, Vol. 98, Springer-Verlag, New York, 1995.
  8. Dessai A., Rigidity theorems for ${\rm Spin}^{\bf C}$-manifolds, Topology 39 (2000), 239-258.
  9. Friedrich T., Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, Vol. 25, Amer. Math. Soc., Providence, RI, 2000.
  10. Friedrich T., Weak Spin(9)-structures on 16-dimensional Riemannian manifolds, Asian J. Math. 5 (2001), 129-160, math.DG/9912112.
  11. Hattori A., ${\rm Spin}^{c}$-structures and $S^{1}$-actions, Invent. Math. 48 (1978), 7-31.
  12. Herrera H., Herrera R., Rigidity and vanishing theorems for almost quaternionic manifolds, Geom. Dedicata 134 (2008), 139-152.
  13. Hirzebruch F., Elliptic genera of level $N$ for complex manifolds, in Differential Geometrical Methods in Theoretical Physics (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 250, Kluwer Acad. Publ., Dordrecht, 1988, 37-63.
  14. Hirzebruch F., Berger T., Jung R., Manifolds and modular forms, Aspects of Mathematics, Vol. E20, Friedr. Vieweg & Sohn, Braunschweig, 1992.
  15. Hirzebruch F., Slodowy P., Elliptic genera, involutions, and homogeneous spin manifolds, Geom. Dedicata 35 (1990), 309-343.
  16. LeBrun C., Salamon S., Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118 (1994), 109-132.
  17. Lichnerowicz A., Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9.
  18. Liu K., On modular invariance and rigidity theorems, J. Differential Geom. 41 (1995), 343-396.
  19. Moroianu A., Semmelmann U., Clifford structure on Riemannian manifolds, Adv. Math. 228 (2011), 940-967, arXiv:0912.4207.
  20. Roe J., Elliptic operators, topology and asymptotic methods, Pitman Research Notes in Mathematics Series, Vol. 395, 2nd ed., Longman, Harlow, 1998.
  21. Salamon S., Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics Series, Vol. 201, Longman, Harlow, 1989.
  22. Taubes C.H., $S^1$ actions and elliptic genera, Comm. Math. Phys. 122 (1989), 455-526.
  23. Witten E., Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987), 525-536.
  24. Witten E., The index of the Dirac operator in loop space, in Elliptic Curves and Modular Forms in Algebraic Topology (Princeton, NJ, 1986), Lecture Notes in Math., Vol. 1326, Springer, Berlin, 1988, 161-181.

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