Conformal structure and spin geometry. (English) Zbl 1109.53051

Branson, Thomas

Bourguignon, Jean-Pierre (ed.) et al., Dirac operators: yesterday and today. Proceedings of the summer school and workshop, Beirut, Lebanon, August 27–September 7, 2001. Somerville, MA: International Press (ISBN 1-57146-175-2/hbk). 163-191 (2005).

Author’s abstract: The study of the Dirac equation on manifolds is grounded in the need to understand physical fields on curved space-times. It is generally believed that the study of Riemannian (positive definite) metrics, or infinitesimal distance functions, will ultimately be relevant to the more directly physical problem of Lorentz signature metrics, via principles of analytic continuation in signature. This adds impetus to the natural mathematical pursuit of studying bundles with spin structure, the Dirac operator, and other related operators on Riemannian manifolds, or on manifolds with a positive definite conformal structure.
In these notes, we attempt to introduce this subject, with an eye toward fundamental issues that are likely to be important in future work. Among these are Stein-Weiss gradients [see E. Stein and G. Weiss, Am. J. Math. 90, 163–196 (1968; Zbl 0157.18303)], Bochner-Weitzenböck formulas [see the author, J. Funct. Anal. 151, 334–383 (1997; Zbl 0904.58054)]; [the author and O. Hijazi, Int. J. Math. 13, 137–182 (2002; Zbl 1109.53306)], the Hijazi inequality [see O. Hijazi, Commun. Math. Phys. 104, 151–162 (1986; Zbl 0593.58040)], and the Penrose local twistor ideas [see R. Penrose and W. Rindler, Spinors and Space-Time, vol. 1 and 2, Cambridge Monographs on Mathematical Physics. (Cambridge) etc.: Cambridge University Press. (1984; Zbl 0538.53024) and (1986; Zbl 0591.53002)]; [T. N. Bailey, M. G. Eastwood, and A. R. Gover, Rocky Mt. J. Math. 24, No. 4, 1191–1217 (1994; Zbl 0828.53012)]; [S. Paneitz, I. E. Segal, and D. Vogan, J. Funct. Anal. 75, 1–57 (1987; Zbl 0682.58022)]; [B. Ørsted and I. E. Segal, J. Funct. Anal. 83, 150–184 (1989; Zbl 0704.55011)], leading to the general theory of tractor bundles see the author and A. R. Gover, Spin-tractors, in preparation]. All of these ideas are intimately connected with the behavior of differential operators under conformal change of metric, or from another viewpoint, operators that need only a conformal structure (and not a metric) to be well-defined.
For the entire collection see [Zbl 1089.53003].

Reviewer: Mircea Craioveanu (Timişoara)

MSC:

53C27	Spin and Spin\({}^c\) geometry
53C28	Twistor methods in differential geometry
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

spin geometry; conformal structure; the Clifford relation; Stein-Weiss gradient; Bochner-Weitzenböck formula; discrete leading symbol; Hijazi inequality; local twistor; spinnor; tractor constructions

Citations:

Zbl 0157.18303; Zbl 0904.58054; Zbl 0593.58040; Zbl 0538.53024; Zbl 0591.53002; Zbl 0828.53012; Zbl 0682.58022; Zbl 0704.55011; Zbl 1109.53306