By J.J. Duistermaat
When traveling M.I.T. for 2 weeks in October 1994, Victor Guillemin made me obsessed with an issue in symplectic geometry which concerned using the so-called spin-c Dirac operator. again in Berkeley, the place I had l spent a sabbatical semester , i attempted to appreciate the elemental evidence approximately this operator: its definition, the most theorems approximately it, and their proofs. This booklet is an outgrowth of the notes during which I labored this out. For me this was once an outstanding studying adventure a result of many appealing mathematical buildings that are concerned. I thank the Editorial Board of Birkhauser, specifically Haim Brezis, for sug gesting the ebook of those notes as a e-book. i'm additionally very thankful for the feedback via the referees, that have resulted in great advancements within the presentation. eventually i need to precise targeted because of Ann Kostant for her support and her prodding me, in her captivating manner, into the best path. J.J. Duistermaat Utrecht, October sixteen, 1995.
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Additional info for The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator
11) shows that r = ±1. 6) is a double covering. Now choose a U(l)-invariant connection in the principal U(l)-bundle U (K*). ) Together with the Levi-Civita connection in SOF M, we get a connection in the product bundle, which then is pulled back by /::). and 'Y to a connection in the spinc frame bundle. That is, the horizontal space in the tangent space at p of Spinc F M consists of the vectors v, such that Tp a( v) is horizontal in the tangent space at a(p) of SOF M, and Tp (3( v) is horizontal in U (K*).
14) This induces an algebra homomorphism CA : C(V) --+ End (AV). • , Vk = VI t\ V2 t\ ... t\ E V: Vk modulo E9 A k - 2l V. 16) I~I It follows that CIA is a linear isomorphism, and preserves the filtration and the grading modulo 2. 2. The Clifford Algebra If a±, b± E C±(V), then one defines the supercommutator of a = a+ and b = b+ + b- by [a, bl s := [a+, b+] + [a+, b-] + [a-, b+] + a- b- + b- a-. 17) If v E V, then the operator ads(v) : a I---t [v, als in C(V) is a superderivation in the sense that ads (v) (a .
2. 4 Choose any U(1)-invariant connection in U (K*), the unit cir- cle bundle in the dual K* of the canonical line bundle K of(M, J). Then the covariant differentiation 'V on E, defined by the Levi-Civita connection in SOFM and the connection in U (K*), is a Clifford connection. If L is a complex vector bundle over M, then the L-valued (0, ·)-forms are the sections of E ® L. We assume that a connection in L is chosen which leaves the Hermitian structure h L in L invariant, and denote the corresponding covariant differentiation of sections of L by 'V L .
The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator by J.J. Duistermaat