Dirac Operators and Spectral Geometry (Cambridge Lecture by Giampiero Esposito

By Giampiero Esposito

The Dirac operator has many beneficial functions in theoretical physics and arithmetic. This booklet offers a transparent, concise and self-contained creation to the worldwide conception of the Dirac operator and to the research of spectral asymptotics with neighborhood or nonlocal boundary stipulations. the speculation is brought at a degree appropriate for graduate scholars. a variety of examples are then given to demonstrate the abnormal houses of the Dirac operator, and the position of boundary stipulations in heat-kernel asymptotics and quantum box thought. themes coated comprise the creation of spin-structures in Riemannian and Lorentzian manifolds; functions of index thought; heat-kernel asymptotics for operators of Laplace style; quark boundary stipulations; one-loop quantum cosmology; conformally covariant operators; and the position of the Dirac operator in a few contemporary investigations of four-manifolds. This quantity offers graduate scholars with a rigorous advent and researchers with a worthy connection with the Dirac operator and its purposes in theoretical physics.

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Piazza 1991, 1993). 6 Pseudo-differential operators Many recent developments in operator theory and spectral asymptotics deal with pseudo-differential operators. e. their parametrices) in the elliptic case, integral and integro-differential operators, including, in particular, the singular integral operators. 1 for the equation 6u = h. Both the operators (6 + 1)-1 and Q are of order 2, but the formalism is so general that one can define operators of any real order. An example presented in Grubb (1996) is the operator (6 + 1)-S, defined in L2(~n) with the help of spectral theory, which is a pseudo-differential operator of order 2s, for any s E ~.

D + d*)2, and the the solutions of (d + d*)u = O. If we harmonic forms are also consider d + d* as an operator nev -+ no dd , its null-space is EB1i 2q , while the null-space of its adjoint is EB1i 2q + l • Hence its index is the Euler characteristic of M. It is now appropriate to define elliptic differential operators and their index. Let us denote again by M a compact oriented smooth manifold, and by E, F two smooth complex vector bundles over M. e. linear operators defined on the spaces of smooth sections and expressible locally by a matrix of partial derivatives.

13) 48 Index problems then S'PE is another state with the same energy. Thus, if 'PE is a bosonic state, S'PE is fermionic, and the other way around, and non-zero energy states in the spectrum appear in Fermi-Bose pairs. This in turn implies that Tr( _1)F e-,6H receives contributions only from zero-energy states. Further to this, the work in Witten (1982) has shown that Tr( _1)F e-,6H is a topological invariant of the quantum theory. 15) Q*'PF = 0, one finds that Index(Q) = dim Ker Q - dim Ker Q* = Tr [(_I)F e-,6H].

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