Explicit Functional Determinants in Four Dimensions

Working on the four-sphere S4, a flat four-torus, S x S, or a compact hyperbolic space, with a metric which is an arbitrary positive function times the standard one, we give explicit formulas for the functional determinants of the conformai Laplacian (Yamabe operator) and the square of the Dirac operator, and discuss qualitative features of the resulting variational problems. Our analysis actually applies in the conformai class of any Riemannian, locally symmetric, Einstein metric on a compact 4-manifold; and to any geometric differential operator which has positive definite leading symbol, and is a positive integral power of a conformally covariant operator. Introduction. Functional determinants in two dimensionsLet (M, g) be an -dimensional compact manifold without boundary, and let A be a formally selfadjoint, geometric partial differential operator with positive definite leading symbol. The order of A is then necessarily a positive even integer 21. We further assume that A scales as its leading term does: if 2 --It g = c g, 0 0, has a small-time asymptotic expansion oo TrL2exp (-M) ~ T t (2i~n) '2ta\, 1 0, M) in which the coefficients a A are integrals of universal local expressions U]. (Addition of universal exact divergences to the Ui does not change the ai, but the Ui are fixed by the requirement that they appear in the pointwise asymptotic expansion of the fiberwise trace of the heat kernel. )