Abstract
We present a canonical construction of the determinant of an elliptic selfadjoint boundary value problem for the Dirac operatorD over an odd-dimensional manifold. For 1-dimensional manifolds we prove that this coincides with the ζ-function determinant. This is based on a result that elliptic self-adjoint boundary conditions forD are parameterized by a preferred class of unitary isomorphisms between the spaces of boundary chiral spinor fields. With respect to a decompositionS 1=X 0∪X 1, we explain how the determinant of a Dirac-type operator overS 1 is related to the determinants of the corresponding boundary value problems overX 0 andX 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atiyah, M.F.: Circular symmetry and stationary phase approximation. Colloquium in honour of Laurent Schwartz, Asterique No.131, Vol. 1, 43–59 (1989)
Atiyah, M.F.: Topological quantum field theories. Publ. Math. I.H.E.S. Paris68, 175–186 (1989)
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry: I. Math. Proc. Camb. Phil. Soc.77, 43–69 (1975)
Bismut, J.M., Cheeger, J.: Remarks on the index theorem for families of Dirac operators on a manifold with boundary. Differential Geometry. A Symposium in Honour of Manfred de Carmo. New York: Longman Scientific and Technical, (1991)
Bismut, J.M., Freed, D.S.: The analysis of elliptic families: (I) Metrics and connections on determinant bundles, (II) Dirac operators, eta invariants, and the holonomy theorem. Commun. Math. Phys.106, 159–76 (1986);107, 103–63 (1986)
Booss, B., Bleecker, D.D.: Topology and Analysis. Berlin, Heidelberg, New York: Springer, 1985
Booss-Bavnbek, B., Wojciechowski, K.P.: Pseudodifferential projections and the topology of certain spaces of elliptic boundary value problems. Commun. Math. Phys.121, 1–9 (1989)
Booss-Bavnbek, B., Wojciechowski, K.P.: Elliptic boundary value problems for Dirac operators. Basel-Boston: Birkhauser, 1993
Burghelea, D., Friedlander, L., Kappeler, T.: On the determinant of elliptic differential and finite difference operators in vector bundles overS 1. Commun. Math. Phys.138, 1–18 (1991)
Douglas, R.G., Wojciechowski, K.P.: Adiabatic limits of the η-invariants. The odd-dimensional Atiyah-Patodi-Singer problem. Commun. Math. Phys.142, 139–68 (1989)
Forman, R.: Functional determinants and geometry. Inv. Math.88, 447–493 (1987)
Gilkey, P.B.: Invariance Theory, The Heat Equation, And The Atiyah-Singer Index Theorem. Berkeley, Ca.: Publish or Perish 1984
Grubb, G., Seeley, R.T.: Preprint (1993)
Lesch, M., Wojciechowski, K.P.: On the η-invariant of generalized Atiyah-Patodi-Singer problems. Preprint (1993)
Maclane, S.: Categories for the working mathematician. Berlin, Heidelberg, New York: Springer, GTM5, 1971
Palais, R.S.: Seminar on the Atiyah-Singer Index Theorem. Princeton, NJ: Princeton Univ. Press, 1965
Pressley, A.N., Segal, G.B.: Loop Groups, Oxford: Oxford Univ. Press, 1988
Quillen, D.G.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funk. Anal. i. ego Prilozhenya19, 37–41 (1985)
Roe, J.: Elliptic Operators, Topology and Asymptotic Methods. Essex, England: Longman Scientific and Technical, 1988
Scott, S.G.: Determinants, Grassmanians and Functorial QFT. Preprint (1995)
Seeley, R.T.: Extension ofC ∞ functions in a half space. Proc. Amer. Math. Soc.15, 625–626 (1964)
Seeley, R.T.: Complex powers of elliptic operators. Proc. Symp. on Singular Integrals, AMS10, 288–307 (1976)
Seeley, R.T.: In: CIME Conference on Pseudo-Differential Operators. Stresa (1968)
Segal, G.B.: Fredholm complexes. Quart. J. Math.21, 385–402 (1970)
Segal, G.B.: The definition of conformal field theory. Preprint
Segal, G.B.: Geometric aspects of quantum field theory. Proc. Int. Cong. Math. Tokyo, 1990
Simon, B.: Trace Ideals and their Applications. London Mathematical Society Lecture Notes Vol.35, Cambridge: Camb. Univ. Press, 1979
Witten, E.: Quantum field theory, Grassmannians and strings. Commun. Math. Phys.113, 529–600 (1988)
Woodhouse, N.M.J.: Geometric Quantization. Oxford: Oxford Univ. Press, 1992
Wojciechowski, K.P.: Smooth, self-adjoint Grassmanian, the η-invariant, and ζ-determinant of boundary problems. IUPUI Preprint (1995)
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
Rights and permissions
About this article
Cite this article
Scott, S.G. Determinants of dirac boundary value problems over odd-dimensional manifolds. Commun.Math. Phys. 173, 43–76 (1995). https://doi.org/10.1007/BF02100181
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02100181