Abstract
We discuss two different applications of the theory of Grassmannians of elliptic boundary value problems to the theory of ς-function determinants of Dirac operators over a manifold with boundary. Our work is motivated by constructions of the determinant in Topological Quantum Field Theory and in Quantum Chromodynamics.
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References
Atiyah, M.F., Patodi, V.K., and Singer, I.M.: 1975, ‘Spectral asymmetry and Riemannian geometry. I’, Math. Proc. Cambridge Phil. Soc. 77, 43–69.
Atiyah, M.F., and Singer, I.M.: 1984, ‘Dirac operators coupled to vector potentials’, Proc. Nat. Acad. Sci. USA 81, 2597–2600.
Booß-Bavnbek, B., Morchio, G., Strocchi, F., and Wojciechowski, K.P.: 1996, Grassmannian and Chiral Anomaly, preprint.
Booß-Bavnbek, B., and Wojciechowski, K.P.: 1993, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston.
Hawking, S.W.: 1977, ‘Zeta function regularization of path integrals in curved space-time’, Comm. Math. Phys. 55, 133–148.
Palais, R.S. (ed.): 1965, Seminar on the Atiyah-Singer Index Theorem, Ann. of Math. Studies 57, Princeton University Press, Princeton.
Quillen, D.G.: 1985, ‘Determinants of Cauchy-Riemann operators over a Riemann surface’, Funk. Anal. i ego Prilozhenya 19, 37–41.
Ray, D., and Singer, I.M.: 1971, ‘R-torsion and the Laplacian on Riemannian manifolds’, Adv. Math. 7, 145–210.
Schmidt, J.R., and Bincer, A.M.: 1987, ‘Chiral asymmetry and the Atiyah-Patodi-Singer index for the Dirac operator on a four-dimensional ball’, Phys. Rev. D (3) 35, 3995–4000.
Scott, S.G.: 1995, ‘Determinants of Dirac boundary value problems over odd-dimensional manifolds’, Comm. Math. Phys. 173, 43–76.
Scott, S.G., and Wojciechowski, K.P.: 1996, Abstract Determinant and ζ-Determinant on the Grassmannian, preprint.
Segal, G.B.: The Definition of Conformai Field Theory, preprint.
Singer, I.M.: 1985, ‘Families of Dirac operators with applications to physics’, Asterisque, hors série, 323–340.
Trautman, A.: 1995, ‘The Dirac operator on hypersurfaces’, Acta Phys. Polon. B 26, 1283–1310.
Wojciechowski, K.P.: 1983, ‘On the spectral flow and the index theorem for flat vector bundles’, in: Partial Differential Equations, Banach Center Publications, vol. 10, PWN-Polish Scientific Publishers, Warsaw, pp. 413–422.
—: 1996, Smooth, Self-Adjoint Grassmannian, the η-Invariant, and the ς-Determinant of Boundary Problems, preprint.
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© 1998 Springer Science+Business Media Dordrecht
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Wojciechowski, K.P., Scott, S.G., Morchio, G., Booss-Bavnbek, B. (1998). Determinants, Manifolds with Boundary and Dirac Operators. In: Dietrich, V., Habetha, K., Jank, G. (eds) Clifford Algebras and Their Application in Mathematical Physics. Fundamental Theories of Physics, vol 94. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5036-1_32
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DOI: https://doi.org/10.1007/978-94-011-5036-1_32
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