Abstract
We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abanov, A.G.: Hopf term induced by fermions. Phys.Lett. B492, 321–323 (2000)
Abanov, A.G., Wiegmann, P.B.: Theta-terms in nonlinear sigma-models. Nucl.Phys. B570, 685–698 (2000)
Agranovich, M.S.: Elliptic operators on closed manifolds. Current problems in mathematics. Fundamental directions, Vol. 63 (Russian), Itogi Nauki i Tekhniki, Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 1990, pp. 5–129
Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Berlin-Heidelberg-New York: Springer-Verlag, 1992
Berry, M.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)
Burghelea, D., Friedlander, L., Kappeler, T.: On the determinant of elliptic differential and finite difference operators in vector bundles over S1. Comm. Math. Phys. 138(1), 1–18 (1991)
Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer-Verlag, 1966
Lidskii, V.B.: Non-selfadjoint operators with a trace. Dokl. Akad. Nauk SSSR 125, 485–487 (1959)
Markus, A.S.: Introduction to the spectral theory of polynomial operator pencils. Translations of Mathematical Monographs, Vol. 71, Providences RI: Amer. Math, Soc. 1998
Ponge, R.: Spectral asymetry, zeta function and the noncommutative residue. Preprint, to appear in J. Funct. Anal.
Redlich, A.N.: Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions. Phys. Rev. Lett. 52, 18–21 (1984)
Redlich, A.N.: Parity violation and gauge noninvariance of the effective gauge field action in three dimensions. Phys. Rev. D 29, 2366–2374 (1984)
Retherford, J.R.: Hilbert space: compact operators and the trace theorem. London Mathematical Society Student Texts, Vol. 27, Cambridge: Cambridge University Press, 1993
Seeley, R.: Complex powers of elliptic operators. Proc. Symp. Pure and Appl. Math. AMS 10, 288–307 (1967)
Shubin, M.A.: Pseudodifferential operators and spectral theory. Berlin, New York: Springer Verlag, 1980
Shubin, M.A.: Semiclassical asymptotics on covering manifolds and Morse inequalities. Geom. Funct. Anal. 6, 370–409 (1996)
Witten, E.: Supersymmetry and Morse theory. J. of Diff. Geom. 17, 661–692 (1982)
Wodzicki, M.: Local invariants of spectral asymmetry. Invent. Math. 75(1), 143–177 (1984)
Wodzicki, M.: Noncommutative residue. I. Fundamentals. K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., Vol. 1289, Berlin: Springer, 1987 pp. 320–399
Wojciechowski, K.P.: Heat equation and spectral geometry. Introduction for beginners. Geometric methods for quantum field theory (Villa de Leyva, 1999), River Edge, NJ: World Sci. Publishing, 2001, pp. 238–292
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Sarnak
The first author was partially supported by the Alfred P. Sloan foundation.
The second author was partially supported by the NSF grant DMS-0204421.
Rights and permissions
About this article
Cite this article
Abanov, A., Braverman, M. Topological Calculation of the Phase of the Determinant of a Non Self-Adjoint Elliptic Operator. Commun. Math. Phys. 259, 287–305 (2005). https://doi.org/10.1007/s00220-005-1394-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1394-6