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Letters in Mathematical Physics

, Volume 73, Issue 2, pp 147–163 | Cite as

Stability Theorems for Chiral Bag Boundary Conditions

  • P. Gilkey
  • K. KirstenEmail author
Article

Abstract

We study asymptotic expansions of the smeared L 2-traces Fet P^2 and FPetP^2, where P is an operator of Dirac type and F is an auxiliary smooth endomorphism. We impose chiral bag boundary conditions depending on an angle θ. Studying the θ-dependence of the above trace invariants, θ-independent pieces are identified. The associated stability theorems allow one to show the regularity of the eta function for the problem and to determine the most important heat kernel coefficient on a four dimensional manifold.

Keywords

bag boundary conditions operator of Dirac type zeta and eta invariants variational formulas 

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References

  1. 1.
    Amsterdamski P., Berkin A., O’Connor D. (1989). b 8 Hamidew coefficient for a scalar field. Class. Quantum Grav. 6: 1981–1991CrossRefzbMATHADSMathSciNetGoogle Scholar
  2. 2.
    Atiyah M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry I, II, III. Math. Proc. Cambr. Phil. Soc. 77, 43–69 (1975); 78, 405–432 (1975); 79, 71–99 (1976).Google Scholar
  3. 3.
    Avramidi I.G. (1990). The covariant technique for the calculation of the heat kernel asymptotic expansion. Phys. Lett. B. 238, 92–97CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Avramidi I.G. (1991). A covariant technique for the calculation of the one-loop effective action. Nucl. Phys. B 355: 712–754CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Beneventano C.G., Gilkey P., Kirsten K., Santangelo E.M. (2003). Strong ellipticity and spectral properties of chiral bag boundary conditions. J. Phys. A: Math. Gen. 36: 11533–11543CrossRefzbMATHADSMathSciNetGoogle Scholar
  6. 6.
    Beneventano C.G., Santangelo E.M., Wipf A. (2002). Spectral asymmetry for bag boundary conditions. J. Phys. A: Math. Gen. 35: 9343–9354CrossRefzbMATHADSMathSciNetGoogle Scholar
  7. 7.
    Botvinnik B., Gilkey P., Stolz S. (1997). The Gromov Lawson Rosenberg conjecture for groups with periodic cohomology. J. Differ. Geom. 46: 374–405zbMATHMathSciNetGoogle Scholar
  8. 8.
    Branson T., Gilkey P. (1990). The asymptotics of the Laplacian on a manifold with boundary. Comm. Partial Diff. Eqs. 15: 245–272zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Branson T., Gilkey P. (1992). Residues of the eta function for an operator of Dirac Type. J. Funct Anal. 108: 47–87CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Branson T., Gilkey P. (1992). Residues of the eta function for an operator of Dirac type with local boundary conditions. Diff. Geom. Appl. 2: 249–267CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Branson T., Gilkey P., Vassilevich D. (1998). Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary. J. Math. Physics 39: 1040–1049CrossRefzbMATHADSMathSciNetGoogle Scholar
  12. 12.
    Branson T., ø rsted B. (1986). Conformal indices of Riemannian manifolds. Compositio Math. 60: 261–293zbMATHMathSciNetGoogle Scholar
  13. 13.
    Branson T., ø rsted B. (1988). Conformal deformation and the heat operator. Indiana Univ. Math. J. 37: 83–110zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dürr S. (1999). Aspects of quasi-phase structure of the Schwinger model on a cylinder with broken chiral symmetry. Ann. Phys. 273: 1–36CrossRefzbMATHADSGoogle Scholar
  15. 15.
    Dürr S., Wipf A. (1997). Finite temperature Schwinger model with chirality breaking boundary conditions. Ann. Phys. 255: 333–361CrossRefzbMATHADSGoogle Scholar
  16. 16.
    Eguchi T., Gilkey P., Hanson A. (1978). Topological invariants and absence of an axial anomaly for a Euclidean Taub-NUT metric. Phys. Rev. D 17: 423–427CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Esposito G., Gilkey P., Kirsten K. (2005). Heat kernel coefficients for chiral bag boundary conditions. J. Phys. A: Math. Gen. 38: 2259–2276CrossRefzbMATHADSMathSciNetGoogle Scholar
  18. 18.
    Esposito G., Kirsten K. (2002). Chiral bag boundary conditions on the ball. Phys. Rev. D 66: 085014CrossRefADSGoogle Scholar
  19. 19.
    Gilkey P. (1975). The boundary integrand in the formula for the signature and Euler characteristic of a Riemannian manifold with boundary. Adv. Math. 15: 334–360CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Gilkey P. (1985). The eta invariant of even dimensional Pin c manifolds. Adv. Math. 58: 243–284CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Gilkey P. (1987). The eta invariant and non-singular bilinear products on \(\mathbb{R}^n\). Canad. Math. Bull. 30: 147–154zbMATHMathSciNetGoogle Scholar
  22. 22.
    Gilkey P. (1995). Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem. Chapman&Hall/CRC, Boca Raton FLzbMATHGoogle Scholar
  23. 23.
    Gilkey P. (2004). Asymptotic Formulae in Spectral Geometry. Chapman&Hall/CRC, Boca Raton FLzbMATHGoogle Scholar
  24. 24.
    Gilkey P., Leahy J.V., Park J.H. (1999). Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture. Chapman&Hall/CRC, Boca Raton FLzbMATHGoogle Scholar
  25. 25.
    Gilkey P., Smith L. (1983). The twisted index problem for manifolds with boundary. J. Diff. Geo. 18: 393–444zbMATHMathSciNetGoogle Scholar
  26. 26.
    Gilkey P., Smith L. (1983). The eta invariant for a class of elliptic boundary value problems. Comm. Pure Appl. Math. XXXVI: 85–131CrossRefMathSciNetGoogle Scholar
  27. 27.
    Goldstone J., Jaffe R.L. (1983). Baryon number in chiral bag models. Phys. Rev. Lett. 51: 1518–1521CrossRefADSGoogle Scholar
  28. 28.
    Greiner P. (1971). An asymptotic expansion for the heat equation. Arch. Rational Mech. Anal. 41: 163–218CrossRefzbMATHADSMathSciNetGoogle Scholar
  29. 29.
    Hrasko P., Balog J. (1984). The fermion boundary condition and the theta angle in QED in two-dimensions. Nucl. Phys. B245: 118–126CrossRefADSGoogle Scholar
  30. 30.
    Kirsten K. (2001). Spectral Functions in Mathematics and Physics. Chapman&Hall/CRC, Boca Raton FLGoogle Scholar
  31. 31.
    Marachevsky V., Vassilevich D. (2004). Chiral anomaly for local boundary conditions. Nucl. Phys. B. 667: 535–552CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Seeley, R.T.: Complex powers of an elliptic operator, singular integrals, Chicago 1966, Proc. Sympos. Pure Math. 10, 288–307 American Mathematics Society, Providence, RI (1968)Google Scholar
  33. 33.
    Seeley, R.T.: Analytic extension of the trace associated with elliptic boundary problems. Amer. J. Math. 91, 963–983 (1969)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Vassilevich, D.V.: Vector fields on a disk with mixed boundary conditions. J. Math. Phys. 36, 3174–3182 (1995)CrossRefzbMATHADSMathSciNetGoogle Scholar
  35. 35.
    Vassilevich, D.V.: Heat kernel expansion: User’s manual. Phys. Rept. 388, 279–360 (2003)CrossRefzbMATHADSMathSciNetGoogle Scholar
  36. 36.
    Wipf, A., Dürr, S.: Gauge theories in a bag. Nucl. Phys. B443, 201–232 (1995)CrossRefADSGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA
  2. 2.Department of MathematicsBaylor UniversityWacoUSA

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