Abstract
Thetranslation Chern-Simons type three-formcoframe∧torsion on a Riemann-Cartan spacetime is related (by differentiation) to the Nieh-Yan fourform. Following Chandia and Zanelli, two spaces with nontrivial translational Chern-Simons forms are discussed. We then demonstrate, first within the classical Einstein-Cartan-Dirac theory and second in the quantum heat kernel approach to the Dirac operator, how the Nieh-Yan form surfaces in both contexts, in contrast to what has been assumed previously.
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References
Y. Dothan, M. Gell-Mann, and Y. Ne'eman, “Series of hadron energy levels as representations of noncompact groups,”Phys. Lett. 17, 148–151 (1965).
Y. Ne'eman, “Gravitational interaction of hadrons: Band-spinor representations ofGL(n, R),”Proc. Natl. Acad. Sci. (USA) 74, 4157–4159 (1977).
L. C. Biedenharn, L. C. Cusson, R. Y. Han, and O. L. Weaver, “Hadronic Regge sequences as primitive realizations ofSL(3, R) symmetry,”Phys. Lett. B 42, 257–260 (1972).
Dj. Šijački, “The unitary irreducible representations of\(\overline {SL} \)(3,R),”J. Math. Phys. 16, 298–311 (1975).
F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne'eman, “Metric-affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance,”Phys. Rep. 258, 1–171 (1995).
O. Chandia and J. Zanelli, “Topological invariants, instantons and chiral anomaly on spaces with torsion,”Phys. Rev. D 55, 7580–7585 (1997), J. Zanelli, Seminar given at Cologne University on 31 December 1996.
A. Mardones and J. Zanelli, “Lovelock-Cartan theory of gravity,”Class. Quantum Gravit. 8, 1545–1558 (1991).
F. W. Hehl, W. Kopczyński, J. D. McCrea, and E. W. Mielke, “Chern-Simons terms in metric-affine spacetime: Bianchi identities as Euler-Lagrange equations,”J. Math. Phys. 32, 2169–2180 (1991).
E. W. Mielke,Geometrodynamics of Gauge Fields—On the Geometry of Yang-Mills and Gravitational Gauge Theories (Akademie-Verlag, Berlin 1987).
Y. N. Obukhov and F. W. Hehl, “On the relation between quadratic and linear curvature Langrangians in Poincaré gauge gravity,”Acta Phys. Pol. B 27, 2685–2693 (1996).
H. T. Nieh and M. L. Yan, “An identity in Riemann-Cartan geometry,”J. Math. Phys. 23, 373–374 (1982).
E. W. Mielke, “Ashtekar's complex variables in general relativity and its teleparallelism equivalent,”Ann. Phys. (N.Y.) 219, 78–108 (1992).
A. Trautman, Private communication to FWH (1995).
J. Budczies, “Dirac operators on Riemann-Cartan spaces of arbitrary dimension” (in German), Diploma thesis, University of Cologne, June 1996.
F. Englert, M. Rooman, and P. Spindel, “Supersymmetry breaking by torsion and the Ricci-flat squashed seven-spheres,”Phys. Lett. B 127, 47–50 (1983).
E. W. Mielke, P. Baekler, F. W. Hehl, A. Macías, and H. A. Morales-Técotl, “Yang-Mills-Clifford form of the chiral Einstein action,” inGravity, Particles and Space-Time, P. Pronin and G. Sardanashvily, eds. (World Scientific, Singapore, 1996), pp. 217–254.
E. W. Mielke, A. Macías, and H. A. Morales-Técotl, “Chiral fermions coupled to chiral gravity,”Phys. Lett. A 215, 14–20 (1996).
Yu. N. Obukhov, “Spectral geometry of the Riemann-Cartan space-time,”Nucl. Phys. B 12, 237–254 (1983).
I. L. Buchbinder, S. D. Odintsov, and L. L. Shapiro,Effective Action in Quantum Gravity, (IOP Publishing, Bristol, 1992); see, in particular, Chap. 4 on “The renormalization group method in curved space-time with torsion”.
S. Yajima, “Evaluation of the heat kernel in Riemann-Cartan space,”Class. Quantum Gravit. 13, 2423–2435 (1996).
C. Wiesendanger, “Poincaré gauge invariance and gravitation in Minkowski spacetime,”Class. Quantum Gravit. 13, 681–699 (1996).
M. Atiyah, “The Dirac equation and geometry,” Prep. Trinity College, Cambridge, 1996.
E. W. Mielke and D. Kreimer, “Chiral anomaly in contorted spacetimes,” Los Alamos e-print Archive gr-qc/9704051.
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Obukhov, Y.N., Mielke, E.W., Budczies, J. et al. On the chiral anomaly in non-Riemannian spacetimes. Found Phys 27, 1221–1236 (1997). https://doi.org/10.1007/BF02551525
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DOI: https://doi.org/10.1007/BF02551525