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The Decomposition of Global Conformal Invariants: Some Technical Proofs. I
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern–Gauss–Bonnet integrand.
Keywords: conormal geometry; renormalized volume; global invariants; Deser–Schwimmer conjecture. 
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Spyros Alexakis. The Decomposition of Global Conformal Invariants: Some Technical Proofs. I. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a18/

[1] Alexakis S., “On the decomposition of global conformal invariants. I”, Ann. of Math. (2), 170 (2009), 1241–1306, arXiv: 0711.1685 | DOI | MR | Zbl

[2] Alexakis S., “On the decomposition of global conformal invariants. II”, Adv. Math., 206 (2006), 466–502, arXiv: 0912.3755 | DOI | MR | Zbl

[3] Alexakis S., The decomposition of global conformal invariants: a conjecture of Deser and Schwimmer, submitted

[4] Alexakis S., The decomposition of global conformal invariants. IV. A proposition on local Riemannian invariants, arXiv: 0912.3761 | MR

[5] Alexakis S., The decomposition of global conformal invariants. V, arXiv: 0912.3764

[6] Atiyah M., Bott R., Patodi V. K., “On the heat equation and the index theorem”, Invent. Math., 19 (1973), 279–330 | DOI | MR | Zbl

[7] Bailey T. N., Eastwood M. G., Gover A. R., “Thomas's structure bundle for conformal, projective and related structures”, Rocky Mountain J. Math., 24 (1994), 1191–1217 | DOI | MR | Zbl

[8] Bailey T. N., Eastwood M. G., Graham C. R., “Invariant theory for conformal and CR geometry”, Ann. of Math. (2), 139 (1994), 491–552 | DOI | MR | Zbl

[9] Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004 | MR | Zbl

[10] Boulanger N., “Algebraic classification of Weyl anomalies in arbitrary dimensions”, Phys. Rev. Lett., 98 (2007), 261302, 4 pp., arXiv: 0706.0340 | DOI | MR

[11] Branson T., Gilkey P., Pohjanpelto J., “Invariants of locally conformally flat manifolds”, Trans. Amer. Math. Soc., 347 (1995), 939–953 | DOI | MR | Zbl

[12] Čap A., Gover A. R., “Tractor calculi for parabolic geometries”, Trans. Amer. Math. Soc., 354 (2002), 1511–1548 | DOI | MR | Zbl

[13] Čap A., Gover A. R., “Standard tractors and the conformal ambient metric construction”, Ann. Global Anal. Geom., 24 (2003), 231–259, arXiv: math.DG/0207016 | DOI | MR | Zbl

[14] Cartan É., “Sur la réduction à sa forme canonique de la structure d'un groupe de transformations fini et continu”, Amer. J. Math., 18 (1896), 1–61 | DOI | MR | Zbl

[15] Deser S., Schwimmer A., “Geometric classification of conformal anomalies in arbitrary dimensions”, Phys. Lett. B, 309 (1993), 279–284, arXiv: hep-th/9302047 | DOI | MR

[16] Fefferman C., “Monge–Ampère equations, the Bergman kernel and geometry of pseudo-convex domains”, Ann. of Math. (2), 103 (1976), 395–416 | DOI | MR | Zbl

[17] Fefferman C., Graham C. R., “Conformal invariants”, The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque, Numero Hors Serie, 1985, 1985, 95–116 | MR | Zbl

[18] Fefferman C., Graham C. R., The ambient metric, arXiv: 0710.0919

[19] Gilkey P. B., “Local invariants of an embedded Riemannian manifold”, Ann. of Math. (2), 102 (1975), 187–203 | DOI | MR | Zbl

[20] Graham C. R., “Extended obstruction tensors and renormalized volume coefficients”, Adv. Math., 220 (2009), 1956–1985, arXiv: 0810.4203 | DOI | MR | Zbl

[21] Graham C. R., “Volume and area renormalizations for conformally compact Einstein metrics”, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999), Rend. Circ. Mat. Palermo (2) Suppl., 63, 2000, 31–42, arXiv: math.DG/9909042 | MR | Zbl

[22] Graham C. R., Hirachi K., “Inhomogeneous ambient metrics”, Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., 144, Springer, New York, 2008, 403–420, arXiv: math.DG/0611931 | DOI | MR | Zbl

[23] Henningson M., Skenderis K., “The holographic Weyl anomaly”, J. High Energy Phys., 1998:7 (1998), 023, 12 pp., arXiv: hep-th/9806087 | DOI | MR | Zbl

[24] Hirachi K., “Construction of boundary invariants and the logarithmic singularity of the Bergman kernel”, Ann. of Math. (2), 151 (2000), 151–191, arXiv: math.CV/0010014 | DOI | MR | Zbl

[25] Thomas T. Y., The differential invariants of generalized spaces, Cambridge University Press, Cambridge, 1934

[26] Weyl H., The classical groups. Their invariants and representations, Princeton University Press, Princeton, 1997 | MR | Zbl