Spectral 𝜁-invariants lifted to coverings

Azzali, Sara; Paycha, Sylvie

doi:10.1090/tran/8067

Spectral $\zeta$-invariants lifted to coverings
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by Sara Azzali and Sylvie Paycha PDF

Trans. Amer. Math. Soc. 373 (2020), 6185-6226 Request permission

Abstract:

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $\zeta$-invariants using lifted defect formulae which express discrepancies of $\zeta$-regularised traces in terms of Wodzicki residues. We derive Atiyah’s $L^2$-index theorem as an instance of the $\mathbb {Z}_2$-graded generalisation of the canonical lift of spectral $\zeta$-invariants and we show that certain lifted spectral $\zeta$-invariants for geometric operators are integrals of Pontryagin and Chern forms.

References

Bernd Ammann, Robert Lauter, Victor Nistor, and András Vasy, Complex powers and non-compact manifolds, Comm. Partial Differential Equations 29 (2004), no. 5-6, 671–705. MR 2059145, DOI 10.1081/PDE-120037329
Sara Azzali, Cyril Lévy, Carolina Neira-Jiménez, and Sylvie Paycha, Traces of holomorphic families of operators on the noncommutative torus and on Hilbert modules, Geometric methods in physics, Trends Math., Birkhäuser/Springer, Cham, 2015, pp. 3–38. MR 3629661, DOI 10.1007/978-3-319-18212-4_{1}
Moulay-Tahar Benameur and Paolo Piazza, Index, eta and rho invariants on foliated bundles, Astérisque 327 (2009), 201–287 (2010) (English, with English and French summaries). MR 2642361
D. Burghelea, L. Friedlander, T. Kappeler, and P. McDonald, Analytic and Reidemeister torsion for representations in finite type Hilbert modules, Geom. Funct. Anal. 6 (1996), no. 5, 751–859. MR 1415762, DOI 10.1007/BF02246786
Alain Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143 (French). MR 548112
Alexander Cardona, Catherine Ducourtioux, and Sylvie Paycha, From tracial anomalies to anomalies in quantum field theory, Comm. Math. Phys. 242 (2003), no. 1-2, 31–65. MR 2018268, DOI 10.1007/s00220-003-0903-8
Yu. V. Egorov, A. I. Komech, and M. A. Shubin, Elements of the modern theory of partial differential equations, Springer-Verlag, Berlin, 1999. Translated from the 1988 Russian original by P. C. Sinha; Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Partial differential equations. II, Encyclopaedia Math. Sci., 31, Springer, Berlin, 1994; MR1364199 (96f:35001)]. MR 1693807
Alexander Engel, $K$-homology classes of elliptic uniform pseudodifferential operators, Ann. Global Anal. Geom. 54 (2018), no. 4, 551–582. MR 3878843, DOI 10.1007/s10455-018-9614-4
Alexander Engel, Index theorems for uniformly elliptic operators, New York J. Math. 24 (2018), 543–587. MR 3855638
Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308
Victor Guillemin, A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. in Math. 55 (1985), no. 2, 131–160. MR 772612, DOI 10.1016/0001-8708(85)90018-0
Maxim Kontsevich and Simeon Vishik, Geometry of determinants of elliptic operators, Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993) Progr. Math., vol. 131, Birkhäuser Boston, Boston, MA, 1995, pp. 173–197. MR 1373003
Y. Kordyukov, Elliptic operators on manifolds with bounded geometry, Ph.D. thesis, Moscow State University (1987)
E. C. Lance, Hilbert $C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694, DOI 10.1017/CBO9780511526206
Matthias Lesch, On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols, Ann. Global Anal. Geom. 17 (1999), no. 2, 151–187. MR 1675408, DOI 10.1023/A:1006504318696
Jouko Mickelsson and Sylvie Paycha, The logarithmic residue density of a generalized Laplacian, J. Aust. Math. Soc. 90 (2011), no. 1, 53–80. MR 2810944, DOI 10.1017/S144678871100108X
G. A. Meladze and M. A. Shubin, Proper uniform pseudodifferential operators on unimodular Lie groups, Trudy Sem. Petrovsk. 11 (1986), 74–97, 244, 246–247 (Russian, with English summary); English transl., J. Soviet Math. 45 (1989), no. 5, 1421–1439. MR 834168, DOI 10.1007/BF01097159
Victor Nistor, Alan Weinstein, and Ping Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152. MR 1687747, DOI 10.2140/pjm.1999.189.117
Sylvie Paycha, Regularised integrals, sums and traces, University Lecture Series, vol. 59, American Mathematical Society, Providence, RI, 2012. An analytic point of view. MR 2987296, DOI 10.1090/ulect/059
Sylvie Paycha and Simon Scott, A Laurent expansion for regularized integrals of holomorphic symbols, Geom. Funct. Anal. 17 (2007), no. 2, 491–536. MR 2322493, DOI 10.1007/s00039-007-0597-8
John Roe, Elliptic operators, topology and asymptotic methods, 2nd ed., Pitman Research Notes in Mathematics Series, vol. 395, Longman, Harlow, 1998. MR 1670907
John Roe, An index theorem on open manifolds. I, II, J. Differential Geom. 27 (1988), no. 1, 87–113, 115–136. MR 918459
Paolo Piazza and Thomas Schick, Bordism, rho-invariants and the Baum-Connes conjecture, J. Noncommut. Geom. 1 (2007), no. 1, 27–111. MR 2294190, DOI 10.4171/JNCG/2
Thomas Schick, $L^2$-index theorems, $KK$-theory, and connections, New York J. Math. 11 (2005), 387–443. MR 2188248
T. Schick, Analysis on $\partial$-Manifolds of Bounded Geometry, Hodge-De Rham Isomorphism and $L^2$-Index Theorem, Dissertation, Universität Mainz 1996
Simon Scott, Traces and determinants of pseudodifferential operators, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2010. MR 2683288, DOI 10.1093/acprof:oso/9780198568360.001.0001
R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR 0237943
M. A. Shubin, Pseudodifferential operators and spectral theory, 2nd ed., Springer-Verlag, Berlin, 2001. Translated from the 1978 Russian original by Stig I. Andersson. MR 1852334, DOI 10.1007/978-3-642-56579-3
M. A. Shubin, Spectral theory of elliptic operators on noncompact manifolds, Astérisque 207 (1992), 5, 35–108. Méthodes semi-classiques, Vol. 1 (Nantes, 1991). MR 1205177
M. A. Shubin, Weak Bloch property and weight estimates for elliptic operators, Séminaire sur les Équations aux Dérivées Partielles, 1989–1990, École Polytech., Palaiseau, 1990, pp. Exp. No. V, 36. With an appendix by Shubin and J. Sjöstrand. MR 1073180
M. A. Shubin, Von Neumann algebras and $L^2$-techniques, book www.mccme.ru/ium/postscript/f02/L2titlepages.ps.gz
Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463, DOI 10.1515/9781400886104
B. Vaillant, Index Theory for Coverings, http://arxiv.org/abs/0806.4043
Stéphane Vassout, Unbounded pseudodifferential calculus on Lie groupoids, J. Funct. Anal. 236 (2006), no. 1, 161–200. MR 2227132, DOI 10.1016/j.jfa.2005.12.027
S. Vassout, Feuillettages et résidu non commutatif longitudinal, Thesis, Université Paris 6, 2001
Mariusz Wodzicki, Noncommutative residue. I. Fundamentals, $K$-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 320–399. MR 923140, DOI 10.1007/BFb0078372