The local index density of the perturbed de Rham complex

Jesús Álvarez López; Peter B. Gilkey

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 3, page 901-932
  • ISSN: 0011-4642

Abstract

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A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian manifold M . We use invariance theory to show that the perturbed index density is independent of Θ ; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on Θ . We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires Θ to be a ∂ ¯ closed 1 -form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on Θ even in the setting of Riemann surfaces.

How to cite

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Álvarez López, Jesús, and Gilkey, Peter B.. "The local index density of the perturbed de Rham complex." Czechoslovak Mathematical Journal 71.3 (2021): 901-932. <http://eudml.org/doc/297970>.

@article{ÁlvarezLópez2021,
abstract = {A perturbation of the de Rham complex was introduced by Witten for an exact 1-form $\Theta $ and later extended by Novikov for a closed 1-form on a Riemannian manifold $M$. We use invariance theory to show that the perturbed index density is independent of $\Theta $; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on $\Theta $. We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires $\Theta $ to be a $\bar\{\partial \}$ closed $1$-form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on $\Theta $ even in the setting of Riemann surfaces.},
author = {Álvarez López, Jesús, Gilkey, Peter B.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Witten deformation; local index density; de Rham complex; Dolbeault complex; equivariant index density},
language = {eng},
number = {3},
pages = {901-932},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The local index density of the perturbed de Rham complex},
url = {http://eudml.org/doc/297970},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Álvarez López, Jesús
AU - Gilkey, Peter B.
TI - The local index density of the perturbed de Rham complex
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 901
EP - 932
AB - A perturbation of the de Rham complex was introduced by Witten for an exact 1-form $\Theta $ and later extended by Novikov for a closed 1-form on a Riemannian manifold $M$. We use invariance theory to show that the perturbed index density is independent of $\Theta $; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on $\Theta $. We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires $\Theta $ to be a $\bar{\partial }$ closed $1$-form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on $\Theta $ even in the setting of Riemann surfaces.
LA - eng
KW - Witten deformation; local index density; de Rham complex; Dolbeault complex; equivariant index density
UR - http://eudml.org/doc/297970
ER -

References

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  1. López, J. A. Álvarez, Kordyukov, Y. A., Leichtnam, E., A trace formula for foliated flow (working paper), Summer Conference on Topology and Its Applications 20 University of Dayton, Dayton (2017), 72 pages. 
  2. Atiyah, M., Bott, R., Patodi, V. K., 10.1007/BF01425417, Invent. Math. 13 (1973), 279-330 errata ibid. 28 1975 277-280. (1973) Zbl0257.58008MR650828DOI10.1007/BF01425417
  3. Atiyah, M., Patodi, V. K., Singer, I. M., 10.1112/blms/5.2.229, Bull. Lond. Math. Soc. 5 (1973), 229-234. (1973) Zbl0268.58010MR331443DOI10.1112/blms/5.2.229
  4. Berline, N., Getzler, E., Vergne, M., 10.1007/978-3-642-58088-8, Grundlehren Text Editions. Springer, Berlin (2004). (2004) Zbl1037.58015MR2273508DOI10.1007/978-3-642-58088-8
  5. Bismut, J.-M., Zhang, W., An Extension of a Theorem by Cheeger and Müller, Astérisque 205. Société Mathématique de France, Paris (1992). (1992) Zbl0781.58039MR1185803
  6. Braverman, M., Farber, M., 10.1017/S0305004197001734, Math. Proc. Camb. Philos. Soc. 122 (1997), 357-375. (1997) Zbl0894.58012MR1458239DOI10.1017/S0305004197001734
  7. Burghelea, D., Haller, S., On the topology and analysis of a closed one form. I. (Novikov's theory revisited), Essays on Geometry and Related Topics. Volume 1 Monographs of L'Enseignement Mathématique 38. Enseignement Mathématique, Geneva (2001), 133-175. (2001) Zbl1017.57013MR1929325
  8. Burghelea, D., Haller, S., 10.1112/jtopol/jtm005, J. Topol. 1 (2008), 115-151. (2008) Zbl1156.57022MR2365654DOI10.1112/jtopol/jtm005
  9. Chern, S.-S., 10.2307/1969302, Ann. Math. (2) 45 (1944), 741-752. (1944) Zbl0060.38103MR11027DOI10.2307/1969302
  10. Gilkey, P. B., 10.1016/0001-8708(73)90014-5, Adv. Math. 11 (1973), 311-325. (1973) Zbl0285.53044MR334290DOI10.1016/0001-8708(73)90014-5
  11. Gilkey, P. B., 10.1016/0001-8708(73)90119-9, Adv. Math. 10 (1973), 344-382. (1973) Zbl0259.58010MR324731DOI10.1016/0001-8708(73)90119-9
  12. Gilkey, P. B., 10.1016/0001-8708(75)90141-3, Adv. Math. 15 (1975), 334-360. (1975) Zbl0306.53042MR368084DOI10.1016/0001-8708(75)90141-3
  13. Gilkey, P. B., Lefschetz fixed point formulas and the heat equation, Partial Differential Equations and Geometry Lecture Notes in Pure and Applied Mathematics 48. Marcel Dekker, New York (1979), 91-147. (1979) Zbl0405.58044MR535591
  14. Gilkey, P. B., Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, Studies in Advanced Mathematics. CRC Press, Boca Raton (1995). (1995) Zbl0856.58001MR1396308
  15. Gilkey, P. B., 10.1201/9780203490464, Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton (2004). (2004) Zbl1080.58023MR2040963DOI10.1201/9780203490464
  16. Gilkey, P. B., Nikčević, S., Pohjanpelto, J., 10.2140/pjm.1997.180.51, Pac. J. Math. 180 (1997), 51-56. (1997) Zbl0885.58091MR1474893DOI10.2140/pjm.1997.180.51
  17. Gilkey, P. B., Park, J. H., Sekigawa, K., 10.1016/j.difgeo.2011.08.005, Differ. Geom. Appl. 29 (2011), 770-778. (2011) Zbl1259.53013MR2846274DOI10.1016/j.difgeo.2011.08.005
  18. Greiner, P., 10.1007/BF00276190, Arch. Ration. Mech. Anal. 41 (1971), 163-218. (1971) Zbl0238.35038MR331441DOI10.1007/BF00276190
  19. Harvey, F. R., Minervini, G., 10.1007/s00208-006-0765-4, Math. Ann. 335 (2006), 787-818. (2006) Zbl1109.57019MR2232017DOI10.1007/s00208-006-0765-4
  20. Helffer, B., Sjöstrand, J., 10.1080/03605308508820379, Commun. Partial Differ. Equations 10 (1985), 245-340 French. (1985) Zbl0597.35024MR780068DOI10.1080/03605308508820379
  21. Hirzebruch, F., 10.1007/978-3-642-62018-8, Die Grundlehren der Mathematischen Wissenschaften 131. Springer, Berlin (1966). (1966) Zbl0138.42001MR0202713DOI10.1007/978-3-642-62018-8
  22. Lee, S.-C., A Lefschetz Formula for Higher Dimensional Fixed Point Sets: Ph.D. Thesis, Brandeis University, Waltham (1976). (1976) MR2625269
  23. H. P. McKean, Jr., I. M. Singer, 10.4310/jdg/1214427880, J. Differ. Geom. 1 (1967), 43-69. (1967) Zbl0198.44301MR217739DOI10.4310/jdg/1214427880
  24. Minervini, G., A current approach to Morse and Novikov theories, Rend. Mat. Appl., VII. Ser. 36 (2015), 95-195. (2015) Zbl1361.58007MR3533253
  25. Novikov, S. P., Multivalued functions and functionals: An analogue of the Morse theory, Sov. Math., Dokl. 24 (1981), 222-226. (1981) Zbl0505.58011MR630459
  26. Novikov, S. P., 10.1070/RM1982v037n05ABEH004020, Russ. Math. Surv. 37 (1982), 1-56. (1982) Zbl0571.58011MR676612DOI10.1070/RM1982v037n05ABEH004020
  27. Novikov, S. P., Bloch homology. Critical points of functions and closed 1-forms, Sov. Math., Dokl. 33 (1986), 551-555. (1986) Zbl0642.58016MR838822
  28. Patodi, V. K., 10.4310/jdg/1214429991, J. Differ. Geom. 5 (1971), 251-283. (1971) Zbl0219.53054MR290318DOI10.4310/jdg/1214429991
  29. Patodi, V. K., 10.4310/jdg/1214429791, J. Differ. Geom. 5 (1971), 233-249. (1971) Zbl0211.53901MR292114DOI10.4310/jdg/1214429791
  30. Pazhitnov, A. V., An analytic proof of the real part of Novikov's inequalities, Sov. Math., Dokl. 35 (1987), 456-457. (1987) Zbl0647.57025MR891557
  31. Seeley, R. T., 10.1090/pspum/010, Singular Integrals Proceedings of Symposia in Pure Mathematics 10. American Mathematical Society, Providence (1968), 288-307. (1968) Zbl0159.15504MR0237943DOI10.1090/pspum/010
  32. Seeley, R. T., 10.2307/2373309, Am. J. Math. 91 (1969), 889-920. (1969) Zbl0191.11801MR265764DOI10.2307/2373309
  33. Weyl, H., 10.1515/9781400883905, Princeton Mathematical Series 1. Princeton University Press, Princeton (1946). (1946) Zbl1024.20502MR1488158DOI10.1515/9781400883905
  34. Witten, E., 10.4310/jdg/1214437492, J. Differ. Geom. 17 (1982), 661-692. (1982) Zbl0499.53056MR683171DOI10.4310/jdg/1214437492

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