The Atiyah-Patodi-Singer index on manifolds with non-compact boundary. (English) Zbl 1464.58008
The paper is well written, well documented and of huge interest in index theory.
The paper deals with an intensive study of an index theory for a Callias-type operator \(\mathcal{D}\) (see Definition 3.4) on a complete Riemannian manifold \(M\) with boundary, \(\partial M\), which is non-compact. More explicitly, as in the work of N. Highson [Int. J. Math. 1, No. 2, 189–210 (1990; Zbl 0714.58054)] on mod k index theory, the authors relate the index of \(\mathcal{D}\), \(index(\mathcal{D})\), to a much more simple index, on a simple manifold whose boundary decomposes into disjoint union of complete manifolds. Moreover, when the dimension of \(M\) is odd, they show that \(index(\mathcal{D})\) is invariant under deformations of the interior of \(M\) (see Theorem 7.5). This allows them to define the relative eta invariant associated to spectral sections for \(\mathcal{D}|_{\partial M}\) (see Definition 8.5 and Proposition 8.8).
The paper deals with an intensive study of an index theory for a Callias-type operator \(\mathcal{D}\) (see Definition 3.4) on a complete Riemannian manifold \(M\) with boundary, \(\partial M\), which is non-compact. More explicitly, as in the work of N. Highson [Int. J. Math. 1, No. 2, 189–210 (1990; Zbl 0714.58054)] on mod k index theory, the authors relate the index of \(\mathcal{D}\), \(index(\mathcal{D})\), to a much more simple index, on a simple manifold whose boundary decomposes into disjoint union of complete manifolds. Moreover, when the dimension of \(M\) is odd, they show that \(index(\mathcal{D})\) is invariant under deformations of the interior of \(M\) (see Theorem 7.5). This allows them to define the relative eta invariant associated to spectral sections for \(\mathcal{D}|_{\partial M}\) (see Definition 8.5 and Proposition 8.8).
Reviewer: Adnane Elmrabty (Guelmim)
MSC:
| 58J28 | Eta-invariants, Chern-Simons invariants |
| 58J30 | Spectral flows |
| 58J32 | Boundary value problems on manifolds |
| 19K56 | Index theory |
Citations:
Zbl 0714.58054References:
| [1] | Anghel, N., \(L^2\)-index formulae for perturbed Dirac operators, Comm. Math. Phys., 128, 1, 77-97 (1990) · Zbl 0697.58048 |
| [2] | Anghel, N., An abstract index theorem on noncompact Riemannian manifolds, Houst. J. Math., 19, 2, 223-237 (1993) · Zbl 0790.58040 |
| [3] | Anghel, N., On the index of Callias-type operators, Geom. Funct. Anal., 3, 5, 431-438 (1993) · Zbl 0843.58116 |
| [4] | Atiyah, MF; Patodi, VK; Singer, IM, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Philos. Soc., 77, 1, 43-69 (1975) · Zbl 0297.58008 |
| [5] | Atiyah, MF; Patodi, VK; Singer, IM, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Camb. Philos. Soc., 79, 1, 71-99 (1976) · Zbl 0325.58015 |
| [6] | Bär, C.; Ballmann, W., Boundary value problems for elliptic differential operators of first order, Surv. Differ. Geom., XVII, 1-78 (2012) · Zbl 1331.58022 |
| [7] | Bär, C.; Ballmann, W., Guide to elliptic boundary value problems for Dirac-type operators, Arbeitstagung Bonn, 2013, 43-80 (2016) · Zbl 1377.58022 |
| [8] | Bär, C.; Strohmaier, A., An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary, Am. J. Math., 141, 5, 1421-1455 (2019) · Zbl 1429.83004 |
| [9] | Bär, C.; Strohmaier, A., A rigorous geometric derivation of the chiral anomaly in curved backgrounds, Commun. Math. Phys., 347, 3, 703-721 (2016) · Zbl 1350.81029 |
| [10] | Berline, N.; Getzler, E.; Vergne, M., Heat Kernels and Dirac Operators (1992), New York: Springer, New York · Zbl 0744.58001 |
| [11] | Bisgard, J., A compact embedding for sequence spaces, MO J. Math. Sci., 24, 2, 182-189 (2012) · Zbl 1260.46027 |
| [12] | Booss-Bavnbek, B.; Lesch, M.; Phillips, J., Unbounded Fredholm operators and spectral flow, Can. J. Math., 57, 2, 225-250 (2005) · Zbl 1085.58018 |
| [13] | Booß-Bavnbek, B.; Wojciechowski, KP, Elliptic Boundary Problems for Dirac Operators, Mathematics: Theory & Applications (1993), Boston: Birkhäuser Boston Inc., Boston · Zbl 0797.58004 |
| [14] | Bott, R.; Seeley, R., Some remarks on the paper of Callias: “Axial anomalies and index theorems on open spaces”, Commun. Math. Phys., 62, 3, 235-245 (1978) · Zbl 0409.58019 |
| [15] | Braverman, M., Index theorem for equivariant Dirac operators on noncompact manifolds, K Theory, 27, 1, 61-101 (2002) · Zbl 1020.58020 |
| [16] | Braverman, M., An index of strongly Callias operators on Lorentzian manifolds with non-compact boundary, Math. Z., 294, 1-2, 229-250 (2020) · Zbl 1431.58010 |
| [17] | Braverman, M.; Cecchini, S., Callias-type operators in von neumann algebras, J. Geom. Anal. (2017) · Zbl 1390.19009 · doi:10.1007/s12220-017-9832-1 |
| [18] | Braverman, M.; Maschler, G., Equivariant APS index for Dirac operators of non-product type near the boundary, Indiana Univ. Math. J., 68, 435-501 (2019) · Zbl 1422.19002 |
| [19] | Braverman, M.; Milatovich, O.; Shubin, M., Essential selfadjointness of Schrödinger-type operators on manifolds, Russ. Math. Surv., 57, 41-692 (2002) · Zbl 1052.58027 |
| [20] | Braverman, M.; Shi, P., Cobordism invariance of the index of Callias-type operators, Commun. Partial Differ. Equ., 41, 8, 1183-1203 (2016) · Zbl 1374.19001 |
| [21] | Braverman, M., Shi, P.: APS index theorem for even-dimensional manifolds with non-compact boundary. Commun. Anal. Geom. (2017). arXiv:1708.08336 |
| [22] | Braverman, M.; Shi, P., The index of a local boundary value problem for strongly Callias-type operators, Arnold Math. J., 5, 1, 79-96 (2019) · Zbl 1444.58005 |
| [23] | Brüning, J.; Moscovici, H., \(L^2\)-index for certain Dirac-Schrödinger operators, Duke Math. J., 66, 2, 311-336 (1992) · Zbl 0765.58029 |
| [24] | Bunke, U., Relative index theory, J. Funct. Anal., 105, 1, 63-76 (1992) · Zbl 0762.58026 |
| [25] | Bunke, U., A K-theoretic relative index theorem and Callias-type Dirac operators, Math. Ann., 303, 2, 241-279 (1995) · Zbl 0835.58035 |
| [26] | Callias, C., Axial anomalies and index theorems on open spaces, Commun. Math. Phys., 62, 3, 213-235 (1978) · Zbl 0416.58024 |
| [27] | Carvalho, C.; Nistor, V., An index formula for perturbed Dirac operators on Lie manifolds, J. Geom. Anal., 24, 4, 1808-1843 (2014) · Zbl 1331.58028 |
| [28] | Dai, X.; Zhang, W., Higher spectral flow, J. Funct. Anal., 157, 2, 432-469 (1998) · Zbl 0932.37062 |
| [29] | Fox, J.; Haskell, P., Heat kernels for perturbed Dirac operators on even-dimensional manifolds with bounded geometry, Int. J. Math., 14, 1, 69-104 (2003) · Zbl 1079.58019 |
| [30] | Fox, J.; Haskell, P., The Atiyah-Patodi-Singer theorem for perturbed Dirac operators on even-dimensional manifolds with bounded geometry, N. Y. J. Math., 11, 303-332 (2005) · Zbl 1102.58011 |
| [31] | Freed, D., Two index theorems in odd dimensions, Commun. Anal. Geom., 6, 2, 317-329 (1998) · Zbl 0920.58044 |
| [32] | Gromov, M.; Lawson, HB Jr, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math., 58, 1983, 83-196 (1984) · Zbl 0538.53047 |
| [33] | Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (1999). MR1688256 · Zbl 0981.58006 |
| [34] | Horava, P.; Witten, E., Heterotic and type I string dynamics from eleven dimensions, Nuclear Phys. B, 460, 3, 506-524 (1996) · Zbl 1004.81525 |
| [35] | Hörmander, L.: The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin (2007). Pseudo-differential operators, Reprint of the 1994 edition. MR2304165 |
| [36] | Kato, T., Perturbation Theory for Linear Operators (1995), New York: Springer, New York · Zbl 0836.47009 |
| [37] | Kottke, C., An index theorem of Callias type for pseudodifferential operators, J. K-Theory, 8, 3, 387-417 (2011) · Zbl 1248.58012 |
| [38] | Kottke, C., A Callias-type index theorem with degenerate potentials, Commun. Partial Differ. Equ., 40, 2, 219-264 (2015) · Zbl 1316.58019 |
| [39] | Lawson, HB; Michelsohn, M-L, Spin Geometry (1989), Princeton: Princeton University Press, Princeton · Zbl 0688.57001 |
| [40] | Melrose, RB; Piazza, P., Families of Dirac operators, boundaries and the \(b\)-calculus, J. Differ. Geom., 46, 1, 99-180 (1997) · Zbl 0955.58020 |
| [41] | Müller, W., Relative zeta functions, relative determinants and scattering theory, Commun. Math. Phys., 192, 2, 309-347 (1998) · Zbl 0947.58025 |
| [42] | Nicolaescu, LI, The Maslov index, the spectral flow, and decompositions of manifolds, Duke Math. J., 80, 2, 485-533 (1995) · Zbl 0849.58064 |
| [43] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics. I (1978), London: Academic Press, London · Zbl 0401.47001 |
| [44] | Shi, P., The index of Callias-type operators with Atiyah-Patodi-Singer boundary conditions, Ann. Glob. Anal. Geom., 52, 4, 465-482 (2017) · Zbl 1467.58013 |
| [45] | Shi, P.: Cauchy data spaces and Atiyah-Patodi-Singer index on non-compact manifolds. J. Geom. Phys. 133, 81-90 (2018). arXiv:1803.01884 · Zbl 1400.58008 |
| [46] | Shubin, MA, Spectral theory of the Schrödinger operators on non-compact manifolds: qualitative results, Spectr. Theory Geom. (Edinburgh, 1998), 273, 226-283 (1999) · Zbl 0985.58015 |
| [47] | Wimmer, R., An index for confined monopoles, Commun. Math. Phys., 327, 1, 117-149 (2014) · Zbl 1288.81151 |
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