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An index formula for perturbed Dirac operators on Lie manifolds. (English) Zbl 1331.58028

This article studies index theory of Callias-type operators (e.g., perturbed Dirac operators) on a broad class of noncompact manifolds, known as Lie manifolds (or manifold with a Lie structure at infinity). A Lie manifold \(M_0\) is assumed to be the interior of a compact manifold \(M\) with corners, equipped with a Lie algebra of vector fields \(\mathcal{V}\subset\Gamma(TM)\) where: (1) Each element of \(\mathcal{V}\) is tangent to all faces of \(M;\) (2) \(\mathcal{V}\) is a finitely generated, projective \(C^{\infty}(M)\)-module, i.e., it is isomorphic to the smooth sections of a vector bundle \(A\) over \(M\); (3) The anchor map \(\rho: A\rightarrow TM\) gives rise to an isomorphism \(\rho|_{M_0}: A|_{M_0}\rightarrow TM_0\). Here \(\rho\) is a vector bundle morphism corresponding the inclusion map \[ \mathcal{V}=\Gamma(A)\rightarrow\Gamma(TM). \]
Lie manifolds were introduced in [B. Ammann et al., Ann. Math. (2) 165, No. 3, 717–747 (2007; Zbl 1133.58020)], extending results by Melrose, Schrohe, Schulze, Vasy and their collaborators, and include many examples such as manifolds with cylindrical ends, or with conical singularities.
For an even-dimensional Lie manifold \((M, \mathcal{V})\) equipped with a metric \(g\) on \(M_0\) that extends to \(A\rightarrow M\), let \(W\) be a Clifford module over \(M\) with a connection \(\nabla^{W}: C^{\infty}(M, W)\rightarrow C^{\infty}(W\otimes A^*)\) and a Clifford multiplication bundle map \(c: A\otimes W\rightarrow W\). The generalised Dirac operator \(D: C^{\infty}(M, W)\rightarrow C^{\infty}(M, W)\) is given by the composition \(c\circ(id\otimes\phi)\circ\nabla^{W}\) where \(\phi: A^*\rightarrow A\) is isomorphism given by the metric \(g\).
Let \(E\) be a Hermitian \(\mathbb{Z}_2\)-graded vector bundle over \(M\) and \(V\in\mathrm{End}(E)\) an unbounded potential (odd self-adjoint, and invertible outside a compact subset of \(M_0\)) of the form \[ V:=f^{-1}V_0=\Pi x_k^{-a_k}V_0,\qquad a_k\in\mathbb{Z}_{+}, \] where \(x_k\) are defining functions on \(M\) where \(\{x_k=0\}\) is a hyperface \(H\) of \(M\) and \(dx_k\neq0\) on \(H,\) and \(V_0\) is a bounded potential extending to a smooth endomorphism of \(E\) and invertible at the boundary of \(M.\)
A Callias-type operator \(T\) on the Lie manifold \((M, \mathcal{V})\) is the closure of an operator on \(C_c^{\infty}(M_0, W\hat\otimes E)\) given by \[ T:=D\hat\otimes 1+1\hat\otimes V. \] Here, \(W\hat\otimes E\) is \(\mathbb{Z}_2\)-graded as \((W\hat\otimes E)^{\pm}=(W^{\pm}\otimes E^{\pm})\oplus(W^{-}\otimes E^{-})\) and \(T: (W\hat\otimes E)^{+}\rightarrow (W\hat\otimes E)^{-}\) is denoted by \(T^+.\) The main result (Theorem 4.13) of this paper states that \(T^+\) is a Fredholm operator and has an Atiyah-Singer type index formula: \[ \mathrm{ind} T^+=\int_{TM_0}\mathrm{ch}[\sigma(D^+)]\mathrm{ch}[\pi^*[V_0]]\pi^*\mathrm{Td}(T_{\mathbb{C}}M). \] Here \(\sigma(D^+)\) stands for the principal symbol of \(D^+\), \(\mathrm{Td}(T_{\mathbb C}M)\) is the Todd class of the complexified tangent bundle of \(M\), and \(\pi: \overline{TM}\rightarrow M\) is the natural projection.
The proof of this theorem uses a few previous achievements in analysis. First, the problem is reduced to the case of a Dirac operator coupled with the bounded potential \(V_0\). This new Callias-type operator falls in a general index problem for fully elliptic operators on a new Lie manifold \((M, \mathcal{W})\), where \(\mathcal{W}=f\mathcal{V}\), possessing a property known as asymptotic commutative. Then, this type of operators is shown to be deformations of pseudo-differential operators (\(\Psi\)DOs) that are asymptotically multiplication on \(M_0\), which can be further approximated by \(\Psi\)DOs that are multiplication outside a compact subset of \(M_0\). Finally, the index formula for \(\Psi\)DOs that are multiplication outside a compact subset of \(M_0\) is studied in [C. Carvalho, \(K\)-Theory 36, No. 1–2, 1–31 (2005; Zbl 1117.58009)].

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
19K56 Index theory
58J40 Pseudodifferential and Fourier integral operators on manifolds
58H05 Pseudogroups and differentiable groupoids
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[1] Albin, P., Rochon, F.: Families index for manifolds with hyperbolic cusp singularities. Int. Math. Res. Not. 4, 625-697 (2009) · Zbl 1184.58008
[2] Amman, B., Lauter, R., Nistor, V.: Pseudodifferential operators on manifolds with a Lie structure at infinity. Ann. Math. (2) 165(3), 717-747 (2007) · Zbl 1133.58020 · doi:10.4007/annals.2007.165.717
[3] Ammann, B., Ionescu, A.D., Nistor, V.: Sobolev spaces and regularity for polyhedral domains. Doc. Math. 11(2), 161-206 (2006) · Zbl 1247.35031
[4] Ammann, B., Lauter, R., Nistor, V.: On the geometry of Riemannian manifolds with a Lie structure at infinity. Int. J. Math. Math. Sci. 2004(1-4), 161-193 (2004) · Zbl 1071.53020 · doi:10.1155/S0161171204212108
[5] Anghel, N.: An abstract index theorem on non-compact Riemannian manifolds. Houst. J. Math. 19, 223-237 (1993) · Zbl 0790.58040
[6] Anghel, N.: On the index of Callias-type operators. Geom. Funct. Anal. 3(5), 431-438 (1993) · Zbl 0843.58116 · doi:10.1007/BF01896237
[7] Atiyah, M.: K-theory. Benjamin, New York (1967)
[8] Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975) · Zbl 0297.58008 · doi:10.1017/S0305004100049410
[9] Atiyah, M., Singer, I.: The index of elliptic operators I. Ann. Math. 87, 484-530 (1968) · Zbl 0164.24001 · doi:10.2307/1970715
[10] Bär, C.: Metrics with harmonic spinors. Geom. Funct. Anal. 6(6), 899-942 (1996) · Zbl 0867.53037 · doi:10.1007/BF02246994
[11] Bär, C., Schmutz, P.: Harmonic spinors on Riemann surfaces. Ann. Glob. Anal. Geom. 10(3), 263-273 (1992) · Zbl 0763.30017 · doi:10.1007/BF00136869
[12] Benameur, M., Oyono-Oyono, H.: Index theory for quasi-crystals I. Computation of the gap-label group. J. Funct. Anal. 252(1), 137-170 (2007) · Zbl 1134.46046 · doi:10.1016/j.jfa.2006.03.029
[13] Bott, R., Seeley, R.: Some remarks on the paper of Callias. Commun. Math. Phys. 62, 235-245 (1978) · Zbl 0409.58019 · doi:10.1007/BF01202526
[14] Brüning, J., Moscovici, H.: L2-index for certain Dirac-Schrödinger operators. Duke Math. J. 66, 311-336 (1992) · Zbl 0765.58029 · doi:10.1215/S0012-7094-92-06609-9
[15] Bunke, U.: A K-theoretic relative index theorem and Callias-type Dirac operators. Math. Ann. 241-279 (1995) · Zbl 0835.58035
[16] Bunke, U.: Index theory, eta forms, and Deligne cohomology. Mem. Am. Math. Soc. 198, 928 (2009) vi+120 · Zbl 1181.58017
[17] Bunke, U., Schick, T.: Smooth K-theory. Astérisque 328, 45-135 (2010) · Zbl 1202.19007
[18] Callias, C.: Axial anomalies and index theorems on open spaces. Commun. Math. Phys. 62, 213-234 (1978) · Zbl 0416.58024 · doi:10.1007/BF01202525
[19] Carrillo-Rouse, P.: Compactly supported analytic indices for Lie groupoids. J. K-Theory 4(2), 223-262 (2009) · Zbl 1188.19004 · doi:10.1017/is008003015jkt069
[20] Carrillo-Rouse, P., Monthubert, B.: An index theorem for manifolds with boundary. C. R. Math. Acad. Sci. Paris 347(23-24), 1393-1398 (2009) · Zbl 1204.58016 · doi:10.1016/j.crma.2009.10.021
[21] Carvalho, C.: A K-theory proof of the cobordism invariance of the index. K-theory 36(1-2), 1-31 (2005) · Zbl 1117.58009
[22] Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994) · Zbl 0818.46076
[23] Cordes, H., McOwen, R.: The C∗-algebra of a singular elliptic problem on a noncompact Riemannian manifold. Math. Z. 153(2), 101-116 (1977) · Zbl 0345.35087 · doi:10.1007/BF01179784
[24] Cordes, H.O.: Spectral Theory of Linear Differential Operators and Comparison Algebras. London Mathematical Society, Lecture Notes Series, vol. 76. Cambridge University Press, Cambridge (1987) · Zbl 0727.35092 · doi:10.1017/CBO9780511662836
[25] Coriasco, S., Schrohe, E., Seiler, J.: Bounded imaginary powers of differential operators on manifolds with conical singularities. Math. Z. 244(2), 235-269 (2003) · Zbl 1160.58305
[26] Crainic, M., Fernandes, R.: Integrability of Lie brackets. Ann. Math. (2) 157(2), 575-620 (2003) · Zbl 1037.22003 · doi:10.4007/annals.2003.157.575
[27] Debord, C.; Lescure, J.-M., Index theory and groupoids, 86-158 (2010), Cambridge · Zbl 1213.81209 · doi:10.1017/CBO9780511712135.004
[28] Debord, C., Lescure, J.-M., Nistor, V.: Groupoids and an index theorem for conical pseudo-manifolds. J. Reine Angew. Math. 628, 1-35 (2009) · Zbl 1169.58005 · doi:10.1515/CRELLE.2009.017
[29] Fox, J., Haskell, P.: Index theory for perturbed Dirac operators on manifolds with conical singularities. Proc. Am. Math. Soc. 123, 2265-2273 (1995) · Zbl 0834.58036 · doi:10.1090/S0002-9939-1995-1243166-4
[30] Fox, J., Haskell, P.: Comparison of perturbed Dirac operators. Proc. Am. Math. Soc. 124(5), 1601-1608 (1996) · Zbl 0849.58063 · doi:10.1090/S0002-9939-96-03263-7
[31] 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
[32] Georgescu, V., Iftimovici, A.: Crossed products of C∗-algebras and spectral analysis of quantum Hamiltonians. Commun. Math. Phys. 228(3), 519-560 (2002) · Zbl 1005.81026 · doi:10.1007/s002200200669
[33] Gesztesy, F., Latushkin, Y., Makarov, K., Sukochev, F., Tomilov, Y.: The index formula and the spectral shift function for relatively trace class perturbations. Adv. Math. 227(1), 319-420 (2011) · Zbl 1220.47017 · doi:10.1016/j.aim.2011.01.022
[34] Gilkey, P.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd edn. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995) · Zbl 0856.58001
[35] Gilkey, P.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Studies in Advanced Mathematics, vol. 16. CRC Press, Boca Raton (1995) · Zbl 0856.58001
[36] Grieser, D., Hunsicker, E.: Pseudodifferential operator calculus for generalized \(\Bbb{Q} \)-rank 1 locally symmetric spaces I. J. Funct. Anal. 257(12), 3748-3801 (2009) · Zbl 1193.58013 · doi:10.1016/j.jfa.2009.09.016
[37] Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1-55 (1974) · Zbl 0284.58016 · doi:10.1016/0001-8708(74)90021-8
[38] Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (1985) · Zbl 0601.35001
[39] Karoubi, M.: K-Theory—An Introduction. Springer, Berlin (1978) · Zbl 0382.55002
[40] Kottke, C.: An index theorem of Callias type for pseudodifferential operators. J. K-Theory 8(3), 387-417 (2011) · Zbl 1248.58012 · doi:10.1017/is010011014jkt132
[41] Lauter, R., Monthubert, B., Nistor, V.: Pseudodifferential analysis on continuous family groupoids. Doc. Math. 5, 625-655 (2000) (electronic) · Zbl 0961.22005
[42] Lauter, R., Moroianu, S.: Fredholm theory for degenerate pseudodifferential operators on manifolds with fibered boundaries. Commun. Part. Diff. Eqs. 26(1-2), 223-283 (2001) · Zbl 0988.58011
[43] Lauter, R.; Nistor, V., Analysis of geometric operators on open manifolds: a groupoid approach, No. 198, 181-229 (2001), Basel · Zbl 1018.58009 · doi:10.1007/978-3-0348-8364-1_8
[44] Lawson, H., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton (1989) · Zbl 0688.57001
[45] Melrose, R.: The Atiyah-Patodi-Singer Index Theorem. Research Notes in Mathematics, vol. 4. Peters, Wellesley (1993) · Zbl 0796.58050
[46] Melrose, R., Rochon, F.: Periodicity and the determinant bundle. Commun. Math. Phys. 274(1), 141-186 (2007) · Zbl 1130.58013 · doi:10.1007/s00220-007-0277-4
[47] Melrose, R. B., Pseudodifferential operators, corners and singular limits, Kyoto, Japan, 1990 · Zbl 0743.58033
[48] Melrose, R.B.: Geometric Scattering Theory. Stanford Lectures. Cambridge University Press, Cambridge (1995) · Zbl 0849.58071
[49] Monthubert, B.; Nistor, V., The K-groups and the index theory of certain comparison C∗-algebras, No. 546, 213-224 (2011), Providence · Zbl 1267.19005 · doi:10.1090/conm/546/10791
[50] Moroianu, S.: Fibered cusp versus d-index theory. Rend. Semin. Mat. Univ. Padova 117, 193-203 (2007) · Zbl 1142.53039
[51] Nistor, V.: On the kernel of the equivariant Dirac operator. Ann. Glob. Anal. Geom. 17(6), 595-613 (1999) · Zbl 0949.58026 · doi:10.1023/A:1006605923605
[52] Nistor, V.: Groupoids and the integration of Lie algebroids. J. Math. Soc. Jpn. 52, 847-868 (2000) · Zbl 0965.58023 · doi:10.2969/jmsj/05240847
[53] Nistor, V.: An index theorem for gauge-invariant families: the case of solvable groups. Acta Math. Hung. 99(1-2), 155-183 (2003) · Zbl 1026.19007 · doi:10.1023/A:1024517714643
[54] Nistor, V., Weinstein, A., Xu, P.: Pseudodifferential operators on groupoids. Pac. J. Math. 189, 117-152 (1999) · Zbl 0940.58014 · doi:10.2140/pjm.1999.189.117
[55] Nistor, V.: Higher index theorems and the boundary map in cyclic cohomology. Doc. Math. 2, 263-295 (1997) (electronic) · Zbl 0893.19002
[56] Parenti, C.: Operatori pseudodifferentiali in ℝn e applicazioni. Ann. Mat. Pura Appl. 93, 391-406 (1972) · Zbl 0291.35071 · doi:10.1007/BF02412029
[57] Rabier, P.: On the index and spectrum of differential operators on \(\Bbb{R}^N\). Proc. Am. Math. Soc. 135(12), 3875-3885 (2007) (electronic) · Zbl 1129.47034 · doi:10.1090/S0002-9939-07-08896-X
[58] Rabier, P.: On the Fedosov-Hörmander formula for differential operators. Integral Equ. Oper. Theory 62(4), 555-574 (2008) · Zbl 1193.47020 · doi:10.1007/s00020-008-1642-1
[59] Rade, J.: Callias’ index theorem, elliptic boundary conditions, and cutting and pasting. Commun. Math. Phys. 161, 51-61 (1994) · Zbl 0797.58081 · doi:10.1007/BF02099412
[60] Renault, J.: A Groupoid Approach to C∗-Algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980) · Zbl 0433.46049
[61] Schulze, B.-W.: Boundary Value Problems and Singular Pseudo-Differential Operators. Wiley, Chichester (1998) · Zbl 0907.35146
[62] Taylor, M., Qualitative studies of linear equations, No. 116 (1996), New York · doi:10.1007/978-1-4757-4187-2
[63] Weinberg, E., Yi, P.: Magnetic monopole dynamics, supersymmetry, and duality. Phys. Rep. 438(2-4), 65-236 (2007) · doi:10.1016/j.physrep.2006.11.002
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