# zbMATH — the first resource for mathematics

The residue determinant. (English) Zbl 1236.58037
Summary: The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators ($$\psi$$-dos) associated to the Guillemin-Wodzicki residue trace. The resulting residue determinant functional is multiplicative, a local invariant, and not defined by a regularization procedure. The residue determinant is consequently a quite different object from the zeta function determinant, which is nonlocal and nonmultiplicative. Indeed, the residue determinant does not arise as the derivative of a trace on the complex power operators and does not depend on a choice of spectral cut. The identification of a certain residue determinant with the index of an elliptic $$\psi$$-do shows the residue determinant to be topologically significant.

##### MSC:
 58J52 Determinants and determinant bundles, analytic torsion 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) 35J46 First-order elliptic systems 47G30 Pseudodifferential operators 58J40 Pseudodifferential and Fourier integral operators on manifolds
##### Keywords:
determinant; multiplicative; residue trace; zeta
Full Text:
##### References:
 [1] Bost J., Asterique 152 pp 113– (1988) [2] DOI: 10.1016/0926-2245(91)90004-S · Zbl 0785.53025 · doi:10.1016/0926-2245(91)90004-S [3] Burak T., Ann. Scusla. Norm. Sup. Pisa 24 pp 209– (1970) [4] Gilkey P., 2nd ed., in: Invariance Theory, the Heat Equation and the Atiyah–Singer Index Theorem (1995) · Zbl 0856.58001 [5] Grubb , G. On the Logarithmic Component in Trace Defect Formulas . Preprint, arXiv: math.AP/0411483 . · Zbl 1087.35100 [6] Grubb G., AMS Contemp. Math. Proc. 366 pp 67– (2005) · doi:10.1090/conm/366/06725 [7] DOI: 10.1007/BF01884310 · Zbl 0851.58043 · doi:10.1007/BF01884310 [8] DOI: 10.1006/aima.1993.1064 · Zbl 0803.58052 · doi:10.1006/aima.1993.1064 [9] Jacobson N., Interscience Tracts in Pure and Applied. Mathematics. 10, in: Lie Algebras (1962) [10] Kassel C., Asterique 177 pp 199– (1989) [11] Kontsevich , M. , Vishik , S. ( 1994, 1995 ).Determinants of Elliptic Pseudodifferential OperatorsarXiv: hep-th/9404046Geometry of Determinants of Elliptic Operators. Functional Analysis on the Eve of the 21st Century 1. Birkhäuser;Progr. Math. 131 : 173 – 197 . [12] McKean H., J. Diff. Geom. 1 pp 43– (1967) [13] DOI: 10.1007/s002200050301 · Zbl 0947.58025 · doi:10.1007/s002200050301 [14] DOI: 10.1215/S0012-7094-95-07918-6 · Zbl 0854.35137 · doi:10.1215/S0012-7094-95-07918-6 [15] DOI: 10.1215/S0012-7094-95-07919-8 · Zbl 0851.58048 · doi:10.1215/S0012-7094-95-07919-8 [16] Paycha , S. ( 2004 ) Anomalies and regularization techniques in mathematics and physics . Preprint (Colombia) . [17] Paycha S., Infinite-dimensional Groups and Manifolds (2002) [18] Paycha , S. , Scott , S. (2004). The Laurent expansion for regularized integrals of holomorphic symbols. preprint . · Zbl 1125.58009 [19] Ponge , R. ( 2005 ). Spectral asymmetry, zeta functions and the non commutative residue . Preprint, arXiv: math.DG/0310102 . [20] DOI: 10.1006/jfan.2001.3893 · Zbl 1007.58018 · doi:10.1006/jfan.2001.3893 [21] Scott , S. , Zagier , D. ( 2004 ). A symbol proof of the local index theorem . Preprint . [22] Seeley , R. T. ( 1967 ).Complex Powers of an Elliptic Operator. AMS Proc. Symp. Pure Math. X, 1966. Providence AMS: pp. 288 – 307 . [23] Shubin M. A., Pseudodifferential Operators and Spectral Theory (1987) · Zbl 0616.47040 · doi:10.1007/978-3-642-96854-9 [24] Simon B., LMS Lecture Notes 35, in: Trace Ideals and Their Applications (1979) [25] DOI: 10.1007/BF01404760 · Zbl 0489.58030 · doi:10.1007/BF01404760 [26] DOI: 10.1007/BF01403095 · Zbl 0538.58038 · doi:10.1007/BF01403095 [27] DOI: 10.1007/BFb0078372 · doi:10.1007/BFb0078372
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.