Abstract
The work of Ray and Singer that introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting to bundles with (1,1) connections, which, by using the Newlander-Nirenberg theorem, are seen to be the bundles with both holomorphic and antiholomorphic structures. The resulting natural generalizations of Laplacians are not always self-adjoint, and the corresponding generalizations of analytic torsions are thus not always real-valued. The Cheeger-Müller theorem on equivalence in a topological setting of analytic torsion to classical topological torsion generalizes to this complex-valued torsion. On the algebraic side the methods introduced include a notion of torsion associated to a complex equipped with both boundary and coboundary maps.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 133-202 |
| Number of pages | 70 |
| Journal | Communications on Pure and Applied Mathematics |
| Volume | 63 |
| Issue number | 2 |
| DOIs | |
| State | Published - Feb 2010 |
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ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
Complex-valued analytic torsion for flat bundles and for holomorphic bundles with (1,1) connections. / Cappell, Sylvain E.; Miller, Edward Y.
In: Communications on Pure and Applied Mathematics, Vol. 63, No. 2, 02.2010, p. 133-202.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Complex-valued analytic torsion for flat bundles and for holomorphic bundles with (1,1) connections
AU - Cappell, Sylvain E.
AU - Miller, Edward Y.
PY - 2010/2
Y1 - 2010/2
N2 - The work of Ray and Singer that introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting to bundles with (1,1) connections, which, by using the Newlander-Nirenberg theorem, are seen to be the bundles with both holomorphic and antiholomorphic structures. The resulting natural generalizations of Laplacians are not always self-adjoint, and the corresponding generalizations of analytic torsions are thus not always real-valued. The Cheeger-Müller theorem on equivalence in a topological setting of analytic torsion to classical topological torsion generalizes to this complex-valued torsion. On the algebraic side the methods introduced include a notion of torsion associated to a complex equipped with both boundary and coboundary maps.
AB - The work of Ray and Singer that introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting to bundles with (1,1) connections, which, by using the Newlander-Nirenberg theorem, are seen to be the bundles with both holomorphic and antiholomorphic structures. The resulting natural generalizations of Laplacians are not always self-adjoint, and the corresponding generalizations of analytic torsions are thus not always real-valued. The Cheeger-Müller theorem on equivalence in a topological setting of analytic torsion to classical topological torsion generalizes to this complex-valued torsion. On the algebraic side the methods introduced include a notion of torsion associated to a complex equipped with both boundary and coboundary maps.
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U2 - 10.1002/cpa.20307
DO - 10.1002/cpa.20307
M3 - Article
VL - 63
SP - 133
EP - 202
JO - Communications on Pure and Applied Mathematics
JF - Communications on Pure and Applied Mathematics
SN - 0010-3640
IS - 2
ER -