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 |
Fingerprint
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.
UR - http://www.scopus.com/inward/record.url?scp=77952485446&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77952485446&partnerID=8YFLogxK
U2 - 10.1002/cpa.20307
DO - 10.1002/cpa.20307
M3 - Article
AN - SCOPUS:77952485446
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 -