Analytic torsion and holomorphic determinant bundles

Bismut, Jean-Michel; Gillet, Henri; Soulé, Christophe

doi:10.1007/bf01466774

Cited by 210 publications

(194 citation statements)

References 14 publications

Supporting

Mentioning

184

Contrasting

Unclassified

Order By: Relevance

“…Consequently, we may apply the differential form of the Atiyah-Singer index theorem for families which is proved in a series of papers [3,4,5] by J.-M. Bismut, D. Freed, H. Gillet, Ch. Soulé.…”

Section: It Is Clear That the Mapmentioning

confidence: 99%

GW invariants and invariant quotients

Halic

2002

Comment. Math. Helv.

View full text Add to dashboard Cite

show abstract

Section: It Is Clear That the Mapmentioning

confidence: 99%

GW invariants and invariant quotients

Halic

2002

Comment. Math. Helv.

View full text Add to dashboard Cite

show abstract

“…Towards this end the work of Bott and Chern [3] and after them, Bismut, Gillet and Soulé [2], is probably relevant. These authors study invariants for chain complexes of vector bundles on a given base.…”

Section: The Case Of a Principal Idealmentioning

confidence: 99%

Some geometric invariants from resolutions of Hilbert modules

Douglas

Misra

Varughese

2001

Systems, Approximation, Singular Integral Operators, and Related Topics

View full text Add to dashboard Cite

“…We emphasize that π is not assumed to be proper. If it is, Grauert's theorem [Gr] describes the holomorphic structure of the direct image, and many papers, including [Be3,BP,BF,BGS,T] reveal some aspects of its Hermitian structure; the most recent related work seems to be [Sch]. However, the chief difficulties we encounter here arise when π is not proper.…”

Section: Introductionmentioning

confidence: 99%

Direct Images, Fields of Hilbert Spaces, and Geometric Quantization

Lempert

Szőke

2014

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract. Geometric quantization often produces not one Hilbert space to represent the quantum states of a classical system but a whole family H s of Hilbert spaces, and the question arises if the spaces H s are canonically isomorphic. [ADW] and [Hi] suggest to view H s as fibers of a Hilbert bundle H, introduce a connection on H, and use parallel transport to identify different fibers. Here we explore to what extent this can be done. First we introduce the notion of smooth and analytic fields of Hilbert spaces, and prove that if an analytic field over a simply connected base is flat, then it corresponds to a Hermitian Hilbert bundle with a flat connection and path independent parallel transport. Second we address a general direct image problem in complex geometry: pushing forward a Hermitian holomorphic vector bundle E → Y along a non-proper map Y → S. We give criteria for the direct image to be a smooth field of Hilbert spaces. Third we consider quantizing an analytic Riemannian manifold M by endowing T M with the family of adapted Kähler structures from [LSz2]. This leads to a direct image problem. When M is homogeneous, we prove the direct image is an analytic field of Hilbert spaces. For certain such M-but not all-the direct image is even flat; which means that in those cases quantization is unique. Introduction.This paper is motivated by a problem in geometric quantization: that of uniqueness. At its simplest, geometric quantization is about associating with a Riemannian manifold M a Hermitian line bundle L → X and a Hilbert space H of its sections. In Kähler quantization, L is a holomorphic Hermitian line bundle and H consists of all square integrable holomorphic sections of L. One often knows how to find L, except that its construction involves choices, so that one really has to deal with a family L s → X s of line bundles and Hilbert spaces H s , parametrized by the possible choices s ∈ S. The problem of uniqueness is to find canonical unitary maps H s → H t corresponding to different choices s = t-or rather projective unitary maps, the natural class of maps, since only the projectivized Hilbert spaces have a physical meaning. There are various solutions to this problem, the first the Stone-von Neumann theorem [St1,vN1], long predating geometric quantization. It applies whenever two Hilbert spaces carry irreducible representations of the canonical commutation relations; if so, there is a unitary map, unique up to a scalar factor, that intertwines the two representations. However, the Hilbert spaces that geometric quantization supplies do not carry such representations unless the manifold to be quantized is an affine space. In geometric quantization there is the Blattner-Kostant-Sternberg pairing [Bl1-2,Ko2], which sometimes gives rise to the sought for unitary map, but even in simple cases it may fail to do so [R].In the early 1990s Hitchin in [Hi] and Axelrod, Della Pietra, and Witten in [ADW] considered a situation when the possible choices s form a complex manifold S. (One has to be careful with ...

show abstract

Analytic torsion and holomorphic determinant bundles

Cited by 210 publications

References 14 publications

GW invariants and invariant quotients

GW invariants and invariant quotients

Some geometric invariants from resolutions of Hilbert modules

Direct Images, Fields of Hilbert Spaces, and Geometric Quantization

Contact Info

Product

Resources

About