About: Quillen metric

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In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by Jean-Michel Bismut and Dan Freed.

Attributes	Values
rdfs:label	Quillen metric (en)
rdfs:comment	In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by Jean-Michel Bismut and Dan Freed. (en)
dcterms:subject	Differential geometry
Wikipage page ID	49946643 (xsd:integer)
Wikipage revision ID	1072508297 (xsd:integer)
Link from a Wikipage to another Wikipage	Anomaly (physics) Holomorphic vector bundle Riemann surface Curvature form Differential geometry Analytic torsion Mathematics Chern–Simons theory Supertrace Edward Witten Eigenvalue Geometric invariant theory Constant scalar curvature Kähler metric Dan Freed Symplectic reduction Simon Donaldson Sobolev space Stable vector bundle Complex manifold Dolbeault operator Toshiki Mabuchi K-theory Line bundle Affine space Algebraic varieties Ample line bundle Daniel Quillen Elliptic equation Kempf–Ness theorem Atiyah–Singer index theorem Jean-Michel Bismut Chern class Karen Uhlenbeck Laplacian Holonomy Moduli space Differential geometry Fredholm operator Kähler manifold Meromorphic continuation Michael Atiyah Raoul Bott Shing-Tung Yau Yang–Mills equations Smooth manifold Vector bundle Compact Riemann surface Narasimhan–Seshadri theorem Hermitian metric Hilbert spaces Zeta function (operator) Topological space Chern–Weil theory Quillen determinant line bundle Hitchin–Kobayashi correspondence Kähler metric Elliptic differential operator Elliptic regularity Inductive proof Almost-complex structure Chern connection Projective algebraic manifold Smooth line bundle Cohomology class Unitary connection K-energy functional
sameAs	Quillen metric Quillen metric
dbp:wikiPageUsesTemplate	dbt:Reflist dbt:Short_description
has abstract	In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by Jean-Michel Bismut and Dan Freed. The Quillen metric was used by Quillen to give a differential-geometric interpretation of the ample line bundle over the moduli space of vector bundles on a compact Riemann surface, known as the Quillen determinant line bundle. It can be seen as defining the Chern–Weil representative of the first Chern class of this ample line bundle. The Quillen metric construction and its generalizations were used by Bismut and Freed to compute the holonomy of certain determinant line bundles of Dirac operators, and this holonomy is associated to certain anomaly cancellations in Chern–Simons theory predicted by Edward Witten. The Quillen metric was also used by Simon Donaldson in 1987 in a new inductive proof of the Hitchin–Kobayashi correspondence for projective algebraic manifolds, published one year after the resolution of the correspondence by Shing-Tung Yau and Karen Uhlenbeck for arbitrary compact Kähler manifolds. (en)
prov:wasDerivedFrom	wikipedia-en:Quillen_metric?oldid=1072508297&ns=0
page length (characters) of wiki page	12710 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Quillen_metric
is rdfs:seeAlso of	Line bundle
is Link from a Wikipage to another Wikipage of	Holomorphic vector bundle Dan Freed Simon Donaldson Daniel Quillen Kobayashi–Hitchin correspondence Jean-Michel Bismut Zeta function (operator) Quillen determinant line bundle
is known for of	Daniel Quillen
is foaf:primaryTopic of	wikipedia-en:Quillen_metric

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