Differential geometry

Quillen metric

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. (Wikipedia).

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Examples: Converting Between Metric Units

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From playlist Fine Measurements

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From playlist Fine Measurements

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Micrometer / diameter of daily used objects

What was the diameter? music: https://www.bensound.com/

From playlist Fine Measurements

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From playlist Unit Conversions: Metric Units

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From playlist Unit Conversions: Metric Units

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Alexander RAHM - Verification of the Quillen conjecture in the rank 2 imaginary quadratic case

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Michael Atiyah | Stable vector bundle | Topological space | Curvature form | Zeta function (operator) | Geometric invariant theory | Analytic torsion | Fredholm operator | Holonomy | K-theory | Quillen determinant line bundle | Supertrace | Sobolev space | Atiyah–Singer index theorem | Complex manifold | Yang–Mills equations | Constant scalar curvature Kähler metric | Ample line bundle | Riemann surface | Raoul Bott | Mathematics | Narasimhan–Seshadri theorem | Dirac operator | Vector bundle | Kähler manifold | Affine space | Holomorphic vector bundle | Differential geometry | Chern class | Line bundle | Moduli space | Kempf–Ness theorem