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[1] for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by Jean-Michel Bismut and Dan Freed.[2]
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.[3][4]
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.[5]
^Quillen, D. (1985), "Determinants of Cauchy-Riemann operators over a Riemann surface", Functional Analysis and Its Applications, 19 (1): 31–34, doi:10.1007/BF01086022, MR 0783704, S2CID 122340883
^Bismut, Jean-Michel; Freed, Daniel S. (1986), "The analysis of elliptic families. I. Metrics and connections on determinant bundles.", Comm. Math. Phys., 106 (1): 159–176, doi:10.1007/BF01210930, MR 0853982, S2CID 55389271
^Bismut, J.M. and Freed, D.S., 1986. The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem. Communications in mathematical physics, 107(1), pp.103-163.
^Witten, E., 1985. Global gravitational anomalies. Communications in Mathematical Physics, 100(2), pp.197-229.
^Donaldson, S.K., 1987. Infinite determinants, stable bundles and curvature. Duke Mathematical Journal, 54(1), pp.231-247.
geometry, the Quillenmetric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen for certain elliptic...
introduced by Quillen (1985). Quillen proved the existence of the Quillenmetric on the determinant line bundle, a Hermitian metric defined using the...
Daniel Gray Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic K-theory...
on supersymmetry, American Mathematical Society 1999 Constructed the Quillenmetric for a family of elliptic operators with Jean-Michel Bismut. Daniel Stuart...
Hermitian–Einstein metric (Donaldson 1987c), proven using an inductive proof and the theory of determinant bundles and Quillenmetrics. A non-singular,...
of the Quillen–Suslin theorem, a result in commutative algebra, first conjectured by Jean-Pierre Serre in 1955, and then proved by Daniel Quillen and Andrei...
spherically symmetric body with mass M {\displaystyle M} (see Schwarzschild metric). The solution contained terms of the form 1 − r s / r {\displaystyle 1-{r_{\text{s}}}/r}...
have no technical significance (e.g. it doesn't measure heating power). Quillen, Paul (1993). "What's RMS Power or RMS Watts?" (PDF). the Voltage that's...
proof. 1976 Daniel Quillen; and independently by Andrei Suslin Serre's conjecture on projective modules polynomial rings Quillen–Suslin theorem 1977...
Archived from the original on December 6, 2010. Retrieved May 29, 2011. Quillen, Dustin (December 10, 2010). "Billy Joel Tracks Headed for Rock Band 3...
Zisman Axiomatizes simplicial homotopy theory 1967 Daniel QuillenQuillen model categories and Quillen model functors: A framework for doing homotopy theory...
Archived from the original on February 19, 2015. Retrieved February 5, 2016. Quillen, Ed (March 18, 2007). "Coloradoan or Coloradan". The Denver Post. Denver...
categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a...
trivial. In particular, every line bundle is trivial. More generally, the Quillen–Suslin theorem implies that every algebraic vector bundle over an affine...
Mighty Jungle Jeff Rosen CBC Television Halifax Film Dirtgirlworld Cate McQuillen Hewey Eustace 2009–2011 Australia ABC Kids Canada CBC Television United...
(producer); Dora V. Smith c-11m June 4, 1952 Background for Reading & Expression The Aztecs I. James Quillen (collaborator) c-11m March 30, 1955 Video...
US men's national team?". MLSsoccer.com. Retrieved February 20, 2020. Quillen, Ian (February 24, 2020). "D.C. United confirm Paul Arriola may miss entire...
answer to this problem is the category of exterior spaces E which admits Quillen model structures and contains as a full subcategory the category of spaces...
manuscript entitled Pursuing Stacks. It began with a letter addressed to Daniel Quillen. This letter and successive parts were distributed from Bangor (see External...
March 2015. Friedlander, Eric; Grayson, Daniel (November 2012). "Daniel Quillen" (PDF). Notices of the AMS. 59 (10): 1392–1406. doi:10.1090/noti903. Archived...
the Bergman kernel near the diagonal. For example, the Catlin-De Angelo-Quillen theorem easily follows from this. In 2002 he was an invited speaker with...
revenue. In addition, the league has seen significant increases in every metric – including team valuations, attendance, sponsorship, and digital and social...
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