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Line bundle - Wikipedia, the free encyclopedia This reminds one of the orientation double cover on a differential manifold: indeed that's a special case in which the line bundle is the determinant bundle (top exterior power) of the tangent bundle. In general if V is a vector bundle on a space X, with constant fibre dimension n, the n-th exterior power of V taken fibre-by-fibre is a line bundle, called the determinant line bundle. The resulting determinant bundle is responsible for the phenomenon of tensor densities, in the sense that for an orientable manifold it has a global section, and its tensor powers with any real exponent may be defined and used to 'twist' any vector bundle by tensor product. en.wikipedia.org /wiki/Line_bundle   (680 words)

Serre duality - Wikipedia, the free encyclopedia In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves. While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent bundle, when V is a manifold, in full generality K cannot just be a single sheaf in the absence of some hypothesis of non-singularity on V. The formulation in full generality uses a derived category and Ext functors, to allow for the fact that K is now represented by a chain complex of sheaves. en.wikipedia.org /wiki/Serre_duality   (419 words)

CRM: Séminaires du laboratoire de Physique Mathématique   (Site not responding. Last check: 2007-11-03) The regularized determinant of the Laplacian on a constant curvature surface forms a function on the moduli space M_g of Riemann surfaces of genus g. Up to "anomaly" factors, it is the modulus square of the determinant of the dbar operator, which is a holomorphic function on M_g and a tau function for an integrable hierarchy. The determinant line bundle associated to a family of elliptic operators was introduced by Quillen. www.crm.umontreal.ca /~physmath/LabPhysMath/seminaire_fr.html   (2634 words)

[No title] In \cite{Sc99} it was shown that the determinant line bundle for such a family has a natural Hermitian metric defined by the Fredholm determinant of a canonically associated operator over the boundary. That the two metrics, and their associated connections, are nevertheless precisely related derives from a certain `relativity principle for determinants', which asserts that for preferred classes of unbounded operators, ratios of $\z$-determinants can be written canonically in terms of Fredholm determinants (Theorem 1). This means that the {\em abstract} relative determinant is a canonical section of the relative determinant line bundle $\DET(\Pf_2,\Pf_1)$. www.univie.ac.at /EMIS/journals/ERA-AMS/2001-01-003/2001-01-003.tex.html   (2872 words)

[No title]   (Site not responding. Last check: 2007-11-03) Given a line-bundle-valued ternary quadratic bundle over any scheme, there is a functorial representation of its group of orthogonal similitudes in the Witt-invariant, which by definition is the degree zero part of the associated generalised Clifford algebra bundle. The various orthogonal groups of a quadratic bundle are canonically determined in terms of the automorphisms of its even Clifford algebra. A specialised algebra arises from a honest quadratic form iff its determinant has a square root and arises from a bilinear form iff the line subbundle generated by 1 is a direct summand. math.rice.edu /~hassett/conferences/Mainz/balaji.html   (522 words)

David Radnell's research page   (Site not responding. Last check: 2007-11-03) An explicit description of the complex structure of the infinite-dimensional moduli space of Riemann surfaces with analytically parametrized boundary components is given and the holomorphicity of the sewing operation is proved. The determinant line bundle is shown to be a holomorphic bundle over this moduli space and the sewing operation is proved to be holomorphic on these bundles. Applications to modular functors, which are high-rank generalizations of the determinant line bundle, are discussed. www.math.lsa.umich.edu /~radnell/Research   (343 words)

Biswas: Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space. The moduli space of stable vector bundles over a moving curve is constructed, and on this a generalized Weil-Petersson form is constructed. Using the local Riemann-Roch formula of Bismut-Gillet-Soulé it is shown that the generalized Weil-Petersson form is the curvature of the determinant line bundle, equipped with the Quillen metric, for a vector bundle on the fiber product of the universal moduli space with the universal curve. www.numdam.org /numdam-bin/item?id=AIF_1997__47_3_885_0   (269 words)

[No title] The new technique arises from an insight that instead of looking at just one moduli space at a time it is important to look at an associated family of moduli spaces and that this family of moduli spaces is then birational to the finite dimensional representations of a (highly) noncommutative ring. Thus vector bundles over a smooth projective curve have a rank and a degree which are discrete invariants; they also have a determinant line bundle which is a continuous invariant. In the case of vector bundles over a smooth projective curve this amounts to considering vector bundles such that the ratio of the rank and degree is fixed. www.maths.bris.ac.uk /~maahs/res.htm   (834 words)

[No title] It also carries an important line bundle, the determinant line bundle, which appears throughout low-dimensional geometry, as far as Arakelov theory (in the description of Quillen metrics). The determinant bundle is defined by identifying LG/LG+ with a subset of a Grassmannian of planes of "half-infinite dimension" in a Hilbert space, and formally considering the determinant="top exterior power" of the tautological bundle.. the point in the moduli space.) In any case the bundles on the moduli space of bundles were known to some extent since the work of Mumford, but to elucidate their full significance we need to introduce the affine Lie algebra g^. www.ma.utexas.edu /~benzvi/math/Langlands4   (465 words)

[No title] The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C As a particular case of this we considered an oblate spheroid and evaluate the drag on it. www.ias.ac.in /mathsci/vol112/aug2002/absaug2002.html   (833 words)

Amazon.com: The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. (MN-44): Books: ...   (Site not responding. Last check: 2007-11-03) Beginning with the groundbreaking work of Donaldson in about 1980 it became clear that gauge-theoretic invariants of principal bundles and connections were an important tool in the study of smooth four-dimensional manifolds. This sets up the discussion of spin bundles in the next chapter, and, even though it is really not the place for it, the author does not prove that a principal SO(V) bundle lifts to a principal Spin(V) if and only if the second Stiefel-Whitney class is equal to zero. The next chapter then moves immediately to the Seiberg-Witten equations and they are viewed as nonlinear generalizations of elliptic partial differential equations in the sense that the linearization of both the Seiberg-Witten equations and the gauge group action is shown to be an ellipic complex. www.amazon.com /exec/obidos/tg/detail/-/0691025975?v=glance   (1196 words)

[No title] The Chern classes of the bundle are obtained by`, ` concatenation of the degree of the class with name. The universal rank-k `, ` quotient bundle on G.c is called Q.c, and its chern classes are`, ` c1, c2,..., c.k. The normal bundle of f must be computable by the `, ` normalbundle procedure, as well as f^* and f-*.`, ` To get the class of the image of the k-fold point locus, push forward `, ` and divide by k! www.mi.uib.no /~stromme/schubert/0.992/schubert/schubertR2   (1509 words)

Memorandum 1744, Abstract   (Site not responding. Last check: 2007-11-03) These Grassmann manifolds are built in such a way that the determinant line bundle and its dual over them still make sense. The same holds for the so-called tau-functions, determinants of certain Fredholm operators that measure the failure of equivariance at lifting the commuting flows of ths hierarchy to these bundles. They are perturbations of the trivial solution with the leading term of the perturbation determining the type. www.math.utwente.nl /publicaties/2004/1744abs.html   (174 words)

[No title]   (Site not responding. Last check: 2007-11-03) determines a projective embedding" seems to imply that the determinant line bundle is very ample, which is false in general without taking tensor powers, see F. Gavioli, Internat. The sentence should read "The space of sections of the pre-quantum line bundle over R_K(X,\mu_1,\ldots,\mu_b) is isomorphic to the space of genus zero conformal blocks....". The error does not affect the statement or proof of any of the results in the paper. www.math.rutgers.edu /~ctw/parab_errat.html   (90 words)

[No title]   (Site not responding. Last check: 2007-11-03) Freed's previous work on the geometry of the determinant bundle associated to the Dirac operator has been very influential. He will continue to study this and in particular calculate the first Chern class of the determinant line bundle. He will also work to gain geometric understanding of the various new cohomology theories which are currently being used to great effect in topology. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a8805684.txt   (177 words)

[No title]   (Site not responding. Last check: 2007-11-03) Abstract: We would be describing the moduli space resulting out of a reduction of the Seiberg-Witten equations to 2 dimensions and its geometric quantization which involves constructing a determinant line bundle over the moduli space whose Quillen curvature is a symplectic form. We generalize the bending deformations of n-gon linkages in Euclidean space (points on the complex line) of Kapovich-Millson (JDG,vol.44,1996,pp.479-513) to points on complex projective m-space. This in turn indicates how to solve the Toda PDE by the factorization method in infinite dimensions, where the unitary group is replaced by the group of area-preserving transformations of the sphere. www.math.psu.edu /nistor/MEETINGS/talks.html   (1116 words)

Geometry-Algebra-Singularities-Combinatorics Seminar Talk   (Site not responding. Last check: 2007-11-03) Deformation of the tangent bundles of moduli spaces of vector bundles Denote by M(r, L) the moduli space of stable vector bundles on X of rank r and determinant a line bundle L whose degree is coprime to r. It is shown that the number of deformations of the tangent bundle of M(r, L) is equal to the genus g. www.math.neu.edu /~suciu/gas/narasimhan.html   (108 words)

UIUC Dept. of Mathematics Seminar Calendar Further we show that the probability density can be thought of as a section of the determinant line bundle over moduli space of Riemann surfaces. Loewner evolutions then have a description in terms of "random walks" in the moduli space, and the thereto associated diffusion equation translates to the Virasoro action of a certain weight-two operator on a uniformised version of the determinant bundle. Abstract: Geometric knot theory is the study of geometric properties of space curves that derive from their topological knottedness. torus.math.uiuc.edu /cal/math/cal?year=2004&month=03&day=30&interval=day   (983 words)

MIT PDE/Analysis Seminar, Fall 2004   (Site not responding. Last check: 2007-11-03) Abstract: If we consider a family of Dirac operators on a manifold with boundary, then there is an associated family of Dirac operators defined on the boundary. It is a relatively well-known result that the determinant line bundle corresponding to this family must be trivial. In this talk, a geometric proof of this result will be given using cusp pseudodifferential operators. www-math.mit.edu /~andras/PDE.html   (345 words)

Talk 2802 data/Fall_1997/0922   (Site not responding. Last check: 2007-11-03) A classical theorem of Quillen relates the curvature of the determinant line bundle of a family of Cauchy-Riemann operators to the determinant (in the sense of zeta-functions) viewed as a function on the space of holomorphic structures. Since the curvature can be thought of as a representative for the first Chern class, Quillen's result raises the question whether there exists an expression for the total Chern class of the index bundle in terms of the determinants of the elliptic operators in the family. In this talk I will present such an expression and explain the proof in a few particular cases. www.math.duke.edu /mcal?abstract-2802   (138 words)

[No title]   (Site not responding. Last check: 2007-11-03) For any $m>0$\, with the sewing operation for vector bundles and the permutations of symmetric groups on local trivializations\, the moduli s pace ${\mathcal E}(m)$ of locally trivialized holomorphic vector bundles o f rank $m$ over the Riemann sphere forms an analytic partial operad. Then\, we discuss the classification and constructions of all $1$-dimensional modu lar functors $\widetilde{\mathcal E}(m)$ over the moduli space ${\mathcal E}(m)$\, including its determinant line bundle. Finally\, we will explai n the main theorem\, which says that the category of integrable algebras over a $1$-dimensional modular functor $\widetilde {\mathcal E}(m)$ is iso morphic to the category of\, what we call\, integrable generalized affine vertex operator algebras. webapps.jhu.edu /eventslist/icalendar.cfm?eventid=6849   (170 words)

ERA - Volume 7 On Noether's bound for polynomial invariants of a finite group Relative zeta determinants and the geometry of the determinant line bundle A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I www.ams.org /era/home-2001.html   (108 words)

papers   (Site not responding. Last check: 2007-11-03) For hard copies of articles not included here (which is generally because I can't find the tex files) send me an email. Geometry of the determinant line bundle in dimension one Zeta-determinant and Quillen determinant on the space of elliptic self-adjoint boundary conditions www.mth.kcl.ac.uk /staff/sscott/papers.html   (104 words)

The LQG Landscape | Musings Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators. The problem with applying the usual arguments for the chiral gauge anomalies to LQG directly is that the methods used, such as calculating the phase of the fermion determinant in the path integral or calculating a triangle diagram are background dependent. I do it because the nontriviality of the Determinant Line Bundle (in the cases above) is a topological fact robust enough (that’s the virtue of topological facts) that any correct formalism ought to see it. golem.ph.utexas.edu /~distler/blog/archives/000855.html   (12814 words)

Recent publications by Jouko Mickelsson   (Site not responding. Last check: 2007-11-03) Carey, J. Mickelsson, and M. Murray: Bundle gerbes applied to quantum field theory. Mickelsson and S. Scott: Functorial QFT, gauge anomalies, and the Dirac determinant line bundle. Second quantization, anomalies, and group extensions, Lectures (as a PS file) at Luminy meeting June 2 - 7, 1997 www.theophys.kth.se /~jouko/publ.html   (152 words)

Gillet, Soulé: Direct images in non-archimedean Arakelov theory We develop a formalism of direct images for metrized vector bundles in the context of the non-archimedean Arakelov theory introduced in our joint work with S. Bloch. We prove a Riemann-Roch-Grothendieck theorem for this direct image. SOULÉ, Characteristic classes for algebraic vector bundles with hermitian metric, Annals of Math., 131 ( www.numdam.org /numdam-bin/item?id=AIF_2000__50_2_363_0   (335 words)

Amazon.com: Gauge Theory and the Topology of Four-Manifolds (Ias/Park City Mathematics Series, V. 4): Books: John ...   (Site not responding. Last check: 2007-11-03) One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kähler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the $SU(2)$-moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. SIPs: based moduli space, parametrized moduli space, rational blowdown, connected sum theorem, smooth principal bundle (more) www.amazon.com /exec/obidos/tg/detail/-/0821805916?v=glance   (632 words)

Atlas: Canonical Representations of Orthogonal Groups of Line-Bundle-Valued Ternary Quadratic Bundles over Schemes with ... Atlas: Canonical Representations of Orthogonal Groups of Line-Bundle-Valued Ternary Quadratic Bundles over Schemes with Applications ---Dedicated to Professor Martin Kneser by Venkata Balaji, Thiruvalloor Eesanaipaadi Vector Bundle, Desingularisation, Azumaya bundle, bilinear form, Clifford algebra, discriminant bundle, line-bundle-valued form, orthogonal group, quadratic bundle, quaternion bundles, semiregular form, similarity, similitude, ternary form, Witt-invariant. The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoz-56. atlas-conferences.com /cgi-bin/abstract/caoz-56   (625 words)

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