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Topic: Determinant bundle

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	Line bundle
		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.
		It carries the universal real line bundle; in terms of homotopy theory that means that any real line bundle on a CW complex determines a classifying map from to, making a bundle isomorphic to the pullback of the universal bundle.
		There are theories of holomorphic line bundles on complex manifolds, and invertible sheaves in algebraic geometry, that work out a line bundle theory in those areas.
	pedia.newsfilter.co.uk /wikipedia/l/li/line_bundle.html (560 words)

	Line bundle - Wikipedia, the free encyclopedia
		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.
		This construction is in particular applied to the tangent bundle of a smooth manifold.
		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 (675 words)

	Line bundle -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		There is an evident difference between one-dimensional real line bundles (as just described) and one-dimensional ((psychoanalysis) a combination of emotions and impulses that have been rejected from awareness but still influence a person's behavior) complex line bundles.
		In general if V is a vector bundle on a space X, with constant fibre dimension n, the n-th (Click link for more info and facts about exterior power) exterior power of V taken fibre-by-fibre is a line bundle, called the determinant line bundle.
		It carries the universal real line bundle; in terms of homotopy theory that means that any real line bundle on a (Click link for more info and facts about CW complex) CW complex determines a classifying map from to, making a bundle isomorphic to the pullback of the universal bundle.
	www.absoluteastronomy.com /encyclopedia/l/li/line_bundle.htm (733 words)

	Station Information - Line bundle (Site not responding. Last check: 2007-11-05)
		In fact the topology of the 1x1 invertible real matrices and complex matrices is entirely different: the first of those is a space homotopy equivalent to a discrete two-point space (positive and negative reals contracted down), while the second has the homotopy type of a circle.
		In the case of the comlex line bundle, we are looking in fact also for circle bundles.
		It carries the universal real line bundle; in terms of homotopy theory that means that any real line bundle L on a CW complex X determines a classifying map from X to RP, making L a bundle isomorphic to the pullback of the universal bundle.
	www.stationinformation.com /encyclopedia/l/li/line_bundle.html (565 words)

	[No title]
		A well known analogous picture concerns a line bundle over the space $\A$ of unitary connections on some hermitian vector bundle over a K\"{a}hler manifold (see \cite{Don2}); in this case, the corresponding line bundle is given by a determinant bundle of certain twisted Dirac operators.
		Define $\L'\rightarrow J$ to be the determinant of the Dirac family coupled to the virtual bundle $(L-L^{-1})^n$, that is, \[ \L' = \bigotimes_i \L_i^{(-1)^i {n\choose i}}, \] where $\L_i$ is the determinant line of the family of twisted Dirac operators $\Gamma(S_J^+ \otimes L^{n-2i}) \rightarrow \Gamma(S_J^- \otimes L^{n-2i})$.
		Second, if we extend a covariant derivative on a $n$-dimensional vector bundle $E \rightarrow M$ to a covariant derivative on $\Lambda^E$ via the signed Leibniz rule, then the trace of the curvature endomorphism $F(v_1,v_2) \in \Gamma(\End(\Lambda^E))$ restricted to $\Lambda^k E$ is ${n-1\choose k-1}$ times the trace of $F(v_1,v_2)$ restricted to $E$.
	math.stanford.edu /~lng/math/dirac.tex (3319 words)

	Citebase - Curvature on determinant bundles and first Chern forms
		The Quillen-Bismut-Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators.
		We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut-Freed connection on the determinant bundle.
		Determinants of invertible pseudo-differential operators (PDOs) close to positive self-adjoint ones are defined throughthe zeta-function regularization.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0009172 (993 words)

	Dictionary of Meaning www.mauspfeil.net (Site not responding. Last check: 2007-11-05)
		There is an evident difference between one-dimensional real line bundles (as just described) and one-dimensional complex number complex line bundles.
		The resulting determinant bundle is responsible for the phenomenon of tensor density 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.
		There are theories of holomorphic line bundles on complex manifolds, and invertible sheaf invertible sheaves in algebraic geometry, that work out a line bundle theory in those areas.
	www.mauspfeil.net /Line_bundle.html (676 words)

	MATHEMATICS COLLOQUIUM
		In this talk I will describe how the tools that Chern and Simons used generalize into higher dimensions, and how the technique can be used to construct a complex line bundle from a family of principal bundles.
		This line bundle is the same as the determinant line bundle defined by D. Quillen in 1985.
		Furthermore, I will give an outline of how the Chern-Simons invariant in the setup of a family of principal bundles appears as a section of the complex line bundle.
	www.math.ku.dk /cal/events/432.htm (118 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 /item?id=AIF_1997__47_3_885_0 (269 words)

	Science Fair Projects - Serre duality
		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.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Serre_duality (563 words)

	colloquium: Gauge Anomalies and the Families Index via Surgery
		The obstruction to finding a global renormalization is measured the Chern class of the determinant bundle which occurs as the degree two component of the index theorem for the family, referred to by physicists as a topological anomaly.
		Thus there are many connections on the determinant bundle, not just the zeta-function connection, and we would like to know how they relate.
		More precisley, it is of interest to know whether the renormalizations of the determinant defined by these connections are the same and how the curvatures compare with the local family's index density constructed from the zeta-function metric.
	www2.mat.dtu.dk /events/uk?id=71 (441 words)

	Tensor (Site not responding. Last check: 2007-11-05)
		A tensor with density r transforms as an ordinary tensor under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian to the r
		This is best explained, perhaps, using vector bundles: where the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles r times.
		is the determinant of the coefficient array a
	www.sciencedaily.com /encyclopedia/tensor (1214 words)

	Immanuel Kant -- Metaphysics [Internet Encyclopedia of Philosophy]
		A central epistemological problem for philosophers in both movements was determining how we can escape from within the confines of the human mind and the immediately knowable content of our own thoughts to acquire knowledge of the world outside of us.
		If we think of ourselves as completely causally determined, and not as uncaused causes ourselves, then any attempt to conceive of a rule that prescribes the means by which some end can be achieved is pointless.
		We are neither wholly determined to act by natural impulse, nor are we free of non-rational impulse.
	www.iep.utm.edu /k/kantmeta.htm (9445 words)

	Atlas: Canonical Representations of Orthogonal Groups of Line-Bundle-Valued Ternary Quadratic Bundles over Schemes with ...
		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.
	atlas-conferences.com /cgi-bin/abstract/caoz-56 (625 words)

	Abstract from Journal of Differential Geometry - 66-2-5 - Xiaotao Sun & I-Hsun Tsai (Site not responding. Last check: 2007-11-05)
		For a family of smooth curves, we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves.
		When the Kodaira--Spencer map of the family of curves is an isomorphism, we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves.
		As an application, the so-called Hitchin connection on the direct image of (powers of) the theta line bundle is derived naturally from the identification theorem.
	www.intlpress.com /JDG/2004/66-2-5nf.htm (139 words)

	AIF : Tome 50 fascicle 5 -- 2000
		Sections of the determinant bundle on the moduli space of rank 2 semi-stable sheaves on the projective plane
		Le Potier's ``Strange Duality" conjecture gives an isomorphism between the space of sections of the determinant bundle on two different moduli spaces of semi-stable sheaves on the projective plane
		We compute in this case the dimension of the space of global sections of the determinant bundle on
	annalif.ujf-grenoble.fr /Vol50/E505_1/E505_1.html (132 words)

	[No title]
		Non-flat algebraic connections for bundles not admitting flat structures on complex projective manifolds are virtually non-existent (we know of none), and a deep theorem of Reznikov \cite{Re} implies that Chern-Simons classes are torsion for flat bundles on such spaces.
		Given a bundle of rank $N$ with connection $(E,\nabla)$ on $X$ and an invariant polynomial $P$ of degree $n$ on the Lie algebra of ${\rm GL}_N$ (cf.
		Given $E$ a vector bundle on $X$, we consider the sequence (Atiyah sequence) \begin{equation} \label{eqn:Atiyah} 0\to E\otimes_{\sO_X}\Omega^1_X \to E\otimes_{\sO_X}\sP_X\to E \to 0 \end{equation} The tensor in the middle is taken using the left $\sO_X$-structure, and then the sequence is viewed as a sequence of $\sO_X$-modules using the {\it right} $\sO_X$-structure.
	www.uni-essen.de /~mat903/preprints/chern_simons.tex (8223 words)

	Line bundle - Enpsychlopedia
		The Möbius band corresponds to a double cover of the circle (the $\Theta\ \rightarrow\ 2\Theta mapping) and can be viewed as we wish as having fibre two points, the unit interval or the real line: the data are equivalent.$
		According to general theory about classifying spaces, we should look for contractible spaces on which there are group actions of the respective groups $C_2 and S^1, that are free actions.$
		It carries the universal real line bundle; in terms of homotopy theory that means that any real line bundle $L on a CW complex X$ determines a classifying map from $X to \mathbb{R}P, making L a$ bundle isomorphic to the pullback of the universal bundle.
	psychcentral.com /psypsych/Determinant_bundle (736 words)

	[No title]
		The determinant is a nonlocal expression, so in order to work only with local expressions, one should replace this determinant with a Gaussian integral over the space of fields times two copies of the odd space of sections of the coadjoint bundle of $E$.
		The basic fields are the connection $A$, the matter fields $\phi$, and the ghosts $c,\bar c$, which are sections of the adjoint bundle to $E$.
		Differentiating the determinant ratio $\text{det}(D_AD_{A_0}-m^2)/\text{det}(D_A)$ in the direction of a gauge trasformation $t\in \hat \g$, we obtain (using the path integral interpretation) that it is equal to $m\
	www.math.ias.edu /QFT/spring/witten16.tex (2092 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		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..
		This was later proven many many times, but all proofs essentially rely on the identification of powers of Det with certain special representations of the affine group g^ of G, making it perhaps the first direct application of conformal field theory techniques to a problem in classical algebraic geometry.
	www.ma.utexas.edu /~benzvi/math/Langlands4 (465 words)

	[No title]
		In contrast to the first, it depends on spectral properties of the Hamiltonian; it is related to (an appropriate regularization of) its determinant.
		\end{eqnarray} The determinant term is an exact two form on $\Phi$, hence it does not contribute to charge transport, and represents conductance fluctuations.
		The determinant of $ D^{} D$ is expressed in terms of Selberg zeta function by \cite{DP}: \begin{eqnarray} \det D^{} D = c_h \, Z(B,\chi), \end{eqnarray} $c_h$ is a constant independent of the fluxes.
	www.ma.utexas.edu /mp_arc/e/94-151.latex (1897 words)

	Citebase - Determinant bundle in a family of curves, after A. Beilinson and V. Schechtman (Site not responding. Last check: 2007-11-05)
		Determinant bundle in a family of curves, after A. Beilinson and V. Schechtman
		Physics 118 (1988), 651-701, A. Beilinson and V. Schechtman define on the total space of a smooth family of curves a so-called trace complex associated to a vector bundle, the 0-th relative cohomology of which is the Atiyah algebra of the determinant bundle.
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
	citebase.eprints.org /cgi-bin/citations?id=oai%3AarXiv%2Eorg%3Amath%2F9910053 (236 words)

	[No title]
		In this paper we extend the result on base point freeness of the powers of the determinant bundle on the moduli space of vector bundles on a curve.
		We describe the parabolic analogues of parabolic theta functions, then we determine a uniform bound depending only on the rank of the parabolic bundles.
		As an application of the theorem on base point freeness, we characterize parabolic semistability on the algebraic stack of quasi-parabolic bundles as the base locus of the linear system of the parabolic determinant bundle.
	dx.doi.org /10.1142/S0129167X04002272 (183 words)

	Curvature On Determinant Bundles And First Chern Forms (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Abstract: The Quillen-Bismut-Freed construction associates a determinant line bundle with connection to an in nite dimensional super vector bundle with a family of Dirac-type operators.
		We de ne the regularized rst Chern form of the in nite dimensional bundle, and relate it to the curvature of the Bismut-Freed connection on the determinant bundle.
		In nite dimensions, these forms agree (up to sign), but in in nite dimensions there is a correction term, which we express in terms of Wodzicki residues.
	citeseer.ist.psu.edu /603469.html (419 words)

	History of GAEL (Site not responding. Last check: 2007-11-05)
		Sections of the determinant bundle on a Jacobian.
		On a smooth compactification of moduli spaces of framed physical instanton bundles.
		Degeneration loci of vector bundle maps and ampleness.
	www-euclid.mathematik.uni-kl.de /~gael/History/History.html (347 words)

	An Atomic Model of Actin Filaments Cross-linked by Fimbrin and Its Implications for Bundle Assembly and Function -- ...
		This view shows the characteristic two crossbands often observed in longitudinal cross-sections in various hexagonally packed in vitro actin bundles including actin–fimbrin bundles in the stereocilia from hair cells of bird cochlea.
		Matsudaira, P., Mandelkow, E., Renner, W., Hesterberg, L.K., and Weber, K. Role of fimbrin and villin in determining the interfilament distances of actin bundles.
		Spudich, J., and Amos, L. Structure of actin filament bundles from microvilli of sea urchin eggs.
	www.jcb.org /cgi/content/full/153/5/947 (7485 words)

	phd
		This refers to the 'determinant' in the usual sense of linear algebra, but for linear operators on infinite dimensional spaces (of functions) whose eigenvalues diverge.
		In this sense, the determinant may be regarded as a refinement of the index, but one which requires a regularization procedure to make sense of it as a number.
		the determinant on the closed manifold is obtained by averaging away the choice of boundary condition.
	www.mth.kcl.ac.uk /staff/sscott/phd.html (727 words)

	research (Site not responding. Last check: 2007-11-05)
		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 (240 words)

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