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Topic: Supermanifold

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Noncommutative geometry

	Supermanifolds - Cambridge University Press
		This is an updated and expanded second edition of a successful and well-reviewed text presenting a detailed exposition of the modern theory of supermanifolds, including a rigorous account of the super-analogs of all the basic structures of ordinary manifold theory.
		This basic material is then applied to the theory of supermanifolds, with an account of super-analogs of Lie derivatives, connections, metric, curvature, geodesics, Killing flows, conformal groups, etc. The book goes on to discuss the theory of super Lie groups, super Lie algebras, and invariant geometrical structures on coset spaces.
		‘Supermanifolds is destined to become the standard work for all serious study of super-symmetric theories of physics.’Nature
	www.cambridge.org /aus/catalogue/catalogue.asp?isbn=0521413206 (258 words)

	Supermanifold - Wikipedia, the free encyclopedia
		In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry.
		In physics, a supermanifold is a manifold with both bosonic and fermionic coordinates.
		A supermanifold is a concept in noncommutative geometry.
	en.wikipedia.org /wiki/Supermanifold (263 words)

	Poisson supermanifold - Wikipedia, the free encyclopedia
		A Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it (let me clarify a bit.
		M isn't a point set space and so, doesn't "really" exist, and really, this algebra is all we have),
		Every symplectic supermanifold is a Poisson supermanifold but not vice versa.
	en.wikipedia.org /wiki/Poisson_supermanifold (105 words)

	Supermanifold - TheBestLinks.com - TheBestLinks.com:Find or fix a stub, Star-algebra, Involution, Grassmann algebra, ...
		Supermanifold - TheBestLinks.com - TheBestLinks.com:Find or fix a stub, Star-algebra, Involution, Grassmann algebra,...
		Supermanifold, TheBestLinks.com:Find or fix a stub, Star-algebra, Involution...
		If M is a real manifold and we define an involution * over the fiber turning it into a * algebra, then the resulting algebra would define a real supermanifold.
	www.thebestlinks.com /Supermanifold.html (218 words)

	[No title]
		One says that a $(m,n)$-dimensional supermanifold $S$ is De Witt if it is locally modeled on $B_L{}^{m,n}$ and has an atlas such that the images of the coordinate maps are open in the De Witt topology of $B_L{}^{m,n}$.
		A {\it complex super line bundle} over a supermanifold can be thought of either as a rank $(1,0)$ or a rank $(0,1)$ super vector bundle since in both cases the standard fiber is $C_L$ while the structure group is $(C_L)_0^* \simeq GL_{1,0}(C_L) \simeq GL_{0,1}(C_L)$, the group of invertible even elements in $C_L$.
		Finally, we remind that, again in contrast with what happens for a general supermanifold, complex super line bundles over a De Witt supermanifold are classified by their {\it obstruction class} and so they are in bijective correspondence with elements in the integer sheaf cohomology group $\check{H}^2(M,\IZ)$.
	www.ma.utexas.edu /mp_arc/papers/99-323 (5301 words)

	Supermanifold: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-20)
		A supermanifold is a concept in noncommutative geometry[For more info, click on this link].
		In mathematics, a sheaf f on a given topological space x gives a set or richer structure f(u) for each open set u of x....
		(they define a supermanifold as a point set space using charts with the "even coordinates" taking values in the linear combination of elements of Λ with even grading and the "odd coordinates" taking values which are linear combinations of elements of Λ with odd grading.
	www.absoluteastronomy.com /encyclopedia/s/su/supermanifold.htm (851 words)

	Kosmann-Schwarzbach, Monterde: Divergence operators and odd Poisson brackets
		We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections.
		Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).
		Leites, “Supermanifold Theory”, Karelia Branch of the USSR Acad.
	aif.cedram.org /aif-bin/item?id=AIF_2002__52_2_419_0 (663 words)

	Citebase - Gauged Linear Sigma Model on Supermanifold
		We formulate a gauged linear sigma model on a supermanifold.
		We find out a constraint for the one-loop divergence to vanish, which is consistent with a Ricci flatness condition for the supermanifold.
		By providing a general correspondence between Landau-Ginzburg orbifolds and non-linear sigma models, we find that the elusive mirror of a rigid manifold is actually a supermanifold.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0503074 (1829 words)

	Splitness \| Musings
		A supermanifold is said to be split if it can be covered in coordinate charts, such that the transition functions are at most linear in the odd coordinates.
		The archetype of a split supermanifold is a vector bundle over an ordinary bosonic manifold, where the fiber directions are taken to be odd.
		As long as the odd dimensionality is greater than one, non-split complex supermanifolds are as common as dirt.
	golem.ph.utexas.edu /~distler/blog/archives/000477.html (2686 words)

	Lifting of Holomorphic Actions on Complex Supermanifolds (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		We study the problem of lifting analytic actions of a Lie group G to a non-split complex analytic supermanifold (M;O) from its retract (M;O gr).
		In the case when G is compact (or complex reductive), two criteria for lifting a Lie group action are found.
		The rst one is invariance of the Cech 1-cocycle with values in a special automorphism sheaf of (M;O gr) determining the non-split supermanifold (M;O), while the second one is invariance of a certain dierential form of a special...
	citeseer.ist.psu.edu /onishchik00lifting.html (355 words)

	Quantum Geometry
		Supergeometry deals with supermanifolds, which are formally defined as dual objects to the extensions S(M).
		A principal goal of supersymmetry was to provide a unifying view of bosonic and fermionic fields, and to establish a framework for a mathematically consistent formulation of quantum theory of gravity.
		The space-time is viewed as a supermanifold, and the symmetry is described by supergroups, which are the supergeometric counterparts of Lie groups.
	www.matem.unam.mx /~micho/qgeom8.html (279 words)

	Supermanifold description of the BRS symmetries of skewsymmetric tensor gauge fields (Site not responding. Last check: 2007-10-20)
		Supermanifold description of the BRS symmetries of skewsymmetric tensor gauge fields
		The authors extend Tulczyjew's geometrical formulation of skewsymmetric tensor gauge fields as connections on generalised principal fibre bundles to the category of supermanifolds.
		Given a smooth d-dimensional manifold M and a k-form field on it, they construct a suitable supersmooth generalised principal fibre bundle P over a (d,2)-dimensional supermanifold M such that the BRS symmetries of the theory have a natural geometrical interpretation.
	stacks.iop.org /0305-4470/16/861 (225 words)

	Re: How do manifolds "grow"
		When we talk about interactions between strings, 1st order branes, certain quantities are conserved and do not change unless there is an interaction with another string.
		But these strings seem to be submanifolds of some supermanifold.
		If there is an overall manifold of which everything else is a submanifold of it, then there does not exist another supermanifold to interact with it, and certain quantities are conserved forever.
	superstringtheory.com /forum/topboard/messages5/79.html (281 words)

	r-Commutative Geometry
		So while the phase space of bosonic system is a manifold, the phase space of a system containing both bosonic and fermionic degrees of freedom is a ``supermanifold'' -- a (particular kind of) algebra which has ``even'' and ``odd'' elements, such that the even, or bosonic, elements commute, while the odd elements anticommute.
		While I personally don't think that bosons and fermions were created equal in the manner postulated by supersymmetry, I do favor an approach to physics which doesn't take bosons, or commuting variables, to be somehow superior to fermions, or anticommuting variables.
		My own personal twist (motivated by the work of many people on anyons, the braid group, quantum groups, etc.) is to try to take a look at what geometry would be like if one wanted to be fair to anyons as well as bosons and fermions.
	math.ucr.edu /home/baez/braids/node6.html (977 words)

	[No title]
		Every supermanifold has an underlying manifold called its "body", and here the "body" is a single point.
		But this ring is not the ring of functions on a supermanifold - basically since the presence of nilpotent elements has nothing to do with exterior algebras here.
		Conversely, most supermanifolds don't have the same relationship to algebraic geometry that my original example did - since the ring of functions on a supermanifold is rarely commutative.
	www.math.niu.edu /~rusin/known-math/00_incoming/schemes (1594 words)

	What IS a superfield, really? \| The String Coffee Table
		In the context of functorial transport over supermanifolds, I mentioned recently that sorting out the true nature of supermanifolds leads to concepts that may look quite sophisticated, compared to the familiar carefree handling of Grassmann coordinates.
		Another important example for the need of “generalized” supermanifolds in physics is the consideration of superstructures on the spaces of maps from one supermanifold to the other.
		But it is certainly not an ordinary supermanifold, but a generalized one, given by a functor from ordinary supermanifolds to sets.
	golem.ph.utexas.edu /string/archives/000763.html (6648 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		We consider the problem of classification of complex analytic supermanifolds with a given reduction $M$.
		As is well known, any such supermanifold is a deformation of its retract, i.e., of a supermanifold $\M$ whose structure sheaf $\Cal O$ is the Grassmann algebra over the sheaf of holomorphic sections of a holomorphic vector bundle $\bold E\to M$.
		For a compact manifold $M$, we apply Hodge theory to construct a finite-dimensional affine algebraic variety which can serve as a moduli variety for our classification problem; it is analogous to the Kuranishi family of complex structures on a compact manifold.
	www.maths.tcd.ie /EMIS/journals/LJM/vol4/onishchik.htm (177 words)

	The International Conference on Secondary Calculus and Cohomological Physics, Moscow, August 24 - August 31, ... (Site not responding. Last check: 2007-10-20)
		WDVV equations and their solutions for a $(3,2)$ and $(3,4)$ supermanifold; the isotropic and nonisotropic case
		Abstract: The WDVV equations for the isotropic and nonisotropic case on a supermanifold of dimension $(3,2)$ and $(3,4)$ will be derived.
		By the method of characteristics, the general solution is constructed after transforming the system of WDVV equations to appropriate coordinates.
	www.emis.de /proceedings/SCCP97/7.html (82 words)

	r-Commutative Geometry
		So while the phase space of bosonic system is a manifold, the phase space of a system containing both bosonic and fermionic degrees of freedom is a ``supermanifold'' -- a (particular kind of) algebra which has ``even'' and ``odd'' elements, such that the even, or bosonic, elements commute, while the odd elements anticommute.
		While I personally don't think that bosons and fermions were created equal in the manner postulated by supersymmetry, I do favor an approach to physics which doesn't take bosons, or commuting variables, to be somehow superior to fermions, or anticommuting variables.
		My own personal twist (motivated by the work of many people on anyons, the braid group, quantum groups, etc.) is to try to take a look at what geometry would be like if one wanted to be fair to anyons as well as bosons and fermions.
	www.math.ucr.edu /home/baez/braids/node6.html (977 words)

	Manchester Geometry Seminar (Site not responding. Last check: 2007-10-20)
		We study various generating operators of a given odd Poisson bracket on a supermanifold.
		They arise as the operators that map a function to the divergence of the associated Hamiltonian derivation, where divergences of derivations can be defined either in terms of Berezin volumes or of graded connections.
		Examples include the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd), the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres) and the "odd Laplacian" (the delta-operator) of Batalin-Vilkovisky quantization.
	www.ma.umist.ac.uk /tv/Seminar/1999-2000/yvette.html (144 words)

	CiteULike: The Supersymmetric Method in Random Matrix Theory and Applications to QCD (Site not responding. Last check: 2007-10-20)
		An elementary introduction of the supersymmetric method in Random Matrix Theory is given in the second and third lecture.
		The main topic of the second last lecture is the recent developments on the relation between the supersymmetric partition function and the Toda lattice hierarchy.
		Also in this case we use symmetry considerations to rewrite the generating function for the resolvent as an integral over a supermanifold.
	www.citeulike.org /user/RMT/article/817655 (580 words)

	ICTP Preprints titles - January 1999
		These variables generate a $q$-deformation of a flag supermanifold of the supergroup $GL(m/n)$, respectively $SL(m/n)$.
		The construction originates from quantization of a classical model of the superparticle we suggest.
		The physical phase space of the classical superparticle is embedded in a symplectic superspace $T^\ast({\rm R}^{1,2})\times{\cal L}^{12}$, where the inner Kahler supermanifold $\rm{\cal L}^{12}\cong OSp(22)/[U(1)\times U(1)]\cong SU(1,12)/[U(22)\times U(1)]$ provides the particle with superspin degrees of freedom.
	ces.iisc.ernet.in /hpg/envis/doc98html/miscictp99126.html (1077 words)

	Lie superalgebra structures in H-center dot (g;g) -- from Mathematica Information Center
		Abstract Let g=vect(M) be the Lie (super)algebra of vector fields on any connected (super)manifold M; let ldquo-rdquo be the change of parity functor, C i and H i the space of i-chains and i-cohomology.
		The Nijenhuis bracket makes into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in L * and for any Lie superalgebra g.
		We use a Mathematica-based package SuperLie (already proven useful in various problems) to explicitly describe the algebras l * for some simple finite dimensional Lie superalgebras g and their ldquorelativesrdquo - the nontrivial central extensions or derivation algebras of the considered simple ones.
	library.wolfram.com /infocenter/Articles/5895 (142 words)

	Halfvalue.com: Supermanifolds (Cambridge Monographs on Mathematical Physics) (ASIN 0521423775), by [Bryce Seligman ... (Site not responding. Last check: 2007-10-20)
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	www.halfvalue.com /displayO.jsp?code=0521423775 (469 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		A set of four even spacetime coordinates x^\mu (\mu = 0, 1, 2, 3) and two odd Grassmannian variables \theta and \bar\theta parametrize this six dimensional supermanifold.
		The new gauge invariant restriction on the above supermanifold, due to the augmented superfield formalism, owes its origin to the (super) covariant derivatives and their intimate relation with the (super) 2-form curvatures (\tilde F^{(2)})F^{(2)} constructed from the (super) 1-form gauge connections (\tilde A^{(1)})A^{(1)}.
		The results obtained separately by exploiting (i) the horizontality condition, and (ii) one of its consistent extensions, are shown to be a simple consequence of this new single restriction on the six (4, 2)-dimensional supermanifold.
	www.thphys.uni-heidelberg.de /cgi-bin/abstracts/hep-th:0510164 (113 words)

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