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108
Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Cited by 101 (22 self)
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
Superconnections and Parallel Transport
, 2006
"... Abstract. This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold M. A superpath in M is, loosely speaking, a path in M together with an odd vector field in M along the path. We also deve ..."
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Cited by 15 (3 self)
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Abstract. This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold M. A superpath in M is, loosely speaking, a path in M together with an odd vector field in M along the path. We also develop a notion of parallel transport associated with a connection (a.k.a. covariant derivative) on a vector bundle over a supermanifold which is a direct generalization of the classical notion of parallel transport for connections over manifolds. 1.
Introduction to supergeometry
, 2011
"... These notes are based on a series of lectures given by the first author at the school of ..."
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Cited by 15 (2 self)
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These notes are based on a series of lectures given by the first author at the school of
From Topological Field Theory to Deformation Quantization and Reduction
 Proceedings of ICM 2006, Vol. III, 339365 (European Mathematical Society
, 2006
"... Abstract. This note describes the functionalintegral quantization of twodimensional topological field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded manifolds and to the B ..."
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Cited by 12 (1 self)
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Abstract. This note describes the functionalintegral quantization of twodimensional topological field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded manifolds and to the Batalin–Vilkovisky formalism is included.
SUPERSYMMETRIC QFT, SUPER LOOP SPACES AND BISMUTCHERN CHARACTER
, 711
"... Abstract. We construct the BismutChern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by Stolz and Teichner [ST07]. We show that this differential form comes up via a loopdeloop process ..."
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Cited by 10 (1 self)
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Abstract. We construct the BismutChern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by Stolz and Teichner [ST07]. We show that this differential form comes up via a loopdeloop process when one goes from 11D theory over a manifold to 01D theory over its loop space by crossing with the standard circle. The super loop space is used in our construction. 1.
The geodesic flow on a Riemannian supermanifold
 Journal of Geometry and Physics
, 2012
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On complex Lie supergroups and split homogeneous supermanifolds
 Transform. Groups
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