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42
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
Ktheory in quantum field theory
 Current Develop. Math
"... Abstract. We survey three different ways in which Ktheory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted Ktheory, and we illustrate with some finite models. Part 2 is a review of pfaffians of ..."
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Cited by 28 (7 self)
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Abstract. We survey three different ways in which Ktheory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted Ktheory, and we illustrate with some finite models. Part 2 is a review of pfaffians of Dirac operators, anomalies, and the relationship to differential Ktheory. Part 3 is a geometric exposition of Dirac charge quantization, which in superstring theories also involves differential Ktheory. Parts 2 and 3 are related by the GreenSchwarz anomaly cancellation mechanism. An appendix, joint with Jerry Jenquin, treats the partition function of RaritaSchwinger fields. Grothendieck invented KTheory almost 50 years ago in the context of algebraic geometry, specifically in his generalization of the Hirzebruch RiemannRoch theorem [BS]. Shortly thereafter, Atiyah and Hirzebruch brought Grothendieck’s ideas into topology [AH], where they were applied to a variety of problems. Analysis entered after it was realized that the symbol of an elliptic operator determines an element of Ktheory. Atiyah and Singer then proved a formula for the index of such an operator (on a compact manifold) in terms of the Ktheory class of the symbol [AS1]. Subsequently, Ktheoretic ideas permeated other areas of linear analysis, algebra, noncommutative geometry, etc. One of the pleasant surprises of the past few years has been the relevance of Ktheory to superstring theory and related parts of theoretical physics. Furthermore, the story involves not only topological Ktheory, but also the Ktheory of C ∗algebras, the Ktheory of sheaves, and other forms of Ktheory. Not surprisingly, this new arena for Ktheory has inspired some developments in mathematics which are the subject of ongoing research. Our exposition here aims to explain three different ways in which topological Ktheory appears in physics, and how this physics motivates the mathematical ideas we are investigating. Part 1 concerns topological quantum field theory. Recall that an ndimensional topological theory assigns a complex number to every closed oriented nmanifold and a complex vector space to every closed oriented (n − 1)manifold. Continuing the superposition principle and ideas of locality to
Landweber: OffShell supersymmetry and filtered Clifford supermodules, arXiv:mathph/0603012v2 (in progress); 12
 Landweber: On Graph Theoretic Identifications of Adinkras, Supersymmetry Representations
, 2007
"... Abstract. An offshell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of enginee ..."
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Cited by 23 (6 self)
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Abstract. An offshell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded offshell representations of onedimensional Nextended supersymmetry, i.e., the super Poincaré algebra p 1N, correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded offshell representations of twodimensional (p, q)supersymmetry, i.e., the super Poincaré algebra p 1,1p,q, correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are the formal deformations of filtered superalgebras and supermodules, which give a onetoone correspondence between filtered spaces and graded spaces with even degreeshifting injections. This generalizes the machinery developed by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends Gerstenhaber’s discussion to the case of filtrations which are compatible with a supersymmetric structure, as well as to filtered modules in addition to filtered algebras. We also describe the analogous constructions for bifiltrations and bigradings. 1.
Supersymmetry of the chiral de Rham complex
, 2006
"... We present a superfield formulation of the chiral de Rham complex (CDR) [MSV99] in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N = 1 structure on CDR (action of the N = 1 super–Vira ..."
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Cited by 19 (4 self)
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We present a superfield formulation of the chiral de Rham complex (CDR) [MSV99] in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N = 1 structure on CDR (action of the N = 1 super–Virasoro, or Neveu–Schwarz, algebra). If the metric is Kähler, and the manifold Ricciflat, this is augmented to an N = 2 structure. Finally, if the manifold is hyperkähler, we obtain an N = 4 structure. The superconformal structures are constructed directly from the LeviCivita connection. These structures provide an analog for CDR of the extended supersymmetries of nonlinear σ–models.
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.
From Minimal Geodesics to Supersymmetric Field Theories
"... In memory of Raoul Bott, friend and mentor. Abstract. There are many models for the Ktheory spectrum known today, each one having its own history and applications. The purpose of this note is to give an elementary description of eight such models (and certain completions of them) and to relate all ..."
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Cited by 12 (3 self)
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In memory of Raoul Bott, friend and mentor. Abstract. There are many models for the Ktheory spectrum known today, each one having its own history and applications. The purpose of this note is to give an elementary description of eight such models (and certain completions of them) and to relate all of them by canonical maps, some of which are homeomorphisms (rather than just homotopy equivalences). Our survey begins with Raoul Bott’s iterated spaces of minimal geodesics in orthogonal groups, whichheusedtoprovehisfamousperiodicity theorem, and includes Milnor’s spaces of Clifford module structures as well as the Atyiah – Singer spaces of Fredholm operators. From these classical descriptions we move via spaces of unbounded operators and supersemigroups of operators to our most recent model, which is given by certain spaces of supersymmetric (11)dimensional field theories. These spaces were introduced by the second two authors for the purpose of generalizing them to spaces of certain supersymmetric (21)dimensional Euclidean field theories that are conjectured to be related to the Hopkins – Miller spectrum TMF of topological modular forms.
Supersymmetric Euclidean field theories and generalized cohomology
, 2008
"... 2 Results and conjectures 3 2.1 Segal’s definition of a conformal field theory.......... 6 ..."
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Cited by 9 (2 self)
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2 Results and conjectures 3 2.1 Segal’s definition of a conformal field theory.......... 6