Here the Kahler form a, cf. (2.24), is viewed as a (1, I)-form. Moreover, the integrand in 0.3) is equal to a universal constant times the Laplacian applied to (3.l), …
Here the Kahler form a, cf. (2.24), is viewed as a (1, I)-form. Moreover, the integrand in 0.3) is equal to a universal constant times the Laplacian applied to (3.l), plus terms involving at most second order derivatives of the metric. This follows from the formulas for E2 and E4 of Gilkey [30, p. 610], using that, in the computation of the supertrace, the linear contributions of the scalar curvature drop out. So the integrand in (1.3) involves fourth order derivatives of a.
In general, if the manifold is not Kähler, then the Dolbeault-Dirac operator D = 2 $$(\bar{\partial} + \bar{\partial}^{*})$$ is not the most suitable one for getting explicit formulas for (2.39) …
In general, if the manifold is not Kähler, then the Dolbeault-Dirac operator D = 2 $$(\bar{\partial} + \bar{\partial}^{*})$$ is not the most suitable one for getting explicit formulas for (2.39) and (2.40). For instance, if M is a complex analytic manifold and n = 2, then Gilkey [29, Thm. 3.7] proved that the difference $$\rm{trace}_{\bf{c}}\,\it{K}^{+}_{1} (\it{x}) - \rm{trace}_{\bf{c}}\,\it{K}^{-}_{1} (\it{x})$$ of the traces of the coefficients of t−1 in the asymptotic expansion (1.2) is equal to a universal constant times 3.1 $$\rm{d}\,\bar{\partial} \sigma\,=\,\partial \bar{\partial} \sigma\,=\,\partial\,\rm{d}\,\sigma.$$
This paper gives a new treatment of the Clifford algebras. We represent the pinor and spinor spaces as subspaces of the Clifford algebras and use these pinors to construct isomorphisms …
This paper gives a new treatment of the Clifford algebras. We represent the pinor and spinor spaces as subspaces of the Clifford algebras and use these pinors to construct isomorphisms between the Clifford algebras and the matrix algebras. In doing these we develop some spinor calculus.
We study bundles over a point, recalling the definition of the Clifford algebra Cℓ(V, q) of a real vector space V of dimension m equipped with a positive definite inner …
We study bundles over a point, recalling the definition of the Clifford algebra Cℓ(V, q) of a real vector space V of dimension m equipped with a positive definite inner product q; the ℤ2-grading of Clifford algebras is shown, followed by an introduction of complex representations of Clifford algebras and the concept of complex Cℓ(V, q)-modules and of Clifford multiplication; the isomorphism classes of irreducible Cℓ(V, q)-modules are studied.
Associated with any Euclidean space ℝi or Minkowski space ℝp,q is a universal Clifford algebra, denoted by and, respectively. Roughly speaking, a Clifford algebra is an associative algebra with unit …
Associated with any Euclidean space ℝi or Minkowski space ℝp,q is a universal Clifford algebra, denoted by and, respectively. Roughly speaking, a Clifford algebra is an associative algebra with unit into which a given Euclidean or Minkowski space may be embedded, in which the corresponding quadratic form may be expressed as the negative of a square. The real numbers ℝ, the complex numbers ℂ, and the quaternions ℝ are the simplest examples.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Abstract For physicists: For supersymmetric quantum mechanics, there are cases when a mod-2 Witten index can be defined, even when a more ordinary $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -valued …
Abstract For physicists: For supersymmetric quantum mechanics, there are cases when a mod-2 Witten index can be defined, even when a more ordinary $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -valued Witten index vanishes. Similarly, for 2d supersymmetric quantum field theories, there are cases when a mod-2 elliptic genus can be defined, even when a more ordinary elliptic genus vanishes. We study such mod-2 elliptic genera in the context of $$\mathcal {N}{=}(0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> supersymmetry, and show that they are characterized by mod-2 reductions of integral modular forms, under some assumptions. For mathematicians: We study the image of the standard homomorphism $$\begin{aligned} \pi _n\textrm{TMF}\rightarrow \pi _n\textrm{KO}((q))\simeq \mathbb {Z}/2((q)) \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mtext>TMF</mml:mtext> <mml:mo>→</mml:mo> <mml:msub> <mml:mi>π</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mtext>KO</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≃</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> for $$n=8k+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> or $$8k+2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , by relating them to the mod-2 reductions of integral modular forms.
Abstract The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that a 2-step nilpotent Lie algebra is isomorphic to …
Abstract The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that a 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with the Lie bracket. This choice of the standard pseudo-metric form allows us to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo H -type algebras have bases with rational structure constants, which implies that the corresponding pseudo H -type groups admit lattices.
Real Clifford algebras for arbitrary numbers of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in …
Real Clifford algebras for arbitrary numbers of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or quaternionic type. Spinors are defined as elements of minimal or quasi-minimal left ideals within the Clifford algebra and as representations of the pin and spin groups. Two types of Dirac adjoint spinors are introduced carefully. The relationship between mathematical structures and applications to describe relativistic fermions is emphasized throughout.
In the present paper we determine the group of automorphisms of pseudo H-type Lie algebras, that are two step nilpotent Lie algebras closely related to the Clifford algebras Cl(Rr,s).
In the present paper we determine the group of automorphisms of pseudo H-type Lie algebras, that are two step nilpotent Lie algebras closely related to the Clifford algebras Cl(Rr,s).
This note provides certain computations with transfer associated with projective bundles of Spin vector bundles. One aspect is to revise the proof of the main result of \cite{B} which says …
This note provides certain computations with transfer associated with projective bundles of Spin vector bundles. One aspect is to revise the proof of the main result of \cite{B} which says that all fourfold products of the Ray classes are zero in symplectic cobordism.
In this paper, we address the problem of constructing a class of representations of Clifford algebras that can be named “alphabetic (re)presentations.” The Clifford algebra generators are expressed as m-letter …
In this paper, we address the problem of constructing a class of representations of Clifford algebras that can be named “alphabetic (re)presentations.” The Clifford algebra generators are expressed as m-letter words written with a three-character or a four-character alphabet. We formulate the problem of the alphabetic presentations, deriving the main properties and some general results. At the end, we briefly discuss the motivations of this work and outline some possible applications.
Abstract Let be an exact Lagrangian submanifold of a cotangent bundle , asymptotic to a Legendrian submanifold . We study a locally constant sheaf of ‐categories on , called the …
Abstract Let be an exact Lagrangian submanifold of a cotangent bundle , asymptotic to a Legendrian submanifold . We study a locally constant sheaf of ‐categories on , called the sheaf of brane structures or . Its fiber is the ‐category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from to the ‐category of sheaves of spectra on with singular support in .
Abstract The first-order Lévy-Leblond differential equations (LLEs) are the non-relativistic analogous of the Dirac equation: they are the “square roots” of the Schrödinger equation in (1 + d ) dimensions …
Abstract The first-order Lévy-Leblond differential equations (LLEs) are the non-relativistic analogous of the Dirac equation: they are the “square roots” of the Schrödinger equation in (1 + d ) dimensions and admit spinor solutions. In this paper we show how to extend to the Lévy-Leblond spinors the real/complex/quaternionic classification of the relativistic spinors (which leads to the notions of Dirac, Weyl, Majorana, Majorana-Weyl, Quaternionic spinors). Besides the free equations, we also consider the presence of potential terms. Applied to a conformal potential, the simplest (1 + 1)-dimensional LLE induces a new differential realization of the osp (1 | 2) superalgebra in terms of first-order differential operators depending on the time and space coordinates.
Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the …
Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the product of two real spectral triples is problematic: among other drawbacks, it is non-commutative, non-associative, does not transform properly under unitaries, and often fails to define a proper spectral triple. In this paper, we explain that these various problems result from using the ungraded tensor product; by switching to the graded tensor product, we obtain a new prescription where all of the earlier problems are neatly resolved: in particular, the new product is commutative, associative, transforms correctly under unitaries, and always forms a well defined spectral triple.
We show that the relation between D-branes and noncommutative tachyons leads very naturally to the relation between D-branes and K-theory. We also discuss some relations between D-branes and K-homology, provide …
We show that the relation between D-branes and noncommutative tachyons leads very naturally to the relation between D-branes and K-theory. We also discuss some relations between D-branes and K-homology, provide a noncommutative generalization of the ABS construction, and give a simple physical interpretation of Bott periodicity. In addition, a framework for constructing Neveu–Schwarz fivebranes as noncommutative solitons is proposed.
In this paper, we identify the Ad-equivariant twisted K-theory of a compact Lie group G with the "Verlinde group" of isomorphism classes of admissible representations of its loop groups.Our identification …
In this paper, we identify the Ad-equivariant twisted K-theory of a compact Lie group G with the "Verlinde group" of isomorphism classes of admissible representations of its loop groups.Our identification preserves natural module structures over the representation ring R(G) and a natural duality pairing.Two earlier papers in the series covered foundations of twisted equivariant K-theory, introduced distinguished families of Dirac operators and discussed the special case of connected groups with free π1.Here, we recall the earlier material as needed to make the paper self-contained.Going further, we discuss the relation to semi-infinite cohomology, the fusion product of conformal field theory, the rôle of energy and a topological Peter-Weyl theorem. PartIII. Computation of twisted K G (G) 965 5.A "Mackey decomposition" lemma 966 6. Computation when the identity component is a torus 967 7. General compact groups 970 Part IV.Loop groups and admissible representations 973 8. Refresher on affine algebras 973 9. Twisted affine algebras 975 10.Representations of L f G 978 Part V. From representations to K-theory 981 11.The affine Dirac operator and its square 982 12.The Dirac family on a simple affine algebra 984 13.Arbitrary compact groups 986 Part VI.Variations and complements 988 14.Semi-infinite cohomology 988 15.Loop rotation, energy and the Kac numerator 990 16.Fusion with G-representations 994 17.Topological Peter-Weyl theorem 997 Appendix A. Affine roots and weights in the twisted case 1002 References 1005 Part I.
An adinkra is a graph-theoretic representation of spacetime supersymmetry. Minimal four-color valise adinkras have been extensively studied due to their relations to minimal 4D, $\cal N$ = 1 supermultiplets. Valise …
An adinkra is a graph-theoretic representation of spacetime supersymmetry. Minimal four-color valise adinkras have been extensively studied due to their relations to minimal 4D, $\cal N$ = 1 supermultiplets. Valise adinkras, although an important subclass, do not encode all the information present when a 4D supermultiplet is reduced to 1D. Eigenvalue equivalence classes for valise adinkra matrices exist, known as $\chi_{\rm o}$ equivalence classes, where valise adinkras within the same $\chi_{\rm o}$ equivalence class are isomorphic in the sense that adinkras within a $\chi_{\rm o}$-equivalence class can be transformed into each other via field redefinitions of the nodes. We extend this to non-valise adinkras, via Python code, providing a complete eigenvalue classification of "node-lifting" for all 36,864 valise adinkras associated with the Coxeter group $BC{}_4$. We term the eigenvalues associated with these node-lifted adinkras Height Yielding Matrix Numbers (HYMNs) and introduce HYMN equivalence classes. These findings have been summarized in a $Mathematica$ notebook that can found at the HEPTHools Data Repository (https://hepthools.github.io/Data/) on GitHub.
Some properties of the Clifford algebras [Formula: see text] and [Formula: see text] are presented, and three isomorphisms between the Dirac–Clifford algebra [Formula: see text] and [Formula: see text] are …
Some properties of the Clifford algebras [Formula: see text] and [Formula: see text] are presented, and three isomorphisms between the Dirac–Clifford algebra [Formula: see text] and [Formula: see text] are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group $pin + (2,4) is also investigated, in the light of a suitable isomorphism between [Formula: see text] and [Formula: see text]. After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of $pin + (2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian ℝ 4,1 spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac–Clifford algebra [Formula: see text] using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose, the Clifford algebra over ℝ 4,1 is also used to describe conformal maps, instead of ℝ 2,4 . Our formalism sheds some new light on the use of the paravector model and generalizations.
Let X be a real algebraic convex 3-manifold whose real part is equipped with a Pin− structure. We show that every irreducible real rational curve with nonempty real part has …
Let X be a real algebraic convex 3-manifold whose real part is equipped with a Pin− structure. We show that every irreducible real rational curve with nonempty real part has a canonical spinor state belonging to {± 1}. The main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real configuration of points, provided that these curves are counted with respect to their spinor states. These invariants provide lower bounds for the total number of such real rational curves independently of the choice of the real configuration of points.
In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi>Pin</mml:mi><mml:mo><!-- --></mml:mo><mml:mo …
In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi>Pin</mml:mi><mml:mo><!-- --></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation></mml:semantics></mml:math></inline-formula>-equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi>Pin</mml:mi><mml:mo><!-- --></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation></mml:semantics></mml:math></inline-formula>-equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B upper P i n left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi>B</mml:mi><mml:mi>Pin</mml:mi><mml:mo><!-- --></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">B\operatorname {Pin}(2)</mml:annotation></mml:semantics></mml:math></inline-formula>and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"><mml:semantics><mml:mi>j</mml:mi><mml:annotation encoding="application/x-tex">j</mml:annotation></mml:semantics></mml:math></inline-formula>-based Atiyah–Hirzebruch spectral sequence.
A bstract We use cobordism theory to analyse anomalies of finite non-abelian symmetries in 4 spacetime dimensions. By applying the method of ‘anomaly interplay’, which uses functoriality of cobordism and …
A bstract We use cobordism theory to analyse anomalies of finite non-abelian symmetries in 4 spacetime dimensions. By applying the method of ‘anomaly interplay’, which uses functoriality of cobordism and naturality of the η -invariant to relate anomalies in a group of interest to anomalies in other (finite or compact Lie) groups, we derive the anomaly for every representation in many examples motivated by flavour physics, including S 3 , A 4 , Q 8 , and SL(2 , 𝔽 3 ). In the case of finite abelian groups, it is well known that anomalies can be ‘truncated’ in a way that has no effect on low-energy physics, by means of a group extension. We extend this idea to non-abelian symmetries. We show, for example, that a system with A 4 symmetry can be rendered anomaly-free, with only one-third as many fermions as naïvely required, by passing to a larger symmetry. As another example, we find that a well-known model of quark and lepton masses utilising the SL(2 , 𝔽 3 ) symmetry is anomalous, but that the anomaly can be cancelled by enlarging the symmetry to a ℤ / 3 extension of SL(2 , 𝔽 3 ).
Techniques have been developed to overcome the fact that orientable manifolds do not necessarily admit a Spinc structure. The authors point out that in the cases of primary interest in …
Techniques have been developed to overcome the fact that orientable manifolds do not necessarily admit a Spinc structure. The authors point out that in the cases of primary interest in physics, orientable manifolds always admit Spinc structures, so that it is not really necessary to consider these more sophisticated techniques.
Nontrivial elements of homotopy groups for unitary, orthogonal, and symplectic groups are given explicitly. In particular, (a) representatives of generators of nontrivial homotopy groups of stable special unitary, orthogonal, and …
Nontrivial elements of homotopy groups for unitary, orthogonal, and symplectic groups are given explicitly. In particular, (a) representatives of generators of nontrivial homotopy groups of stable special unitary, orthogonal, and symplectic groups are constructed using Clifford algebras; (b) the values for ‘‘winding numbers’’ for stable SU, SO, and Sp are calculated for generators of homotopy groups; and (c) representatives of generators of homotopy groups Πn−2(O(n−1)), Π2n−2(U(n−1)), Π4n−2(Sp(n−1)) are given.
We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a …
We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This constructs the complexified string orientation of elliptic cohomology, which determines a pushfoward for families of rational string manifolds. A second pushforward is constructed from quantizing a supersymmetric $\sigma$-model. These two pushforwards agree, giving a precise physical interpretation for the elliptic index theorem with complex coefficients. This both refines and supplies further evidence for the long-conjectured relationship between elliptic cohomology and 2-dimensional quantum field theory. Analogous methods in supersymmetric mechanics recover path integral constructions of the Mathai--Quillen Thom form in complexified ${\rm KO}$-theory and a cocycle representative of the $\hat{A}$-class for a family of oriented manifolds.
Abstract We determine the spectrum of the sub-Laplacian on pseudo H -type nilmanifolds and present pairs of isospectral but non-homeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these …
Abstract We determine the spectrum of the sub-Laplacian on pseudo H -type nilmanifolds and present pairs of isospectral but non-homeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-homeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.
Abstract Real geometric algebras distinguish between space and time; complex ones do not. Space-times can be classified in terms of number n of dimensions and metric signature s (number of …
Abstract Real geometric algebras distinguish between space and time; complex ones do not. Space-times can be classified in terms of number n of dimensions and metric signature s (number of spatial dimensions minus number of temporal dimensions). Real geometric algebras are periodic in s , but recursive in n . Recursion starts from the basis vectors of either the Euclidean plane or the Minkowskian plane. Although the two planes have different geometries, they have the same real geometric algebra. The direct product of the two planes yields Hestenes’ space-time algebra. Dimensions can be either open (for space-time) or closed (for the electroweak force). Their product yields the eight-fold way of the strong force. After eight dimensions, the pattern of real geometric algebras repeats. This yields a spontaneously expanding space-time lattice with the physics of the Standard Model at each node. Physics being the same at each node implies conservation laws by Noether’s theorem. Conservation laws are not pre-existent; rather, they are consequences of the uniformity of space-time, whose uniformity is a consequence of its recursive generation.
The question of vector fields on spheres arises in homotopy theory and in the theory of fibre bundles, and it presents a classical problem, which may be explained as follows. …
The question of vector fields on spheres arises in homotopy theory and in the theory of fibre bundles, and it presents a classical problem, which may be explained as follows. For each n, let Sn-I be the unit sphere in euclidean n-space Rn. A vector field on Sn-1 is a continuous function v assigning to each point x of Sn-1 a vector v(x) tangent to Sn-1 at x. Given r such fields v1, v2, ..., Vr, we say that they are linearly independent if the vectors v1(x), v2(x), *--, vr(x) are linearly independent for all x. The problem, then, is the following: for each n, what is the maximum number r of linearly independent vector fields on Sn-i? For previous work and background material on this problem, we refer the reader to [1, 10, 11, 12, 13, 14, 15, 16]. In particular, we recall that if we are given r linearly independent vector fields vi(x), then by orthogonalisation it is easy to construct r fields wi(x) such that w1(x), w2(x), *I * , wr(x) are orthonormal for each x. These r fields constitute a cross-section of the appropriate Stiefel fibering. The strongest known positive result about the problem derives from the Hurwitz-Radon-Eckmann theorem in linear algebra [8]. It may be stated as follows (cf. James [13]). Let us write n = (2a + 1)2b and b = c + 4d, where a, b, c and d are integers and 0 < c < 3; let us define p(n) = 2c + 8d. Then there exist p(n) 1 linearly independent vector fields on Sn-1. It is the object of the present paper to prove that the positive result stated above is best possible.
yield the recurrence relation p(a + I)eis g(a; a + 1) = 1-q(a + 1)el8g(a + 1; a + 2) (6.4) T. Discussion.-The foregoing techniques can be extended to treat …
yield the recurrence relation p(a + I)eis g(a; a + 1) = 1-q(a + 1)el8g(a + 1; a + 2) (6.4) T. Discussion.-The foregoing techniques can be extended to treat the case where there is a general distribution of jumps at each stage, and the case where we introduce time dependence by requiring the probability that the particle emerge at a on or before a given time T.These matters, as well as those mentioned in Section 1, will be discussed in subsequent papers.
RAOUL BOTT( Ann Arbor ) ( 1 ). 1. Introduction.-Les récents travaux de ATIYAH et HIRZEBRUCH [3] ont inspiré cette note.Son modeste but est de discuter les propriétés de naturalité …
RAOUL BOTT( Ann Arbor ) ( 1 ). 1. Introduction.-Les récents travaux de ATIYAH et HIRZEBRUCH [3] ont inspiré cette note.Son modeste but est de discuter les propriétés de naturalité des équivalences d'homotopie (faibles) définies dans [2].ATIYAH interprète l'équivalence d'homotopie faible (l.i) -C.BU-^^BU où Bu désigne l'espace de base universel du groupe unitaire infini comme une formule de Kunneth dans le domaine des complexes C. W. de dimension finie.Il obtient cette transformation de la manière suivante.Soit EW=7:{X-^Bu) les classes d'homotopie d'applications de X dans Bu ( 2 ).On voit sans peine que E(X) est un groupe abélien pour la somme de Whitney, [ou, si l'on veut, en vertu de (l.i)].De plus, il existe un couplage naturel de E ^) ®zE( Y) dans E{X# Y\ ou X' ^ F désigne l'espace obtenu à partir de JTx F en identifiant X\j Y (accotement de X et Vpar leur points-bases) au point base de X ^ Y. Ce couplage sera appelé le produit tensoriel réduit.(Pour les détails, voir le paragraphe 2.) Nous avons donc, en particulier, un couplage naturel de E(X) (^)z^(S 2 ) dans E(^^S 2 ).L'adjointe de y est une application C 1 -2 ) ^-.S^BU-^BU ( l ) L'auteur bénéficie d'un Sloan Fellowship.( 2 ) Dans tout ceci, X et Y désigneront des complexes C. W. connexes de dimension finie avec un point base.