Coeffective Numbers of Riemannian 8-Manifolds with Holonomy in Spin(7)

We introduce a differential complex of coeffective type for anySpin(7)-manifold M locally conformal to aRiemannian 8-manifold with holonomy contained in Spin(7).Local properties of this complex, such as ellipticity and acyclicity,are studied. The relationship between the coeffective cohomology ofM and the topology of the manifold is discussed in the caseof M having a subgroup of Spin(7) as aholonomy group.     

A construction of Spin(7)-instantons

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi–Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi–Yau four-orbifold divided by an anti-holomorphic involution fixing only the singular points. On the other hand, there are higher-dimensional analogues of anti-self-dual instantons in four dimensions on Spin(7)-manifolds, which are called Spin(7)-instantons. They are minimizers of the Yang–Mills action, and the Spin(7)-instanton equation together with a gauge fixing condition forms an elliptic system. In this article, we construct Spin(7)-instantons on the examples of compact Spin(7)-manifolds above, starting with Hermitian–Einstein connections on the Calabi–Yau four-orbifolds and ALE spaces. Under some assumptions on the Hermitian–Einstein connections, we glue them together to obtain Spin(7)-instantons on the compact Spin(7)-manifolds. We also give a simple example of our construction.     

R-Matrices for SU(2)-Invariant Two-Leg Spin Ladders

A list of R-matrices for the set of eight integrable spin ladders is presented. Four of these matrices are new. Bibliography: 13 titles.     

Spin(9) geometry of the octonionic Hopf fibration

We deal with Riemannian properties of the octonionic Hopf fibration S ¹⁵ → S ⁸, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S ¹ subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.     

Sechzehndimensionale lokalkompakte Translationsebenen mit Spin (7) als Kollineationsgruppe

Ohne Zusammenfassung     

试题研讨(8)

题1(2002年上海市春季高考题)如图1,若从点O所作的两条射线OM、ON上分别有点M1、M2与点N1、N2,则三角形面积之比S△OM1N1/S△OM2N2=OM1/OM2·ON1/ON2.如图2,若从点O所作的不在同一平面内的三条射线OP、OQ和OR上,分别有点P1、P2,点Q1、Q2和点R1、R2,则类似的结论为_______.     

New examples of compact manifolds with holonomy Spin(7)

We find new examples of compact Spin(7)-manifolds using a construction of Joyce (J. Differ. Geom., 53:89–130, 1999; Compact manifolds with special holonomy. Oxford University Press, Oxford, 2000). The essential ingredient in Joyce’s construction is a Calabi–Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi–Yau 4-orbifolds.     

Spin Top (Toward a Theorem on the Spin-Statistics Relation)

Theoretical and Mathematical Physics - We propose a model of a spin particle as an analogue of a quantum mechanical top. We show that for this model, we can prove the theorem on the relation...     

Spin(9) and almost complex structures on 16-dimensional manifolds

For a Spin(9)-structure on a Riemannian manifold M ¹⁶ we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form Φ_Spin(9). We then prove that Φ_Spin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Φ_Spin(9) and \({\Phi_{\rm Spin(9)}^2}\) in the special case of holonomy Spin(9).     

On the Elliptic Genera of Manifolds of Spin(7) Holonomy

Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The \({\mathcal{N} = 1}\) superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a c = 12 realization of the symmetry group \({\mathcal{S}W(3/2,2)}\). In this paper, we compute the characters of this supergroup and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. We find suggestive relations to various sporadic groups, which are made more precise in a companion paper.     

不动点集为RP(8)(U) P(8,2n-1)的对合

设(M,T)是一个带有光滑对合T的光滑闭流形,T的不动点集为RP(8)() P(8,2n-1).本文证明了(M,T)必协边于(RP(8)×RP(8),twist)和(P(8,RP(2n)),T')之一.     

Existence of (2, 8) GWhD( v) and (4, 8) GWhD( v) with $${ v \equi v 0,1 (mod 8)}$$

(2, 8) Generalized Whist tournament Designs (GWhD) on v players exist only if . We establish that these necessary conditions are sufficient for all but a relatively small number of (possibly) exceptional cases. For there are at most 12 possible exceptions: {177, 249, 305, 377, 385, 465, 473, 489, 497, 537, 553, 897}. For there are at most 98 possible exceptions the largest of which is v = 3696. The materials in this paper also enable us to obtain four previously unknown (4, 8)GWhD(8n+1), namely for n = 16,60,191,192 and to reduce the list of unknown (4, 8) GWhD(8n) to 124 values of v the largest of which is v = 3696.      

The odd primary homology of SU(n), Sp(n) and Spin(n)-gauge groups

For an odd prime $p$, we calculate the mod-$p$ homology of $SU\left(n\right)$-gauge groups over a simply-connected, closed 4-manifold for all $n\ge 2$. Similar calculations are obtained for the structure groups $Sp\left(n\right)$ when $n\ge 1$ and $Spin\left(n\right)$ for $n\ge 3$ (except for some cases when $n$ is even and $p=3$).     

Spin chain related to the quantum superalgebra slq(1¦1)

We consider an integrable system with R-matrix related to the algebra sl _q(1 | 1). The Hamiltonian of the system is constructed, and its spectrum is found by means of the algebraic Bethe ansatz. The symmetry algebra of the chain is written out. The partition function of the model on the lattice with domain wall boundary conditions is calculated. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 146–162.