Spinor Bundle Research Articles

HOMOLOGY OF BASED GAUGE GROUPS ASSOCIATED WITH SPINOR GROUP

We study the homology of based gauge groups associated with the principal Spin(n) bundles over the four-sphere using the Eilenberg-Moore spectral sequence and the Serre spectral sequence with the Dyer-Lash of operations.

Holomorphic spectrum of twisted Dirac operators on compact Riemann surfaces

Given a Hermitian line bundle L with a harmonic connection over a compact Riemann surface ( S , g ) of constant curvature, we study the spectral geometry of the corresponding twisted Dirac operator D . This problem is analyzed in terms of the natural holomorphic structures of the spinor bundles E ± defined by the Cauchy–Riemann operators associated with the spinorial connection. By means of two elliptic chains of line bundles obtained by twisting E ± with the powers of the canonical bundle K S , we prove that there exists a certain subset Spec hol ( D ) of the spectrum such that the eigensections associated with λ ∈ Spec hol ( D ) are determined by the holomorphic sections of a certain line bundle of the elliptic chain. We give explicit expressions for the holomorphic spectrum and the multiplicities of the corresponding eigenvalues according to the genus p of S , showing that Spec hol ( D ) does not depend on the spin structure and depends on the line bundle L only through its degree. This technique provides the whole spectrum of D for genus p = 0 and 1, whereas for genus p > 1 we obtain a finite number of eigenvalues.

Calibrated lifts of minimal submanifolds

A canonical real line bundle associated to a minimal Lagrangian submanifold in a Kähler–Einstein manifold X is known to be special Lagrangian when considered as a subset of the canonical line bundle of X with a natural Calabi–Yau structure. We first verify this result by standard moving frame computation, and obtain a uniform lower bound for the mass of compact minimal Lagrangian submanifolds in C P n . Similar correspondence is then proved for integrable G 2 and Spin ( 7 ) structures on the bundle of anti self dual 2-forms and a Spin bundle respectively of a self dual Einstein 4-manifold N constructed by Bryant and Salamon. In this case, analogues of tangent and normal bundles of certain minimal surfaces in N are calibrated, i.e., associative, coassociative, or Cayley.

Local Lagrangian formalism and discretization of the Heisenberg magnet model

In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (HM) model which are then used as the basis for geometric discretizations of HM. Despite a topological obstruction to the existence of a global Lagrangian density, a local variational formulation allows one to derive local conservation laws using a version of Nöther’s theorem from the formal variational calculus of Gelfand-Dikii. Using the local Lagrangian form we extend the method of Marsden, Patrick and Schkoller to derive local multisymplectic discretizations directly from the variational principle. We employ a version of the finite element method to discretize the space of sections of the trivial magnetic spin bundle N = M × S 2 over an appropriate space-time M. Since sections do not form a vector space, the usual FEM bases can be used only locally with coordinate transformations intervening on element boundaries, and conservation properties are guaranteed only within an element. We discuss possible ways of circumventing this problem, including the use of a local version of the method of characteristics, non-polynomial FEM bases and Lie-group discretization methods.

Formula omitted]-structures and Dirac operators on contact manifolds

Any contact metric manifold has a Spin c -structure. Thus, we study on any Spin c -spinor bundle of a contact metric manifold, Dirac type operators associated to the generalized Tanaka–Webster connection. Bochner–Lichnerowicz type formulas are derived in this setting and vanishing theorems are obtained.

On the equivalence ofN=1 brane worlds and geometric singularities with flux

We consider Kaluza Klein reductions of M-theory on the N orbifold of the spin bundle over S3 along two different U(1) isometries. The first one gives rise to the familiar ``large-N duality'' of the = 1 SU(N) gauge theory in which the UV is realized as the world-volume theory of N D6-branes wrapped on S3, whereas the IR involves N units of RR flux through an S2. The second reduction gives an equivalent version of this duality in which the UV is realized geometrically in terms of a 1 of AN−1 singularities, with one unit of RR flux through the 1. The IR is reached via a geometric transition and involves a single D6 brane on a lens space S3/N or, alternatively, a singular background (S2 × 4)/N, with one unit of RR flux through S2 and, localized at the singularities, an action of their stabilizer group in the U(1) RR gauge bundle, so that no massless twisted states occur. We also consider linear σ-model descriptions of these backgrounds.

The general form of supersymmetric solutions of N = (1, 0) U(1) and SU(2) gauged supergravities in six dimensions

We obtain necessary and sufficient conditions for a supersymmetric field configuration in the N = (1, 0) U(1) or SU(2) gauged supergravities in six dimensions, and impose the field equations on this general ansatz. It is found that any supersymmetric solution is associated with an structure. The structure is characterized by a null Killing vector which induces a natural 2 + 4 split of the six-dimensional spacetime. A suitable combination of the field equations implies that the scalar curvature of the four-dimensional Riemannian part, referred to as the base, obeys a second-order differential equation; surprisingly, for a large class of solutions the equation in the SU(2) theory requires the vanishing of the Weyl anomaly of N = 4 SYM on the base. Bosonic fluxes introduce torsion terms that deform the structure away from a covariantly constant one. The most general structure can be classified into terms of its intrinsic torsion. For a large class of solutions the gauge field strengths admit a simple geometrical interpretation: in the U(1) theory the base is Kähler, and the gauge field strength is the Ricci form; in the SU(2) theory, the gauge field strengths are identified with the curvatures of the left-hand spin bundle of the base. We employ our general ansatz to construct new solutions; we show that the U(1) theory admits a symmetric Cahen–Wallach4 × S2 solution together with a compactifying pp-wave. The SU(2) theory admits a black string, whose near horizon limit is AdS3 × S3, which is supported by a self-dual 3-form flux and a meron on the S3. In the limit of the zero 3-form flux we obtain the Yang–Mills analogue of the Salam–Sezgin solution of the U(1) theory, namely R1,2 × S3. Finally we obtain the additional constraints implied by enhanced supersymmetry, and discuss Penrose limits in the theories.

GLOBAL SOLUTIONS OF EINSTEIN-DIRAC EQUATION

The conformal space \frac M was introduced by Dirac in 1936. It is an algebraic manifold with a spin structure and possesses naturally an invariant Lorentz metric. By carefully studying the birational transformations of \frac M, we obtain explicitly the transition functions of the spin bundle over \frac M. Since the transition functions are closely related to the propagation in physics, we get a kind of solutions of the Dirac equation by integrals constructed from the propagation. Moreover, we prove that the invariant Lorentz metric together with one of such solutions satisfies the Einstein-Dirac combine equation

The noncommutative Lorentzian cylinder as an isospectral deformation

We present a new example of a finite-dimensional noncommutative manifold, namely, the noncommutative cylinder. It is obtained by isospectral deformation of the canonical triple associated with the Euclidean cylinder. We discuss Connes’ character formula for the cylinder. In the second part, we discuss noncommutative Lorentzian manifolds. Here, the definition of spectral triples involves Krein spaces and operators on Krein spaces. A central role is played by the admissible fundamental symmetries on the Krein space of square integrable sections of a spin bundle over a Lorentzian manifold. Finally, we discuss isospectral deformation of the Lorentzian cylinder and determine all admissible fundamental symmetries of the noncommutative cylinder.

New cohomogeneity one metrics with Spin(7) holonomy

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A 8 , is complete and non-singular on R 8 . The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S 4, and are denoted by B 8 + , B 8 − and B 8 . The metrics on B 8 + and B 8 − occur in families with a non-trivial parameter. The metric on B 8 arises for a limiting value of this parameter, and locally this metric is the same as the one for A 8 . The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP 3 . We construct the covariantly constant spinor and calibrating 4-form. We also obtain an L 2-normalisable harmonic 4-form for the A 8 manifold, and two such 4-forms (of opposite dualities) for the B 8 manifold.

Moduli of vector bundles in characteristic 2

Let X be a nonsingular curve of genus 2 over an algebraically closed field of characteristic 2. We show that the moduli space of semistable rank two vector bundles with a fixed determinant of even degree on X is isomorphic to the three dimensional projective space. We prove results on orthogonal and spin bundles on hyperelliptic curves generalising this result.

The sigma orientation for analytic circle-equivariant elliptic cohomology

We construct a canonical Thom isomorphism in Grojnowski's equivariant elliptic cohomology, for virtual T-oriented T-equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the complex-analytic case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga's weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems of Witten-Bott-Taubes including generalizations by Kefeng Liu follow.

Band Asymptotics for Operators of Dirac Type on S n

In this article, the author computes spectral invariants of the sum of the Dirac operator and a zeroth-order potential operator Vacting on sections of the spinor bundle over S n . The invariants computed here arise as the coefficients of asymptotic expansions associated with the singularities of the wave trace for an operator system. Because of the non-commutativity of the symbol calculus for operator systems, the computations of these asymptotic expansions are more complicated than those for a scalar operator. The non-commutativity of the symbol calculus necessitates the scaling of the potential Vby a small parameter and the use of semiclassical techniques to extract the invariants. The band invariants computed here depend on the parallel transport around closed geodesics, and the traces of certain endomorphisms which are related to parallel transport and the potential term averaged around closed geodesics.

DIRAC EQUATION IN SPACETIMES WITH NON-METRICITY AND TORSION

Dirac equation is written in a non-Riemannian spacetime with torsion and non-metricity by lifting the connection from the tangent bundle to the spinor bundle over spacetime. Foldy–Wouthuysen transformation of the Dirac equation in a Schwarzschild background spacetime is considered and it is shown that both the torsion and non-metricity couples to the momentum and spin of a massive, spinning particle. However, the effects are small to be observationally significant.

The equivariant {$J$}-homomorphism

May’s J-theory diagram is generalized to an equivariant setting. To do this, equivariant orientation theory for equivariant periodic ring spectra (such as KOG) is developed, and classifying spaces are constructed for this theory, thus extending the work of Waner. Moreover, Spin bundles of dimension divisible by 8 are shown to have canonical KOG-orientations, thus generalizing work of Atiyah, Bott, and Shapiro. Fiberwise completions for equivariant spherical fibrations are constructed, also on the level of classifying spaces. When G is an odd order p-group, this allows for a classifying space formulation of the equivariant Adams conjecture. It is also shown that the classifying space for stable fibrations with fibers being sphere representations completed at p is a delooping of the 1-component of QG(S 0 )ˆ p. The “Adams-May square,” relating generalized characteristic classes and Adams operations, is constructed and shown to be a pull-back after completing at p and restricting to G-connected covers. As a corollary, the canonical map from the p-completion of J k G to the G-connected cover of QG(S 0 )ˆ p is shown to split after restricting to G-connected covers.

An index theorem for open manifolds whose ends are warped products

We give an index formula for the Dirac operator acting onL2-sections of the spinor bundle of an open spin manifold with a co-compact cylindrical end over which the metric is a warped product.

Clifford residues and charge quantization

We derive the quantization of action, particle number, andelectric charge in a Lagrangian spin bundle over\({\mathbb{M}} \equiv {\mathbb{M}}_\# \backslash \cup D_J \) Penrose’s conformal compactification of Minkowsky space, with the world tubes of massive particles removed.Our Lagrangian density,\({\mathcal{L}}_g \), is the spinor factorization of the Maurer-Cartan 4-form Ω4; it’s action,S g , measures the covering number of the 4internal u (1)×su (2) phases over external spacetime\({\mathbb{M}}\). UnderPTC symmetry,\({\mathcal{L}}_g \) reduces to the second Chern formTrK L ⋏K R for a left ⊕ right chirality spin bundle. We prove aresidue theorem forgl (2, ℂ)-valued forms, which says that, when we “sew-in” singular lociD J over which theu (1)×su (2) phases of the matter fields have some extra twists compared to the8 vacuum modes, the additional contributions to the action, electric charge, lepton and baryon numbers are alltopologically quantized. Because left and right chirality 2-forms arechiral dual, forms are quantized over theirdual cycles. Thus it is the interactionc 2 (E), with a globally nontrivialmagnetic field, that forceselectric fields to be topologically quantized overspatial 2 cycles,\(\int_{{\mathbb{S}}^2 } { K_{or} } e^\theta \wedge e^\varphi = 4\pi {\rm N}\).

Comments on M theory dynamics onG2holonomy manifolds

We study the dynamics of M-theory on G2 holonomy manifolds, and consider in detail the manifolds realized as the quotient of the spin bundle over S^3 by discrete groups. We analyse, in particular, the class of quotients where the triality symmetry is broken. We study the structure of the moduli space, construct its defining equations and show that three different types of classical geometries are interpolated smoothly. We derive the N=1 superpotentials of M-theory on the quotients and comment on the membrane instanton physics. Finally, we turn on Wilson lines that break gauge symmetry and discuss some of the implications.

Quaternionic holomorphic geometry: Pl�cker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given.

New complete noncompact Spin(7) manifolds

We construct new explicit metrics on complete noncompact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by A 8 , is topologically R 8 and another, which we denote by B 8 , is the bundle of chiral spinors over S 4. Unlike the previously-known complete noncompact metric of Spin(7) holonomy, which was also defined on the bundle of chiral spinors over S 4, our new metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP 3 . We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L 2-normalisable harmonic 4-form for the A 8 manifold, and two such 4-forms (of opposite dualities) for the B 8 manifold. We use the metrics to construct new supersymmetric brane solutions in M-theory and string theory. In particular, we construct resolved fractional M2-branes involving the use of the L 2 harmonic 4-forms, and show that for each manifold there is a supersymmetric example. An intriguing feature of the new A 8 and B 8 Spin(7) metrics is that they are actually the same local solution, with the two different complete manifolds corresponding to taking the radial coordinate to be either positive or negative. We make a comparison with the Taub-NUT and Taub-BOLT metrics, which by contrast do not have special holonomy. In Appendix A we construct the general solution of our first-order equations for Spin(7) holonomy, and obtain further regular metrics that are complete on manifolds B 8 + and B 8 − similar to B 8 .