Clifford Modules Research Articles

K-symplectic structures and absolutely trianalytic subvarieties in hyperkähler manifolds

Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ a complex subvariety in $(M,I)$. We say that $Z$ is trianalytic if it is complex analytic with respect to $J$ and $K$, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures $(M,I,J',K')$ containing $I$. For a generic complex structure $I$ on $M$, all complex subvarieties of $(M,I)$ are absolutely trianalytic. It is known that a normalization $Z'$ of a trianalytic subvariety is smooth; we prove that $b_2(Z')$ is no smaller than $b_2(M)$ when $M$ has maximal holonomy (that is, $M$ is IHS). To study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of the hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold $X$ is a k-dimensional space $R$ of closed 2-forms on $X$ which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in $R$ is a smooth, non-degenerate quadric hypersurface in $R$. We consider absolutely trianalytic tori in a hyperkahler manifold $M$ of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where $k=b_2(M)$. We show that the tangent bundle $TX$ of a k-symplectic manifold is a Clifford module over a Clifford algebra $Cl(k-1)$. Then an absolutely trianalytic torus in a hyperkahler manifold $M$ with $b_2(M)\geq 2r+1$ is at least $2^{r-1}$-dimensional.

On the Spectral Flow for Dirac Operators with Local Boundary Conditions

Let M be an even dimensional compact Riemannian manifold with boundary and let D be a Dirac operator acting on the sections of the Clifford module E over M. We impose certain local elliptic boundary conditions for D obtaining a selfadjoint extension D_F of D. For a smooth U(n)--valued function g:M -> U(n) we establish a formula for the spectral flow along the straight line between D_F and g^{-1} D_F g. This spectral flow is motivated by index theory: in odd dimensions it gives the natural pairing between the K--homology class of the operator and the K--theory class of g. In our situation, with dim M having the "wrong" parity, the answer can be expressed in terms of the natural spectral flow pairing on the odd--dimensional boundary. Our result generalizes a recent paper by M. Prokhorova in which the two-dimensional case is treated. Furthermore, our paper may be seen as an odd-dimensional analogue of a paper by D. Freed. As an application we obtain a new proof of the cobordism invariance of the spectral flow.

On Dirac operators on Lie algebroids

Dirac operators on Lie algebroids are defined and investigated. The Lie algebroid is equipped with a structure of a Clifford module. The quadratic form generating the Clifford algebra structure need not to be nondegenerated. Weitzenböck type formulas for the square of the Dirac operators are derived.

Properties of Modules Over Clifford Algebras and Variational Problems

We establish some properties of normed modules over Clifford algebras and develop an approach to the study of generally invertible differential operators in Sobolev–Clifford modules. A similar approach was earlier proposed by the author for Sobolev spaces. Bibliography: 17 titles.

关于Atiyah-Singer指标定理的随机方法证明

According to Watanabe [6], by generalized Wiener functional we could give a simple stochastic proof of the index theorem for twisted Dirac operator (the Atiyah-Singer index theorem) on Clifford module.

Inverse Boundary Problems for Systems in Two Dimensions

We prove identification of coefficients up to gauge equivalence by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of \({\mathbb{C}}\) . In the geometric setting, we fix a Riemann surface with boundary and consider both a Dirac-type operator plus potential acting on sections of a Clifford module and a connection Laplacian plus potential (i.e. Schrodinger Laplacian with external Yang–Mills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determine both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of \({\mathbb{C}}\) , we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.

Trivializable sub-Riemannian structures on spheres

We classify the trivializable sub-Riemannian structures on odd-dimensional spheres SN that are induced by a Clifford module structure of RN+1. The underlying bracket generating distribution is of step two and spanned by a set of global linear vector fields X1,…,Xm. As a result we show that such structures only exist in the cases where N=3,7,15. The corresponding hypo-elliptic sub-Laplacians Δsub are defined as the (negative) sum of squares of the vector fields Xj. In the case of a trivializable rank four distribution on S7 and a trivializable rank eight distribution on S15 we obtain a part of the spectrum of Δsub. We also remark that in both cases there is a relation between the eigenfunctions and Jacobi polynomials.

CLIFFORD MODULES AND SYMMETRIES OF TOPOLOGICAL INSULATORS

We complete the classification of symmetry constraints on gapped quadratic fermion hamiltonians proposed by Kitaev. The symmetry group is supposed compact and can include arbitrary unitary or antiunitary operators in the Fock space that conserve the algebra of quadratic observables. We analyze the multiplicity spaces of real irreducible representations of unitary symmetries in the Nambu space. The joint action of intertwining operators and antiunitary symmetries provides these spaces with the structure of Clifford module: we prove a one-to-one correspondence between the ten Altland–Zirnbauer symmetry classes of fermion systems and the ten Morita equivalence classes of real and complex Clifford algebras. The antiunitary operators, which occur in seven classes, are projectively represented in the Nambu space by unitary "chiral symmetries". The space of gapped symmetric hamiltonians is homotopically equivalent to the product of classifying spaces indexed by the dual object of the group of unitary symmetries.

Clifford modules and invariants of quadratic forms

AbstractWe construct new invariants of quadratic forms over commutative rings, using ideas from Topology. More precisely, we define a hermitian analog of the Bott class with target algebraic K-theory, based on the classification of Clifford modules. These invariants of quadratic forms go beyond the classical invariants defined via the Clifford algebra. An appendix by J.-P. Serre, of independent interest, describes the “square root” of the Bott class in the general framework of lambda rings.

Clifford Algebras and Matrix Factorizations

The purpose of this short note is to connect the two parts of the title, on one hand Clifford algebras and Clifford modules, on the other hand matrix factorizations of a non degenerate quadratic homogeneous polynomial. The main result states the category of Clifford modules is equivalent to a suitably defined category of Matrix factorizations of the quadratic polynomial. This shed a new light about the category of Clifford modules.

Representation Rings of Lie Superalgebras

Given a Lie superalgebra \g, we introduce several variants of the representation ring, built as subrings and quotients of the ring R_{\Z_2}(\g) of virtual \g-supermodules (up to even isomorphisms). In particular, we consider the ideal R_{+}(\g) of virtual \g-supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring SR(\g) on which the parity reversal operator takes the class of a virtual \g-supermodule to its negative. We also construct representation groups built from ungraded \g-modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring SR^{*}(\g), including all degree shifts, is then a \Z_{2}-graded ring in the complex case and a \Z_{8}-graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings R^{*}_{\Z_2}(\g), R^{*}_{+}(\g), and SR^{*}(\g). We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring SR^{*}(\g). In the real case, we obtain the expected 24-term version, as well as a surprising six-term version of this periodic exact sequence.

Gerbes, Clifford Modules and the Index Theorem

The use of bundle gerbes and bundle gerbe modules is considered as a replacement for the usual theory of Clifford modules on manifolds that fail to be spin. It is shown that both sides of the Atiyah-Singer index formula for coupled Dirac operators can be given natural interpretations using this language and that the resulting formula is still an identity.

(Fermionic) mass meets (intrinsic) curvature

Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space–time, defines the total curvature of the Clifford module bundle representing a “free” fermion within the geometrical setup of spontaneously broken Yang–Mills–Higgs gauge theories. The geometrical frame discussed here gives rise to a natural class of Lagrangian densities. It is shown that the geometry of the Clifford module bundle representing a free fermion is described by a canonical spectral invariant Lagrangian density.

Variétés homogènes d'Einstein de courbure scalaire négative: Construction à l'aide de certains modules de Clifford

Using certain Clifford modules, we give a method for constructing many examples of homogeneous Einstein spaces with negative scalar curvature.

Scattering Theory and Adiabatic Decomposition of the $\z$-determinant of the Dirac Laplacian

In this note we announce the adiabatic decomposition formula for the ζ-determinant of the Dirac Laplacian. Theorem 1.1 of this paper extends the re- sult of our earlier work (see (8) and (9)), which covered the case of the invertible tangential operator. The presence of the non-trivial kernel of the tangential op- erator requires careful analysis of the small eigenvalues of the Dirac Laplacian, which employs elements of scattering theory. 1. Statement of the Result Let D : C ∞ (M; S) → C ∞ (M; S) denote a compatible Dirac operator acting on sections of a bundle of Clifford modules S over a closed manifold M of di- mension 2k +1.Assume that we have a decomposition of M as M1 ∪M2 , where M1 and M2 are compact manifolds with boundary so that M = M1 ∪ M2 ,M 1 ∩ M2 = Y = ∂M1 = ∂M2 .

H-type groups and Clifford modules

2-step nilpotent Lie groups are non-commutative groups which in a certain sense are very close to commutative groups. The simplest of them is the Heisenberg group, whose Lie algebra can be described with a single non-trivial bracket. TheH-type groups are modelled on this example. Their bracket structure is quite simple yet structurally well organized so that they become very rigid. From the definning properties it follows, thatH-type groups are characterized by Clifford algebra representations. Geometrically they are tied to (generalized) contact transformations. Analytic methods are used in order to establish rigidity for groups whose center is of dimension at least three.

Scalar products on clifford modules and pseudo-H-type lie algebras

In 1980 A. Kaplan introduced the so called generalised Heisenberg algebras, which are two step nilpotent algebras endowed with an inner product satisfying a compatibility condition with the Lie algebra structure. In this paper we generalize the definition of A. Kaplan to the case of a nonpositive definite scalar product. In the non-positive definite case the proof of the existence and the classification raise entirely new problems. The natural setting to solve them is that of the theory of Clifford modules.

Nonabelian monopoles from matrices

We study the expectation value of (the product) of the one-particle projector(s) in the reduced matrix model and matrix quantum mechanics in general. This quantity is given by the nonabelian Berry phase: we discuss the relevance of this with regard to the spacetime structure. The case of the USp matrix model is examined from this respect. Generalizing our previous work, we carry out the complete computation of this quantity which takes into account both the nature of the degeneracy of the fermions and the presence of the spacetime points belonging to the antisymmetric representation. We find the singularities as those of the SU(2) Yang monopole connection as well as the pointlike singularities in 9+1 dimensions coming from its SU(8) generalization. The former type of singularities, which extend to four of the directions lying in the antisymmetric representations, may be regarded as seeds of our four-dimensional spacetime structure and is not shared by the IIB matrix model. From a mathematical viewpoint, these connections can be generalizable to arbitrary odd space dimensions due to the nontrivial nature of the eigenbundle and the Clifford module structure.

On the Kernel of the Equivariant Dirac Operator

We prove a vanishing theorem for certain isotypical components of the kernel of the S1-equivariant Dirac operator with coefficients in an admissible Clifford module. The method is based on changing the metric by a conformal (generally unbounded) factor and studying the effect of this change on the Dirac operator and its kernel. In the cases relevant to S1-actions we find that the kernel of the new operator is naturally isomorphic to the kernel of the original operator.

On the cobordism invariance of the index of Dirac operators

We describe a “tunneling” proof of the cobordism invariance of the index of Dirac operators. The goal of this note is to present a very short proof of the cobordism invariance of the index. More precisely, if D is a Dirac operator on an odd dimensional manifold M with boundary ∂M = M then we show that the index of its restriction D to M is zero. The novelty of this proof consists in the fact that we provide an explicit isomorphism between the kernel and the cokernel of D. This map can be viewed as a sort of “propagator” (see Sect. 4). 1. The setting Consider the following collection of data. (a) A compact, oriented, (2n + 1)-dimensional Riemann manifold (M, ĝ) with boundary ∂M = M such that ĝ is a product metric near the boundary. We denote by s the longitudinal coordinate on a collar neighborhood of M . The various orientations are defined as in Figure 1. (b) A bundle of complex self-adjoint Clifford modules E → M (in the sense of [BGV]). The Clifford multiplication is denoted by ĉ : T ∗M → End (E).