Family Of Dirac Operators Research Articles

The Fermi gerbe of Weyl semimetals

In the gap topology, the unbounded self-adjoint Fredholm operators on a Hilbert space have third homotopy group the integers. We realise the generator explicitly, using a family of Dirac operators on the half-line, which arises naturally in Weyl semimetals in solid-state physics. A "Fermi gerbe" geometrically encodes how discrete spectral data of the family interpolate between essential spectral gaps. Its non-vanishing Dixmier-Douady invariant protects the integrity of the interpolation, thereby providing topological protection of the Weyl semimetal's Fermi surface.

The Batalin–Vilkovisky formalism and the determinant line bundle

Given a smooth family of massless free fermions parametrized by a base manifold B, we show that the (mathematically rigorous) Batalin–Vilkovisky quantization of the observables of this family gives rise to the determinant line bundle for the corresponding family of Dirac operators.

Factorization of Dirac operators on toric noncommutative manifolds

We factorize the Dirac operator on the Connes–Landi 4-sphere in unbounded KK-theory. We show that a family of Dirac operators along the orbits of the torus action defines an unbounded Kasparov module, while the Dirac operator on the principal orbit space –an open quadrant in the 2-sphere – defines a half-closed chain. We show that the tensor sum of these two operators coincides up to unitary equivalence with the Dirac operator on the Connes–Landi sphere and prove that this tensor sum is an unbounded representative of the internal Kasparov product in bivariant K-theory. We also generalize our results to Dirac operators on all toric noncommutative manifolds subject to a condition on the principal stratum. We find that there is a curvature term that arises as an obstruction for having a tensor sum decomposition in unbounded KK-theory. This curvature term can however not be detected at the level of bounded KK-theory.

Families of spectral triples and foliations of space(time)

We study a noncommutative analog of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a “Lorentzian spectral triple,” which can also be viewed as the “reverse Wick rotation” of a product spectral triple. This construction of “Lorentzian spectral triples” fits well into the Krein space approach to noncommutative Lorentzian geometry.

Dirac families for loop groups as matrix factorizations

We identify the category of integrable lowest-weight representations of the loop group LG of a compact Lie group G with the category of twisted, conjugation-equivariant curved Fredholm complexes on the group G: namely, the twisted, equivariant matrix factorizations of a super-potential built from the loop rotation action on LG. This lifts the isomorphism of K-groups of [3–5] to an equivalence of categories. The construction uses families of Dirac operators.

An index theorem of Callias type for pseudodifferential operators

AbstractWe prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + iΦ, where D is elliptic pseudodifferential with Hermitian symbols, and Φ is a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Melrose in order to prove our results using methods of topological K-theory, and we devote special attention to the case in which D is a family of Dirac operators, in which case our theorem specializes to give family versions of the previously known index formulas.

Dirac operators coupled to instantons on positive definite four-manifolds

We study the moduli space of instantons on a simply connected, positive definite, closed four manifold by analyzing the classifying map of the index bundle of a family of Dirac operators parameterized by the moduli space. As an application, we compute the cohomology rings for the charge 1 and 2 moduli spaces in the rank stable limit.

The η-form and a generalized Maslov index

Given a family {L0(b), L1(b)}b∈B of pairs of transverse Lagrangian subspaces of a hermitean symplectic vector space we define a family of Dirac operators on the unit interval and consider its η-form η(L0, L1). To a family {L0(b), L1(b), L2(b)}b∈B of pairwise transverse Lagrangian subspaces we associate the cocycle η(L0, L1) + η(L1, L2) + η(L2, L1) which is a closed form. We identify its cohomology class with a generalization to families of the triple Maslov index.

Families of Dirac operators, boundaries and the $b$-calculus

Families of Dirac operators, boundaries and the $b$-calculus

On spectral flow of transversal dirac operators and a theorem of Vafa-Witten

We outline the proof of a theorem of Vafa-Witten type on uniform bounds for the eigenvalues of a family of transversal Dirac operators relative to a Riemannian foliation. The family in question is parameterized by a moduli space of basic connections with respect to the foliation modulo a suitable group of foliation preserving gauge transformations. The proof is based on the concept of spectral flow, applied to the suspension of suitable gauge transformations to periodic families of Dirac operators.

A note on families of Dirac operators

The index theorem for Dirac families is applied to deduce a result concerning isomorphisms of homotopy groups of certain spaces.

?/k-manifolds and families of Dirac operators

The index of a family of a family of Dirac operators is aK-Theory element in the parameter space. Sullivan'sℤ/k-manifolds are used to detect this index completely. For the first Chern class this gives a topological interpretation of Witten's global anomaly. The relationship with the geometry of the index bundle is considered.

Quantization of fermions interacting with point-like external fields

A second quantization of a four-parameter family of Dirac operators interacting with point-like fields leads to inequivalent representations of the CAR. Implementability of gauge transformations gives quantization conditions. We determine the algebra of charges, identify the Schwinger term, and work out a connection between the Fredholm index and the winding number.

A proof of Bismut local index theorem for a family of Dirac operators

On donne une demonstration du theoreme de Bismut basee sur le developpement classique des noyaux de la chaleur. On exprime le noyau de la chaleur du laplacien d'un fibre vectoriel comme une moyenne sur le groupe d'holonomie

Spectral density and a family of Dirac operators

The spectral density for a class of Dirac operators is investigated by relating its even and odd parts to the Riemann ζ-function and to the η-invariant by Atiyah, Patodi and Singer. Asymptotic expansions are studied and a “hidden” supersymmetry is revealed and used to relate the Dirac operator to a supersymmetric quantum mechanics. A general method for the computation of the odd spectral density is developed, and various applications are discussed. In particular the connection to the fermion number and a relation between the odd spectral density and some ratios of Jost functions and relative phase shifts are pointed out. Chiral symmetry breaking is investigated using methods analogous to those applied in the investigation of the fermion number, and related to supersymmetry breaking in the corresponding quantum mechanical model.