Jouko Mickelsson

Kac-moody Groups, Topology of the Dirac Determinant Bundle and Fermionization

The relation between Kac-Moody groups and algebras and the determinant line bundle of the massless Dirac operator in two dimensions is clarified. Analogous objects are studied in four space-time dimensions and a generalization of Wittens fermionization mechanism is presented in terms of the topology of the Dirac determinant bundle.

On topological boundary characteristics in nonabelian gauge theory

We study a topological classification of gauge potentials based on an examination of the Chern–Simons surface term C(U) at appropriate boundary components of the space‐time manifold when the potential approaches a pure gauge dU U−1 at the boundary. We derive an explicit local formula for a 2‐form H(U) such that C(U)=dH(U).

Families Index Theorem in Supersymmetric WZW Model and Twisted K-Theory: The SU(2) Case

The construction of twisted K-theory classes on a compact Lie group is reviewed using the supersymmetric Wess-Zumino-Witten model on a cylinder. The Quillen superconnection is introduced for a family of supercharges parametrized by a compact Lie group and the Chern character is explicitly computed in the case of SU(2). For large euclidean time, the character form is localized on a D-brane.

Labeling and Multiplicity Problem in the Reduction U(n + m) ↓ U(n) × U(m)

A labeling for the basis vectors in a general unitary irreducible representation of U(n + m) is introduced with the Casimir operators of the subgroup U(n) × U(m) diagonal. The multiplicities are calculated in the reduction U(n + m) ↓ U(n) × U(m) for some special cases.

THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN

We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in four dimensions and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell-Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.

Third group cohomology and gerbes over Lie groups

The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space H is given by the third cohomology H3(H,Z). When H is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of H. We shall study in more detail this relation in the case of a group extension 1→N→G→H→1 when the gerbe is defined by an abelian extension 1→A→Nˆ→N→1 of N. In particular, when Hs1(N,A) vanishes we shall construct a transgression map Hs2(N,A)→Hs3(H,AN), where AN is the subgroup of N-invariants in A and the subscript s denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper.

The Phase of the Scattering Operator from the Geometry of Certain Infinite-Dimensional Groups

We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson (J Math Phys 37(8):3933–3953, 1996) is that we can avoid the somewhat arbitrary choice in the regularization of the time evolution for intermediate times using a natural choice of the connection form on the space of appropriate unitary operators.

The vector form of the neutrino equation and the photon neutrino duality

A new form of the vector wave equation for a neutrino is derived that makes the relation to the massive case more transparent than that given by Reifler. Some new aspects of the photon neutrino duality are discussed.

FAMILIES OF DIRAC OPERATORS AND QUANTUM AFFINE GROUPS

Twisted K-theory classes over compact Lie groups can be realized as families of Fredholm operators using the representation theory of loop groups. In this talk I want to show how to deform the Fredholm family, in the sense of quantum groups. The family of Dirac type operators is parametrized by vectors in the adjoint module for a quantum affine algebra and transform covariantly under a (central extension of) the algebra.

Extensions of lattice groups, gerbes and chiral fermions on a torus

Abstract Motivated by the topological classification of hamiltonians in condensed matter physics (topological insulators) we study the relations between chiral Dirac operators coupled to an abelian vector potential on a torus in 3 and 1 space dimensions. We find that a large class of these hamiltonians in three dimensions is equivalent, in K theory, to a family of hamiltonians in just one space dimension but with a different abelian gauge group. The moduli space of U(1) gauge connections over a torus with a fixed Chern class is again a torus up to a homotopy. Gerbes over a n -torus can be realized in terms of extensions of the lattice group acting in a real vector space. The extension comes from the action of the lattice group (thought of as “large” gauge transformations, homomorphisms from the torus to U(1)) in the Fock space of chiral fermions. Interestingly, the K theoretic classification of Dirac operators coupled to vector potentials in this setting in 3 dimensions can be related to families of Dirac operators on a circle with gauge group the 3-torus.