arXiv subjects

Jouko Mickelsson

Publications and source records attributed to Jouko Mickelsson.

At least 19 recordsLinked to original sources

arXiv2024

A crossed module representation of a $2$-group constructed from the $3$-loop group $Ω^3 G$

The quantization of chiral fermions on a 3-manifold in an external gauge potential is known to lead to an abelian extension of the gauge group. In this article we concentrate on the case of $Ω^3 G$ of based smooth maps on a 3-sphere taking values in a compact Lie group $G.$ There is a crossed module constructed from an abelian extension $\widehat{Ω^3 G}$ of this group and a group of automorphims acting on it as explained in a recent article by Mickelson and Niemimäki. We shall construct a representation of this crossed module in terms of a repesentation of $\widehat{Ω^3 G}$ on a space of functions of gauge potentials with values in a fermionic Fock space and a representation of the automorphism group of $\widehat{Ω^3 G}$ as outer automorphisms of the canonical anticommutation relations algebra in the Fock space.

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arXiv2020

Non associative magnetic translations from parallel transport in projective Hilbert bundles

The non-associativity of translations in a quantum system with magnetic field background has received renewed interest in association with topologically trivial gerbes over $\mathbb{R}^n.$ The non-associativity is described by a 3-cocycle of the group $\mathbb{R}^n$ with values in the unit circle $S^1.$ The gerbes over a space $M$ are topologically classified by the Dixmier-Douady class which is an element of $\mathrm{H}^3(M,\mathbb{Z}).$ However, there is a finer description in terms of local differential forms of degrees $d=0,1,2,3$ and the case of the magnetic translations for $n=3$ the 2-form part is the magnetic field $B$ with non zero divergence. In this paper we study a quantum field theoretic construction in terms of $n$-component fermions on a circle. The non associativity arises when trying to lift the translation group action on the 1-particle system to the second quantized system.

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arXiv2019

Non-associative magnetic translations: A QFT construction

The non-associativity of translations in a quantum system with magnetic field background has received renewed interest in association with topologically trivial gerbes over $\mathbb{R}^n.$ The non-associativity is described by a 3-cocycle of the group $\mathbb{R}^n$ with values in the unit circle $S^1.$ The gerbes over a space $M$ are topologically classified by the Dixmier-Douady class which is an element of $\mathrm{H}^3(M,\mathbb{Z}).$ However, there is a finer description in terms of local differential forms of degrees $d=0,1,2,3$ and the case of the magnetic translations for $n=3$ the 2-form part is the magnetic field $B$ with non zero divergence. In this paper we study a quantum field theoretic construction in terms of $n$-component fermions on a real line or a unit circle. The non associativity arises when trying to lift the translation group action on the 1-particle system to the second quantized system.

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arXiv2018

A 2-group construction from an extension of the 3-loop group $Ω^3G$

We define a 3-loop group $Ω^3G$ as a subgroup of smooth maps from a 3-ball to a Lie group $G$, and then construct a 2-group based on an automorphic action on the Mickelsson-Faddeev extension of $Ω^3G$. In this we follow the strategy of Murray et al., who earlier described a similar construction in one dimension. The three-dimensional situation presented here is further complicated by the fact that the 3-loop group extension is not central.

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arXiv2018

3D Current Algebra and Twisted K Theory

Equivariant twisted K theory classes on compact Lie groups $G$ can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra $LG$ using a supersymmetric Wess-Zumino-Witten model. The aim of the present article is to extend the construction to higher loop algebras using an abelian extension of a $3D$ current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to $3D$ gauge transformations but do not define true Hilbert space operators.

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arXiv2017

Extensions of Lattice Groups, Gerbes and Chiral Fermions on a Torus

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 classication 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.

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arXiv2016

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 $\text{H}^3(H, \Bbb Z)$. 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 $1\to N \to G \to H \to 1$ when the gerbe is defined by an abelian extension $1\to A \to \hat N \to N \to 1$ of $N$. In particular, when $\text{H}_s^1(N,A)$ vanishes we shall construct a transgression map $\text{H}^2_s(N, A) \to \text{H}^3_s(H, A^N)$, where $A^N$ 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.

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arXiv2014

Projective Families of Dirac operators on a Banach Lie Groupoid

We introduce a Banach Lie group $G$ of unitary operators subject to a natural trace condition. We compute the homotopy groups of $G$, describe its cohomology and construct an $S^1$-central extension. We show that the central extension determines a non-trivial gerbe on the action Lie groupoid $G\ltimes \mathfrak{k}$, where $\mathfrak{k}$ denotes the Hilbert space of self-adjoint Hilbert-Schmidt operators. With an eye towards constructing elements in twisted K-theory, we prove the existence of a cubic Dirac operator $\mathbb{D}$ in a suitable completion of the quantum Weil algebra $\mathcal{U}(\mathfrak{g}) \otimes Cl(\mathfrak{k})$, which is subsequently extended to a projective family of self-adjoint operators $\mathbb{D}_A$ on $G\ltimes \frak{k}$. While the kernel of $\mathbb{D}_A$ is infinite-dimensional, we show that there is still a notion of finite reducibility at every point, which suggests a generalized definition of twisted K-theory for action Lie groupoids.

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arXiv2014

Twisted K-theory constructions in the case of a decomposable Dixmier-Douady class

Twisted K-theory on a manifold X, with twisting in the 3rd integral cohomology, is dis- cussed in the case when X is a product of a circle T and a manifold M. The twist is assumed to be decomposable as a cup product of the basic integral one form on T and an integral class in H2(M,Z). This case was studied some time ago by V. Mathai, R. Melrose, and I.M. Singer. Our aim is to give an explicit construction for the twisted K-theory classes using a quantum field theory model, in the same spirit as the supersymmetric Wess-Zumino-Witten model is used for constructing (equivariant) twisted K-theory classes on compact Lie groups.

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arXiv2014

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, [LM], 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.

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arXiv2012

The Chern Character of Certain Infinite Rank Bundles arising in Gauge Theory

A cocycle $Ω: P \times G \to H$ taking values in a Lie group $H$ for a free right action of $G$ on $P$ defines a principal bundle $Q$ with the structure group $H$ over $P/G.$ The Chern character of a vector bundle associated to $Q$ defines then characteristic classes on $X.$ This observation becomes useful in the case of infinite dimensional groups. It typically happens that a representation of $G$ is not given by linear operators which differ from the indentity by a trace-class operator. For this reason the Chern character of a vector bundle associated to the principal fibration $P \to P/G$ is ill-defined. But it may happen that the Lie algebra representations of the group $H$ are given in terms of trace-class operators and therefore the Chern character is well-defined; this observation is useful especially if the map $g\mapsto Ω(p;g)$ is a homotopy equivalence on the image for any $p\in P.$ We apply this method to the case $P= \Cal A,$ the space of gauge connections in a finite-dimensional vector bundle, and $G= \Cal G$ is the group of (based) gauge transformations. The method for constructing the appropriate cocycle $Ω$ comes from ideas in quantum field theory, used to define the renormalized gauge currents in a Fock space.

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arXiv2012

Fractional Loop Group and Twisted K-Theory

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on $G$ is discussed.

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arXiv2010

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.

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arXiv2010

The logarithmic residue density of a generalised Laplacian

We show that the residue density of the logarithm of a generalised 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 formulae provide a pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in dimension $4$ and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by S. Scott and D. Zagier announced in \cite{Sc2} and to appear in \cite{Sc3}. In our approach, which is of perturbative nature, we use either a Campbell-Hausdorff formula derived by Okikiolu or a non commutative Taylor type formula.

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arXiv2008

From gauge anomalies to gerbes and gerbal actions

The purpose of this contribution is to point out connections between recent ideas about gerbes and gerbal actions (as higher categorical extension of representation theory) and old discussion in quantum field theory on commutator anomalies, gauge group extensions, and 3-cocycles. The unifying concept is the classical obstruction theory for group extensions as explained in the reference [ML].

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arXiv2008

Differential Twisted K-theory and Applications

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.

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arXiv2006

Star products and central extensions

The purpose of the present note is two-fold. First, to show that deformations of algebras of smooth functions can be used to construct topologically nontrivial standard central extensions of loop groups. Second, to use noncommutative geometry as a regularization of current algebras in higher dimensions with the aim of constructing representations of current algebras.

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arXiv2006

Renormalised Chern-Weil forms associated with families of Dirac operators

We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in work by S. Scott and later work by S. Scott and the second author. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern-Weil forms in as far as they coincide with regularised Chern-Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature formula. We interpret the odd degree Chern-Weil type forms built from superconnections as Wodzicki residues and establish a transgression formula along the lines of known transgression formulae for eta-forms.

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