0000000000256143
AUTHOR
Jouko Mickelsson
Current Algebras as Hilbert Space Operator Cocycles
Aspects of a generalized representation theory of current algebras in 3 + 1 dimensions axe discussed. Rules for a systematic computation of vacuum expectation values of products of currents are described. Their relation to gauge group actions in bundles of fermionic Fock spaces and to the sesquilinear form approach of Langmann and Ruijsenaars is explained. The regularization for a construction of an operator cocycle representation of the current algebra is explained. An alternative formula for the Schwinger terms defining gauge group extensions is written in terms of Wodzicki residue and Dixmier trace.
The Chiral Anomaly
The Dirac operator on a manifold M is a first order partial differential operator acting on sections of a spin bundle over M. The Dirac operator is elliptic when the metric of M is positive definite. The main task in this chapter is to study properties of the determinant of the Dirac operator.
Semisimple Lie Algebras
Let F be the field of real or complex numbers. A Lie algebra is a vector space g over F with a Lie product (or commutator) [·,·]: g × g → g such that $$x \mapsto \left[ {x,y} \right]\;is\;linear\;for\;any\;y \in g,$$ (1) $$\left[ {x,y} \right] =- \left[ {y,x} \right],$$ (2) $$\left[ {x,\left[ {y,z} \right]} \right] + \left[ {y,\left[ {z,x} \right]} \right] + \left[ {z,\left[ {x,y} \right]} \right] = 0.$$ (3) The last condition is called the Jacobi identity. From (1) and (2) it follows that also y ↦ [x,y] is linear for any x ∈ g. In this chapter we shall consider only fini te-dimensional Lie algebras. In any vector space g one can always define a trivial Lie product [x,y] = 0. A Lie algebra …
Representations of Affine Kac-Moody Algebras
In the first chapter we explained how simple finite-dimensional Lie algebras can be completely characterized in terms of their Cartan matrices or Dynkin diagrams. The same holds for an arbitrary semisim-ple finite-dimensional Lie algebra. A semisimple Lie algebra is a direct sum of simple ideals which are pairwise orthogonal with respect to the Killing form. It follows that the Cartan matrix of a semisimple Lie algebra decomposes to a block diagonal form, each block representing a simple ideal. Similarly, the Dynkin diagram is a disconnected union of Dynkin diagrams of simple Lie algebras. Next we shall study certain infinite-dimensional Lie algebras which have many similarities with the si…
The Virasoro Algebra
In this chapter we shall study the Lie algebra Vect S1 of vector fields on a circle and some of its generalizations. The Lie algebra Vect S1 has a central extension, the Virasoro algebra. The representation theory of the Virasoro algebra is closely related to the representation theory of affine Lie algebras. In fact, through the Sugawara construction, to be defined below, a highest weight representation of an affine Lie algebra carries always a highest weight representation of the Virasoro algebra. All the irreducible highest weight representations of the Virasoro algebra are known and they can be exponentiated to representations of associated infinite-dimensional Lie groups. The representa…
Chiral anomalies in even and odd dimensions
Odd dimensional Yang-Mills theories with an extra ‘topological mass” term, defined by the Chern-Simons secondary characteristic, are discussed. It is shown in detail how the topological mass affects the equal time charge commutation relations and how the modified commutation relations are related to non-abelian chiral anomalies in even dimensions. We also study the SU(3) chiral model (Wess-Zumino model) in four dimensions and we show how a gauge invariant interaction with an external SU(3) vector potential can be defined with the help of the Chern-Simons characteristic in five dimensions.
The Fock Bundle of a Dirac Operator and Infinite Grassmannians
In the earlier chapters we have studied representations of current algebras in fermionic Fock spaces. A (fermionic) Fock space is determined by a single Dirac operator D. To set up a Fock space we need a splitting of a complex Hilbert space H to the subspaces H± corresponding to positive and negative frequencies of D. However, in an interacting quantum field theory one really should consider a bundle of Fock spaces parametrized by different Dirac operators. For example, in Yang-Mills theory any smooth vector potential defines a Dirac operator and one must consider the whole bunch of these operators and associated Fock spaces if one wants to describe the interaction of the vector potential w…
On a relation between massive Yang-Mills theories and dual string models
The relations between mass terms in Yang-Mills theories, projective representations of the group of gauge transformations, boundary conditions on vector potentials and Schwinger terms in local charge algebra commutation relations are discussed. The commutation relations (with Schwinger terms) are similar to the current algebra commutation relations of the SU(N) extended dual string model.
Sato's universal Grassmannian and group extensions
An extension \(\widehat{GL}\) of the symmetry group GL of Sato's universal Grassmannian GM is constructed. The extension plays a similar role to that of the central extension \(\widehat{GL}_{{\text{res}}}\) in the approach of Segal and Wilson to τ functions and KP hierarchy. Our group G contains GLres as a subgroup and the associated τ function is a deformation of the usual τ function, leading to a deformed KP hierarchy. A relation to current algebra of Yang-Mills theory in 3+1 dimension is discussed.
Holomorphic Aspects of String Theory
A string is a piecewise smooth map of the interval to a manifold M. A closed string is a map of the circle S1 into M. In string theory the strings replace the points of the manifold M as fundamental objects. The enormous amount of work done on quantized string models in physics has been motivated by the hope that the quantum string theory would produce a finite quantized theory of gravity, free of the divergences of the ordinary quantized Einstein theory of gravitation. So far the proof is missing but work is continuing. It has been proposed that some kind of string theory would be the unified theory of all fundamental interactions in physics. However, the fundamental principles of string t…
The Boson Fermion Correspondence
In 1932 Louis de Broglie suggested that photons could be constructed from pairs of neutrinos [de Broglie, 1932]. Both are massless particles (except for some recent unconfirmed experiments according to which the neutrino could have a very small mass) and are electrically neutral. The main difference between free photons and free neutrinos is that the former obeys Bose statistics and the second Fermi statistics. The spin of a photon is 1 and the spin of a neutrino is 1/2 and therefore kinematically it should be possible to think of a photon as a neutrino pair. However, this old formulation of boson fermion equivalence has not been very fruitful in particle physics. Instead, there has been a …
The Kp Hierarchy
As an application of the theory of infinite-dimensional Grassmannians and the representation theory of gl1 we shall study in this chapter certain nonlinear “exactly solvable” systems of differential equations. Exactly solvable means here that the nonlinear system can be transformed to an (infinite-dimensional) linear problem. A prototype of the equations is the Korteweg-de Vries equation $$\frac{{\partial u}}{{\partial t}} = \frac{3}{3}u\frac{{\partial u}}{{\partial x}} + \frac{1}{4}\frac{{{\partial ^3}u}}{{\partial {x^3}}}$$ . It turns out that it is more natural to consider an infinite system of equations like that above, for obtaining explicit solutions. The set of equations is called th…
Extensions of Groups of Gauge Transformations
In this chapter we shall discuss the structure of the infinite-dimensional Lie groups associated to the affine Kac-Moody algebras. We shall also construct the group of the current algebra of a gauge field theory in 3+1 space-time dimensions and we shall study the implications of the commutation relations for the spin-statistics relation in 3+1 dimensions.
The Nonlinear σ Model
The nonlinear (principal) σ model has been for a long time a theoretical laboratory to test different approaches for quantizing classical field theories. Here we shall discuss as an application of the current algebra representation theory a construction of the quantized σ model.
On the hamiltonian approach to commutator anomalies in (3+1) dimensions
Abstract The quantization of Weyl fermions in the presence of an external nonabelian vector potential is discussed in the case of spacetime dimension (3+1). The hamiltonian approach is used, in the temporal gauge A 0 = 0. In particular, it is explicitly shown how one can lift the action of (an extension of) the group of gauge transformations to the bundle of Fock spaces parametrized by smooth vector potentials.
Commutator anomalies and the Fock bundle
We show that the anomalous finite gauge transformations can be realized as linear operators acting on sections of the bundle of fermionic Fock spaces parametrized by vector potentials, and more generally, by splittings of the fermionic one-particle space into a pair of complementary subspaces. On the Lie algebra level we show that the construction leads to the standard formula for the relevant commutator anomalies.
Infinite sets of conservation laws for linear and nonlinear field equations
The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of the space-time symmetry group is established. It is shown that each symmetric element of the enveloping algebra of the space-time symmetry group of a linear field equation generates a one-parameter group of symmetries of the field equation. The cases of the Maxwell and Dirac equations are studied in detail. Then it is shown that (at least in the sense of a power series in the ‘coupling constant’) the conservation laws of the linear case can be deformed to conservation laws of a nonlinear field equation which is obtained from the linear one by adding a nonlinear term invariant u…
Determinant Bundles over Grassmannians
Denoting by H the Hilbert space of square-integrable Dirac spinor fields on a manifold M, transforming according to a unitary representation p of a gauge group G, we have a linear representation of the group g of gauge transformations in the space H. If ρ is faithful we can consider g as a subgroup of the general linear group GL(H). By constructing representations of GL(H) we automatically obtain representations of g. It turns out that in the case when the dimension d of M is odd, g is contained in a smaller group GLp ⊂ GL(H) which has the property that it perturbs the subspace H+ ⊂ H consisting of eigenvectors of a Dirac operator belonging to positive eigenvalues, by an operator A for whic…
Deformation of current algebras in 3+1 dimensions
It was shown in an earlier paper that there is an Abelian extension \(\widehat{{\text{gl}}}_2 \) of the general linear algebra gl2, that contains the current algebra with anomaly in 3+1 dimensions. We construct a three-parameter family of deformations \(\widetilde{{\text{gl}}}_2 (t)\) of \(\widehat{{\text{gl}}}_2 \). For certain choices of the deformation parameters, we can construct unitary representations. We also construct highest-weight nonunitary representations for all choices of the parameters.