David I. Olive

A Class of Lorentzian Kac-Moody algebras

Abstract We consider a natural generalisation of the class of hyperbolic Kac–Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they possess a real principal so(1,2) subalgebra. Our class of algebras include the Lorentzian Kac–Moody algebras that have recently been proposed as symmetries of M-theory and the closed bosonic string.

Exact electromagnetic duality

Electromagnetic duality is an idea with a long pedigree that addresses a number of old questions, for example: n n nWhy does space-time possess four dimensions? n n nWhy is electric charge quantised? n n nWhat is the origin of mass? n n nWhat is the internal structure of the elementary particles? n n nHow are quarks confined?

Solitons and vertex operators in twisted affine Toda field theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.

Spin and Abelian Electromagnetic Duality¶on Four-Manifolds

Abstract: We investigate the electromagnetic duality properties of an Abelian gauge theory on a compact oriented four-manifold by analysing the behaviour of a generalised partition function under modular transformations of the dimensionless coupling constants. The true partition function is invariant under the full modular group but the generalised partition function exhibits more complicated behaviour depending on topological properties of the four-manifold concerned. It is already known that there may be “modular weights” which are linear combinations of the Euler number and Hirzebruch signature of the four-manifold. But sometimes the partition function transforms only under a subgroup of the modular group (the Hecke subgroup). In this case it is impossible to define real spinor wave-functions on the four-manifold. But complex spinors are possible provided the background magnetic fluxes are appropriately fractional rather than integral.nThis gives rise to a second partition function which enables the full modular group to be realised by permuting the two partition functions, together with a third. Thus the full modular group is realised in all cases. The demonstration makes use of various constructions concerning integral lattices and theta functions that seem to be of intrinsic interest.

Charges and Fluxes in Maxwell Theory on Compact Manifolds with Boundary

We investigate the charges and fluxes that can occur in higher-order Abelian gauge theories defined on compact space-time manifolds with boundary. The boundary is necessary to supply a destination to the electric lines of force emanating from brane sources, thus allowing non-zero net electric charges, but it also introduces new types of electric and magnetic flux. The resulting structure of currents, charges, and fluxes is studied and expressed in the language of relative homology and de Rham cohomology and the corresponding abelian groups. These can be organised in terms of a pair of exact sequences related by the Poincaré-Lefschetz isomorphism and by a weaker flip symmetry exchanging the ends of the sequences. It is shown how all this structure is brought into play by the imposition of the appropriately generalised Maxwell’s equations. The requirement that these equations be integrable restricts the world-volume of a permitted brane (assumed closed) to be homologous to a cycle on the boundary of space-time. All electric charges and magnetic fluxes are quantised and satisfy the Dirac quantisation condition. But through some boundary cycles there may be unquantised electric fluxes associated with quantised magnetic fluxes and so dyonic in nature.

The Dirac Quantisation Condition for Fluxes on Four-Manifolds

Abstract:A systematic treatment is given of the Dirac quantisation condition for electromagnetic fluxes through two-cycles on a four-manifold space-time which can be very complicated topologically, provided only that it is connected, compact, oriented and smooth. This is sufficient for the quantised Maxwell theory on it to satisfy electromagnetic duality properties. The results depend upon whether the complex wave function needed for the argument is scalar or spinorial in nature. An essential step is the derivation of a “quantum Stokes theorem” for the integral of the gauge potential around a closed loop on the manifold. This can only be done for an exponentiated version of the line integral (the “Wilson loop”) and the result again depends on the nature of the complex wave functions, through the appearance of what is known as a Stiefel–Whitney cohomology class in the spinor case. A nice picture emerges providing a physical interpretation, in terms of quantised fluxes and wave-functions, of mathematical concepts such as spin structures, spinC structures, the Stiefel–Whitney class and Wus formula. Relations appear between these, electromagnetic duality and the Atiyah–Singer index theorem. Possible generalisation to higher dimensions of space-time in the presence of branes are mentioned.