Weak form of electric-magnetic duality and its implications

The notion of electric-magnetic duality was proposed first by Olive and Montonen and is central in N=4 supersymmetric gauge theories. It states that magnetic monopoles and ordinary particles are two different phases of theory and that the description in terms of monopoles can be applied at the limit when the running gauge coupling constant becomes very large and perturbation theory fails to converge. The notion of electric-magnetic self-duality is more natural since for CP₂ geometry Kähler form is self-dual and Kähler magnetic monopoles are also Kähler electric monopoles and Kähler coupling strength is by quantum criticality renormalization group invariant rather than running coupling constant. The notion of electric-magnetic (self-)duality emerged already two decades ago in the attempts to formulate the Kähler geometric of world of classical worlds. Quite recently a considerable step of progress took place in the understanding of this notion. What seems to be essential is that one adopts a weaker form of the self-duality applying at partonic 2-surfaces.

Every new idea must be of course taken with a grain of salt but the good sign is that this concept leads to precise predictions. Also a dramatic progress in the understanding of the mathematical structure of quantum TGD has taken place so that this notion has become basic piece of TGD during last two months.

1. The first implication is a new view about electro-weak massivation reducing it to weak confinement in TGD framework. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement realized in terms of magnetic confinement.

2. This picture generalizes to the case of color confinement. Also quarks correspond to pairs of magnetic monopoles but the charges need not vanish now. Rather, valence quarks would be connected by flux tubes of length of order hadron size such that magnetic charges sum up to zero. For instance, for baryonic valence quarks these charges could be (2,-1,-1) and could be proportional to color hyper charge.

3. The highly non-trivial prediction making more precise the earlier stringy vision is that elementary particles are string like objects in electro-weak scale: this should become manifest at LHC energies.

4. The weak form electric-magnetic duality together with Beltrami flow property of Kähler current leads to the reduction of Kähler action to Chern-Simons action so that TGD reduces to almost topological QFT and Kähler function is explicitly calculable. This has enormous impact concerning practical calculability of the theory.

5. The requirement that WCW Kähler metric is non-trivial in M⁴ degrees of freedom forces to replace CP₂ Kähler form with the sum of CP₂ and S² Kähler forms. The latter defines a magnetic monopole field of a monopole residing at the time-like line connecting the tips of CD. The non-vacuum extremals remain extremals and the vacuum extremals representable as graphs M⁴→ CP₂ are replaced with vacuum extremals for which the induced Kähler forms of CP₂ sum up to zero. The most general extremals of this kind have 3-D CP₂ projection which is a good news from the point of view of TGD based description of the classical gravitation.

6. One ends up also to a general solution ansatz for field equations from the condition that the theory reduces to almost topological QFT. The solution ansatz is inspired by the idea that all isometry currents are proportional to Kähler current which is integrable in the sense that the flow parameter associated with its flow lines defines a global coordinate (Beltrami flow). The proposed solution ansatz would describe a hydrodynamical flow with the property that isometry charges are conserved along flow lines. A general ansatz satisfying the integrability conditions is found. The solution ansatz applies also to the extremals of Chern-Simons action and to the conserved currents associated with the modified Dirac equation defined as contractions of the modified gamma matrices between the solutions of the modified Dirac equation. The strongest form of the solution ansatz states that various classical and quantum currents flow along flow lines of the Beltrami flow defined by Kähler current (Kähler magnetic field associated with Chern-Simons action). Intuitively this picture is attractive. A more general ansatz would allow several Beltrami flows meaning multi-hydrodynamics. The integrability conditions boil down to two scalar functions: the first one satisfies massless d'Alembert equation in the induced metric and the the gradients of the scalar functions are orthogonal. The interpretation in terms of momentum and polarization directions is natural.

For details see the article Weak form of electric-magnetic duality and its implications.