Weak form of electric magnetic duality and duality between Minkowskian and Euclidian space-time regions

The reduction of the Kähler action for the space-time sheets with Minkowskian signature of the induced metric follows from the assumption that Kähler current is proportional to instanton current and from the weak form of electric-magnetic duality. The first property implies a reduction to a 3-D term associated with wormhole throats and the latter property reduces this term to Abelian Chern-Simons term. I have not explicitly considered whether the same happens in the 4-D regions of Euclidian signature representing wormhole contacts.

If these assumptions are made also in the Euclidian region, the outcome is that one obtains a difference of two Chern-Simons terms coming from Minkowskian and Euclidian regions at light-like wormhole throats. This difference can be non-trivial since Kähler form for CP₂ defines non-trivial U(1) bundle. This however suggests that the total Kähler action is quantized in integer multiples of the Kähler action for CP₂ type vacuum extremal so that one would have effectively sum over n-instanton configurations.

If the Kähler function of the "world of classical worlds" (WCW) is identified as total Kähler action, this implies the vanishing of the Kähler metric of WCW, which is a catastrophe. Should one modify the definition of Kähler function by considering only the contribution from either Minkowskian or Euclidian regions? What about vacuum functional: should one identify it as the exponent of Kähler function or of Kähler action in this case?

1. The Kähler metric of WCW must be non-trivial. If Kähler function is piecewise constant in WCW, and if its second functional derivatives with respect to complex WCW coordinates indeed define WCW Kähler metric, then this metric must vanish identically almost everywhere. This is a catastrophe.

2. To understand how the problem can be cured, notice that WCW metric receives a non-trivial contribution from the two Lagrange multiplier terms stating the weak form of electric-magnetic duality at Minkowskian and Euclidian sides. Neither Lagrangian multiplier term contributes to Kähler action. Either of them would guarantee that the theory does not reduce to a mere topological QFT and would give rise to a non-trivial Kähler metric. If both are taken into account their contributions cancel each other and the metric is trivial. The conjectured duality between the descriptions based on space-time regions of Minkowskian and Euclidian signature suggests that one should define Kähler function and Kähler metric using only either of the two contributions at wormhole throats.

   This duality could be very useful practically since the two expansions could correspond to weakly and strongly interacting phases analogous to those encountered in the case of electric-magnetic duality. For Euclidian side of the duality one would have power series in powers exp(-n8π²/g_K²) multiplied by the exponential of the Minkowskian contribution with negative sign. At the Minkowskian side of the duality one would have exponent of the Minkowskian contribution with positive sign. p-Adicization suggests strongly that the exponent exp(-8π²/g_K²) defining the Kähler action of CP₂ is a rational number.

3. Usually vacuum functional is identified as exponent of Kähler function but could one identify vacuum functional as exponent of total Kähler action giving a discrete spectrum for its values? The answer seems to be negative. There are excellent reasons for the identification of the vacuum functional as exponent of Kähler function. For instance, Gaussian and metric determinants cancel each other and the constant curvature spaced property with vanishing Ricci scalar implies that the curvature scalar giving otherwise divergent loop contributions vanishes. If one modifies the vacuum functional from Kähler function to total Kähler action, there is no kinetic term in the exponent of the vacuum functional, and one must give up the idea about perturbative definition of WCW functional integral using the (1,1) part of the contravariant WCW metric as propagator.

4. The symmetric space property of WCW is what gives hopes about a practical definition of functional integral which is number theoretically universal making therefore sense also in the p-adic context. The reduction of the functional integral to harmonic analysis in infinite-dimensional symmetric spaces allowing to define integrals group theoretically would allow to define functional integrals non-perturbatively without propagator expansion. However, if the functional integral fails perturbatively, the hopes that it makes sense physically, are meager.

The overall conclusion is that the only reasonable definitions of Kähler function of WCW and vacuum functional realize the conjecture duality between Minkowskian and Euclidian regions of space-time surfaces. The overall conclusion is that the only reasonable definitions of Kähler function of WCW and vacuum functional realize the conjecture duality between Minkowskian and Euclidian regions of space-time surfaces. This duality would have also number theoretical interpretation. Minkowskian regions of the space-time surface would correspond to hyper-quaternionic and Eulidian regions to quaternionic regions. In hyper-quaternionic regions the modified gamma matrices would span hyper-quaternionic plane of complexified octonions (imaginary units multiplied by commutative imaginary unit). In quaternionic regions the modified gamma matrices multiplied by a product of fixed octonionic imaginary unit and commutative imaginary unit would span a quaternionic plane of complexified octonions (see this).

For background see the chapter Does the Modified Dirac Equation Define the Fundamental Variational Principle.