What about counterparts of T, S, and U dualities in TGD framework?

The natural question is what could be the TGD counterparts of S-, T- and U-dualities. If one accepts the identification of U-duality as product U=ST and the proposed counterpart of T duality as a strong form of general coordinate invariance discussed in previous posting, it remains to understand the TGD counterpart of S-duality - in other words electric-magnetic duality - relating the theories with gauge couplings g and 1/g. Quantum criticality selects the preferred value of g_K: Kähler coupling strength is very near to fine structure constant at electron length scale and can be equal to it. Since there is no coupling constant evolution associated with α_K, it does not make sense to say that g_K becomes strong and is replaced with its inverse at some point. One should be able to formulate the counterpart of S-duality as an identity following from the weak form of electric-magnetic duality and the reduction of TGD to almost topological QFT. This seems to be the case.

TGD based view about S-duality

The following arguments suggests that in TGD framework S duality is realized for each preferred extremal of Kähler action separately whereas in standard view the duality would be realized only at the level of path integral defining the partition function.

1. For preferred extremals the interior parts of Kähler action reduces to a boundary term because the term j^μA_μ vanishes. The weak form of electric-magnetic duality requires that Kähler electric charge is proportional to Kähler magnetic charge, which implies reduction to abelian Chern-Simons term: the Kähler coupling strength does not appear at all in Chern-Simons term. The proportionality constant beween the electric and magnetic parts J_E and J_B of Kähler form however enters into the dynamics through the boundary conditions stating the weak form of electric-magnetic duality. At the Minkowskian side the proportionality constant must be proportional to g_K² to guarantee a correct value for the unit of Kähler electric charge - equal to that for electric charge in electron length scale- from the assumption that electric charge is proportional to the topologically quantized magnetic charge. It has been assumed that

   J_E= α_K J_B

   holds true at both sides of the wormhole throat but this is an un-necessarily strong assumption at the Euclidian side. In fact, the self-duality of CP₂ Kähler form stating

   J_E=J_B

   favours this boundary condition at the Euclidian side of the wormhole throat. Also the fact that one cannot distinguish between electric and magnetic charges in Euclidian region since all charges are magnetic can be used to argue in favor of this form. The same constraint arises from the condition that the action for CP₂ type vacuum extremal has the value required by the argument leading to a prediction for gravitational constant in terms of the square of CP₂ radius and α_K the effective replacement g_K² → 1 would spoil the argument.

2. Minkowskian and Euclidian regions should correspond to a strongly/weakly interacting phase in which Kähler magnetic/electric charges provide the proper description. In Euclidian regions associated with CP₂ type extremals there is a natural interpretation of interactions between magnetic monopoles associated with the light-like throats: for CP₂ type vacuum extremal itself magnetic and electric charges are actually identical and cannot be distinguished from each other. Therefore the duality between strong and weak coupling phases seems to be trivially true in Euclidian regions if one has J_B= J_E at Euclidian side of the wormhole throat. This is however an un-necessarily strong condition as the following argument shows.

3. In Minkowskian regions the interaction is via Kähler electric charges and elementary particles have vanishing total Kähler magnetic charge consisting of pairs of Kähler magnetic monopoles so that one has confinement characteristic for strongly interacting phase. Therefore Minkowskian regions naturaly correspond to a weakly interacting phase for Kähler electric charges. One can write the action density at the Minkowskian side of the wormhole throat as

   (J_E²-J_B²)/α_K= α_KJ_B² - J_B²/α_K .

   The exchange J_E↔ J_B accompanied by g_K²→ -1/g_K² leaves the action density invariant. Since only the behavior of the vacuum functional infinitesimally near to the wormhole throat matters by almost topological QFT property, the duality is realized. Note that the argument goes through also in Euclidian regions so that it does not allow to decide which is the correct form of weak form of electric-magnetic duality.

4. S-duality could correspond geometrically to the duality between partonic 2-surfaces responsible for magnetic fluxes and string worlds sheets responsible for electric fluxes as rotations of Kähler gauge potentials around them and would be very closely related with the counterpart of T-duality implied by the strong form of general coordinate invariance and saying that space-like 3-surfaces at the ends of space-time sheets are equivalent with light-like 3-surfaces connecting them.

Comparison with standard view about dualities

One can compare the proposed realization of T-, S and U-duality to the more general dualities defined by the modular group SL(2,Z), which in QFT framework can hold true for the path integral over all possible gauge field configurations. In the resent case the dualities hold true for every preferred extremal separately and the functional integral is only over the space-time projections of fixed Kähler form of CP₂. Modular invariance for Maxwell action was discussed by E. Verlinde for Maxwell action with θ term for a general 4-D compact manifold with Euclidian signature of metric. In this case one has path integral giving sum over infinite number of extrema characterized by the cohomological equivalence class of the Maxwell field the action exponential to a high degree. Modular invariance is broken for CP₂: one obtains invariance only for τ→ τ+2 whereas S induces a phase factor to the path integral.

1. In the recent case these homology equivalence classes would correspond to homology equivalence classes of holomorphic partonic 2-surfaces associated with the critical points of K\"ahler function with respect to zero modes.

2. In the case that the Euclidian contribution to the Kähler action is expressible solely in terms of wormhole throat Chern-Simons terms, and one can neglect the measurement interaction terms, the exponent of Kähler action can be expressed in terms of Chern-Simons action density as

   L= τ L_C-S ,

   L_C-S=J∧ A ,

   τ=1/g_K² +ik/4π , k=1 .

   Here the parameter τ transforms under full SL(2,Z) group as

   τ→ (aτ+b)/(cτ+d) .

   The generators of SL(2,Z) transformations are T: τ → τ+1, S:τ→-1/τ. The imaginary part in the exponents corresponds to Kac-Moody central extension k=1.

   This form corresponds also to the general form of Maxwell action with CP breaking θ term given by

   L= 1/g_K² J∧^*J +i(θ/8π²) J∧J , θ=2π .

   Hence the Minkowskian part mimicks the θ term but with a value of θ for which the term does not give rise to CP breaking in the case that the action is full action for CP₂ type vacuum extremal so that the phase equals to 2π and phase factor case is trivial. It would seem that the deviation from the full action for CP₂ due to the presence of wormhole throats reducing the value of the full Kähler action for CP₂ type vacuum extremal gives rise to CP breaking. One can visualize the excluded volume as homologically non-trivial geodesic spheres with some thickness in two transverse dimensions. At the limit of infinitely thin geodesic spheres CP breaking would vanish. The effect is exponentially sensitive to the volume deficit.

CP breaking and ground state degeneracy

Ground state degeneracy due to the possibility of having both signs for Minkowskian contribution to the exponent of vacuum functional provides a general view about the description of CP breaking in TGD framework.

1. In TGD framework path integral is replaced by inner product involving integral over WCV. The vacuum functional and its conjugate are associated with the states in the inner product so that the phases of vacuum functionals cancel if only one sign for the phase is allowed. Minkowskian contribution would have no physical significance. This of course cannot be the case. The ground state is actually degenerate corresponding to the phase factor and its complex conjugate since kenosqrtg can have two signs in Minkowskian regions. Therefore the inner products between states associated with the two ground states define 2× 2 matrix and non-diagonal elements contain interference terms due to the presence of the phase factor. At the limit of full CP₂ type vacuum extremal the two ground states would reduce to each other and the determinant of the matrix would vanish.

2. A small mixing of the two ground states would give rise to CP breaking and the first principle description of CP breaking in systems like K-Kbar and of CKM matrix should reduce to this mixing. K⁰ mesons would be CP even and odd states in the first approximation and correspond to the sum and difference of the ground states. Small mixing would be present having exponential sensitivity to the actions of CP₂ type extremals representing wormhole throats. This might allow to understand qualitatively why the mixing is about 50 times larger than expected for B⁰ mesons.

3. There is a strong temptation to assign the two ground states with two possible arrows of geometric time. At the level of M-matrix the two arrows would correspond to state preparation at either upper or lower boundary of CD. Do long- and shortlived neutral K mesons correspond to almost fifty-fifty orthogonal superpositions for the two arrow of geometric time or almost completely to a fixed arrow of time induced by environment? Is the dominant part of the arrow same for both or is it opposite for long and short-lived neutral measons? Different lifetimes would suggest that the arrow must be the same and apart from small leakage that induced by environment. CP breaking would be induced by the fact that CP is performed only K⁰ but not for the environment in the construction of states. One can probably imagine also alternative interpretations.

Remark: The proportionality of Minkowskian and Euclidian contributions to the same Chern-Simons term implies that the critical points with respect to zero modes appear for both the phase and modulus of vacuum functional. The Kähler function property does not allow extrema for vacuum functional as a function of complex coordinates of WCW since this would mean Kähler metric with non-Euclidian signature. If this were not the case the stationary values of phase factor and extrema of modulus of the vacuum functional would correspond to different configurations.

For details see the new chapter Motives and Infinite Primes of "Physics as a Generalized Number Theory" or the article with same title.