Absolute extremum property for Kähler action implies dynamical Kac-Moody and super conformal symmetries

The absolute extremization of Kähler action in the sense that the value of the action is maximal or minimal for a space-time region where the sign of the action density is definite, is a very attractive idea. Both maxima and minima seem to be possible and could correspond to quaternionic (associative) and co-quaternionic (co-associative) space-time sheets emerging naturally in the number theoretic approach to TGD.

It seems now clear that the fundamental formulation of TGD is as an almost-topological conformal field theory for lightlike partonic 3-surfaces. The action principle is uniquely Chern-Simons action for the Kähler gauge potential of CP₂ induced to the space-time surface. This approach predicts basic super Kac Moody and superconformal symmetries to be present in TGD and extends them. The quantum fluctuations around classical solutions of these field equations break these super-symmetries partially.

The Dirac determinant for the modified Dirac operator associated with Chern-Simons action defines vacuum functional and the guess is that it equals to the exponent of Kähler action for absolute extremal. The plausibility of this conjecture would increase considerably if one could show that also the absolute extrema of Kähler action possess appropriately broken super-conformal symmetries. This has been a long-lived conjecture but only quite recently I was able to demonstrate it by a simple argument.

The extremal property for Kähler action with respect to variations of time derivatives of initial values keeping h^k fixed at X³ implies the existence of an infinite number of conserved charges assignable to the small deformations of the extremum and to H isometries. Also infinite number of local conserved super currents assignable to second variations and to covariantly constant right handed neutrino are implied. The corresponding conserved charges vanish so that the interpretation as dynamical gauge symmetries is appropriate. This result provides strong support that the local extremal property is indeed consistent with the almost-topological QFT property at parton level.

The starting point are field equations for the second variations. If the action contain only derivatives of field variables one obtains for the small deformations δh^k of a given extremal

∂_α J^α_k = 0 ,

J^α_k = (∂² L/∂ h^k_α∂ h^l_β) δ h^l_β ,

where h^k_α denotes the partial derivative ∂_α h^k. A simple example is the action for massless scalar field in which case conservation law reduces to the conservation of the current defined by the gradient of the scalar field. The addition of mass term spoils this conservation law.

If the action is general coordinate invariant, the field equations read as

D_αJ^α,k = 0

where D_α is now covariant derivative and index raising is achieved using the metric of the imbedding space.

The field equations for the second variation state the vanishing of a covariant divergence and one obtains conserved currents by the contraction this equation with covariantly constant Killing vector fields j_A^k of M⁴ translations which means that second variations define the analog of a local gauge algebra in M⁴ degrees of freedom.

∂_αJ^A,α_n = 0 ,

J^A,α_n = J^α,k_n j^A_k .

Conservation for Killing vector fields reduces to the contraction of a symmetric tensor with D_kj_l which vanishes. The reason is that action depends on induced metric and Kähler form only.

Also covariantly constant right handed neutrino spinors Ψ_R define a collection of conserved super currents associated with small deformations at extremum

J^α_n = J^α,k_nγ_kΨ_R .

Second variation gives also a total divergence term which gives contributions at two 3-dimensional ends of the space-time sheet as the difference

Q_n(X³_f)-Q_n(X³) = 0 ,

Q_n(Y³) = ∫_Y³ d³x J_n ,

J_n = J^tk h_klδh^l_n .

The contribution of the fixed end X³ vanishes. For the extremum with respect to the variations of the time derivatives ∂_th^k at X³ the total variation must vanish. This implies that the charges Q_n defined by second variations are identically vanishing

Q_n(X³_f) = ∫_X³fJ_n = 0 .

Since the second end can be chosen arbitrarily, one obtains an infinite number of conditions analogous to the Virasoro conditions. The analogs of unbroken loop group symmetry for H isometries and unbroken local super symmetry generated by right handed neutrino result. Thus extremal property is a necessary condition for the realization of the gauge symmetries present at partonic level also at the level of the space-time surface. The breaking of super-symmetries could perhaps be understood in terms of the breaking of these symmetries for light-like partonic 3-surfaces which are not extremals of Chern-Simons action.

For more details see the chapter TGD and Astrophysics .