Still about the symmetries of WCW

I have been analyzing the basic visions of TGD trying to identify weak points. WCW geometry exists only if it has maximal isometries. I have proposed that WCW could be regarded as a union of generalized symmetric spaces labelled by zero modes which do not contribute to the metric. The induced Kähler field is invariant under symplectic transformations of CP₂ and would therefore define zero mode degrees of freedom if one assumes that WCW metric has symplectic transformations as isometries. In particular, Kähler magnetic fluxes would define zero modes and are quantized closed 2-surfaces. The induced metric appearing in Kähler action is however not zero mode degree of freedom. If the action contains volume term, the assumption about union of symmetric spaces is not well-motivated.

Symplectic transformations are not the only candidates for the isometries of WCW. The basic picture about what these maximal isometries could be, is partially inspired by string models.

1. A weaker proposal is that the symplectomorphisms define only symplectomorphisms of WCW. Extended conformal symmetries define also a candidate for isometry group. Remarkably, light-like boundary has an infinite-dimensional group of isometries which are in 1-1 correspondence with conformal symmetries of S²⊂ S²× R₊= δ M⁴₊.
2. Extended Kac Moody symmetries induced by isometries of δ M⁴₊ are also natural candidates for isometries. The motivation for the proposal comes from physical intuition deriving from string models. Note they do not include Poincare symmetries, which act naturally as isometries in the moduli space of causal diamonds (CDs) forming the “spine” of WCW.
3. The light-like orbits of partonic 2-surfaces might allow separate symmetry algebras. One must however notice that there is exchange of charges between interior degrees of freedom and partonic 2-surfaces. The essential point is that one can assign to these surface conserved charges when the dual light-like coordinate defines time coordinate. This picture also assumes a slicing of space-time surface by by the partonic orbits for which partonic orbits associated with wormrhole throats and boundaries of the space-time surface would be special. This slicing would correspond to Hamilton-Jacobi structure.
4. Fractal hierarchy of symmetry algebras with conformal weights, which are non-negative integer multiples of fundamental conformal weights, is essential and distinguishes TGD from string models. Gauge conditions are true only the isomorphic subalgebra and its commutator with the entire algebra and the maximal gauge symmetry to a dynamical symmetry with generators having conformal weights below maximal value. This view also conforms with p-adic mass calculations.
5. The realization of the symmetries for 3-surfaces at the boundaries of CD and for light-like orbits of partonic 2-surfaces is known. The problem is how to extend the symmetries to the interior of the space-time surface. It is natural to expect that the symmetries at partonic orbits and light-cone boundary extend to the same symmetries.

This picture is rather speculative, allows several variants, and is not proven. There is now however a rather convincing ansatz for the general form of preferred extremals. This proposal relies on the realization of holography as generalized 4-D holomorphy. Could it help to make the picture more precise?

1. Explicit solution of field equations in terms of the generalized holomorphy is now known. The solution ansatz is independent of action as long it is general coordinate invariance depending only on the induced geometric structures. Space-time surfaces would be minimal surfaces apart from lower-dimensional singular surfaces at which the field equations involve the entire action. Only the singularities, classical charges and positions of topological interaction vertices depend on the choice of the action (see this). Kähler action plus volume term is the choice of action forced by twistor lift making the choice of H unique.
2. Hamilton-Jacobi structures emerge naturally as generalized conformal structures of space-time surfaces and M⁴ (see this). This inspires a proposal for a generalization of modular invariance and of moduli spaces as subspaces of Teichmüller spaces.
3. One can assign to holomorphy conserved Noether charges. The conservation reduces to the algebraic conditions satisfied for the same reason as field equations, i.e. the conservation conditions involving contractions of complex tensors of type (1,1) with tensors of type (2,0) and (0,2). The charges have the same form as Noether charges but it is not completely clear whether the action remains invariant under these transformations. This point is non-trivial since Noether theorem says that invariance of the action implies the existence of conserved charges but not vice versa. Could TGD represent a situation in which the equivalence between symmetries of action and conservation laws fails?

Also string models have conformal symmetries but in this case 2-D area form suffers conformal scaling. Also the fact that holomorphic ansatz is satisfied for such a large class of actions apart from singularities suggests that the action is not invariant.

Here one must be very cautious since by holography the variations of the space-time surface are induced by those of 3-surface defining holographic data so that the entire space-time surface is modified and the action can change. The presence of singularities, analogous to poles and cuts of an analytic function and representing particles, suggests that the action represents the interactions of particles and must change. Therefore the action might not be invariant under holomorphies. The parameters characterizing the singularities should affect the value of the action just as the positions of these singularities in 2-D electrostatistics affect the Coulomb energy.

Generalized conformal charges and supercharges define a generalization of Super Virasoro algebra of string models. Also Kac-Moody algebras assignable to the isometries of δ M⁴₊× CP₂ and light-like 3 surfaces generalize trivially. These findings make it possible to challenge and perhaps sharpen the existing speculations concerning the metric and isometries of WCW.

1. Could generalized conformal and KM charges and supercharges give all relevant isometry charges? Are super symplectic charges needed at all?
2. An absolutely essential point could be that generalized holomorphisms are not symmetries of action so that Kähler metric involving second derivatives with respect to complex coordinates of WCW is non-trivial if defined by these symmetry generators as differential operators. Also the symplectomorphisms of δ M⁴₊× CP₂ fail to leave the action invariant. This as it should be and they should allow a continuation to transformations of space-time surface acting as isometries of WCW.

With this assumption, the anticommutators of fermionic superchargers and WCW gamma matrices defined by the 4-D analogs of conformal and Kac-Moody generators should give a non-trivial contribution to the WCW metric. Is the super symplectic contribution to the metric needed at all? This raises a bundle of questions. Could the conformal and associated Kac-Moody gamma matrices determine the entire WCW metric? Is super symplectic algebra needed at all? Or are both needed? Or could these two symmetries define the same WCW metric and define isometry algebras which are somehow dual?

This question is inspired by the fact that symplectomorphisms and holomorphisms are very closely related. Kähler manifolds are symplectic manifolds and complex manifolds. Kähler metric is proportional to the symplectic form in complex coordinates. In superstring models there was an entire industry about Calabi-Yau spaces, which are dual in that they give rise to the same physics: this meant that the Kähler and complex structures for these pairs were somehow dual.

1. How do the symplectic and generalized conformal symmetries relate? Consider CP₂ as a simple example. For CP₂, isometries are both holomorphisms and symplectomorphisms. Could the isometries of WCW be both holomorphisms and symplectomorphisms of WCW? Could the isometries of WCW correspond to generalized holomorphisms and Kac Moody symmetries for M⁴× CP₂ or for δ M⁴₊× CP₂ acting also as symplectomorphisms of WCW? Could the conserved charges be corresponding Hamiltonians at the WCW level. This inspires the question whether the symplectomorphisms of δ M⁴₊× CP₂ are needed at all.
2. Also the string model picture forces to ask whether the generalized conformal and Kac-Moody algebras could act as gauge symmetries. Conformal invariance in string model view would suggest that the generators of the generalized conformal and KM symmetries act as gauge transformations annihilate the physical states. This does not however make sense physically. This also suggests that the components of the metric defined by supergenerators of generalized conformal and Kac Moody transformations vanish. If so, the symplectomorphisms δ M⁴₊× CP₂ localized with respect to the light-like radial coordinate acting as isometries would be needed.

TGD however leads to the proposal that only subalgebras of both conformal and Kac-Moody half algebras, isomorphic to the full algebras, act as gauge algebras. Here it is essential that only the half algebra with non-negative multiples of the fundamental conformal weights is allowed. For the subalgebra annihilating the states the conformal weights would be fixed integer multiples of those for the full algebra. The gauge property would be true for all algebras involved. The remaining symmetries would be genuine dynamical symmetries of the reduced WCW and this would reflect the number theoretically realized finite measurement resolution. In this picture symplectic algebra δ M⁴₊× CP₂ would not be necessary. The reduction of degrees of freedom would also be analogous to the basic property of hyperfinite assumed to play a key role in thee definition of finite measurement resolution.

Could generalized holomorphic and KM charges at partonic orbits be dual to the symplectic charges at the boundary of light-cone? Either algebra would be enough to determine the WCW metric. Note that the light-likeness partonic orbits allows to assign conserved holomorphic charges to them with a time coordinate identified as the dual light-like coordinate. The action would be Kähler-Chern-Simons action which does not contain the induced metric. To sum up, this suggests two basic options.

1. The simplest option is that the isometry algebra of δ M⁴₊× CP₂ forms consists of generalized conformal and KM algebras at both the 3-D partonic orbits and the 3-surfaces at δ M⁴₊× CP₂. For the strongest 2→ 4 form of holography these representations would be equivalent. This would be true at least when there is no breaking of classical determinism for the Bohr orbits. The isomorphic subalgebras labelled by a positive integer and their commutators with the entire algebra would annihilate the physical states. These two representations would generalize the notions of inertial and gravitational mass and their equivalence would generalize the Equivalence Principle.

What makes this so interesting is that, due to the light-likeness of light-cone boundary, the algebra of isometries of δ M⁴₊× CP₂ corresponds to the infinite-dimensional algebra of holomorphisms of S² localized with respect to the light-like radial coordinate δ M⁴₊! Radially localized holomorphisms would act as isometries of the light-cone boundary and induce isometries of WCW! Same is true at the light-like orbits.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.