Symmetries and Geometry of the "World of Classical Worlds"

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 of H 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.

Could generalized holomorphy allow to sharpen the existing views?

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.

4. The action should define Kähler function for WCW identified as the space of Bohr orbits. WCW Kähler metric is defined in terms of the second derivatives of the Kähler action of type (1,1) with respect to complex coordinates of WCW. Does the invariance of the action under holomorphies imply a trivial Kähler metric and constant Kähler function?

   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.

5. An absolutely essential point is that generalized holomorphisms are not symmetries of Kähler function since otherwise Kähler metric involving second derivatives of type (1,1) with respect to complex coordinates of WCW is non-trivial if defined by these symmetry generators as differential operators. If Kähler function is equal to Kähler action, as it seems, Kähler action cannot be invariant under generalized holomorphies.

   Noether's theorem states that the invariance of the action under a symmetry implies the conservation of corresponding charge but does not claim that the existence of conserved Noether currents implies invariance of the action. Since Noether currents are conserved now, one would have a concrete example about the situation in which the inverse of Noether's theorem does not hold true. In a string model based on area action, conformal transformations of complex string coordinates give rise to conserved Noether currents as one easily checks. The area element defined by the induced metric suffers a conformal scaling so that the action is not invariant in this case.

Challenging the existing picture of WCW geometry

These findings make it possible to challenge and perhaps sharpen the existing speculations concerning the metric and isometries of WCW.

I have considered the possibility that also the symplectomorphisms of δ M⁴+× CP₂ could define WCW isometries. This actually the original proposal. One can imagine two options.

1. The continuation of symplectic transformations to transformations of the space-time surface from the boundary of light-cone or from the orbits partonic 2-surfaces should give rise to conserved Noether currents but it is not at all obvious whether this is the case.
2. One can assign conserved charges to the time evolution of the 3-D boundary data defining the holographic data: the time coordinate for the evolution would correspond to the light-like coordinate of light-cone boundary or partonic orbit. This option I have not considered hitherto. It turns out that this option works!

The conclusion would be that generalized holomorphies give rise to conserved charges for 4-D time evolution and symplectic transformations give rise to conserved charged for 3-D time evolution associated with the holographic data.

About extremals of Chern-Simons-Kähler action

Let us look first the general nature of the solutions to the extremization of Chern-Simons-Kähler action.

1.  The light-likeness of the partonic orbits requires Chern-Simons action, which is equivalent to the topological action J∧ J, which is total divergence and   is a symplectic in variant.  The field equations at the boundary cannot involve  induced metric so that only induced symplectic structure remains. The 3-D holographic data   at partonic orbits would extremize Cherns-Simons-Kähler action. Note that at the ends of the space-time surface about boundaries of CD one cannot pose any dynamics.
2. If the induced Kähler form has only the CP₂ part, the variation of Chern-Simons-Kähler form would give equations  satisfied if the CP₂ projection is at most 2-dimensional and Chern-Simons action would vanish and imply that instanton number vanishes.
3. If the action is the sum of M⁴ and CP₂  parts, the field equations in M⁴ and CP₂ degrees of freedom would give the same result. If the induced Kähler form is  identified as the sum of the M⁴ and CP₂ parts, the equations also allow solutions for which the induced M⁴ and CP₂ Kähler forms sum up to zero.  This phase would involve a map identifying M⁴ and CP₂ projections and force induce Kähler forms to be identical. This would force magnetic charge in M⁴ and the question is whether the line connecting the tips of the CD makes non-trivial homology possible.  The homology charges and the 2-D ends of the partonic orbit cancel each other so that partonic surfaces can have monopole charge.

   The conditions at the partonic orbits do not pose conditions on the interior and should allow generalized holomorphy. The following considerations show that besides homology charges as Kähler magnetic fluxes also Hamiltonian fluxes are conserved in Chern-Simons-Kähler dynamics.

Can one assign conserved charges with symplectic transformations or partonic orbits and 3-surfaces at light-cone boundary?

The geometric picture is that symplectic symmetries are Hamiltonian flows along the light-like partonic orbits generated by the projection A_t of the Kähler gauge potential in the direction of the light-like time coordinate. The physical picture is that the partonic 2-surface is a Kähler charged particle that couples to the Hamilton H=A_t. The Hamiltonians H_A are conserved in this time evolution and give rise to conserved Noether currents. The corresponding conserved charge is integral over the 2-surface defined by the area form defined by the induced Kähler form.

Let's examine the change of the Chern-Simons-Kähler action in a deformation that corresponds, for example, to the CP₂ symplectic transformation generated by Hamilton H_A. M⁴ symplectic transformations can be treated in the same way:here however M⁴ Kähler form would be involved, assumed to accompany Hamilton-Jacobi structure as a dynamically generated structure.

1. Instanton density for the induced Kähler form reduces to a total divergence and gives Chern-Simons-Kähler action, which is TGD analog of topological action. This action should change in infinitesimal symplectic transformations by a total divergence, which should vanish for extremals and give rise to a conserved current. The integral of the divergence gives a vanishing charge difference between the ends of the partonic orbit. If the symplectic transformations define symmetries, it should be possible to assign to each Hamiltonian H_A a conserved charge. The corresponding quantal charge would be associated with the modified Dirac action.

2. The conserved charge would be an integral over X². The surface element is not given by the metric but by the symplectic structure, so that it is preserved in symplectic transformations. The 2-surface of the time evolution should correspond to the Hamiltonian time transformation generated by the projection A_α=A_k ∂_αs^k of the Kähler gauge potential A_k to the direction of light-like time coordinate xα== t.

3. The effect of the generator j_A^k= J^kl∂_lH_A on the Kähler potential A_l is given by j^k_A∂_kA_l. This can be written as ∂_kA_l=J_kl + ∂_lA_k. The first term gives the desired total divergence ∂_α (ε^αβγJ_βγ H_A).

   The second term is proportional to the term ∂_αH_A- {A_α,H}. Suppose that the induced Kähler form is transversal to the light-like time coordinate t, i.e. the induced Kähler form does not have components of form J_tμ. In this kind of situation the only possible choice for α corresponds to the time coordinate t. In this situation one can perform the replacement ∂_αH_A-{A_α,H}→ dH_A/dt-{A_t,H}. This corresponds to a Hamiltonian time evolution generated by the projection A_t acting as a Hamiltonian. If this is really a Hamiltonian time evolution, one has dH_A/dt-{A,H}=0. Because the Poisson bracket represents a commutator, the Hamiltonian time evolution equation is analogous to the vanishing of a covariant derivative of H_A along light-like curves: ∂_tH_A +[A,H_A]= 0. The physical interpretation is that the partonic surface develops like a particle with a Kähler charge. As a consequence the change of the action reduces to a total divergence.

   An explicit expression for the conserved current J_A^α=H_A ε^αβγJ_βγ can be derived from the vanishing of the total divergence. Symplectic transformations on X² generate an infinite-dimensional symplectic algebra. The charge is given by the Hamiltonian flux Q_A =∫ H_A J_αβdx^α∧dx^β.

4. If the projection of the partonic path CP₂ or M⁴ is 2-D, then the light-like geodesic line corresponds to the path of the parton surface. If A_l can be chosen parallel to the surface, its projection in the direction of time disappears and one has A_t=0. In the more general case, X² could, for example, rotate in CP₂. In this case A_t is nonvanishing. If J is transversal (no Kähler electric field), charge conservation is obtained.

Do the above observations apply at the boundary of the light-cone?

1. Now the 3-surface is space-like and Chern-Simons-Kähler action makes sense. It is not necessary but emerges from the "instanton density" for the Kähler form. The symplectic transformations of δ M⁴₊× CP₂ are the symmetries. The most time evolution associated with the radial light-like coordinate would be from the tip of the light-cone boundary to the boundary of CD. Conserved charges as homological invariants defining symplectic algebra would be associated with the 2-D slices of 3-surfaces. For closed 3-surfaces the total charges from the sheets of 3-space as covering of δ M⁴₊ must sum up to zero.
2. Interestingly, the original proposal for the isometries of WCW was that the Hamiltonian fluxes assignable to M⁴ and CP₂ degrees of freedom at light-like boundary act define the charges associated with the WCW isometries as symplectic transformations so that a strong form of holography would have been be realized and space-time surface would have been effectively 2-dimensional. The recent view is that these symmetries pose conditions only on the 3-D holographic data. The holographic charges would correspond to additional isometries of WCW and would be well-defined for the 3-surfaces at the light-cone boundary.

To sum up, one can imagine many options but the following picture is perhaps the simplest one and is supported by physical intuition and mathematical facts. The isometry algebra ofδ M⁴₊× CP₂ consists of generalized conformal and KM algebras at 3-surfaces in δ M⁴₊× CP₂ and symplectic algebras at the light cone boundary and 3-D light-like partonic orbits. The latter symmetries give constraints on the 3-D holographic data. It is still unclear whether one can assign generalized conformal and Kac-Moody charges to Chern-Simons-K\"ahler action. The isomorphic subalgebras labelled by a positive integer and their commutators with the entire algebra would annihilate the physical states.

The TGD counterparts of the gauge conditions of string models

The string model picture forces to ask whether the symplectic algebras and the generalized conformal and Kac-Moody algebras could act as gauge symmetries.

1. In string model picture conformal invariance would suggest that the generators of the generalized conformal and KM symmetries act as gauge transformations annihilate the physical states. In the TGD framework, 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. The half-algebras of both symplectic and conformal generators are labelled by a non-negative integer defining an analog of conformal weight so there is a fractal hierarchy of isomorphic subalgebras in both cases.
2. TGD forces to ask whether only subalgebras of both conformal and Kac-Moody half algebras, isomorphic to the full algebras, act as gauge algebras. This applies also to the symplectic case. 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. The reduction of degrees of freedom would also be analogous to the basic property of hyperfinite factors assumed to play a key role in thee definition of finite measurement resolution.
3. For strong holography, the orbits of partonic 2-surfaces and boundaries of the spacetime surface at δ M⁴₊ would be dual in the information theoretic sense. Either would be enough to determine the space-time surface.

See the articles About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe, Holography and Hamilton-Jacobi Structure as 4-D generalization of 2-D complex structure, Symmetries and Geometry of the "World of Classical Worlds" and the chapter Recent View about K\"ahler Geometry and Spin Structure of "World of Classical Worlds".

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.