In the following I describe the basic picture about lifting space-time surfaces of M4× CP2 to twistor spaces in CP3 × F3 getting their twistor structure via induction like process. This approach means generalizing the induction procedure from the level of space-time surfaces and imbedding space to the level of twistor spaces of space-time surfaces and twistor space of imbedding space (some time ago I wrote about induced second quantization). The outcome is that the magnificient mathematical knowhow about algebraic geometry utilized in super string theories becomes available in TGD framewok and in principle any string theorist can start doing TGD. Of course, Calabi-Yaus are replaced with twistor spaces and physical interpretation changes dramatically.
Conditions for twistor spaces as sub-manifolds
Consider the conditions that must be satisfied using local trivializations of the twistor spaces identified as sub-manifolds of CP3× F3 with induced twistor structure. Before continuing let us introduce complex coordinates zi=xi+iyi resp. wi=ui+ivi for CP3 resp. F3.
- 6 conditions are required and they must give rise by bundle projection to 4 conditions relating the coordinates in the Cartesian product of the base spaces of the two bundles involved and thus defining 4-D surface in the Cartesian product of compactified M4 and CP2.
- One has Cartesian product of two fiber spaces with fiber CP1 giving fiber space with fiber CP11× CP12. For the 6-D surface the fiber must be CP1. It seems that one must identify the two spheres CP1i. Since holomorphy is essential, holomorphic identification w1 = f(z1) or z1=f(w1) is the first guess. A stronger condition is that the function f is meromorphic having thus only finite numbers of poles and zeros of finite order so that a given point of CP1i is covered by CP1i+1. Even stronger and very natural condition is that the identification is bijection so that only Möbius transformations parametrized by SL(2,C) are possible.
- Could the Möbius transformation f: CP11→ CP12 depend parametrically on the coordinates z2,z3 so that one would have w1= f1(z1,z2,z3), where the complex parameters a,b,c,d (ad-bc=1) of Möbius transformation depend on z2 and z3 holomorphically? Does this mean the analog of local SL(2,C) gauge invariance posing additional conditions? Does this mean that the twistor space as surface is determined up to SL(2,C) gauge transformation?
What conditions can one pose on the dependence of the parameters a,b,c,d of the Möbius transformation on (z2,z3)? The spheres CP1 defined by the conditions w1= f(z1,z2,z3) and z1= g(w1,w2,w3) must be identical. Inverting the first condition one obtains z1= f-1(w1,z2,z3). If one requires that his allows an expression as z1= g(w1,w2,w3), one must assume that z2 and z3 can be expressed as holomorphic functions of (w2,w3): zi= fi(wk), i=2,3, k=2,3. Of course, non-holomorphic correspondence cannot be excluded.
- Further conditions are obtained by demanding that the known extremals - at least non-vacuum extremals - are allowed. The known extremals can be classified into CP2 type vacuum
extremals with 1-D light-like curve as M4 projection, to vacuum extremals with CP2 projection, which is Lagrangian sub-manifold and thus at most 2-dimensional, to massless extremals with 2-D CP2 projection such that CP2 coordinates depend on arbitrary manner on light-like coordinate defining local propagation direction and space-like coordinate defining a local polarization direction, and to string like objects with string world sheet as M4 projection (minimal surface) and 2-D complex sub-manifold of CP2 as CP2 projection. There are certainly also other extremals such as magnetic flux tubes
resulting as deformations of string like objects. Number theoretic vision relying on classical number fields suggest a very general construction based on the notion of associativity of tangent space or co-tangent space.
- The conditions coming from these extremals reduce to 4 conditions expressible in the holomorphic case in terms of the base space coordinates (z2,z3) and (w2,w3) and in the more general case in terms of the corresponding real coordinates. It seems that holomorphic ansatz is not consistent with the existence of vacuum extremals, which however give vanishing contribution to transition amplitudes since WCW ("world of classical worlds") metric is completely degenerate for them.
The mere condition that one has CP1 fiber bundle structure does not force field equations since it leaves the dependence between real coordinates of the base spaces free. Of course, CP1 bundle structure alone does not imply twistor space structure. One can ask whether non-vacuum extremals could correspond to holomorphic constraints between (z2, z3) and (w2, w3).
- The metric of twistor space is not Kähler in the general case. However, if it allows complex structure there is a Hermitian form ω, which defines what is called balanced Kähler form
satisfying d(ω∧ω)=2ω∧ dω=0: ordinary Kähler form satisfying dω=0 is special case about this. The natural metric of compact 6-dimensional twistor space is therefore balanced. Clearly, mere CP1 bundle structure is not enough for the twistor structure. If the the Kähler and symplectic forms are induced from those of CP3× Y3, highly non-trivial conditions are obtained for the imbedding of the twistor space, and one might hope that they are equivalent with those implied by Kähler action at the level of base space.
- Pessimist could argue that field equations are additional conditions completely independent of the conditions realizing the bundle structure! One cannot exclude this possibility. Mathematician could easily answer the question about whether the proposed CP1 bundle structure with some added conditions is enough to produce twistor space or not and whether field equations could be the additional condition and realized using the holomorphic ansatz.
- Canonical imbeddings of M4 and CP2 and their disjoint unions are certainly the natural starting point and correspond to canonical imbeddings of CP3 and F3 to CP3× F3.
- Deformations of M4 correspond to space-time sheets with Minkowskian signature of the induced metric and those of CP2 to the lines of generalized Feynman diagrams. The simplest deformations of M4 are vacuum extremals with CP2 projection which is Lagrangian manifold.
Massless extremals represent non-vacuum deformations with 2-D CP2 projection. CP2 coordinates depend on local light-like direction defining the analog of wave vector and local polarization direction orthogonal to it.
The simplest deformations of CP2 are CP2 type extremals with light-like curve as M4 projection and have same Kähler form and metric as CP2. These space-time regions have Euclidian signature of metric and light-like 3-surfaces separating Euclidian and Minkowskian regions define parton orbits.
String like objects are extremals of type X2× Y2, X2 minimal surface in M4 and Y2 a complex sub-manifold of CP2. Magnetic flux tubes carrying monopole flux are deformations of these.
Elementary particles are important piece of picture. They have as building bricks wormhole contacts connecting space-time sheets and the contacts carry monopole flux. This requires at least two wormhole contacts connected by flux tubes with opposite flux at the parallel sheets.
- Space-time surfaces are constructed using as building bricks space-time sheets, in particular massless exrremals, deformed pieces of CP2 defining lines of generalized Feynman diagrams as orbits of wormhole contacts, and magnetic flux tubes connecting the lines. Space-time surfaces have in the generic case discrete set of self intersections and it is natural to remove them by connected sum operation. Same applies to twistor spaces as sub-manifolds of CP3× F3 and this leads to a construction analogous to that used to remove singularities of Calabi-Yau spaces.
Twistor spaces by adding CP1 fiber to space-time surfaces
Physical intuition suggests that it is possible to find twistor spaces associated with the basic building bricks and to lift this engineering procedure to the level of twistor space in the sense that the twistor projections of twistor spaces would give these structure. Lifting would essentially mean assigning CP1 fiber to the space-time surfaces.
- Twistor spaces should decompose to regions for which the metric induced from the CP3× F3 metric has different signature. In particular, light-like 5-surfaces should replace the light-like 3-surfaces as causal horizons. The signature of the Hermitian metric of 4-D (in complex sense) twistor space is (1,1,-1,-1). Minkowskian variant of CP3 is defined as the projective space SU(2,2)/SU(2,1)×U(1). The causal diamond (CD) (intersection of future and past directed light-cones) is the key geometric object in zero energy ontology (ZEO) and the generalization to the intersection of twistorial light-cones is suggestive.
- Projective twistor space has regions of positive and negative projective norm, which are 3-D complex manifolds. It has also a 5-dimensional sub-space consisting of null twistors analogous to light-cone and has one null direction in the induced metric. This light-cone has conic singularity analogous to the tip of the light-cone of M4.
These conic singularities are important in the mathematical theory of Calabi-You manifolds since topology change of Calabi-Yau manifolds via the elimination of the singularity can be associated with them. The S2 bundle character implies the structure of S2 bundle for the base of the singularity (analogous to the base of the ordinary cone).
- Null twistor space corresponds at the level of M4 to the light-cone boundary (causal diamond has two light-like boundaries). What about the light-like orbits of partonic 2-surfaces whose light-likeness is due to the presence of CP2 contribution in the induced metric? For them the determinant of induced 4-metric vanishes so that they are genuine singularities in metric sense. The deformations for the canonical imbeddings of this sub-space (F3 coordinates constant) leaving its metric degenerate should define the lifts of the light-like orbits of partonic 2-surface. The singularity in this case separates regions of different signature of induced metric.
It would seem that if partonic 2-surface begins at the boundary of CD, conical singularity is not necessary. On the other hand the vertices of generalized Feynman diagrams are 3-surfaces at which 3-lines of generalized Feynman digram are glued together. This singularity is completely analogous to that of ordinary vertex of Feynman diagram. These singularities should correspond to gluing together 3 deformed F3 along their ends.
- These considerations suggest that the construction of twistor spaces is a lift of construction space-time surfaces and generalized Feynman diagrammatics should generalize to the level of twistor spaces. What is added is CP1 fiber so that the correspondence would rather concrete.
- For instance, elementary particles consisting of pairs of monopole throats connected buy flux tubes at the two space-time sheets involved should allow lifting to the twistor level. This means double connected sum and this double connected sum should appear also for deformations of F3 associated with the lines of generalized Feynman diagrams. Lifts for the deformations of magnetic flux tubes to which one can assign CP3 in turn would connect the two F3s.
- A natural conjecture inspired by number theoretic vision is that Minkowskian and Euclidian space-time regions correspond to associative and co-associative space-time regions. At the level of twistor space these two kinds of regions would correspond to deformations of CP3 and F3. The signature of the twistor norm would be different in this regions just as the signature of induced metric is different in corresponding space-time regions.
These two regions of space-time surface should correspond to deformations for disjoint unions of CP3s and F3s and multiple connected sum form them should project to multiple connected sum (wormhole contacts with Euclidian signature of induced metric) for deformed CP3s. Wormhole contacts could have deformed pieces of F3 as counterparts.
Twistor spaces as analogs of Calabi-Yau spaces of super string models
CP3 is also a Calabi-Yau manifold in the strong sense that it allows Kähler structure and complex structure. Witten's twistor string theory considers 2-D (in real sense) complex surfaces in twistor space CP3. This inspires tome questions.
- Could TGD in twistor space formulation be seen as a generalization of this theory?
- General twistor space is not Calabi-Yau manifold because it does does not have Kähler structure. Do twistor spaces replace Calabi-Yaus in TGD framework?
- Could twistor spaces be Calabi-Yau manifolds in some weaker sense so that one would have a closer connection with super string models.
Consider the last question.
- One can indeed define non-Kähler Calabi-Yau manifolds by keeping the hermitian metric and giving up symplectic structure or by keeping the symplectic structure and giving up hermitian metric (almost complex structure is enough). Construction recipes for non-Kähler Calabi-Yau manifold are discussed in here. It is shown that these two manners to give up Kähler structure correspond to duals under so called mirror symmetry, which maps complex and symplectic structures to each other. This construction applies also to the twistor spaces and is especially natural for them because of the fiber space structure.
- For the modification giving up symplectic structure, one starts from a smooth Kähler Calabi-Yau 3-fold Y, such as CP3. One assumes a discrete set of disjoint rational curves diffeomorphic to CP1. In TGD framework work they would correspond to special fibers of twistor space.
One has singularities in which some rational curves are contracted to point - in twistorial case the fiber of twistor space would contract to a point - this produces double point singularity which one can visualize as the vertex at which two cones meet (sundial should give an idea about what is involved). One deforms the singularity to a smooth complex manifold. One could interpret this as throwing away the common point and replacing it with connected sum contact: a tube connecting the holes drilled to the vertices of the two cones. In TGD one would talk about wormhole contact.
- Suppose the topology looks locally like S3× S2× R+/- near the singularity, such that two copies analogous to the two halves of a cone (sundial) meet at single point defining double point singularity. In the recent case S2 would correspond to the fiber of the twistor space. S3 would correspond to 3-surface and R+/- would correspond to time coordinate in past/future direction. S3 could be replaced with something else.
The copies of S3× S2 contract to a point at the common end of R+ and R- so that both the based and fiber contracts to a point. Space-time surface would look like the pair of future and past directed light-cones meeting at their tips.
For the first modification giving up symplectic structure only the fiber S2 is contracted to a point and S2× D is therefore replaced with the smooth "bottom" of S3. Instead of sundial one has two balls touching. Drill small holes two the two S3s and connect them by connected sum contact (wormhole contact). Locally one obtains S3× S3 with k connected sum contacts.
For the modification giving up Hermitian structure one contracts only S3 to a point instead of S2. In this case one has locally two CP3:s touching (one can think that CPn is obtained by replacing the points of Cn at infinity with the sphere CP1). Again one drills holes and connects them by a connected sum contact to get k-connected sum of CP3.
For k CP1s the outcome looks locally like to a k-connected sum of S3 × S3 or CP3 with k≥ 2. In the first case one loses
symplectic structure and in the second case hermitian structure. The conjecture is that the two manifolds form a mirror pair.
The general conjecture is that all Calabi-Yau manifolds are obtained using these two modifications. One can ask whether this conjecture could apply also the construction of twistor spaces representable as surfaces in CP3× F3 so that it would give mirror pairs of twistor spaces.
- This smoothing out procedures isa actually unavoidable in TGD because twistor space is sub-manifold. The 6-D twistor spaces in 12-D CP3× F3 have in the generic case self intersections consisting of discrete points. Since the fibers CP1 cannot intersect and since the intersection is point, it seems that the fibers must contract to a point. In the similar manner the 4-D base spaces should have local foliation by spheres or some other 3-D objects with contract to a point. One has just the situation described above.
One can remove these singularities by drilling small holes around the shared point at the two sheets of the twistor space and connected the resulting boundaries by connected sum contact. The preservation of fiber structure might force to perform the process in such a manner that local modification of the topology contracts either the 3-D base (S3 in previous example or fiber CP1 to a point.
To sum up, the construction of space-times as surfaces of H lifted to that of (almost) complex sub-manifolds in CP3× F3 with induced twistor structure shares the spirit of the vision that induction procedure is the key element of classical and quantum TGD. It also gives deep connection with the mathematical methods applied in super string models and these methods should be of direct use in TGD.