What's new inTGD: Physics as InfiniteDimensional GeometryNote: Newest contributions are at the top! 
Year 2011 
Algebraic braids, submanifold braid theory, and generalized Feynman diagramsUlla send me a link to an article by Sam Nelson about very interesting newtome notion known as algebraic knots, which has initiated a revolution in knot theory. This notion was introduced 1996 by Louis Kauffmann so that it is already 15 year old concept. While reading the article I realized that this notion fits perfectly the needs of TGD and leads to a progress in attempts to articulate more precisely what generalized Feynman diagrams are. The challenge is to understand how the notion of algebraic knot could be applied to generalized Feynman diagrams. The algebraic structrures kei, quandle, rack, and biquandle and their algebraic modifications as such are not enough. The lines of Feynman graphs are replaced by braids and in vertices braid strands redistribute. This poses several challenges: the crossing associated with braiding and crossing occurring in nonplanar Feynman diagrams should be integrated to a more general notion; braids are replaced with submanifold braids; braids of braids ....of braids are possible; the redistribution of braid strands in vertices should be algebraized. One challenge is to abstract the basic operations which should be algebraized in the case of generalized Feynman diagrams. One should be also able to concretely identify braids and 2braids (string world sheets) as well as partonic 2surfaces and I have discussed several identifications during last years. Legendrian braids turn out to be very natural candidates for braids and their duals for the partonic 2surfaces. String world sheets in turn could correspond to the analogs of Lagrangian submanifolds or to minimal surfaces of spacetime surface satisfying the weak form of electricmagnetic duality. The latter option turns out to be more plausible. Finite measurement resolution would be realized as symplectic invariance with respect to the subgroup of the symplectic group leaving the end points of braid strands invariant. In accordance with the general vision TGD as almost topological QFT would mean symplectic QFT. The identification of braids, partonic 2surfaces and string world sheets  if correct  would solve quantum TGD explicitly at string world sheet level in other words in finite measurement resolution. Irrespective of whether the algebraic knots are needed, the natural question is what generalized Feynman diagrams are. It seems that the basic building bricks can be identified so that one can write rather explicit Feynman rules already now. Of course, the rules are still far from something to be burned into the spine of the first year graduate student. For details and background see the article Algebraic braids, submanifold braid theory, and generalized Feynman diagrams or the chapter Knots and TGD. 
Is Kähler action expressible in terms of areas of minimal surfaces?The general form of ansatz for preferred extremals implies that the Coulombic term in Kähler action vanishes so that it reduces to 3dimensional surface terms in accordance with general coordinate invariance and holography. The weak form of electricmagnetic duality in turn reduces this term to ChernSimons terms. The strong form of General Coordinate Invariance implies effective 2dimensionality (holding true in finite measurement resolution) so that also a strong form of holography emerges. The expectation is that ChernSimons terms in turn reduces to 2dimensional surface terms. The only physically interesting possibility is that these 2D surface terms correspond to areas for minimal surfaces defined by string world sheets and partonic 2surfaces appearing in the solution ansatz for the preferred extremals. String world sheets would give to Kähler action an imaginary contribution having interpretation as Morse function. This contribution would be proportional to their total area and assignable with the Minkowskian regions of the spacetime surface. Similar but real string world sheet contribution defining Kähler function comes from the Euclidian spacetime regions and should be equal to the contribution of the partonic 2surfaces. A natural conjecture is that the absolute values of all three areas are identical: this would realize duality between string world sheets and partonic 2surfaces and duality between Euclidian and Minkowskian spacetime regions. Zero energy ontology combined with the TGD analog of large N_{c} expansion inspires an educated guess about the coefficient of the minimal surface terms and a beautiful connection with padic physics and with the notion of finite measurement resolution emerges. The t'Thooft coupling λ should be proportional to padic prime p characterizing particle. This means extremely fast convergence of the counterpart of large N_{c} expansion in TGD since it becomes completely analogous to the pinary expansion of the partition function in padic thermodynamics. Also the twistor description and its dual have a nice interpretation in terms of zero energy ontology. This duality permutes massive wormhole contacts which can have off mass shell with wormhole throats which are always massive (also for the internal lines of the generalized Feynman graphs). For details and background see the article Is Kähler action expressible in terms of areas of minimal surfaces? or the chapter Identification of configuration space Kähler function. 
Weak form of electric magnetic duality and duality between Minkowskian and Euclidian spacetime regionsThe reduction of the Kähler action for the spacetime sheets with Minkowskian signature of the induced metric follows from the assumption that Kähler current is proportional to instanton current and from the weak form of electricmagnetic duality. The first property implies a reduction to a 3D term associated with wormhole throats and the latter property reduces this term to Abelian ChernSimons term. I have not explicitly considered whether the same happens in the 4D regions of Euclidian signature representing wormhole contacts. If these assumptions are made also in the Euclidian region, the outcome is that one obtains a difference of two ChernSimons terms coming from Minkowskian and Euclidian regions at lightlike wormhole throats. This difference can be nontrivial since Kähler form for CP_{2} defines nontrivial U(1) bundle. This however suggests that the total Kähler action is quantized in integer multiples of the Kähler action for CP_{2} type vacuum extremal so that one would have effectively sum over ninstanton configurations. If the Kähler function of the "world of classical worlds" (WCW) is identified as total Kähler action, this implies the vanishing of the Kähler metric of WCW, which is a catastrophe. Should one modify the definition of Kähler function by considering only the contribution from either Minkowskian or Euclidian regions? What about vacuum functional: should one identify it as the exponent of Kähler function or of Kähler action in this case?
The overall conclusion is that the only reasonable definitions of Kähler function of WCW and vacuum functional realize the conjecture duality between Minkowskian and Euclidian regions of spacetime surfaces. The overall conclusion is that the only reasonable definitions of Kähler function of WCW and vacuum functional realize the conjecture duality between Minkowskian and Euclidian regions of spacetime surfaces. This duality would have also number theoretical interpretation. Minkowskian regions of the spacetime surface would correspond to hyperquaternionic and Eulidian regions to quaternionic regions. In hyperquaternionic regions the modified gamma matrices would span hyperquaternionic plane of complexified octonions (imaginary units multiplied by commutative imaginary unit). In quaternionic regions the modified gamma matrices multiplied by a product of fixed octonionic imaginary unit and commutative imaginary unit would span a quaternionic plane of complexified octonions (see Does the Modified Dirac Equation Define the Fundamental Variational Principle. 
Is the effective metric defined by modified gamma matrices effectively one or twodimensional?The following argument suggests that the effective metric defined by the anticommutators of the modified gamma matrices is effectively one or twodimensional. Effective onedimensionality would conform with the observation that the solutions of the modified Dirac equations can be localized to onedimensional world lines in accordance with the vision that finite measurement resolution implies discretization reducing partonic manyparticle states to quantum superpositions of braids. This localization to 1D curves occurs always at the 3D orbits of the partonic 2surfaces. The argument is based on the following assumptions.

Witten's physical view about Khovanov homology translated to TGD frameworkKhovanov homology generalizes the Jones polynomial as knot invariant. The challenge is to find a quantum physical construction of Khovanov homology analous to the topological QFT defined by ChernSimons action allowing to interpret Jones polynomial as vacuum expectation value of Wilson loop in nonAbelian gauge theory. Witten's approach to Khovanov homology relies on fivebranes as is natural if one tries to define 2knot invariants in terms of their cobordisms involving violent unknottings. Despite the difference in approaches it is very useful to try to find the counterparts of this approach in quantum TGD since this would allow to gain new insights to quantum TGD itself as almost topological QFT identified as symplectic theory for 2knots, braids and braid cobordisms. An essentially unique identification of string world sheets and therefore also of the braids whose ends carry quantum numbers of many particle states at partonic 2surfaces emerges if one identifies the string word sheets as singular surfaces in the same manner as is done in Witten's approach. Even more, the conjectured slicings of preferred extremals by 3D surfaces and string world sheets central for quantum TGD can be identified uniquely. The slicing by 3surfaces would be interpreted in gauge theory in terms of Higgs= constant surfaces with radial coordinate of CP_{2} playing the role of Higgs. The slicing by string world sheets would be induced by different choices of U(2) subgroup of SU(3) leaving Higgs=constant surfaces invariant. Also a physical interpretation of the operators Q, F, and P of Khovanov homology emerges. P would correspond to instanton number and F to the fermion number assignable to right handed neutrinos. The breaking of M^{4} chiral invariance makes possible to realize Q physically. The finding that the generalizations of Wilson loops can be identified in terms of the gerbe fluxes ∫ H_{A} J supports the conjecture that TGD as almost topological QFT corresponds essentially to a symplectic theory for braids and 2knots. I do not bother to type the details but give a link to the article Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2knots?. See also the new chapter Knots and TGD. 
Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2knots?Lubos gave a link to a recent talk of Witten about knots and quantum physics. While listening the lecture one senses the enormous respect and I dare say love that the audience feels towards this silently talking genius completely free of all what might be called ego. Warmly recommended. Witten manages to explain in rather comprehensible manner both the construction recipe of Jones polynomial and the idea about how Jones polynomial emerges from topological quantum field theory as a vacuum expectation of so called Wilson loop defined by path integral with weighting coming from ChernSimons action. Witten also tells that during the last year he has been working with an attempt to understand in terms of quantum theory the so called Khovanov polynomial associated with a much more abstract link invariant whose interpretation and real understanding remains still open. This kind of talks are extremely inspiring and lead to a series of questions unavoidably culminating to the frustrating "Why I do not have the brain of Witten making perhaps possible to answer these questions?". This one must just accept. In the following I summarize some thoughts inspired by the associations of the talk of Witten with quantum TGD and with the model of DNA as topological quantum computer. In my own childish manner I dare believe that these associations are interesting and dare also hope that some more brainy individual might take them seriously. An idea inspired by TGD approach which also main streamer might find interesting is that the Jones invariant defined as vacuum expectation for a Wilson loop in 2+1D spacetime generalizes to a vacuum expectation for a collection of Wilson loops in 2+2D spacetime and could define an invariant for 2D knots and for cobordisms of braids analogous to Jones polynomial. As a matter fact, it turns out that a generalization of gauge field known as gerbe is needed and that in TGD framework classical color gauge fields defined the gauge potentials of this field. Also topological string theory in 4D spacetime could define this kind of invariants. Of course, it might well be that this kind of ideas have been already discussed in literature. As reader have noticed, the posting has gradually evolved during last days as I have noticed elementary errors and inaccuracies. My apologies for possible inconvenience. 1. Some TGD background What makes quantum TGD interesting concerning the description of braids and braid cobordisms is that braids and braid cobordisms emerge both at the level of generalized Feynman diagrams and in the model of DNA as a topological quantum computer. 1.1 Timelike and spacelike braidings for generalized Feynman diagrams
Time like braidings induces spacelike braidings and one can speak of timelike or dynamical braiding and even duality of timelike and spacelike braiding. What happens can be understood in terms of dance metaphor.
1.3 DNA as topological quantum computer The model for topological quantum computation is based on the idea that timelike braidings defining topological quantum computer programs. These programs are robust since the topology of braiding is not affected by small deformations.
2. Could braid cobordisms define more general braid invariants? Witten says that one should somehow generalize the notion of knot invariant. The above described framework indeed suggests a very natural generalization of braid invariants to those of braid cobordisms reducing to braid invariants when the braid at the other end is trivial. This description is especially natural in TGD but allows a generalization in which Wilson loops in 4D sense describe invariants of braid cobordisms. 2.1 Difference between knotting and linking Before my modest proposal of a more general invariant some comments about knotting and linking are in order.
2.2 Topological strings in 4D spacetime define knot cobordisms What makes the 4D braid cobordisms interesting is following.
3. Invariants 2knots as vacuum expectations of Wilson loops in 4D spacetime? The interpretation of string world sheets in terms of Wilson loops in 4dimensional spacetime is very natural. This raises the question whether Witten's a original identification of the Jones polynomial as vacuum expectation for a Wilson loop in 2+1D space might be replaced with a vacuum expectation for a collection of Wilson loops in 3+1D spacetime and would characterize in the general case (multi)braid cobordism rather than braid. If the braid at the lower or upped boundary is trivial, braid invariant is obtained. The intersections of the Wilson loops would correspond to the violent unknotting operations and the boundaries of the resulting holes give an additional Wilson loop. An alternative interpretation would be as the analog of Jones polynomial for 2D knots in 4D spacetime generalizing Witten's theory. This description looks completely general and does not require TGD at all. The following considerations suggest that Wilson loops are not enough for the description of general 2knots and that Wilson loops must be replaced with 2D fluxes. This requires a generalization of gauge field concept so that it corresponds to a 3form instead of 2form is needed. In TGD framework this kind of generalized gauge fields exist and their gauge potentials correspond to classical color gauge fields. 3.1 What 2knottedness means concretely? It is easy to imagine what ordinary knottedness means. One has circle imbedded in 3space. One projects it in some plane and looks for crossings. If there are no crossings one knows that unknot is in question. One can modify a given crossing by forcing the strands to go through each other and this either generates or removes knottedness. One can also destroy crossing by reconnection and this always reduces knottedness. Since knotting reduces to linking in 3D case, one can find a simple interpretation for knottedness in terms of linking of two circles. For 2knots linking is not what gives rise to knotting. One might hope to find something similar in the case of 2knots. Can one imagine some simple local operations which either increase of reduce 2knottedness?
Whether all possible 2knots are allowed for stringy world sheets, is not clear. In particular, if they are dynamically determined it might happen that many possibilities are not realized. For instance, the condition that the signature of the induced metric is Minkowskian could be an effective killer of 2knottedness not reducing to braid cobordism.
Suppose that the spacelike braid strands connecting partonic 2surfaces at given boundary of CD and lightlike braids connecting partonic 2surfaces belonging to opposite boundaries of CD form connected closed strands. The collection of closed loops can be identified as boundaries of Wilson loops and the expectation value is defined as the product of traces assignable to the loops. The definition is exactly the same as in 2+1D case. Is this generalization of Wilson loops enough to describe 2knots? In the spirit of the proposed philosophy one could ask whether there exist twoknots not reducible to cobordisms of 1knots whose knot invariants require cobordisms of 2knots and therefore 2braids in 5D spacetime. Could it be that dimension D=4 is somehow very special so that there is no need to go to D=5? This might be the case since for ordinary knots Jones polynomial is very faithful invariant. Innocent novice could try to answer the question in the following manner. Let us study what happens locally as the 2D closed surface in 4D space gets knotted.
In the sequel the considerations are restricted to TGD and to a comparison of Witten's ideas with those emerging in TGD framework. 4.1 Weak form of electricmagnetic duality and duality of spacelike and timelike braidings Witten notices that much of his work in physics relies on the assumption that magnetic charges exist and that rather frustratingly, cosmic inflation implies that all traces of them disappear. In TGD Universe the nontrivial topology of CP_{2}makes possible Kähler magnetic charge and inflation is replaced with quantum criticality. The recent view about elementary particles is that they correspond to string like objects with length of order electroweak scale with Kähler magnetically charged wormhole throats at their ends. Therefore magnetic charges would be there and LHC might be able to detect their signatures if LHC would get the idea of trying to do this. Witten mentions also electricmagnetic duality. If I understood correctly, Witten believes that it might provide interesting new insights to the knot invariants. In TGD framework one speaks about weak form of elecric magnetic duality. This duality states that Kähler electric fluxes at spacelike ends of the spacetime sheets inside CDs and at wormhole throats are proportional to Kähler magneic fluxes so that the quantization of Kähler electric charge quantization reduces to purely homological quantization of Kähler magnetic charge. The weak form of electricmagnetic duality fixes the boundary conditions of field equations at the lightlike and spacelike 3surfaces. Together with the conjecture that the Kähler current is proportional to the corresponding instanton current this implies that Kähler action for the preferred extremal sof Kähler action reduces to 3D ChernSimons term so that TGD reduces to almost topological QFT. This means an enormous mathematical simplification of the theory and gives hopes about the solvability of the theory. Since knot invariants are defined in terms of Abelian ChernSimons action for induced Kähler gauge potential, one might hope that TGD could as a byproduct define invariants of braid cobordisms in terms of the unitary Umatrix of the theory between zero energy states and having as its rows the nonunitary Mmatrices analogous to thermal Smatrices. Electric magnetic duality is 4D phenomenon as is also the duality between spacelike and time like braidings essential also for the model of topological quantum computation. Also this suggests that some kind of topological string theory for the spacetime sheets inside CDs could allow to define the braid cobordism invariants. 4.2 Could Kähler magnetic fluxes define invariants of braid cobordisms? Can one imagine of defining knot invariants or more generally, invariants of knot cobordism in this framework? As a matter fact, also Jones polynomial describes the process of unknotting and the replacement of unknotting with a general cobordism would define a more general invariant. Whether the Khovanov invariants might be understood in this more general framework is an interesting question.
4.3 Classical color gauge fields and their generalizations define gerbe gauge potentials allowing to replace Wilson loops with Wilson sheets As already noticed, the description of 2knots seems to necessitate the generalization of gauge field to 3form and the introduction of a gerbe structure. This seems to be possible in TGD framework.
This picture is very speculative and sounds too good to be true but follows if one consistently applies holography. 5. Summing up Let us summarize the ideas discussed above.
What is interesting that twistorial considerations lead to a conjecture that 4D spacetime surfaces in 8D imbedding space have a dual description in terms of certain 6D homomorphic surfaces which are sphere bundles in 12D CP_{3}× CP_{3} and effectively 4D. This suggests a connection between descriptions based on topological strings in 6D space and Wilson loops in 4D spacetime. Could it really be that these completely trivial observations of a mad Finnish scientist are not a standard part of knot theory? Addition. I found from web an article by Dror BarNatan with title Khovanov's homology for tangles and cobordisms. The article states that the Khovanov Homology theory for knots and links generalizes to tangles, cobordisms and 2knots. The articles says nothing explicit about Wilson loops but talks about topological QFTs. Addition. An article of Witten about his physical approach to Khovanov homology has appeared in arXiv. The article is more or less abracadabra for anyone not working with Mtheory but the basic idea is simple. Witten reformulates 3D ChernSimons theory as a path integral for N=4 super YM theory in the 4D half space W×R. This allows him to use dualities and bring in the machinery of Mtheory and branes. The basic structure of TGD forces a highly analogous appproach: replace 3surfaces with 4surfaces, consider knot cobordisms and also 2knots, introduce gerbes, and be happy with symplectic instead of topological QFT, which might more or less be synonymous with TGD as almost topological QFT. Symplectic QFT would obviously make possible much more refined description of knots. This posting can be found also as a more organized article Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2knots?. See also the new chapter Knots and TGD. 