Year 2011

Algebraic braids, sub-manifold braid theory, and generalized Feynman diagrams

Ulla send me a link to an article by Sam Nelson about very interesting new-to-me 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 non-planar Feynman diagrams should be integrated to a more general notion; braids are replaced with sub-manifold 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 2-braids (string world sheets) as well as partonic 2-surfaces 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 2-surfaces. String world sheets in turn could correspond to the analogs of Lagrangian sub-manifolds or to minimal surfaces of space-time surface satisfying the weak form of electric-magnetic 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 2-surfaces 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, sub-manifold 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 3-dimensional surface terms in accordance with general coordinate invariance and holography. The weak form of electric-magnetic duality in turn reduces this term to Chern-Simons terms.

The strong form of General Coordinate Invariance implies effective 2-dimensionality (holding true in finite measurement resolution) so that also a strong form of holography emerges. The expectation is that Chern-Simons terms in turn reduces to 2-dimensional surface terms.

The only physically interesting possibility is that these 2-D surface terms correspond to areas for minimal surfaces defined by string world sheets and partonic 2-surfaces 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 space-time surface. Similar but real string world sheet contribution defining Kähler function comes from the Euclidian space-time regions and should be equal to the contribution of the partonic 2-surfaces. A natural conjecture is that the absolute values of all three areas are identical: this would realize duality between string world sheets and partonic 2-surfaces and duality between Euclidian and Minkowskian space-time 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 p-adic physics and with the notion of finite measurement resolution emerges. The t'Thooft coupling λ should be proportional to p-adic 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 p-adic 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 space-time regions

The reduction of the Kähler action for the space-time 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 electric-magnetic duality. The first property implies a reduction to a 3-D term associated with wormhole throats and the latter property reduces this term to Abelian Chern-Simons term. I have not explicitly considered whether the same happens in the 4-D 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 Chern-Simons terms coming from Minkowskian and Euclidian regions at light-like wormhole throats. This difference can be non-trivial since Kähler form for CP₂ defines non-trivial U(1) bundle. This however suggests that the total Kähler action is quantized in integer multiples of the Kähler action for CP₂ type vacuum extremal so that one would have effectively sum over n-instanton 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?

1. The Kähler metric of WCW must be non-trivial. If Kähler function is piecewise constant in WCW, and if its second functional derivatives with respect to complex WCW coordinates indeed define WCW Kähler metric, then this metric must vanish identically almost everywhere. This is a catastrophe.

2. To understand how the problem can be cured, notice that WCW metric receives a non-trivial contribution from the two Lagrange multiplier terms stating the weak form of electric-magnetic duality at Minkowskian and Euclidian sides. Neither Lagrangian multiplier term contributes to Kähler action. Either of them would guarantee that the theory does not reduce to a mere topological QFT and would give rise to a non-trivial Kähler metric. If both are taken into account their contributions cancel each other and the metric is trivial. The conjectured duality between the descriptions based on space-time regions of Minkowskian and Euclidian signature suggests that one should define Kähler function and Kähler metric using only either of the two contributions at wormhole throats.

   This duality could be very useful practically since the two expansions could correspond to weakly and strongly interacting phases analogous to those encountered in the case of electric-magnetic duality. For Euclidian side of the duality one would have power series in powers exp(-n8π²/g_K²) multiplied by the exponential of the Minkowskian contribution with negative sign. At the Minkowskian side of the duality one would have exponent of the Minkowskian contribution with positive sign. p-Adicization suggests strongly that the exponent exp(-8π²/g_K²) defining the Kähler action of CP₂ is a rational number.

3. Usually vacuum functional is identified as exponent of Kähler function but could one identify vacuum functional as exponent of total Kähler action giving a discrete spectrum for its values? The answer seems to be negative. There are excellent reasons for the identification of the vacuum functional as exponent of Kähler function. For instance, Gaussian and metric determinants cancel each other and the constant curvature spaced property with vanishing Ricci scalar implies that the curvature scalar giving otherwise divergent loop contributions vanishes. If one modifies the vacuum functional from Kähler function to total Kähler action, there is no kinetic term in the exponent of the vacuum functional, and one must give up the idea about perturbative definition of WCW functional integral using the (1,1) part of the contravariant WCW metric as propagator.

4. The symmetric space property of WCW is what gives hopes about a practical definition of functional integral which is number theoretically universal making therefore sense also in the p-adic context. The reduction of the functional integral to harmonic analysis in infinite-dimensional symmetric spaces allowing to define integrals group theoretically would allow to define functional integrals non-perturbatively without propagator expansion. However, if the functional integral fails perturbatively, the hopes that it makes sense physically, are meager.

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 space-time 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 space-time surfaces. This duality would have also number theoretical interpretation. Minkowskian regions of the space-time surface would correspond to hyper-quaternionic and Eulidian regions to quaternionic regions. In hyper-quaternionic regions the modified gamma matrices would span hyper-quaternionic 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 two-dimensional?

The following argument suggests that the effective metric defined by the anti-commutators of the modified gamma matrices is effectively one- or two-dimensional. Effective one-dimensionality would conform with the observation that the solutions of the modified Dirac equations can be localized to one-dimensional world lines in accordance with the vision that finite measurement resolution implies discretization reducing partonic many-particle states to quantum superpositions of braids. This localization to 1-D curves occurs always at the 3-D orbits of the partonic 2-surfaces.

The argument is based on the following assumptions.

1. The modified gamma matrices for Kähler action are contractions of the canonical momentum densities T^α_k with the gamma matrices of H.

2. The strongest assumption is that the isometry currents

   J^Aα =T^α _kj^Ak

   for the preferred extremals of Kähler action are of form

   J^{A α}= Ψ^A (∇Φ)^α

   with a common function Φ guaranteeing that the flow lines of the currents integrate to coordinate lines of single global coordinate variables (Beltrami property). Index raising is carried out by using the ordinary induced metric.

3. A weaker assumption is that one has two functions Φ₁ and Φ₂ assignable to the isometry currents of M⁴ and CP₂ respectively.:

   J^{A α}₁ = Ψ₁^A (∇Φ₁)^α ,

   J^{A α}₂ = Ψ₂^A (∇Φ₂)^α .

   The two functions Φ₁ and Φ₂ could define dual light-like curves spanning string world sheet. In this case one would have effective 2-dimensionality and decomposition to string world sheets (for the concrete realization see this). Isometry invariance does not allow more that two independent scalar functions Φ_i.>.

1. One can multiply both sides of this equation with j^Ak and sum over the index A labeling isometry currents for translations of M⁴ and SU(3) currents for CP₂. The tensor quantity ∑_A j^Akj^Al is invariant under isometries and must therefore satisfy

   ∑_A η_ABj^Akj^Al= h^kl ,

   where η_AB denotes the flat tangent space metric of H. In M⁴ degrees of freedom this statement becomes obvious by using linear Minkowski coordinates.

   In the case of CP₂ one can first consider the simpler case S²=CP₁= SU(2)/U(1). The coset space property implies in standard complex coordinate transforming linearly under U(1) that only the the isometry currents belonging to the complement of U(1) in the sum contribute at the origin and the identity holds true at the origin and by the symmetric space property everywhere. Identity can be verified also directly in standard spherical coordinates. The argument generalizes to the case of CP₂=SU(3)/U(2) in an obvious manner.

2. In the most general case one obtains

   T^{α k}₁ =∑_AΨ₁^A j^Ak × (∇Φ₁)^α == f^k₁ (∇Φ₁)^α ,

   T^{α k}₂ =∑_AΨ₁^A j^Ak × (∇Φ₂)^α ≡ f^k₂ (∇Φ₂)^α .

   Here i=1 refers to M⁴ part of energy momentum tensor and i=2 to its CP₂ part.

3. The effective metric given by the anti-commutator of the modified gamma matrices is in turn is given by

   G^{α β} = m_klf^k₁f^l₁ (∇Φ₁)^α(∇Φ₁)^β +s_kl f^k₂ f^l₂ (∇Φ₂)^α(∇Φ₂)^β .

   The covariant form of the effective metric is effectively 1-dimensional for Φ₁=Φ₂ in the sense that the only non-vanishing component of the covariant metric G_{α β} is diagonal component along the coordinate line defined by Φ≡ Φ₁=Φ₂. Also the contravariant metric is effectively 1-dimensional since the index raising does not affect the rank of the tensor but depends on the other space-time coordinates. This would correspond to an effective reduction to a dynamics of point-like particles for given selection of braid points. For Φ₁≠ Φ₂ the metric is effectively 2-dimensional and would correspond to stringy dynamics.

Witten's physical view about Khovanov homology translated to TGD framework

Khovanov 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 Chern-Simons action allowing to interpret Jones polynomial as vacuum expectation value of Wilson loop in non-Abelian gauge theory.

Witten's approach to Khovanov homology relies on fivebranes as is natural if one tries to define 2-knot invariants in terms of their cobordisms involving violent un-knottings. 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 2-knots, 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 2-surfaces 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 3-D surfaces and string world sheets central for quantum TGD can be identified uniquely. The slicing by 3-surfaces would be interpreted in gauge theory in terms of Higgs= constant surfaces with radial coordinate of CP₂ 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 M⁴ 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 2-knots.

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 2-knots?. 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 2-knots?

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 Chern-Simons 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+1-D space-time generalizes to a vacuum expectation for a collection of Wilson loops in 2+2-D space-time and could define an invariant for 2-D 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 4-D space-time 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 Time-like and space-like braidings for generalized Feynman diagrams

1. In TGD framework space-times are 4-D surfaces in 8-D imbedding space. Basic objects are partonic 2-surfaces at the two ends of causal diamonds CD (intersections of future and past directed light-cones of 4-D Minkowski space with each point replaced with CP₂). The light-like orbits of partonic 2-surfaces define 3-D light-like 3-surfaces identifiable as lines of generalized Feynman diagrams. At the vertices of generalized Feynman diagrams incoming and outgoing light-like 3-surfaces meet. These diagrams are not direct generalizations of string diagrams since they are singular as 4-D manifolds just like the ordinary Feynman diagrams.

   By strong form of holography one can assign to the partonic 2-surfaces and their tangent space data space-time surfaces as preferred exrtremals of Kähler action. This guarantees also general coordinate invariance and allows to interpret the extremals as generalized Bohr orbits.

2. One can assign to the partonic 2-surfaces discrete sets of points carrying quantum numbers. As a matter fact, these sets of points seem to emerge from the solutions of the Chern-Simons Dirac equation rather naturally. These points define braid strands as the partonic 2-surface moves and defines a light-like 3-surface as its orbit as a surface of 4-D space-time surface. In the generic case the strands get tangled in time direction and one has linking and knotting giving rise to a time-like braiding.

3. Also space-like braidings are possible. One can imagine that the partonic 2-surfaces are connected by space-like curves defining TGD counterparts for strings and that in the initial state these curves define space-like braids whose ends belong to different partonic 2-surfaces. Quite generally, the basic conjecture is that the preferred extremals define orbits of string-like objects with their ends at the partonic 2-surfaces. One would have slicing of space-time surfaces by string world sheets one one hand and by partonic 2-surface on one hand. This string model is very special due to the fact that the string orbits define what could be called braid cobordisms representing which could represent unknotting of braids. String orbits in higher dimensional space-times do not allow this topological interpretation.

Time like braidings induces space-like braidings and one can speak of time-like or dynamical braiding and even duality of time-like and space-like braiding. What happens can be understood in terms of dance metaphor.

1. One can imagine that the points carrying quantum numbers are like dancers at parquettes defined by partonic 2-surfaces. These parquettes are somewhat special in that it is moving and changing its shape: dancers like me would probably get sea sick at this kind of parquette.

2. Space-like braidings means that the feet of the dancers at different parquettes are connected by threads. As the dance continues, the threads connecting the feet of different dancers at different parquettes get tangled so that the dance is coded to the braiding of the threads. Time-like braiding induce space-like braiding. One has what might be called a cobordism for space-like braiding transforming it to a new one.

1.3 DNA as topological quantum computer

The model for topological quantum computation is based on the idea that time-like braidings defining topological quantum computer programs. These programs are robust since the topology of braiding is not affected by small deformations.

1. The first key idea in the model of DNA as topological quantum computer is based on the observation that the lipids of cell membrane form a 2-D liquid whose flow defines the dance in which dancers are lipids which define a flow pattern defining a topological quantum computation. Lipid layers assignable to cellular and nuclear membranes are the parquettes. This 2-D flow pattern can be induced by the liquid flow near the cell membrane or in case of nerve pulse transmission by the nerve pulses flowing along the axon. This alone defines topological quantum computation.

2. In DNA as topological quantum computer model one however makes a stronger assumption motivated by the vision that DNA is the brain of cell and that information must be communicated to DNA level wherefrom it is communicated to what I call magnetic body. It is assumed that the lipids of the cell membrane are connected to DNA nucleotides by magnetic flux tubes defining a space-like braiding. It is also possible to connect lipids of cell membrane to the lipids of other cell membranes, to the tubulins at the surfaces of microtubules, and also to the aminoadics of proteins. The spectrum of possibilities is really wide.

   The space-like braid strands would correspond to magnetic flux tubes connecting DNA nucleotides to lipids of nuclear or cell membrane. The running of the topological quantum computer program defined by the time-like braiding induced by the lipid flow would be coded to a space-like braiding of the magnetic flux tubes. The braiding of the flux tubes would define a universal memory storage mechanism and combined with 4-D view about memory provides a very simple view about how memories are stored and how they are recalled.

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 4-D 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.

1. One must distinguish between internal knotting of each braid strand and linking of 2 strands. They look the same in the 3-D case but in higher dimensions knotting and linking are not the same thing. Codimension 2 surfaces get knotted in the generic case, in particular the 2-D orbits of the braid strands can get knotted so that this gives additional topological flavor to the theory of strings in 4-D space-time. Linking occurs for two surfaces whose dimension d₁ and d₂ satisfying d₁+d₂= D-1, where D is the dimension of the imbedding space.

2. 2-D orbits of strings do not link in 4-D space-time but do something more radical since the sum of their dimensions is D=4 rather than only D-1=3. They intersect and it is impossible to eliminate the intersection without a change of topology of the stringy 2-surfaces: a hole is generated in either string world sheet. With a slight deformation intersection can be made to occur generically at discrete points.

2.2 Topological strings in 4-D space-time define knot cobordisms

What makes the 4-D braid cobordisms interesting is following.

1. The opening of knot by using brute force by forcing the strands to go through each other induces this kind of intersection point for the corresponding 2-surfaces. From 3-D perspective this looks like a temporary cutting of second string, drawing the string ends to some distance and bringing them back and gluing together as one approaches the moment when the strings would go through each other. This surgical operation for either string produces a pair of non-intersecting 2-surfaces with the price that the second string world sheet becomes topologically non-trivial carrying a hole in the region were intersection would occur. This operation relates a given crossing of braid strands to its dual crossing in the construction of Jones polynomial in given step (string 1 above string 2 is transformed to string 2 above string 1).

2. One can also cut both strings temporarily and glue them back together in such a manner that end a/b of string 1 is glued to the end c/d of string 2. This gives two possibilities corresponding to two kinds of reconnections. Reconnections appears as the second operation in the construction of Jones invariant besides the operation putting the string above the second one below it or vice versa. Jones polynomial relates in a simple manner to Kauffman bracket allowing a recursive construction. At a given step a crossing is replaced with a weighted sum of the two reconnected terms. Reconnection represents the analog of trouser vertex for closed strings replaced with braid strands.

3. These observations suggest that stringy diagrams describe the braid cobordisms and a kind of topological open string model in 4-D space-time could be used to construct invariants of braid cobordisms. The dynamics of strand ends at the partonic 2-surfaces would partially induce the dynamics of the space-like braiding. This dynamics need not induce the un-knotting of space-like braids and simple string diagrams for open strings are enough to define a cobordism leading to un-knotting. The holes needed to realize the crossover for braid strands would contribute to the Wilson loop an additional term corresponding to the rotation of the gauge potential around the boundary of the hole (non-integrable phase factor). In Abelian case this gives simple commuting phase factor.

3. Invariants 2-knots as vacuum expectations of Wilson loops in 4-D space-time?

The interpretation of string world sheets in terms of Wilson loops in 4-dimensional space-time 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+1-D space might be replaced with a vacuum expectation for a collection of Wilson loops in 3+1-D space-time 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 un-knotting 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 2-D knots in 4-D space-time 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 2-knots and that Wilson loops must be replaced with 2-D fluxes. This requires a generalization of gauge field concept so that it corresponds to a 3-form instead of 2-form 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 2-knottedness means concretely?

It is easy to imagine what ordinary knottedness means. One has circle imbedded in 3-space. One projects it in some plane and looks for crossings. If there are no crossings one knows that un-knot 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 3-D case, one can find a simple interpretation for knottedness in terms of linking of two circles. For 2-knots linking is not what gives rise to knotting.

One might hope to find something similar in the case of 2-knots. Can one imagine some simple local operations which either increase of reduce 2-knottedness?

1. To proceed let us consider as simple situation as possible. Put sphere in 3-D time= constant section E³ of 4-space. Add a another sphere to the same section E³ such that the corresponding balls do not intersect. How could one build from these two spheres a knotted 2-sphere?

2. From two spheres one can build a single sphere in topological sense by connecting them with a small cylindrical tube connecting the boundaries of disks (circles) removed from the two spheres. If this is done in E³, a trivial 2-knot results. One can however do the gluing of the cylinder in a more exotic manner by going temporarily to "hyper-space", in other words making a time travel. Let the cylinder leave the second sphere from the outer surface, let it go to future or past and return back to recent but through the interior. This is a good candidate for a knotted sphere since the attempts to deform it to self-non-intersecting sphere in E³ are expected to fail since the cylinder starting from interior necessarily goes through the surface of sphere if wants to the exterior of the sphere.

3. One has actually 2× 2 manners to perform the connected sum of 2-spheres depending on whether the cylinders leave the spheres through exterior or interior. At least one of them (exterior-exterior) gives an un-knotted sphere and intuition suggests that all the three remaining options requiring getting out from the interior of sphere give a knotted 2-sphere. One can add to the resulting knotted sphere new spheres in the same manner and obtain an infinite number of them. As a matter fact, the proposed 1+3 possibilities correspond to different versions of connected sum and one could speak of knotting and non-knotting connected sums. If the addition of knotted spheres is performed by non-knotting connected sum, one obtains composites of already existing 2-knots.

   Connected sum composition is analogous to the composition of integer to a product of primes. One indeed speaks of prime knots and the number of prime knots is infinite. Of course, it is far from clear whether the connected sum operation is enough to build all knots. For instance it might well be that cobordisms of 1-braids produces knots not producible in this manner. In particular, the effects of time-like braiding induce braiding of space-like strands and this looks totally different from local knotting.

Whether all possible 2-knots 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 2-knottedness not reducing to braid cobordism.

1. One must start from string world sheets with Minkowskian signature of the induced metric. In other words, in the previous construction one must E³ with 3-dimensional Minkowski space M³ with metric signature 1+2 containing the spheres used in the construction. Time travel is replaced with a travel in space-like hyper dimension. This is not a problem as such. The spheres however have at least one two special points corresponding to extrema at which the time coordinate has a local minimum or maximum. At these points the induced metric is necesssarily degenerate meaning that its determinant vanishes. If one allows this kind of singular points one can have elementary knotted spheres. This liberal attitude is encouraged by the fact that the light-like 3-surfaces defining the basic dynamical objects of quantum TGD correspond to surfaces at which 4-D induced metric is degenerate. Otherwise 2-knotting reduces to that induced by cobordisms of 1-braids. If one allows only the 2-knots assignable to the slicings of the space-time surface by string world sheets and even restricts the consideration to those suggested by the duality of 2-D generalization of Wilson loops for string world sheets and partonic 2-surfaces, it could happen that the string world sheets reduce to braidings.

2. The time=constant intersections define a representation of 2-knots as a continuous sequence of 1-braids. For critical times the character of the 1-braids changes. In the case of braiding this corresponds to the basic operations for 1-knots having interpretation as string diagrams (reconnection and analog of trouser vertex). The possibility of genuine 2-knottedness brings in also the possibility that strings pop up from vacuum as points, expand to closed strings, are fused to stringy words sheet temporarily by the analog of trouser vertex, and eventually return to the vacuum. Essentially trouser diagram but second string open and second string closed and beginning from vacuum and ending to it is in question. Vacuum bubble interacting with open string would be in question. The believer in string model might be eager to accept this picture but one must be cautious.

Suppose that the space-like braid strands connecting partonic 2-surfaces at given boundary of CD and light-like braids connecting partonic 2-surfaces 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+1-D case.

Is this generalization of Wilson loops enough to describe 2-knots? In the spirit of the proposed philosophy one could ask whether there exist two-knots not reducible to cobordisms of 1-knots whose knot invariants require cobordisms of 2-knots and therefore 2-braids in 5-D space-time. 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 2-D closed surface in 4-D space gets knotted.

1. In 1-D case knotting reduces to linking and means that the first homotopy group of the knot complement is changed so that the imbedding of first circle implies that the there exists imbedding of the second circle that cannot be transformed to each other without cutting the first circle temporarily. This phenomenon occurs also for single circle as the connected sum operation for two linked circles producing single knotted circle demonstrates.

2. In 2-D case the complement of knotted 2-sphere has a non-trivial second homotopy group so that 2-balls have homotopically non- equivalent imbeddings, which cannot be transformed to each other without intersection of the 2-balls taking place during the process. Therefore the description of 2-knotting in the proposed manner would require cobordisms of 2-knots and thus 5-D space-time surfaces. However, since 3-D description for ordinary knots works so well, one could hope that the generalization the notion of Wilson loop could allow to avoid 5-D description altogether. The generalized Wilson loops would be assigned to 2-D surfaces and gauge potential A would be replaced with 2-gauge potential B defining a three-form F= dB as the analog of gauge field.

3. This generalization of bundle structure known as gerbe structure has been introduced in algebraic geometry (see this and this) and studied also in theoretical physics. 3-forms appear as analogs of gauge fields also in the QFT limit of string model. Algebraic geometer would see gerbe as a generalization of bundle structure in which gauge group is replaced with a gauge groupoid. Essentially a structure of structures seems to be in question. For instance, the principal bundles with given structure group for given space defines a gerbe. In the recent case the space of gauge fields in space-time could be seen as a gerbe. Gerbes have been also assigned to loop spaces and WCW can be seen as a generalization of loop space. Lie groups define a much more mundane example about gerbe. The 3-form F is given by F(X,Y,Z)= B(X,[Y,Z]) , where B is Killing form and for U(n) reduces to (g^-1dg)³. It will be found that classical color gauge fields define gerbe gauge potentials in TGD framework in a natural manner.

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 electric-magnetic duality and duality of space-like and time-like 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 non-trivial topology of CP₂makes 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 electro-weak 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 electric-magnetic 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 space-like ends of the space-time 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 electric-magnetic duality fixes the boundary conditions of field equations at the light-like and space-like 3-surfaces. 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 3-D Chern-Simons 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 Chern-Simons action for induced Kähler gauge potential, one might hope that TGD could as a by-product define invariants of braid cobordisms in terms of the unitary U-matrix of the theory between zero energy states and having as its rows the non-unitary M-matrices analogous to thermal S-matrices.

Electric magnetic duality is 4-D phenomenon as is also the duality between space-like 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 space-time 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.

1. One can assign to the 2-dimensional stringy surfaces defined by the orbits of space-like braid strands Kähler magnetic fluxes as flux integrals over these surfaces and these integrals depend only on the end points of the space-like strands so that one deform the space-like strands in an arbitrarily manner. One can in fact assign these kind of invariants to pairs of knots and these invariants define the dancing operation transforming these knots to each other. In the special case that the second knot is un-knot one obtains a knot-invariant (or link- or braid- invariant).

2. The objection is that these invariants depend on the orbits of the end points of the space-like braid strands. Does this mean that one should perform an averaging over the ends with the condition that space-like braid is not affected topologically by the allowed deformations for the positions of the end points?

3. Under what conditions on deformation the magnetic fluxes are not affect in the deformation of the braid strands at 3-D surfaces? The change of the Kähler magnetic flux is magnetic flux over the closed 2-surface defined by initial non-deformed and deformed stringy two-surfaces minus flux over the 2-surfaces defined by the original time-like and space-like braid strands connected by a thin 2-surface to their small deformations. This is the case if the deformation corresponds to a U(1) gauge transformation for a Kähler flux. That is diffeomorphism of M⁴ and symplectic transformation of CP₂ inducing the U(1) gauge transformation.

   Hence a natural equivalence for braids is defined by these transformations. This is quite not a topological equivalence but quite a general one. Symplectic transformations of CP₂ at light-like and space-like 3-surfaces define isometries of the world of classical worlds so that also in this sense the equivalence is natural. Note that the deformations of space-time surfaces correspond to this kind of transformations only at space-like 3-surfaces at the ends of CDs and at the light-like wormhole throats where the signature of the induced metric changes. In fact, in quantum TGD the sub-spaces of world of classical worlds with constant values of zero modes (non-quantum fluctuating degrees of freedom) correspond to orbits of 3-surfaces under symplectic transformations so that the symplectic restriction looks rather natural also from the point of view of quantum dynamics and the vacuum expectation defined by Kähler function be defined for physical states.

4. A further possibility is that the light-like and space-like 3-surfaces carry vanishing induced Kähler fields and represent surfaces in M⁴× Y², where Y² is Lagrangian sub-manifold of CP₂ carrying vanishing Kähler form. The interior of space-time surface could in principle carry a non-vanishing Kähler form. In this case weak form of self-duality cannot hold true. This however implies that the Kähler magnetic fluxes vanish identically as circulations of Kähler gauge potential. The non-integrable phase factors defined by electroweak gauge potentials would however define non-trivial classical Wilson loops. Also electromagnetic field would do so. It would be therefore possible to imagine vacuum expectation value of Wilson loop for given quantum state. Exponent of Kähler action would define for non-vacuum extremals the weighting. For 4-D vacuum extremals this exponent is trivial and one might imagine of using imaginary exponent of electroweak Chern-Simons action. Whether the restriction to vacuum extremals in the definition of vacuum expectations of electroweak Wilson loops could define general enough invariants for braid cobordisms remains an open question.

5. The quantum expectation values for Wilson loops are non-Abelian generalizations of exponentials for the expectation values of Kähler magnetic fluxes. The classical color field is proportional to the induced Kähler form and its holonomy is Abelian which raises the question whether the non-Abelian Wilson loops for classical color gauge field could be expressible in terms of Kähler magnetic fluxes.

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 2-knots seems to necessitate the generalization of gauge field to 3-form and the introduction of a gerbe structure. This seems to be possible in TGD framework.

1. Classical color gauge fields are proportional to the products B_A= H_AJ of the Hamiltonians of color isometries and of Kähler form and the closed 3-form F_A= dB_A= dH_A∧ J could serve as a colored 3-form defining the analog of U(1) gauge field. What would be interesting that color would make F non-vanishing. The "circulation" h_A= ∫ H_AJ over a closed partonic 2-surface transforms covariatly under symplectic transformations of CP₂, whose Hamiltonians can be assigned to irreps of SU(3): just the commutator of Hamiltonians defined by Poisson bracket appears in the infinitesimal transformation. One could hope that the expectation values for the exponents of the fluxes of B_A over 2-knots could define the covariants able to catch 2-knottedness in TGD framework. The exponent defining Wilson loop would be replaced with exp(iQ^A h_A), where Q^A denote color charges acting as operators on particles involved.

2. Since the symplectic group acting on partonic 2-surfaces at the boundary of CD replaces color group as a gauge group in TGD, one can ask whether symplectic SU(3) should be actually replaced with the entire symplectic group of U_+/-δ M⁴_+/-× CP₂ with Hamiltonians carrying both spin and color quantum numbers. The symplectic fluxes ∫ H_AJ are indeed used in the construction of both quantum states and of WCW geometry. This generalization is indeed possible for the gauge potentials B_AJ so that one would have infinite number of classical gauge fields having also interpretation as gerbe gauge potentials.

3. The objection is that symplectic transformations are not symmetries of Kähler action. Therefore the action of symplectic transformation induced on the space-time surface reduces to a symplectic transformation only at the partonic 2-surfaces. This spoils the covariant transformation law for the 2-fluxes over stringy world sheets unless there exist preferred stringy world sheets for which the action is covariant. The proposed duality between the descriptions based on partonic 2-surfaces and stringy world sheets realized in terms of slicings of space-time surface by string world sheets and partonic 2-surfaces suggests that this might be the case.

   This would mean that one can attach to a given partonic 2-surface a unique collection string world sheets. The duality suggests even stronger condition stating that the total exponents exp(iQ^Ah_A) of fluxes are the same irrespective whether h_A evaluated for partonic 2-surfaces or for string world sheets defining the analog of 2-knot. This would mean an immense calculational simplification! This duality would correspond very closely to the weak form of electric magnetic duality whose various forms I have pondered as a must for the geometry of WCW. Partonic 2-surfaces indeed correspond to magnetic monopoles at least for elementary particles and stringy world sheets to surfaces carrying electric flux (note that in the exponent magnetic charges do not make themselves visible so that the identity can make sense also for H_A=1).

4. Quantum expectation means in TGD framework functional integral over the symplectic orbits of partonic 2-surfaces plus 4-D tangent space data assigned to the upper and lower boundaries of CD. Suppose that holography fixes the space-like 3-surfaces at the ends of CD and light-like orbits of partonic 2-surfaces.
   1. In completely general case the braids and the stringy space-time sheets could be fixed using a representation in terms of space-time coordinates so that the representation would be always the same but the imbedding varies as also the values of the exponent of Kähler function, of the Wilson loop, and of its 2-D generalization. The functional integral over symplectic transforms of 3-surfaces implies that Wilson loop and its 2-D generalization varies.

   2. The proposed duality however suggests that both Wilson loop and its 2-D generalization are actually fixed by the dynamics. One can ask whether the presence of 2-D analog of Wilson loop has a direct physical meaning bringing into almost topological stringy dynamics associated with color quantum numbers and coding explicit information about space-time interior and topology of field lines so that color dynamics would also have interpretation as a theory of 2-knots. If the proposed duality suggested by holography holds true, only the data at partonic 2-surfaces would be needed to calculate the generalized Wilson loops. Maybe TGD as almost topological QFT could be seen as a symplectic -rather than topological- QFT for 1-braids and 2-knots!

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.

1. Instead of knots, links, and braids one could study knot and link cobordisms, that is their dynamical evolutions concretizable in terms of dance metaphor and in terms of interacting string world sheets. Each space-like braid strand can have purely internal knotting and braid strands can be linked. TGD could allow to identify uniquely both space-like and time-like braid strands and thus also the stringy world sheets more or less uniquely and it could be that the dynamics induces automatically the temporary cutting of braid strands when knot is opened violently so that a hole is generated. Gerbe gauge potentials defined by classical color gauge fields could make also possible to characterize 2-knottedness in symplectic invariant manner in terms of color gauge fluxes over 2-surfaces.

   The weak form of electric-magnetic duality would reduce the situation to almost topological QFT in general case with topological invariance replaced with symplectic one which corresponds to the fixing of the values of non-quantum fluctuating zero modes in quantum TGD. In the vacuum sector it would be possible to have the counterparts of Wilson loops weighted by 3-D electroweak Chern-Simons action defined bythe induced spinor connection.

2. One could also leave TGD framework and define invariants of braid cobordisms and 2-D analogs of braids as vacuum expectations of Wilson loops using Chern-Simons action assigned to 3-surfaces at which space-like and time-like braid strands end. The presence of light-like and space-like 3-surfaces assignable to causal diamonds could be assumed also now.

What is interesting that twistorial considerations lead to a conjecture that 4-D space-time surfaces in 8-D imbedding space have a dual description in terms of certain 6-D homomorphic surfaces which are sphere bundles in 12-D CP₃× CP₃ and effectively 4-D. This suggests a connection between descriptions based on topological strings in 6-D space and Wilson loops in 4-D space-time. 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 Bar-Natan 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 2-knots. 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 M-theory but the basic idea is simple. Witten reformulates 3-D Chern-Simons theory as a path integral for N=4 super YM theory in the 4-D half space W×R. This allows him to use dualities and bring in the machinery of M-theory and branes. The basic structure of TGD forces a highly analogous appproach: replace 3-surfaces with 4-surfaces, consider knot cobordisms and also 2-knots, 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 2-knots?. See also the new chapter Knots and TGD.

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