Floer homology and TGD

TGD can be seen as almost topological quantum field theory. This could have served as a motivation for spending most of last months to the attempt to learn some of the mathematics related to various kind of homologies and cohomologies. The decisive stimulus came from the attempt to understand the basic ideas of motivic cohomology. I am not a specialist and do not have any ambition or abilities to become such. My goals is to see whether these ideas could be applied in quantum TGD.

Documentation is the best manner to develop ideas and the learning process has materialized as a new chapter entitled Infinite Primes and Motives of "Physics as Generalized Number Theory". It soon became clear that much of the mathematics needed by TGD has existed for decades and developing all the time. The difficult task is to understand the essentials of this mathematics and translate to the language that I talk and understand. Also generalization is unavoidable. Those who think that new physics can be done by taking math as such are wasting their time.

Among another things I have been learning about various cohomologies and homologies - about quantum cohomology, about Floer homology and topological string theories, about Gromov-Witten invariants,... It would be very naive to think that these notions would work as such in TGD framework. It looks however very plausible that the their generalizations to TGD exist, and could be very useful in the more detailed formulation of quantum TGD. The crucially important notion is finite measurement resolution making everything almost topological and highly number theoretic. In this brain-stormy spirit I have even become a proud father of my own pet homology, which I have christened as braided Galois homology. It is based on the correspondence between infinite primes and polynomials of several variables and is formulated in braided group algebras with braidings realized as symplectic flows and generalizing somewhat the usual notion of homology meaning that the square of boundary operation gives something in commutator group reducing to unit element of ordinary homology only in the factor group obtained by dividing with the commutator group.

Floer homology in its original form replaces Morse function in symplectic manifold M in the loop space LM of M. The loops can be seen as homotopies of Hamiltonians and paths in loops space describe cylinders in M. With an appropriate choice of symplectic action these cylinderes can be regarded as (pseudo-)holomorphic surface completely analogous to string orbits. By combining Floer's theory with Witten's discovery about the connection between the Morse theory and supersymmetry one ends up with topological QFTs as a manner to formulate Floer homology and various variants of this notion- in particular topological QFTs characterizing topology of three-manifolds.

This kind of learning periods are very useful as a rule since they allow to improve bird's eye of view about TGD and its problems. The understanding of both quantum TGD and its classical counterpart is still far from from comprehensive.

For instance, the view about the physical and mathematical roles of Kähler actions for Euclidian and Minkowskian space-time regions is far from clear. Do they provide dual descriptions as suggested or are both needed? Kähler action for preferred extremal in Euclidian regions defines naturally positive definite Kähler function. But can one regard the Kähler action in Minkowskian regions as equivalent definition for Kähler function or should one regard it as imaginary as the presence of square root of metric determinant would suggest? What could be the interpretation in this case? The basic ideas about Floer homology suggest and answer to these questions.

1. Since quantum fluctuating WCW degrees of freedom correspond to a symmetric space assignable to the symplectic group in TGD framework symplectic geometry is of special interest from TGD point of view. Floer homology is indeed about symplectic geometry as also Gromov-Witten invariants and topological string theories developed for the purpose of calculating these invariants. Hence the question whether Floer homology could have a generalization to TGD framework is highly relevant.

2. As such Floer homology for M⁴× CP₂ is deadly boring since it reduces to ordinary singular homology. The correspondence between canonical momentum densities of Kähler action and time derivatives of imbedding space coordinates is however one-to-many- and inspires the replacement of the imbedding space with its singular covering with different space-time regions corresponding to different number of sheets for the covering. The effective hierarchy of Planck constants emerges as a result. The homology in WCW could be mapped to the homology of this structure just as the homology of loop space of M is mapped to that of M in Floer theory.

3. The obvious question is how to generalize Floer homology to TGD framework and the obvious guess is that Kähler action for preferred extremals must take the role of symplectic action for pseudo-holomorphic surfaces which could in fact be replaced with hyper-quaternionic space-time surfaces containing string world sheets whose ends defined braid strands carrying quantum numbers and which intersect partonic 2-surfaces at the future and past light-like boundaries of CDs. This actually suggests an obvious generalization for quantum cohomology based on quantal notion of intersection: partonic surfaces intersect if there exist a string world sheets connecting them. Fuzzy intersection has interpretation in terms of causal dependence: by effective 2-dimensionality this causal dependence is along light-like 3-surfaces and along space-like 3-surfaces at the boundaries of CDs. The notion of quantum intersection is so beautiful that one an almost forgive for the theoricians who have begun to take seriously the idea about branes connected by strings.

4. The question providing the new insight is simple. Could Kähler function allow to define Morse theory? The answer is negative. Kähler metric must be positive definite so that the Hessian associated with it in quantum fluctuating degrees of freedom must have positive signature: no saddle points are possible in quantum fluctuating degrees of freedom although in zero modes they are allowed. Second counter argument is that quantum Morse theory is based on path integral rather than functional integral.

   How could one circumvent this difficulty? Could Kähler action in Minkowskian regions- naturally imaginary by negative sign of metric determinant- give an imaginary contribution to the vacuum functional and define Morse function so that both Kähler and Morse would find a prominent role in the world order of TGD? Maybe! The presence of Kähler function and Morse function in the vacuum functional would give much more direct connection with the path integral approach and Kähler function would also make path integral well-defined since one integrates only over preferred extremals of Kähler action for which Kähler action reduces to Chern-Simons term coming from Minkowskian region and contribution from Euclidian region (generalized Feynman graph).

Should one assume that the reduction to Chern-Simons terms occurs for the preferred extremals in both Minkowskian and Euclidian regions or only in Minkowskian regions?

1. All arguments for this have been represented for Minkowskian regions involve local light-like momentum direction which does not make sense in the Euclidian regions. This does not however kill the argument: one can have non-trivial solutions of Laplacian equation in the region of CP₂ bounded by wormhole throats: for CP₂ itself only covariantly constant right-handed neutrino represents this kind of solution and at the same time supersymmetry. In the general case solutions of Laplacian represent broken super-symmetries and should be in one-one correspondences with the solutions of the modified Dirac equation. The interpretation for the counterparts of momentum and polarization would be in terms of classical representation of color quantum numbers.

   If the reduction occurs in Euclidian regions, it gives in the case of CP₂ two 3-D terms corresponding to two 3-D gluing regions for three coordinate patches needed to define coordinates and spinor connection for CP₂ so that one would have two Chern-Simons terms. Without any other contributions the first term would be identical with that from Minkowskian region apart from imaginary unit. Second Chern-Simons term would be however independent of this. For wormhole contacts the two terms could be assigned with opposite wormhole throats and would be identical with their Minkowskian cousins from imaginary unit. This looks a little bit strange.

2. There is however a very delicate issue involved. Quantum classical correspondence requires that the quantum numbers of partonic states must be coded to the space-time geometry, and this is achieved by adding to the action a measurement interaction term which reduces to what is almost a gauge term present only in Chern-Simons-Dirac equation but not at space-time interior. This term would represent a coupling to Poincare quantum numbers at the Minkowskian side and to color and electro-weak quantum numbers at CP₂ side. Therefore the net Chern-Simons contributions and would be different.

3. There is also a very beautiful argument stating that Dirac determinant for Chern-Simons-Dirac action equals to Kähler function, which would be lost if Euclidian regions would not obey holography. The argument obviously generalizes and applies to both Morse and Kähler function.

In any case, it is still too early to give up the possibility that these two parts of Kähler action (real and positive- imaginary) provide dual descriptions as functional integral and path integral: Wick rotations is what comes in mind. Certainly, the rigorous definition of the path integral would be as difficult -should one say hopeless- as in ordinary QFT.

Floer homology and Gromov-Witten invariants provide also other insights about quantum TGD. For more details see the new chapter Infinite Primes and Motives or the article with same title.