Intersection forms, exotic smooth structures, and particle vertices in the TGD framework

Gary Ehlenberg sent an email telling about his discussion with GPT  related to exotic smooth structures.     The timing was perfect.  I was just yesterday evening pondering what the exotic smooth structures as ordinary smooth  structures with defects could really mean and how to get a concrete grasp of them.    

Here are some facts about the exotic smooth structures. I do not count myself as a real mathematician but the results give a very useful perspective.

Summary of the basic findings about exotic smooth structures

The study of exotic R⁴'s   has led to numerous significant mathematical developments, particularly in the fields of differential topology, gauge theory, and 4-manifold theory. Here are some key developments:

1. Donaldson's Theorems

   Simon Donaldson's groundbreaking work in the early 1980s revolutionized the study of smooth 4-manifolds. His theorems provided new invariants, known as Donaldson polynomials, which distinguish between different smooth structures on 4-manifolds.

   Donaldson's Diagonalization Theorem: This theorem states that the intersection form of a smooth, simply connected 4-manifold must be diagonalizable over the integers, provided the manifold admits a smooth structure. This result was crucial in showing that some topological 4-manifolds cannot have a smooth structure.

   Donaldson s Polynomial Invariants: These invariants help classify and distinguish different smooth structures on 4-manifolds, particularly those with definite intersection forms.

2. Freedman's Classification of Topological 4-Manifolds

   Michael Freedman's work, which earned him a Fields Medal in 1986, provided a complete classification of simply connected topological 4-manifolds. His results showed that every such manifold is determined by its intersection form up to homeomorphism.

   h-Cobordism and the Disk Embedding Theorem: Freedman's proof of the h-cobordism theorem in dimension 4 and the disk embedding theorem were instrumental in his classification scheme.

3.  Seiberg-Witten Theory

   The development of Seiberg-Witten invariants provided a new set of tools for studying smooth structures on 4-manifolds, complementing and sometimes simplifying the methods introduced by Donaldson.

   Seiberg-Witten Invariants: These invariants are simpler to compute than Donaldson invariants and have been used to prove the existence of exotic smooth structures on 4-manifolds.

4.  Gauge Theory and 4-Manifolds

   Gauge theory, particularly through the study of solutions to the Yang-Mills equations, has provided deep insights into the structure of 4-manifolds.

   Instantons: The study of instantons (solutions to the self-dual Yang-Mills equations) has been crucial in understanding the differential topology of 4-manifolds. Instantons and their moduli spaces have been used to define Donaldson and Seiberg-Witten invariants.

5.  Symplectic and Complex Geometry

   The interaction between symplectic and complex geometry with 4-manifold theory has led to new discoveries and techniques.

   Gompf's Construction of Symplectic 4-Manifolds: Robert Gompf's work on constructing symplectic 4-manifolds provided new examples of exotic smooth structures. His techniques often involve surgeries and handle decompositions that preserve symplectic structures.

   Symplectic Surgeries: Techniques such as symplectic sum and Luttinger surgery have been used to construct new examples of 4-manifolds with exotic smooth structures.

6.  Floer Homology

   Floer homology, originally developed in the context of 3-manifolds, has been extended to 4-manifolds and provides a powerful tool for studying their smooth structures.

   Instanton Floer Homology: This theory associates a homology group to a 3-manifold, which can be used to study the 4-manifolds that bound them. It has applications in understanding the exotic smooth structures on 4-manifolds.

7.  Exotic Structures and Topological Quantum Field Theory (TQFT)

   The study of exotic R⁴'s  has also influenced developments in TQFT, where the smooth structure of 4-manifolds plays a crucial role.  TQFTs are sensitive to the smooth structures of the underlying manifolds, and exotic R⁴'s provide interesting examples for testing and developing these theories.

To sum up, the exploration of exotic R⁴'s has led to significant advances across various areas of mathematics, particularly in the understanding of smooth structures on 4-manifolds. Key developments include Donaldson and Seiberg-Witten invariants, Freedman s topological classification, advancements in gauge theory, symplectic and complex geometry, Floer homology, and topological quantum field theory. These contributions have profoundly deepened our understanding of the unique and complex nature of 4-dimensional manifolds.

How exotic smooth structures appear in TGD

The recent TGD view of particle vertices    relies  on exotic smooth structures emerging in D=4.  For a  background see this, this , and this .

1. In TGD string world sheets are replaced with 4-surfaces in H=M⁴xCP₂ which allow generalized complex structure as also M⁴  and H.

2. The notion of generalized complex structure.

   The generalized complex structure is introduced  for M⁴, for  H=M⁴× CP₂  and for the space-time surface X⁴ ⊂ H.

   1. The generalized complex structure of M⁴  is a fusion of hypercomplex structure and complex structure involving slicing of M⁴ by string world sheets and partonic 2-surfaces transversal to each other.  String world sheets allow  hypercomplex structure and  partonic 2-surface complex structure. Hypercomplex coordinates  of M⁴ consist of a pair of  light-like coordinates as a generalization of a light-coordinate of M² and complex coordinate as a generalization of a complex coordinate for E².  
   2. One obtains a generalized complex structure for H=M⁴×CP₂ with 1 hypercomplex coordinate and 3 complex coordinates.
   3. One  can use a  suitably selected  hypercomplex coordinate and a complex coordinate of H as generalized complex coordinates for X⁴ in regions where the induced metric is Minkowskian. In regions where it is Euclidean one has two complex coordinates for X⁴.    

3. Holography= generalized holomorphy  

   This conjecture gives a general solution of classical field equations.  Space-time surface X⁴ is defined as a zero locus for  two functions of generalized complex coordinates of H,  which are generalized-holomorphic and thus depend on 3 complex coordinates and one light-like coordinate. X⁴ is a minimal surface  apart from singularities at which the minimal surface property fails.  This irrespective of action assuming that it is constructed in terms of the induced geometry.   X⁴   generalizes the complex submanifold of algebraic geometry.  

   At  X⁴ the trace of the second fundamental form, H^k, vanishes. Physically this means that the generalized acceleration for a 3-D particle vanishes i.e one has free massless particle. Equivalently, one has a geometrization of a massless field.  This means particle-field duality.

4. What happens at the interfaces between Euclidean and Minkowskian regions of X⁴ are light-like 3-surfaces X³?  

   The light-like surface  X³ is topologically 3-D but metrically 2-D  and corresponds  to a  light-like orbit of a partonic  2-surface  at which the  induced metric of X⁴ changes its signature from Minkowskian to Euclidean.  At X³  a  generalized complex structure of X⁴ changes from Minkowskian to  its Euclidean variant.

   If the embedding is generalized-holomorphic, the induced  metric  of X⁴ degenerates to an  effective 2-D metric at at X³ so that   the topologically  4-D tangent space is effectively  2-D metrically.

5. Identification of the 2-D singularities (vertices) as regions at which the minimal surface property fails.

   At 2-D  singularities X², which I propose to be counterparts of 4-D smooth structure,  the minimal surface property fails. X² is  a hypercomplex analog  of a pole of complex functions and 2-D.   It is analogous to a source of a massless field.

   At X²  the generalized complex structure fails such that the trace of the second fundamental form generalizing acceleration for a point-like particle develops a delta function like singularity.  This singularity develops for the hypercomplex part  of the generalized complex structure and one has  as an  analog   a pole of analytic function at  which  analyticity fails. At X² the tangent space is 4-D rather than 2-D as elsewhere at the partonic orbit.

   At X²   there is an  infinite generalized acceleration. This  generalizes Brownian motion  of a point-like particle as a piecewise free motion. The partonic orbits could  perform Brownian motion and the 2-D singularities correspond to vertices for particle reactions.

   At least the creation of a   fermion-antifermion pair   occurs at this kind of singularity.    Fermion turns backwards in time. Without  these singularities fermion and antifermion number would be separately conserved and TGD  would be trivial as a physical theory.

6. One can identify the singularity X²  as a defect of the ordinary smooth structure.  

   This is the conjecture that I would like to understand better and here my limitations as a mathematician are the problem.

   I can  only ask questions inspired by the result that the intersection form I (X⁴)  for 2-D homologically non-trivial surfaces of X⁴  detects  the defects of the ordinary smooth structure,  which should correspond to surfaces X², i.e.  vertices for a pair creation.

   1. In  homology, the  defect  should correspond  to an intersection point  of homologically non-trivial 2-surfaces identifiable as wormhole throats, which  correspond to homologically non-trivial 2-surfaces of CP₂. This suggests that  I(X⁴₁)  for X⁴₁ containing the singularity/vertex differs from I(X⁴₂) when X⁴ does  not contain the vertex.  
   2. Singularities  contribute to the intersection form. The creation of  fermion-antifermion pair  has an interpretation in terms of closed monopole flux tubes. A closed monopole flux tube with wormhole contacts at its "ends" splits into two by reconnection.  The  vertex at which the particle is created, should contribute to  the intersection form: the fermion-antifermion vertex as  the intersection point?
   For a summary of earlier postings see Latest progress in TGD.

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.