Sunday, February 13, 2022

Homology of "world of classical worlds" in relation to Floer homology and quantum homology

One of the mathematical challenges of TGD is the construction of the homology of "world of classical worlds" (WCW). With my rather limited mathematical skills, I had regarded this challenge as a mission impossible. The popular article in Quanta Magazine with title "Mathematicians transcend the geometric theory of motion" (see this) however stimulated the attempts to think whether it might be possible to say something interesting about WWC homology.

The article told about a generalization of Floer homology by Abouzaid and Blumberg (see this) published as 400 page article with the title "Arnold Conjecture and Morava K-theory". This theory transcends my mathematical skills but the article stimulated the idea WCW homology might be obtained by an appropriate generalization of the basic ideas of Floer homology (see this).

The construction of WCW homology as a generalization of Floer homology looks rather straightforward in the zero ontology (ZEO) based view about quantum TGD. The notions of ZEO and causal diamond (CD)(see this and this), the notion of preferred extremal (PE) (see this and this), and the intuitive connection between the failure of strict non-determinism and criticality pose strong conditions on the possible generalization of Floer homology.

WCW homology group could be defined in terms of the free group formed by preferred extremals PE(X3,Y3) for which X3 is a stable maximum of Kähler function K associated with the passive boundary of CD and Y3 associated with the active boundary of CD is a more general critical point.

The stability of X3 conforms with the TGD view about state function reductions (SFRs) (see this). The sequence of "small" SFRs (SSFRs) at the active boundary of CD as a locus of Y3 increases the size of CD and gradually leads to a PE connecting X3 with stable 3-surface Y3. Eventually "big" SFR (BSFR) occurs and changes the arrow of time and the roles of the boundaries of the CD changes. The sequence of SSFRs is analogous to a decay of unstable state to a stable final state.

The identification of PEs as minimal surfaces with lower-dimensional singularities as loci of instabilities implying non-determinism allows to assign to the set PE(X3,Y3i) numbers n(X3,Y3i→ Y3j) as the number of instabilities of singularities leading from Y3i to Y3j and define the analog of criticality index (number of negative eigenvalues of Hessian of function at critical point) as number n(X3,Y3i)= ∑jn(X3,Y3i→ Y3j). The differential d defining WCW homology is defined in terms of n(X3,3i→ Y3j) for pairs Y3i,Y3j such that n(X3,Y3j)-n(X3,Y3i)=1 is satisfied. What is nice is that WCW homology would have direct relevance for the understanding of quantum criticality.

The proposal for the WCW homology also involves a generalization of the notion of quantum connectivity crucial for the definition of Gromow-Witten invariants. Two surfaces (say branes) can be said to intersect if there is a string world sheet connecting them generalizes. In ZEO quantum connectivity translates to the existence of a preferred extremal (PE), which by the weak form of holography is almost unique, such that it connects the 3-surfaces at the opposite boundaries of causal diamond (CD).

See the article Homology of "world of classical worlds" in relation to Floer homology and quantum homology or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD. 

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