Periodic table of topological insulators and topological superconductors and discrete symmetries at the space-time level

At the level of the embedding space H=M⁴× CP₂, the TGD view about discrete symmetries T, P, and C is essentially the same as in the standard model except that C has a geometric interpretation as a complex conjugation in CP₂ and has therefore a geometric meaning (see this). At the space-time level holography = holomorphy principle (see this, this, and this) forces us to reconsider the TGD based view of discrete symmetries. They would relate naturally to transformations relating different holomorphic structures differing by a conjugation of some in some complex coordinates of H or of a hypercomplex coordinate of M⁴.

How do these discrete symmetries relate to those of condensed matter physics? According to Wikipedia (see this), the periodic table of topological insulators and topological superconductors, also called tenfold classification of topological insulators and superconductors, is an application of topology to condensed matter physics. It indicates the mathematical group for the topological invariant of the topological insulators and topological superconductors, given a dimension and discrete symmetry class. The ten possible discrete symmetry families are classified according to three main symmetries: particle-hole symmetry, time-reversal symmetry and chiral symmetry. The table was developed between 2008 2010 by the collaboration of Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki and Andreas W. W. Ludwig; and independently by Alexei Kitaev. The table applies to topological insulators and topological superconductors with an energy gap, when particle-particle interactions are excluded. The table is no longer valid when interactions are included. The table applies in dimensions D=1, 2, 3. The table also gives the value set of the topological invariant with the associated symmetry, which can be Z₂, Z or 2Z in these dimensions.

Three basic symmetries appear in the 10-fold periodic table for topological insulators and superconductors and various cases are classified by the symmetries T, C, and S for which the squares of these symmetries are well-defined and by the broken symmetries. T acts as time reversal, C is the analog of charge conjugation, which transforms particles and holes to each other, and chiral transformation S transforms left-and right-handed chiralities to each other. C and T are antiunitary operators involving hermitian conjugation at the level of the Hilbert space and expressible as products of a unitary operator with a complex conjugation operator K. CPT is therefore unitary and it is possible to have CPT=1. These transformations are idempotent: T²=Id, C²=Id, S²=Id. In quantum theory, one can consider the possibility that their representations are projective so that one can have T²=-Id and C²=-Id. S²=1 holds always true.

In the table of the Wikipedia article, the violation of these symmetries is expressed using the symbol X. My understanding is that for instance, the charge conjugate need not belong to the space of allowed states. The TGD counterparts of T, S, and C at the space-time level would be naturally realized in terms of various conjugations of the complex coordinates and hypercomplex coordinate of H.

1. In the TGD framework, holography = holomorphy vision involves the notion of Hamilton-Jacobi structure for M⁴ ⊂ M⁴× CP₂ (see this) as a combination of longitudinal hypercomplex structure and transversal complex structure. The hypercomplex coordinate u has coordinate lines with light-like tangent vectors and the hypercomplex conjugation transforms u to its conjugate v (both coordinates are real).

   The complex coordinate w is transversal to (u and v). The coordinate lines define local polarization direction and light-like direction and characterize massless extremals as analogs of modes of massless fields. To simplify the situation, it is convenient to consider the simplex possible hypercomplex structure with (u= t-z,v= t+z) and w identifiable as the transversal planar complex coordinate w=x+iy.

2. There are two complex CP₂ coordinates ξⁱ but the conjugation as a symmetry applies simultaneously to both of them. CP₂ complex conjugation is naturally related to C, or at least its particle physics counterpart mapping fermions with antifermions and vice versa. In condensed physics C maps creation and annihilation operators of fermions (electrons and holes) to each other and one expects that CP₂ complex conjugation is involved.

   The complex conjugation in M⁴ cannot be involved with C. Does C involve HC? Geometric picture does not favor this. CPT =1 at the space-time level does not allow the presence of HC in C since HC is involved with both P and T realized at the space-time level assuming holography= holomorphy principle.

3. Chiral transformation S changed the chirality of say DNA strand and actd as reflection P. S would therefore naturally correspond to the HC: u→ v combined with w→ -w? Also no hypercomplex structure goes to its conjugate. This definition of S and the relation S=TC which by S²=1, is assumed in the 10-fold periodic table, corresponds to CPT=1. Together with S²=1, which is always true in the periodic table, it guarantees CST=1 as a counterpart of CPT=1. Note that neither C nor P are symmetries at the level of the Dirac equation at the embedding space level while CP is.
4. For the simplest option (u,v)=(t-z,t+z), HC combined with T corresponds to u→ -u and changes the functional form of f. This does not affect the generalized complex structure but affects the space-time spacetime surface. T corresponds to HC followed but u→ -u and conjugation of the HC structure and change of the analytic form. For P the presence of HC conjugation changes the hypercomplex structure.
5. w→ -w and (u,v)→ -(u,v) as analog of PT does not affect the generalized complex structure but affect the space-time surface since the functional form of the functions (f₁,f₂) obeying generalized holomorphy is changes. P, T, and CP affect the generalized complex structure and are always accompanied by HC. These symmetries cannot leave space-time surfaces invariant unless they consist of regions with different generalized complex structures glued together.
6. The minus sign in T² and C² could be also due an additional multiplication of the complex coordinate of H with an imaginary unit. In the case of T, -1 sign cannot appear in this way but would come from the unitary operator.

A possible interpretation of these symmetries is as space-time analogs of the basic symmetries of H representable in terms of complex and hypercomplex conjugations allowing to transform space-time surfaces satisfying holography= holomorphy principle to new such surfaces.

1. If all 3 basic conjugations are performed one obtains a new preferred extremal if longitudinal and transversal M⁴ degrees are independent. Same is true if one performs either both M⁴ conjugations or both CP₂ conjugations.
2. What is new is that the holomorphy violates these symmetries locally at the level of a single space-time surface. For instance, matter and matter could correspond to space-time regions with holomorphic structures related by CP₂ conjugation C and the formation of large space-time regions with a given holomorphic structure could correspond to matter antimatter asymmetry. This globalization of C and CP could relate the mystery of antimatter symmetry. These regions could correspond to the field bodies with large size induced antimatter asymmetry at the smaller space-time sheets associated with them. Quantum coherence in astrophysical and even cosmic scales with a quantum coherence scale characterizing the space-time surface, would be essential (see this).
3. The model for elementary particles (see this) involves in an essential manner a pair of parallel Minkoskian space-time sheets connected by wormhole contacts. How could the complex structures of the Minkowskian space-time sheets relate to each other?

   Ordinary complex surfaces are invariant under complex conjugation although the complex structure changes in the conjugation. Also the space-time surface can remain invariant under the complex conjugation although it is holomorphic with respect to the complex coordinates. What about the hypercomplex conjugation? Could the parallel space-time sheets be related by a hypercomplex conjugation HC and possibly also by the conjugation w→ w and C? If they are related by charge conjugation C (see this), fermions and antifermion lines would reside at the wormhole throats of opposite space-time sheets, which has been assumed.

4. One can also consider quantum superpositions of the space-time surfaces with different complex and hypercomplex structures. The expectation is that fermion and antifermion lines reside at the opposite wormhole throats of the space-time sheets connected by wormhole contacts. Do the space-time sheets have conjugate complex structures in both CP₂ and M⁴? Interesting questions relate to the CP breaking of K⁰ and B⁰ mesons. CP involves HC conjugating the hypercomplex structure and affecting the space-time surface: at the level H and Dirac equation this kind CP violation does not occur. Could the presence of HC explain the CP violation?

See the article TGD and Condensed matter or the chapter with the same title.

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.