Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy

Abstract

The Kummer surface was constructed in 1864. It corresponds to the desingularization of the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in some models of quantum gravity. Following our recent model of the DNA genetic code based on the irreducible characters of the finite group G5:=(240,105)≅Z5⋊2O (with 2O the binary octahedral group), we now find that groups G6:=(288,69)≅Z6⋊2O and G7:=(336,118)≅Z7⋊2O can be used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some biological functions. Groups G6 and G7 are found to involve the Kummer surface in the structure of their character table. An analogy between quantum gravity and DNA/RNA packings is suggested.

Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface

“…In Section 5, we investigate the nucleosome complex which is 8-fold symmetric. Following our previous work in [4,5], we find that the nucleosome complex allows to define another group theoretical model of the genetic code based on the characters of the group G 8 . In addition, one can map the DNA double helix scaffold of the nucleosome complex to the 16 singular points of a Kummer surface.…”

“…In our previous paper [5], it was shown that 7-fold symmetry may be mirrored in the finite group G 7 = Z 7 × 2O (with 2O the binary octahedral group) whose characters may be mapped to the amino acids of the genetic code. Such a mapping is reproduced in Table 4.…”

“…We found in a previous work that the approach of quantum computation based on magic states [1][2][3] may also be used to explore the symmetries and the structure of the genetic code [4][5][6]. Given an appropriate finite group G with d conjugacy classes, one takes an irreducible character κ = κ r and a corresponding r-dimensional representation in the conjugacy class.…”

“…Given an appropriate finite group G with d conjugacy classes, one takes an irreducible character κ = κ r and a corresponding r-dimensional representation in the conjugacy class. For the application to the genetic code, one takes the finite group G n := Z n 2O (with n = 5 or 7 and 2O the binary octahedral group) [4,5]. For such a group, the dimension r may be 1, 2, 3, 4, or 6 and the relevant conjugacy classes may be mapped to the amino acids of degeneracy r in their relation to codons.…”

“…Our first investigation points out the possible role of two algebraic building blocks. The first one is the hyperbolic (unoriented) 3-manifold of smallest volume known as the Gieseking manifold [11], when the secondary code only consists of two letters H and C. The second one is the oriented hypercartographic group H + 2 [12][13][14] (alias the two-generator free group), when the secondary code needs the three letters H, E, and C. The consistency of the (primary) genetic code and the secondary code is studied under the light of the Kummer surface that we already assumed to play a role in the quaternary structure of protein complexes [5].…”