Finite groups for the Kummer surface: the genetic code and a quantum gravity analogy

: The Kummer surface was constructed in 1864. It corresponds to the desingularisation of 1 the quotient of a 4-torus by 16 complex double points. Kummer surface is kwown to play a role in 2 some models of quantum gravity. Following our recent model of the DNA genetic code based on the 3 irreducible characters of the ﬁnite group G 5 : = ( 240, 105 ) ∼ = Z 5 (cid:111) 2 O (with 2 O the binary octahedral 4 group), we now ﬁnd that groups G 6 : = ( 288, 69 ) ∼ = Z 6 (cid:111) 2 O and G 7 : = ( 336, 118 ) ∼ = Z 7 (cid:111) 2 O can be 5 used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some 6 biological functions. Groups G 6 and G 7 are found to involve the Kummer surface in the structure of 7 their character table. An analogy between quantum gravity and DNA/RNA packings is suggested. 8

In this paper, in order to approach these biological issues -the hexamer and pentamer rings-, we 45 generalize our previous model of the DNA/RNA, which has been based on the five-fold symmetry 46 group G 5 , to models of DNA/RNA complexes, based on the six-fold symmetry group G 6 := (288, 69 Cte Cte z 3 z 3 z 3 z 3 z 3 z 3 Cte Cte Table 1. For the group G 6 := (288, 69) ∼ = Z 6 2O, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 30 -dimensional Pauli group and the entries involved in the characters. All characters are neither faithful nor informationally complete. The notation is I = exp(2iπ/4), z 1 = − √ 2, z 2 = I √ 2 and z 3 = −2 * cos(π/9).

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The structure of the character table is shown in Table 1. All characters are neither faithful nor 119 informationally complete since the rank of the Gram matrix is never d 2 = 30 2 for any character.

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Some characters contain entries with complex numbers I or z 2 = I √ 2. There are 12 characters 121 containing entries with z 3 = −2 * cos(π/9) featuring the angle π/9. We now show an important 122 characteristics of such characters. As an example, let us write the character number 11 as obtained where # denotes the algebraic conjugation, that is #k indicates replacing the root of unity w by w k .
127 Following our approach in [1], we construct an hyperelliptic curve C 6 of the form y 2 = ∏ ±l (x − 128 k l ) = f (x). In an explicit way, it is 129 a genus 2 hyperelliptic curve. Using Magma [13], one gets the polynomial definition of the The desingularisation of the Kummer surface is obtained in a simple way by restricting the 132 product f (x) = ∏ ±l (x − k l ) to the five first factors with indices ±1, ±2 and 4.

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One embeds C 6 in a weighted projective plane, with weights 1, g + 1, and 1, respectively on 134 coordinates x, y and z. Therefore, point triples are such that (x : y : z) = (µx : µy : µz), µ in the field of 135 definition, and the points at infinity take the form (1 : y : 0). Below, the software Magma is used for 136 the calculation of points of C 6 [13]. For the points of C 6 , there is a parameter called 'bound' that loosely 137 follows the heights of the x-coordinates found by the search algorithm.

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Character Kummer   Table 3. The algebra for the character table of group G 7 := (336, 118). In column 1 are the characters in question. Column 2 provides the powers of the entries z i , i = 1, 2, 3 or 5. The z i are z 1 = 2 cos(2π/7), z 2 = 2z 1 , z 3 = −6 cos(π/7), z 4 = √ 2 and z 5 = 2 cos(2π/21). Column 3 explicits the polynomial f (x) whose roots are the powers of a selected z i . When f (x) is an elliptic curve defined over the rationals the Cremona reference is in column 4. If f (x) is a sextit polynomial it leads to a Kummer surface.
A summary of the elliptic and genus 2 hyperelliptic curves that can be defined from G 7 is in Table   159 3.

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The defining polynomial can be given an explicit expression over the rational field leading to the Kummer surface A section at constant x 4 of this Kummer surface is given in Fig. 2b using the MathMod software  be processed afterwards. This cutting process is called splicing. There is a heptamer ring, called the 199 Lsm 1-7 complex, displaying a 7-fold symmetry in its protein constituents, as shown in Fig. 1c [7]. The group G 6 is not an appropriate candidate for modeling the degeneracies of amino acids in the 204 genetic code since none irreducible character of G 6 is informationally complete, as shown in Table 1.

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In Table 5 of the appendix, we reproduce the structure of the character table of the group G 5 206 and the assignments of its conjugacy classes to the proteinogenic amino acids as given in Ref. [1].

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One drawback of the model is that there are only 2 sextets in the table while 3 of them are needed to 208 fit the 3 sextets of the genetic code. In Table 4, this problem is solved since there are precisely three 209 slots of degeneracy 6 in the character table of G 7 . Table 4 shows entries proportional to the cosines 210 of angles involved in the characters as z 1 = 2 cos(2π/7), z 2 = 2z 1 , z 3 = −6 cos(π/7), z 4 = √ 2 and Table 4. For the group G 7 := (336, 118) ∼ = Z 7 2O, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 29 -dimensional Pauli group, the order of a group element in the class, the angles involved in the character and a good assignment to an amino acid according to its polar requirement value. Bold characters are for faithful representations. All characters are informationally complete except for the trivial character and the one assigned to 'Met'. The entries involved in the characters are z 1 = 2 cos(2π/7), z 2 = 2z 1 , z 3 = −6 cos(π/7), z 4 = √ 2 and z 5 = 2 cos(2π/21) featuring the angles 2π/8 (in z 4 ), 2π/7 and 2π/21. [27] for another view of the latter topic.

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In the forefront of differential geometry, there is a connection of K 3 surfaces to quantum gravity It is a challenging question that we are not able to solve. Mathematics offers clues for models of nature.

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Biology is not an unified field as is quantum physics of elementary particles or the general relativity 281 for the universe at large scales. We offered relationships between the n-fold symmetries (n = 5, 6 and