Article
Finite groups for the Kummer surface: the genetic
code and a quantum gravity analogy
Michel Planat 1 0000-0001-5739-546X, David Chester 2, Raymond Aschheim
3 0000-0001-6953-8202, Marcelo M. Amaral 4 0000-0002-0637-1916, Fang Fang 5and Klee Irwin
6 0000-0003-2938-3941
1 Université de Bourgogne/Franche-Comté, Institut FEMTO-ST CNRS UMR 6174, 15 B Avenue des
Montboucons, F-25044 Besançon, France.; michel.planat@femto-st.fr
2 Quantum Gravity Research, Los Angeles, CA 90290, USA; DavidC@QuantumGravityResearch.org
3 Quantum Gravity Research, Los Angeles, CA 90290, USA; raymond@QuantumGravityResearch.org
4 Quantum Gravity Research, Los Angeles, CA 90290, USA; Marcelo@quantumgravityresearch.org
5 Quantum Gravity Research, Los Angeles, CA 90290, USA; fang@quantumgravityresearch.org
6 Quantum Gravity Research, Los Angeles, CA 90290, USA;Klee@quantumgravityresearch.org
* Correspondence: michel.planat@femto-st.fr
Version January 6, 2021 submitted to Journal Not Specified
Abstract: 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 G5 := (240, 105) ∼= Z5 o 2O (with 2O the binary octahedral
4
group), we now find that groups G6 := (288, 69) ∼= Z6 o 2O and G7 := (336, 118) ∼= Z7 o 2O can be
5
used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some
6
biological functions. Groups G6 and G7 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
Keywords: Kummer surface, DNA genetic code, hexamers and pentamers, informationally complete
9
characters, finite groups, hyperelliptic curve
10
PACS: 02. 20.-a, 02.10.-v, 03.65.Fd, 82.39.Rt, 87.10.-e, 87.14.gk
11
1. Introduction
12
In a recent paper we found that the 22 irreducible characters of the group G5 := (240, 105) ∼=
13
Z5 o 2O, with 2O the binary octahedral group, could be made in one-to-one correspondence with
14
the DNA multiplets encoding the proteinogenic amino acids [1]. The cyclic group Z5 features the
15
five-fold symmetry of the constituent bases, see Fig. 1a. An important aspect of this approach is that
16
the irreducible characters of G5 may be seen as ‘magic’ quantum states carrying minimal and complete
17
quantum information, see [1]-[3] for the meaning of these concepts. It was also shown that the physical
18
structure of DNA was reflected in some of the entries of the character table including the Golden ratio,
19
the irrational number
√
2, as well as the four roots of a quartic polynomial.
20
In molecular biology, there exists an ubiquitous family of RNA-binding proteins called LSM
21
proteins whose function is to serve as scaffolds for RNA oligonucleotides, assisting the RNA to
22
maintain the proper three-dimensional structure. Such proteins organize as rings of six or seven
23
subunits. The Hfq protein complex was discovered in 1968 as an Escherichia coli host factor that was
24
essential for replication of the bacteriophage Qβ [4], it displays an hexameric ring shape shown in Fig.
25
1b.
26
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It is known that, in the process of transcription of DNA to proteins through messenger RNA
27
sequences (mRNAs), there is an important step performed in the spliceosome [5]. It consists of
28
removing the non-coding intron sequences for obtaining the exons that code for the proteinogenic
29
amino acids. A ribonucleoprotein (RNP) –a complex of ribonucleic acid and RNA-binding protein–
30
plays a vital role in a number of biological functions that include transcription, translation, the
31
regulation of gene expression and the metabolism of RNA. Individual LSm proteins assemble into
32
a six or seven member doughnut ring which usually binds to a small RNA molecule to form a
33
ribonucleoprotein complex.
34
Thus, while fivefold symmetry is inherent to the bases A, T, G, C –the building blocks of DNA–,
35
six-fold and seven-fold symmetries turn out to be the rule at the level of the spliceosome [6]. Of the five
36
small ribonucleoproteins, four of them called U1, U2, U4 and U5, contain an heptamer ring, whereas
37
the U6 contains a specific Lsm2-8 heptamer with seven-fold symmetry. A specific Lsm heptameric
38
complex Lsm1-7 playing a role in mRNA decapping is shown in Fig. 1c [7].
39
Observe that six-fold rings are also present in other biological functions such as genomic DNA
40
replication [8,9]. The minichromosome maintenance complex (MCM) hexameric complex (Mcm2–7)
41
forms the core of the eukaryotic replicative helicase. Eukaryotic MCM consists of six gene products,
42
Mcm2–7, which form a heterohexamer [9]. Deregulation of MCM function has been linked to genomic
43
instability and a variety of carcinomas.
44
Figure 1. (a) Five-fold symmetry in the DNA, (b) six-fold symmetry in the LSM protein complex Hfq
[4], (c) seven-fold symmetry of the Lsm1-7 complex in the spliceosome [7].
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 G5, to models of DNA/RNA complexes, based on the six-fold symmetry group G6 := (288, 69) ∼=
47
Z6 × 2O and the seven-fold symmetry group G7 := (336, 118) ∼= Z7 × 2O.
48
What corresponds to the quartic curve, derived from some of the entries in the character table
49
of G5, is a genus 2 hyperelliptic curve derived from the character table of G6 or G7, underlying the
50
so-called Kummer surface, a gem of algebraic geometry [11]. The Kummer surface is a prototypic
51
exemple of a K3 surface – a Calabi-Yau manifold of complex dimension two– and as such it is part of
52
models in string theory and/or quantum gravity.
53
Sec. 2.1 is a brief introduction to elliptic and hyperelliptic curves defined over any field. In Sec.
54
2.2 and 2.3, the main objective is to describe a construction of the Kummer surface K based on the
55
character table of the groups G6 or G7. One identifies the 16 double points of K as the 16 two-torsion
56
points of a genus two hyperelliptic curve C and one provides an explicit description of the group law
57
and of the Kummer surface.
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In Sec. 3, a new encoding of the proteinogenic amino acids by the irreducible characters and the
59
corresponding representations of the group G7 is described. It improves the description obtained in [1]
60
from the 22 irreducible characters of group G5.
61
In Sec. 4, we browse over some applications of the Kummer surface to models of quantum gravity.
62
2. The hyperelliptic curve and the attached Kummer surface from groups G6 and G7
63
Let us first recall an important aspect of our previous work. Let G be a finite group with d
64
conjugacy classes. An irreducible character κ = κr corresponding to a r-dimensional representation
65
of G carries quantum information [1]-[3]. It may be calculated thanks to the action of elements of a
66
d-dimensional Pauli group Pd acting on κ. In other words, the character κ may be seen as the ‘magic
67
state’ of a quantum computation [2,10].
68
In a concrete way, one defines d2 one-dimensional projectors Πi = |ψi〉 〈ψi|, where the |ψi〉 are
69
the d2 states obtained from the action of Pd on κ, and one calculates the rank of the Gram matrix
70
G with elements tr(ΠiΠj). A Gram matrix G with rank equal to d2 is the signature of a minimal
71
informationally complete quantum measurement (or MIC), see e.g. [1, Sec. 3] for more details.
72
As in our previous work, in a character table, we will display the measure of quantum information
73
of a character κr = κ as the rank of the attached Gram matrix G.
74
2.1. Excerpts about elliptic and hyperelliptic curves
75
Let us consider a curve C defined with the algebraic equation y2 + h(x)y = f (x) where h(x) and
76
f (x) are finite degree polynomials with elements in a field K.
77
For an elliptic curve –let us use the notation E instead od C for this case–, polynomials h(x) and
78
f (x) are of degrees 1 and 2, respectively and the genus of E is g = 1. In the Weierstrass form of an
79
elliptic curve, one takes h(x) = a1x + a3 and f (x) = x3 + a2x2 + a4x + a6 so that E is specified with the
80
sequence [a1, a2, a3, a4, a5] of elements of K. There is a rich literature about elliptic curves defined over
81
rational fields Q, complex fields C, number fields or general fields. Explicit results are documented in
82
tables such as the Cremona table [12] or can be obtained from a mathematical software such as Magma
83
[13].
84
An elliptic curve (as well as a hyperelliptic curve) may be viewed as embedded in a weighted
85
projective space, with weights 1, g + 1 and 1, in which the points at infinity are non singular. In the
86
present work, one meets genus 2 curves for which there exists a set of 16 double points leading to the
87
construction of a Kummer surface. All curves of genus 2 are hyperelliptic but generic curves of genus
88
g > 2 are not. Again references [12] and [13] are basic references for explicit results.
89
There are plenty known invariants of a (hyper)elliptic curve C over a field. One of them is the
90
conductor N of an elliptic curve E seen as an abelian variety A. For A defined over Q, the conductor is
91
the positive integer whose prime factors are the primes where A has a bad reduction. The conductor
92
characterizes the isogeny class of A so that the curves E over Q may be classified according to the
93
isogeny classes. Another important invariant is the Mordell-Weil group of A which is the group A(Q)
94
of Q-rational points of E . Weil proved that A(Q) is finitely generated with a unique decomposition
95
of the form A(Q) ∼= ZrA ⊕ T, where the finite group T is the torsion subgroup and rA is the rank of
96
A. This result was later generalized to the elliptic curves defined over any field K. For a hyperelliptic
97
curve, the invariant in the Weierstrass equation of C called the discriminant ∆ may be defined over any
98
field K. For an elliptic curve, this leads to the j-invariant j = c34/∆ with c4 a polynomial function of the
99
coefficients in the Weierstrass form.
100
The main applications of elliptic curves are in the field of public-key cryptography. For
101
hyperelliptic curves, one makes use of the Jacobian as the abelian group in which to do arithmetic, as
102
one uses the group of points on an elliptic curve.
103
From now we describe how genus 1 curves (elliptic curves) and genus 2 curves (hyperelliptic
104
curves of the Kummer type) arise in the character table of groups of signature Gi ∼= Zi o 2O, i = 5, 6
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and 7. Other finite groups and curves of genus g > 2 built from the character table of a finite group G
106
are worthwhile to be investigated in the future.
107
Let us illustrate our description with the genus 1 hyperelliptic curve introduced in the context
108
of our model of the genetic code based on the group G5
:= (240, 105) in which h(x) = 0 and
109
f (x) = x4 − x3 − 4x2 + 4x + 1 [1, Sec 5]. Seeing this curve over the rationals, one learns from Magma
110
[13] that the conductor is N = 300 and the discriminant is ∆ = 18000. A look at the Cremona table
111
for elliptic curves [12] allows us to put our curve in the isogeny class of Cremona reference ‘300d1’.
112
The Weierstass form is y2 = x3 − x2 − 13x + 22 and the Mordell-Weil group is the group of infinite
113
cardinality Z×Z2.
114
Now we use this knowledge to investigate some properties of the character tables of group G6
115
and G7.
116
2.2. The group G6 := (288, 69) ∼= Z6 o 2O
117
(288,69)
dimension
1
1
1
1
2
2
2
2
2
2
Z6 o (Z2.S4) d-dit, d=30
31
796
867
867
882
882
880
897
897
880
char
Cte
Cte
I
I
Cte
Cte
z1
z2
z2
z1
(288,69)
dimension
2
2
2
2
2
2
4
4
4
4
d-dit, d=30
885
885
885
885
885
885
876
878
899
899
char
z3
z3
z3
z3
z3
z3
Cte
Cte
I
I
(288,69)
dimension
4
4
4
4
4
4
6
6
6
6
d-dit, d=30
877
878
885
885
885
885
885
885
880
880
char
Cte
Cte
z3
z3
z3
z3
z3
z3
Cte
Cte
Table 1. For the group G6
:= (288, 69) ∼= Z6 o 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), z1 = −
√
2, z2 = I
√
2 and z3 = −2 ∗ cos(π/9).
One first considers the group G6 := (288, 69) ∼= Z6 o 2O, with 2O the binary octahedral group.
118
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 d2 = 302 for any character.
120
Some characters contain entries with complex numbers I or z2 = I
√
2. There are 12 characters
121
containing entries with z3 = −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
123
from Magma [13]
124
κ11 = [2, 2,−2,−2,−1, 2,−2, 0, 0, 1,−1, 1, 0, 0, 0, 0, z3, z3#2, z3#4,−1,
1,−z3#2,−z3#4,−z3#4, z3#2, z3,−z3, z3#4,−z3,−z3#2]
where # denotes the algebraic conjugation, that is #k indicates replacing the root of unity w by wk.
125
The non constant (but real) entries are k±l = ±z3#l, with l = 1, 2 or 4. We obtain k1 = −2 ∗ cos(π/9)
126
and k2,4 = cos(π/9)∓ cos(2π/9)± cos(4π/9) and k1 + k2 + k4 = 0.
127
Following our approach in [1], we construct an hyperelliptic curve C6 of the form y2 = ∏±l(x−
128
kl) = f (x). In an explicit way, it is
129
C6 : y2 = x6 − 6x4 + 9x2 − 1,
(1)
a genus 2 hyperelliptic curve. Using Magma [13], one gets the polynomial definition of the
130
Kummer surface S(x1, x2, x3, x4) as
131
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Figure 2. (a) A standard plot of the Kummer surface in its 3-dimensional projection, (b) a section at
constant x4 of the Kummer surface defined in Sec. 2.3.
S(x1, x2, x3, x4) = 36x41 + 4x
3
1x4 − 24x21x22 + 220x21x23 − 36x21x3x4 − 8x1x22x3
+ 24x1x23x4 − 4x1x3x24 + 4x42 − 36x22x23 + x22x24 + 24x43 − 4x33x4.
The desingularisation of the Kummer surface is obtained in a simple way by restricting the
132
product f (x) = ∏±l(x− kl) to the five first factors with indices ±1, ±2 and 4.
133
One embeds C6 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 C6 [13]. For the points of C6, there is a parameter called ‘bound’ that loosely
137
follows the heights of the x-coordinates found by the search algorithm.
138
It is found that the corresponding Jacobian of C6 has 16 = 6 + 10 points as follows
139
* the 6 points bounded by the modulus 1:
140
Id := (1, 0, 0), K±1 := (x− k±1, 0, 1), K±2 := (x− k±2, 0, 1) and K4 = (k4, 0, 1).
141
* the 10 points of modulus > 1:
142
a1 := K−1 + K4, a2 := K1 + K−1, a3 := K1 + K−2, a4 := K1 + K−1 + K2, a5 := K1 + K−1 + K−2,
143
a6 := K1 + K4, a7 := K−1 + K2, a8 := K1 + K−1 + K4, a9 := K−1 + K−2 and a10 := K1 + K−1 + K2,
144
The 16 points organize as a commutative group isomorphic to the maximally abelian group Z42 as
145
shown in the following Jacobian addition table
146
where the blocks are given explicitely as
147
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A B
C D
B A D C
C D A B
D C
B A
Table 2. The structure of the addition table for the 16 singular Jacobian points of the hyperelliptic
curves C6 and C7.
A :

Id
K1 K−1
a2
K1
Id
a2 K−1
K−1
a2
Id
K1
a2 K−1 K1
Id
 , B :

K2 a10
a7
a4
a10 K2
a4
a7
a7
a4 K2 a10
a4
a7
a10 K2
 ,
C :

K−2
a3
a9
a5
a3 K−2
a5
a3
a9
a5 K−2
a3
a5
a9
a3 K−2
 , D :

a8
a1
a6 K4
a1
a8 K4
a6
a6 K4
a8
a1
K4
a6
a1
a8
 .
As a whole, one can check that there are only 48 points in the Jacobian J(C6). They
148
organize at the group Z3 ×Z32, i.e. three copies of the group of singular points.
149
2.3. The group G7 := (336, 118) ∼= Z7 × 2O
150
Let us consider the group G7
:= (336, 118) ∼= Z7 o 2O, with 2O the binary
151
octahedral group. The structure of the character table is shown in the table of section
152
3 about a refined model of the genetic code. Except for the singlets, the irreducible
153
characters of G7 are informationally complete (with rank of the Gram matrix equal
154
to d2 = 292 for any character). Only the first two singlets are exceptions. The entries
155
involved in the characters are z1 = 2 cos(2π/7), z2 = 2z1, z3 = −6 cos(π/7), z4 =
√
2
156
and z5 = 2 cos(2π/21) featuring the angles 2π/8 (in z4), 2π/7 and 2π/21. There are
157
9 faithful characters over the 10 quartets.
158
Character
zi powers
f (x) polynomial
Cremona ref.
4-6
z1 : [1, 2, 3]
x3 + x2 − 2x− 1
784i1
18-20
z1 : [1, 2, 3]
.
.
.
z2 : [1, 2, 3]
x3 + 2x2 − 8x− 8
3136x1
.
z1,2
x6 + 3x5 − 8x4 − 21x3 + 6x2 + 24x + 8
Kummer
27-29
z1 : [1, 2, 3]
.
.
.
z3 : [1, 2, 3]
x3 + 3x2 − 18x− 27
1764j1
.
z1,3 : [1, 2, 3]
x6 + 4x5 − 17x4 − 52x3 + 6x2 + 72x + 27
Kummer
9-14&21-26
z1,2
.
.
z5 : [1, 2, 4, 5, 8, 10]
x6 − x5 − 6x4 + 6x3 + 8x2 − 8x + 1
Kummer
Table 3. The algebra for the character table of group G7 := (336, 118). In column 1 are the characters in
question. Column 2 provides the powers of the entries zi, i = 1, 2, 3 or 5. The zi are z1 = 2 cos(2π/7),
z2 = 2z1, z3 = −6 cos(π/7), z4 =
√
2 and z5 = 2 cos(2π/21). Column 3 explicits the polynomial f (x)
whose roots are the powers of a selected zi. 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.
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A summary of the elliptic and genus 2 hyperelliptic curves that can be defined from G7 is in Table
159
3.
160
For instance, characters 4 to 6 as obtained from Magma [13] contains non constant entries with
161
z1, z1#2 and z1#3. With the polynomial f (x) = (x− z1)(x− z1#2)(x− z1#3) one defines the elliptic
162
curve y2 = f (x) over the rationals whose conductor N and discriminant ∆ are equal to 784 and whose
163
j-invariant equals 1792. It corresponds to the isogeny class of the curve 784i1 in the Cremona table [12].
164
There are 12 characters containing entries with z5. We now show an important characteristics of
165
such characters. As an example, let us write the character number 9 as obtained from Magma [13]
166
κ9 = [2, 2,−1, 2, 0,−1, z1#3, z1#2, z1, 0, 0, z1#2, z1#3, z1, z5, z5#4, z5#8,
z5#10, z5#2, z5#5, z1#2, z1, z1#3, z5#8, z5#5, z5#2, z5, z5#4, z5#10]
The hyperelliptic curve C7 : y2 = f (x) attached to the Kummer surface defined over the group
167
G7 := (336, 118) is
168
y2 = f (x) = (x− k)(x− l)(x−m)(x− n)(y− o)(y− p),
(2)
with k = 2 cos(10π/21), l = 2 cos(4π/21), m = 2 cos(16π/21), n = 2 cos(2π/21) and o =
169
−2 cos(π/21) (as above) and p = cos(π/21) + cos(8π/21)− cos(6π/21). The sum of roots of the
170
sextic curve f (x) equals 1.
171
The defining polynomial can be given an explicit expression over the rational field
172
f (x) = x6 − x5 − 6x4 + 6x3 + 8x2 − 8x + 1,
leading to the Kummer surface
173
K3(x1, x2, x3, x4) = 32x41 − 24x31x2 + 96x31x3 − 4x31x4 + 24x21x22
− 196x21x2x3 + 16x21x2x4 + 240x21x23 − 32x21x3x4 + 4x1x32 − 24x1x22x3
− 12x1x2x3x4 + 12x1x33 + 24x1x23x4− 4x1x3x24 − 4x42 + 32x32x3 − 32x22x23
+ x22x
2
4 − 24x2x33 + 2x2x23x4 + 25x43.
A section at constant x4 of this Kummer surface is given in Fig. 2b using the MathMod software
174
[14].
175
The desingularisation of the Kummer surface is obtained by restricting the product in the
176
polynomial f (x) to the five first factors. It is found that the corresponding Jacobian of C7 has 16 = 6+ 10
177
points as follows
178
* the 6 points bounded by the modulus 1:
179
Id := (1, 0, 0), K := (x − k, 0, 1), L := (x − l, 0, 1), N := (x − n, 0, 1), M := (x − m, 0, 1) and
180
O := (x− o, 0, 1).
181
* the 10 points of modulus > 1: a1 := K + L, a2 := K + M, a3 := K + N, a4 := K +O, a5 := L + N,
182
a6 := K + L + M, a7 := K + L + N, a8 := K + L + O, a9 := 2K + L + M and a10 := 2K + L + O.
183
More explicitly, for instance, K + L = (x2 − (k + l)x + 2 cos(2π/7), 0, 2).
184
The 16 points organize as a commutative group isomorphic to the maximally abelian group Z42 as
185
shown in table 2 with the entries
186
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A :

Id K
L
a1
K
Id
a1
L
L
a1
Id K
a1
L K
Id
 , B :

N a3
a5
a7
a3 N a7
a5
a5
a7 N a3
a7
a5
a3 N

C :

M a2
a9
a6
a2 M a6
a9
a9
a6 M a2
a6
a9
a2 M
 , D :

a8
a10
a4 O
a10
a8 O
a4
a4 O
a8
a10
O
a4
a10
a8
 .
There are 12 points in the Jacobian of C7 bounded by the modulus 1: the 6 points in the Jacobian
187
of C as above and 6 extra points, (1 : 1 : 0) (an extra point at infinity), (0 : 1 : 1), (0 : −1 : 1), (1 : 1 : 1),
188
(1 : −1 : 1) (4 rational points) and the point P := (x− p, 0, 1).
189
One finds 70 points bounded by the modulus 2 or 3, 694 points bounded by the modulus 4 and so
190
on.
191
3. The genetic code revisited
192
Our theory of the genetic code takes its inspiration in the symmetries observed in the DNA
193
double helix and the biological steps leading to the conversion of transfer RNA (tRNA) into the
194
amino acids that code for proteins. While 5-fold symmetry is inherent to the DNA double helix, being
195
present in all its constituents, as shown in Fig. 1a, the way to transcription into proteins needs an
196
extra step in the spliceosome. The spliceosome is found within the nucleus of eukaryotic cells. Its
197
role is to remove introns from the primary form of messenger RNA (mRNA) leaving the exons to
198
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].
200
Accounting for this observation, it is tempting to generalize our theory of the genetic code based on the
201
22 irreducible characters of the group G5 := (240, 105) ∼= Z5 o 2O, displaying the 5-fold symmetry, to
202
the 29 irreducible characters of the group G7 := (336, 118) ∼= Z7 o 2O, displaying the 7-fold symmetry.
203
The group G6 is not an appropriate candidate for modeling the degeneracies of amino acids in the
204
genetic code since none irreducible character of G6 is informationally complete, as shown in Table 1.
205
In Table 5 of the appendix, we reproduce the structure of the character table of the group G5
206
and the assignments of its conjugacy classes to the proteinogenic amino acids as given in Ref. [1].
207
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 G7. Table 4 shows entries proportional to the cosines
210
of angles involved in the characters as z1 = 2 cos(2π/7), z2 = 2z1, z3 = −6 cos(π/7), z4 =
√
2 and
211
z5 = 2 cos(2π/21). Let us first concentrate on the 11 classes of degeneracy 2 of the group G. The
212
character table contains the angle 2π/8 through the 2 entries with z4 = 2 cos(2π/8) =
√
2 as well as
213
the angles 2π/7 and 2π/21 through the entries with z1 and z1,5, respectively. We choose not to assign
214
amino acids to the 2 conjugacy classes with degeneracy 2 and 2π/8 angle. The entries containing z5
215
correspond to the hyperelliptic curve y2 = f (x) in (2) and the related Kummer surface. Then, there
216
are two conjugacy classes with degeneracy 1 (ignoring the class with a trivial character) and 3, as
217
expected. There are 3 classes with degeneracy 6 with entries containing z3 and thus the angle 2π/7, as
218
we would expect. Finally, there are 10 slots for quartets but only 5 slots with entries containing the
219
entry z5 related to the Kummer surface. The first 4 slots are assigned to the 4 (degeneracy 4) amino
220
acids and a slot is left empty. This leaves the freedom to assign this slot to the 21st proteinogenic acid
221
Sec and the 22nd amino acid Pyl (compare to [1] or Table 5).
222
Now comes a question. Is the Kummer surface an attribute of RNA packings or is the proposed
223
theory just another musing about the biological reality? We are not aware of any experiment featuring
224
the Kummer surface in the biological realm. The physical link between messenger RNA (mRNA) and
225
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9 of 12
(336,118)
dimension
1
1
1
2
2
2
2
2
2
2
Z7 o (Z2.S4) d-dit, d=29
29
785
d2
d2
d2
d2
d2
d2
d2
d2
∼= Z7 o 2O
amino acid
.
Met
Trp
Cys
Phe
Tyr
.
.
His
Gln
order
1
2
3
4
4
6
7
7
7
8
char
Cte
Cte
Cte
z1
z1
z1
z4
z4
z1,5
z1,5
polar req.
.
5.3
5.2
4.8
5.0
5.4
.
.
8.4
8.6
(336,118)
dimension
2
2
2
2
3
3
4
4
4
4
d-dit, d=29
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
amino acid
Asn
Lys
Glu
Asp
Ile
Stop
.
.
.
.
order
14
14
14
21
21
21
21
21
21
21
char
z1,5
z1,5
z1,5
z1,5
Cte
Cte
Cte
z1,2
z1,2
z1,2
polar req.
10.0
10.1
12.5
13.0
10
15
.
.
.
.
(336,118)
dimension
4
4
4
4
4
4
6
6
6
d-dit, d=29
d2
d2
d2
d2
d2
d2
d2
d2
d2
amino acid
Val
Pro
Thr
Ala
Gly
.
Leu
Ser
Arg
order
28
28
28
42
42
42
42
42
42
char
z2,5
z2,5
z2,5
z2,5
z2,5
z2,5
z1,3
z1,3
z1,3
polar req.
5.6
6.6
6.6
7.0
7.9
.
4.9
7.5
9.1
Table 4. For the group G7
:= (336, 118) ∼= Z7 o 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 z1 = 2 cos(2π/7), z2 = 2z1,
z3 = −6 cos(π/7), z4 =
√
2 and z5 = 2 cos(2π/21) featuring the angles 2π/8 (in z4), 2π/7 and 2π/21.
the amino acid sequence of proteins is a transfer RNA (tRNA). Corresponding to the three bases of
226
an mRNA codon is an anticodon. Each tRNA has a distinct anticodon triplet sequence that can form
227
three complementary base pairs to one or more codons for an amino acid. Some anticodons pair with
228
more than one codon due to so-called wobble base pairing [15]-[18]. Considering the secondary and
229
tertiary structure of tRNA, as well as the fact that the third position in the codon is not strictly red by
230
the anticodon according to Watson-Crick pairing rules, Crick hypothesized that codon translation into
231
a proteins is mainly due to the first two positions of the codon [15]. There are 16 groups of codons
232
specified by the first two codonic positions and the level of degeneracy can be dermined by them
233
according to Lagerkvist’s rules [16,17]. Our bet is that the 16 groups of codons correspond to the 16
234
singularities (double points) of the Kummer surface.
235
In the next section, we discuss the relevance of the Kummer surface in the context of 4-dimensional
236
(space-time) quantum gravity.
237
4. Kummer surface and quantum gravity
238
The Kummer surface first made its appearance in the Fresnel wave equation for light in a biaxial
239
crystal [19,20]. The four singularities corresponding to the two shells in the Fresnel surface for a biaxial
240
crystal lead to internal conical refraction as predicted by Hamilton in 1832. It is also known that, for a
241
magnetoelectric biaxial crystal, the Fresnel surface may display 16 real singular points, the maximal
242
number permitted for a linear material whose dispersion relation is quartic in the frequency and/or
243
wave number [20]. Although Kummer surface is relevant to electromagnetism, Ref. [21] discusses how
244
it may also rely on gravity.
245
A theory of quantum gravity needs to conciliate our view of space-time as described by the
246
general relativity and our view of particles and fields as described by quantum mechanics or quantum
247
field theory. How is the Kummer surface related to attempts of formulating a theory of quantum
248
gravity?
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If one follows the historical perspective, our derivation of the Kummer surface in Sec. 2 relies
250
on the work of Felix Klein in 1870 [19,22]. He introduces a quadratic line complex as the intersection
251
X = Gr(2, 4) ∩W of the Grassmann quadric Gr(2, 4) in the five-dimensional Plücker space with
252
another quadratic hypersurface W. The set of lines in X is parametrized by the Jacobian Jac(C) of a
253
Riemann surface of genus 2 ramified along 6 points corresponding to 6 singular quadrics. See [23] for
254
the relation to string dualities.
255
Nowadays, in the classification by algebraic geometry, the Kummer surface is an exemple of a K3
256
surface built from the quotient of an abelian variety A by the action from a point a to its opposite −a,
257
resulting in 16 singularities [24,25]. The minimal resolution is the Kummer surface. There are many
258
constructions of a K3 surface Y and it is known that all of them are diffeomorphic to each other. A K3
259
surface is a compact connected complex manifold of dimension 2 with trivial first Chern class c1(Y) = 0
260
so that the second Chern class (which corresponds to the topological Euler characteristic) is c2(Y) = 24.
261
Another important topological invariant of topological spaces is that of a Betti number bk. Roughly, b0 is
262
the number of connected components, b1 is the number of one-dimensional ‘holes’, b2 is the number of
263
two-dimensional ‘voids’, and so on. For a K3 surface, one has b0=b4=1, b1=b3=0 and b2=22. This defines
264
Y as the unique unimodular even quadratic lattice of signature (3, 19) isomorphic to E8(−1)⊕2 ⊕U⊕3,
265
where U the integral hyperbolic plane and E8 is the well known E8 lattice. It is also known that
266
the elliptic genus of a K3 surface has a decomposition in terms of the dimensions of irreducible
267
representations of the largest Mathieu group M24 [26], a concept named ‘umbral moonshine’. See also
268
[27] for another view of the latter topic.
269
In the forefront of differential geometry, there is a connection of K3 surfaces to quantum gravity
270
in the concept of a Kähler manifold (with a Kähler metric). Such a manifold possesses a complex
271
structure, a Riemannian structure and a symplectic structure. A K3 surface admits a Kähler ‘Ricci-flat
272
metric’ although it is not known how to write it in an explicit way. It is worthwhile to mention that a
273
K3 surface appears in string theory with the concept of ‘string duality’– how distinct string theories are
274
related–, see Ref. [23,28]. Another work relating quantum gravity and K3 surfaces is in Ref. [29]-[31].
275
5. Discussion
276
Since the Kummer surface appears in our models of DNA/RNA packings of some protein
277
complexes such as the hexamer and pentamer rings (the LSMs, MCMs and other biological complexes
278
not given here) one can ask the question whether quantum gravity is relevant in such biological realms.
279
It is a challenging question that we are not able to solve. Mathematics offers clues for models of nature.
280
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
282
7) found in DNA and some proteins and the mathematics of Kummer surfaces. It is time to quote
283
earlier work devoted to the possible relation between the microtubules of cytoskeleton and the field of
284
quantum consciousness, e.g. [32]. The 13-fold symmetry is found in tubulin complexes [33]. Using the
285
same approach than the one for DNA and hexamer/pentamer complexes, we can associate a finite
286
group G13 := (624, 134) ∼= Z13 × 2O to such a 13-fold symmetric complex. Such a group possesses
287
50 conjugacy classes and the dimensions of representations are 1, to 2, 3, 4 and 6 as expected for this
288
series of groups with factors Zn and 2O in the semidirect product. It is found that a Kummer surface
289
may be derived from some characters of G13, as expected.
290
To conclude, one can observe a mathematical analogy between the way DNA/RNA organize and
291
some theories of quantum cosmology based on string dualities. Lessons from one field may lead to
292
progress in the other field. Should we talk about a new paradigm of ‘biologic quantum cosmology’
293
and revisit the philosophical foundations of quantum theory? A few papers are already written in this
294
direction [34,35] and [36].
295
Author Contributions: Conceptualization, M.P., F.F. and K.I.; methodology, D.C., M.P. and R.A.; software, M.P..;
296
validation, D.C., R.A., F.F. and M.A..; formal analysis, M.P. and M.A.; investigation, D.C., M.P., F.F. and M.A.;
297
writing–original draft preparation, M.P.; writing–review and editing, M.P.; visualization, D.C., F.F. and R.A.;
298
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11 of 12
supervision, M.P. and K.I.; project administration, K.I..; funding acquisition, K.I. All authors have read and agreed
299
to the published version of the manuscript.
300
Funding: Funding was obtained from Quantum Gravity Research in Los Angeles,CA
301
Conflicts of Interest: The authors declare no conflict of interest.
302
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369
6. Appendix
370
The table below was found in our paper [1]. An introduction to the DNA genetic code and the
371
mention to some mathematical theories proposed before is discussed in this paper and not duplicated
372
here.
373
(240,105)
dimension
1
1
2
2
2
2
2
2
2
2
2
Z5 o (Z2.S4) d-dit, d=22
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
∼= Z5 o 2O
amino acid
Met
Trp
Cys
Phe
Tyr His
Gln Asn
Lys
Glu Asp
order
1
2
3
4
4
5
5
6
8
8
10
char
Cte
Cte
Cte
z1
z1
z4
z4
z1,5
z1,5
z1,5
z1,5
polar req.
5.3
5.2
4.8
5.0
5.4
8.4
8.6
10.0
10.1
12.5
13.0
(240,105)
dimension
3
3
4
4
4
4
4
4
4
6
6
d-dit, d=22
d2
475
483
480
d2
d2
d2
d2
d2
d2
d2
amino acid
Ile
Stop
Leu,Pyl,Sec
Leu Val
Pro
Thr
Ala
Gly
Ser
Arg
order
10
15
15
15
15
20
20
30
30
30
30
char
Cte
Cte
Cte
z1,2
z1,2
z2,5
z2,5
z2,5
z2,5
z1,3
z1,3
polar req.
4.9
4.9
5.6
6.6
6.6
7.0
7.9
7.5
9.1
Table 5. For the group G5
:= (240, 105) ∼= Z5 o 2O, the table provides the dimension of the
representation, the rank of the Gram matrix obtained under the action of the 22 -dimensional Pauli
group, the order of a group element in the class, the entries involved in the character and a good
assignment to an amino acid according to its polar requirement value. Bold characters are for faithful
representations. There is an ‘exception’ for the assignment of the sextet ‘Leu’ that is assumed to occupy
two 4-dimensional slots. All characters are informationally complete except for the ones assigned to
‘Stop’, ‘Leu’, ‘Pyl’ and ‘Sec’. The notation in the entries is as follows: z1 = −(
√
5 + 1)/2, z2 =
√
5− 1,
z3 = 3(1 +
√
5)/2, z4 =
√
2, z5 = −2 cos(π/15), compare [1, Table 7].
c© 2021 by the authors.
Submitted to
Journal Not Specified
for possible open access
374
publication under the terms and conditions of the Creative Commons Attribution (CC BY) license
375
(http://creativecommons.org/licenses/by/4.0/).
376
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