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

• Published in: Quantum Reports
• Main Authors: Michel Planat, David Chester, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, Klee Irwin
• Format: Article
• Language: English
• Published: MDPI AG 2021-01-01
• Subjects: kummer surface; DNA genetic code; hexamers and pentamers; informationally complete characters; finite groups; hyperelliptic curve
• Online Access: https://www.mdpi.com/2624-960X/3/1/5

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>5</mn></msub><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mn>240</mn><mo>,</mo><mn>105</mn><mo>)</mo></mrow><mo>≅</mo><msub><mi mathvariant="double-struck">Z</mi><mn>5</mn></msub><mo>⋊</mo><mn>2</mn><mi>O</mi></mrow></semantics></math></inline-formula> (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>O</mi></mrow></semantics></math></inline-formula> the binary octahedral group), we now find that groups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>6</mn></msub><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mn>288</mn><mo>,</mo><mn>69</mn><mo>)</mo></mrow><mo>≅</mo><msub><mi mathvariant="double-struck">Z</mi><mn>6</mn></msub><mo>⋊</mo><mn>2</mn><mi>O</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>7</mn></msub><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mn>336</mn><mo>,</mo><mn>118</mn><mo>)</mo></mrow><mo>≅</mo><msub><mi mathvariant="double-struck">Z</mi><mn>7</mn></msub><mo>⋊</mo><mn>2</mn><mi>O</mi></mrow></semantics></math></inline-formula> can be used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some biological functions. Groups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>6</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>7</mn></msub></semantics></math></inline-formula> 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.