The dark side of the moon - new connections in geometry, number theory and physics - Daniel Persson Mathematical Sciences Colloquium January 29, 2018 Symmetries Transformations that leave a geometric object invariant A circle has infinitely many rotational symmetries Symmetries Transformations that leave a geometric object invariant A circle has infinitely many rotational symmetries Infinite rotational Finite reflection symmetry symmetry Symmetry groups The set of symmetries of an object form a mathematical structure called a group. Two consecutive transformations is also a symmetry There exists an identity transformation Each transformation has an inverse Consecutive transformations are associative Permutation symmetries The set {1,2,3} has 6 permutations (1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1) These belong to the permutation group of 3 elements. “Ile de feu 2” (Island of fire) M12 Olivier Messiaen 1949 This piece of music has the symmetry group of a 45-dim algebraic structure “Ile de feu 2” (Island of fire) Olivier Messiaen 1949 M12 This piece of music has the symmetry group of a 45-dim algebraic structure This corresponds to the following permutation: P1 =(1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8) This corresponds to the following permutation: P1 =(1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8) Messiaen also uses the following permutation in another part: P2 =(1, 6, 9, 2, 7, 3, 5, 4, 8, 10, 11)(12) This corresponds to the following permutation: P1 =(1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8) Messiaen also uses the following permutation in another part: P2 =(1, 6, 9, 2, 7, 3, 5, 4, 8, 10, 11)(12) Theorem (Diaconis, Graham, Kantor 1985): M = P ,P 12 h 1 2i Of course, Messiaen did not use all 95040 permutations in the Mathieu group! Finite simple groups Finite groups that can not be divided into smaller pieces are called simple. These are like building blocks of symmetries Finite simple groups Finite groups that can not be divided into smaller pieces are called simple. These are like building blocks of symmetries Prime numbers Elementary particles 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Classification of finite simple groups 1832: Galois discovers the first infinite family (alternating groups) Classification of finite simple groups 1832: Galois discovers the first infinite family (alternating groups) 1861-1873: Mathieu discovers M11,M12,M22,M23,M24 Classification of finite simple groups 1832: Galois discovers the first infinite family (alternating groups) 1861-1873: Mathieu discovers M11,M12,M22,M23,M24 Classification of finite simple groups 1832: Galois discovers the first infinite family (alternating groups) 1861-1873: Mathieu discovers M11,M12,M22,M23,M24 1972-1983: Gorenstein program to classify all finite simple groups The monster Conjecture (Fischer & Griess, 1973): The monster There exists a huge finite simple group of order Conjecture (Fischer & Griess, 1973): The monster There exists a huge finite simple group of order Theorem (Griess 1982): The monster group exists. It is the symmetry group of a certain 196884-dimensional algebraic structure. (commutative, non-associative algebra) Conjecture (Fischer & Griess, 1973): The monster There exists a huge finite simple group of order Theorem (Griess 1982): The monster group exists. It is the symmetry group of a certain 196884-dimensional algebraic structure. (commutative, non-associative algebra) 1983: The classification program is completed! This was a heroic joint effort of 100’s of mathematicians. [https://irandrus.files.wordpress.com/2012/06/periodic-table-of-groups.pdf] Classification Several infinite families: cyclic, alternating, Lie type 26 sporadic cases Classification Several infinite families: cyclic, alternating, Lie type 26 sporadic cases Monster M Classification Several infinite families: cyclic, alternating, Lie type 26 sporadic cases The happy family (all related to the monster) Classification Several infinite families: cyclic, alternating, Lie type 26 sporadic cases The happy family (all related to the monster) Pariah groups (not related to the monster) Classification Several infinite families: cyclic, alternating, Lie type 26 sporadic cases The happy family (all related to the monster) Pariah groups (not related to the monster) Why do these groups exist? The first hints of moonshine… The first hints of moonshine… In 1978 John McKay was taking a break from the classification program of finite groups and was doing some recreational reading in number theory. The first hints of moonshine… In 1978 John McKay was taking a break from the classification program of finite groups and was doing some recreational reading in number theory. He then stumbled upon the following series expansion: 1 J(q)= + 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + q ··· The first hints of moonshine… In 1978 John McKay was taking a break from the classification program of finite groups and was doing some recreational reading in number theory. He then stumbled upon the following series expansion: 1 J(q)= + 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + q ··· Being a finite group theorist he immediately opened up the Atlas… 196884 = 1 + 196883 McKay’s equation 196884 = 1 + 196883 McKay’s equation 21493760 = 1 + 196883 + 21296876 Thompson’s equation 196884 = 1 + 196883 McKay’s equation 21493760 = 1 + 196883 + 21296876 Thompson’s equation What does this really mean? Fun with the monster 196884 = 1 + 196883 21493760 = 1 + 196883 + 21296876 We can think of 196883 as the dimension of a vector space on which the monster acts This is the smallest non-trivial irreducible representation of M The factor 1 then corresponds to the trivial reps. Fun with the monster 196884 = 1 + 196883 21493760 = 1 + 196883 + 21296876 We can think of 196883 as the dimension of a vector space on which the monster acts This is the smallest non-trivial irreducible representation of M The factor 1 then corresponds to the trivial reps. 1 V dim V =1 ! 1 R 1 196883 V dim V = 196883 ! 2 R 2 21296876 V dim V = 21296876 ! 3 R 3 Fun with the monster 196884 = 1 + 196883 21493760 = 1 + 196883 + 21296876 We can think of 196883 as the dimension of a vector space on which the monster acts This is the smallest non-trivial irreducible representation of M The factor 1 then corresponds to the trivial reps. 1 V dim V =1 =Tr e V1 ! 1 R 1 196883 V2 dim V2 = 196883 =Tr eV2 ! R 21296876 V dim V = 21296876 =Tr eV3 ! 3 R 3 Fun with the monster 196884 = 1 + 196883 21493760 = 1 + 196883 + 21296876 We can think of 196883 as the dimension of a vector space on which the monster acts This is the smallest non-trivial irreducible representation of M The factor 1 then corresponds to the trivial reps. 1 V dim V =1 =Tr e V1 ! 1 R 1 196883 V2 dim V2 = 196883 =Tr eV2 ! R 21296876 V dim V = 21296876 =Tr eV3 ! 3 R 3 characters of the identity element e M 2 McKay and Thompson conjectured that there exists an infinite-dimensional, graded monster module, called V \ , such that 1 V \ = V (n) n= 1 M− McKay and Thompson conjectured that there exists an infinite-dimensional, graded monster module, called V \ , such that 1 V \ = V (n) n= 1 M− ( 1) V − = V1 V (0) =0 and the J-function is the graded dimension (1) 1 V = V1 V2 J(q)= Tr(e V (n))qn ⊕ | n= 1 V (2) = V V V X− 1 ⊕ 2 ⊕ 3 McKay and Thompson conjectured that there exists an infinite-dimensional, graded monster module, called V \ , such that 1 V \ = V (n) n= 1 M− ( 1) V − = V1 V (0) =0 and the J-function is the graded dimension (1) 1 V = V1 V2 J(q)= Tr(e V (n))qn ⊕ | n= 1 V (2) = V V V X− 1 ⊕ 2 ⊕ 3 This suggests to also consider for each g M the McKay-Thompson series 2 1 T (q)= Tr(g V (n))qn g | n= 1 X− Modular forms 1 J(q)= + 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + q ··· Modular forms 1 J(q)= + 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + q ··· The J-function has a very special invariance property. If we set q = e2⇡i⌧ ⌧ H = z C (z) > 0 2 { 2 | = } then we have a⌧ + b ab f = f(⌧) SL(2, Z) c⌧ + d cd 2 ✓ ◆ ✓ ◆ Modular forms 1 J(q)= + 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + q ··· The J-function has a very special invariance property. If we set q = e2⇡i⌧ ⌧ H = z C (z) > 0 2 { 2 | = } then we have a⌧ + b ab f = f(⌧) SL(2, Z) c⌧ + d cd 2 ✓ ◆ ✓ ◆ This is a modular form of weight 0 (or a modular function). Up to a constant there is a unique such function which is holomorphic away from a simple pole at the cusp (⌧) = !1 It generates the field of rational functions on the sphere (Hauptmodul) SL(2, Z) H \ ⇠ polynomial in J(⌧) any modular function = polynomial in J(⌧) It generates the field of rational functions on the sphere (Hauptmodul) SL(2, Z) H \ ⇠ polynomial in J(⌧) any modular function = polynomial in J(⌧) Monstrous moonshine conjecture (Conway-Norton 1979): For all elements g M The McKay-Thompson series Tg(⌧) 2 are hauptmoduls with respect to some genus zero Γg SL(2, R) ⇢ Γ genus zero Γg H g \ ⇠ “The stuff we were getting was not supported by logical argument. It had the feeling of mysterious moonbeams lighting up dancing Irish leprechauns. Moonshine can also refer to illicitly distilled spirits, and it seemed almost illicit to be working on this stuff.” - John Conway Monster group M ???? Modular function J(⌧) Monster group Enter physics! M 2d conformal field theory (vertex operator algebra) Modular function J(⌧) 1988: Frenkel, Lepowsky, Meurman constructed the moonshine module V \ It is a 2-dimensional orbifold conformal field theory (string theory), a.