The dark side of the moon - new connections in geometry, number theory and physics -

Transcription

1 The dark side of the moon - new connections in geometry, number theory and physics - Daniel Persson Mathematical Sciences Colloquium January 29, 2018

2 Symmetries Transformations that leave a geometric object invariant A circle has infinitely many rotational symmetries

3 Symmetries Transformations that leave a geometric object invariant A circle has infinitely many rotational symmetries

4 Infinite rotational symmetry Finite reflection symmetry

5 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

6 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.

7 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

8 Ile de feu 2 Olivier Messiaen 1949 M 12 (Island of fire) This piece of music has the symmetry group of a 45-dim algebraic structure

9 This corresponds to the following permutation: P 1 =(1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8)

10 This corresponds to the following permutation: P 1 =(1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8) Messiaen also uses the following permutation in another part: P 2 =(1, 6, 9, 2, 7, 3, 5, 4, 8, 10, 11)(12)

11 This corresponds to the following permutation: P 1 =(1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8) Messiaen also uses the following permutation in another part: P 2 =(1, 6, 9, 2, 7, 3, 5, 4, 8, 10, 11)(12) Theorem (Diaconis, Graham, Kantor 1985): M 12 = hp 1,P 2 i Of course, Messiaen did not use all permutations in the Mathieu group!

12 Finite simple groups Finite groups that can not be divided into smaller pieces are called simple. These are like building blocks of symmetries

13 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

14 Classification of finite simple groups 1832: Galois discovers the first infinite family (alternating groups)

15 Classification of finite simple groups 1832: Galois discovers the first infinite family (alternating groups) : Mathieu discovers M 11,M 12,M 22,M 23,M 24

16 Classification of finite simple groups 1832: Galois discovers the first infinite family (alternating groups) : Mathieu discovers M 11,M 12,M 22,M 23,M 24

17 Classification of finite simple groups 1832: Galois discovers the first infinite family (alternating groups) : Mathieu discovers M 11,M 12,M 22,M 23,M : Gorenstein program to classify all finite simple groups

18 The monster

19 Conjecture (Fischer & Griess, 1973): There exists a huge finite simple group of order The monster

20 Conjecture (Fischer & Griess, 1973): There exists a huge finite simple group of order The monster Theorem (Griess 1982): The monster group exists. It is the symmetry group of a certain dimensional algebraic structure. (commutative, non-associative algebra)

21 Conjecture (Fischer & Griess, 1973): There exists a huge finite simple group of order The monster Theorem (Griess 1982): The monster group exists. It is the symmetry group of a certain dimensional algebraic structure. (commutative, non-associative algebra) 1983: The classification program is completed! This was a heroic joint effort of 100 s of mathematicians.

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24 Classification Several infinite families: cyclic, alternating, Lie type 26 sporadic cases

25 Classification Several infinite families: cyclic, alternating, Lie type 26 sporadic cases Monster M

26 Classification Several infinite families: cyclic, alternating, Lie type 26 sporadic cases The happy family (all related to the monster)

27 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)

28 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?

29 The first hints of moonshine

30 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.

31 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: J(q) = 1 q q q q q 4 +

32 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: J(q) = 1 q q q q q 4 + Being a finite group theorist he immediately opened up the Atlas

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36 = McKay s equation

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38 = McKay s equation = Thompson s equation

39 = McKay s equation = Thompson s equation What does this really mean?

40 Fun with the monster = = We can think of 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.

41 Fun with the monster = = We can think of 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 1 dim R V 1 = ! V 2 dim R V 2 = ! V 3 dim R V 3 =

42 Fun with the monster = = We can think of 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 1 dim R V 1 = ! V 2 dim R V 2 = ! V 3 dim R V 3 = =Tr e V 1 =Tr e V 2 =Tr e V 3

43 Fun with the monster = = We can think of 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 1 dim R V 1 = ! V 2 dim R V 2 = ! V 3 dim R V 3 = =Tr e V 1 =Tr e V 2 =Tr e V 3 characters of the identity element e 2 M

44 McKay and Thompson conjectured that there exists an infinite-dimensional, graded monster module, called V \, such that V \ = 1M n= 1 V (n)

45 McKay and Thompson conjectured that there exists an infinite-dimensional, graded monster module, called V \, such that V \ = 1M n= 1 V (n) V ( 1) = V 1 V (0) =0 V (1) = V 1 V 2 V (2) = V 1 V 2 V 3 and the J-function is the graded dimension 1X J(q) = Tr(e V (n) )q n n= 1

46 McKay and Thompson conjectured that there exists an infinite-dimensional, graded monster module, called V \, such that V \ = 1M n= 1 V (n) V ( 1) = V 1 V (0) =0 V (1) = V 1 V 2 V (2) = V 1 V 2 V 3 and the J-function is the graded dimension 1X J(q) = Tr(e V (n) )q n n= 1 This suggests to also consider for each g 2 M the McKay-Thompson series T g (q) = 1X n= 1 Tr(g V (n) )q n

47 Modular forms J(q) = 1 q q q q q 4 +

48 Modular forms J(q) = 1 q q q q q 4 + The J-function has a very special invariance property. If we set q = e 2 i 2 H = {z 2 C =(z) > 0} then we have f a + b c + d = f( ) a c b d 2 SL(2, Z)

49 Modular forms J(q) = 1 q q q q q 4 + The J-function has a very special invariance property. If we set q = e 2 i 2 H = {z 2 C =(z) > 0} then we have f a + b c + d = f( ) a c b d 2 SL(2, Z) 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

50 It generates the field of rational functions on the sphere (Hauptmodul) SL(2, Z)\H any modular function = polynomial in J( ) polynomial in J( )

51 It generates the field of rational functions on the sphere (Hauptmodul) SL(2, Z)\H any modular function = polynomial in J( ) polynomial in J( ) Monstrous moonshine conjecture (Conway-Norton 1979): For all elements g 2 M The McKay-Thompson series T g ( ) are hauptmoduls with respect to some genus zero g SL(2, R) g genus zero g\h

52 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

53 Monster group M???? Modular function J( )

54 Monster group Enter physics! M 2d conformal field theory (vertex operator algebra) Modular function J( )

55 1988: Frenkel, Lepowsky, Meurman constructed the moonshine module V \ It is a 2-dimensional orbifold conformal field theory (string theory), a. k. a. vertex operator algebra, whose automorphism group is M A quantum field theory with conformal symmetry defined on a Riemann surface

56 1988: Frenkel, Lepowsky, Meurman constructed the moonshine module V \ It is a 2-dimensional orbifold conformal field theory (string theory), a. k. a. vertex operator algebra, whose automorphism group is M A quantum field theory with conformal symmetry defined on a Riemann surface To explain moonshine we need a special class of CFT s: holomorphic CFT = self-dual Vertex Operator Algebra (VOA) In general a VOA describes only part of a CFT

57 1988: Frenkel, Lepowsky, Meurman constructed the moonshine module V \ It is a 2-dimensional orbifold conformal field theory (string theory), a. k. a. vertex operator algebra, whose automorphism group is M A quantum field theory with conformal symmetry defined on a Riemann surface To explain moonshine we need a special class of CFT s: holomorphic CFT = self-dual Vertex Operator Algebra (VOA) In general a VOA describes only part of a CFT 1992: Borcherds proved the full moonshine conjecture, earning him the Fields medal

58 Do we understand everything now? Why genus zero? Proven by Borcherds but no understanding of its origin. 2016: Explanation proposed by Paquette-D.P.-Volpato using T-duality in string theory. (part of which is a mathematical proof using VOAs)

59 Do we understand everything now? Why genus zero? Proven by Borcherds but no understanding of its origin. 2016: Explanation proposed by Paquette-D.P.-Volpato using T-duality in string theory. (part of which is a mathematical proof using VOAs) Long-standing conjecture by Frenkel-Lepowsky-Meurman: The moonshine module V \ is unique. Partial results by Dong-Li-Mason, Tuite, Paquette-D.P.-Volpato, Carnahan-Komuro-Urano,

60 Do we understand everything now? Why genus zero? Proven by Borcherds but no understanding of its origin. 2016: Explanation proposed by Paquette-D.P.-Volpato using T-duality in string theory. (part of which is a mathematical proof using VOAs) Long-standing conjecture by Frenkel-Lepowsky-Meurman: The moonshine module V \ is unique. Partial results by Dong-Li-Mason, Tuite, Paquette-D.P.-Volpato, Carnahan-Komuro-Urano, Generalized moonshine conjecture by Norton in 1987: Moonshine for orbifolds of V \ Given as a PhD project by Borcherds to his student Carnahan around Complete proof announced by Carnahan in 2017.

61 Generalized moonshine includes several more of the sporadic groups occurring as centralizers of elements in the monster. Frenkel-Lepowsky-Meurman: The sporadic groups exist because of moonshine.

62 What does the future have in store? New moonshine!

63 New moonshine!

64 Moonshine revolution 2010: New moonshine for M 24 conjectured by Eguchi, Ooguri, Tachikawa The role of the J-function is now played by the elliptic genus of string theory on a K3-surface (weak Jacobi form)

65 Moonshine revolution 2010: New moonshine for M 24 conjectured by Eguchi-Ooguri-Tachikawa 2012: Proven by Gannon But no analogue of the monster module is constructed! 2013: Generalized Mathieu moonshine established by Gaberdiel-D.P.-Volpato The role of the J-function is now played by the elliptic genus of string theory on a K3-surface (weak Jacobi form)

66 Moonshine revolution 2010: New moonshine for M 24 conjectured by Eguchi-Ooguri-Tachikawa 2012: Proven by Gannon But no analogue of the monster module is constructed! 2013: Generalized Mathieu moonshine established by Gaberdiel-D.P.-Volpato But why?? The role of the J-function is now played by the elliptic genus of string theory on a K3-surface (weak Jacobi form) There is no string theory on K3 with Mathieu symmetry (Gaberdiel-Hohenegger-Volpato) Also a mathematical formulation of no-go theorem by Huybrechts using autoequivalences of derived categories of coherent sheaves on K3-surfaces.

67 But it doesn t stop there! 2012: Umbral moonshine proposed by Cheng-Duncan-Harvey A class of 23 moonshines, including Mathieu moonshine. Classified by Niemeier lattices. Involves also non-sporadic groups!

68 But it doesn t stop there! 2012: Umbral moonshine proposed by Cheng-Duncan-Harvey A class of 23 moonshines, including Mathieu moonshine. Classified by Niemeier lattices. Involves also non-sporadic groups! 2015: Umbral moonshine conjecture proven by Duncan-Griffin-Ono 2016: Generalized Umbral moonshine established by Cheng-de Lange-Whalen

69 But it doesn t stop there! 2012: Umbral moonshine proposed by Cheng-Duncan-Harvey A class of 23 moonshines, including Mathieu moonshine. Classified by Niemeier lattices. Involves also non-sporadic groups! 2015: Umbral moonshine conjecture proven by Duncan-Griffin-Ono 2016: Generalized Umbral moonshine established by Cheng-de Lange-Whalen Naturally formulated using mock modular forms This goes back to Ramanujan s last letter to Hardy

70 The Last Letter... Towards the end of his stay in England, Ramanujan fell very ill and he returned to India in months before his death, Ramanujan wrote a final letter to Hardy, including a list of 17 mysterious functions: Mock Theta Functions It took nearly 100 years, and the efforts of generations of mathematicians before Ramanujan s theory was finally understood... (Zwegers)

71 The Last Letter... Towards the end of his stay in England, Ramanujan fell very ill and he returned to India in months before his death, Ramanujan wrote a final letter to Hardy, including a list of 17 mysterious functions: Mock Theta Functions The mock theta functions arise as McKay-Thompson series in Umbral moonshine!

72 So what are mock modular forms? Def (Zwegers, Zagier): A mock modular form of weight k is the first member of a pair (h, g) h : H! C is holomorphic with at most exponential growth at cusps g : H! C is a holomorphic cusp form with weight 2 k (the shadow)

73 So what are mock modular forms? Def (Zwegers, Zagier): A mock modular form of weight k is the first member of a pair (h, g) h : H! C is holomorphic with at most exponential growth at cusps g : H! C is a holomorphic cusp form with weight 2 k (the shadow) The completion h + g with g =(4i) k 1 Z 1 (z + ) k g( z)dz is a non-holomorphic modular form (harmonic Maass form) of weight k

74 So what are mock modular forms? Def (Zwegers, Zagier): A mock modular form of weight k is the first member of a pair (h, g) h : H! C is holomorphic with at most exponential growth at cusps g : H! C is a holomorphic cusp form with weight 2 k (the shadow) The completion h + g with g =(4i) k 1 Z 1 (z + ) k g( z)dz is a non-holomorphic modular form (harmonic Maass form) of weight k Umbral moonshine arises for weight k =1/2 Note: modular forms are mock modular forms with zero shadow g =0

75 Pariah moonshine 2017: Duncan, Mertens, Ono proposes moonshine for two of the Pariahs

76 Pariah moonshine 2017: Duncan, Mertens, Ono proposes moonshine for two of the Pariahs F ( ) = q q q q 7 + Mock modular form of weight k =3/2

77 Pariah moonshine 2017: Duncan, Mertens, Ono proposes moonshine for two of the Pariahs F ( ) = q q q q 7 + Mock modular form of weight k =3/2 New connections to number theory F ( ) = q X D<0 a(d)q D a(d) The coefficients encode information about elliptic curves of discriminant D Implications for the Birch-Swinnerton-Dyer conjecture on the asymptotics of L-functions?

78 So what is moonshine, really?

79 So what is moonshine, really? Moonshine is not a well defined term, but everyone in the area recognizes it when they see it. Richard E. Borcherds

80 So what is moonshine, really?

81 Roles of vertex operator algebras/cft? M Monstrous Moonshine J( ) =q q + modular function holomorphic VOA V \ monster Lie algebra (Borcherds algebra) m

82 Roles of vertex operator algebras/cft? M 24 Mathieu Moonshine mock modular forms string theory on K3? VOA? chiral de Rham complex? (Song) Borcherds Lie algebra???

83 Roles of vertex operator algebras/cft? The fact that generalized moonshine holds for Umbral moonshine strongly suggest some kind of underlying VOA structure (Gaberdiel-D.P.-Volpato, Cheng-de Lange-Whalen)

84 Rigidity (Harvey) Consider the case of the J-function: w =0 modular form for SL(2, Z) If it had a shadow it would have weight 2. But there are no weight 2 holomorphic modular forms for. This is due to the genus zero property. SL(2, Z)

85 Rigidity (Harvey) Consider the case of the J-function: w =0 modular form for SL(2, Z) If it had a shadow it would have weight 2. But there are no weight 2 holomorphic modular forms for. This is due to the genus zero property. SL(2, Z) Umbral moonshine Thompson moonshine Pariah moonshine weight 1/2 mock modular forms weight 3/2 mock modular forms no genus zero property

86 Rigidity (Harvey) Consider the case of the J-function: w =0 modular form for SL(2, Z) If it had a shadow it would have weight 2. But there are no weight 2 holomorphic modular forms for. This is due to the genus zero property. SL(2, Z) Umbral moonshine Thompson moonshine Pariah moonshine weight 1/2 mock modular forms weight 3/2 mock modular forms no genus zero property What is the analogue of genus zero? A consequence of T-duality in string theory (Paquette-D.P.-Volpato) McKay-Thompson series should be Rademacher summable (Duncan-Frenkel, Cheng-Duncan)

87 Quantum black holes? Witten proposed that the monster VOA describes the quantum properties of black holes in 3d gravity (via the AdS/CFT correspondence) McKay-Thompson series are partition functions of black hole microstates

88 Quantum black holes? Witten proposed that the monster VOA describes the quantum properties of black holes in 3d gravity (via the AdS/CFT correspondence) McKay-Thompson series are partition functions of black hole microstates A similar statement holds for Mathieu moonshine (Dabholkar-Murthy-Zagier) (Cheng, Raum, D.P.-Volpato) Black hole microstates organized into conjugacy classes of the sporadic group

89 Freeman Dyson in 1987: The mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure, analogous to the structure of modular forms... This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include mock theta-functions...

90 What else will we discover on the dark side of the moon?