Mock and quantum modular forms

Transcription

1 Mock and quantum modular forms Amanda Folsom (Amherst College) 1

2 Ramanujan s mock theta functions 2

3 Ramanujan s mock theta functions

4 Ramanujan s mock theta functions

5 History S. Ramanujan Encountered math at a young age Ramanujan failed out of school Worked as a shipping clerk, pursued mathematics on his own 5

6 History Ramanujan wrote letters to mathematics professors in Cambridge, England Initially, all were ignored 6

7 History G.H. Hardy recognized Ramanujan s talent Hardy invited Ramanujan to Cambridge Ramanujan produced nearly 4000 original and deep results G.H. Hardy 7

8 Ramanujan s last letter Dear Hardy, I am extremely sorry for not writing you a single letter up to now. I discovered very interesting functions recently which I call Mock ϑ-functions...i am sending you with this letter some examples. - S. Ramanujan, January 12,

9 Ramanujan s mock theta functions Examples. q n2 f (q) := ( q; q) 2 n 0 n 9

10 Ramanujan s mock theta functions Examples. q n2 f (q) := ( q; q) 2 n 0 n = 1 + q (1 + q) 2 + q 4 (1 + q) 2 (1 + q 2 ) 2 + 9

11 Ramanujan s mock theta functions Examples. q n2 f (q) := ( q; q) 2 n 0 n = 1 + q (1 + q) 2 + q 4 (1 + q) 2 (1 + q 2 ) 2 + n 1 Def. (a; q) n := (1 aq j ), n N, (a; q) 0 := 1. j=0 9

12 Ramanujan s mock theta functions Examples. q n2 f (q) := ( q; q) 2 n 0 n = 1 + q (1 + q) 2 + q 4 (1 + q) 2 (1 + q 2 ) 2 + ω(q) := n 0 q 2n(n+1) (q; q 2 ) 2 n = 1 + q 4 (1 q) 2 + q 12 (1 q) 2 (1 q 3 ) 2 + n 1 Def. (a; q) n := (1 aq j ), n N, (a; q) 0 := 1. j=0 9

13 Ramanujan s mock theta functions Examples. q n2 f (q) := ( q; q) 2 n 0 n = 1 + q (1 + q) 2 + q 4 (1 + q) 2 (1 + q 2 ) 2 + ω(q) := n 0 F 2 (q) := n 0. q 2n(n+1) (q; q 2 ) 2 n = 1 + q n(n+1) (q n+1 ; q) n+1 = q 4 (1 q) 2 + q 12 (1 q) 2 (1 q 3 ) (1 q) + q 2 (1 q 2 )(1 q 3 ) + n 1 Def. (a; q) n := (1 aq j ), n N, (a; q) 0 := 1. j=0 9

14 10 Ramanujan s mock theta functions Atkin, Andrews, Dyson, Hardy, Ramanujan, Selberg, Swinnerton-Dyer, Watson, etc. studied asymptotic behaviors analytic properties combinatorial properties q-hypergeometric identites

15 11 Ramanujan s mock theta functions Atkin, Andrews, Dyson, Hardy, Ramanujan, Selberg, Swinnerton-Dyer, Watson, etc. studied asymptotic behaviors analytic properties combinatorial properties q-hypergeometric identites

16 12 Partition numbers Definition A partition of a natural number n is any way to write n as a non-increasing sum of natural numbers.

17 12 Partition numbers Definition A partition of a natural number n is any way to write n as a non-increasing sum of natural numbers. The partition function p(n) := number of partitions of n.

18 13 Integer partitions 1 = 1 p(1) = 1

19 13 Integer partitions 1 = 1 p(1) = 1 2 = 2, p(2) = 2

20 13 Integer partitions 1 = 1 p(1) = 1 2 = 2, p(2) = 2 3 = 3, 2 + 1, p(3) = 3

21 13 Integer partitions 1 = 1 p(1) = 1 2 = 2, p(2) = 2 3 = 3, 2 + 1, p(3) = 3 4 = 4, 3 + 1, 2 + 2, , p(4) = 5..

22 14 Integer partitions: 1700s - present Euler Hardy Watson Ramanujan Rademacher Dyson Atkin Swinnerton-Dyer Andrews Ono

23 15 The partition function p(n)

24 15 The partition function p(n) 1

25 15 The partition function p(n) 1, 2

26 15 The partition function p(n) 1, 2, 3

27 15 The partition function p(n) 1, 2, 3, 5

28 15 The partition function p(n) 1, 2, 3, 5, 7

29 15 The partition function p(n) 1, 2, 3, 5, 7, 11

30 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15

31 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22

32 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30

33 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42

34 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56

35 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77

36 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101

37 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135

38 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176

39 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231

40 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297

41 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385

42 15 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

43 The partition function p(n) 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , p(115) 15

44 16 Generating functions The generating function A(q) for a sequence {a(n)} n 0 is A(q) := n 0 a(n)q n = a(0) + a(1)q + a(2)q 2 + a(3)q 3 +

45 17 The partition generating function The partition generating function P(q) := n 0 p(n)q n = 1 + q + 2q 2 + 3q 3 + 5q 4 + 7q 5 +

46 18 The partition generating function Theorem (Euler, 1700s) For q < 1, P(q) := n 0 p(n)q n = m q m.

47 18 The partition generating function Theorem (Euler, 1700s) For q < 1, P(q) := n 0 p(n)q n = m q m. Geometric Series: j 0 q j = 1 1 q

48 19 Partitions and modular forms A key to answering certain questions about P(q), hence p(n), is its relationship to modular forms.

49 20 Modular forms Modular forms g : H := {τ C Im(τ) > 0} C satisfy: y

50 20 Modular forms Modular forms g : H := {τ C Im(τ) > 0} C satisfy: y g(γ τ) = testingtesting g(τ)

51 20 Modular forms Modular forms g : H := {τ C Im(τ) > 0} C satisfy: y g(γ τ) = testingtesting g(τ) γ = ( ) a b c d SL2 (Z) = {2 2 integer matrices, determinant 1} γ τ := aτ+b cτ+d H

52 21 Modular forms Modular forms g : H := {τ C Im(τ) > 0} C satisfy: y g(γ τ) = ɛ(γ)(cτ + d) k g(τ) k 1 2 Z ɛ(γ) = 1

53 22 Modular forms Definition (Dedekind s η-function) Let q = q τ := e 2πiτ, where τ H. η(τ) := q 1/24 n 1(1 q n ).

54 23 Modular forms Proposition The η-function is a modular form of weight 1/2. In particular, η (( ) τ) = η(τ + 1) = ζ 24 η(τ), η (( ) ) τ = η( 1/τ) = iτ η(τ).

55 23 Modular forms Proposition The η-function is a modular form of weight 1/2. In particular, η (( ) τ) = η(τ + 1) = ζ 24 η(τ), η (( ) ) τ = η( 1/τ) = iτ η(τ). Notation: ζ N := e 2πi N is an Nth root of unity.

56 24 Modular forms To understand aspects of the combinatorial function p(n), we can exploit the modularity of the weight 1 2 form η(τ) := q 1/24 n 1 (1 q n ) Euler = q 1/24 p(n)q n n 0 1

57 25 The partition function Remark. A combinatorial proof shows that we may also rewrite q n2 P(q) = p(n)q n = (q; q) 2. n 0 n 0 n

58 26 q-hypergeometric series Remark. Series of this shape are called q-hypergeometric series.

59 27 q-hypergeometric series Therefore: q n2 q 1/24 η 1 (τ) = (q; q) 2 n 0 n

60 27 q-hypergeometric series Therefore: q n2 q 1/24 η 1 (τ) = (q; q) 2 n 0 n Recall: Ramanujan s mock theta function q n2 f (q) := ( q; q) 2 n 0 n

61 28 Partition ranks Definition (Dyson) Let λ be a partition. Then rank(λ) := largest part of λ number of parts of λ.

62 28 Partition ranks Definition (Dyson) Let λ be a partition. Then rank(λ) := largest part of λ number of parts of λ. Example: the partition λ = has rank(λ) = 3 4 = 1.

63 29 Partition ranks Lemma The mock theta function f satisfies q n2 f (q) := ( q; q) 2 n 0 n = (p e (n) p o (n)) q n n 0 where p e (n) := p(n even rank), p o (n) := p(n odd rank).

64 30 Combinatorial modular(?) q-hypergeometric series q 1/24 η 1 (τ) = p(n)q n q n2 = (q; q) 2 n 0 n 0 n modular combinatorial q-hypergeometric f (q) = e (n) p 0 (n))q n 0(p n = q n2 ( q; q) 2 n 0 n modular??? combinatorial q-hypergeometric

65 31 Modular forms and mock theta functions Question What roles do mock theta functions play within the theory of modular forms?

66 32 Modular forms and mock theta functions Let τ H := {τ C Im(τ) > 0}, and q = e 2πiτ.

67 32 Modular forms and mock theta functions Let τ H := {τ C Im(τ) > 0}, and q = e 2πiτ.

68 33 Modular forms and mock theta functions Ramanujan s last letter (1920). Asymptotics, near roots of unity, of Eulerian (i.e. q-hypergeometric) series.

69 33 Modular forms and mock theta functions Ramanujan s last letter (1920). Asymptotics, near roots of unity, of Eulerian (i.e. q-hypergeometric) series. i.e. q-series similar in shape to f (q) = q n2 q ( q; q) 2 = 1 + n 0 n (1 + q) 2 + q 4 (1 + q) 2 (1 + q 2 ) 2 +

70 34 Modular forms and mock theta functions Question (Ramanujan) Must Eulerian series with similar asymptotics be the sum of a modular form and a function which is O(1) (i.e. bounded) at all roots of unity?

71 34 Modular forms and mock theta functions Question (Ramanujan) Must Eulerian series with similar asymptotics be the sum of a modular form and a function which is O(1) (i.e. bounded) at all roots of unity? Ramanujan s Answer The answer is it is not necessarily so...i have not proved rigorously that it is not so. But I have constructed a number of examples...

72 35 Modular forms and mock theta functions Define the modular form b(q) by b(q) := (1 q 2m 1 ) ( 1) n q n2 m 1 n Z

73 35 Modular forms and mock theta functions Define the modular form b(q) by b(q) := (1 q 2m 1 ) ( 1) n q n2 m 1 n Z = q 1 24 η 3 (τ) η 2 (2τ) (q = e 2πiτ )

74 36 Modular forms and mock theta functions Claim (Ramanujan) If q radially approaches an even order 2k root of unity, then f (q) ( 1) k b(q) = O(1).

75 37 F. Dyson (1987)

76 37 F. Dyson (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 which Hecke built around the old theta-functions of Jacobi....This remains a challenge for the future...

77 38 Modular forms and mock theta functions Theorem (Zwegers, 02) Ramanujan s mock theta functions may be completed to obtain vector valued, non-holomorphic modular forms.

78 39 A completion Example: F (τ) := (q 1 24 f (q), 2q 1 3 ω(q 1 2 ), 2q 1 3 ω( q 1 2 )) T

79 39 A completion Example: F (τ) := (q 1 24 f (q), 2q 1 3 ω(q 1 2 ), 2q 1 3 ω( q 1 2 )) T q n2 f (q) := ( q; q) 2 n 0 n ω(q) := n 0 q 2n(n+1) (q; q 2 ) 2 n

80 40 A completion Example (cont.): G(τ) := 2i 3 ( i τ g 1 (z) dz i, i(τ + z) τ g 0 (z) dz i, i(τ + z) τ ) T g 2 (z) dz i(τ + z)

81 40 A completion Example (cont.): G(τ) := 2i 3 ( i τ g 1 (z) dz i, i(τ + z) τ g 0 (z) dz i, i(τ + z) τ ) T g 2 (z) dz i(τ + z) g j (z) = weight 3 2 modular Θ functions

82 41 Some results of Zwegers Theorem (Zwegers) The function H(τ) := F (τ) G(τ) is a vector-valued weight 1 2 non-holomorphic modular form. In particular, H(τ) transforms as H(τ + 1) = H( 1/τ) = iτ ζ ζ 3 0 ζ H(τ), H(τ).

83 42 Harmonic Maass forms Definition (Bruinier-Funke, 04)

84 42 Harmonic Maass forms Definition (Bruinier-Funke, 04) A harmonic Maass form of weight k 1 2 Z on Γ 0(4N) is a smooth M : H C satisfying

85 42 Harmonic Maass forms Definition (Bruinier-Funke, 04) A harmonic Maass form of weight k 1 2 Z on Γ 0(4N) is a smooth M : H C satisfying (1) transformation law: A Γ 0 (4N), τ H, {( c ) 2k M(Aτ) = d ɛ 2k d (cτ + d) k M(τ), k 1 2 Z Z (cτ + d) k M(τ), k Z

86 42 Harmonic Maass forms Definition (Bruinier-Funke, 04) A harmonic Maass form of weight k 1 2 Z on Γ 0(4N) is a smooth M : H C satisfying (1) transformation law: A Γ 0 (4N), τ H, {( c ) 2k M(Aτ) = d ɛ 2k d (cτ + d) k M(τ), k 1 2 Z Z (cτ + d) k M(τ), k Z (2) harmonic: ( k M = 0, where ) k := y iky x 2 y 2 (τ = x + iy) ( ) x + i y

87 42 Harmonic Maass forms Definition (Bruinier-Funke, 04) A harmonic Maass form of weight k 1 2 Z on Γ 0(4N) is a smooth M : H C satisfying (1) transformation law: A Γ 0 (4N), τ H, {( c ) 2k M(Aτ) = d ɛ 2k d (cτ + d) k M(τ), k 1 2 Z Z (cτ + d) k M(τ), k Z (2) harmonic: ( k M = 0, where ) k := y iky x 2 y 2 (τ = x + iy) ( ) x + i y (3) M satisfies a suitable growth condition in the cusps.

88 43 Harmonic Maass forms Technical remarks. 1 Γ 0 (N) := {( ) a b c d SL2 (Z) c 0 (mod N) },

89 43 Harmonic Maass forms Technical remarks. 1 Γ 0 (N) := {( ) a b c d SL2 (Z) c 0 (mod N) }, 2 We require 4 N if k 1 2 Z Z.

90 43 Harmonic Maass forms Technical remarks. 1 Γ 0 (N) := {( ) a b c d SL2 (Z) c 0 (mod N) }, 2 We require 4 N if k 1 2 Z Z. 3 Kronecker symbol: ( )

91 43 Harmonic Maass forms Technical remarks. 1 Γ 0 (N) := {( ) a b c d SL2 (Z) c 0 (mod N) }, 2 We require 4 N if k 1 2 Z Z. 3 Kronecker symbol: ( ) 4 ɛ d := { 1, if d 1 (mod 4), i, if d 3 (mod 4).

92 43 Harmonic Maass forms Technical remarks. 1 Γ 0 (N) := {( ) a b c d SL2 (Z) c 0 (mod N) }, 2 We require 4 N if k 1 2 Z Z. 3 Kronecker symbol: ( ) 4 ɛ d := { 1, if d 1 (mod 4), i, if d 3 (mod 4). 5 We take the principal branch of the holomorphic

93 44 Harmonic Maass forms Condition (3): P M C[q 1 ] such that for some ɛ > 0 as y. M(τ) P M (τ) = O(e ɛy ),

94 45 Harmonic Maass forms Lemma Harmonic Maass forms M decompose into two parts:

95 45 Harmonic Maass forms Lemma Harmonic Maass forms M decompose into two parts: where M = M + + M,

96 45 Harmonic Maass forms Lemma Harmonic Maass forms M decompose into two parts: where M + := M = M + + M, n r M c + M (n)qn holomorphic part (r M Z)

97 45 Harmonic Maass forms Lemma Harmonic Maass forms M decompose into two parts: where M + := M = M + + M, n r M c + M (n)qn holomorphic part (r M Z) M := n<0 c M (n)γ(1 k, 4π n y)qn non-holomorphic part

98 Harmonic Maass forms Lemma Harmonic Maass forms M decompose into two parts: where M + := M = M + + M, n r M c + M (n)qn holomorphic part (r M Z) M := n<0 c M (n)γ(1 k, 4π n y)qn non-holomorphic part Γ(a, x) := x t a 1 e t dt 45

99 46 Mock theta functions and mock modular forms Fact: Ramanujan s mock theta functions are examples of holomorphic parts M + of harmonic Maass forms.

100 46 Mock theta functions and mock modular forms Fact: Ramanujan s mock theta functions are examples of holomorphic parts M + of harmonic Maass forms. Ex. M + (τ) = q 1 q n2 24 ( q; q) 2 n n 0

101 46 Mock theta functions and mock modular forms Fact: Ramanujan s mock theta functions are examples of holomorphic parts M + of harmonic Maass forms. Ex. M + (τ) = q 1 q n2 24 ( q; q) 2 n n 0 Definition (Zagier). part of a HMF. A mock modular form := a holomorphic

102 47 Applications q-series modular L-functions combinatorics generalized Borcherds products Moonshine Donaldson invariants mathematical physics quantum modular forms.

103 48 Applications q-series modular L-functions combinatorics generalized Borcherds products Moonshine Donaldson invariants mathematical physics quantum modular forms.

104 49 Applications q-series modular L-functions (combinatorics) generalized Borcherds products Moonshine Donaldson invariants mathematical physics quantum modular forms.

105 50 Quantum modular forms Let τ H := {τ C Im(τ) > 0}, and q = e 2πiτ.

106 50 Quantum modular forms Let τ H := {τ C Im(τ) > 0}, and q = e 2πiτ.

107 51 Quantum modular forms Let g : H C, γ := ( ) a b c d Γ SL2 (Z), τ H.

108 51 Quantum modular forms Let g : H C, γ := ( ) a b c d Γ SL2 (Z), τ H. Modular transformation: ( ) aτ + b g = ɛ(γ)(cτ + d) k g(τ) cτ + d or rather, g(τ) ɛ(γ) 1 (cτ + d) k g ( ) aτ + b = 0 cτ + d

109 52 Quantum modular forms Let g : Q C, γ := ( ) a b c d Γ SL2 (Z), x Q. Modular transformation: g(x) ɛ(γ) 1 (cx + d) k g ( ) ax + b =? cx + d

110 53 Quantum modular forms Definition (Zagier 10) A quantum modular form of weight k (k 1 2Z) is function g : Q \ S C, for some discrete subset S, such that

111 53 Quantum modular forms Definition (Zagier 10) A quantum modular form of weight k (k 1 2Z) is function g : Q \ S C, for some discrete subset S, such that for all γ = ( ) a b c d Γ, the functions

112 53 Quantum modular forms Definition (Zagier 10) A quantum modular form of weight k (k 1 2Z) is function g : Q \ S ) C, for some discrete subset S, such that for all Γ, the functions γ = ( a b c d h γ (x) = h g,γ (x) := g(x) ɛ(γ) 1 (cx + d) k g ( ) ax + b cx + d satisfy a suitable property of continuity or analyticity in R.

113 54 Quantum modular forms Example. Kontsevich s strange function (x Q): φ(x) := e πix 12 ( e 2πix ; e 2πix) n n=0

114 54 Quantum modular forms Example. Kontsevich s strange function (x Q): φ(x) := e πix 12 ( e 2πix ; e 2πix) n n=0 Note. φ converges for no open subset of C.

115 54 Quantum modular forms Example. Kontsevich s strange function (x Q): φ(x) := e πix 12 ( e 2πix ; e 2πix) n n=0 Note. φ converges for no open subset of C. Exercise: φ converges for any rational number x.

116 55 Quantum modular forms Theorem (Zagier) The function φ is a quantum modular form of weight 3/2,

117 55 Quantum modular forms Theorem (Zagier) The function φ is a quantum modular form of weight 3/2, i.e. φ(x + 1) = ζ 24 φ(x), φ(x) ζ 8 x 3 2 φ( 1/x) = h(x), where h is a real analytic function (except at 0).

118 56 Quantum modular forms Definition A sequence {a j } s j=1 of integers is called strongly unimodal of size n if a 1 + a a s = n,

119 56 Quantum modular forms Definition A sequence {a j } s j=1 of integers is called strongly unimodal of size n if a 1 + a a s = n, 0 < a 1 < a 2 < < a r > a r+1 > a s > 0 for some r.

120 56 Quantum modular forms Definition A sequence {a j } s j=1 of integers is called strongly unimodal of size n if a 1 + a a s = n, 0 < a 1 < a 2 < < a r > a r+1 > a s > 0 for some r. The rank equals s 2r + 1 (difference between # terms after and before the peak ).

121 56 Quantum modular forms Definition A sequence {a j } s j=1 of integers is called strongly unimodal of size n if a 1 + a a s = n, 0 < a 1 < a 2 < < a r > a r+1 > a s > 0 for some r. The rank equals s 2r + 1 (difference between # terms after and before the peak ). Ex. {2, 5, 8, 1} is a s.u.s. of size = 16

122 56 Quantum modular forms Definition A sequence {a j } s j=1 of integers is called strongly unimodal of size n if a 1 + a a s = n, 0 < a 1 < a 2 < < a r > a r+1 > a s > 0 for some r. The rank equals s 2r + 1 (difference between # terms after and before the peak ). Ex. {2, 5, 8, 1} is a s.u.s. of size = 16 and rank = 1.

123 56 Quantum modular forms Definition A sequence {a j } s j=1 of integers is called strongly unimodal of size n if a 1 + a a s = n, 0 < a 1 < a 2 < < a r > a r+1 > a s > 0 for some r. The rank equals s 2r + 1 (difference between # terms after and before the peak ). Ex. {2, 5, 8, 1} is a s.u.s. of size = 16 and rank = 1. {1, 2, 3, 5, 3, 2} is also a s.u.s. of size 16

124 56 Quantum modular forms Definition A sequence {a j } s j=1 of integers is called strongly unimodal of size n if a 1 + a a s = n, 0 < a 1 < a 2 < < a r > a r+1 > a s > 0 for some r. The rank equals s 2r + 1 (difference between # terms after and before the peak ). Ex. {2, 5, 8, 1} is a s.u.s. of size = 16 and rank = 1. {1, 2, 3, 5, 3, 2} is also a s.u.s. of size 16 and rank = 1.

125 57 Quantum modular forms A combinatorial generating function. u(m, n) := #{size n strongly unimodal sequences of rank m}.

126 57 Quantum modular forms A combinatorial generating function. u(m, n) := #{size n strongly unimodal sequences of rank m}. U(τ) := q 1 24 n=0 (q; q) 2 nq n+1 = q 1 24 u(m, n)( 1) m q n. m Z n 1

127 58 Quantum modular forms Theorem (Bryson-Ono-Pittman-Rhoades) Let x Q. We have that φ( x) = U(x).

128 58 Quantum modular forms Theorem (Bryson-Ono-Pittman-Rhoades) Let x Q. We have that φ( x) = U(x). Theorem (Bryson-Ono-Pittman-Rhoades) For x H Q, the function U satisfies U(x) + ( ix) 3 2 U( 1/x) = h(x), where h is a real analytic function (except at 0).

129 59 Quantum modular forms Theorem (Bryson-Ono-Pittman-Rhoades) Let x Q. We have that φ( x) = U(x). Theorem (Bryson-Ono-Pittman-Rhoades) For x H Q, the function U satisfies U(x) + ( ix) 3 2 U( 1/x) = h(x), where h is a real analytic function (except at 0).

130 60 Modular forms and mock theta functions Ramanujan s claim revisited (F - Ono - Rhoades)

131 61 Modular forms and mock theta functions Claim (Ramanujan) If q radially approaches an even order 2k root of unity, then f (q) ( 1) k b(q) = O(1).

132 61 Modular forms and mock theta functions Claim (Ramanujan) If q radially approaches an even order 2k root of unity, then f (q) ( 1) k b(q) = O(1). b(q) := (1 q 2m 1 ) m 1 n Z ( 1) n q n2 = q 1 24 η 3 (τ) η 2 (2τ) q n2 f (q) := ( q; q) 2 n 0 n

133 62 Numerics As q 1, we computed f ( 0.994) , f ( 0.996) , f ( 0.998)

134 63 Numerics (cont.) Ramanujan s claim gives: q f (q) + b(q)

135 63 Numerics (cont.) Ramanujan s claim gives: q f (q) + b(q) This suggests that lim (f (q) + b(q)) = 4. q 1

136 64 Numerics (cont.) q 0.992i 0.994i 0.996i f (q) i i i f (q) b(q) i i i

137 64 Numerics (cont.) q 0.992i 0.994i 0.996i f (q) i i i f (q) b(q) i i i This suggests that lim(f (q) b(q)) = 4i. q i

138 65 Questions Questions What are the O(1) constants in How do they arise? lim (f (q) q ζ ( 1)k b(q)) = O(1)?

139 66 Ramanujan s radial limits Theorem (F-Ono-Rhoades) If ζ is an even 2k order root of unity, then lim (f k 1 q ζ (q) ( 1)k b(q)) = 4 (1+ζ) 2 (1+ζ 2 ) 2 (1+ζ n ) 2 ζ n+1. n=0

140 67 Radial limits Remark. We prove this as a special case of a more general theorem involving:

141 Radial limits Remark. We prove this as a special case of a more general theorem involving: R(w; q) := N (m, n)w m q n q n2 = (wq; q) n 0 n=0 n (w 1, q; q) n m Z mock modular rank function [Bringmann-Ono] C(w; q) := M(m, n)w m q n (q; q) = (wq; q) (w 1, q; q) n 0 U(w; q) := n 1 m Z modular crank function u(m, n)( w) m q n = m Z (wq; q) n (w 1 q; q) n q n+1. n=0 quantum modular unimodal function [B-O-P-R] 67

142 68 Combinatorial modular forms Here, N (m, n) := #{partitions λ of n rank(λ) = m}, M(m, n) := #{partitions λ of n crank(λ) = m}, u(m, n) := #{size n strongly unimodal sequences with rank m}.

143 69 Radial limits Theorem (F-Ono-Rhoades) If ζ b = e 2πi b and 1 a < b, then for every suitable root of unity ζ there is an explicit integer c for which ( lim R(ζ a b ; q) ζ c q ζ b C(ζ a 2 b ; q)) = (1 ζb a )(1 ζ a b )U(ζa b ; ζ).

144 70 Radial limits Remark The first theorem is the special case a = 1, b = 2, using the facts that R( 1; q) = f (q) and C( 1; q) = b(q).

145 70 Radial limits Remark The first theorem is the special case a = 1, b = 2, using the facts that R( 1; q) = f (q) and C( 1; q) = b(q). Remark With the specialization (w; q) (ζb a; ζ), the function U(ζa b ; ζ) is a finite sum.

146 71 Radial limits Theorems = lim (Mock ϑ Mod. Form) = Quantum MF. q ζ

147 71 Radial limits Theorems = lim (Mock ϑ Mod. Form) = Quantum MF. q ζ lim (rank function crank function) = unimodal function. q ζ

148 72 Upper and lower half-planes

149 72 Upper and lower half-planes Proof philosophy.

150 72 Upper and lower half-planes Proof philosophy. Example For Ramanujan s f (q), remarkably we have f (q 1 q n ) = ( q; q) 2 n n=0 = 1 + q q 2 + 2q 3 4q

151 72 Upper and lower half-planes Proof philosophy. Example For Ramanujan s f (q), remarkably we have f (q 1 q n ) = ( q; q) 2 n n=0 = 1 + q q 2 + 2q 3 4q Remark. Under τ q = e 2πiτ, f (q) is defined on both H H.

152 73 Upper and lower half-planes

153 74 Upper and lower half-planes A larger framework:

154 75 Radial limits

155 76 Radial limits Poll to rank most fascinating formulas in the lost notebook: 1 Dyson s Ranks. 2 Mock ϑ-functions. 3 Andrews-Garvan Crank. 4 Continued fraction with three limit points. 5 Early QMFs: Sums of Tails of Euler s Products.

156 Related work: Bajpai-Kimport-Liang-Ma-Ricci Bringmann-Creutzig-Rolen Bringmann-F-Rhoades Bringmann-Rolen Bryson-Ono-Pittman-Rhoades F-Ki-Truong Vu-Yang F-Ono-Rhoades Hikami-Lovejoy Joo-Löbrich Rolen-Schneider Zagier etc. 77

157 78 Thank you