Quantum Jacobi and quantum modular forms

Quantum Jacobi and quantum modular forms

Amanda Folsom (Amherst College)

1 Modular transformation:  aτ + b  f (τ) − −1(γ)(cτ + d)−k f = 0 cτ + d

Quantum Jacobi and quantum modular forms

Quantum Jacobi and quantum modular forms

Let a b
f : H → C, γ := c d ∈ Γ ⊆ SL2(Z), τ ∈ H := {τ ∈ C | Im(τ) > 0}

2 Quantum Jacobi and quantum modular forms

Quantum Jacobi and quantum modular forms

Let a b
f : H → C, γ := c d ∈ Γ ⊆ SL2(Z), τ ∈ H := {τ ∈ C | Im(τ) > 0}

Modular transformation:  aτ + b  f (τ) − −1(γ)(cτ + d)−k f = 0 cτ + d

2 Quantum Jacobi and quantum modular forms

Quantum modular forms

a b
Let f : Q → C, γ := c d ∈ Γ ⊆ SL2(Z), x ∈ Q.

Modular transformation: ax + b  f (x) − −1(γ)(cx + d)−k f =? cx + d

3 for all a b
γ = c d ∈ Γ, the functions ax + b  h (x) = h (x) := f (x) − −1(γ)(cx + d)−k f γ f ,γ cx + d

extend to suitably continuous or analytic functions in R (or R \ T ).

Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition (Zagier ’10) 1 A quantum modular form of weight k (k ∈ 2 Z) is function f : Q \ S → C, for some discrete subset S, such that

4 ax + b  h (x) = h (x) := f (x) − −1(γ)(cx + d)−k f γ f ,γ cx + d

extend to suitably continuous or analytic functions in R (or R \ T ).

Quantum Jacobi and quantum modular forms

Quantum modular forms

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

4 Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition (Zagier ’10) 1 A quantum modular form of weight k (k ∈ 2 Z) is function f : Q \ S → C, for some discrete subset S, such that for all a b
γ = c d ∈ Γ, the functions ax + b  h (x) = h (x) := f (x) − −1(γ)(cx + d)−k f γ f ,γ cx + d

extend to suitably continuous or analytic functions in R (or R \ T ).

4 Quantum Jacobi and quantum modular forms

Quantum modular forms

Example. Kontsevich’s “strange” function (x ∈ Q):

∞ πix X 12 2πix 2πix
φ(x) := e e ; e n n=0 πix 2πix 2πix 4πix
= e 12 1 + (1 − e ) + (1 − e )(1 − e ) + ···

5 Quantum Jacobi and quantum modular forms

Quantum modular forms

Here, for n ∈ N0,

n−1 Y j 2 n−1 (a; q)n := (1 − aq ) = (1 − a)(1 − aq)(1 − aq ) ··· (1 − aq ). j=0

is the q-Pochhammer symbol.

6 i.e.

3 −1 − 2 φ(x) − ζ24 φ(x + 1) = 0, φ(x) ∓ ζ8|x| φ(−1/x) = h(x),

where h is a real analytic function (except at 0).

Quantum Jacobi and quantum modular forms

Quantum modular forms

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

7 Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (Zagier) The function φ is a quantum modular form of weight 3/2, i.e.

3 −1 − 2 φ(x) − ζ24 φ(x + 1) = 0, φ(x) ∓ ζ8|x| φ(−1/x) = h(x),

where h is a real analytic function (except at 0).

7 2πiτ Let q = e , τ ∈ H. ∞ ∞   2 X 12 n 1 Y n Let η(τ) := n q 24 , η(τ) := q 24 (1 − q ). e n n=0 n=1

Proposition (Zagier) We have that

∞ ∞ n ! X  1  1 1 X q q 24 (q; q) − η(τ) = − η(τ) + η(τ) − . n 2 e 2 1 − qn n=0 n=1

Quantum Jacobi and quantum modular forms

Quantum modular forms

Proof ingredient:

8 ∞ ∞   2 X 12 n 1 Y n Let η(τ) := n q 24 , η(τ) := q 24 (1 − q ). e n n=0 n=1

Proposition (Zagier) We have that

∞ ∞ n ! X  1  1 1 X q q 24 (q; q) − η(τ) = − η(τ) + η(τ) − . n 2 e 2 1 − qn n=0 n=1

Quantum Jacobi and quantum modular forms

Quantum modular forms

2πiτ Proof ingredient: Let q = e , τ ∈ H.

8 Proposition (Zagier) We have that

∞ ∞ n ! X  1  1 1 X q q 24 (q; q) − η(τ) = − η(τ) + η(τ) − . n 2 e 2 1 − qn n=0 n=1

Quantum Jacobi and quantum modular forms

Quantum modular forms

2πiτ Proof ingredient: Let q = e , τ ∈ H. ∞ ∞   2 X 12 n 1 Y n Let η(τ) := n q 24 , η(τ) := q 24 (1 − q ). e n n=0 n=1

8 Quantum Jacobi and quantum modular forms

Quantum modular forms

2πiτ Proof ingredient: Let q = e , τ ∈ H. ∞ ∞   2 X 12 n 1 Y n Let η(τ) := n q 24 , η(τ) := q 24 (1 − q ). e n n=0 n=1

Proposition (Zagier) We have that

∞ ∞ n ! X  1  1 1 X q q 24 (q; q) − η(τ) = − η(τ) + η(τ) − . n 2 e 2 1 − qn n=0 n=1

8 Quantum Jacobi and quantum modular forms

Quantum modular forms

2πiτ Proof ingredient: Let q = e , τ ∈ H. ∞ ∞   2 X 12 n 1 Y n Let η(τ) := n q 24 , η(τ) := q 24 (1 − q ). e n n=0 n=1

Proposition (Zagier) We have that

∞ ∞ n ! X  1  1 1 X q q 24 (q; q) − η(τ) = − η(τ)+ η(τ) − . n 2 e 2 1 − qn n=0 n=1

9 Define

n(n+1) X q 2 X n−1 σ(q) := =: T (n)q 24 , (−q; q)n n≥0 n≥0 n n2 X (−1) q X |n|−1 ∗ 24 σ (q) := 2 2 =: T (n)q . (q; q )n n≥0 n<0 (Andrews, Cohen, Ramanujan)

Quantum Jacobi and quantum modular forms

Quantum modular forms

Example.

10 n(n+1) X q 2 X n−1 σ(q) := =: T (n)q 24 , (−q; q)n n≥0 n≥0 n n2 X (−1) q X |n|−1 ∗ 24 σ (q) := 2 2 =: T (n)q . (q; q )n n≥0 n<0 (Andrews, Cohen, Ramanujan)

Quantum Jacobi and quantum modular forms

Quantum modular forms

Example. Define

10 (Andrews, Cohen, Ramanujan)

Quantum Jacobi and quantum modular forms

Quantum modular forms

Example. Define

n(n+1) X q 2 X n−1 σ(q) := =: T (n)q 24 , (−q; q)n n≥0 n≥0 n n2 X (−1) q X |n|−1 ∗ 24 σ (q) := 2 2 =: T (n)q . (q; q )n n≥0 n<0

10 Quantum Jacobi and quantum modular forms

Quantum modular forms

Example. Define

n(n+1) X q 2 X n−1 σ(q) := =: T (n)q 24 , (−q; q)n n≥0 n≥0 n n2 X (−1) q X |n|−1 ∗ 24 σ (q) := 2 2 =: T (n)q . (q; q )n n≥0 n<0 (Andrews, Cohen, Ramanujan)

10 1
∗ −1 Ex. f k = ζ24k · σ(ζk ) = −ζ24k · σ (ζk ).

Quantum Jacobi and quantum modular forms

Quantum modular forms

For x ∈ Q, define

πix 2πix
πix ∗ −2πix f (x) := e 12 σ e = −e 12 σ (e ).

11 Quantum Jacobi and quantum modular forms

Quantum modular forms

For x ∈ Q, define

πix 2πix
πix ∗ −2πix f (x) := e 12 σ e = −e 12 σ (e ).

1
∗ −1 Ex. f k = ζ24k · σ(ζk ) = −ζ24k · σ (ζk ).

11 i.e.

 x  f (x + 1) = ζ f (x), f (x) − ζ−1(2x + 1)−1f = h(x), 24 24 2x + 1

where h is a real analytic function (except at −1/2).

Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (Zagier) The function f is a quantum modular form of weight 1,

12 Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (Zagier) The function f is a quantum modular form of weight 1, i.e.

 x  f (x + 1) = ζ f (x), f (x) − ζ−1(2x + 1)−1f = h(x), 24 24 2x + 1

where h is a real analytic function (except at −1/2).

12 πi i.e., u(τ + 1) = e 12 u(τ), −1 u = u(τ), 2τ  ∂2 ∂2  1 −y 2 + u(τ) = u(τ). ∂x2 ∂y 2 4

Quantum Jacobi and quantum modular forms

Quantum modular forms

Proof ingredient: Theorem (Cohen) Let τ = x + iy ∈ H. The function   1 X 2π|n|y πinx u(τ) := y 2 T (n)K e 12 0 24 n∈1+24Z is a Maass wave form.

13 Quantum Jacobi and quantum modular forms

Quantum modular forms

Proof ingredient: Theorem (Cohen) Let τ = x + iy ∈ H. The function   1 X 2π|n|y πinx u(τ) := y 2 T (n)K e 12 0 24 n∈1+24Z

πi is a Maass wave form. i.e., u(τ + 1) = e 12 u(τ), −1 u = u(τ), 2τ  ∂2 ∂2  1 −y 2 + u(τ) = u(τ). ∂x2 ∂y 2 4

13 Quantum Jacobi and quantum modular forms

Harmonic Maass forms

Definition (Bruinier-Funke) 1 A harmonic Maass form of weight k ∈ 2 Z on Γ0(4N) is a smooth Mb : H → C satisfying a b
(1) ∀A = c d ∈ Γ0(4N), τ ∈ H,

( c
2k −2k k 1  (cτ + d) Mb(τ), k ∈ − M(Aτ) = d d 2 Z Z b k (cτ + d) Mb(τ), k ∈ Z 2  ∂2 ∂2   ∂ ∂  (2)∆ k Mb = 0, where ∆k := −y ∂x2 + ∂y 2 + iky ∂x + i ∂y (τ = x + iy)

(3) Mb has linear exponential growth at all cusps.

14 Quantum Jacobi and quantum modular forms

Harmonic Maass forms

Decomposition: Mb = M + M−

X n M := cM (n)q “holomorphic part” = mock modular form

n≥rM

− X − n M := cM (n)Γ(1 − k, 4π|n|y)q “non-holomorphic part” n<0

Z ∞ Γ(a, x) := ta−1e−t dt x

15 Example:

∞ 2 X qn q q4 f (q) := = 1 + + + ··· (−q; q)2 (1 + q)2 (1 + q)2(1 + q2)2 n=0 n

Quantum Jacobi and quantum modular forms

Mock theta functions and mock modular forms

Zwegers: Ramanujan’s mock theta functions are mock modular.

16 Quantum Jacobi and quantum modular forms

Mock theta functions and mock modular forms

Zwegers: Ramanujan’s mock theta functions are mock modular.

Example:

∞ 2 X qn q q4 f (q) := = 1 + + + ··· (−q; q)2 (1 + q)2 (1 + q)2(1 + q2)2 n=0 n

16 Quantum Jacobi and quantum modular forms

Quantum and mock modular forms

Question: What is the connection between quantum & mock modular forms?

17 Quantum Jacobi and quantum modular forms

Quantum modular forms

Example. Let q = e2πiτ .

∞ − 1 X 2 n+1 − 1 X m n U(τ) := q 24 (q; q)nq = q 24 u(m, n)(−1) q , n=0 m∈Z n≥1

u(m, n) := #{size n strongly unimodal sequences of rank m}.

18 0 < a1 < a2 < ··· < ar > ar+1 > ··· as > 0 for some r. The rank equals s − 2r + 1 (difference between # terms after and before the “peak”).

Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition s A sequence {aj }j=1 of integers is called strongly unimodal of size n if

a1 + a2 + ··· + as = n,

19 The rank equals s − 2r + 1 (difference between # terms after and before the “peak”).

Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition s A sequence {aj }j=1 of integers is called strongly unimodal of size n if

a1 + a2 + ··· + as = n,

0 < a1 < a2 < ··· < ar > ar+1 > ··· as > 0 for some r.

19 Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition s A sequence {aj }j=1 of integers is called strongly unimodal of size n if

a1 + a2 + ··· + as = n,

0 < a1 < a2 < ··· < ar > ar+1 > ··· as > 0 for some r. The rank equals s − 2r + 1 (difference between # terms after and before the “peak”).

19 Theorem (Bryson-Ono-Pittman-Rhoades) For x ∈ H ∪ Q, the function U satisfies

− 3 U(x) + (−ix) 2 U(−1/x) = g(x),

where g is a real analytic function (except at 0).

Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (Bryson-Ono-Pittman-Rhoades) Let x ∈ Q. We have that

φ(−x) = U(x).

20 Quantum Jacobi and quantum modular forms

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

− 3 U(x) + (−ix) 2 U(−1/x) = g(x),

where g is a real analytic function (except at 0).

20 Quantum Jacobi and quantum modular forms

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

− 3 U(x) + (−ix) 2 U(−1/x) = g(x),

where g is a real analytic function (except at 0).

21 Note. −23πix e 12 F1(x) = φ(x), 23πix e 12 U1(x) = U(x)

Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition (Hikami, Hikami-Lovejoy) For t ∈ N, define t−1   t X Y kj (kj +1) kj+1 Ft (τ) := q (q; q)kt q , kj kt ≥···k1≥0 j=1 q t−1 j−1 2  P  −t X 2 kt Y kj kj+1 + kj − j + 2 s=1 ks Ut (τ) := q (q; q)kt −1q q . kj+1 − kj kt ≥···k1≥1 j=1 q

22 Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition (Hikami, Hikami-Lovejoy) For t ∈ N, define t−1   t X Y kj (kj +1) kj+1 Ft (τ) := q (q; q)kt q , kj kt ≥···k1≥0 j=1 q t−1 j−1 2  P  −t X 2 kt Y kj kj+1 + kj − j + 2 s=1 ks Ut (τ) := q (q; q)kt −1q q . kj+1 − kj kt ≥···k1≥1 j=1 q

Note. −23πix e 12 F1(x) = φ(x), 23πix e 12 U1(x) = U(x)

22 Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (Hikami-Lovejoy) Let k ∈ N. We have that

1
1
Ft − k = Ut k .

Moreover, Ft and Ut are quantum modular forms.

23 Definition Let w = e2πiz , q = e2πiτ . Define the two-variable function

∞ X −1 n+1 X m n U(z; τ) := (wq; q)n(w q; q)nq = u(m, n)(−w) q . n=0 m∈Z n≥1

Note. 1 U(0; τ) = q 24 U(τ).

Quantum Jacobi and quantum modular forms

Quantum modular forms

∞ 1 X 1 X − 24 2 n+1 − 24 m n Previously, U(τ) := q (q; q)nq = q u(m, n)(−1) q . n=0 m∈Z n≥1

24 Define the two-variable function

∞ X −1 n+1 X m n U(z; τ) := (wq; q)n(w q; q)nq = u(m, n)(−w) q . n=0 m∈Z n≥1

Note. 1 U(0; τ) = q 24 U(τ).

Quantum Jacobi and quantum modular forms

Quantum modular forms

∞ 1 X 1 X − 24 2 n+1 − 24 m n Previously, U(τ) := q (q; q)nq = q u(m, n)(−1) q . n=0 m∈Z n≥1 Definition Let w = e2πiz , q = e2πiτ .

24 Note. 1 U(0; τ) = q 24 U(τ).

Quantum Jacobi and quantum modular forms

Quantum modular forms

∞ 1 X 1 X − 24 2 n+1 − 24 m n Previously, U(τ) := q (q; q)nq = q u(m, n)(−1) q . n=0 m∈Z n≥1 Definition Let w = e2πiz , q = e2πiτ . Define the two-variable function

∞ X −1 n+1 X m n U(z; τ) := (wq; q)n(w q; q)nq = u(m, n)(−w) q . n=0 m∈Z n≥1

24 Quantum Jacobi and quantum modular forms

Quantum modular forms

∞ 1 X 1 X − 24 2 n+1 − 24 m n Previously, U(τ) := q (q; q)nq = q u(m, n)(−1) q . n=0 m∈Z n≥1 Definition Let w = e2πiz , q = e2πiτ . Define the two-variable function

∞ X −1 n+1 X m n U(z; τ) := (wq; q)n(w q; q)nq = u(m, n)(−w) q . n=0 m∈Z n≥1

Note. 1 U(0; τ) = q 24 U(τ).

24 Note. −πix e 12 F (0; x) = φ(x).

Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition (F-Ki-TruongVu-Yang) Define the two-variable function ∞ X n+1 F (z; τ) := w (wq; q)n n=0

25 Quantum Jacobi and quantum modular forms

Quantum modular forms

Definition (F-Ki-TruongVu-Yang) Define the two-variable function ∞ X n+1 F (z; τ) := w (wq; q)n n=0

Note. −πix e 12 F (0; x) = φ(x).

25 Moreover, for suitable fixed z, these functions are quantum modular forms when viewed as functions of x ∈ Q.

Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (F-Ki-TruongVu-Yang) Let k ∈ N, and h ∈ Z such that gcd(h, k) = 1. For suitable z ∈ C, h we have for x = k that F (z; −x) = U (z; x) .

26 Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (F-Ki-TruongVu-Yang) Let k ∈ N, and h ∈ Z such that gcd(h, k) = 1. For suitable z ∈ C, h we have for x = k that F (z; −x) = U (z; x) .

Moreover, for suitable fixed z, these functions are quantum modular forms when viewed as functions of x ∈ Q.

26 Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (F-Ki-TruongVu-Yang) Let k ∈ N, and h ∈ Z such that gcd(h, k) = 1 For suitable z ∈ C, h we have for x = k that F (z; −x) = U (z; x) .

Moreover, for suitable fixed z, these functions are quantum modular forms when viewed as functions of x ∈ Q.

27 Quantum Jacobi and quantum modular forms

Quantum modular forms

Question: Can we view U(z; x) as a two variable quantum modular form?

28 hγ(z; τ) := φ(z; τ) 2   −1 −k −2πimcz z aτ + b −  (γ)(cτ + d) e cτ+d φ ; , 1 cτ + d cτ + d

g(λ,µ)(z; τ) := φ(z; τ) −1 2πim(λ2τ+2λz) − 2 (λ, µ)e φ(z + λτ + µ; τ),

2 satisfy a “suitable” property of continuity or analyticity in R .

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Definition (Bringmann-F) 1 1 A weight k ∈ 2 Z and index m ∈ 2 Z quantum Jacobi form is a b
function φ : Q × Q → C such that ∀γ = c d ∈ SL2(Z) and (λ, µ) ∈ Z × Z, the functions

29 g(λ,µ)(z; τ) := φ(z; τ) −1 2πim(λ2τ+2λz) − 2 (λ, µ)e φ(z + λτ + µ; τ),

2 satisfy a “suitable” property of continuity or analyticity in R .

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Definition (Bringmann-F) 1 1 A weight k ∈ 2 Z and index m ∈ 2 Z quantum Jacobi form is a b
function φ : Q × Q → C such that ∀γ = c d ∈ SL2(Z) and (λ, µ) ∈ Z × Z, the functions

hγ(z; τ) := φ(z; τ) 2   −1 −k −2πimcz z aτ + b −  (γ)(cτ + d) e cτ+d φ ; , 1 cτ + d cτ + d

29 2 satisfy a “suitable” property of continuity or analyticity in R .

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Definition (Bringmann-F) 1 1 A weight k ∈ 2 Z and index m ∈ 2 Z quantum Jacobi form is a b
function φ : Q × Q → C such that ∀γ = c d ∈ SL2(Z) and (λ, µ) ∈ Z × Z, the functions

hγ(z; τ) := φ(z; τ) 2   −1 −k −2πimcz z aτ + b −  (γ)(cτ + d) e cτ+d φ ; , 1 cτ + d cτ + d

g(λ,µ)(z; τ) := φ(z; τ) −1 2πim(λ2τ+2λz) − 2 (λ, µ)e φ(z + λτ + µ; τ),

29 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Definition (Bringmann-F) 1 1 A weight k ∈ 2 Z and index m ∈ 2 Z quantum Jacobi form is a b
function φ : Q × Q → C such that ∀γ = c d ∈ SL2(Z) and (λ, µ) ∈ Z × Z, the functions

hγ(z; τ) := φ(z; τ) 2   −1 −k −2πimcz z aτ + b −  (γ)(cτ + d) e cτ+d φ ; , 1 cτ + d cτ + d

g(λ,µ)(z; τ) := φ(z; τ) −1 2πim(λ2τ+2λz) − 2 (λ, µ)e φ(z + λτ + µ; τ),

satisfy a “suitable” property of continuity or analyticity in 2. R 29 and the “error” function

2 Z πiτt −4πzt Z 2 2πt
i ϑ(z; τ) e i πiτt −2πzt sinh 3 H(z; τ) := dt − √ e 3 dt. 2 η(τ) R cosh(πt) 3 R cosh(πt)

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

We define using the unimodal function

+ − 1 Y (z; τ) := −2i sin(πz)q 24 U(z; τ),

30 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

We define using the unimodal function

+ − 1 Y (z; τ) := −2i sin(πz)q 24 U(z; τ),

and the “error” function

2 Z πiτt −4πzt Z 2 2πt
i ϑ(z; τ) e i πiτt −2πzt sinh 3 H(z; τ) := dt − √ e 3 dt. 2 η(τ) R cosh(πt) 3 R cosh(πt)

30 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

2 We define the following subset Q2 of Q  a h   Q := , ∈ 2 : b, k ∈ , gcd(a, b) = gcd(h, k) = 1, b | k . 2 b k Q N

31 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Theorem (Bringmann-F) The following transformation properties hold.

(i) For (z, τ) ∈ (C × H) ∪ Q2, we have that

+ πi + Y (z; τ) − e 12 Y (z; τ + 1) = 0, (1)

2   + 3πiz − 1 + z 1 Y (z; τ) + ie τ (−iτ) 2 Y ; − = −H(z; τ), (2) τ τ Y +(z; τ) + Y +(z + 1; τ) = 0, (3) Y +(z; τ) + e−6πiz−3πiτ Y +(z + τ; τ) (4)

−5πiz− 25πiτ 4πiz+2πiτ
iϑ(z; τ) −2πiz− πiτ −6πiz− 9πiτ  = e 12 1 − e − e 4 −e 4 . η(τ)

32 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Here,

2 Z πiτt −2πzt Z 2 2πt
i ϑ(z; τ) e i πiτt −2πzt sinh 3 H(z; τ) := dt−√ e 3 dt. 2 η(τ) R cosh πt 3 R cosh(πt)

33 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Theorem (Bringmann-F) The following transformation properties hold.

(i) For (z, τ) ∈ (C × H) ∪ Q2, we have that

+ πi + Y (z; τ) − e 12 Y (z; τ + 1) = 0, (5)

2   + 3πiz − 1 + z 1 Y (z; τ) + ie τ (−iτ) 2 Y ; − = −H(z; τ), (6) τ τ Y +(z; τ) + Y +(z + 1; τ) = 0, (7) Y +(z; τ) + e−6πiz−3πiτ Y +(z + τ; τ) (8)

−5πiz− 25πiτ 4πiz+2πiτ
iϑ(z; τ) −2πiz− πiτ −6πiz− 9πiτ  = e 12 1 − e − e 4 −e 4 . η(τ)

34 The function on the right-hand side of (9) extends to a C ∞ × function on (R \ (±1/6 + Z)) × R , and the function on the ∞ 2 right-hand-side of (10) extends to a C function on R .

Quantum Jacobi and quantum modular forms

Theorem (Bringmann-F, cont.)

(ii) In particular, for (z, τ) ∈ Q2, we have that

2   + 3πiz − 1 + z 1 Y (z; τ) + ie τ (−iτ) 2 Y ; − (9) τ τ

Z 2 2πt
i πiτt −2πzt sinh 3 = √ e 3 dt, 3 R cosh(πt) Y +(z; τ) + e−6πiz−3πiτ Y +(z + τ; τ) (10) −5πiz− 25πiτ 4πiz+2πiτ
= e 12 1 − e .

35 Quantum Jacobi and quantum modular forms

Theorem (Bringmann-F, cont.)

(ii) In particular, for (z, τ) ∈ Q2, we have that

2   + 3πiz − 1 + z 1 Y (z; τ) + ie τ (−iτ) 2 Y ; − (9) τ τ

Z 2 2πt
i πiτt −2πzt sinh 3 = √ e 3 dt, 3 R cosh(πt) Y +(z; τ) + e−6πiz−3πiτ Y +(z + τ; τ) (10) −5πiz− 25πiτ 4πiz+2πiτ
= e 12 1 − e .

The function on the right-hand side of (9) extends to a C ∞ × function on (R \ (±1/6 + Z)) × R , and the function on the ∞ 2 right-hand-side of (10) extends to a C function on R .

35 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Theorem (Bringmann-F, cont.) In particular, Y +(z; τ) is a quantum Jacobi form of weight 1/2 and index −3/2 on SL2(Z).

36 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Theorem (Bringmann-F)

Let τ = h/k and z = a/b be such that (z, τ) ∈ Q2. Then

Y + (z; τ) k−1 1 X  h(2j+1) 2 = − ζ−5aζ−25h (−1)j+1ζ−5hj 1 − ζ2aζ ζ−3jaζ−3j h. 2 2b 24k 2k b k b 2k j=0

37 R(w; q) = partition rank g.f., C(w; q) = partition crank g.f.

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Corollary (Bringmann-F) Let 1 ≤ a < b, 1 ≤ h < k with gcd(a, b) = gcd(h, k) = 1, b|k and 0 0 h h ∈ Z with hh ≡ −1 (mod k). Then, as q → ζk radially within the unit disc, we have that

 a −a2h0k a  lim R (ζ ; q) − ζ 2 C (ζ ; q) h b b b q→ζk k−1 1 X  h(2j+1) 2 = 1 − ζ−a
ζ−2aζ−h (−1)j+1ζ−5hj 1 − ζ2aζ ζ−3jaζ−3j h. 2 b b k 2k b k b 2k j=0

38 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Corollary (Bringmann-F) Let 1 ≤ a < b, 1 ≤ h < k with gcd(a, b) = gcd(h, k) = 1, b|k and 0 0 h h ∈ Z with hh ≡ −1 (mod k). Then, as q → ζk radially within the unit disc, we have that

 a −a2h0k a  lim R (ζ ; q) − ζ 2 C (ζ ; q) h b b b q→ζk k−1 1 X  h(2j+1) 2 = 1 − ζ−a
ζ−2aζ−h (−1)j+1ζ−5hj 1 − ζ2aζ ζ−3jaζ−3j h. 2 b b k 2k b k b 2k j=0

R(w; q) = partition rank g.f., C(w; q) = partition crank g.f.

38 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Corollary (Bringmann-F)

For τ = h/k and z = a/b such that (z, τ) ∈ Q2, the Eichler integral Z 2 2πt
i πiτt −2πzt sinh 3 √ e 3 dt 3 R cosh(πt) a h is given as an explicit 2-variable polynomial in (ζb , ζk ).

39 Note. V is a rank g.f. for odd balanced unimodal sequences. W is a g.f. for partitions without repeated odd parts.

Quantum Jacobi and quantum modular forms

Combinatorial q-series

Definitions (Kim-Lim-Lovejoy)

∞ −1 n X (−wq; q)n(−qw ; q)nq V(z; τ) := (q; q2) n=0 n+1 ∞ 2 −1 2 2n X (wq; q )n(w q; q )nq W(z; τ) := (−q; q) n=0 2n+1

40 Quantum Jacobi and quantum modular forms

Combinatorial q-series

Definitions (Kim-Lim-Lovejoy)

∞ −1 n X (−wq; q)n(−qw ; q)nq V(z; τ) := (q; q2) n=0 n+1 ∞ 2 −1 2 2n X (wq; q )n(w q; q )nq W(z; τ) := (−q; q) n=0 2n+1

Note. V is a rank g.f. for odd balanced unimodal sequences. W is a g.f. for partitions without repeated odd parts.

40 Quantum Jacobi and quantum modular forms

Quantum modular forms

Theorem (Kim-Lim-Lovejoy) 14πix 1 6πix The functions v(x) := e V( 2 ; 8x) and w(x) := e W(0; x) are quantum modular forms.

41 Note. As was the case with Y +, V+ and W+ exhibit suitable transformation properties on( C × H) ∪ Q2, where Q2 ⊆ Q × Q.

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Theorem (Barnett-F-Ukogu-Wesley-Xu) The functions

+ 7 V (z; τ) = 2 cos(πz)q 8 V(w; q),

and + 3 1 W (z; τ) := 2 cos(πz)q 8 W(w; q 2 ), 1 1 are quantum Jacobi forms of weight 2 and index − 2 on Γ0(4) and Γ0(2), respectively.

42 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Theorem (Barnett-F-Ukogu-Wesley-Xu) The functions

+ 7 V (z; τ) = 2 cos(πz)q 8 V(w; q),

and + 3 1 W (z; τ) := 2 cos(πz)q 8 W(w; q 2 ), 1 1 are quantum Jacobi forms of weight 2 and index − 2 on Γ0(4) and Γ0(2), respectively.

Note. As was the case with Y +, V+ and W+ exhibit suitable transformation properties on( C × H) ∪ Q2, where Q2 ⊆ Q × Q.

42 Note. As was the case with Y +, we obtain explicit evaluations of (2-variable) Eichler integrals at pairs of rationals as (2-variable) polynomials in roots of unity.

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Theorem (Barnett-F-Ukogu-Wesley-Xu) a h
For b , k ∈ Q2 ⊂ Q × Q, we have that

k−1 1 X −hj(1+j) a(k−j) V+ a ; h
= − ζ−aζ7h (−1)j ζ ζ , b k 2 2b 8k 2k b j=0 k + a h
X j+1 a(2j−1) h(2j−1)(4k−2j+1) W b ; k = (−1) ζ2b ζ8k , j=1

2πi where ζN := e N .

43 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Theorem (Barnett-F-Ukogu-Wesley-Xu) a h
For b , k ∈ Q2 ⊂ Q × Q, we have that

k−1 1 X −hj(1+j) a(k−j) V+ a ; h
= − ζ−aζ7h (−1)j ζ ζ , b k 2 2b 8k 2k b j=0 k + a h
X j+1 a(2j−1) h(2j−1)(4k−2j+1) W b ; k = (−1) ζ2b ζ8k , j=1

2πi where ζN := e N .

Note. As was the case with Y +, we obtain explicit evaluations of (2-variable) Eichler integrals at pairs of rationals as (2-variable) polynomials in roots of unity. 43 Analytic continuation: transformation properties C × H → Q × Q. Elliptic Quantum Jacobi transformation properties: ⇒ functional equation ⇒ polynomial evaluations.

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Proof ingredients.

Re-write functions in terms of Appell-lerch sums A` (Zwegers, Mortenson).

44 Elliptic Quantum Jacobi transformation properties: ⇒ functional equation ⇒ polynomial evaluations.

Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Proof ingredients.

Re-write functions in terms of Appell-lerch sums A` (Zwegers, Mortenson). Analytic continuation: transformation properties C × H → Q × Q.

44 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Proof ingredients.

Re-write functions in terms of Appell-lerch sums A` (Zwegers, Mortenson). Analytic continuation: transformation properties C × H → Q × Q. Elliptic Quantum Jacobi transformation properties: ⇒ functional equation ⇒ polynomial evaluations.

44 Quantum Jacobi and quantum modular forms

Quantum Jacobi Forms

Thank you

45