Quantum Mock Modular Forms Arising From eta-theta Functions

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1 Quantum Mock Modular Forms Arising From eta-theta Functions Holly Swisher CTNT 2016 Joint with Amanda Folsom, Sharon Garthwaite, Soon-Yi Kang, Stephanie Treneer (AIM SQuaRE) and Brian Diaz, Erin Ellefsen (NSF REU) August 12, 2016

2 Mock Modular Forms Ramanujan s last letter to Hardy indicated a fascination wth what he called mock theta functions, whose modular properties remained unknown for nearly a century. Zwegers showed that these (holomorphic) mock theta functions could be completed to form a harmonic Maass form, which is modular but no longer holomorphic. Brunier and Funke defined these harmonic Maass forms, which in addition to being modular, also satisfy bounded growth conditions and are annihilated by a weighted Laplacian. Zagier defined mock modular form to mean a holomorphic part of a harmonic Maass form.

3 Zwegers and shadows For τ H and u, v C\(Zτ + Z), Zwegers defines µ(u, v; τ) := eπiu ϑ(v; τ) n Z ( 1) n e 2πinv q n(n+1) 2 1 e 2πiu q n, where ϑ(v; τ) is the weight 1/2 theta function given by ϑ(v; τ) := n Z e 2πi(n+ 1 2 )(v+ 1 2 ) q 1 2 (n+ 1 2 )2. Zwegers shows that µ(u, v; τ) can be completed to form a non-holomorphic, two-variable modular Jacobi form µ(u, v; τ) of weight 1/2.

4 Current landscape of mmf research Since the fundamental work of Zwegers, there has been an incredible blossoming of new work on mock modular forms, including connections to Partition Theory (Garvan, Jennings-Shaffer, Bringmann, Ono, and others) Geometry and p-adic analysis (Candelori, Castella) Moonshine (Duncan, Griffin, Ono) Plus much more!

5 Shadows A harmonic Maass form f of weight κ is mapped to a classical modular form of weight 2 κ by the differential operator ξ κ := 2iy κ τ The image of f under ξ κ is called the shadow of f. Our work was motivated by an interest in relating shadows and mock modular forms.

6 Work of Zwegers For appropriate choices of u, v, a, b the Jacobi form µ(u, v; τ) has shadow related to the unary theta function defined for τ H by g a,b (τ) := n Z (n + a)e 2πib(n+a) q (n+a)2 2. In particular, when a and b are rational g a,b is a modular form of weight 3/2.

7 Work of Lemke Oliver Lemke Oliver establishes which eta-quotients of weight 1/2 and 3/2 are theta functions (or linear combinations). Weight 3/2: E 1 (τ) = η(8τ) 3 = ( ) 4 nq n2, n n 1 η(16τ) 9 E 2 (τ) = η(8τ) 3 η(32τ) 3 = ( ) 2 nq n2, n n 1 E 3 (τ) = η(3τ)2 η(12τ) 2 = ( n ) nq n2, η(6τ) 3 n 1 η(48τ) 13 ( ) 6 E 4 (τ) = nq n2, η(24τ) 5 η(96τ) 5 = n n 1 E 5 (τ) = η(24τ)5 η(48τ) 2 = ( n 12 n 1 ( 2 E 6 (τ) = η(6τ)5 η(3τ) 2 = n 1 ) nq n2, ( n 12) ( n 3 )) nq n2.

8 Constructing Examples For τ H, we define for a, b R and u, v C \ (Zτ + Z) the function M a,b (τ) := 2e 2πia(b+ 1 2) q a2 2 µ(u, v; τ). (1) Proposition Let τ H, and u, v C \ (Zτ + Z). If u v = aτ b for some a, b R, then the function M a,b (τ) satisfies (i) ξ 1 2 ( Ma,b (τ)) (ii) 1 ( M a,b (τ)) = 0. 2 = g c a+ 1 2,b+ 1 2 (τ), We denote the holomorphic part of M a,b by M a,b : M a,b (τ) := 2e 2πia(b+ 1 2) q a2 2 µ(u, v; τ).

9 Constructing Examples We can express each weight 3/2 eta-theta E m in term of Zwegers g a,b functions. For example, E 1 (τ) = 4g 1 4,0(32τ). We can express each weight 1/2 eta-theta e n in terms of ϑ(v; τ). For example, ϑ( τ 2 ; τ) = iq 1 8 e 1 ( τ 2 ). Thus for each pair E m, e n (up to a few exceptions) we can construct a holomorphic function V mn = µ(u, v; τ) that includes the factor e n such that u v = aτ b so that the shadow is related to E m (up to normalizations).

10 Mock Modular Result Theorem (Folsom, Garthwaite, Kang, S-, Treneer) The functions V mn are mock modular forms of weight 1/2 with respect to the congruence subgroups A mn. Moreover, the shadow of V mn is given by a constant multiple of the odd eta-theta ). In particular, the functions V mn may be ( function E 2τ m c 2 m completed to form harmonic Maass forms V mn of weight 1/2 on A mn, which satisfy for all γ mn = ( amn b mn c mn d mn ) A mn, and τ H, V mn (γ mn τ) = ν(γ mn ) 3 (c mn τ + d mn ) 1 2 Vmn (τ), where is an explicit root of unity, and ν(γ mn ) is the usual eta-function multiplier.

11 Table for m = 1 Table: Mock theta functions with normalized shadow E 1 (τ). ( ) V 1n (τ) Series w 1 q t1 µ u (1n) τ, v τ (1n) ; τ q 9/32 ( 1) n q (n+1)2 /2 V 11 (τ) q 1/32 µ( τ e 1 (τ/2) 1 + q n+1/ , τ 2 ; τ) V 12 (τ) V 13 (τ) V 14 (τ) V 15 (τ) n Z q 9/32 e 2 (τ/2) q 9/32 e 3 (τ/72) n Z q 9/32 e 4 (τ/72) q 9/32 e 5 (τ/32) n Z q (n+1)2 /2 1 q n+1/4 q 1/32 µ( τ 4, τ ; τ) ( 1) n q (n+5/6)2 /2 1 + q n+1/12 q 1/32 µ( τ , τ 3 ; τ) n Z q (n+5/6)2 /2 1 q n+1/12 q 1/32 µ( τ 12, τ ; τ) ( 1) n q (n+3/4)2 /2 1 + q n q 1/32 µ( 1 2, τ 4 ; τ) n Z V 16 (τ) q 1/32 µ(0, τ ; τ)

12 Comparison with known Mock Theta Functions q 1/24 V 41 (12τ) = ψ(q), q 2/3 V 64 (6τ) = ρ(q), q 1/8 V 12 (4τ) = U 1 (q), q 1/24 V 58 (3τ) = χ(q), q 1/8 V 21 (4τ) = A(q), 2q 1/8 V 15 (4τ) = U 0 (q), where ψ(q), χ(q), and ρ(q) are Ramanujan s third order mock thetas, A(q) is Ramanujan s second order mock theta, and U 1 (q) and U 0 (q) are Gordon and McIntosh s eighth order mock thetas.

13 Quantum Sets We call a subset S Q a quantum set for a function F with respect to the group G SL 2 (Z) if both F (x) and F (Mx) exist (are non-singular) for all x S and M G.

14 Quantum Modular Forms For k 1 2Z, a quantum modular form of weight k on the set S for the group G is a complex-valued function f such that S is a quantum set for f with respect to the group G SL 2 (Z), and for all γ = ( a b c d ) G, and for all x S (x d c h f,γ (x) := f(x) ɛ(γ)(cx + d) k f ), the functions ( ax + b cx + d are suitably continuous or analytic in (a subset of) R. We will take this to mean real analytic. (The ɛ(γ) are appropriate complex numbers, like those in the theory of half-integer weight modular forms.) )

15 Current landscape of qmf research Zagier first defined quantum modular forms in 2010 and demonstrated some interesting examples Quantum modular forms have naturally arisen in several places in work by Andrews, Dyson, Hickerson; Bryson, Ono, Pitman, Rhoades; Bringmann, Creutzig, Rolen; Folsom, Ono, Rhoades; as well as others. They arise, for example, in the study of Kontsevich s strange function, unimodal sequences, and false theta functions.

16 Quantum Modular Result Theorem (Folsom, Garthwaite, Kang, S-, Treneer) The functions V mn are quantum modular forms of weight 1/2 on the sets S mn for the groups G mn. In particular, (i) For all x H S mn \ { } 1 2, V mn (x) + ζ4 lm (2x + 1) 1 2 Vmn (M 2 x) = i ( i E 2u ) m c 2 m du. c m 1 2 i(u + x) (ii) For all x H S mn, V mn (x) ζ κmn a m V mn (x + κ mn b m ) = 0.

17 Curious Application We obtain interesting corollaries which give closed expressions for the Eichler integrals of the eta-theta functions E m in terms of the following truncated q-hypergeometric series F h,k (z 1, z 2 ) := k 1 n=0 ( ζ2k h ; ζh 2k ) nζ n(n+1)h 4k (z 1 ; ζ2k h ) n+1(z 2 ; ζ2k h ). n+1 A particularly pretty result is that for m {1, 2, 5}, F h,k ( i lm 3 ζ h c mk, i3 lm ζ dmh a mk ) + F h,k(i lm 3 ζ h c mk, i3 lm ζ dmh a mk ) = 0.

18 Example of Corollaries The Eichler integral of the eta-quotient E 1 may be evaluated as i 8 i 1/2 E 1 (z/32)dz i(z + 13 ) = ζ 7 ( ) ζ n= n=0 ( ζ 6 ; ζ 6 ) n ζ n(n+1) 12 (iζ 24 ; ζ 6 ) n+1 ( iζ 8 ; ζ 6 ) n+1 ( ζ 10 ; ζ 10 ) n ζ n(n+1) 20 (iζ 40 ; ζ 10 ) n+1 ( iζ 3 40 ; ζ 10) n i.

19 Example of Corollaries We also have the following curious algebraic identity. 2 n=0 ( ζ 6 ; ζ 6 ) n ζ n(n+1) 12 (iζ 24 ; ζ 6 ) n+1 ( iζ 8 ; ζ 6 ) n n=0 ( ζ 6 ; ζ 6 ) n ζ n(n+1) 12 ( iζ 24 ; ζ 6 ) n+1 (iζ 8 ; ζ 6 ) n+1 = 0. While the above may appear elementary, term by term the two sums appearing are quite different.

20 Lower Half Plane It is natural to ask if the functions V mn also extend into the lower half-plane H := {z C Im(z) < 0}. For z H, define Ẽ 1 (z) = ( 4 n n 1 Ẽ 2 (z) = ( 2 n n 1 Ẽ 3 (z) = n 1 ) e 2πizn2, ) e 2πizn2, ( n 3 ) e 2πizn2, Ẽ 4 (z) = n 1 Ẽ 5 (z) = n 1 Ẽ 6 (z) = n 1 ( ) 6 e 2πizn2, n ( n e 12) 2πizn2, ( ( n ( n )) 2 e 12) 2πizn2. 3

21 Lower Half Plane Proposition The functions Ẽm are quantum modular forms of weight 1/2. In particular, for any x S mn, up to multiplication by a constant, the functions Ẽm ( 2x/c 2 m ) satisfy the transformation laws given in the quantum modularity theorem for the functions V mn (x).

22 Outline of Proof when n = 1 Theorem (Kang) If α C such that α 1 2 Zτ + 1 2Z, then µ (2α, τ ) 2 ; τ = iq 1 8 g2 (e(α); q 1 1 η(τ) 4 2 ) e( α)q 8 η( τ 2 )2 ϑ(2α; τ), where g 2 is the universal mock theta function defined by g 2 (z; q) := n=0 ( q) n q n(n+1)/2 (z; q) n+1 (z 1 q; q) n+1.

23 Outline of Proof when n = 1 Using Kang s theorem, determine a quantum set S m1 and group G m1 for which V m1 is well-defined for all x S m1, and Mx for any M G m1. Use Zwegers transformations for µ on H to obtain explicit transformation properties for V m1 on H with respect to G m1. A couple of error terms will arise. Using Zwegers, convert the error terms to Eichler integrals and see them beautifully reduce to a single integral (when set up properly). Put everything together to obtain explicit results in quantum modularity theorem.

24 Definitions for Quantum Sets S := {h/k Q h Z, k N, gcd(h, k) = 1, h 1 (mod 2)} S := {h/k S h ±1 (mod 6)} S ev := {h/k S k 0 (mod 2)} S od := {h/k S k 1 (mod 2)}

25 Quantum Sets S 11, S 17, S 21, S 27, S 41, S 45, S 47 := S S 12, S 18, S 22, S 28, S 42, S 52 := S ev S 13, S 23, S 31, S 34, S 35, S 36, S 43, S 53, S 61, S 63, S 64, S 65, S 66 := S S 14, S 15, S 24, S 26 := S od S 32, S 33, S 62 := S S ev S 37, S 68 := S S od S 44 := S S od S 46, S 48, S 51, S 55, S 56, S 57, S 58 := S S ev.

26 Groups for Quantum Modularity G 12, G 15, G 22, G 26 := ( ), ( ) Γ 0(2) Γ 0 (4), G 13, G 14, G 17, G 18, G 23, G 24, G 27, G 28, G 35, G 36, G 4n, G 55, G 56, G 65, G 66 := ( ), ( ) Γ 0(2) Γ 0 (12), G 32, G 33, G 34, G 37, G 52, G 53, G 57, G 58, G 62, G 63, G 64, G 68 := ( ), ( ) Γ 0(2) Γ 0 (6).

27 Question for REU Students Are there patterns here for the quantum sets and groups appearing? How special is it that these V mn functions are quantum modular? Could one prove a more general result that captures the quantum modularity properties shown for this catalog?

28 REU Definition of Quantum Sets Define S := { h k } Q h Z, k N, gcd(h, k) = 1, and h odd. Given α = A 2C τ + a 4, where 0 a 3, gcd(a, C) = 1, and 0 < A C < 1, we define S α = {{ h k S C h} if a even, { h k S C 2h} { h k S C h and k even} if a odd.

29 REU Definition of Quantum Groups Recall α = A 2C τ + a 4. We define ( ) ( ) C,, if a and C even ( 1 1 ) ( 0 1 ) C,, if a odd and C even G α = ( ) ( ) C,, if a even and C odd ( 1 1 ) ( 0 1 ) C,, if a and C odd

30 REU Results Theorem (Diaz, Ellefsen, S-) Let α = A 2C τ + a 4, as before. Then the functions V α (τ) := i a+1 q (2A C)2 8C 2 µ (2α, τ ) 2 ; τ. are quantum modular forms on the sets S α for the groups G α, with explicit transformations that hold on all of H S α.

31 REU Results For example, when a is odd, we have that for all τ H S α, V α (τ) (2τ + 1) 1 2 Vα (M 2 τ) = i 2 where M 2 = ( ). i 1 2 g A C,0(z) i(z + τ) dz,

32 Further Work and Open Questions It remains to obtain explicit transformations for V α and show it is mock modular; this is work in progress. It may be possible to do a similar shifting process to obtain results for a wider scope of functions which encompass the entire table from Folsom, et al. Further generalization seems possible perhaps sacrificing some optimization of quantum sets and groups. A truly general result seems out of reach due to an expansion of cases based on divisibility.

33 Thank you!