FURTHER TYPES OF INDEFINITE THETA FUNCTIONS AND
THEIR CONNECTIONS TO MAASS WAVE FORMS, QUANTUM
MODULAR FORMS, AND KNOT THEORYLARRY ROLEN

1. Indefinite theta series related to Maass waveforms
In the last three lectures, we saw many examples of theta series for general indefinite
quadratic forms. These were constructed to be completions of holomorphic functions(for example, built out of products of differences of sign functions) which arise in many
natural areas of combinatorics, representation theory, and physics. Here, we will describe
another type of indefinite theta function. Although it looks at first glance to be similar
to the functions in Zwegers’ thesis, we will see that it has rather different modularcompletions.
The first motivating example is the following q-hypergeometric series from Ramanujan’s Lost Notebook:σ(q) :=X q n(n+1)
2n≥0(−q)n=:

XS(n)q n ,n≥0

Andrews, Dyson, and Hickerson showed that the coefficients S(n) satisfy striking properties; in particular, they showed that lim sup |S(n)| = ∞, but at the same time S(n) = 0
for infinitely many n. Their proof hinged on the fact that σ can be written as anindefinite theta series:
XX
n+j 3 (n+1/6)2 −j 2(1)σ(q) = +.
 (−1) q 2n+j≥0n−j≥0n+j<0n−j<0This looks very similar to the holomorphic parts of type (1, 1) indefinite
theta series fromP
P
Zwegers’ thesis. Indeed, if we change a single sign and replace
byn+j<0n+j≥0 +n−j<0n−j≥0
PP
n+j<0 , we essentially get a sixth-oder mock theta function of Ramanujan.
n+j≥0 −n−j≥0

n−j<0

In Section 2, we will describe an alternative theory, also due to Zwegers, for suchindefinite theta series. The first instance of such a theory was given by Cohen. He used12

LARRY ROLEN

σ, together with its complementary function
X (−1)n q n2?
σ (q) := 2,2)
(q;
qnn≥1
to explicitly build a Maass waveform. Namely, he showed that if (n ∈ 24Z + 1)
XX
T (n)q −n ,T (n)q n ,q −1 σ ? q 24 =:qσ q 24 =:n<0n≥0then (where K0 is a Bessel function, τ = x + iy)  X
12π|n|ynxu(τ ) := y 2T (n)K0
e2424n∈24Z+1
transforms as a modular form on the congruence subgroup Γ0 (2). and is an eigenfunctionof the hyperbolic Laplacian ∆0 . with eigenvalue 1/4.
Further examples of functions analogous to σ(q) were constructed by a number ofauthors, such as Corson-Favero-Liesinger-Zubairy, Bringmann-Kane. and Lovejoy. Li,Ngo, and Rhoades later studied many properties of these functions, and conjectured
that they all can be related to Maass waveforms like Cohen showed for σ(q) (the maindifficulty being that Cohen’s method involved finding certain Hecke characters whichbecomes computationally infeasible for most examples). This was proven by Krauel,
Woodbury, and the author by using Zwegers’ theory of indefinite theta series like (1)(cf. Section 2).
It turns out that these examples also all fit into several natural, infinite families ofq-hypergeometric series. This realization makes use of polynomials Hn (k, `; b; q) (definedin (2)), which were studied in relation to torus knots and quantum modular forms (which
are also described in Section 3). Specifically, the n-th coefficient in Habiro’s cyclotomic
expansion of the colored Jones polynomial of the left-handed torus knot T (2, 2k + 1) was
shown to be q n+1−k Hn+1 (k, 1; 1; q), and the general Hn (k, `; 1; q) were used to constructa class of q-hypergeometric series with interesting behavior both at roots of unity andinside the unit circle. These functions, denoted by Fj (k, `; q) (j ∈ {1, 2, 3, 4}), are built
from Hikami-Lovejoy’s polynomials(2)Pjk−1XY 2(2nr + χ`>r )nj +(1−b)nj nj+1 − nj − bj +r=1Hn (k, `; b; q) :=q
nj+1 − njqn=nk ≥nk−1 ≥···≥n1 ≥0 j=1

and, for example,F1 (k, `; q) :=X

n+1(q)n (−1)n q ( 2 ) Hn (k, `; 0; q).

n≥0The relationship between thees series involves the following “period integral” map (cf.
Exercise 3 for an explanation of this terminology), which was studied by Lewis andOTHER INDEFINITE THETA FUNCTIONS AND QUANTUM MODULAR FORMS

3Zagier and used by Zagier to show that functions like σ(q) are quantum modular forms(cf. Section 3). Specifically, if f a Maass waveform with eigenvalue 1/4 on a congruencesubgroup which is cuspidal at i∞, it has a Fourier expansion of the form  X
1
2π|n|vnu
f (τ ) =: v 2
A(n)K0e,NN
n6=0and we let(3)f + (τ ) :=X

nA(n)q N .

n>0Theorem (Bringmann-Lovejoy-R.). The functions Fj k, `; q d are the images of Maasscusp forms under the map (3).
Remark. The cuspidality of the associated Maass waveforms isn’t obvious. In fact, thespecific shapes of the q-hypergeometric representations of F are essential.
The idea behind the proof is that after using the Bailey pair machinery on the qhypergeometric series defining F1 (k, `; q), such functions become indefinite theta functions of the shape discussed in the next section. A specific example of applying Zwegers’
framework to functions like Fj is sketched in Exercises 4 and 5.
2. Zwegers’ Mock Maass Theta Functions
Zwegers beautifully contextualized Cohen’s observations using indefinite theta series
with a similar flavor as the ones in his thesis. As per our discussions in the last twolectures, finding modular completions for these series boils down to finding nice functions
which satisfy Vignéras’ differential equation, include the desired sums of sign functions
as canonical pieces, and which yield convergent series. Of course, since the sign of series
like (1) is flipped, relative to those in Zwegers’ thesis, we require different functions forconvergence.
As in Zwegers’ thesis, the general situation begins by taking a pair of vectors c1 , c2 ∈CQ . Then Q splits over R as a product of linear factors Q(r) = Q0 (P r) for some (nonunique) P ∈ GL2 (R); namely, Q0 (r) := r1 r2 . Then, for each c ∈ CQ , there is a uniquet ∈ R such that
et−1c = c(t) := P.
−e−tt We also set c⊥ = c⊥ (t) := P −1 ee−t . The following function then takes the role that
the E function played for our original type (1, 1) indefinite theta series when we applyVignéras’ theorem (you will be asked to show this in Exercise 1):
Z t22(4)p(v) :=e−πB(v,c(t)) dt.
t1

4LARRY ROLEN

Using this function, Zwegers was able to complete the indefinite theta series
X
1
1 ,c2
Φa,b (τ ) = Φca,b(τ ) : = sgn(t2 − t1 )v 2
ρA (r)e(Q(r)u + B(r, b))K0 (2πQ(r)v)
r∈a+Z21

X+ sgn(t2 − t1 )v 2

ρ⊥
A (r)e(Q(r)u + B(r, b))K0 (−2πQ(r)v),r∈a+Z2where1
ρA (r) := ρcA1 ,c2 (r) := (1 − sgn (B (r, c1 ) B (r, c2 ))) ,2c⊥ ,c⊥1 2and ρ⊥.
A := ρA
Assuming convergence, it is immediate from the differential equation satisfied by K0
that Φa,b is an eigenfunction of ∆0 with eigenvalue 1/4. Zwegers’ modular completionof Φa,b splits in terms of the functions
 1
X
1ϕca,b (τ ) := v 2
αt rv 2 q Q(r) e(B(r, b)),r∈a+Z2whereαt (r) :=Z ∞2
e−πB(r,c(x)) dx tZt2

e−πB(r,c(x)) dx−


−∞
0


if B (r, c) B r, c⊥ > 0,
if B (r, c) B r, c⊥ < 0,otherwise.

Zwegers then showed thatb c1 ,c2 := Φc1 ,c2 + ϕc1 − ϕc2Φ
a,ba,ba,ba,btransforms like a modular form. Conveniently, the functions ϕc satisfy nice relationships
under the action of matrices fixing the quadratic form Q, and this can be used in manyb cancel out. In general,
examples to show that the differences of two ϕ functions in Φ
this is no longer an eigenfunction for ∆0 , and hence not a Maass waveform. However,there is still a “mock” picture. Recall that completing indefinite theta functions of type(r − 1, 1) forces one to sacrifice holomorphicity (and so are no longer in the kernel of theshadow operator ξk ), but that the images under ξk of these indefinite theta functions areb also loses its eigenvalue property“nicer.” In this situation, the completed function Φ
under a differential operator (∆0 ), but Zwegers showed that there is a “shadow” definedusing the fact that1 b c1 ,c2− 23v∆0 −Φa,b ∈ S 3 ⊗ S 3 .
224

OTHER INDEFINITE THETA FUNCTIONS AND QUANTUM MODULAR FORMS

53. Quantum Modular Forms
We conclude this series of lectures by giving one more way in which the above indefinite
theta functions are related to modular-type objects. This is facilitated by the seeminglyinnocuous map (3). It turns out that this map is a special kind of integral transform, notunlike “period integrals” which allow one to build non-holomorphic parts of harmonic
Maass forms by integrating their shadows. If one carefully filters out what remnants ofmodularity survive this integral transformation, one discovers examples of new objects
known as quantum modular forms. These were recently defined by Zagier and have beenconnected to many important combinatorial generating functions, as well as to knot and3-manifold invariants. Roughly speaking, a quantum modular form is a function which
is defined on Q and which fails to transform modular by a particularly “nice” function.
“Definition”. A quantum modular form of weight k is a function f : Q → C suchthat for all γ in a congruence subgroup Γ, the “error of modularity” cocycle
f |k (1 − γ)
is “nice.”
Clearly, this isn’t a well-defined notion as stated. However, Zagier intentionally left
the definition open-ended to include a wide variety of different examples, and quantum
modular forms are something which you “know when you see them.” To paraphrase
Tolstoy, all modular-type functions (modular forms, harmonic Maass forms, etc.) arealike; each quantum modular form is quantum modular in its own way.
Zagier showed that the image of Maass waveforms under the map (3) are indeed
quantum modular forms. This uses the formulation of this map as a special integral
transform, and this transform and the resulting quantum modular cocycle relations arestated in Exercise 3. In the case of σ, σ ? (as well as for the families Fj above using thespecial properties of the polynomials of Hikami and Lovejoy), it is possible to explicitly
see the values of these quantum modular forms as finite sums. To see this, we first need
the identities due to Andrews and Cohen (respectively):(5)

σ(q) = 1 +∞Xn n+1(−1) q(q)n ,?

σ (q) = −2

n=0∞
X


q n+1 q 2 ; q 2 n .

n=0

The right hand sides of these two identities converge not only when |q| < 1 but also atroots of unity (as eventually (q; q)n becomes 0. Cohen further showed that (cf. Exercise2)
(6)σ(ζ) = −σ ? ζ −1 .
Zagier then proved the following transformation properties for function (x ∈ Q)
πix

πix

f (x) := e 12 σ(e2πix ) = −e 12 σ ? (e−2πix ).

6LARRY ROLEN

Theorem (Zagier). The function f : Q → C satisfies
f (x + 1) = ζ24 f (x),−1f (x) − ζ24(2x + 1)−1 fx2x + 1= h(x),

where h : R → C is C ∞ on R, and analytic on R \ {−1/2}. In particular, f is a quantum
modular form of weight 1 on the subgroup of SL2 (Z) generated by ( 10 11 ) and ( 12 01 ).
It turns out that related quantum modularity results also hold for period integralsarising in the theory of mock modular forms. The interested reader is referred to Zagier’s
original survey “Quantum Modular Forms” for a description of other such types ofquantum modular forms and their relations to period integrals, Eichler integrals, andknot theory.
4. Exercises(1) Show that the function p in (4) satisfies Vignéras’ differential equation.
(2) Prove (6) using the following argument from Zagier’s paper on quantum modularforms. Show that the Laurent serieskXn(n−1)Sk :=q − 2 (−q)k−nn=1

is a solution to the recursion
Sk+1 − Sk = q k+1 (Sk (q) − (−q)k ) − q −k(k+1)
2.

Conclude thatk−1
k−1XX−1−nq n+1 (q 2 ; q 2 )n = (q)k Sk .
(q − 1) . . . (q − 1) −
n=0

n=0

Deduce the claimed result (6) by noting that if q is a root of unity of order at
most k, then the right hand side of this equation becomes 0 and by using (5).
(3) The explicit formulation of the map (3) in terms of a “period integral”, as Lewis
in Zagier showed, is thatZ
2 i∞+F (τ ) = −[F (z), Rτ (z)] .
π τHere, for general functions f, g which are eigenfunctions of ∆ with eigenvalue
1/4, [f, g] is the Green’s form∂f∂g
[f, g] :=gdz + f dz,∂z∂zand (z = x + iy with x, y ∈ R)
1y2Rτ (z) := p.
(x − τ )2 + y 2

OTHER INDEFINITE THETA FUNCTIONS AND QUANTUM MODULAR FORMS

7Show this identity. This requires some significant rewriting and a few integral
transforms; a detailed proof can also be found in Proposition 3.5 of Li, Ngo, andRhoades’ paper “Renormalization and quantum modular forms, Part I: Maass
wave forms.” Conclude that for a Maass form F and for γ = ( ac db ) in thecongruence subgroup F is modular on, we have
Z i∞+−1 +[F (z), Rτ (z)] .
F (τ ) − (cτ + d) F (γτ ) = −γ −1 i∞After ensuring convergence towards appropriate sets of rational numbers, thisformula gives the quantum modularity transformations for the quantum modularforms discussed above.
(4) Consider the q-hypergeometric seriesn(n+1)q 2
f1 (q) :=.
2n+1 )(−q)(1−qnn≥0X

This series was shown by Bringmann and Kane to have the following representation:XX2222
f1 (q) =q 4n +5n+1−2j −2j 1 + q 6n+6 +q 4n +n−2j 1 + q 6n+3 .
−n−1≤j≤n−n≤j≤nShow that f1 fits into Zwegers’ framework, where (in the notation for Φa,b above)0
the quadratic form has associated matrix A = ( 80 −4), and where
 1
1
 
1
5
0
−2
88, c1 =, b=, and c2 = 2 .
a1 = 1 , a2 =00
1
1
2(5) Zwegers showed the following relations between the functions ϕcj appearing inthe completions of the functions Φc1 ,c2 , where a, b ∈ R2 :ϕca+λ,b+µ = e(B(a, µ))ϕca,bfor all λ ∈ Z2 and µ ∈ A−1 Z2 ,ϕc−a,−b = ϕca,b ,andc
ϕγcγa,γb = ϕa,bfor all γ ∈ SO+ (Q, Z),

where
SO+ (Q, Z) := γ ∈ SL2 (Z) | Q(γr) = Q(r) for all r ∈ R2 , γ(CQ ) = CQ .
Use these relations to show that for the indefinite theta series f1 from Exercise 4,
the completion terms in the associated Maass form you found in that case cancel
out, so that the coefficients of f1 (q) encode the positive coefficients of a Maass
waveform. (Hint: Use the matrix γ = ( 34 23 ).)