INDEFINITE THETA FUNCTIONS OF MORE GENERAL TYPE:
NEW RESULTS AND APPLICATIONSLARRY ROLEN

1. Higher-type indefinite theta functions: Motivation and applications
In the last two lectures, we described several types of theta functions. In particular,
we discussed how to form convergent, modular functions by summing terms of the form
p(n)q Q(n) (where p is a function possibly depending on v = Im(τ )) as n ranges over anr-dimensional lattice. We saw that the determination of which functions p are allowed
is dictated by a second-order differential equation of Vignéras. The main difficulty is tofind choices of p which are both interesting (or include as pieces predetermined examples)and yield convergent sums. If the quadratic form Q is positive definite, then we havegreat freedom in picking p, as the terms q Q(n) decay rapidly in all directions. As soon
as there is a negative value of Q(n), however, there will be exponential growth in someof the terms q Q(n) , and so we need to introduce decay or cancellation of terms using thefunctions p. Zwegers gave very important examples of (r−1, 1) forms which are built outof differences of particularly natural solutions to Vignéras’ equation. This realization
has led to many applications throughout number theory, combinatorics, knot theory,
representation theory, and physics.
It is very natural to ask, therefore, whether an analogous theory exists for indefinite
theta functions associated to quadratic forms of arbitrary type. In the search of sucha theory, it is a good idea to look for motivating examples from related fields, such as
physics and representation theory. These alternative perspectives are a guiding force
directing one towards objects which are both natural and are built with applications inmind. Zwegers’ original motivations in fact did come from such areas, and in particular
the first hints of such a theory were given by Göttsche and Zagier in their study of
Donaldson invariants for certain 4-manifolds, as well as by the work of Andrews, Watson,and many others on Ramanujan’s mock theta functions.
A theory generalizing the indefinite theta functions of Zwegers has begun to blossom
in just the past year. This is very exciting, as we are now in a similar situation as wewere in directly after Zwegers’ thesis was written; namely, there are many new examples
of modular-type objects to study which are natural on the number theory side and whichhave similarly abuntant appearances in other areas of mathematics. Here, we would liketo describe a few instances of where these higher-type functions appear, and give someindication as to what shapes they take.
12

LARRY ROLEN

Some early candidates for natural generalizations of Zwegers’ functions were certain
generalized Appell-Lerch sums which were noticed to be connected to partition functions
coming from N = 4 supersymmetric gauge theory. Such partition functions also encode
Poincaré polynomials of moduli spaces of semi-stable sheaves on the projective plane P2
with arbitrary rank r, and Manschot showed how these algebro-geometric objects areconnected to general Appell-Lerch sums. Another instance where general Appell-Lerchsums were highlighted was in work of Kac and Wakimoto. Extending their work ons`(m|n)∧ highest weight modules (mentioned in the first lecture in relation to mocktheta functions), Kac and Wakimoto also connected functions of the shape in (1) torepresentations of affine superalgebras.
Definition. Let Q be a positive-dimensional quadratic form for an n+ -dimensional lattice Λ. Now suppose that we choose m1 , . . . , mn− in the dual lattice Λ∗ . Then for
u = (u1 , . . . , un− ) ∈ Cn− , and for v ∈ Λ∗ ⊗ C ∼= Cn+ , we define the type (n+ , n− )Appell-Lerch sum byQ(k)(1)

q 2 e(v · k)
Qn−.
AQ,{mj } (u, v; τ ) :=k·m` e(u ))``=1 (1 − qk∈ΛX

Kac and Wakimoto required the vectors mj to be pairwise orthogonal (which, as weshall hint at below, often makes the functions much simpler), but the applications ingauge theory require more general sets of vectors. As discussed in the first lecture,
via geometric series expansions we can essentially think of these functions as indefinite
theta series for a quadratic form of type (n+ , n− ). As just two more examples of naturalcontexts for such indefinite theta functions, we refer the reader to important connections
to knot theory made by Hikami and Lovejoy (discussed further in the fourth lecture), as
well as series related to Gromov-Witten theory discussed in Section 4.
2. New Realizations
We now describe the recent progress which has been made in understanding the generalized versions of Zwegers’ functions. Early modularity results were proven in severalcases. For example, Westerholt-Raum used the theory of H-harmonic Maass-Jacobi
forms to describe and characterize certain higher-type indefinite theta functions which
are modular but of a more complicated type than harmonic Maass forms (these arehigher depth forms; cf. Section 3). As we shall discuss in Section 4, Bringmann, Zwegers,
and the author showed modularity results for special examples with an application toGromov-Witten theory (and this program set forth by Lau and Zhou was completed
later by Bringmann, Kaszian, and the author). More hints were also offered in work byBringmann, Manschot, and the author, where certain suggestive elliptic transformationsof higher-type Appell-Lerch sums were shown, and which gave alternative, direct proofsof certain blow-up formulas.

HIGHER-TYPE INDEFINITE THETA FUNCTIONS

3A few months ago, a major breakthrough was achieved by Alexandrov, Banerjee,Manschot, and Pioline. Their work made the general picture of Zwegers’-type resultsfor indefinite quadratic forms of type (r − 2, 2) clear, and offered a path to a completelygeneral picture of (r − s, s)-type indefinite theta functions. As discussed above, one of
the main obscurities in the field before this breakthrough was how to find an appropriate
function p which satisfies Vignéras’ equation which can play the role that the specialfunction E did in Zwegers’ case. In this situation, it turns out that the “right” functions
are what they call double error functions. These arose from string-theoretic considerations, and are certain Penrose-type integrals. These functions are two-dimensional
integrals, but also involve integrating square exponential terms. Explicitly, (although
this is a somewhat rewritten form from the original definition, and is stated in their
Theorem 3.11), we define for α ∈ R, (u1 , u2 ) ∈ R2 with u1 6= 0, u2 6= αu1 the function
Z22e−π(u1 −z1 ) −π(u2 −z2 ) sign(z2 )sign(z1 + αz2 )dz1 dz2 .
E2 (α; u1 , u2 ) :=R2Remark. Constructions such as this can also be considered to be special cases of related
results Borcherds used to construct Siegel theta functions.
Remark. It is also possible to write the function E from Zwegers’ thesis in a similar
shape, namely:
Z2E(u) =e−π(u−t) sign(t)dt.
RThis function has the following desirable properties (Exercise 1 will ask you to show
the first one):(1) The function E2 satisfies Vignéras’ equation for Q(u1 , u2 ) = u21 + u22 with λ = 0.
(2) Towards infinity, we have
E2 (α; u1 , u2 ) ∼ sgn(u2 ) sgn(u1 + αu2 ).
Now, given a quadratic form Q of type (r − 2, 2) and vectors c1 , c2 with Q(c1 ), Q(c2 ) < 0and ∆(c1 , c2 ) := Q(c1 )Q(c2 )−B(c1 , c2 )2 > 0, we can define the boosted error function!
B(c1 , c2 ) B(c1⊥2 , x) B(c2 , x)
E2 (c1 , c2 ; x) := E2 p;p,p,∆(c1 , c2 )Q(c1⊥2 )Q(c2 )
whereB(c1 , c2 )c2 .
Q(c2 )The point is that this “boosted” function now satisfies Vignéras’ equation with λ = 0 for
our quadratic form Q, and towards infinity it grows like a product of two sign functions:c1⊥2 := c1 −(2)E(c1 , c2 ; x) ∼ sgn(B(c1 , x)) sgn(B(c2 , x)).

4LARRY ROLEN

Similar to the case in Zwegers’ thesis, convergent indefinite theta series of type (r − 2, 2)
can be build out of pairs of these boosted functions; for example (2) can be used to showthat these indefinite theta series have “holomorphic parts” with a shapeX(sgn(B(c1 , n)) − sgn(B(c01 , n))) (sgn(B(c2 , n)) − sgn(B(c02 , n))) q Q(n) .
n∈ZrAs another example, Alexandrov, Banerjee, Manschot, and Pioline gave modular completions of generalized µ-functions such as (ζj := e(zj ))1X ζ n1 +2n2 q n21 +n22 +n1 n2 +2n1 n21
ζ122µ2,1 (z1 , z2 ; τ ) := +.
2 ϑ(z2 ; τ ) n ,n ∈Z
1 − ζ1 q 2n1 +n212

Following these results of Alexandrov, Banerjee, Manschot, and Pioline, using aroadmap which they described at the end of their work, general Zwegers’-type indefinite
theta functions were elucidated by Nazaroglu for (n − r, r) quadratic forms (althoughthe “cuspidal” case hasn’t been worked out yet). Exercises 4 and 5 will ask you to workout some properties of some higher-type error functions which can be used to constructgeneralized examples.
3. Higher depth mock modular forms
The functions constructed by Alexandrov, Banerjee, Manschot, and Pioline shouldplay an important role in number theory, since they have many applications, as discussed above. It is natural to ask what spaces these modular-type objects live in. The
results described above show that these functions transform like modular forms, but likeharmonic Maass forms, they should be “completions” of holomorphic functions which
are built out of “simpler” functions. It turns out that the correct definition for such
spaces is the following.
Definition (Zagier-Zwegers,Westerholt-Raum). Let Mk0 := Mk , the space of modular
forms of weight k. For d > 0, define the space Mkd of depth d forms to be the space offunctions which transform like modular forms and whose images under ξk lie inXd−1M` ⊗ Mk−`.
`For example, the depth 1 forms are just the mixed mock modular forms, briefly
discussed in the last two lectures. The reason that such indefinite theta functions built
using Vignéras’ theorem often lie in such spaces follows by a useful fact noticed byVignéras. Namely, the Maass lowering operator∂Lk := −2iv 2∂τis essentially the shadow operator ξk and intertwines in a beautiful way with indefinite
theta functions. In particular, if p satisfies Vigneras’ equation with eigenvalue λ, then

HIGHER-TYPE INDEFINITE THETA FUNCTIONS

5E − λ sends p to an eigenfunction with eignvalue λ − 2, and the theta function built outof this function is (up to a fixed constant) the lowering operator applied on the indefinite
theta function for p. Exercise 2 will ask you to compute the action of this operator onthe double error function E2 , which directly shows how the type (r − 2, 2) indefinite
theta functions above are depth 2 forms.
4. An example from Gromov-Witten theory
Several examples which arises from Gromov-Witten theory of elliptic orbifolds werestudied by Bringmann-R.-Zwegers and Bringmann-Kaszian-R., and are motivated bytheir connections to mirror symmetry and applications of modularity results in extending
certain potentials to global moduli spaces. Most of the functions which arise can be
written very neatly in terms of the simple indefinite theta function!
XX
1 −1 1 1
1 2 1
F (z1 , z2 , z3 ; τ ) := q − 8 ζ1 2 ζ22 ζ32(−1)k q 2 k + 2 k+k`+km+`m ζ1k ζ2` ζ3m .
+k>0, `,m≥0k≤0, `,m<0

The key result which makes the applications to Gromov-Witten theory possible is thenthe following curious identity (|Im(z2 )|, |Im(z3 )| < Im(τ )):(3)
F (z1 , z2 , z3 ; τ ) = iϑ(z1 ; τ )µ(z1 , z2 ; τ )µ(z1 , z3 ; τ ) −η 3 (τ )ϑ(z2 + z3 ; τ )µ(z1 , z2 + z3 ; τ ).
ϑ(z2 ; τ )ϑ(z3 ; τ )Although this identity can be proven using calculations of the corresponding elliptictransformations together with Liouville’s theorem (Exercise 3 will ask you to explore
some of the steps of these calculations), such “elementary” considerations prove mostof the main results in Chapter 1 of Zwegers’ thesis as well. As was the case for theµ-function, the most interesting task it to find such a decomposition in the first place.
Note that this example is also of a much simpler form than the general higher-typeindefinite theta functions defined above; indeed, it splits into products of modular andmock modular forms. The general philosophy about such situations is that these special
scenarios typically happen when the quadratic form is degenerate (as in this case), thevectors such as those in (1) have orthogonality relations, and the quadratic form hasinordinately many symmetries in its variables.
5. Exercises(1) Show that the double error function E2 (α; u1 , u2 ) satisfies Vignéras’ equation forQ(u1 , u2 ) = u21 + u22 with λ = 0. That is, show22∂w1 + ∂w2 + 2π (w1 ∂w1 + w2 ∂w2 ) E2 (α; w1 , w2 ) = 0,6LARRY ROLEN(2) Show thatπ(w2 +αw1 )22αw
2 − w1−21+α∂w2 E2 (α; w1 , w2 ) = √e
E √, and1 + α2
1 + α2π(w2 +αw1 )2αw−w
2α21−−πw121+α2E √∂w1 E2 (α; w1 , w2 ) = 2e
E (w2 ) + √e.
1 + α2
1 + α2Using the discussion in Section 3, deduce what the shadows (or lowering operatorsof) indefinite theta functions built out of the boosted functions E2 are.
(3) In this problem, you will consider some of the first steps towards proving (3).
The idea is to consider both sides of the equation as functions of z3 , and compute
their elliptic transformations and locations of poles and residues. Showing thatthese all match will then show, by a standard application of Liouville’s theorem(as it is commonly used in elliptic function theory) that the identity holds.
Here you will prove the desired transformation for the left-hand-side. First,show, using geometric series, that for |y3 | < v (z3 = x3 + iy3 ), we can write theleft-hand-side fL (z3 ) as
1 −1 1 1 q − 8 ζ1 2 ζ22 ζ32 

Xk,`∈Zk>0, `≥0X  (−1)k q 12 k2 + 21 k+k` ζ k ζ `1 2−k+`
1 − ζ3 qk,`∈Z
k≤0, `<0

1 −1 1 1= q − 8 ζ1 2 ζ22 ζ32

1 21(−1)k q 2 k + 2 k+k` ζ1k ζ2`ρ(k − 1, `),k+`1−ζ3qk,`∈ZXwhere
if k, ` ≥ 0,1
ρ(k, `) := −1 if k, ` < 0,0otherwise.
For 0 < y2 < v, show that−1
ζ2 2 ϑ(z1 )µ(z1 , z2 ) =1 2 1
1 2 3
X(−1)k q 2 k + 2 k+k` ζ1k ζ2`(−1)k q 2 k + 2 k+1+k`+` ζ1k ζ2`ρ(k, `)− ζ3
ρ(k, `).
k+`+1k+`+1
1−
ζ1−ζ3q3qk,`∈Zk,`∈Z

X

Using the easily checked identity
ρ(k, `) = ρ(k − 1, `) + δk ,(δk = 1 if k = 0 and δk = 0 otherwise) in the first sum and replacing k by k − 1
in the second, compute that−13

1−1 −1ζ2 2 ϑ(z1 )µ(z1 , z2 ) = q − 8 ζ12 ζ2 2 ζ3 2 fL (z3 + τ ) − iζ2−1

1 −1 −1 1η 3 ϑ(z2 + z3 )+ q 8 ζ1 2 ζ2 2 ζ32 fL (z3 ).
ϑ(z2 )ϑ(z3 )HIGHER-TYPE INDEFINITE THETA FUNCTIONS

7(Hint: use the identityX3
ζ2`−1 η ϑ(z2 + z3 ).)=−iζ
21 − ζ3 q `+1
ϑ(z2 )ϑ(z3 )`∈ZConclude that
11

1−11

1

3− 1 − 1 η ϑ(z2 + z3 )fL (z3 ) + q − 2 ζ1 ζ3−1 fL (z3 + τ ) = q − 8 ζ12 ζ3 2 ϑ(z1 )µ(z1 , z2 ) + iq − 8 ζ12 ζ2 2 ζ3 2(4) Define a generalized error function (N ≥ 3) EN : RN (N −1)2

ϑ(z2 )ϑ(z3 ).

× RN → R byEN (α; w)Z2sign(t1 )sign(t2 + α1 t1 )sign(t3 + α2 t1 + α3 t2 ) · · · sign tN + . . . + α N (N −1) tN −1 e−πkt−wk2 dt,:=2RN√
where kak2 := aT a denotes the Euclidian norm. Show that, as λ → ∞,E3 (α; λw) ∼ sign (w1 ) sign (w2 + α1 w1 ) sign (w3 + α2 w1 + α3 w2 ) .
Hint: Change variables to show
ZT
E3 (α; w1 , w2 , w3 ) =e−πt M t sign (t1 + w1 ) sign (t2 + v2 ) sign (t3 + v3 ) dt,R3
T

where t := (t1 , t2 , t3 ) , v2 := w2 + α1 w1 , v3 := w3 + α2 w1 + α3 w2 , and
1 + (α1 α3 − α2 )2 + α12 −α1 − α3 (α1 α3 − α2 ) α1 α3 − α2−α3  .
α32 + 1M :=  −α1 − α3 (α1 α3 − α2 )α1 α3 − α2−α31Now consider the difference
E3 (α; λw1 , λw2 , λw3 ) − sign (w1 ) sign (w2 + α1 w1 ) sign (w3 + α3 w2 + α2 w1 )Z
T=e−πt M t sign (t1 + λv1 ) sign (t2 + λv2 ) sign (t3 + λv3 ) − sign (v1 ) sign (v2 ) sign (v3 ) dt.
R3(5) Prove the analogue of the Vignéras equation in Exercise 1 for E3 . Specifically,show that3 X∂w2 j + 2wj ∂wj E3 (α; w) = 0.
j=1