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K. Bringmann et al. Res Math Sci (2021) 8:54
https://doi.org/10.1007/s40687-021-00284-1
RESEARCH
Integral representations of rank two false
theta functions and their modularity
properties
Kathrin Bringmann1*, Jonas Kaszian2, Antun Milas3and Caner Nazaroglu1
*Correspondence:
kbringma@math.uni-koeln.de
1Department of Mathematics
and Computer Science,
University of Cologne, Weyertal
86-90, 50931 Cologne, Germany
Full list of author information is
available at the end of the article
Abstract
False theta functions form a family of functions with intriguing modular properties and
connections to mock modular forms. In this paper, we take the first step towards
investigating modular transformations of higher rank false theta functions, following
the example of higher depth mock modular forms. In particular, we prove that under
quite general conditions, a rank two false theta function is determined in terms of
iterated, holomorphic, Eichler-type integrals. This provides a new method for
examining their modular properties and we apply it in a variety of situations where rank
two false theta functions arise. We first consider generic parafermion characters of
vertex algebras of type A2and B2. This requires a fairly non-trivial analysis of Fourier
coefficients of meromorphic Jacobi forms of negative index, which is of independent
interest. Then we discuss modularity of rank two false theta functions coming from
superconformal Schur indices. Lastly, we analyze ˆ
Z-invariants of Gukov, Pei, Putrov, and
Vafa for certain plumbing H-graphs. Along the way, our method clarifies previous
results on depth two quantum modularity.
1 Introduction and statement of results
Modular forms and their variations provide a rich source of interaction between physics
and mathematics. More recently, functions with more general forms of modular proper-
ties, such as mock modular forms, have gathered attention in both areas. In this paper,
we focus on such a family of functions with generalized modularity properties called false
theta functions. These are functions that are similar to ordinary theta functions on lattices
with positive definite signature, except for certain extra sign functions, which prevent
them from having the same simple modular properties as ordinary theta functions. For
false theta functions over rank one lattices, one approach to understand them is by noting
that they can be realized as holomorphic Eichler integrals of unary theta functions. This
representation can be used to study the modular transformations of such functions and
helps one understand why their limit to rational numbers yield quantum modular forms
[35]. An alternative approach to modularity of false theta functions in [17,18] is moti-
vated by the concept of the S-matrix in conformal field theory. In this setup, false theta
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54 Page 2 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
functions are “regularized” (defined on C×H, where His the complex upper half-plane)
and transform with integral kernels under the modular group. The S-kernel can be used
to formulate a continuous version of the Verlinde formula [17]. Yet another approach is
to follow the example of mock modular forms and form a modular completion as done in
[11], where elliptic variables can also be naturally understood. The modular completion
now depends on two complex variables in the upper half-plane (τ,w)∈H×H, which
transform in the same way under modular transformations,1and similar to mock modular
forms, differentiating in wyields a modular form in w.
One of the main goals in this paper is to generalize the considerations from [11]torank
two false theta functions. As for rank one false theta functions, to study the modular trans-
formations we follow the lead of higher depth mock modular forms, which were defined in
unpublished work of Zagier and Zwegers and were recently developed through signature
(n, 2) indefinite theta functions by [2].2In particular, the double error functions intro-
duced by [2] show how double products of sign functions can be replaced to give modular
completions. In Lemma 3.1 we give a particularly useful form to understand this fact in a
shape suitable for our context. This result then suggests a notion of false theta functions at
“depth two”, where we find a modular completion again depending on two complex vari-
ables (τ,w)∈H×H\{τ=w}and where the derivative in wleads to modular completions
of the kind studied in [11], which are at “depth one”. More specifically, our result leads
us to modular completions
f(τ,w) which transform like modular forms under simultane-
ous modular transformations (τ,w)→ (aτ+b
cτ+d,aw+b
cw+d) for ab
cd
∈SL2(Z) and reproduce
the rank two false theta functions we are studying through the limit limw→τ+i∞
f(τ,w).
Moreover, their derivatives with respect to wappear in the form
∂
f(τ,w)
∂w=
j
(i(w−τ))rj
gj(τ,w)hj(w),
where rj∈Z
2,hjis a weight 2 +rjmodular form (with an appropriate multiplier system),
and
gj(τ,w) is a modular completion of the sort studied in [11]. This is a structure that
closely resembles those of depth two mock modular forms. It would be interesting to
elaborate on the details here and form an appropriate notion of “higher depth false mod-
ular forms” by mirroring the structure of higher depth mock modular forms. We leave
this problem as future work and restrict our attention to answering concrete modularity
questions about rank two false theta functions arising in a variety of mathematical fields.
A rich source of false theta functions that is studied in this paper is through the Fourier
coefficients of meromorphic Jacobi forms with negative index or their multivariable gener-
alizations [7,12].3Such meromorphic Jacobi forms naturally arise in representation theory
of affine Lie algebras and in conformal field theory. In vertex algebra theory, important
examples of meromorphic Jacobi forms come from characters of irreducible modules for
the simple affine vertex operator algebra Vk(g) at an admissible level k.Ataboundary
admissible level [26], these characters admit particularly elegant infinite product form.
1A similar picture is obtained for mock modular forms by complexifying the complex conjugate of the modular variable
τso that we have a pair of complex variables (τ,w)one living in the upper half-plane and one in the lower half-plane
with both transforming in the same way under modular transformations.
2A notion that is similar to higher depth mock modular forms is that of polyharmonic Maass forms [5,29].
3Here and in the rest of this paper, whenever we say Fourier coefficients of (meromorphic) Jacobi forms, we mean
Fourier coefficients with respect to the elliptic variables.
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 3 of 31 54
Modular properties of their Fourier coefficients are understood only for g=sl2and
V−3
2(sl3). For the latter, the Fourier coefficients are essentially rank two false theta func-
tions (see [7] for more details). On the very extreme, if the level is generic, the character
of Vk(g) is given by
ch[Vk(g)](ζ;q)] =q−dim(g)k
24(k+h∨)
(q;q)n
∞α∈+(ζαq;q)∞α∈+(ζ−αq;q)∞
,(1.1)
where nis the rank of g,h∨the dual Coxeter number, and as usual, (a;q)r:=r−1
j=0(1−aqj)
for r∈N0∪{∞}. Moreover ζare variables parametrizing the set of positive roots +of
gand throughout this paper we use bold letters to denote vectors. Although (1.1)isnota
Jacobi form, a slight modification in the Weyl denominator gives a genuine Jacobi form of
negative index. The Fourier coefficients of (1.1) are important because they are essentially
characters for the parafermion vertex algebra Nk(g)[19,20,25] (see also Sect. 5), whose
character is given by
(q;q)n
∞CT[ζ](ch[Vk(g)](ζ;q)),(1.2)
where CT[ζ]denotes the constant term in the expansion in ζj. The character can be
expressed as linear combinations of coefficients of Jacobi forms. One of the goals of this
paper is to investigate modular properties of (1.2) for types A2and B2, which leads us to
the following result.
Theorem 1.1 Characters of the parafermion vertex algebras of type A2and B2can be
written as linear combinations of (quasi-)modular forms and false theta functions of rank
one and two. The rank two pieces in these decompositions can be written as iterated holo-
morphic Eichler-type integrals, which yields the modular transformation properties of these
functions.
Note that more precise versions of this result are given in Propositions 5.1,5.5,5.6,6.1,6.6,
and 6.7 . Independent of modular properties, we expect that the analysis we make on
the characters ch[Vk(g)] in these two cases will also shed some light on the nature of
coefficients of meromorphic, multivariable Jacobi forms of negative definite index. We
furthermore hope that our techniques can be extended to study parafermionic characters
at boundary admissible levels.
Meromorphic Jacobi forms closely related to characters of affine Lie algebras at bound-
ary admissible levels also show up in the computation of the Schur index I(q)of4dN=2
superconformal field theories (SCFTs) [4,13]. If refined by flavor symmetries, the Schur
index is denoted by I(q, z1, .., zn). In this paper, we are only interested in the Schur index
of some specific SCFTs, called Argyres–Douglas theories of type (A1,D
2k+2), whose index
with two flavors was first computed in [13](seealso[15]) and later identified with certain
vertex algebra characters in [16]. In particular, for k=1 the index coincides with the
character of the aforementioned vertex algebra V−3
2(sl3). Our second main result deals
with modularity of Fourier coefficients of these indices; for a more precise statement see
Sect. 7.
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54 Page 4 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
Theorem 1.2 The Fourier coefficients of the Schur indices of Argyres–Douglas theories of
type (A1,D
2k+2)are essentially rank two false theta functions. Moreover, the constant terms
in these Fourier expansions can be expressed as double Eichler-type integrals.
The third main result concerns the ˆ
Z-invariants, called homological blocks, of plumbed
3-invariants introduced recently by Gukov, Pei, Putrov, and Vafa [24] and further studied
from several viewpoints in [9,14,21,23,24,27,32]. For Seifert homology spheres, it is well-
known that they can be expressed as linear combinations of derivatives of unary false theta
functions, whose modular properties are known. Further computations of ˆ
Z-invariants
for certain non-Seifert integral homology spheres were given in [9]. Our next result is an
integral representation of these invariants. Compared to [9], Theorem 1.3 gives a more
direct relationship between iterated Eichler integrals and ˆ
Z-invariants.
Theorem 1.3 Let M be a plumbed 3-manifold obtained from a unimodular Hgraph as
in [9]. Then the ˆ
Z-invariant of M has a representation of the shape
ˆ
Z(τ)=τ+i∞
τw1
τ
1(w1,w
2)
i(w1−τ)i(w2−τ)dw2dw1+2(τ),
where 1(w1,w
2)is a linear combination of products of derivatives of unary theta functions
in w1and w2and 2(τ)is a rank two theta function. Moreover, there is a completion of ˆ
Z
which transforms like a weight one modular form.4
Importantly, Theorems 1.1,1.2,and1.3 completely determine the modular properties
of the functions under investigation. These results in turn pave the way for studying
“precision asymptotics” for the relevant functions within all the contexts stated above,
i.e., characters of parafermionic algebras, supersymmetric Schur indices, and homologi-
cal invariants of 3-manifolds. In the case of classical modular forms, this is accomplished
by studying Poincaré series and by using the Circle Method. The most classical example is
the exact formula for the integer partition function found by Rademacher [33], whose con-
vergent formula extended the asymptotic results of Hardy and Ramanujan significantly.
In fact, such results are intimately related to the finite-dimensionality of the associated
vector spaces of modular objects and this property forms the basis for many of the remark-
able applications of modularity to different fields of mathematics. The Circle Method has
already been applied to a case involving rank one false theta functions in [11] and to one
involving depth two mock modular forms in [10]. It would be interesting to extend these
results to the class of functions studied in this paper and explore the implications to the
different fields considered here.
Finally, the outline of the paper is as follows: In Sect. 2, we gather several facts on certain
classical modular forms, Jacobi theta functions, and a number of meromorphic Jacobi
forms of two complex variables used in the paper. In Sect. 3, we prove Lemma 3.1, which
is the main technical tool used to study rank two false theta functions as we demonstrate in
the rest of the section. Then in Sect. 4, we collect several technical results used in studying
Fourier coefficients of meromorphic Jacobi forms. In Sect. 5, we turn our attention to
parafermionic characters of type A2and show that one can write them in terms of modular
4In this paper, we employ “hats” to denote modular completions as is common in the literature for mock modular
forms. This should not be confused with the hat that appears in ˆ
Zfor homological blocks, which is also a standard
notation in literature.
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 5 of 31 54
forms and a rank two false theta function. We then find the modular transformations of
the rank two piece using tools from Sect. 3. In Sect. 6, we apply the same type of analysis on
generic parafermionic characters of type B2. In Sect. 7, we demonstrate how the tools used
in this paper also applies to rank two false theta functions coming from superconformal
Schur indices and ˆ
Z-invariants of 3-manifolds. We conclude in Sect. 8with final remarks
and comments on future prospects.
2 Preliminaries
We start by recalling several functions which we require in this paper. Firstly, let
η(τ):=q1
24 ∞
n=1
(1−qn)
be Dedekind’s η-function, where q:=e2πiτ. It satisfies the modular transformations
η(τ+1) =eπi
12 η(τ),η−1
τ=√−iτη(τ).
Note that these two transformations imply that for M=ab
cd
∈SL2(Z)wehave
ηaτ+b
cτ+d=νη(M)(cτ+d)1
2η(τ),
where νηdenotes the multiplier system for the η-function. We furthermore use the identity
η(τ)3=
n∈Z
(−1)nn+1
2q
1
2n+1
22
.
We also require the Jacobi theta function defined by (ζ:=e2πiz)
ϑ(z;τ):=
n∈Z+1
2
eπinqn2ζn.
By the Jacobi triple product formula, we have the product expansion
ϑ(z;τ)=−iq 1
8ζ−1
2(q;q)∞(ζ;q)∞ζ−1q;q∞.(2.1)
The Jacobi theta function transforms like a Jacobi form of weight and index 1
2:
ϑ(z;τ+1) =eπi
4ϑ(z;τ),ϑz
τ;−1
τ=−i√−iτeπiz2
τϑ(z;τ),(2.2)
ϑ(z+1; τ)=−ϑ(z;τ),ϑ(z+τ;τ)=−q−1
2ζ−1ϑ(z;τ).(2.3)
Moreover, we have
∂
∂zϑ(z;τ)z=0=−2πη(τ)3.(2.4)
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We also need the unary theta functions
ϑm,r (z;τ):=⎧
⎪
⎪
⎨
⎪
⎪
⎩
n∈Z+r
2m
qmn2ζ2mn if m∈Z,
n∈Z+r
2m+1
2
(−1)n−r+m
2mqmn2ζ2mn if m∈Z+1
2.
They satisfy the following elliptic and modular transformations.
Lemma 2.1 (1) For m ∈Zand r ∈Z/2mZ,wehave:
ϑm,r (z;τ+1) =eπir2
2mϑm,r (z;τ),
ϑm,r z
τ;−1
τ=e2πimz2
τ
√−iτ
√2m
(mod 2m)
e−πir
mϑm,(z;τ).
(2) For m ∈Z+1
2and r ∈Z/2mZ,wehave:
ϑm,r (z;τ+1) =eπi(r+m)2
2mϑm,r (z;τ),
ϑm,r z
τ;−1
τ=e2πimz2
τe−πim√−iτ
√2m
(mod 2m)
(−1)r+e−πir
mϑm,(z;τ).
We denote the derivatives of ϑm,r (z;τ) with respect to zas:
ϑ[k]
m,r (τ):=1
4πim
∂
∂zk
ϑm,r (z;τ)z=0
.
Note that we drop the superscript if k=0.
Another function we use is the quasimodular Eisenstein series
E2(τ):=1−24 ∞
n=1
d|n
dqn,
which satisfies the (quasi)modular transformations
E2(τ+1) =E2(τ),E
2−1
τ=τ2E2(τ)+6τ
πi.
This function is used in the definition of the Ramanujan–Serre derivative,
Dk:=1
2πi
∂
∂τ −k
12 E2(τ),
which maps modular forms of weight kto modular forms of weight k+2.
Finally, in Sects. 5and 6, we analyze Fourier coefficients of two multivariable mero-
morphic Jacobi forms defined as follows:
TA(z;τ):=1
ϑ(z1;τ)ϑ(z2;τ)ϑ(z1+z2;τ),T
B(z;τ):=TA(z;τ)
ϑ(2z1+z2;τ).(2.5)
Here we recall that a Jacobi form f:CN×H→Cof weight k∈1
2Zand matrix index
M∈1
4ZN×Nsatisfies the following transformation laws (with multipliers ν1,ν2):
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 7 of 31 54
(1) For ab
cd
∈SL2(Z)wehave
fz
cτ+d;aτ+b
cτ+d=ν1ab
cd
(cτ+d)ke2πic
cτ+dzTMzf(z;τ).
(2) For (m,)∈ZN×ZNwe have
f(z+mτ+;τ)=ν2(m,)q−mTMme−4πimTMzf(z;τ).
From (2.2)and(2.3) we easily see that TAand TBtransform like Jacobi forms with weights
−3
2and −2, and matrix indices −1
221
12and −1
263
33, respectively (with some multipli-
ers). We also consider in Sect. 7for k∈N,
Tk(z;τ):=ϑ(z1;(k+1)τ)ϑ(z2;(k+1)τ)ϑ(z1+z2;(k+1)τ)
ϑ(z1;τ)ϑz2;k+1
2τϑz1+z2;k+1
2τ.
The function Tk((k+1)z;τ) with rescaled elliptic variables is a Jacobi form of weight zero
and matrix index −k+1
2k+11
12
.
3 Products of sign functions and iterated integrals
A key technical result in this paper is the following lemma which allows one to write
products of sign functions in terms of iterated integrals. This lemma essentially follows
from Proposition 3.8 of [2], which gives an expression that allows efficient numeric eval-
uation of double error functions developed there. These double error functions play a
fundamental role in understanding modular properties of indefinite theta functions for
lattices of signature (n, 2). The double error functions become signs towards infinity and
this is what we express in the next lemma. It is further processed and cast into a form
from which the modular properties of false theta functions are manifest.
Lemma 3.1 For 1,2∈R,κ∈R,with(1,2+κ1)= (0,0),wehave
sgn(1)sgn(2+κ1)q
2
1
2+2
2
2
=τ+i∞
τ
1eπi2
1w1
i(w1−τ)w1
τ
2eπi2
2w2
i(w2−τ)dw2dw1
+τ+i∞
τ
m1eπim2
1w1
i(w1−τ)w1
τ
m2eπim2
2w2
i(w2−τ)dw2dw1+2
πarctan(κ)q
2
1
2+2
2
2,
where sgn(x):=x
|x|for x = 0,sgn(0) :=0,m
1:=2+κ1
√1+κ2,andm
2:=1−κ2
√1+κ2.
Remark 3.2 We use τ+i∞in the upper limits of these integrals to indicate that all such
integrals are taken along the vertical path from τto i∞and we use the principal value of
the square root.
Proof of Lemma 3.1 We first assume that both 1,2+κ1= 0. Shifting wj→ iwj+τ
the first term on the right-hand side of the lemma equals
−12q
2
1
2+2
2
2∞
0
e−π2
1w1
√−w1w1
0
e−π2
2w2
√−w2
dw2dw1.(3.1)
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54 Page 8 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
On the path of integration, we have −wj=i√wj. Changing wj→ w2
j,Eq.(3.1) thus
equals
412q
2
1
2+2
2
2∞
0
e−π2
1w2
1w1
0
e−π2
2w2
2dw2dw1.
We then employ the following integral identity, which is straightforward to verify
412∞
0
e−π2
1w2
1w1
0
e−π2
2w2
2dw2dw1=2
πarctan 2
1.
Using that m2
1+m2
2=2
1+2
2, the statement of the lemma is equivalent to
2
πarctan 2
1+arctan m2
m1+arctan(κ)=sgn(1)sgn(2+κ1).
This identity may be deduced using general properties of arctangent. The cases in which
one of 1,2+κ1vanishes can be shown similarly.
Now, consider a general rank two false theta function
n∈Z2+α
sgn(n1)sgn(n2)q1
2(an2
1+2bn1n2+cn2
2),
where a,b,andcare integers such that the quadratic form in the exponent is positive
definite, and α=(α1,α2)∈Q2. Moreover define the theta functions
1(w):=
n∈Z2+α
n1n2+b
cn1eπi
cn2
1w1+πicn2+b
cn12w2,
2(w):=
n∈Z2+α
n2n1+b
an2eπi
an2
2w1+πian1+b
an22w2,
where :=ac −b2>0, and the modular theta function
(τ):=
n∈Z2+α
q1
2(an2
1+2bn1n2+cn2
2).
Then we have the following:
Proposition 3.3 We have
n∈Z2+α
sgn(n1)sgn(n2)q1
2(an2
1+2bn1n2+cn2
2)−2
πδα∈Z2arctan b
√
=√τ+i∞
τw1
τ
1(w)+2(w)
i(w1−τ)i(w2−τ)dw2dw1−2
πarctan b
√(τ),
where δC=1if a condition C holds and zero otherwise.
Proof Letting 1=
cn1,2=√cn2+b
√cn1,andκ=− b
√,weget
sgn(1)sgn(2+κ1)q
2
1
2+2
2
2=sgn(n1)sgn(n2)q1
2(an2
1+2bn1n2+cn2
2).
Summing over Z2+αusing Lemma 3.1, noting that m1=
an2and m2=1
√a(an1+bn2)
and including a correction for the case (1,2+κ1)=(0,0) which occurs if α∈Z2yields
the claim.
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 9 of 31 54
Remark 3.4 We may modify the above construction to get a family of functions for which
both the modular part including (τ) and the correction term including δα∈Z2vanish. For
this purpose, consider false theta functions of the form
n∈Z2+(0,α2)
(−1)n1sgn(n1)sgn(n2)q1
2(an2
1+2bn1n2+cn2
2),
such that a|band b
aα2≡1
2(mod 1). In particular, we have α2/∈Zand hence the
correction term, with δα∈Z2, vanishes. Note that this condition is satisfied if α2=1
2r,
where r=b
a. Some series of this form are discussed in Chapter 5. As in Proposition 3.3,
we can represent these q-series as iterated Eichler-type integrals with 1,2,andnow
picking up an additional (−1)n1factor. Because b
aα2≡1
2(mod 1), the corresponding
-part is vanishing as
n∈Z2+(0,α2)
(−1)n1q1
2(an2
1+2bn1n2+cn2
2)=
n2∈Z+α2
q
2an2
2
n1∈Z
(−1)n1q
a
2n1+bn2
a2
=0.
4 Decomposition formulas for meromorphic Jacobi forms
Before moving to examples, we collect a few auxiliary results used in decomposing mul-
tivariable meromorphic Jacobi forms and extracting their Fourier coefficients. We start
with a basic result involving two Jacobi theta functions. Besides its use in Sect. 5,the
methods employed in its proof are employed as a blueprint for more complex variations
that we need in sections below. Here and throughout we sometimes drop dependencies
on τif they are clear from the context; e.g. we often write ηinstead of η(τ). The next result
was suggested to us by S. Zwegers.
Lemma 4.1 For r ∈Zand w /∈Zτ+Zwe have
ζr
ϑ(z)ϑ(z+w)=i
η3ϑ(w)
n∈Z
qn2−rne−2πinw
1−ζqn−ie−2πirw
η3ϑ(w)
n∈Z
qn2−rne2πinw
1−ζe2πiwqn.
Proof Define
h(z):=e2πirz
ϑ(z)ϑ(z+w),g(z, z):=
n∈Z
qn2−rne−2πin(2z+w)
1−ζe−2πizqn.
Using (2.3) gives that z→ h(z)g(z, z) is elliptic. Let Pδ:=δ+[0,1]+[0,1]τbe a fundamental
parallelogram with δin a small neighborhood of 0 such that z→ h(z)g(z, z)hasnopoles
on the boundary. Moreover, we assume that zand −ware in Pδand prove the proposition
statement for such values; the result generalizes to the whole complex plane by analytic
continuation. If we integrate h(z)g(z, z) around Pδcounterclockwise, then the integral
vanishes by ellipticity of the function and we have, by the Residue Theorem
0=∂Pδ
h(z)g(z, z)dz=2πi
w∈Pδ
Resz=w(h(z)g(z, z)).
Using that Resz=z(g(z, z)) =1
2πi,weget
h(z)=−2πig(z, 0) Resz=0(h(z)) −2πig(z, −w)Res
z=−w(h(z)).
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54 Page 10 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
We compute, using (2.4)
Resz=0(h(z)) =− 1
2πη3ϑ(w),Resz=−w(h(z)) =e−2πirw
2πη3ϑ(w),
which then gives the claim.
We next state two variations of this result involving three Jacobi theta functions, which
we need in Sect. 6and whose proofs follow the same method as the one used in Lemma 4.1.
Lemma 4.2 For w1,w
2,w
1−w2/∈Zτ+Z,andr∈Z+1
2,wehave
ζr
ϑ(z)ϑ(z+w1)ϑ(z+w2)=i
η3ϑ(w1)ϑ(w2)
n∈Z
(−1)nq3n2
2−rne−2πin(w1+w2)
1−ζqn
+ie−2πirw1
η3ϑ(w1)ϑ(w1−w2)
n∈Z
(−1)nq3n2
2−rne−2πin(w2−2w1)
1−ζe2πiw1qn
+ie−2πirw2
η3ϑ(w2)ϑ(w2−w1)
n∈Z
(−1)nq3n2
2−rne−2πin(w1−2w2)
1−ζe2πiw2qn.
Lemma 4.3 For w1,w
2/∈Zτ
2+Z1
2,w
1−w2/∈Zτ+Z,andr∈Z,wehave
ζr
ϑ(2z)ϑ(z+w1)ϑ(z+w2)
=ie−2πirw1
η3ϑ(2w1)ϑ(w1−w2)
n∈Z
q3n2−rne2πin(5w1−w2)
1−ζe2πiw1qn
+ie−2πirw2
η3ϑ(2w2)ϑ(w2−w1)
n∈Z
q3n2−rne2πin(5w2−w1)
1−ζe2πiw2qn
+i
2η3
1,2∈{0,1}
(−1)1+2+r2q1(1+r)
2
ϑw1+1τ+2
2ϑw2+1τ+2
2
n∈Z
q3n2−(31+r)ne−2πin(w1+w2)
1−(−1)2ζqn−1
2
.
5 Generic parafermionic characters of type A2
5.1 Parafermions and parfermion algebras
The parafermionic conformal field theories first appeared in the famous article of Fateev
and Zamolodchikov on Zk-parafermions [36]. The fields in such theories have fractional
conformal weight and are not necessarily local to each other, which thereby generalizes
the familiar bosonic and fermionic free fields.
In mathematics literature, parafermions and parafermionic spaces originally appeared
in the ground-breaking work of Lepowsky and Wilson on Z-algebras and Rogers–
Ramanujan identities [30]. This concept was later formalized by Dong and Lepowsky
in [19], where parafermionic spaces [36] were viewed as examples of generalized vertex
algebras. Although [30,36] dealt only with sl2parafermions at positive integral levels,
parafermions can be defined for any affine Lie algebra gand any level k.Inthisgeneral-
ity, the parafermionic space k(g) consists of highest weight vectors for the Heisenberg
vertex subalgebra inside the affine vertex algebra Vk(g). The parafermion (vertex) algebra,
denoted by Nk(g)⊂k(g), is defined as the charge zero subspace of the parafermionic
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 11 of 31 54
space. It has a natural vertex operator algebra structure of central charge c=kdim(g)
k+h∨−n.
Then the parafermionic character is defined by
ch[Nk(g)](q):=tr|Nk(g)qL(0)−c
24 ,
where L(0) is the degree operator. This can in turn be expressed as the constant term
(1.2) discussed in the introduction. To illustrate this concept, let us consider the simplest
non-trivial case of V2(sl2). The parafermionic space 2(sl2) is simply the free fermion
vertex superalgebra and N2(sl2) is the even part thereof, also known as the c=1
2Ising
model. Therefore,
ch[N2(sl2)](q)=q−1
48 ⎛
⎜
⎝−q1
2;q∞
2+q1
2;q∞
2⎞
⎟
⎠.
For other levels, k∈N,k≥3, the algebraic structure of Nk(sl2) is more complicated
and involves non-linear W-algebras. Parafermionic characters of sl2for positive integral
levels are well-understood [3,25] and they transform as vector-valued modular forms of
weight zero. Similar results persist for higher rank algebras.
For generic k, that is if Vk(g) is the universal affine vertex algebra (e.g. k/∈Q), properties
of Nk(g) are quite different. The structure of the parafermion algebra is known explicitly
only in a handful of examples and their parafermionic characters are not modular.
5.2 Parafermionic character of A2
We are finally at a point where we can work out our first example involving generic
parafermionic characters of type A2. As a warm up to this discussion, we first consider
the simplest example, which is the generic parafermionic characters of type A1.
Example. For g=sl2, the parafermionic character is known to be (see for instance [1,3])
CT[ζ]1
(ζq;q)∞(ζ−1q;q)∞=1
(q;q)2
∞−1+2∞
n=0
(−1)nqn(n+1)
2
=−q1
12
η(τ)2+2q−1
24 ψ(τ)
η(τ)2,
where ψ(τ):=n∈Zsgn(n+1
4)q2(n+1
4)2is Rogers’ false theta function. The modular
properties of ψ(τ)
η(τ)2were studied and used in [11] to give a Rademacher type exact formula
for its coefficients in the q-expansion. The constant term in the above example splits into
two q-series with different modular behaviors (note the different q-powers). Our goal is
to obtain a similar decomposition for the A2vacuum character.
5.3 Expression in terms of false theta functions
Specializing Eqs. (1.1)and(1.2)tothecaseofA2with positive roots
+:= α1=1
0,α2=0
1,α1+α2!,
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54 Page 12 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
the goal in this section is to study the constant term of
G(ζ):=q8k
24(k+3) (q;q)2
∞ch[Vk(sl3)](ζ;q)=1
ζ1q, ζ−1
1q, ζ2q, ζ−1
2q, ζ1ζ2q, ζ−1
1ζ−1
2q;q∞
,
where (a1,...,a
;q)n:=
j=1(aj;q)n.Using (2.1) we rewrite it as (ζj:=e2πizj)
G(ζ)=iq 1
4η3ζ−1
1ζ−1
2(1 −ζ1)(1 −ζ2)(1 −ζ1ζ2)
ϑ(z1)ϑ(z2)ϑ(z1+z2).(5.1)
Then, to state our result on the constant term of G(ζ), we introduce the following
functions:
G0(τ):=1+3
n∈Z|n|qn2−6q−1
4
n∈Z+1
2
|n|qn2,
(τ):=
n∈Z2+1
3,1
3sgn(n1)sgn(n2)n1qQA(n),where QA(n):=n2
1+n1n2+n2
2.
Proposition 5.1 For |q|<|ζ1|,|ζ2|,|ζ1ζ2|<1we have
CT[ζ](G(ζ)) =q1
4
η(τ)6G0(τ)+9q−1
12
η(τ)6(τ)
=1+3q2+8q3+21q4+48q5+116q6+252q7+555q8+1156q9+Oq10.
To prove Proposition 5.1, we employ Lemma 4.1 and another auxiliary result stated
below, which itself is a corollary of Lemma 4.1.
Lemma 5.2 For r ∈Zwe have
ζr
ϑ(z)2=−1
η6
n∈Z
qn2−rn 2n−r−1
1−ζqn+1
(1 −ζqn)2.
Proof Using (2.4) and the fact that ϑis odd, we find that for a function Fthat is holomor-
phic in a neighborhood of w=0, we have
F(w)
ϑ(w)=− 1
2πη3F(0)
w+F(0)+O(w)asw→0.
Thus taking the limit w→0 in Lemma 4.1 yields (noting that F(0) =0inthiscase)
ζr
ϑ(z)2=− i
2πη6
n∈Z
qn2−rn ∂
∂we−2πinw
1−ζrqn−e2πi(n−r)w
1−ζre2πiwqnw=0
.
The result follows, using that
i
2π∂
∂we−2πinw
1−ζqn−e2πi(n−r)w
1−ζe2πiwqnw=0=2n−r−1
1−ζqn+1
(1 −ζqn)2.
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 13 of 31 54
We are now ready to compute Fourier coefficients of the meromorphic Jacobi form
appearing in Eq. (5.1). To state our result, we define
D(r):=CT[ζ]iη9ζr1
1ζr2
2
ϑ(z1)ϑ(z2)ϑ(z1+z2),
D1(r):=
n∈N2
0
(n1+2n2−r1)qn2
1+n1n2+n2
2−r2n1−r1n2,
D2(r):=
n∈N2
0
(n1−2n2+r1−r2)qn2
1−n1n2+n2
2−r2n1+(r2−r1)n2.
Corollary 5.3 For |q|<|ζ1|,|ζ2|,|ζ1ζ2|<1and for r∈Z2we have
D(r)=D1(r)+D2(r).
Proof Using Lemma 4.1 with (r, z, w)→ (r2,z
2,z
1) we find that with TAdefined in (2.5),
TA(z)ζr2
2=i
η3ϑ(z1)2⎛
⎝
n1∈Z
qn2
1−r2n1ζ−n1
1
1−ζ2qn1−
n1∈Z
qn2
1−r2n1ζn1−r2
1
1−ζ1ζ2qn1⎞
⎠.
Next, we use Lemma 5.2 with (r, z)→ (r1−n1,z
1)and(r, z)→ (r1−r2+n1,z
1) to write
iη9TA(z)ζr1
1ζr2
2
=
n∈Z2
qn2
1+n1n2+n2
2−r2n1−r1n2
1−ζ2qn12n2+n1−r1−1
1−ζ1qn2+1
(1 −ζ1qn2)2
−
n∈Z2
qn2
1−n1n2+n2
2−r2n1+(r2−r1)n2
1−ζ1ζ2qn12n2−n1+r2−r1−1
1−ζ1qn2+1
(1 −ζ1qn2)2.
The claim now follows using the identity
CT[ζ]1
(1 −ζqn)k=⎧
⎨
⎩
1ifn≥0,
0ifn<0,(5.2)
which holds for zsufficiently close to 0 with |ζ|<1andk∈N.
We are now ready to prove Proposition 5.1.
Proof of Proposition 5.1 Using (5.1) and Corollary 5.3, for |q|<|ζ1|,|ζ2|,|ζ1ζ2|<1we
have
CT[ζ](G(ζ)) =q1
4
η6
r∈SA
εA(r)D(r),
where
SA:="(1,0),(0,1),(−1,−1),(−1,0),(0,−1),(1,1)#,
εA(r):=⎧
⎨
⎩
1ifr∈{(1,0),(0,1),(−1,−1)},
−1ifr∈{(−1,0),(0,−1),(1,1)}.
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54 Page 14 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
Defining Q∗
A(n):=QA(−n1,n
2), we rewrite D1(r)andD2(r)as
D1(r)=q−Q∗
A(r)
3
n∈N2
0
(n1+2n2−r1)qQAn1+r1−2r2
3,n2+r2−2r1
3,
D2(r)=q−Q∗
A(r)
3
n∈N2
0
(n1−2n2+r1−r2)qQ∗
An1−r1+r2
3,n2+r2−2r1
3.
Then,
q1
3
r∈SA
εA(r)D1(r)=
n∈N2
0
⎛
⎝(n1+2n2−1)qQAn1+1
3,n2−2
3+(n1+2n2)qQAn1−2
3,n2+1
3
+(n1+2n2+1)qQAn1+1
3,n2+1
3−(n1+2n2+1)qQAn1−1
3,n2+2
3
−(n1+2n2)qQAn1+2
3,n2−1
3−(n1+2n2−1)qQAn1−1
3,n2−1
3⎞
⎠.
Shifting either n1or n2by one while collecting the one-dimensional boundary terms yields
r∈SA
εA(r)D1(r)
=3q−1
3
n∈N2
0(n1+2n2+1)qQAn1+1
3,n2+1
3−(n1+2n2+2)qQAn1+2
3,n2+2
3
+∞
n=0(n−1)qn2+2nqn2−(2n+1)qn(n+1) −nqn(n+1) −(2n−1)qn(n−1) −nqn(n+1).
Changing n→−(1,1) −nfor the second two-dimensional term and shifting n→ n+1
in the one-dimensional contribution with the factor qn(n−1) we find that
r∈SA
εA(r)D1(r)=3
2q−1
3
n∈Z2+1
3,1
3(1 +sgn(n1)sgn(n2))(n1+2n2)qQA(n)
+1+∞
n=0(3n−1)qn2−2(3n+1)qn(n+1).
A similar computation gives
r∈SA
εA(r)D2(r)=3
2q−1
3
n∈Z2+1
3,1
3(−1+sgn(n1)sgn(n2))(n1+2n2)qQA(n)
+∞
n=0(3n+1)qn2−2(3n+2)qn(n+1).
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 15 of 31 54
Then, combining the two terms we find
r∈SA
εA(r)D(r)=3q−1
3
n∈Z2+1
3,1
3sgn(n1)sgn(n2)(n1+2n2)qQA(n)
+1+∞
n=06nqn2−6(2n+1)qn+1
22−1
4.
Noting the symmetry between n1and n2of the two-dimensional sum and antisymmetry
of the two one-dimensional sums under n→−nand n→−n−1, respectively, (as well
as the vanishing of the first one-dimensional summand for n=0) yields the result.
Remark 5.4 Note that for r=(r1,r
2), such that r1+r2≡0(mod 3), the coefficient D(r)
is a finite sum of one-dimensional false theta functions. Specifically for k∈N0,wehave
D(3k, 3k)=⎛
⎝
k−1
j=0−
3k
j=k+1⎞
⎠qj2−3kj ∞
n=0
(2n+j−3k)qn2+(j−3k)n
+
k
j=−k
qj2−3k2∞
n=0
qn2nqjn +q−jn
2+jqjn −q−jn.
In particular, D(0,0) =∞
n=1nqn2. This leads to the new q-hypergeometric representa-
tion
D(0,0)
(q)6
∞=
n∈N4
0
qn1+n2+n4
(q)n1+n4−n3−1(q)n2+n4−n3−1(q)n1(q)n2(q)n3(q)n4
,
which easily follows from applying Euler’s identity 1
(a)∞=∞
n=0an
(q)nto TA(z;τ)sixtimes.
Another consequence of the formula for D(0,0) is the following q-series identity
∞
n=1
nqn2
=
n∈N4
0
(−1)n1+n2+n3q1
2(n2
1+n2
2+n2
3+n1+n2+n3)+(n1+n2+n3+2)(n4+1)−n11+qn1−n2−n3−n4−1,
which follows after three applications of another well-known identity [3]
(q)2
∞
(ζ)∞(ζ−1q)∞=
∈Z
ζ
n≥0
(−1)nqn2+n
2+n||+1
2(||−).
5.4 Modular properties of the parafermion character
We now study the modular transformations of appearing in the A2parafermion
character. This contains a two-dimensional false theta function and is the more interesting
part of the character. The first step is to apply Lemma 3.1 and rewrite in a more
appropriate form to analyze modular properties. To give this statement, we consider the
function
h(w):=ϑ[1]
3,1(w1)ϑ1,1(w2)−ϑ[1]
3,2(w1)ϑ1,0(w2)
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54 Page 16 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
and also define the following regularized integral for w1∈H\{τ}:
w1
∗τ
f(w2)
(i(w2−τ)) 3
2
dw2:=lim
z→τw1
z
f(w2)
(i(w2−τ)) 3
2
dw2+2if(τ)
i(z−τ),
where both the integral and the one-sided limit are taken along the hyperbolic geodesic
from τto w. Now, one could deform the path of integration away from the hyperbolic
geodesic and provided that the contour does not cross the branch point at w2=τ, the value
of the regularized integral is maintained thanks to the holomorphy of the integrand. The
choice for the path here gives a concrete way to compute the integral while working with
the principal value of the square root and moreover is quite convenient for studying the
modular transformation properties we encounter in this paper. In fact, for the remainder of
this paper we assume that all similar (iterated) integrals in the upper half-plane, including
the one-sided limits involved in the regularization, are taken along hyperbolic geodesics.
Proposition 5.5 We have
(τ)=√3
2πτ+i∞
τw1
∗τ
h(w)
i(w1−τ)(i(w2−τ)) 3
2
dw2dw1.
Proof The claim follows directly from Lemma 3.1 and integration by parts noting that
h(w1,w
1)=0.
We now define the completion of as a function on H×Hby
(τ,w):=√3
2πw
τw1
∗τ
h(w)
i(w1−τ)(i(w2−τ)) 3
2
dw2dw1,
so that, with the limit taken to be vertical
(τ)=lim
w→τ+i∞
(τ,w).
Note that, unlike the one-dimensional false theta functions studied in [11] (where a cut-
plane is used for the domain of w), the integral to i∞can be taken in any direction as long
as the same branch of square-root is used for both half-integral powers in the integrand.
Proposition 5.6 For M =ab
cd
∈SL2(Z)we have
aτ+b
cτ+d,aw +b
cw +d=νη(M)8(cτ+d)2
(τ,w).
Proof It suffices to prove the statement for translation and inversion, in which case the
claim is
(τ+1,w+1) =e2πi
3
(τ,w)and
−1
τ,−1
w=τ2
(τ,w
).(5.3)
We first recall that the integrals in w1and w2(as well as the one-sided limit used in
regularizing the integral) are taken along the hyperbolic geodesic from τto w, i.e., along
the unique circle with a real center containing τand wor along the straight vertical line
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 17 of 31 54
from τto wif Im(τ)=Im(w). Then, we modify
(τ,w) in the following way without
changing its value
(τ,w)=√3
2πw
τ
1
i(w1−τ)lim
z→τ
w1
z
h(w)
(i(w2−τ)) 3
2
dw2+2ih(w1,z)
i(z−τ)
z−w1
τ−w1dw1.(5.4)
In this form, the modular transformation properties may be concluded by the following
modular transformations for h(w), which can be deduced from Lemma 2.1:
h(w+(1,1)) =e2πi
3h(w)andh−1
w1
,−1
w2=w
3
2
1w
1
2
2h(w).
In fact, to work with
(−1
τ,−1
w), we change variables as w1→−
1
w1,w
2→−
1
w2,
z→−
1
z,z→−
1
zin Eq. (5.4). Note that integration and limit are originally taken along
the geodesic from −1
τto −1
w, and the transformations map them to be on the geodesic
from τto w.Wethenfind
−1
τ,−1
w=√3
2πw
τ
χw1,τ√w1√τ
i(w1−τ)
×lim
z→τw1
z
χw2,τ
w
3
2
2τ3
2h−1
w1,−1
w2
(i(w2−τ)) 3
2
dw2
w2
2
+2iχz,τ
z1
2τ1
2h−1
w1,−1
z
i(z−τ)
z−w1
τ−w1
τ
zdw1
w2
1
,
where
χτ1,τ2:=$i(τ1−τ2)
τ1τ2
√τ1√τ2
i(τ1−τ2)∈{−1,+1}for τ1,τ2∈H
keeps track of signs required to work with the principal value of the square-root. Crucially,
along the geodesic from τto wwe have χw1,τ=χw2,τ=χz,τ=χw,τ. Using this fact as
well as the inversion properties of hwe obtain the second identity in (5.3).
A similar and easier computation yields the translation property.
The one-dimensional false theta functions appearing in G0can be treated as in [11]by
following a similar strategy to Propositions 5.5 and 5.6 and using the regularized inte-
gral defined there. Its quantum modularity can be studied by a slight adjustment of the
argument in [11, Theorem 1.5].
6 Generic parafermionic characters of type B2
6.1 Expression in terms of false functions
Our next goal is to study parafermionic characters associated to the affine Lie algebra of
type B2, which has the positive roots
α1=1
0,α2=0
1,α1+α2,2α1+α2!.
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54 Page 18 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
In particular, specializing Eqs. (1.1)and(1.2)tothecaseofB2, so that g=so5,westudy
the constant term of
F(ζ):=q10k
24(k+3) (q;q)2
∞ch[Vk(so5)](ζ;q)
=1
ζ1q, ζ−1
1q, ζ2q, ζ−1
2q, ζ1ζ2q, ζ−1
1ζ−1
2q, ζ2
1ζ2q, ζ−2
1ζ−1
2q;q∞
.
Using Eq. (2.1)weobtain
F(ζ)=q1
3η4ζ−2
1ζ−3
2
2(1 −ζ1)(1 −ζ2)(1 −ζ1ζ2)1−ζ2
1ζ2
ϑ(z1)ϑ(z2)ϑ(z1+z2)ϑ(2z1+z2).
Now, to state our main result for the associated character, we first require some notation.
Using the quadratic form QB(n):=3
2n2
1+3n1n2+3n2
2,wedefine
(τ):=1(τ)+2(τ),
where
1(τ):=
n∈Z2+1
3,1
6(−1)n1−1
3(sgn(n2)+sgn(n1+n2))sgn(n1)(n1+2n2)2−E2(τ)
18 qQB(n),
2(τ):=
n∈Z2+1
3,1
6(−1)n1−1
3sgn(n1+n2)sgn(n2)n1(n1+2n2)qQB(n).
Then for a∈Z2,welet
a(τ):=
n∈Z2+1
3,a1
2+1
6(−1)(a2+1)n1−1
3sgn(n1)sgn(n1+2n2)qQB(n).
We also need the following one-dimensional false theta functions:
φr(τ):=
n∈Z+r
6
sgn(n)q3n2and ωr(τ):=
n∈Z+r
3+1
2
(−1)n−r
3−1
2sgn(n)q3n2
2
to define
F0(τ):=E2(τ)+2
4+η(τ)6
ϑ1
2;τ2+6q−1
24 D1
2(ω1(τ)) −6q−3
8D1
2(ω0(τ))
+q−1
12 6D1
2−η(τ)6
ϑ1
2;τ2+q−1
2η(τ)6
ϑτ
2;τ2−q−1
2η(τ)6
ϑτ+1
2;τ2(φ1(τ))
−q−1
36D1
2+η(τ)6
ϑ1
2;τ2+η(τ)6
ϑτ
2;τ2+η(τ)6
ϑτ+1
2;τ2(φ2(τ)).
With these definitions at hand, we can give our result.
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 19 of 31 54
Proposition 6.1 In the range |q|<|ζ2
1|,|ζ2|,|ζ1ζ2|,|ζ2
1ζ2|<1,wehave
CT[ζ](F(ζ))=q1
3
η(τ)8F0(τ)+9q−1
12
2η(τ)8(τ)+q−1
12
η(τ)20,1(τ)
ϑ1
2;τ2+q−1
41,0(τ)
ϑτ
2;τ2−q−1
41,1(τ)
ϑτ+1
2;τ2
=1+4q2+12q3+38q4+100q5+276q6+688q7+1709q8+4020q9+Oq10.
To prove Proposition 6.1, we require several preliminary results based on Lemmas 4.2
and 4.3 . The first of these is an auxiliary statement that helps us study various limits of
the two lemmas and it follows from Eq. (2.4) and the identity ϑ(3)(0) =2π3η3E2.
Lemma 6.2 For a function F that is holomorphic in a neighborhood of w =0and for
a, b ∈Rwe have, as w →0
F(w)
ϑ(aw)ϑ(bw)=1
4π2abη6F(0)
w2+F(0)
w+a2+b2
ab
E2
24η6F(0) +F(0)
8π2abη6+O(w).
The next two results are then two particular limits of Lemmas 4.2 and 4.3 , respectively,
that appear in the proof of Proposition 6.1. Taking w=(w, −w) in Lemma 4.2 and then
letting w→0 using Lemma 6.2, yields the following statement.
Lemma 6.3 For r ∈Z+1
2we have
ζr
ϑ(z)3=−i
η9
n∈Z
(−1)nq3n2
2−rn 4(3n−r−1)2−E2
8(1 −ζqn)+6n−2r−3
2(1 −ζqn)2+1
(1 −ζqn)3.
Plugging in w=(w, −w) in Lemma 4.3 and taking w→0 using Lemma 6.2,weobtain
the following result.
Lemma 6.4 For r ∈Zwe have
ζr
ϑ(z)2ϑ(2z)=−i
η9
n∈Z
q3n2−rn 2(6n−r−1)2−E2
8(1−ζqn)+12n−2r−3
4(1 −ζqn)2+1
2(1 −ζqn)3
−η6
2
1,2∈{0,1}
=(0,0)
1
ϑ1τ+2
22
(−1)1+2+r2q1(1−r)
2+31n
1−(−1)2ζqn+1
2
⎞
⎟
⎟
⎠.
We are now ready to prove Proposition 6.1.
Proof of Proposition 6.1 Let for r1∈Z,r
2∈Z+1
2and TBdefined in (2.5)
C(r):=CT[ζ]TB(z)η12ζr1
1ζr2
2.
Using Lemma 4.2 with (r, z, w1,w
2)→ (r2,z
2,z
1,2z1), Lemma 6.4 with (z, r)→ (z1,r
1−3k)
and (z, r)→ (z1,r
1−2r2+3k), and Lemma 6.3 with (z, r)→ (z1,r
1−r2)weobtain
TB(z)η12ζr1
1ζr2
2
=
n∈Z2
(−1)n1q
3n2
1
2+3n1n2+3n2
2−r2n1−r1n2
1−ζ2qn1⎛
⎜
⎜
⎝
2(3n1−r1+6n2−1)2−E2
8(1−ζ1qn2)+6n1+12n2−2r1−3
4(1−ζ1qn2)2
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54 Page 20 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
+1
2(1−ζ1qn2)3−
1,2∈{0,1}
=(0,0)
η6
2ϑ1τ+2
22
(−1)1+2+(1+r1+n1)2q1(1+3n1−r1)
2+31n2
1−(−1)2ζ1qn2+1
2
⎞
⎟
⎟
⎠
+
n∈Z2
(−1)n1q
3n2
1
2−3n1n2+3n2
2−r2n1+(2r2−r1)n2
1−ζ2
1ζ2qn1⎛
⎜
⎜
⎝
2(3n1−6n2+r1−2r2+1)2−E2
8(1−ζ1qn2)
−6n1−12n2+2r1−4r2+3
4(1−ζ1qn2)2+1
2(1 −ζ1qn2)3
−
1,2∈{0,1}
=(0,0)
η6
2ϑ1τ+2
22
(−1)1+2+(1+r1+2r2+n1)2q1(1−3n1−r1+2r2)
2+31n2
1−(−1)2ζ1qn2+1
2
⎞
⎟
⎟
⎠
−
n∈Z2
(−1)n1+n2q
3n2
1
2+3n2
2
2−r2n1+(r2−r1)n2
1−ζ1ζ2qn1
×4(3n2−r1+r2−1)2−E2
8(1 −ζ1qn2)+6n2−2r1+2r2−3
2(1 −ζ1qn2)2+1
(1 −ζ1qn2)3.
In the range |q|1
2<|ζ1|<1, |q|<|ζ2|,|ζ1ζ2|,|ζ2
1ζ2|<1, we use (5.2) to find the
constant term as
C(r)=
6
=1
C(r)+
1,2∈{0,1}
=(0,0)
(C7,(r)+C8,(r)),
where
C1(r):=1
4
n∈N2
0
(−1)n1(6n2+3n1−r1)2q
3n2
1
2+3n1n2+3n2
2−r2n1−r1n2,
C2(r):=1
4
n∈N2
0
(−1)n1(6n2−3n1+2r2−r1)2q
3n2
1
2−3n1n2+3n2
2−r2n1+(2r2−r1)n2,
C3(r):=−1
2
n∈N2
0
(−1)n1+n2(3n2+r2−r1)2q
3n2
1
2+3n2
2
2−r2n1+(r2−r1)n2,
C4(r):=−E2
8
n∈N2
0
(−1)n1q
3n2
1
2+3n1n2+3n2
2−r2n1−r1n2,
C5(r):=−E2
8
n∈N2
0
(−1)n1q
3n2
1
2−3n1n2+3n2
2−r2n1+(2r2−r1)n2,
C6(r):=E2
8
n∈N2
0
(−1)n1+n2q
3n2
1
2+3n2
2
2−r2n1+(r2−r1)n2,
C7,(r):=−η6(−1)1+(r1+1)2
2ϑ1τ+2
22q1(1−r1)
2
n∈N2
0
(−1)(2+1)n1q
3n2
1
2+3n1n2+3n2
2+31
2−r2n1+(31−r1)n2,
C8,(r):=−η6(−1)1+r12
2ϑ1τ+2
22q1(1−r1)
2+1r2
n∈N2
0
(−1)(1+2)n1q
3n2
1
2−3n1n2+3n2
2−31
2+r2n1+(31+2r2−r1)n2.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
K. Bringmann et al. Res Math Sci (2021) 8:54 Page 21 of 31 54
Then we may write
CT[ζ](F(ζ)) =q1
3
η8
r∈SB
εB(r)C(r),
where
SB:=%−2,−3
2,−1,1
2,1,−1
2,2,3
2,−1,−3
2,−2,−1
2,1,3
2,2,1
2&,
εB(r):= 1ifr∈"−2,−3
2,−1,1
2,1,−1
2,2,3
2#,
−1ifr∈"−1,−3
2,−2,−1
2,1,3
2,2,1
2#.
Next, we simplify the individual terms in the decomposition of C(r). We start with
C6(r)=E2
8q−1
6(r2
1−2r1r2+2r2
2)
n∈N2
0
(−1)n1+n2q
3
2(n1−r2
3)2+3
2n2−r1−r2
32
.
Note that the sum in C6(r) is invariant if n1and n2are interchanged together with their
respective shifts. The terms C6(r) cancel in pairs and we have
r∈SB
εB(r)C6(r)=0.
For the remaining pieces, the details are quite lengthy. Therefore, we carry them out
only for one of the terms and leave the remaining ones to the reader. We focus on
C3(r)=−9
2q−1
6(r2
1−2r1r2+2r2
2)
n∈N2
0
(−1)n1+n2n2+r2−r1
32
q
3
2(n1−r2
3)2+3
2n2+r2−r1
32
.
We have
C31,−1
2−C3−2,−1
2
=−9
2q−5
12
n∈N2
0
(−1)n1+n2q
3
2n1+1
62n2−1
22
q
3
2n2−1
22
−n2+1
22
q
3
2n2+1
22
=−9
8q−1
24 ∞
n1=0
(−1)n1q
3
2n1+1
62
+9q−5
12
n∈N2
0
(−1)n1+n2n2+1
22
q
3
2n1+1
62+3
2n2+1
22
,
where for the last equality we split off the n2=0 contribution from the first term and
then shift n2→ n2+1 there.
Next, we have
C3−1,1
2−C32,1
2
=−9
2q−5
12
n∈N2
0
(−1)n1+n2q
3
2n1−1
62n2+1
22
q
3
2n2+1
22
−n2−1
22
q
3
2n2−1
22
=9
8q−1
24 ∞
n1=0
(−1)n1q
3
2n1−1
62
−9q−5
12
n∈N2
0
(−1)n1+n2n2+1
22
q
3
2n1−1
62+3
2n2+1
22
,
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54 Page 22 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
where for the last equality we split off the n2=0 part from the second term and then shift
n2→ n2+1 there. Splitting off the terms with n1=0 from the second sum and then
changing n1→−n1and n2→−n2−1weget
C3−1,1
2−C32,1
2
=9
8q−1
24 ∞
n1=0
(−1)n1q
3
2n1−1
62
−9q−3
8∞
n2=0
(−1)n2n2+1
22
q
3
2n2+1
22
+9q−5
12
n∈−N2
(−1)n1+n2n2+1
22
q
3
2n1+1
62+3
2n2+1
22
.
Next, we study
C3−2,−3
2−C31,3
2
=−9
2q−5
12
n∈N2
0
(−1)n1+n2n2+1
62
q
3
2n2+1
62q
3
2n1+1
22
−q
3
2n1−1
22
=9
2q−1
24 ∞
n2=0
(−1)n2n2+1
62
q
3
2n2+1
62
−9q−5
12
n∈N2
0
(−1)n1+n2n2+1
62
q
3
2n1+1
22+3
2n2+1
62
,
where for the last equality we split off the n1=0 contribution from the second term and
then shift n1→ n1+1 there. Finally, we study
C32,3
2−C3−1,−3
2
=−9
2q−5
12
n∈N2
0
(−1)n1+n2n2−1
62
q
3
2n2−1
62q
3
2n1−1
22
−q
3
2n1+1
22
=−9
2q−1
24 ∞
n2=0
(−1)n2n2−1
62
q
3
2n2−1
62
+9q−5
12
n∈N2
0
(−1)n1+n2n2−1
62
q
3
2n1+1
22+3
2n2−1
62
,
where for the last equality we split off the n1=0 part from the first term and then shift
n1→ n1+1 there. Splitting off the n2=0 contribution from the double sum and then
changing n2→−n2and n1→−n1−1weget
C32,3
2−C3−1,−3
2
=−9
2q−1
24 ∞
n2=0
(−1)n2n2−1
62
q
3
2n2−1
62
+1
4q−3
8∞
n1=0
(−1)n1q
3
2n1+1
22
−9q−5
12
n∈−N2
(−1)n1+n2n2+1
62
q
3
2n1+1
22+3
2n2+1
62
.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
K. Bringmann et al. Res Math Sci (2021) 8:54 Page 23 of 31 54
Therefore, the two-dimensional contribution in r∈SBεB(r)C3(r)is
9q−5
12
2
n∈Z2+1
2,1
6(−1)n1−1
2+n2−1
6(1+sgn(n1)sgn(n2))n2
1−n2
2q
3n2
1
2+3n2
2
2.
Under n1→−n1the contribution of the “1” in parentheses picks a minus sign (note that
(−1)2n1=−1) and hence vanishes. Then we can rewrite the two-dimensional part as
9q−5
12
2
n∈Z2+1
2,1
6(−1)n1−1
2+n2−1
6sgn(n1)sgn(n2)n2
1−n2
2q
3n2
1
2+3n2
2
2.
Mapping n→ (−n1−n2,n
2)weget
9q−5
12
2
n∈Z2+1
3,1
6(−1)n1−1
3sgn(n1+n2)sgn(n2)n1(n1+2n2)qQB(n).
The one-dimensional pieces are
−9
8q−1
24
n1∈Z
(−1)n1sgn(n1)q
3
2n1+1
62
+1
8q−3
8
n1∈Z
(−1)n1sgn n1+1
2q
3
2n1+1
22
+9
2q−1
24
n2∈Z
(−1)n2sgn(n2)n2+1
62
q
3
2n2+1
62
−9
2q−3
8
n2∈Z
(−1)n2sgn n2+1
2n2+1
22
q
3
2n2+1
22
.
We next consider the remaining pieces following similar computations.
•Firstly,wehave
r∈SB
εB(r)C1(r)
=9
4q−5
12 ⎛
⎜
⎜
⎝
n∈Z2+1
3,1
6−
n∈Z2+1
3,1
2⎞
⎟
⎟
⎠(−1)n1−1
3(1 +sgn(n1)sgn(n2))(n1+2n2)2qQB(n)
−9
4
n2∈Z
sgn(n2)2n2+2
32
q3n2
2+2n2+9
4
n2∈Z
sgn(n2)2n2+1
32
q3n2
2+n2
−9
2
∞
n1=0
(−1)n1n1+1
32
q
3n2
1
2+3n1
2−9
4
∞
n1=0
(−1)n1n1−2
32
q
3n2
1
2−n1
2
+9
4
∞
n1=0
(−1)n1n1−1
32
q
3n2
1
2+n1
2,
r∈SB
εB(r)C2(r)
=9
4q−5
12 ⎛
⎜
⎜
⎝
n∈Z2+1
3,1
2−
n∈Z2+1
3,1
6⎞
⎟
⎟
⎠(−1)n1−1
3(1 −sgn(n1)sgn(n2))(n1+2n2)2qQB(n)
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54 Page 24 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
+9
4
n2∈Z
sgn(n2)2n2+1
32
q3n2
2+n2−9
4
n2∈Z
sgn(n2)2n2+2
32
q3n2
2+2n2
−9
4
∞
n1=1
(−1)n1n1+1
32
q
3n2
1
2−n1
2+9
4
∞
n1=1
(−1)n1n1+2
32
q
3n2
1
2+n1
2
−9
2
∞
n1=0
(−1)n1n1+2
32
q
3n2
1
2+3n1
2.
Combining the contributions from C1,C
2,andC3, we then find that
r∈SB
εB(r)(C1(r)+C2(r)+C3(r))
=9
2q−5
12
n∈Z2+1
3,1
6(−1)n1−1
3sgn(n1)(sgn(n2)+sgn(n1+n2))(n1+2n2)2qQB(n)
+9
2q−5
12
n∈Z2+1
3,1
6(−1)n1−1
3sgn(n1+n2)sgn(n2)n1(n1+2n2)qQB(n)
+1
2+18q−1
12
n∈Z+1
6
sgn(n)n2q3n2−18q−1
3
n∈Z+1
3
sgn(n)n2q3n2
+9q−1
24
n∈Z+1
6
(−1)n−1
6sgn(n)n2q3n2
2−9q−3
8
n∈Z+1
2
(−1)n−1
2sgn(n)n2q3n2
2.
• Next, we have
r∈SB
εB(r)C4(r)=
r∈SB
εB(r)C5(r)
=−E2
8q−5
12
n∈Z2+1
3,1
6(−1)n1−1
3sgn(n1)(sgn(n2)+sgn(n1+n2))qQB(n)
+E2
8−E2
8q−1
12
n∈Z+1
6
sgn(n)q3n2+E2
8q−1
3
n∈Z+1
3
sgn(n)q3n2
−E2
8q−1
24
n∈Z+1
6
(−1)n−1
6sgn(n)q3n2
2+E2
8q−3
8
n∈Z+1
2
(−1)n−1
2sgn(n)q3n2
2
so that we get
r∈SB
εB(r)(C4(r)+C5(r)+C6(r))=2
r∈SB
εB(r)C4(r).
• We next consider the contributions from C7,(0,1) and C8,(0,1) to find that
−2ϑ1
22
η6
r∈SB
εB(r)C7,(0,1)(r)
=−q−5
12
n∈Z2+1
3,1
6sgn(n1)(sgn(n2)+sgn(n1+n2))qQB(n)
−1+q−1
12
n∈Z+1
6
sgn(n)q3n2+q−1
3
n∈Z+1
3
sgn(n)q3n2
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K. Bringmann et al. Res Math Sci (2021) 8:54 Page 25 of 31 54
+q−1
24
n∈Z+1
6
q3n2
2−q−3
8
n∈Z+1
2
q3n2
2,
−2ϑ1
22
η6
r∈SB
εB(r)C8,(0,1)(r)
=−q−5
12
n∈Z2+1
3,1
6sgn(n1)(sgn(n2)+sgn(n1+n2))qQB(n)
−1+q−1
12
n∈Z+1
6
sgn(n)q3n2+q−1
3
n∈Z+1
3
sgn(n)q3n2
+q−3
8
n∈Z+1
2
q3n2
2−q−1
24
n∈Z+1
6
q3n2
2.
They combine as
ϑ1
22
η6
r∈SB
εB(r)C7,(0,1)(r)+C8,(0,1) (r)
=q−5
12
n∈Z2+1
3,1
6sgn(n1)(sgn(n2)+sgn(n1+n2))qQB(n)
+1−q−1
12
n∈Z+1
6
sgn(n)q3n2−q−1
3
n∈Z+1
3
sgn(n)q3n2.
• Finally, we consider the case =(1,2)(2∈{0,1}) and determine
2ϑτ+2
22
η6
r∈SB
εB(r)C7,(1,2)(r)=
2ϑτ+2
22
η6
r∈SB
εB(r)C8,(1,2)(r)
=(−1)2q−2
3
n∈Z2+1
3,2
3(−1)(2+1)n1−1
3sgn(n1)(sgn(n2)+sgn(n1+n2))qQB(n)
−q−1
3
n∈Z+1
3
sgn(n)q3n2+(−1)2q−7
12
n∈Z+1
6
sgn(n)q3n2.
The claim of the theorem now follows by a direct calculation using the following sign-
identity on the two-dimensional contributions from C7and C8together with the change
of variables n→ (−n1−2n2,n
2) which leaves both QBand the respective lattice shifts
invariant:
sgn(n1)(sgn(n2)+sgn(n1+n2)) −sgn(n1+2n2)(sgn(n2)−sgn(n1+n2))
=2sgn(n1+2n2)sgn(n1).
6.2 Modular properties of the vacuum character
As in the case of parafermionic characters of type A2, we focus on the rank two con-
tributions (τ)anda(τ). For a(τ), the two vectors determining the factors inside the
sign functions are orthogonal with respect to the quadratic form QB. Therefore, a(τ)
can be written in terms of products of two rank one false theta functions. This leaves us
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54 Page 26 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
with (τ) as the only nontrivially rank two piece in the decomposition, which we study
next. We start by defining
f0(w):=ϑ[1]
3,1(w1)ϑ[1]
3,2(w2)−ϑ[1]
3,2(w1)ϑ[1]
3,1(w2),
f1(w):=ϑ[1]
3,1(w1)ϑ[3]
3,2(w2)−ϑ[1]
3,2(w1)ϑ[3]
3,1(w2),
g0(w):=ϑ[1]
3
2,1(w1)ϑ[1]
3
2,0(w2)−ϑ[1]
3
2,0(w1)ϑ[1]
3
2,1(w2),
g1(w):=ϑ[1]
3
2,1(w1)ϑ[3]
3
2,0(w2)−ϑ[1]
3
2,0(w1)ϑ[3]
3
2,1(w2).
A direct calculation, using Lemma 3.1 then gives:
Lemma 6.5 We have
(τ)=2
3τ+i∞
τw1
τ
72f1(w)−E2(τ)f0(w)
i(w1−τ)i(w2−τ)dw2dw1
+1
6τ+i∞
τw1
τ
36g1(w)−E2(τ)g0(w)
i(w1−τ)i(w2−τ)dw2dw1.
Using integration by parts, while noting that we have ϑ[3]
m,r (τ)=1
2πim
∂
∂τ ϑ[1]
m,r (τ)and
f0(w1,w
1)=g0(w1,w
1)=0, we obtain the following:
Proposition 6.6 We have
(τ)=1
πτ+i∞
τw1
∗τ
(4f0(w)+g0(w))1−πi
6(w2−τ)E2(τ)
i(w1−τ)(i(w2−τ)) 3
2
dw2dw1.
In parallel with Sect. 5.2, we define the completion of as
(τ,w):=1
πw
τw1
∗τ
(4f0(w)+g0(w))1−πi
6(w2−τ)E2(τ)
i(w1−τ)(i(w2−τ)) 3
2
dw2dw1.
The following proposition shows the transformation law of
.
Proposition 6.7 For M =ab
cd
∈SL2(Z),wehave
aτ+b
cτ+d,aw +b
cw +d=νη(M)10(cτ+d)3
(τ,w).
Proof It is enough to verify the claim for translation and inversion, in which case it reads
(τ+1,w+1) =e5πi
6
(τ,w),
−1
τ,−1
w=−iτ3
(τ,w).
As in Proposition 5.6, these transformations follow from the following modular transfor-
mations for f0and g0:
f0(w+(1,1)) =e5πi
6f0(w),f
0−1
w1
,−1
w2=−iw
3
2
1w
3
2
2f0(w),
g0(w+(1,1)) =e5πi
6g0(w),g
0−1
w1
,−1
w2=−iw
3
2
1w
3
2
2g0(w).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
K. Bringmann et al. Res Math Sci (2021) 8:54 Page 27 of 31 54
7 Higher rank false theta functions from Schur indices and ˆ
Z-invariants
In this section, we study further examples of rank two false theta functions coming from
Schur’s indices [13,16]and ˆ
Z-invariants [24].
7.1 False theta functions from Schur indices
A remarkable correspondence (or duality) between four-dimensional N=2 superconfor-
mal field theories (SCFTs) and vertex operator algebras was recently found in [4]. Accord-
ingto[4], the Schur index of a N=2 SCFT agrees with the character of a vertex operator
algebra. As mentioned above, the Schur index of the (A1,D
2k+2) Argyres–Douglas the-
ory is a meromorphic Jacobi form of negative index in two variables z=(z1,z
2). It was
demonstrated in [13,16] that the index agrees with the character of a certain affine W-
algebra. Up to some Euler factors and change of variables, this character is given by the
Jacobi form Tk(z;τ) defined in Sect. 2(see [16]). Here we analyze its Fourier coefficients.
For k∈N, letting
Fk(τ):=1
2
n∈Z2+0,1
2(−1)n1sgn(n1)sgn (n2)q
n2
1
2+n1n2+(k+1)n2
2,
it is not hard to prove the following result using slight adjustments of [7, Lemma 3.5].
Proposition 7.1 For |q|<|ζ1|,|ζ2|,|ζ1ζ2|<1and r∈Z2,ther-th Fourier coefficient of
η(τ)3η(k+1
2τ)2
η((k+1)τ)Tk(z;τ)equals
qk+1
4(2r2+1)
n∈Z2
(−1)n1n1,n2+r1n2+r2,n2qn1(n1+1)
2+n1(n2+r1)+(k+1)n2
2+(k+1)(r2+1)n2,
where m,n :=1
2(sgn(m+1
2)+sgn(n+1
2)). In particular,
Fk(τ)=
η(τ)3ηk+1
2τ2
η((k+1)τ)CT[ζ](Tk(z;τ)).
Proof We first let
h(z;τ):=− iη((k+1)τ)3ζ−1
2q−k+1
2ϑ(z1;(k+1)τ)
ϑz2+k+1
2τ;(k+1)τϑz1+z2+k+1
2τ;(k+1)τ.
As in the proof of [7, Lemma 3.5], we obtain an expansion
h(z;τ)=
(n3,n4)∈Z2
n3,n4q(k+1)n3n4+k+1
2n3+k+1
2n4ζ−n4
1ζn3−n4
2.
This combined with the well-known formula (see [3, formula (2.1)])
−iζ−1
2
1η(τ)3
ϑ(z1;τ)=
n∈Z2
n1,n2(−1)n1qn1(n1+1)
2+n1n2ζn1
1,
easily implies the statement.
A direct calculation using Lemma 3.1 then shows the following.
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54 Page 28 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
Proposition 7.2 We have
2Fk(τ)
√2k+1=τ+i∞
τ
η((2k+1)w1)3
i(w1−τ)w1
τ
η(w2)3
i(w2−τ)dw2dw1
+2(k+1)
2k+1
j=0
(−1)jτ+i∞
τ
ϑ[1]
k+1,j((2k+1)w1)
i(w1−τ)w1
τ
ϑ[1]
k+1,j+k+1(w2)
i(w2−τ)dw2dw1.
We now specialize to k=1. This recovers the A2false theta function entering the
character formula of the W0(2)A2vertex algebra studied in [1,6,7]. In this case, the right-
hand side in Proposition 7.2 simplifies and we obtain an elegant integral representation
F1(τ)=3√3
4τ+i∞
τ
η(3w1)3
i(w1−τ)w1
τ
η(w2)3
i(w2−τ)dw2dw1,
as a consequence of the identity
3
j=0
(−1)jϑ[1]
2,j (3w1)ϑ[1]
2,j+2(w2)=1
8η(3w1)3η(w2)3.
This integral admits a modular completion
F1(τ,w) (see Sect. 5) whose modular transfor-
mation properties under SL2(Z) can be easily analyzed.
7.2 ˆ
Z-invariants of 3-manifolds from unimodular H-graphs
The methods of this paper can also be used in analyzing the modular properties of ˆ
Z-
invariants or homological blocks of 3-manifolds. These are certain q-series with integer
coefficients proposed by [24] as a new class of 3-manifold invariants. Remarkably, these
q-series, which are convergent on the unit disk, are designed and expected to produce
the WRT (Witten–Reshetikhin–Turaev) invariants of the relevant manifolds through the
radial limits of the parameter qto the roots of unity.
More concretely, we restrict our attention to plumbed 3-manifolds whose plumbing
graphs are trees. The vertices of the plumbing graph, which we label by {vj}1≤j≤N,are
decorated with a set of integers mjj for 1 ≤j≤N. This data then determines the linking
matrix M=(mjk )1≤j,k≤Nby setting the off-diagonal entries mjk to −1 if the associated
vertices vjand vkare connected by an edge in the graph and by setting it to 0 otherwise.5
We further restrict to cases in which the matrix Mis positive definite. Finally, we define the
shift vector δ:=(δj)1≤j≤Nwhere δj≡deg(vj)(mod 2)and deg(vj) denotes the degree of
the vertex vj. Then the ˆ
Z-invariant is defined for each equivalence class a∈2coker(M)+δ
by
ˆ
Za(q):=q−3N+tr(M)
4
(2πi)NP.V.|w1|=1
...|wN|=1
N
j=1wj−w−1
j2−deg(vj)
−M,a(q;w)dwN
wN
... dw1
w1
,
where
−M,a(q;w):=
∈2MZN+a
q1
4TM−1w
5Here we follow the conventions of [9] and switch the sign of the linking matrix Mcompared to [24].
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
K. Bringmann et al. Res Math Sci (2021) 8:54 Page 29 of 31 54
Fig. 1 The H-graph
and the integrals are defined using the Cauchy principal value (as indicated by the nota-
tion P.V.) and performed in counterclockwise direction. If more specifically the linking
matrix is invertible (unimodular), in which case we also call the associated plumbing graph
unimodular, then coker(M)=0 and there is only one ˆ
Z-invariant.
The ˆ
Z-invariants are conjectured to yield quantum modular forms, which for example
can be verified in the case of unimodular, 3-star plumbing graphs for which the relevant
invariants can be written in terms of unary false theta functions [23, Proposition 4.8]
(see also [9,14]) The simplest plumbing graph for which the corresponding homological
block can not be written in terms of one-dimensional false theta functions is the H-
graph (see Fig. 1). For unimodular H-graph, two of the authors and Mahlburg computed
the ˆ
Z-invariants and studied their higher depth quantum modular properties in [9]. We
explain how the method of Sect. 3can be used to analyze their modular properties. As
demonstrated in [9], the relevant ˆ
Z-invariants can be expressed as a difference of two
series of the form
FS,Q,ε(τ):=
α∈S
ε(α)
n∈N2
0
qKQ(n+α).
Here Q(n)=:σ1n2
1+2σ2n1n2+σ3n2
2with σ1,σ2,σ3∈Zdefines a positive definite quadratic
form and S⊂Q2
>0is a finite set with the property that (1,1)−α,(1−α1,α2)∈Sfor α∈S,
ε(α)=ε((1,1) −α)=ε((1 −α1,α2)), and K∈Nis minimal such that A:=KS⊂N2.
For explicit formulas for Qand Ssee [9]. We can use the symmetry in the sum over αto
obtain that
FS,Q,ε(τ)=1
4
α∈S
ε(α)
n∈Z2+α
sgn(n1)(sgn(n1)+sgn(n2))qKQ(n).
The contribution from sgn(n1)sgn(n1)=1 yields a theta function which is a modular
form. For the contribution from sgn(n1)sgn(n2) we proceed as in Sect. 3to obtain a
representation of ˆ
Zin terms of double integrals and ordinary theta functions.
Proposition 7.3 We have
FS,Q,ε(τ)
=Kσ3√D
2
α∈S,
r(mod σ3)
ε(α)τ+i∞
τ
ϑ[1]
KDσ3,2KD(α1+r)(w1)
i(w1−τ)w1
τ
ϑ[1]
Kσ3,2K(σ2(α1+r)+σ3α2)(w2)
i(w2−τ)dw2dw1
+Kσ1√D
2
α∈S,
r(mod σ1)
ε(α)τ+i∞
τ
ϑ[1]
KDσ1,2KD(α2+r)(w1)
i(w1−τ)w1
τ
ϑ[1]
Kσ1,2K(σ2(α2+r)+σ1α1)(w2)
i(w2−τ)dw2dw1
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54 Page 30 of 31 K. Bringmann et al. Res Math Sci (2021) 8:54
+1
41−2
πarctan σ2
√D
α∈S,
r(mod σ3)
ε(α)ϑKDσ3,2KD(α1+r)(τ)ϑKσ3,2K(σ2(α1+r)+σ3α2)(τ),
where D :=σ1σ3−σ2
2.
8 Conclusion and future work
In this paper, modular properties of rank two false theta functions are studied following
the recent developments in depth two mock modular forms. These results are then used to
study characters of parafermionic vertex algebras of type A2and B2. A natural question is
then how our results extend to parafermions associated to other simple Lie algebras. The
only remaining rank two simple Lie algebra G2is a natural setting where our approach
would directly apply. A more interesting problem is the extension to higher rank Lie alge-
bras such as A3. The approach we use in Sects. 5and 6to compute the constant term
of meromorphic Jacobi forms would still be applicable, albeit becoming computation-
ally more expensive as the number of roots increases. Although being straightforward,
computations of the linear combinations that give the characters of parafermionic vertex
algebras were a particularly strenuous part of the calculations. Therefore, it would be
desirable to streamline this part of the computation ahead of the generalizations.
The modular properties for these higher rank cases, on the other hand, can in principle
be studied again following the corresponding structure for mock modular forms. The
details on higher depth mock modular forms are developed in [2,22,28,31,34]andwe
leave it as future work to form this connection. Another interesting prospect would be
to understand and make predictions on these modular behaviors (weights, multiplier
systems, etc.) through either physical or algebraic methods.
A slightly different direction would be studying the modular properties of Fourier coef-
ficients of the character of Vk(sl3) at the boundary admissible levels k=−3+3
j, where
j≥2andgcd(j, 3) =1, generalizing the results for j=2obtainedin[7] (see also Sect. 7).
This problem essentially requires analyzing the Fourier coefficients of the Jacobi form (see
[26])
ϑ(z1;jτ)ϑ(z2;jτ)ϑ(z1+z2;jτ)
ϑ(z1;τ)ϑ(z2;τ)ϑ(z1+z2;τ),
which can be handled using the methods of this paper.
Acknowledgements
The authors thank Sander Zwegers for fruitful discussions. The research of the first author is supported by the Alfried
Krupp Prize for Young University Teachers of the Krupp foundation and has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 101001179). The third author was partially supported by the NSF grant DMS-1601070 and a Simons
Collaboration Grant for Mathematicians. The research of the fourth author is supported by the SFB/TRR 191 “Symplectic
Structures in Geometry, Algebra and Dynamics”, funded by the DFG (Projektnummer 281071066 TRR 191). Finally, we
thank the referees for providing useful comments.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author details
1Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany,
2Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany, 3Department of Mathematics and Statistics,
SUNY-Albany, Albany, NY 12222, USA.
Received: 22 January 2021 Accepted: 9 July 2021 Published online: 5 September 2021
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
K. Bringmann et al. Res Math Sci (2021) 8:54 Page 31 of 31 54
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