Higher Rank $\hat{Z}$ and $F_{K}$

Higher Rank $\boldsymbol{\hat{Z}}$ and $\boldsymbol{F_{K}}$

Sunghyuk PARK

S. Park

California Institute of Technology, Pasadena, USA<EMAIL_ADDRESS>
Received January 15, 2020, in final form May 11, 2020; Published online May
24, 2020

We study $q$-series-valued invariants of $3$-manifolds that depend on the
choice of a root system $G$. This is a natural generalization of the earlier
works by Gukov–Pei–Putrov–Vafa [arXiv:1701.06567] and Gukov–Manolescu
[arXiv:1904.06057] where they focused on $G={\rm SU}(2)$ case. Although a full
mathematical definition for these “invariants” is lacking yet, we define
$\hat{Z}^{G}$ for negative definite plumbed $3$-manifolds and $F_{K}^{G}$ for
torus knot complements. As in the $G={\rm SU}(2)$ case by Gukov and Manolescu,
there is a surgery formula relating $F_{K}^{G}$ to $\hat{Z}^{G}$ of a Dehn
surgery on the knot $K$. Furthermore, specializing to symmetric
representations, $F_{K}^{G}$ satisfies a recurrence relation given by the
quantum $A$-polynomial for symmetric representations, which hints that there
might be HOMFLY-PT analogues of these $3$-manifold invariants.

$3$-manifold; knot; quantum invariant; complex Chern–Simons theory; TQFT;
$q$-series; colored Jones polynomial; colored HOMFLY-PT polynomial

57K16; 57K31; 81R50

##  1 Introduction

Categorification of the Chern–Simons theory is one of the most exciting open
questions in quantum topology. While homology theories categorifying quantum
link invariants are fairly well-understood by now, whether there is a homology
theory categorifying the Witten–Reshetikhin–Tureav (WRT) invariants is still
widely open. One approach to this problem comes from physics. Using physical
intuition, Gukov, Putrov and Vafa [15] and Gukov, Pei, Putrov and Vafa [14]
conjectured that WRT invariants can be decomposed into categorifiable
“homological blocks” often denoted by $\hat{Z}$. These are integer coefficient
$q$-series and are supposed to be the graded Euler characteristic of a
conjectural homological invariant $\mathcal{H}^{j,k}_{b,{\rm BPS}}$ so that

$\displaystyle\hat{Z}_{b}(Y;q)=\sum_{j,k}(-1)^{j}q^{k}\operatorname{rank}\mathcal{H}^{j,k}_{b,{\rm
BPS}}(Y).$

Furthermore, in [14], they gave a definition of $\hat{Z}$ for negative
definite plumbed $3$-manifolds.

More recently, Gukov and Manolescu [13] studied an analog of $\hat{Z}$ for
knot complements, denoted by $F_{K}(x,q):=\hat{Z}\big{(}S^{3}\setminus
K\big{)}$, where $x$ parametrizes the boundary condition, namely the holonomy
eigenvalue along the meridian in the complex Chern–Simons theory. Their
motivation was to study $\hat{Z}$ of $3$-manifolds described as Dehn surgery
on knots, and in fact they demonstrated that $\hat{Z}$ behaves well under
gluing pieces of $3$-manifolds along their toral boundaries.

As these exciting developments were focused on ${\rm SU}(2)$ as the gauge
group, a natural question is whether they can be generalized to other gauge
groups. One strong motivation for this question comes from the fact that,
quantum ${\rm SU}(N)$ link invariants (and their categorifications) exhibit a
nice regularity under the change of rank, which often can be lifted to an
independent variable $a=q^{N}$ (the third grading), as in HOMFLY-PT
polynomials (and HOMFLY-PT homology, as conjectured in [10]). As we show in
this paper, $\hat{Z}$ and $F_{K}$ have higher rank analogues, and moreover
there is a certain regularity under the change of rank, which hints that there
might be HOMFLY-PT analogues of these $3$-manifold invariants.

### Summary of the paper

The purpose of this paper is to extend these previous works [13, 14] to
arbitrary root system $G$, thereby studying the higher rank analogues of
$\hat{Z}$ and $F_{K}$.

We start by studying the higher rank analogue of $\hat{Z}$ in Section 2. Our
main result is Definition 2.2, where we give a definition of $\hat{Z}_{b}^{G}$
for (weakly) negative definite plumbed $3$-manifolds. We show, in Theorem 2.3,
that $\hat{Z}_{b}^{G}$ is indeed an invariant by proving that it is invariant
under Neumann moves. We also provide some examples of Seifert manifolds and
express their $\hat{Z}_{b}^{G}$ in terms of higher rank false theta functions.

We turn our attention to the higher rank analogue of $F_{K}$ in Section 3.
Since torus knot complements can be expressed as plumbings, we can deduce
$F_{K}^{G}$ for torus knots from Definition 2.2. An explicit expression is
given in Theorem 3.3. A conjectural higher rank surgery formula is also
presented.

Then in Section 4, we specialize our higher rank $F_{K}$ to symmetric
representations. For simplicity we set $G={\rm SU}(N)$. Upon specialization to
symmetric representations, we experimentally check that $F_{K}^{{\rm
SU}(N),{\rm sym}}$ is annihilated by the quantum $A$-polynomial for symmetric
representations, the $a$-deformed quantum $A$-polynomial with specialization
$a=q^{N}$. From this observation, we are naturally led to speculate the
existence of the HOMFLY-PT analogue of $F_{K}$. Some of the expected
properties of the HOMFLY-PT analogue of $F_{K}$ are stated in Conjecture 4.2.

### Notations and conventions

We follow the convention used in [13] for knots, 3-manifolds, and colored
Jones polynomials. Throughout this article, $G$ is a connected, simply
connected111While we believe that our work can be further generalized to non-
simply connected gauge groups by considering the lattice of characters of the
maximal torus instead of the weight lattice $P$, in this article we focus on
simply connected case for simplicity. semisimple Lie group with the root
system $\Delta\subset\mathfrak{h}^{*}$, $Q\subset\mathfrak{h}^{*}$ is the root
lattice, $Q^{\vee}\subset\mathfrak{h}$ is the coroot lattice,
$P\subset\mathfrak{h}^{*}$ is the weight lattice, $W$ is the Weyl group,
$\Delta^{+}$ is the set of positive roots, $\rho$ denotes the Weyl vector
(half-sum of positive roots), and the letters $\alpha$ and $\omega$ will be
reserved for roots and fundamental weights. The inner product $(\cdot,\cdot)$
on $\mathfrak{h}^{*}$ is the standard one normalized such that
$(\alpha,\alpha)=2$ for short roots $\alpha$. The length of a Weyl group
element $w\in W$ will be denoted by $l(w)$. We use the letter $\rm B$ for the
linking matrix of a plumbed 3-manifold, and $\sigma=\sigma({\rm B})$ and
$\pi=\pi({\rm B})$ denote the signature and the number of positive eigenvalues
of ${\rm B}$, respectively. For a multi-index monomial, we use the following
notation

$x^{\beta}:=\prod_{1\leq j\leq r}x_{j}^{(\beta,\omega_{j})},$

where $r=\operatorname{rank}G$ and $\beta\in P$. When it comes to $q$-series,
often we do not bother to fix the overall power of $q$, and just use the
notation $\cong$ for equivalence up to sign and overall power of $q$.

##  2 Higher rank $\boldsymbol{\hat{Z}}$

###  2.1 The set of labels $\boldsymbol{\mathcal{B}}$

Before getting into the definition of $\hat{Z}^{G}_{b}$ for negative definite
plumbings, we have to first describe what the labels $b$ are. In case of
$G={\rm SU}(2)$, these labels $b$ were ${\rm Spin}^{c}$-structures (at least
for plumbings on trees), as clarified by [13]. That is, conjecturally
$\hat{Z}^{{\rm SU}(2)}$ is an invariant for 3-manifolds decorated by ${\rm
Spin}^{c}$-structures. From this, it is natural to expect that, as we
generalize $G$ to an arbitrary root system, $\hat{Z}^{G}$ will be an invariant
of 3-manifolds decorated by structures analogous to ${\rm
Spin}^{c}$-structures.

######  Definition 2.1.

For a plumbed 3-manifold $Y=Y(\Gamma)$, define

$\displaystyle\mathcal{B}^{G}(Y):=\big{(}Q^{V}+\delta\big{)}/{\rm B}Q^{V},$

where $V=V(\Gamma)$ is the set of vertices, and $\delta_{v}=(2-\deg v)\rho$.

This is essentially a generalization of ${\rm Spin}^{c}$-structures, in a
sense that $\mathcal{B}^{{\rm SU}(2)}(Y)\cong{\rm Spin}^{c}(Y)$ canonically.
Recall that ${\rm Spin}^{c}(Y)$ is affinely isomorphic to $H^{2}(Y)$ and
admits a $\mathbb{Z}_{2}$ action by conjugation. Similarly, two of the main
features of $\mathcal{B}^{G}(Y)$ are that it is affinely isomorphic to
$H^{2}(Y;Q)$ and that it admits an action by the Weyl group $W$ (and hence
carries an action by $H^{2}(Y;Q)\rtimes W$).

###  2.2 Higher rank $\boldsymbol{\hat{Z}}$ for negative definite plumbings

Plumbed $3$-manifolds are $3$-manifolds naturally associated to each graph
whose vertices are decorated by integers. Assume for simplicity that the
plumbing graph $\Gamma$ is a tree. Given a decorated tree $\Gamma$, one can
make a framed link $L_{\Gamma}$ by replacing each vertex by an unknot whose
framing is the decoration of the vertex, and by linking any two of the unknots
(in the simplest possible way) whenever the corresponding vertices are
connected by an edge. The plumbed $3$-manifold with plumbing graph $\Gamma$ is
the $3$-manifold obtained as the Dehn surgery on $L_{\Gamma}$. Following [13],
we call the linking matrix ${\rm B}$ of $L_{\Gamma}$ _weakly negative
definite_ if ${\rm B}^{-1}$ is negative definite on the subspace spanned by
vertices of degree $\geq 3$.

We present here a formula for $\hat{Z}$ for (weakly) negative definite plumbed
manifolds, with arbitrary root system $G$. The motivation for this definition
is a heuristic decomposition of WRT invariants which relies heavily on the
Gauss sum reciprocity formula; see Appendix A.

######  Definition 2.2 (higher rank $\hat{Z}$ for negative-definite
plumbings).

For a plumbed 3-manifold $Y$ with a weakly negative definite linking matrix
${\rm B}$222We follow the convention of [13] where every edge is positively
oriented (i.e., the linking is $+1$). Changing orientation of the edges will
change the overall sign. The prefactor $(-1)^{|\Delta^{+}|\pi}$ takes care of
the change of orientation of the edges under Neumann moves. and a choice of
$b\in\mathcal{B}^{G}(Y)$, define

$\displaystyle\hat{Z}_{b}^{G}(Y;q):=(-1)^{|\Delta^{+}|\pi}q^{\frac{3\sigma-\operatorname{Tr}{\rm
B}}{2}(\rho,\rho)}$
$\displaystyle\hphantom{\hat{Z}_{b}^{G}(Y;q):=}{}\times{\rm
v.p.}\int_{|x_{vj}|=1}\prod_{v\in V}\prod_{1\leq j\leq r}\frac{{\rm
d}x_{vj}}{2\pi{\rm i}x_{vj}}\left(\sum_{w\in
W}(-1)^{l(w)}x_{v}^{w(\rho)}\right)^{2-\deg v}\\!\\!\Theta_{b}^{-{\rm
B}}\big{(}x^{-1},q\big{)},\\!\\!\\!$ (2.1) $\displaystyle\Theta_{b}^{-{\rm
B}}\big{(}x^{-1},q\big{)}:=\sum_{\ell\in{\rm
B}Q^{V}+b}q^{-\frac{1}{2}(\ell,{\rm B}^{-1}\ell)}\prod_{v\in
V}x_{v}^{-\ell_{v}}.$

In particular, in case $G={\rm SU}(N)$, this takes the following simple form:

$\displaystyle\hat{Z}_{b}^{{\rm
SU}(N)}(Y;q)=(-1)^{\frac{N(N-1)}{2}\pi}q^{\frac{3\sigma-\operatorname{Tr}{\rm
B}}{2}\frac{N^{3}-N}{12}}$ $\displaystyle\hphantom{\hat{Z}_{b}^{{\rm
SU}(N)}(Y;q)=}{}\times{\rm v.p.}\oint_{|x_{vj}|=1}\prod_{v\in V}\prod_{1\leq
j\leq N-1}\frac{{\rm d}x_{vj}}{2\pi{\rm
i}x_{vj}}F_{3d}(x)\Theta_{2d}^{b}(x,q)$

with

$\displaystyle F_{3d}(x):=\prod_{v\in V}\left(\sum_{w\in
W}(-1)^{l(w)}\prod_{1\leq j\leq
N-1}x_{vj}^{(\omega_{j},w(\rho))}\right)^{2-\deg v}$
$\displaystyle\hphantom{F_{3d}(x)}{}=\prod_{v\in V}\left(\prod_{1\leq j<k\leq
N}\big{(}y_{vj}^{1/2}y_{vk}^{-1/2}-y_{vj}^{-1/2}y_{vk}^{1/2}\big{)}\right)^{2-\deg
v},$ $\displaystyle\Theta_{2d}^{b}(x,q):=\sum_{\ell\in{\rm
B}Q^{V}+b}q^{-\frac{1}{2}(\ell,{\rm B}^{-1}\ell)}\prod_{v\in V}\prod_{1\leq
j\leq N-1}x_{vj}^{-(\omega_{j},\ell_{v})},$

where $x_{j}=\frac{y_{j}}{y_{j+1}}$.

Here “v.p.” denotes the principal value integral. That is, taking the average
over $W$ number of deformed contours, each corresponding to a Weyl chamber.
For instance, for $G={\rm SU}(N)$, the deformed contour corresponding to a
permutation $\sigma\in W=S_{N}$ is given by

$\displaystyle|y_{\sigma(1)}|<|y_{\sigma(2)}|<\cdots<|y_{\sigma(N)}|.$

In practice, this means that there are $W$ number of ways to expand the
integrand into power series, and the contour integral simply picks out the
constant term in the average of these power series.

$m_{1}\pm 1$$\pm 1$$m_{2}\pm 1$$\simeq$$m_{1}$$m_{2}$$m\pm 1$$\pm
1$$\simeq$$m$$m_{1}$$0$$m_{2}$$\simeq$$m_{1}+m_{2}$ Figure 1: Neumann moves on
plumbing trees.

It is known by Neumann that two plumbed manifolds $Y(\Gamma)$ and
$Y(\Gamma^{\prime})$ are the same if and only if the plumbing graphs $\Gamma$
and $\Gamma^{\prime}$ are related by a sequence of Neumann moves in Fig. 1.
Therefore the following result implies that $\hat{Z}_{b}^{G}$ is a topological
invariant:

######  Theorem 2.3.

The $q$-series $\hat{Z}_{b}^{G}$ defined above is invariant under Neumann
moves.

The proof is straightforward and is basically the same as the proof of
Proposition 4.6 in [13]. So here we only give a sketch of the proof.

###### Proof.

Consider the first move. Under this move $3\sigma-\operatorname{Tr}{\rm B}$
remains unchanged. When the signs on the top are $-1$, $\pi$ does not change,
and the contribution of the vector
$\ell=\big{(}\vec{\ell}_{l},0,\vec{\ell}_{r}\big{)}$ for the top graph to the
theta function is the same as the contribution of the vector
$\ell^{\prime}=\big{(}\vec{\ell}_{l},\vec{\ell}_{r}\big{)}$ for the bottom
graph. That is,

$\displaystyle\big{(}\ell,{\rm B}^{-1}\ell\big{)}=\big{(}\ell^{\prime},{\rm
B}^{\prime-1}\ell^{\prime}\big{)}.$ (2.2)

When the signs on the top are $+1$, the sign of $(-1)^{\pi}$ changes, and the
contribution of the vector
$\ell=\big{(}\vec{\ell}_{l},0,\vec{\ell}_{r}\big{)}$ for the top graph is the
same as the contribution of the vector
$\ell^{\prime}=\big{(}\vec{\ell}_{l},-\vec{\ell}_{r}\big{)}$ for the bottom
graph in a sense of (2.2). The effect of change of sign in $\vec{\ell}_{r}$ is
compensated by the change of sign in $(-1)^{|\Delta^{+}|\pi}$. Hence
$\hat{Z}^{G}$ is invariant under the first Neumann move.

Consider the second move. When the signs on the top are $-1$, $\pi$ does not
change, and $3\sigma-\operatorname{Tr}{\rm B}$ increases by $1$. The
contribution of the vector
$\ell=\big{(}\vec{\ell}_{l},\ell_{0},w(\rho)\big{)}$ for the top graph is
related to that of the vector
$\ell^{\prime}=\big{(}\vec{\ell}_{l},\ell_{0}+w(\rho)\big{)}$ for the bottom
graph via

$\big{(}\ell,{\rm B}^{-1}\ell\big{)}=\big{(}\ell^{\prime},{\rm
B}^{\prime-1}\ell^{\prime}\big{)}-(\rho,\rho).$

The extra factor of $q^{-\frac{(\rho,\rho)}{2}}$ due to this change is
cancelled out by the change in $q^{\frac{3\sigma-\operatorname{Tr}{\rm
B}}{2}(\rho,\rho)}$. When the signs on the top are $+1$, both $\pi$ and
$3\sigma-\operatorname{Tr}{\rm B}$ decrease by $1$. The contribution of the
vector $\ell=\big{(}\vec{\ell}_{l},\ell_{0},w(\rho)\big{)}$ for the top graph
is related to that of the vector
$\ell^{\prime}=\big{(}\vec{\ell}_{l},\ell_{0}-w(\rho)\big{)}$ for the bottom
graph via

$\big{(}\ell,{\rm B}^{-1}\ell\big{)}=\big{(}\ell^{\prime},{\rm
B}^{\prime-1}\ell^{\prime}\big{)}+(\rho,\rho).$

The extra factor of $q^{\frac{(\rho,\rho)}{2}}$ due to this change is
cancelled by the change in $q^{\frac{3\sigma-\operatorname{Tr}{\rm
B}}{2}(\rho,\rho)}$, and the effect of change of sign in $w(\rho)$ is
compensated by the change of sign in $(-1)^{|\Delta^{+}|\pi}$. Hence
$\hat{Z}^{G}$ is invariant under the second Neumann move.

Consider the third move. This time $3\sigma-\operatorname{Tr}{\rm B}$ is
preserved, and the sign of $(-1)^{\pi}$ changes. Under this move, the
contribution of the vector
$\ell=\big{(}\vec{\ell}_{l},\ell_{l},0,\ell_{r},\vec{\ell}_{r}\big{)}$ for the
top graph is the same as the contribution of the vector
$\ell^{\prime}=\big{(}\vec{\ell}_{l},\ell_{l}-\ell_{r},-\vec{\ell}_{r}\big{)}$
for the bottom graph, in a sense of (2.2). Again, the effect of change of sign
in $\vec{\ell}_{r}$ is cancelled out the change of sign in
$(-1)^{|\Delta^{+}|\pi}$. Hence $\hat{Z}^{G}$ is invariant under the third
Neumann move. ∎

###  2.3 Some examples and higher rank false theta functions

####  2.3.1 $\boldsymbol{Y=S_{0}^{3}(K_{n})}$

The 0-surgery on twist knots are probably the simplest examples. They admit
simple plumbing diagrams given in Fig. 2.333Although we have assumed for
simplicity in Definition 2.2 that the plumbing graph is a tree, we can extend
this definition to plumbings with loops, as in [7]. For instance,

$G$ |  $\hat{Z}_{0}\big{(}S_{0}^{3}(\mathbf{5}_{2})\big{)}\cong$2pt
---|---
${\rm SU}(2)$ |  $\frac{1}{2!}\big{(}1-q+q^{3}-q^{6}+q^{10}-q^{15}+q^{21}-q^{28}+q^{36}-q^{45}+q^{55}-q^{66}+q^{78}-\cdots\big{)}$2pt2pt
${\rm SU}(3)$ |  $\frac{1}{3!}\big{(}1-2q+2q^{3}+q^{4}-4q^{6}+2q^{9}+2q^{10}+q^{12}-2q^{13}-4q^{15}+2q^{18}+2q^{19}+\cdots\big{)}$2pt2pt
${\rm SU}(4)$ |  $\frac{1}{4!}\big{(}1-3q+q^{2}+4q^{3}-2q^{4}+q^{5}-5q^{6}-2q^{7}+3q^{8}+2q^{9}+9q^{10}-2q^{11}-\cdots\big{)}$2pt2pt
${\rm SU}(5)$ | $\frac{1}{5!}\big{(}1-4q+3q^{2}+6q^{3}-7q^{4}-2q^{5}+2q^{7}-2q^{8}+6q^{9}+15q^{10}-12q^{11}-23q^{12}+\cdots\big{)}$

Indeed, for every positive twist knot $K_{p}$ the following is easy to deduce
from (2.1).

######  Proposition 2.4.

For the $0$-surgery on the twist knot $K_{p}$, its $\hat{Z}^{G}$ is given by

$\displaystyle\hat{Z}_{0}^{G}(S_{0}^{3}(K_{p}))\cong\frac{1}{|W|}\sum_{\ell\in
P_{+}\cap(Q+\rho)}N_{\ell}\sum_{w\in
W}(-1)^{l(w)}q^{\frac{1}{2}||\sqrt{p}\ell-\frac{1}{\sqrt{p}}w(\rho)||^{2}}=:\frac{1}{|W|}\chi_{p,\rho},$
(2.3)

where

$\displaystyle N_{\ell}:=\sum_{w\in W}(-1)^{l(w)}K(w(\ell)),$

and $K(\beta)$ denotes the Kostant partition function.444For example,
$N_{\ell}$ is $\operatorname{sgn}((\ell,\alpha_{1}))$ for ${\rm SU}(2)$, and
$\operatorname{sgn}\big{(}\prod\limits_{\alpha\in\Delta_{+}}(\ell,\alpha)\big{)}\min\\{|(\ell,\alpha_{1})|,|(\ell,\alpha_{2})|\\}$
for ${\rm SU}(3)$.

Note that $\chi_{p,\rho}$ is exactly the higher rank false theta function (a
character of the log-VOA $W^{0}(p)_{Q}$) given in equation (1.2) of [5]!
Similarly for double twist knots $K_{m,n}$ with $m,n>0$,555In our notation,
$K_{m,n}$ denotes the double twist knot with $m$ and $n$ full twists.

$\hat{Z}_{0}^{G}(S_{0}^{3}(K_{m,n}))\cong\frac{1}{|W|}\chi_{m,\rho}\chi_{n,\rho}.$
(2.4)

###### Proof of Proposition 2.4.

The 0-surgery on $K_{p}$ has a simple plumbing description as shown in Fig. 2.

$m$$0$$0$$n$$+$$-$$-1$$0$$p$$+$$-$ Figure 2: Plumbing diagrams for the
0-surgery on $K_{m,n}$ and $K_{p}=K_{1,p}$.

The linking matrix and its inverse are

${\rm B}=\begin{pmatrix}-1&0&0\\\ 0&0&1\\\ 0&1&p\end{pmatrix},\qquad{\rm
B}^{-1}=\begin{pmatrix}-1&0&0\\\ 0&-p&1\\\ 0&1&0\end{pmatrix}.$

There is a single trivalent vertex with $0$ framing. This contributes the
following factor in $F_{3d}(x)$:

$\bigg{(}\sum_{w\in
W}(-1)^{l(w)}x_{0}^{w(\rho)}\bigg{)}^{-1}=\frac{1}{|W|}\sum_{\ell_{0}\in
P_{+}\cap(Q+\rho)}N_{\ell_{0}}\sum_{w\in W}(-1)^{l(w)}x_{0}^{w(\ell_{0})}.$

For $\ell=(0,\ell_{0},\ell_{p})^{t}$,

$q^{-\frac{1}{2}(\ell,{\rm
B}^{-1}\ell)}=q^{\frac{1}{2}||\sqrt{p}\ell_{0}-\frac{1}{\sqrt{p}}\ell_{p}||^{2}-\frac{1}{2p}||\ell_{p}||^{2}}.$

Applying (2.1), it is straightforward to get (2.3).

Using a plumbing description of the 0-surgery on $K_{m,n}$ (Fig. 2), it is
easy to derive (2.4) as well. ∎

####  2.3.2 $\boldsymbol{Y=\Sigma(p_{1},p_{2},p_{3})}$

For convenience let us define the following notation for higher rank false
theta functions:

$\displaystyle\chi_{p,\beta}^{G}:=\sum_{\ell\in
P_{+}\cap(Q+\rho)}N_{\ell}\sum_{w\in
W}(-1)^{l(w)}q^{\frac{1}{2}||\sqrt{p}\ell-\frac{1}{\sqrt{p}}w(\beta)||^{2}}.$

When $G={\rm SU}(2)$, this becomes

$\displaystyle\chi_{p,n\rho}^{{\rm SU}(2)}=\Psi_{p,p-n},\qquad\text{for}\quad
n=1,\dots,p-1,$

where

$\displaystyle\Psi_{p,r}:=\sum_{\begin{subarray}{c}\ell\in\mathbb{Z}\\\
\ell=r\text{ mod }2p\end{subarray}}\operatorname{sgn}(\ell)q^{\ell^{2}/4p}$

is the usual false theta function, and in this sense $\chi^{G}_{p,\beta}$ is
the higher rank generalization of the false theta functions.

######  Proposition 2.5.

For the Brieskorn sphere $Y=\Sigma(p_{1},p_{2},p_{3})$ with
$0<p_{1}<p_{2}<p_{3}$ pairwise relatively prime, we have

$\displaystyle\hat{Z}_{0}^{G}(\Sigma(p_{1},p_{2},p_{3}))\cong\sum_{(w_{1},w_{2})\in
W^{2}}(-1)^{l(w_{1}w_{2})}\chi_{p_{1}p_{2}p_{3},p_{2}p_{3}\rho+p_{1}p_{3}w_{1}(\rho)+p_{1}p_{2}w_{2}(\rho)}.$

That is, it is a sum of $|W|^{2}$ number of higher rank false theta
functions.666That $\hat{Z}$’s for Brieskorn spheres should be expressed as
sums of higher rank false theta functions was envisaged earlier in [6].

###### Proof.

The proof is analogous to that of Proposition 4.8 in [13]. ∎

Note that we did not have to treat $\Sigma(2,3,5)$ separately. In this sense,
using $\chi_{p,\beta}$ as false theta functions is more natural than using
$\Psi_{p,n}$.

####  2.3.3
$\boldsymbol{Y=M\big{(}a_{0};\frac{a_{1}}{b_{1}},\frac{a_{2}}{b_{2}},\frac{a_{3}}{b_{3}}\big{)}}$

Let $b_{1},b_{2},b_{3}>0$ and assume that $Y$ has negative orbifold number,
i.e.,

$e=a_{0}+\sum_{j=1}^{3}\frac{a_{j}}{b_{j}}<0.$ (2.5)

Assume further that the central meridian is trivial in homology, i.e.,

$\displaystyle e\operatorname{lcm}(b_{1},b_{2},b_{3})=-1.$

Then their $\hat{Z}_{b}$’s can be expressed as signed sum of higher rank false
theta functions:

######  Proposition 2.6.

Under the assumptions as above, $\hat{Z}^{G}$ for
$Y=M\big{(}a_{0};\frac{a_{1}}{b_{1}},\frac{a_{2}}{b_{2}},\frac{a_{3}}{b_{3}}\big{)}$
is given by

$\displaystyle\hat{Z}_{b}^{G}\left(M\left(a_{0};\frac{a_{1}}{b_{1}},\frac{a_{2}}{b_{2}},\frac{a_{3}}{b_{3}}\right)\right)$
$\displaystyle\qquad{}\cong\sum_{(w_{1},w_{2})\in
W^{2}}\mathbf{1}_{b}(w_{1},w_{2})(-1)^{l(w_{1}w_{2})}\chi_{\frac{b_{1}b_{2}b_{3}}{|H_{1}|},\frac{b_{2}b_{3}}{|H_{1}|}\rho+\frac{b_{1}b_{3}}{|H_{1}|}w_{1}(\rho)+\frac{b_{1}b_{2}}{|H_{1}|}w_{2}(\rho)},$
(2.6)

where

$\displaystyle\mathbf{1}_{b}(w_{1},w_{2}):=\begin{cases}1,&\text{if
}\ell(\rho,\rho,w_{1}(\rho),w_{2}(\rho))\in{\rm B}Q^{V}+b,\\\
0,&\text{otherwise.}\end{cases}$

Observe that Proposition 2.6 is a slight generalization of Proposition 2.5.

###### Proof.

Since $Y$ is a Seifert manifold with $3$ singular fibers, it can be described
as a star-shaped plumbing with 3 legs. The only vertices whose degree is not 2
are the central vertex and the terminal vertices. Denote by
$\ell(\ell_{0},\ell_{1},\ell_{2},\ell_{3})$ an element $\ell\in{\rm B}Q^{V}+b$
such that

$\ell_{v}=\begin{cases}\ell_{0},&v\text{ is the central vertex},\\\
\ell_{1},\ell_{2},\ell_{3},&v\text{ is the corresponding terminal vertex},\\\
0,&\text{otherwise}.\end{cases}$

Then for any $\ell$ with $\ell_{1},\ell_{2},\ell_{3}\in W(\rho)$,

$q^{-\frac{1}{2}(\ell,{\rm
B}^{-1}\ell)}=q^{\frac{1}{2|H_{1}|}||\sqrt{b_{1}b_{2}b_{3}}\ell_{0}-\frac{1}{\sqrt{b_{1}b_{2}b_{3}}}(b_{2}b_{3}\ell_{1}+b_{3}b_{1}\ell_{2}+b_{1}b_{2}\ell_{3})||^{2}+C}$

for some constant $C$ independent of $\ell$. Applying (2.1), it is
straightforward to obtain (2.6). Note that the assumption
$e\operatorname{lcm}(b_{1},b_{2},b_{3})=-1$ was introduced so that

$\displaystyle\ell(\rho,\rho,w_{1}(\rho),w_{2}(\rho))\in{\rm
B}Q^{V}+b\Leftrightarrow\ell(\rho+Q,\rho,w_{1}(\rho),w_{2}(\rho))\in{\rm
B}Q^{V}+b.$ ∎

We end this section by posing some interesting open questions arising from our
definition of higher rank $\hat{Z}$. When $G={\rm SU}(2)$, the study of
modular-like properties of the $q$-series $\hat{Z}$ was initiated in [3, 4,
6]. It is proved in special cases and conjectured for general negative
definite plumbings that, although $\hat{Z}$ is not modular in the traditional
sense, it exhibits a more exotic modular property called _quantum modularity_
à la Zagier [20]. Naturally, we can ask if the same is true for higher rank
$\hat{Z}$.

######  Question 2.7.

What are the quantum modular properties of $\hat{Z}^{G}$?

Another open question is to extend our definition to drop the condition of
(weakly) negative definiteness. In particular, the orientation reversal of a
negative definite plumbed manifold is positive definite. While for polynomial
invariants, such as Jones polynomials, the orientation reversal is governed
simply by the transformation $q\leftrightarrow q^{-1}$, for $q$-series
invariants such transformation is ill-defined, and the orientation reversal is
more subtle. This issue, in $G={\rm SU}(2)$ case, was first tackled in [6,
Section 7], where it is observed that for some negative definite plumbed
manifolds $Y$, the $\hat{Z}$ of the orientation reversal $-Y$ can be obtained
by taking the mock theta function associated to the false theta function
$\hat{Z}(Y)$. It would be very interesting to study this orientation reversal
problem in our higher rank setting.

######  Question 2.8.

How does $\hat{Z}^{G}$ behave under orientation reversal? What are the higher
rank mock theta functions associated to the higher rank false theta functions?

##  3 Higher rank $\boldsymbol{F_{K}}$

###  3.1 Review of $\boldsymbol{{\rm SU}(2)}$ case

Recently Gukov and Manolescu [13] conjectured the existence of a knot
invariant $F_{K}(x,q)$, which is the analogue of $\hat{Z}$ for knot
complements.

######  Conjecture 3.1 (Gukov–Manolescu [13]).

For any knot $K$, there exists a series

$\displaystyle F_{K}(x,q)=\frac{1}{2!}\sum_{\begin{subarray}{c}m\geq 1\\\
\text{odd}\end{subarray}}\big{(}x^{m/2}-x^{-m/2}\big{)}f_{m}(q),$

whose coefficients $f_{m}(q)$ are Laurent series with integer coefficients,
such that the asymptotic expansion of $F_{K}\big{(}x,{\rm e}^{\hbar}\big{)}$
is the same as the Melvin–Morton–Rozansky expansion [2, 17, 19] for colored
Jones polynomials:

$\displaystyle F_{K}\big{(}x,{\rm
e}^{\hbar}\big{)}=\big{(}x^{1/2}-x^{-1/2}\big{)}\sum_{j\geq
0}\frac{P_{j}(x)}{\Delta_{K}(x)^{2j+1}}\frac{\hbar^{j}}{j!},$

where $P_{j}(x)\in\mathbb{Z}\big{[}x,x^{-1}\big{]}$, $P_{0}=1$, and
$\Delta_{K}(x)=\nabla_{K}\big{(}x^{1/2}-x^{-1/2}\big{)}$ is the Alexander
polynomial for $K$. In particular, in the semi-classical limit we should have

$\displaystyle\lim_{q\rightarrow
1}F_{K}(x,q)=\frac{x^{1/2}-x^{-1/2}}{\Delta_{K}(x)}.$

Moreover, this series is annihilated by the quantum $A$-polynomial:

$\displaystyle\hat{A}_{K}(\hat{x},\hat{y},q)F_{K}(x,q)=0.$

In [13], $F_{K}$ is computed for torus knots and the figure-8 knot, and more
recently, in [18], $F_{K}$ is computed for a bigger class of knots, including
positive braid knots, positive double twist knots and the Whitehead link.

###  3.2 Higher rank $\boldsymbol{F_{K}}$

Let us study the higher rank generalization of $F_{K}(x,q)$. As a natural
generalization of Conjecture 3.1, we make the following conjecture.

######  Conjecture 3.2 (higher rank $F_{K}$).

For any knot $K$ and a choice of a root system $G$, there exists a series

$\displaystyle F_{K}^{G}(\mathbf{x},q)=\frac{1}{|W|}\sum_{\beta\in
P_{+}\cap(Q+\rho)}f_{\beta}^{G}(q)\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)},$

where $\mathbf{x}=(x_{1},\dots,x_{r})$ and the coefficients $f_{\beta}^{G}(q)$
are Laurent series with integer coefficients, such that its asymptotic
expansion agrees with the higher rank Melvin–Morton–Rozansky expansion for the
higher rank colored Jones polynomials

$\displaystyle F_{K}^{G}\big{(}\mathbf{x},{\rm
e}^{\hbar}\big{)}=\prod_{\alpha\in\Delta^{+}}\big{(}x^{\alpha/2}-x^{-\alpha/2}\big{)}\sum_{j\geq
0}\frac{P_{j}(\mathbf{x})}{\Big{(}\prod\limits_{\alpha\in\Delta^{+}}\Delta_{K}(x^{\alpha})\Big{)}^{2j+1}}\frac{\hbar^{j}}{j!},$

where
$P_{j}(\mathbf{x})\in\mathbb{Z}\big{[}x_{1},x_{1}^{-1},\dots,x_{r},x_{r}^{-1}\big{]}$
and $P_{0}=1$. In particular, in the semi-classical limit we should have

$\displaystyle\lim_{q\rightarrow
1}F_{K}^{G}(\mathbf{x},q)=\prod_{\alpha\in\Delta^{+}}\frac{x^{\alpha/2}-x^{-\alpha/2}}{\Delta_{K}(x^{\alpha})}.$

Moreover, this series should be annihilated by the $($higher rank$)$ quantum
$A$-polynomial:

$\displaystyle\hat{A}_{K}\big{(}\hat{x}_{1},\hat{y}_{1},\dots,\hat{x}_{r},\hat{y}_{r}\big{)}F_{K}^{G}(\mathbf{x},q)=0.$

Our main result in this section is an explicit expression for
$F_{K}^{G}(\mathbf{x},q)$ for torus knots.

######  Theorem 3.3 (higher rank $F_{K}$ for torus knots).

For $K=T_{s,t}$, $f_{\beta}^{G}(q)$ is a monomial of degree
$\frac{(\beta,\beta)}{2st}$, up to an overall $q$-power. More precisely,

$\displaystyle F_{T_{s,t}}^{G}\cong\frac{1}{|W|}\sum_{\beta\in
P_{+}\cap(Q+\rho)}\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}$
$\displaystyle\hphantom{F_{T_{s,t}}^{G}\cong}{}\times\sum_{(w_{1},w_{2})\in
W^{2}}(-1)^{l(w_{1}w_{2})}\mathbf{1}(\beta,w_{1},w_{2})N_{\frac{1}{st}(\beta+tw_{1}(\rho)+sw_{2}(\rho))}q^{\frac{(\beta,\beta)}{2st}},$
(3.1)

where

$\displaystyle\mathbf{1}(\beta,w_{1},w_{2}):=\begin{cases}1,&\text{if
}\frac{1}{st}(\beta+tw_{1}(\rho)+sw_{2}(\rho))\in P_{+}\cap(Q+\rho),\\\
0,&\text{otherwise}.\end{cases}$

###### Proof.

This can be derived either directly from (2.1) by using plumbing description
or by reverse-engineering using the higher rank surgery formula that we
discuss below. Here we present a direct derivation. Recall from [13] that the
complement of $T_{s,t}$ has a plumbing description as in Fig. 3, where
$0<t^{\prime}<t$, $0<s^{\prime}<s$ are chosen such that
$st^{\prime}\equiv-1~{}(\text{mod }t)$ and $ts^{\prime}\equiv-1~{}(\text{mod
}s)$.

$-\frac{t}{t^{\prime}}$$-1$$-\frac{s}{s^{\prime}}$$-st$ Figure 3: Complement
of $T_{s,t}$.

The linking matrix is

${\rm B}=\begin{pmatrix}-st&1&0&0\\\ 1&-1&1&1\\\
0&1&\big{(}{-}\frac{t}{t^{\prime}}\big{)}&0\\\
0&1&0&\big{(}{-}\frac{s}{s^{\prime}}\big{)}\end{pmatrix},$

where $\big{(}{-}\frac{t}{t^{\prime}}\big{)}$ and
$\big{(}{-}\frac{s}{s^{\prime}}\big{)}$ should be understood as block matrices
corresponding to the continued fractions. To compute the integral (2.1) with
$x_{-st}$ left unintegrated, we just have to replace the theta function
$\Theta^{-{\rm B}}\big{(}x^{-1},q\big{)}$ with

$\Theta^{-{\rm B}^{\prime}}\big{(}x^{-1},q\big{)}\cong\sum_{\alpha\in
Q^{V^{\prime}}}q^{-\frac{1}{2}(\alpha,{\rm
B}^{\prime}\alpha)-(\alpha,\delta)}\prod_{v\in V^{\prime}}x_{v}^{-({\rm
B}^{\prime}\alpha+\delta)}\cdot x_{-st}^{-\alpha_{-1}-\rho},$

where $V^{\prime}=V\setminus\\{v_{-st}\\}$ and ${\rm B}^{\prime}$ is the
corresponding sub-linking matrix. Set $\beta=-\alpha_{-1}-\rho$. We need to
multiply $\Theta^{-{\rm B}^{\prime}}\big{(}x^{-1},q\big{)}$ with

$\prod_{v\in V^{\prime}}\bigg{(}\sum_{w\in
W}(-1)^{l(w)}x_{v}^{w(\rho)}\bigg{)}^{2-\deg v}$

and take the constant term with respect to variables $x_{v}$, $v\in
V^{\prime}$. As $2-\deg v$ is non-zero for only 3 vertices (the central vertex
$v_{-1}$ and the 2 terminal vertices) it is pretty easy to compute. The only
contributions come from those $\alpha$’s such that ${\rm
B}^{\prime}\alpha+\delta$ takes values $w_{1}(\rho)$, $w_{2}(\rho)$ on the
terminal vertices for some $w_{1},w_{2}\in W$, a value in $Q+\rho$ in the
central vertex, and $0$ on all the other vertices. Using simple linear
algebra, it is easy to check that for those $\alpha$’s,

$q^{-\frac{1}{2}(\alpha,{\rm
B}^{\prime}\alpha)-(\alpha,\delta)}=q^{\frac{(\beta,\beta)}{2st}+C}$

for some constant $C$ independent of $\alpha$, and that
$\frac{1}{st}(\beta+tw_{1}(\rho)+sw_{2}(\rho))$ is the value of ${\rm
B}^{\prime}\alpha+\delta$ on the central vertex. This proves (3.1). ∎

######  Example 3.4 (right-handed trefoil with $G={\rm SU}(3)$).

The first few $f_{\beta}^{{\rm SU}(3)}$ are (up to overall sign and $q$-power)

$\displaystyle f_{(1,1)}=-q,\qquad f_{(4,1)}=-q^{2},\qquad
f_{(5,2)}=-2q^{3},\qquad f_{(7,1)}=-q^{4},\qquad f_{(5,5)}=q^{5},$
$\displaystyle f_{(7,4)}=2q^{6},\qquad f_{(10,1)}=-q^{7},\qquad
f_{(8,5)}=q^{8},\qquad f_{(11,2)}=-2q^{9},\qquad f_{(7,7)}=-q^{9},$
$\displaystyle f_{(13,1)}=-q^{11},\qquad f_{(11,5)}=q^{12},\qquad\dots,$

where we have written $\beta$ in the fundamental weights basis. (Because
$f_{(m,n)}=f_{(n,m)}$, we have only written those terms with $m\geq n$.) The
$q$-power of this $f_{\beta}$ is, up to overall constant,

$\displaystyle\frac{(\beta,\beta)}{12}.$

In the $q\rightarrow 1$ limit we have, as expected,

$\displaystyle F_{\mathbf{3}_{1}^{r}}^{{\rm
SU}(3)}(x_{1},x_{2},1)=\frac{x_{1}^{1/2}-x_{1}^{-1/2}}{x_{1}+x_{1}^{-1}-1}\frac{x_{2}^{1/2}-x_{2}^{-1/2}}{x_{2}+x_{2}^{-1}-1}\frac{x_{1}^{1/2}x_{2}^{1/2}-x_{1}^{-1/2}x_{2}^{-1/2}}{x_{1}x_{2}+x_{1}^{-1}x_{2}^{-1}-1}.$

Just as in ${\rm SU}(2)$ case [13], we conjecture the following surgery
formula (analogous to Conjecture 1.7 of [13]) relating $F_{K}^{G}$ to
$\hat{Z}^{G}_{b}\big{(}S_{p/r}^{3}(K)\big{)}$:

######  Conjecture 3.5 (higher rank surgery formula).

Let $K\subset S^{3}$ be a knot. Then

$\displaystyle\hat{Z}_{b}^{G}\big{(}S_{p/r}^{3}(K)\big{)}\cong\mathcal{L}_{p/r}^{(b)}\left[\prod_{\alpha\in\Delta^{+}}\big{(}x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}}\big{)}F_{K}^{G}(\mathbf{x},q)\right],$

whenever the RHS makes sense.

This is a theorem for knots and 3-manifolds represented by negative-definite
plumbings, as a straightforward generalization of Theorem 1.2 of [13]. For
instance, surgery on $\mathbf{3}_{1}^{r}$ gives us the following
$\hat{Z}^{{\rm SU}(3)}$’s:

$r$ | $S_{-1/r}^{3}(\mathbf{3}_{1}^{r})$ |  $\hat{Z}_{0}^{{\rm SU}(3)}\big{(}S_{-1/r}^{3}(\mathbf{3}_{1}^{r})\big{)}$2pt
---|---|---
$1$ | $\Sigma(2,3,7)$ |  $1-2q+2q^{3}+q^{4}-2q^{5}-2q^{8}+4q^{9}+2q^{10}-4q^{11}+2q^{13}-6q^{14}+\cdots$2pt2pt
$2$ | $\Sigma(2,3,13)$ |  $1-2q+2q^{3}-q^{4}+2q^{10}-2q^{11}-2q^{14}+2q^{16}+2q^{19}-2q^{20}+4q^{21}-\cdots$2pt2pt
$3$ | $\Sigma(2,3,19)$ |  $1-2q+2q^{3}-q^{4}+2q^{16}-2q^{17}-2q^{20}+2q^{22}+2q^{25}-2q^{26}+4q^{33}-\cdots$2pt2pt
$4$ | $\Sigma(2,3,25)$ |  $1-2q+2q^{3}-q^{4}+2q^{22}-2q^{23}-2q^{26}+2q^{28}+2q^{31}-2q^{32}+4q^{45}-\cdots$2pt2pt
$5$ | $\Sigma(2,3,31)$ |  $1-2q+2q^{3}-q^{4}+2q^{28}-2q^{29}-2q^{32}+2q^{34}+2q^{37}-2q^{38}+4q^{57}-\cdots$2pt2pt
$r$ | $\Sigma(2,3,6r+1)$ | $\sum\limits_{(w_{1},w_{2})\in W^{2}}(-1)^{l(w_{1}w_{2})}\chi_{36r+6,3(6r+1)w_{1}(\rho)+2(6r+1)w_{2}(\rho)+6\rho}$

In fact it is easy to check that for $K=T_{s,t}$,

$\displaystyle\mathcal{L}_{-1/r}\left[\prod_{\alpha\in\Delta^{+}}\big{(}x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}}\big{)}F_{K}(\mathbf{x},q)\right]$
$\displaystyle\qquad{}\cong\sum_{(w_{1},w_{2})\in
W^{2}}(-1)^{l(w_{1}w_{2})}\chi_{st(rst+1),t(rst+1)w_{1}(\rho)+s(rst+1)w_{2}(\rho)+st\rho}$
$\displaystyle\qquad{}\cong\hat{Z}_{0}^{G}(\Sigma(s,t,rst+1)),$

which is consistent with what we have seen in Proposition 2.5.

Furthermore we conjecture that our 0-surgery formula in [7] holds for higher
rank as well:

######  Conjecture 3.6 (higher rank 0-surgery formula).

Let $K\subset S^{3}$ be a knot. Then

$\displaystyle\hat{Z}_{0}^{G}\big{(}S_{0}^{3}(K)\big{)}\cong\frac{1}{|W|}f_{\rho}^{G}(K).$

In particular, for positive double twist knots $K_{m,n}$,

$\displaystyle f_{\rho}^{G}(K_{m,n})\cong\chi_{m,\rho}\chi_{n,\rho}.$

##  4 Symmetric representations and large $\boldsymbol{N}$

###  4.1 Specialization to symmetric representations

In this section we study a specialization of $F_{K}^{G}({\bf x},q)$ to
symmetric representations. We restrict our attention to $G={\rm SU}(N)$. We
start from the reduced version of $F_{K}$:

$\displaystyle F_{K}^{\rm red}(\mathbf{x},q):=\frac{1}{|W|}\sum_{\beta\in
P_{+}\cap(Q+\rho)}f_{\beta}(q)\frac{\sum\limits_{w\in
W}(-1)^{l(w)}x^{w(\beta)}}{\sum\limits_{w\in W}(-1)^{l(w)}x^{w(\rho)}}.$

Then the (reduced) symmetrically colored $F_{K}$ corresponds to the following
specialization:

$\displaystyle F_{K}^{\rm sym}(x,q):=F_{K}^{\rm red}((x,q,\dots,q),q).$

That is, we set $x_{2}=\cdots=x_{r}=q$. A version of quantum volume conjecture
[12] states that this should be annihilated by the symmetrically colored
quantum $A$-polynomial:

$\displaystyle\hat{A}_{K}\big{(}\hat{x},\hat{y},a=q^{N},q\big{)}F_{K}^{{\rm
SU}(N),{\rm sym}}(x,q)=0.$ (4.1)

######  Example 4.1 (right-handed trefoil).

For the right-handed trefoil, $F_{\mathbf{3}_{1}^{r}}^{{\rm SU}(N),{\rm
sym}}(x,q)$ for the first few values of $N$ look like the following:

  * •

For ${\rm SU}(2)$,

$\displaystyle F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong\frac{1}{2}\big{[}\big{(}{-}q+q^{2}+q^{3}-q^{6}-q^{8}+q^{13}+q^{16}-\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}x+x^{-1}\big{)}\big{(}q^{2}+q^{3}-q^{6}-q^{8}+q^{13}+q^{16}-\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}x^{2}+x^{-2}\big{)}\big{(}q^{2}+q^{3}-q^{6}-q^{8}+q^{13}+q^{16}-\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}x^{3}+x^{-3}\big{)}\big{(}q^{3}-q^{6}-q^{8}+q^{13}+q^{16}-\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}x^{4}+x^{-4}\big{)}\big{(}{-}q^{6}-q^{8}+q^{13}+q^{16}-\cdots\big{)}+\cdots\big{]}.$

  * •

For ${\rm SU}(3)$,

$\displaystyle F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong\frac{1}{2}\big{[}\big{(}{-}2q-2q^{2}+2q^{4}+4q^{5}+4q^{6}+4q^{7}+2q^{8}-2q^{10}-4q^{11}-\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}q^{1/2}x+q^{-1/2}x^{-1}\big{)}q^{1/2}\big{(}{-}1-2q-q^{2}+q^{3}+3q^{4}+4q^{5}+4q^{6}+\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}qx^{2}+q^{-1}x^{-2}\big{)}\big{(}{-}q-q^{2}+2q^{4}+3q^{5}+4q^{6}+3q^{7}+2q^{8}-2q^{10}+\cdots\big{)}\\!$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}q^{3/2}x^{3}+q^{-3/2}x^{-3}\big{)}q^{1/2}\big{(}q^{3}+2q^{4}+3q^{5}+3q^{6}+2q^{7}+q^{8}+\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}q^{2}x^{4}+q^{-2}x^{-4}\big{)}\big{(}q^{3}+q^{4}+2q^{5}+2q^{6}+2q^{7}+q^{8}+\cdots\big{)}+\cdots\big{]}.$

  * •

For ${\rm SU}(4)$,

$\displaystyle F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong\frac{1}{2}\big{[}\big{(}q^{-2}+q^{-1}-2-4q-8q^{2}-7q^{3}-7q^{4}+\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}qx+q^{-1}x^{-1}\big{)}\big{(}q^{-2}-1-5q-6q^{2}-8q^{3}-5q^{4}-2q^{5}+\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}q^{2}x^{2}+q^{-2}x^{-2}\big{)}\big{(}{-}2-3q-6q^{2}-5q^{3}-5q^{4}+4q^{6}+\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}q^{3}x^{3}+q^{-3}x^{-3}\big{)}\big{(}{-}q^{-1}-1-3q-3q^{2}-4q^{3}-2q^{4}+5q^{6}+9q^{7}+\cdots\\!\big{)}\\!$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\big{(}q^{4}x^{4}+q^{-4}x^{-4}\big{)}\big{(}{-}1-q-2q^{2}-q^{3}-q^{4}+2q^{5}+4q^{6}+8q^{7}+\cdots\big{)}$
$\displaystyle\hphantom{F_{\mathbf{3}_{1}^{r}}^{\rm
sym}(x,q)\cong}{}+\cdots\big{]}.$

Note that the overall factor is $\frac{1}{2}$ instead of $\frac{1}{N!}$. This
is due to reduction of the Weyl symmetry to $\mathbb{Z}_{2}$ as we specialize
to symmetric representations.

It is easy to experimentally check (4.1) term by term in this case, using the
$a$-deformed quantum $A$-polynomial for the right-handed trefoil

$\hat{A}_{\mathbf{3}_{1}^{r}}(\hat{x},\hat{y},a,q)=a_{0}+a_{1}\hat{y}+a_{2}\hat{y}^{2},$

where

$\displaystyle
a_{0}=-\frac{(-1+\hat{x})\big{(}{-}1+aq\hat{x}^{2}\big{)}}{a\hat{x}^{3}(-1+a\hat{x})\big{(}{-}q+a\hat{x}^{2}\big{)}},$
$\displaystyle
a_{1}=\frac{\big{(}{-}1+a\hat{x}^{2}\big{)}\big{(}{-}a^{2}\hat{x}^{2}+aq^{3}\hat{x}^{2}+aq\hat{x}(1+\hat{x}+a(-1+\hat{x})\hat{x})-q^{2}\big{(}1+a^{2}\hat{x}^{4}\big{)}\big{)}}{a^{2}q\hat{x}^{3}(-1+a\hat{x})\big{(}{-}q+a\hat{x}^{2}\big{)}},$
$\displaystyle a_{2}=1$

with $a$ specialized to $q^{N}$.

###  4.2 Future direction: large $\boldsymbol{N}$

From (4.1), we are naturally led to the following conjecture:

######  Conjecture 4.2 (HOMFLY-PT analogue of $F_{K}$).

For each knot $K$, there exists a function $F_{K}(x,a,q)$ such that

$\displaystyle\hat{A}_{K}(\hat{x},\hat{y},a,q)F_{K}(x,a,q)=0$

and

$\displaystyle F_{K}\big{(}x,q^{N},q\big{)}=F_{K}^{{\rm SU}(N),{\rm
sym}}(x,q).$ (4.2)

Moreover, this function should have the following Weyl symmetry:

$\displaystyle
F_{K}\big{(}x^{-1},a,q\big{)}=F_{K}\big{(}a^{-1}q^{2}x,a,q\big{)}.$

In particular, (4.2) implies

$\displaystyle\lim_{q\rightarrow
1}F_{K}\big{(}x,q^{N},q\big{)}=\Delta_{K}(x)^{1-N}.$

The study of this HOMFLY-PT analogue of $F_{K}$ is the subject of [11].

We end with a more speculative question regarding homology theories:

######  Question 4.3.

What is $\mathcal{H}_{b,{\rm BPS}}^{G}$ categorifying $\hat{Z}_{b}^{G}(q)$? Is
there a family of differentials analogous to that of [10]?

We will pursue these questions in our future work.

##  Appendix A Motivation for Definition 2.2

In this section we motivate Definition 2.2 through a heuristic decomposition
of WRT invariants. This is essentially a generalization of Appendix A of [14].
Let $Y$ be a plumbing on a connected tree $\Gamma$, each vertex $v$ decorated
with an integer $a_{v}$, with a negative definite linking matrix ${\rm B}$.
Then the WRT invariant $Z_{G_{k}}(Y)$ can be computed by777Here we ignored a
framing factor.

$\displaystyle Z_{G_{k}}(Y)\cong\sum_{\rm colorings}\prod_{v\in
V}\mathcal{V}_{v}\prod_{e\in E}\mathcal{E}_{e},$ (A.1)

where the vertex and the edge factors are

$\displaystyle\mathcal{V}=t_{\lambda\lambda}^{a_{v}}s_{\rho\lambda}^{2-\deg
v},\qquad\mathcal{E}=s_{\mu\lambda}.$

Here the $s$, $t$ matrices are as usual888See [1, Theorem 3.3.20].

$\displaystyle s_{\lambda\mu}=\frac{{\rm
i}^{|\Delta_{+}|}}{|P/kQ^{\vee}|^{1/2}}\sum_{w\in
W}(-1)^{l(w)}q^{(w(\lambda),\mu)},\qquad
t_{\lambda\mu}=\delta_{\lambda\mu}q^{\frac{1}{2}(\lambda,\lambda)}q^{-\frac{1}{2}(\rho,\rho)}$

with $q={\rm e}^{\frac{2\pi{\rm i}}{mk}}$ and set of (shifted) colors being

$\lambda,\mu\in C=\big{\\{}\lambda\in
P_{+}+\rho\,|\,\big{(}\lambda,\theta^{\vee}\big{)}<k\big{\\}}.$

Then these $s$, $t$ matrices are invariant (up to sign) under the action of
the affine Weyl group

$W^{a}=W\ltimes kQ^{\vee}.$

In fact, $C$ is simply the fundamental domain $P/W^{a}$. Using this fact, we
can manipulate the form of $Z_{G_{k}}(Y)$ to write it in a Gauss sum
reciprocity-friendly way:

$\displaystyle\text{\eqref{eq2.1}}=\frac{1}{|W^{V}|}\sum_{\text{coloring}\in
W(C)^{V}}\prod_{v\in V}\mathcal{V}_{v}\prod_{e\in E}\mathcal{E}_{e}$
$\displaystyle\hphantom{\text{\eqref{eq2.1}}}{}=\frac{1}{|W^{V}|}q^{-\frac{\sum
a_{j}}{2}(\rho,\rho)}\left(\frac{{\rm
i}^{|\Delta_{+}|}}{|P/kQ^{\vee}|^{1/2}}\right)^{|V|+1}\sum_{\lambda\in
W(C)^{V}}\prod_{v\in V}\left(\sum_{w\in
W}(-1)^{l(w)}q^{(\lambda_{v},w(\rho))}\right)^{2-\deg v}$
$\displaystyle\hphantom{\text{\eqref{eq2.1}}=}{}\times
q^{\frac{a_{v}}{2}(\lambda_{v},\lambda_{v})}\prod_{(u_{1},u_{2})\in
E}\sum_{w\in W}(-1)^{l(w)}q^{(w(\lambda_{u_{1}}),\lambda_{u_{2}})}$
$\displaystyle\hphantom{\text{\eqref{eq2.1}}}{}=\frac{1}{|W|}q^{-\frac{\sum
a_{j}}{2}(\rho,\rho)}\left(\frac{{\rm
i}^{|\Delta_{+}|}}{|P/kQ^{\vee}|^{1/2}}\right)^{|V|+1}$
$\displaystyle\hphantom{\text{\eqref{eq2.1}}=}{}\times\sum_{\lambda\in
W(C)^{V}}\prod_{v\in V}\left(\sum_{w\in
W}(-1)^{l(w)}q^{(\lambda_{v},w(\rho))}\right)^{2-\deg
v}q^{\frac{1}{2}(\lambda,{\rm B}_{s}\lambda)}.$ (A.2)

To extend the range of summation $W(C)$ to $P/kQ^{\vee}$, i.e., to make sense
of the summation even for colors upon which the action of $W^{a}$ is not free,
we need to regularize the linear term $\prod\limits_{v\in
V}\big{(}\sum\limits_{w\in
W}(-1)^{l(w)}q^{(\lambda_{v},w(\rho))}\big{)}^{2-\deg v}$. Let
$\omega_{1},\dots,\omega_{r}\in P$ be a $\mathbb{Z}$-linear basis of $P$
(e.g., fundamental weights). We can then write
$\lambda_{v}=\sum\limits_{j=1}^{r}n_{vj}\omega_{j}$ for some
$n_{v1},\dots,n_{vr}\in\mathbb{Z}$. Thanks to Weyl denominator formula, we
have

$\displaystyle\prod_{v\in V}\left(\sum_{w\in
W}(-1)^{l(w)}q^{(\lambda_{v},w(\rho))}\right)^{2-\deg v}$
$\displaystyle\qquad{}=\prod_{v\in
V}\left(\prod_{\alpha\in\Delta_{+}}\big{(}q^{\frac{(\lambda_{v},\alpha)}{2}}-q^{-\frac{(\lambda_{v},\alpha)}{2}}\big{)}\right)^{2-\deg
v}$ $\displaystyle\qquad{}=\prod_{v\in
V}\left(\prod_{\alpha\in\Delta_{+}}\left(\prod_{1\leq j\leq
r}x_{vj}^{\frac{(\omega_{j},\alpha)}{2}}-\prod_{1\leq j\leq
r}x_{vj}^{-\frac{(\omega_{j},\alpha)}{2}}\right)\right)^{2-\deg
v}\bigg{|}_{x_{vj}=q^{n_{vj}}}.$ (A.3)

In case $\deg v>2$, this expression can be singular only when

$\left|\prod_{1\leq j\leq r}y_{j}^{\frac{(\omega_{j},\alpha)}{2}}\right|=1$

for some $\alpha\in\Delta_{+}$. In terms of new variables
$z_{j}:=\log|x_{j}|$, this is simply

$\sum_{1\leq j\leq r}(\omega_{j},\alpha)z_{j}=0.$

These are precisely the walls (hyperplanes) for Weyl reflections. Deforming
the origin $z_{1}=\cdots=z_{r}=0$ to a complement of these walls is the same
as a choice of a Weyl chamber. Moreover, for each choice of such a Weyl
chamber, we can expand (A.3) as a geometric series. Therefore, we can
regularize the linear term by taking an average of $|W|$ number of $q$-series,
each determined by a choice of a Weyl chamber. To sum up, we can re-express
the linear term in the following form:

$\displaystyle\prod_{v\in V}\left(\sum_{w\in
W}(-1)^{l(w)}q^{(\lambda_{v},w(\rho))}\right)^{2-\deg
v}\overset{\text{regularize}}{=\joinrel=\joinrel=\joinrel\Rightarrow}\frac{1}{|W^{V}|}\sum_{\ell\in\delta+Q^{V}}n_{\ell}q^{(\lambda,\ell)}$
(A.4)

with $\delta_{v}=(2-\deg v)\rho\mod Q$ and $n_{\ell}\in\mathbb{Z}$.

With this regularization in hand, we extend the range of summation in (A.2)
and apply the Gauss sum reciprocity.999See [9] for the version of Gauss sum
reciprocity formula we use. Then, if we choose $n\in\mathbb{Z}^{+}$ to be such
that $nP\subseteq Q^{\vee}\subseteq P$, we get

$\displaystyle\frac{1}{|W|}q^{-\frac{\sum
a_{j}}{2}(\rho,\rho)}\left(\frac{{\rm
i}^{|\Delta_{+}|}}{|P/kQ^{\vee}|^{1/2}}\right)^{|V|+1}\frac{1}{|(Q^{\vee}/nP)^{V}|}$
$\displaystyle\qquad\quad{}\times\sum_{\lambda\in
P^{V}/nkP^{V}}q^{\frac{1}{2}(\lambda,{\rm
B}\lambda)}\cdot\left(\frac{1}{|W^{V}|}\sum_{\ell\in\delta+Q^{V}}n_{\ell}q^{(\lambda,\ell)}\right)$
$\displaystyle\qquad{}=\frac{(-1)^{|\Delta_{+}||V|}}{|W|^{|V|+1}}\left(\frac{{\rm
i}^{|\Delta_{+}|}}{|P/Q^{\vee}|^{1/2}}\right)^{-|V|+1}q^{-\frac{\sum
a_{j}}{2}(\rho,\rho)}k^{-\frac{r}{2}}\frac{{\rm e}^{\frac{\pi{\rm
i}}{4}\sigma({\rm B})}}{|\det{\rm B}|^{1/2}}$
$\displaystyle\qquad\quad{}\times\sum_{a\in(P^{\vee})^{V}/{\rm
B}(P^{\vee})^{V}}{\rm e}^{-\pi{\rm i}k(a,{\rm
B}^{-1}a)}\sum_{b\in(Q^{V}+\delta)/{\rm B}Q^{V}}{\rm e}^{-2\pi{\rm i}(a,{\rm
B}^{-1}b)}\sum_{\ell\in{\rm B}Q^{V}+b}n_{\ell}q^{-\frac{1}{2}(\ell,{\rm
B}^{-1}\ell)}.$

Note that $a$ and $b$ takes values in different sets:
$a\in\big{(}P^{\vee}\big{)}^{V}/{\rm B}\big{(}P^{\vee}\big{)}^{V}$ while
$b\in\big{(}Q^{V}+\delta\big{)}/{\rm B}Q^{V}$. The $a$ labels should be
understood as ‘Abelian flat connections’ and the $b$ labels are
‘$\mathcal{B}^{G}$-structures’ of Section 2.1.

To summarize, we have heuristically decomposed the WRT invariant
$Z_{G_{k}}(Y)$ into some linear combinations of $q$-series. When compared with
Conjecture 2.1 of [14], this suggests the following expression of
$\hat{Z}^{G}$ for negative definite plumbed manifolds:101010Our decomposition
was only a heuristic. Although we believe that this heuristic can be made
rigorously to prove Conjecture 2.1 of [14] for negative definite plumbings on
trees, we do not bother to do so, because Conjecture 2.1 of [14] should be
modified anyway for general $3$-manifolds, see [6, 7, 13].

$\displaystyle\hat{Z}_{b}^{G}(Y;q)\cong|W|^{-|V|}q^{-\frac{\operatorname{Tr}{\rm
B}}{2}(\rho,\rho)}\sum_{\ell\in{\rm
B}Q^{V}+b}n_{\ell}q^{-\frac{1}{2}(\ell,{\rm
B}^{-1}\ell)}\;\in|W|^{-|V|}q^{\Delta_{b}}\mathbb{Z}[[q]],$

where

$\displaystyle b\in\big{(}Q^{V}+\delta\big{)}/{\rm
B}Q^{V},\qquad\Delta_{b}=-\frac{\operatorname{Tr}{\rm
B}}{2}(\rho,\rho)+\min_{\ell\in{\rm B}Q^{V}+b}-\frac{1}{2}\big{(}\ell,{\rm
B}^{-1}\ell\big{)}\in\mathbb{Q},$

and the integers $n_{\ell}$ are determined as in (A.4). Moreover the higher
rank analog of the $S_{ab}$ matrix in Conjecture 2.1 of [14] is $S_{ab}={\rm
e}^{-2\pi{\rm i}(a,{\rm B}^{-1}b)}$ before folding by Weyl symmetry. Note that
because we have ignored the framing factor in (A.1), $\Delta_{b}$ as defined
above is only meaningful up to overall shift; only the differences
$\Delta_{b}-\Delta_{b^{\prime}}$ are meaningful values. It is easy to check
that the expression we have just arrived is equivalent to Definition 2.2.

##  Appendix B Comparison with Chung’s paper

In [8], Chung studied $\hat{Z}$ for some Seifert manifolds, with $G={\rm
SU}(N)$. His approach was to start from Marino’s integral expression for
$Z_{a}=\sum_{b}S_{ab}\hat{Z}_{b}$ in [16] and decompose it into
$\hat{Z}_{b}$’s by collecting terms whose $q$-degrees differ by an integer. In
this section we compare our result with Chung’s result in some examples.

  * •

$Y=M\big{(}{-}1;\frac{1}{2},\frac{1}{3},\frac{1}{7}\big{)}$. In this case
$|H_{1}(Y)|=1$ and there’s only one homological block.

$G$ | $\hat{Z}_{0}(Y)\cong$
---|---
${\rm SU}(2)$ |  $1-q-q^{5}+q^{10}-q^{11}+q^{18}+q^{30}-q^{41}+q^{43}-q^{56}-q^{76}+q^{93}-\cdots$2pt2pt
${\rm SU}(3)$ |  $1-2q+2q^{3}+q^{4}-2q^{5}-2q^{8}+4q^{9}+2q^{10}-4q^{11}+2q^{13}-6q^{14}+2q^{15}-\cdots$2pt2pt
${\rm SU}(4)$ | $1-3q+q^{2}+3q^{3}-3q^{5}-q^{6}-q^{7}-5q^{8}+15q^{9}+5q^{10}-11q^{11}-q^{12}+\cdots$
  * •

$Y=M\big{(}{-}1;\frac{1}{2},\frac{1}{5},\frac{2}{7}\big{)}$. Again,
$|H_{1}(Y)|=1$ and there is only one homological block.

$G$ | $\hat{Z}_{0}(Y)\cong$
---|---
${\rm SU}(2)$ |  $1-q^{3}-q^{5}+q^{12}-q^{23}+q^{36}+q^{42}-q^{59}+q^{81}-q^{104}-q^{114}+q^{141}-\cdots$2pt2pt
${\rm SU}(3)$ |  $1-2q^{3}-2q^{5}+2q^{6}+2q^{9}+q^{12}-2q^{14}+2q^{15}-2q^{18}-3q^{20}+6q^{21}-4q^{23}-\cdots$2pt2pt
${\rm SU}(4)$ | $1-3q^{3}-3q^{5}+5q^{6}-q^{7}+2q^{8}+3q^{9}-q^{10}-q^{12}+2q^{13}-6q^{14}+2q^{15}-\cdots$
  * •

$Y=M\big{(}{-}1;\frac{1}{3},\frac{1}{5},\frac{3}{7}\big{)}$. In this case
$|H_{1}(Y)|=4$.

$G$ | $\hat{Z}_{b}(Y)\cong$
---|---
${\rm SU}(2)$ |  $1+q^{4}+q^{16}-q^{68}+q^{144}-q^{260}-q^{320}-q^{356}+q^{484}+q^{528}+q^{612}-q^{832}+\cdots$2pt
|
$-q^{15/4}\big{(}1+q^{6}+q^{10}+q^{12}-q^{44}-q^{48}-q^{58}-q^{88}+q^{122}+q^{164}+q^{182}+\cdots\big{)}$
|
$q^{13/2}\big{(}1-q^{32}-q^{56}-q^{72}+q^{136}+q^{160}+q^{208}-q^{344}+q^{496}-q^{696}-q^{792}-\cdots\big{)}$2pt
${\rm SU}(3)$ |  $1+3q^{4}+2q^{12}+3q^{16}+2q^{28}+2q^{48}+2q^{52}+q^{64}+4q^{68}+4q^{80}+4q^{92}-\cdots$2pt
|
$-q^{-7/4}\big{(}2+2q+2q^{3}-4q^{5}+2q^{6}-2q^{7}+4q^{9}+2q^{10}+4q^{12}+2q^{13}-2q^{14}+\cdots\big{)}$
|
$-q^{-7}\big{(}1+2q^{6}-2q^{8}+2q^{12}-2q^{14}+2q^{18}-2q^{20}+q^{24}-2q^{26}+2q^{28}+\cdots\big{)}$
|
$q^{-21/4}\big{(}1+q-q^{2}+q^{4}-q^{5}+q^{7}+q^{10}+q^{13}-2q^{14}-q^{15}+2q^{16}+\cdots\big{)}\times
2$2pt
${\rm SU}(4)$ |  $1+q^{12}+8q^{16}+3q^{20}+16q^{24}+11q^{28}+15q^{32}+4q^{36}+26q^{40}+5q^{44}+\cdots$2pt
|
$-q^{-15/4}\big{(}1+2q-2q^{2}+2q^{4}-2q^{5}+q^{6}+4q^{7}-2q^{8}+3q^{10}+6q^{13}-\cdots\big{)}$
|
$q^{-1}\big{(}2-2q^{2}-2q^{4}-3q^{6}-2q^{8}-5q^{10}-14q^{12}-5q^{14}-4q^{16}-12q^{18}+\cdots\big{)}$
|
$q^{-1/4}\big{(}2+2q-2q^{2}+2q^{3}+q^{4}-2q^{5}+6q^{6}+2q^{7}+8q^{8}+6q^{9}-\cdots\big{)}\times
2$
|
$-q^{3/2}\big{(}1+3q^{4}+2q^{8}+6q^{12}+6q^{16}+4q^{20}+9q^{24}+9q^{28}+11q^{32}+\cdots\big{)}\times
2$
|
$q^{-2}\big{(}1+2q^{4}+3q^{8}+6q^{16}+6q^{20}+11q^{24}+17q^{32}+10q^{36}+9q^{40}+14q^{48}+\cdots\big{)}$
|
$-q^{-15/4}\big{(}2+q-2q^{2}+q^{3}+2q^{4}-q^{5}+4q^{6}+4q^{7}+2q^{8}-2q^{9}+\cdots\big{)}$
|
$q^{-11/2}\big{(}1+3q^{6}-4q^{8}+7q^{12}-2q^{14}-q^{16}+6q^{18}+3q^{20}+7q^{22}+4q^{24}+\cdots\big{)}$

Observe that [8] agrees with our example computations except in the ${\rm
SU}(4)$ case of the last example. Our $\hat{Z}$’s are more refined in a sense
that Chung’s $\hat{Z}_{1}$ is the sum of our 2nd and 7th ones and Chung’s
$\hat{Z}_{3}$ is the sum of our 1st and 6th ones. This example illustrates
that in general we cannot fully decompose $Z_{a}$ into $\hat{Z}_{b}$’s by just
collecting terms whose $q$-powers differ by an integer as done in
[8].111111Still, it is possible to derive our formula (2.1) from Mariño’s
Chern–Simons matrix model or vice versa in case of Seifert manifolds. This is
because Gaussian measure is the same as Laplace transform accompanied by the
$S_{ab}$ matrix.

### Acknowledgements

I would like to thank my advisor Sergei Gukov for his invaluable guidance, as
well as Francesca Ferrari, Sarah Harrison, Ciprian Manolescu and Nikita
Sopenko for helpful conversations. Special thanks go to Nikita Sopenko for his
kind help with Mathematica coding. I would also like to thank the anonymous
referees for useful comments that helped to improve the paper.

The author was supported by Kwanjeong Educational Foundation.

## References

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# On the boundary behavior of bounded analytic functions in the unit disc

Spyros Pasias

###### Abstract

In this paper we will deal with problems in approximation theory of bounded
analytic functions on the unit disc and their boundary behavior on the unit
circle. We will attempt to unify two known such theorems to create a stronger
theorem. We will solve various special cases of the theorem, and in particular
one case extends the main result found in [5] regarding the boundary behavior
of Blaschke products. The necessity part of the proof uses a classical theorem
of Baire. We will see that the proof of the necessity part of the extension
theorem provides a simplification and a more elegant approach for the
necessity part of the main result found in [5]. Lastly, we will prove an
analogue of a classical theorem by Kolesnikov for Blaschke products provided
an extra requirement is satisfied and use the main result in [5] to simplify
the proof of a result found in [6].

We start by stating some important theorems about the boundary behavior of
bounded analytic functions. Throughout the paper we let $D$ represent the unit
disc and $T$ the unit circle. The following important theorem is due to S.V
Kolesnikov [8].

######  Theorem 1 (Kolesnikov’s Theorem).

Let $E\subset T$ be of type $G_{\delta\sigma}$ and of measure $0$. Then there
exists a bounded analytic function in the unit disc $D$ which fails to have
radial limits exactly at the points of $E$.

It should be noted that the converse statement of Kolesnikov’s Theorem which
states:
The set in which the radial limits of a bounded analytic function is
necessarily $G_{\delta\sigma}$, is elementary and has been known long before
Kolesnikov’s Theorem was introduced. A proof can be found in ([10], p. 23).

######  Theorem 2 (Fatou’s interpolation theorem, 1906).

Let $K$ be a closed subset of $T$ such that $m(K)=0$. Then there exists a
function $f$ in the disc algebra which vanishes precisely on $K$.

A proof of Fatou’s interpolation theorem, (that shows the real part of $f$ is
positive on $D$) can be found in ([8], p. $29-30$). A new proof may also be
found in [13]. The construction of Fatou’s function is widely used and has
many applications. In particular it is used to prove an important result by
_F. and M. Riesz_ which states that any analytic measure is absolutely
continuous with respect to the Lebesgue measure.
The necessity part of the following theorem is well known and its proof is
elementary. The sufficiency part is proven in [12] and its proof is using the
method of the paper [15] and is based on Fatou’s interpolation theorem.

######  Theorem 3.

Let $E$ be a set on $T$. Then there exists a function $f\in H^{\infty}$ which
has no radial limits on $E$ but has unrestricted limits at each point of
$T\setminus E$ if and only if $E$ is an $F_{\sigma}$ set of measure zero.

The following theorem is a solution to a problem proposed by Rubel in 1973 and
was solved in [11] with an affirmative answer to the problem. The problem asks
whether for any $G_{\delta}$ set $F\subset T$ of measure zero, there exists a
non vanishing bounded analytic function $f$ on $D$ such that $f=0$ precisely
of $F$ and such that the radial limits of $f$ exist everywhere on $T$. The
following stronger theorem found in [9] provides an affirmative answer to
Rubel’s problem as well.

######  Theorem 4.

Let $F$ be a $G_{\delta}$ of measure zero on $T$. Then there exists a
nonvanishing bounded analytic function $g$ such that:

  1. 1.

$g$ has non zero radial limits everywhere on $T\setminus F$.

  2. 2.

$g$ has vanishing unrestricted limits at each point of $F$.

Now we use the three theorems above to solve a particular case of Problem $1$
stated in [12]. Although the problem stated in [12] is still an open problem
and a good research opportunity, we have solved several particular cases of it
in this thesis. The problem basically is an attempt to unify both Kolesnikov’s
Theorem and Theorem 3 to a single stronger theorem. The problem we are
interested in (See [11], p. 4) is the following:

Unification Problem. _Let $E_{1}\subset E_{2}$ be subsets of the unit circle
$T\equiv\\{z\mid|z|=1\\}$. Find necessary and sufficient conditions for the
existence of an $f\in H^{\infty}$ such that the radial limits of $f$ fail to
exist precisely on $E_{1}$ and it has unrestricted limits precisely at
$T\setminus E_{2}$._

Obvious necessary conditions are that $E_{1}$ is of type $G_{\delta\sigma}$
and $E_{2}$ is of type $F_{\sigma}$. The case $E_{1}=\emptyset$ is also dealt
with by the following theorem.

######  Theorem 5 (Brown, Gauthier and Hengartner).

Let $E$ be a set on $T$. Then there exists a function $f\in H^{\infty}$ which
has no unrestricted limits on $E$ but has unrestricted limits at every point
of $T\setminus E$ if and only if $E$ is of type $F_{\sigma}$.

The proof of the theorem above is not difficult. The idea is to express $E$ as
a countable union of closed sets $E=\bigcup_{n}C_{n}$, and then for each of
the sets $C_{n}$ pick a respective Blaschke product $B_{n}$ with zeros
accumulating precisely on $C_{n}$. Then, since Blaschke products are bounded
by modulus by $1$, the infinite sum $\sum_{n=1}^{\infty}\frac{B_{n}}{2^{n}}$
converges to the function with the desired properties.

Now we will proceed and solve some particular cases of the Unification
Problem. In the proofs that follow, we note that each function of the unit
disc is extended to take the value of its radial limits on the unit circle $T$
when they exist.

######  Theorem 6.

Let $E_{1}\subset E_{2}$ be subsets of the unit circle
$T\equiv\\{z\mid|z|=1\\}$ and suppose that the measure of $E_{2}$ is $2\pi$.
The following conditions are necessary and sufficient for the existence of an
$f\in H^{\infty}$ such that the radial limits of $f$ fail to exist precisely
at $E_{1}$ and it has vanishing unrestricted limits precisely at $T\setminus
E_{2}$:

  1. 1.

$E_{1}$ is of type $G_{\delta\sigma}$ and of measure zero.

  2. 2.

$E_{2}$ is of type $F_{\sigma}$.

###### Proof.

(Necessity) It is well known that the exceptional set of points in $T$ such
that $f$ fails to have radial limits is of type $G_{\delta\sigma}$ and by
Fatou’s Theorem is of measure zero. Also if $f$ has unrestricted limits on
$T\setminus E_{2}$ then $E_{2}=\bigcup_{n=1}^{\infty}F_{n}$ where $F_{n}$ are
the closed sets consisting of all points in $E_{2}$ any neighborhood of which
contains points $t_{1},t_{2}\in D$ such that
$|f(t_{1})-f(t_{2})|>\frac{1}{n}$. Since clearly each $F_{n}$ is closed it
follows that $E_{2}$ is of type $F_{\sigma}$ and the proof of necessity is
complete.
(Sufficiency) Since $E_{1}$ is of type $G_{\delta\sigma}$ and of measure zero
by Kolesnikov’s Theorem there exists a bounded analytic function $f_{1}$ that
fails to have radial limits exactly on $E_{1}$. By adding a suitable constant
we may assume that $f_{1}$ does not have vanishing radial limits anywhere on
$T$.
Now since $E_{2}$ is $F_{\sigma}$ it follows $T\setminus E_{2}$ is of type
$G_{\delta}$ and therefore by Theorem 4 there exists a non zero bounded
analytic function $f_{2}$ such that $f_{2}=0$ exactly on $T\setminus E_{2}$
and such that the radial limits of $f_{2}$ exist at all points of $T$. The
function $F\equiv f_{1}f_{2}$ satisfies the conditions of the theorem hence
the proof is complete. ∎

Now recall that by the factorization Theorem (see p.67, [14]) every function
in $H^{1}$ can be factored uniquely into the product of three different types
of functions.

Definition 1. An _Singular inner function_ is an analytic zero free function
$S$ in the unit disc, such that $|S(z)|\leq 1$, $S(0)>0$ and
$|S(e^{i\theta}|=1$ almost everywhere on the unit circle. An _outer function_
is an analytic function $F$ in the unit disc of the form

$F(z)=\lambda\exp\left[\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}k(\theta)d\theta\right]$

where $k$ is a real valued integrable function on the circle and $\lambda$ is
a complex number of modulus $1$.

The factorization theorem states that every $f\in H^{1}$ can be factored
uniquely as $f=BSF$ where $B$ is a Blaschke product, $S$ a singular inner
function and $F$ an outer function. Factoraization Theorem is of great
significance because it allows us to analyze a function in $H^{1}$ by looking
at each of its factors independently. We will attempt to solve a problem
proposed in [12] for one of the three types of functions comprising the
factorization, namely Blaschke products. One might then continue the research
by examining the problem under the lens of the remaining types of functions in
the factorization theorem. Hopefully, by examining the problem with each type
of functions one will indeed manage to solve the problem for a general
function in $H^{\infty}$.

The following theorem is well known in analytic function theory.

######  Theorem 7 (Fatou’s Theorem).

Let $f\in H^{\infty}$. Then the radial limits of $f$ exist on $T$ except
perhaps for a subset $E$ of measure zero.

The following theorem has been proven recently and its proof can be found in
[1].

######  Theorem 8.

Let E be a set on $T$. Then there exists a Blaschke product which has no
radial limits on $E$ but has unrestricted limit at each point of $T\setminus
E$ if and only if $E$ is a closed set of measure zero.

The proof of the above mentioned Theorem uses Fatou’s Theorem and some results
on the boundary behavior of Blaschke products due to R.D. Berman [2] and A.
Nicolau [3].

We follow the presentation of Nicolau [3] to formulate the results of Berman
[2] and Nicolau [3], respectively. The next theorem is due to Berman ([2], p.
250).

######  Theorem 9 (R.Berman).

Let $E$ be a subset of the unit circle of zero Lebesgue measure and of type
$F_{\sigma}$ and $G_{\delta}$. Then there exist Blaschke products $B_{0}$ and
$B_{1}$ such that:
(i) $B_{0}$ extends analytically to $T\setminus\overline{E}$ and
$\lim_{r\rightarrow 1}B_{0}(re^{it})=0$ if and only if $e^{it}\in E$;
(ii) $\lim_{r\rightarrow 1}B_{1}(re^{it})=1$ if and only if $e^{it}\in E$.

Now we formulate a theorem of Nicolau (see [3], Proposition on p. 251).

######  Theorem 10 (A. Nicolau).

Let $E$ be a subset of the unit circle. Assume that there exist a Blaschke
product $B_{0}$ that extends analytically to $T\setminus\overline{E}$ with
$\lim_{r\rightarrow 1}B_{0}(re^{it})=0$ for $e^{it}\in E$, and an analytic
function $f_{1}$ in the unit ball of $H^{\infty}$, $f_{1}\neq 1$, such that
$\lim_{r\rightarrow 1}f_{1}(re^{it})=1$ for $e^{it}\in E$. Then for each
analytic function $g$ in the unit ball of $H^{\infty}$, there exists a
Blaschke product $I$ that extends analytically to $T\setminus\overline{E}$,
such that

$\lim_{r\rightarrow 1}[I(re^{it})-g(re^{it})]=0\text{ for }e^{it}\in E$

The following theorem is a corollary of the necessity part of the of Rene-
Louis Baire’s topological theorem for functions in the Baire first class (see
[7], p. 462, Baire’s Theorem) which we formulate for functions on the unit
circle $T$.

######  Theorem 11 (Baire’s Theorem).

Let $f$, $f_{n}$ be defined on $T$ and let $f_{n}$ be continuous on $T$. If
$\\{f_{n}\\}$ converges to $f$ at every point of $T$, then $f$ is continuous
on a dense subset of $T$.

Definition 2. Let $T$ be a complete metric space and $Z$ a Banach space. A
function $f:T\rightarrow Z$ is said to be of the _Baire first class_ , if $f$
is the pointwise limit of a sequence of continuous functions.

The following variation of Baire’s Theorem also holds (See [4], p. 986).

######  Theorem 12.

Let $T$ and $Z$ be as above and $f:T\rightarrow Z$, then for every closed set
$C\subset T$, the function $f\mid C$ has a point of continuity in $C$.

From that, it follows as a corollary that the points of discontinuity of $f$
are nowhere dense in $T$.

We will use Theorem 8 from above and the corollary of Baire’s Theorem
mentioned above to prove a special case of the unification problem which
extends the main theorem proven by Collwel in [5].

######  Theorem 13 (Collwel’s Theorem).

Let $E\subset T$. There exists a Blaschke product $B(z;A)$ for which
$B(e^{i\theta})$ is defined and of modulus one at every point of $T$ and
$A^{\prime}=E$ where $A$ is the set of zeros of our Blaschke product $B(z;A)$
if and only if $E$ is closed and nowhere dense.

Remark 1. Now we note that Collwel’s Theorem can be used to provide a
significant simplification of the proof of the sufficiency part of Theorem 1
found in ([6], p.5). The sufficiency part can be stated as a Theorem as
follows:

######  Theorem 14 (Lohwater and Piranian).

Let $K$ be an $F_{\sigma}$ set of first category in the unit circle $T$. Then
there exists a bounded analytic function $\Phi(z)\in H^{\infty}$ such that the
radial limits of $\Phi(z)$ exists everywhere on $T$ and has unrestricted
limits exactly on $T\setminus K$.

###### Proof.

Since $K$ is $F_{\sigma}$ and of first category, it follows
$K=\bigcup_{i=1}^{\infty}K_{i}$ where $K_{i}$ are closed nowhere dense subsets
of $C$. Then for each $i$ by the theorem in Collwel’s paper mentioned above
there exists a Blaschke product $B(z,A_{i})$ that has radial limits of modulus
$1$ everywhere on $T$ and has unrestricted limits of modulus $1$ exactly on
$T\setminus K_{i}$.
Now since for all $z\in D$ $\mid B(z,A_{i})\mid<1$, it follows that
$\sum_{i=1}^{\infty}\frac{\mid B(z,A_{i})\mid}{i^{2}}<\frac{\pi^{2}}{6}$, thus
the infinite sum $\sum_{i=1}^{\infty}(\frac{B(z,A_{i})}{i^{2}})$ converges
uniformly to an analytic function $\Phi(z)$ on $D$. Now since each
$B(z,A_{i})$ has radial limits everywhere on $T$ and unrestricted limits of
modulus $1$ exactly on $T\setminus K_{i}$, it follows that $\Phi(z)$ has
radial limits everywhere on $T$ and unrestricted limits exactly on the
$G_{\delta}$ set $T\setminus K$. The proof is complete. ∎

Now we are ready to state and prove the following theorem which extends
Collwel’s theorem and also provides a different and more elegant approach in
proving both the necessity.

######  Theorem 15 (Extension Theorem).

Let $E_{1}\subset E_{2}$ be subsets of the unit circle
$T\equiv\\{z\mid|z|=1\\}$. Moreover assume that $E_{1}$ is closed and that
$E_{1}$ and $E_{2}\setminus E_{1}$ are separated. The following conditions are
necessary and sufficient for the existence of a Blaschke product $B(z)$ such
that the radial limits of $B(z)$ fail to exist precisely at $E_{1}$ and it has
unrestricted limits precisely at $T\setminus E_{2}$:

  1. 1.

$E_{1}$ is of measure zero.

  2. 2.

$E_{2}$ is closed and nowhere dense.

###### Proof.

(Necessity.) The proof that $E_{1}$ is of measure zero follows from Fatou’s
Theorem and that $E_{2}$ is closed follows since $E_{2}$ is clearly the
accumulation of zeros of $B(z)$. Moreover, since $E_{2}$ and $E_{1}$ are
closed and separated by hypothesis, it follows that $E_{2}\setminus E_{1}$ is
closed as well. Now to prove that $E_{2}$ is nowhere dense, we consider the
closed sets $T\setminus O_{n}$, where

$O_{n}\equiv\bigcup_{e^{i\theta}\in
E_{1}}(e^{i(\theta-\frac{1}{n})},e^{i(\theta+\frac{1}{n})}).$

Note that for all $n$, the radial function $B(e^{i\theta})$ is of the Baire
first class on the closed set $T\setminus O_{n}$. The points of discontinuity
of our function $B(e^{i\theta})$ restricted on $T\setminus O_{n}$ is the set
$E_{2}\setminus O_{n}$. It follows by Baire’s theorem that the set
$E_{2}\setminus O_{n}$ is nowhere dense on $T\setminus O_{n}$. Since
$E_{2}\setminus O_{n}$ is closed, it follows that it is nowhere dense in the
standard topology on $T$ as well. Therefore the closed set $E_{2}\setminus
E_{1}=\bigcup_{n=1}^{\infty}E_{2}\setminus O_{n}$ is nowhere dense as well.
Lastly, since $E_{1}$ is a closed set of measure zero, of course it is nowhere
dense. Now notice that $E_{2}=(E_{2}\setminus E_{1})\cup E_{1}$, hence it
follows that $E_{2}$ is nowhere dense as well and the proof of necessity is
complete.
(Sufficiency.) Since $E_{1}$ is a closed set we know by Theorem 8 there exists
a Blaschke product $B_{1}(z,A_{1})$ whose radial limits fail to exist exactly
on ${A_{1}}^{{}^{\prime}}=E_{1}$ and it’s unrestricted limits exist everywhere
on $T\setminus E_{1}$. Now since $E_{2}\setminus E_{1}$ is closed and nowhere
dense, by (_Colwell’s Theorem_ there exist a Blaschke product $B_{2}(z,A_{2})$
with ${A_{2}}^{{}^{\prime}}=E_{2}\setminus E_{1}$ and such that
$B_{2}(z,A_{2})$ has radial limits everywhere on $T$ and has unrestricted
limits exactly only on $T\setminus(E_{2}\setminus E_{1})$. Then the Blaschke
product $B(z,A_{1}\cup A_{2})$ satisfies the conditions of the theorem and the
proof is complete. ∎

Remark 2. If $E_{1}=\emptyset$ in the Extension Theorem (Theorem 15) becomes
Theorem 13 (Collwel’s Theorem). Furthermore in the original proof of the
necessity part of Collwel’s Theorem the author’s proof is more complex and it
can be simplified using a more elegant and natural approach by utilizing
Theorem 11 (Baire’s Theorem). To this end let $E\subset T$ and $B(z;A)$ be as
in Collwel’s Theorem, then the necessity part claims that for a Blascke
product whose radial limits exist everywhere on $T$, the points of
discontinuity of the extended radial function on $T$ occur on a closed nowhere
dense set. Indeed, since $B(e^{i\theta})$ is of the Baire first class by
Baire’s Theorem it follows $B(e^{i\theta})$ is continuous on a dense subset of
$T$. Now, since the points of discontinuity of $B(e^{i\theta})$ is precisely
the set $E=A^{\prime}$ it follows that $E$ must be nowhere dense. Moreover,
$E$ must be a closed set because since is the accumulation of the zeros of
$B(z;A)$. Thus $E$ is a closed and nowhere dense set as required and the proof
is complete.

The sufficiency part of Collwel’s Theorem in [5], that is for each $E\subset
T$ that is closed and nowhere dense there exists a Blaschke product $B(z;A)$
with $A^{\prime}=E$ is based on the following result due to Frostman.

######  Theorem 16 (Frostman’s condition).

Let $A$ be a sequence of points inside the unit disc $D$ and let $B(z,A)$
denote the Blaschke product whose zero set is $A$. Then a necessary and
sufficient condition that $\lim_{r\rightarrow 1}B(re^{i\theta^{0}})=L$ where
$|L|=1$ is that

$\sum_{A}\frac{1-|a|}{|e^{i\theta}-a|}<\infty.$ (1)

Now we will use Frostman’s condition $(1)$ and prove the following:

######  Proposition 1.

Let $B(z,A)$ be a Blaschke product whose radial limits exist everywhere on the
unit circle with zero set $A=\\{a_{n}\\}$, then the subset of $A^{\prime}$ in
the unit circle in which $B(z,A)$ has radial limits of modulus $1$ is a set
that is both $G_{\delta}$ and $F_{\sigma}$.

###### Proof.

Indeed if we define the following sequence of functions $\\{f_{n}\\}$

$\displaystyle f_{n}:[0,2$ $\displaystyle\pi]\rightarrow\mathbb{R}_{>0}$
$\displaystyle\theta\longmapsto\sum_{k=1}^{n}\frac{1-|a_{k}|}{|e^{i\theta}-a_{k}|}.$

We note that the sequence $\\{f_{n}\\}$ is a sequence of continuous functions
on $[0,2\pi]$ and the sequence is unbounded exactly at the values of $\theta$
where Frostman’s condition $(3.1)$ from above is not satisfied. That is
exactly at the points where the Blaschke product $B(z,A)$ fails to have radial
limits of modulus $1$. Now by a well known theorem the set points where the
sequence $\\{f_{n}\\}$ is unbounded is of type $G_{\delta}$. It follows that
the set points where the radial limit exists and is of modulus $1$ is of type
$F_{\sigma}$. Now let $r_{n}=1-\frac{1}{n}$ and consider the following set of
continuous functions

$B^{*}_{n}(r_{n}e^{i\theta})=\frac{1}{1-|B_{{r_{n}}}(e^{i\theta})|}.$

Then, clearly the set of points $\\{e^{i\theta}\\}$ for $\theta\in[0,2\pi]$
where the sequence of functions $B^{*}_{n}$ is unbounded is exactly the set of
point where $\limsup_{r\rightarrow 1}B(re^{i\theta})=1$. But since out
Blaschke product $B(z)$ is in the class $B_{L}$, this is exactly the set of
points where $\lim_{r\rightarrow 1}B(re^{i\theta})=1$. Therefore by the well
known theorem this set is of type $G_{\delta}$ and the proof is complete. ∎

Now we will prove another theorem regarding the boundary behavior of Blaschke
products. To prove the theorem we use the following theorem found in [3].

######  Theorem 17.

Let $E$ be a subset of the unit circle of zero Lebesgue measure and of type
$F_{\sigma}$ and $G_{\delta}$. Let $\phi$ be a function defined on $E$ with
$\sup\\{|\phi(e^{it})|:e^{it}\in E\\}\leq 1$ and such that for each open set
$U$ in the complex plane, $\phi^{-1}(U)$ is of type $F_{\sigma}$ and
$G_{\delta}$. Then there exists a Blaschke product $I$ extending analytically
on $T\setminus\overline{E}$ such that

$\lim_{r\rightarrow 1}I(re^{it})=\phi(e^{it})\text{ for }e^{it}\in E.$

The Theorem we want to prove is the following:

######  Theorem 18.

Let $E_{1}\subset E_{2}$ be subsets of the unit circle
$T\equiv\\{z\mid|z|=1\\}$. Moreover, assume that $E_{2}$ is of measure zero.
Then there exists a Blaschke product $B$ such that the radial limits of $B$
fail to exist precisely at $E_{1}$ and it has unrestricted limits precisely at
$T\setminus E_{2}$ if and only if $E_{2}$ is closed and $E_{1}$ is of type
$G_{\delta\sigma}$ and of measure zero.

###### Proof.

(Necessity.) The proof that $E_{2}$ is closed follows since any Blaschke
product has unrestricted limits only on the set complimentary to the
accumulation of its zeros. This implies that the set $E_{2}$ is the closed set
in which the zeros of our Blaschke product accumulate. The fact that $E_{1}$
is $G_{\delta\sigma}$ is well known and as noted earlier its proof may be
found in [10].
(Sufficiency.) Since $E_{1}$ is a $G_{\delta\sigma}$ of measure zero we know
by Kolesnikov’s Theorem that there exists a bounded analytic function $f_{1}$
with no radial limits exactly on $E_{1}$. Moreover, since $E_{2}$ is a closed
zero measure set we can apply Theorem 2.4.8 to find a Blaschke product $B_{2}$
such that the radial limits of $B_{2}$ are equal to $\frac{1}{2}$ for all
$e^{it}\in E_{2}$. Moreover, since $E_{2}$ is closed the conditions of
Nicolau’s Theorem are satisfied as well. It follows that for every bounded
analytic function $g$ there exists a Blaschke product $I$ mimicking its
boundary behavior on the set $E_{2}$. Now the product function $g(z)\equiv
f_{1}(z)\cdot B_{2}(z)$ has the desired boundary behavior restricted on
$E_{2}$; that is $g$ has no radial limits exactly on $E_{1}$ and has radial
but not unrestricted on $E_{2}\setminus E_{1}$ . Thus, by Nicolau’s Theorem we
can find a Blaschke product $I$ that extends analytically on $T\setminus
E_{2}$ and such that

$\lim_{r\rightarrow 1}[I(re^{it})-g(re^{it}]=0$

Then the Blaschke product $I$ has the desired boundary behavior and the proof
is complete.

Remark 3. Recall that Kolesnikov’s Theorem states that for any set $E$ of type
$G_{\delta\sigma}$ on the unit circle there exists a bounded analytic function
which fails to have radial limits precisely on that set. Thus the Theorem
above can be thought of as an analogue of Kolesnikov’s Theorem for Blaschke
products, but with the extra requirement that $\overline{E}$ is of measure
zero.

∎

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