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# The Schwinger function, confinement and positivity violation in pure gauge
QED

Lee C. Loveridge1,2<EMAIL_ADDRESS>Orlando Oliveira1<EMAIL_ADDRESS>Paulo J. Silva1<EMAIL_ADDRESS>
1 CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra,
Portugal

2 Los Angeles Pierce College, 6201 Winnetka Ave., Woodland Hills CA 91371, USA

###### Abstract

The lattice regularized pure gauge compact U(1) theory is an ideal laboratory
to explore how confinement is realized as its phase diagram has a confined and
a deconfined phase that depends on the value of the coupling constant, i.e. on
$\beta$. Herein, the connection between confinement and positivity violation
through the Schwinger function associated with the Landau gauge photon
propagator is investigated. The simulations reported show a very clear link
between the realization of confinement and positivity violation of the photon
Schwinger function and, therefore, of the photon Källén-Lehmann spectral
density. Furthermore, a mass scale that characterizes the decay of the
Schwinger function for small time separations is computed and used to
distinguish the two phases of the theory.

The quanta associated with the fundamental fields of QCD have never been
observed as free particles. However, in QCD, the hadron spectra and
phenomenology are explained based on the dynamics of quarks and gluons.
Despite all efforts, we still do not have a non-perturbative solution for QCD
and, therefore, the mechanism that prevents the observation of free quarks and
gluons remains to be understood Alkofer:2010ue . Some authors suggest that
confinement of quarks is linked with particular classes of gluon
configurations Greensite:2011zz  such as non-Abelian monopoles Kronfeld:1987ri
; Maedan:1988yi ; Kondo:1998nw ; DiGiacomo:1998pm ; Kondo:2018edh ;
Greensite:2014gra , center vortices Mandelstam:1974pi ; Auzzi:2003fs ;
Cornwall:1979hz ; Mandelstam:1974vf ; Fujimoto:2021wsr ; Eto:2021nle ;
Oxman:2018dzp , etc., with some of these configurations suggesting the
emergence of Abelian dominance in QCD Nambu:1974zg ; Mandelstam:1974pi ;
tHooft:1981bkw ; Sakumichi:2015rfa . In general, these gluon configurations
are used to give support to a (linearly) rising static potential that
certainly provides an intuitive picture for confinement that builds on the
study of non-relativistic systems. Explaining the quark dynamics using a
potential is physically appealing, and static potentials for systems other
than meson and baryons have been computed. As mentioned, these potentials are
typically rising functions of the distance between constituents, for
sufficiently large distances between those constituents. However, the
description of the quark dynamics through a static potential is questionable,
specially for the lighter quarks where quantum fluctuations have a leading
role. Despite its usefulness, the introduction of a static potential freezes
the gluon dynamics and it cannot give the full picture of the QCD dynamics.

In a quantum field theory the information on the dynamics, such as the
spectra, form factors, transport properties, etc., is summarized in its Green
functions. These functions are not necessarily easy to compute, this is
particularly true outside the perturbative solution of the theory, and the
extraction of the required information can be a complex task. Returning to the
problem of the confinement mechanism in QCD, based on BRST invariance of the
QCD action, Kugo and Ojima Kugo:1979gm  suggested a confinement mechanism that
strongly constrains the infrared properties of gluon and ghost propagators the
two-point correlation functions of pure Yang-Mills theory. Interestingly, the
same type of constrains were predicted using the localised Gribov-Zwanziger
action for QCD Zwanziger:1991gz ; Zwanziger:1992qr  whose starting point is
completely different, namely the reduction of the functional integration
space. Lattice simulations of pure Yang-Mills theory to access the gluon and
ghost propagators, see e.g. Cucchieri:2007md ; Bogolubsky:2009dc ;
Dudal:2010tf ; Duarte:2016iko  and references therein, did not confirm the
predictions of these approaches for the propagators. Some of these issues have
been solved by extending the formulation of Gribov-Zwanziger actions; see e.g.
Dudal:2007cw ; Dudal:2008sp . The properties of the pure Yang-Mills gluon and
ghost propagators are now well established and there is a vast literature on
the subject; see, for example, Leinweber:1998uu ; Frasca:2007uz ;
Aguilar:2008xm ; Fischer:2008uz ; Boucaud:2010gr ; Bornyakov:2010nc ;
Ilgenfritz:2010gu ; Maas:2011ez ; Tissier:2017fqf ; Siringo:2018uho ;
Aguilar:2019uob ; Reinosa:2020skx  and references therein.

The predictions of Kugo-Ojima mechanism and of the Gribov-Zwanziger action for
the gluon propagator imply the violation of positivity for the Källén-Lehmann
gluon spectral representation, that necessarily invalidate a quantum
mechanical probabilistic interpretation for one-gluon particle states, meaning
that the Hilbert space of QCD does not include one-gluon particle states. The
corresponding quanta can only appear as components of the wave function of
composite states and, in this sense, they are confined and cannot be observed
as free particles. In principle, that is what happens with quarks and gluons
that contribute only to the internal structure of hadrons but are at the heart
of the hadronic dynamics. In this sense positivity violations imply confined
quanta. Positivity violation has been observed both for pure Yang-Mills gluon
and ghost propagators Aubin:2003ih ; Fischer:2003rp ; Aubin:2004av ;
Cucchieri:2004mf ; Bowman:2007du ; Dudal:2019gvn . Independently of the
confinement mechanism it turns out that, in non-Abelian Yang-Mills theories,
positivity violation of the Källén-Lehmann spectral function is linked also
with asymptotic freedom Cornwall:2013zra . The violation of positivity
condition for the spectral function associated with two-points functions has
also been reported for $O(N)$ scalar theories Matsubara:1983zt  and for QCD at
finite temperature Silva:2013rrr ; Su:2014rma .

Denoting the propagator in (Euclidean) momentum space by $D(p^{2})$, its
Källén-Lehmann integral representation, see e.g. Peskin:1995ev , is given by

$D(p^{2})=\int^{+\infty}_{\mu_{0}}d\mu\leavevmode\nobreak\
\frac{\rho(\mu)}{p^{2}+\mu}\ ,$ (1)

where the spectral function $\rho(\mu)$ reads

$\rho(\mu)=\sum_{n}\,\delta(\mu-m^{2}_{n})\,\left|\langle
0|\,\mathcal{O}\,|n_{0}\rangle\right|^{2}$ (2)

with $\mathcal{O}$ being the field operator associated with the corresponding
quanta, $m_{n}$ are the masses of the physical states and $|n_{0}\rangle$ are
the physical states at rest, i.e. with $\vec{p}=0$, that are eigenstates of
the four momentum operator. It follows from the definition that $\rho(\mu)$ is
positive definite. The Schwinger function

$\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{-\infty}\frac{dp_{4}}{2\,\pi}\leavevmode\nobreak\
\left.\frac{}{}D(p)\right|_{\vec{p}=0}\leavevmode\nobreak\ e^{-\,i\,p_{4}\,t}$
(3) $\displaystyle=$
$\displaystyle\int^{+\infty}_{\mu_{0}}\frac{d\mu}{2\,\sqrt{\mu}}\leavevmode\nobreak\
\rho(\mu)\leavevmode\nobreak\ e^{-\sqrt{\mu}\,t}$

is given by a sum of positive terms and, therefore, is also positive definite
for all $t$. From the point of view of testing confinement through positivity
violation, the Schwinger function allows to check for the positivity of
$\rho(\mu)$ without having to invert the integral equation (1). On the other
hand, if $C(t)$ is positive for all $t$, it does not provide any information
on the sign for the spectral function $\rho(\mu)$. Indeed, in this case, the
contributions of the negative $\rho(\mu)$ can be cancelled by the
contributions where $\rho(\mu)>0$ resulting in a positive Schwinger function.
In the following, we will always assume that the propagators have an integral
representation of the Källén-Lehmann type. Note that it is not clear that this
assumption holds for confined particles; see e.g. Binosi:2019ecz ; Li:2019hyv
; Falcao:2020vyr ; Li:2021wol ; Hayashi:2021nnj ; Hayashi:2021jju  and
references therein.

Compact pure gauge QED allows us to check the connection between confinement
and positivity violation. Indeed, recently, first principles lattice
simulations of the compact formulation of a pure gauge U(1) theory
Wilson:1974sk  addressed the computation of the Landau gauge photon propagator
Loveridge:2021qzs ; Loveridge:2021wav ; Loveridge:2021zke

$\langle
A_{\mu}(k)\,A_{\nu}(p)\rangle=V\,\delta(k+p)\,\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\,D(p^{2})\
,$ (4)

where $\langle\cdots\rangle$ stands for gauge average and $V$ is the lattice
volume, confirm that pure gauge compact QED has two phases. At low
$\beta=1/e^{2}$, where $e$ is the bare coupling constant and $\beta$ is the
coupling constant that appears in the lattice version of the action, the
theory is confining, in the sense that the corresponding static potential
grows linearly with the distance between the fermions, while at high $\beta$
values the theory approaches a free field theory in the thermodynamic limit.
Moreover, these studies also show that in the confined phase compact QED has a
mass gap and a non-trivial topological structure, while in the deconfined
phase the theory becomes massless and the topology of the gauge theory becomes
trivial111The issues related to the topology of QED and its compact
formulation on the lattice are discussed in Loveridge:2021qzs ;
Loveridge:2021wav . as it should be for a free field theory. These properties
are translated into a Landau gauge propagator $D(p^{2})$ that is rather
different in both phases; see e.g. Figs. 1 and 2 in Loveridge:2021wav . The
investigation of the Landau gauge photon propagator confirms previous studies
who show that the transition between the two phases occurs at $\beta$ slightly
above unity and that the transition from the confined to the deconfined phase
is first order Polyakov:1975rs ; Polyakov:1976fu ; Banks:1977cc ; Glimm:1977gz
; Fradkin:1978th ; Creutz:1979zg ; Guth:1979gz ; Lautrup:1980xr ;
Frohlich:1982gf ; Berg:1983is ; Kogut:1987cd ; Jersak:1996mj ; Panero:2005iu ;
Gubarev:2000eu ; Arnold:2002jk ; Coddington:1987yz ; Nakamura:1991ww .

The question addressed in the current work is to explore how the connection
between the transition from the confined to the deconfined phase translates
into the photon spectral function. In particular, we aim to investigate the
relation between confinement and positivity violation for the case of pure
gauge compact QED.

Figure 1: Bare Schwinger function for $\beta=0.8$ and for $\beta=1.2$ computed
using a simulation on a $48^{4}$ lattice.

The 4D lattice simulations of the compact U(1) gauge theory considered here
use hypercubic lattices with $N$ points in each direction. For details on the
sampling, gauge fixing, the computation of the gauge fields from the links and
the propagator we refer the reader to Loveridge:2021qzs ; Loveridge:2021wav .
Note that for the computation of the Schwinger function only the so-called
naive momenta are considered. Moreover, the Schwinger function can be
extracted from the Euclidean momentum propagator considering only time-like
momenta. The lattice Schwinger function is then given by

$C(t)=\frac{1}{N}\sum^{N-1}_{n=0}\leavevmode\nobreak\
\left.\frac{}{}D(p)\right|_{\vec{p}=0}\,e^{-i\,\frac{2\,\pi\,n}{N}\,t}\ .$ (5)

For pure gauge non-Abelian theories the Schwinger function has been
investigated for lattice regularized SU(2) Cucchieri:2004mf  and SU(3)
Silva:2007sj ; Silva:2006bs  Yang-Mills theories. Further, for pure gauge
theories the Schwinger function was also computed within a class of solutions
of Dyson-Schwinger equations AlkoferPPNP03 ; Alkofer:2003jj . In all cases it
was observed that $C(t)$ takes positive and negative values, i.e. the spectral
function is not positive definite. At small $t$ the Schwinger function
decreases with time, followed by oscillations at larger $t$ often times are
around zero. The Schwinger function can be described by two mass scales that
are associated the initial decay and with the period of oscillation observed
at larger $t$.

Figure 2: Bare Schwinger function close to the confinement/deconfinement
transition.

The computation of the spectral function directly from the propagator is an
ill-defined problem that is soluble after calling for regularization
techniques or by introducing a bias. For non-Abelian gauge theories, direct
attempts to measure the gluon or the ghost spectral density Iritani:2009mp ;
Dudal:2013yva ; Oliveira:2016stx ; Cyrol:2018xeq ; Tripolt:2018qvi ;
Dudal:2019gvn ; Binosi:2019ecz ; Li:2019hyv ; Horak:2021pfr ; Horak:2021syv
give $\rho(\mu)$ that take positive and negative values for different ranges
of $\mu$ and, therefore, confirm positivity violation in pure gauge Yang-Millg
theories.

In Loveridge:2021qzs ; Loveridge:2021wav  the Landau gauge photon propagator
in compact pure gauge U(1) theory was computed for various lattice volumes and
$\beta$ values. Here we calculate the Schwinger function using the outcome of
the simulations reported before. All the quantities that are shown are bare
quantities. To define a renormalized Schwinger function one would have to
renormalize the photon propagator. The difficulties of renormalizing the two
point correlation function have been discussed in Loveridge:2021wav . However,
the renormalization of the Schwinger function translates into a global rescale
and has no impact on the observed positivity violation as a function of
$\beta$. The Schwinger function for the smallest $\beta=0.8$, where the theory
is in the confined phase, and largest $\beta=1.2$, that is associated with an
essentially free theory, is reported in Fig. 1. In the confined phase
($\beta=0.8$), that shows a mass gap and a non-trivial topological structure,
$C(t)$ is positive defined for $t\lesssim 3$, it becomes negative for the
first time after $t=3$ and then it oscillates for larger times. This result
implies that for $\beta=0.8$ the spectral function $\rho(\mu)$ is not a
positive function and, therefore, there are no one-photon states belonging to
the Hilbert space of the physical states in this phase. Positivity violation
of $C(t)$ is not observed in the deconfined phase, where the theory is
massless and topologically trivial, and, therefore, no statement about the
positivity of the spectral function can be made. The results of Fig. 1
suggests that for the compact formulation of QED there is a connection between
confinement and positivity violation.

In order to investigate this connection deeper, the Schwinger function for
various $\beta$ values computed around the transition between the two phases
is reported in Fig. 2. Positivity violation of $C(t)$ and, therefore, of
$\rho(\mu)$ is seen for $\beta<1.005$. For larger $\beta$ the function $C(t)$
is always positive definite and/or becomes compatible with zero at larger $t$,
within one standard deviation, and no positivity violation can be claimed. In
the confined phase, the oscillations of $C(t)$ at large $t$’s are difficult to
observe but the data is still compatible with an oscillatory behaviour at
larger times. This could be a limitation of using relatively small lattices in
the simulation. Hopefully, an increase in statistics and in the lattice volume
will help identifying the oscillations with $t$. This oscillatory behaviour is
no longer present for $\beta\geqslant 1.005$. For the largest $\beta$
reported, the Schwinger function is always a smooth positive function without
oscillations; see also the bottom plot in Fig. 1. Furthermore, we have also
checked that for $\beta=0.8$ (confined phase) and $\beta=1.2$ (deconfined
phase) the Schwinger functions for the largest lattice volumes considered in
Loveridge:2021qzs , i.e. the simulations using $96^{4}$ lattices, reproduce
the same qualitative behaviour as observed in Figs 1 and 2.

Figure 3: Effective mass associated with the decay of $C(t)$ at $t=0$ as a
function of $\beta$. The errors were computed assuming Gaussian error
propagation. Figure 4: Effective mass $m(t)$ associated with the decay of
$C(t)$ for the same $\beta$ as in Fig. 2. In all cases, the errors on $m(t)$
were computed assuming Gaussian error propagation and the mass was defined
only when $C(t)$ is positive definite.

At small times the data for $C(t)$ decreases with $t$ both for the confined
and deconfined phase. An effective mass scale associated with the decay can be
measured taking the definition

$m(t)=-\ln\frac{C(t+1)}{C(t)}\ .$ (6)

In the continuum formulation this effective mass scale is proportional to the
first derivative of $C(t)$ at $t$ and it follows from Eq. (3) that

$m(t)\propto\frac{1}{2}\,\int^{+\infty}_{\mu_{0}}d\mu\leavevmode\nobreak\
\rho(\mu)\leavevmode\nobreak\ e^{-\sqrt{\mu}\,t}\ .$ (7)

In particular $m=m(0)$ is proportional to the integral of the photon spectral
density. The above mass scale has been considered previously and studied
within U(1) and SU(2) gauge theories and its variation related to positivity
violation of the spectral density Aubin:2003ih ; Aubin:2004av ;
Cucchieri:2004mf . Indeed, as demonstrated in these previous works an increase
of $m(t)$ signals positivity violation. As discussed below, for compact U(1)
lattice pure gauge theory, we observe an increase of $m(t)$ at small $t$ in
the confined phase and, therefore, positivity violation occurs in this phase.
Our results for the Schwinger function confirm this result.

In all cases that we have computed the Schwinger function, $C(t)$ is a
decreasing function of $t$ for small $t$ and, therefore, $m$ is always
positive definite for small $t$. $m=m(0)$ being positive definite is not in
contradiction with the observed positivity violation in the confined phase.
Indeed, a $m>0$ means that the area under the curve $y=\rho(\mu)$ for $\rho>0$
is larger than the area under the curve $y=-\rho(\mu)$ where $\rho<0$. The
Schwinger function is a weighted combination of these two areas and it is the
balance of the two contributions that allows or not to observe positivity
violation.

In Fig. 3 this mass for $t=0$, given in lattice units, is reported as a
function of the coupling constant $\beta$. $m$ is large in the confined phase
and shows a sudden drop in the vicinity of the transition towards the
deconfined phase. In the deconfined phase and for the larger $\beta$’s $m$ is
compatible, within one standard deviation, with zero. As Fig. 3 shows $m$ can
be used to distinguish the two phases and it takes large values (in lattice
units) in the confined phase. Recall that the study of the photon propagator
also shows that the confined phase has a mass gap. Further, in Fig. 4 the mass
function $m(t)$ is given for $t$’s where the Schwinger function $C(t)$ has the
same sign as $C(0)$, and for the same $\beta$ given in Fig. 2. Fig. 4
complements the content of Fig. 3 and it also shows that, in general, $m(t)$
is an increasing function of $t$ for small $t$ in the confined phase, i.e.
positivity violation of the spectral function occurs in the confined phase,
while in the deconfined phase $m(t)$ is essentially constant and is compatible
with zero within errors. The study of $C(t)$ and $m(t)$ also shows clearly
that, for the confined phase, even for small $t$ the Schwinger function $C(t)$
deviates from a simple exponential decay. This can be understood from the
behaviour of $m(t)$, see Fig. 4, where $m(t)$ increases with $t$ for small
times.

The reported Schwinger function shows an oscillatory behaviour at large $t$
that appears only in the confined phase. These oscillations define another
mass that vanishes in the deconfined phase, as no oscillations are observed.
We made no attempt to access this second mass scale. Our data suggests then
that the confined phase is characterized by two mass scales that are
associated with the observed decay of $C(t)$ at small $t$ and with the
oscillations observed at larger times. On the other hand for the deconfined
phase, that includes the continuum limit, the theory is conformal invariant
and has no mass scales associated with it.

In summary, the simulations reported here show, once more, that the nature of
the Landau gauge photon propagator and also its Källén-Lehmann spectral
density depend on the coupling of the compact version of the pure U(1) gauge.
In the confined phase, the Schwinger function and, therefore, the Källén-
Lehmann spectral density are not always positive definite. On the other hand,
the Schwinger function does not exhibit positivity violation in the deconfined
phase, where the continuum theory is realized. Our results illustrate,
clearly, for the pure gauge compact QED the link between positivity violation
and confinement that is expected to occur in other theories when confinement
occurs.

## Acknowledgments

This work was partly supported by the FCT – Fundação para a Ciência e a
Tecnologia, I.P., under Projects Nos. UIDB/04564/2020 and UIDP/04564/2020. P.
J. S. acknowledges financial support from FCT (Portugal) under Contract No.
CEECIND/00488/2017. The authors acknowledge the Laboratory for Advanced
Computing at the University of Coimbra (http://www.uc.pt/lca) for providing
access to the HPC resource Navigator.

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# Bailey pairs, radial limits of $q$-hypergeometric false theta functions, and
a conjecture of Hikami

Jeremy Lovejoy  CNRS, Université Paris Cité, Bâtiment Sophie Germain, Case
Courier 7014, 8 Place Aurélie Nemours, 75205 Paris Cedex 13, FRANCE
<EMAIL_ADDRESS>and  Rishabh Sarma  Department of Mathematics, University
of Florida, Gainesville, FL, 32611, USA<EMAIL_ADDRESS>
###### Abstract.

In the first part of this paper we prove a conjecture of Hikami on the values
of the radial limits of a family of $q$-hypergeometric false theta functions.
Hikami conjectured that the radial limits are obtained by evaluating a
truncated version of the series. He proved a special case of his conjecture by
computing the Kashaev invariant of certain torus links in two different ways.
We prove the full conjecture using Bailey pairs. In the second part of the
paper we show how the framework of Bailey pairs leads to further results of
this type.

###### Key words and phrases:

false theta functions, radial limits, Bailey pairs

###### 2020 Mathematics Subject Classification:

33D15

##  1\. Introduction

The first main goal of this paper is to prove a conjecture of Hikami [3] on
the values of the radial limits of certain $q$-hypergeometric series. To state
Hikami’s conjecture, first recall the $q$-Pochhammer symbol, defined for
$n\geq 0$ by

$(x)_{n}=(x;q)_{n}=(1-x)(1-xq)\cdots(1-xq^{n-1}),$ (1.1)

and the $q$-binomial coefficient (or Gaussian polynomial)

$\begin{bmatrix}n\\\ k\end{bmatrix}=\begin{bmatrix}n\\\
k\end{bmatrix}_{q}=\begin{cases}\frac{(q;q)_{n}}{(q;q)_{n-k}(q;q)_{k}},&\text{if
$0\leq k\leq n$},\\\ 0,&\text{otherwise}.\end{cases}$ (1.2)

Next, for $m\geq 2$ and $0\leq a\leq m-2$, define the $q$-hypergeometric
series $\widetilde{\Phi}_{m}^{(a)}(q)$ by111Hikami included the prefactor $m$
for convenience. For consistency, we preserve it here.

$\displaystyle\widetilde{\Phi}_{m}^{(a)}(q)=mq^{\frac{(m-1-a)^{2}}{4m}}\sum_{n_{1},\dots,n_{m-1}\geq
0}$
$\displaystyle(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+n_{1}^{2}+\cdots+n_{m-2}^{2}+n_{a+1}+\cdots+n_{m-2}}$
(1.3)
$\displaystyle\times\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\\\
n_{i}\end{bmatrix}.$

Hikami [3, Corollary 7] showed that the series $\widetilde{\Phi}_{m}^{(a)}(q)$
is the false theta function

$\widetilde{\Phi}_{m}^{(a)}(q)=m\sum_{n\geq
0}\chi_{2m}^{(a)}(n)q^{\frac{n^{2}}{4m}},$ (1.4)

where $\chi_{2m}^{(a)}(n)$ is the odd periodic function defined by

$\chi_{2m}^{(a)}(n)=\begin{cases}1&\mbox{if $n\equiv m-a-1\pmod{2m}$},\\\
-1&\mbox{if $n\equiv m+a+1\pmod{2m}$},\\\ 0,&\mbox{otherwise}.\end{cases}$
(1.5)

It is known that such false theta functions have well-defined limiting values
as $q$ approaches a root of unity radially from inside the unit disk and that
the resulting function is a quantum modular form [2]. Hikami’s conjecture
states that in the case of $\widetilde{\Phi}_{m}^{(a)}(q)$ these radial limits
are given by evaluating a truncated version of the series. Namely, define the
polynomial $Y_{m,N}^{(a)}(q)$ by

$Y_{m,N}^{(a)}(q)=\sum_{n_{1},\dots,n_{m-1}=0}^{N-1}(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+n_{1}^{2}+\cdots+n_{m-2}^{2}+n_{a+1}+\cdots+n_{m-2}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\\\
n_{i}\end{bmatrix}.$ (1.6)

Then we have the following.

######  Conjecture 1.1 (Hikami [3]).

Let $q=e^{2\pi i/N}$. For any $m\geq 2$ and $0\leq a\leq m-2$ we have

$\widetilde{\Phi}_{m}^{(a)}(q)=q^{\frac{(m-1-a)^{2}}{4m}}Y_{m,N}^{(a)}(q).$
(1.7)

Hikami proved his conjecture in the case $a=0$ by showing that both sides are
essentially the Kashaev invariant of the torus link $T(2,2m)$. He writes that
it is unclear whether these expressions for $a\neq 0$ correspond to any
quantum invariant, perhaps hinting that another method is needed to prove the
full conjecture. Here we show that the full conjecture follows from the theory
of Bailey pairs. In fact, it does so quite easily once the Bailey pair
framework is set up. Moreover, it holds for any root of unity.

######  Theorem 1.2.

Let $q=e^{2\pi iM/N}$ be a primitive $N$th root of unity with $N>0$. Then for
any $m\geq 2$ and $0\leq a\leq m-2$ we have

$\widetilde{\Phi}_{m}^{(a)}(q)=q^{\frac{(m-1-a)^{2}}{4m}}Y_{m,N}^{(a)}(q).$
(1.8)

In the second part of the paper we show that a similar phenomenon occurs for
other families of $q$-hypergeometric false theta functions constructed using
Bailey pairs. That is, the radial limits at roots of unity are obtained by
evaluating the truncated series. See Theorems 3.3, 3.7, and 3.11.

The paper is organized as follows. In the next section we prove Hikami’s
conjecture. In Section 3 we review the computation of radial limits of false
theta functions and present three more results along the lines of Theorem 1.2.
We conclude with some remarks on quantum $q$-series identities in the spirit
of [4] that arise from our work.

##  2\. Proof of Theorem 1.2

In this section we prove Theorem 1.2. Before proceeding to the proof we note
two facts. First, for any primitive $N$th root of unity $q$, Hikami [3,
Proposition 10] computed the limiting values $\widetilde{\Phi}_{m}^{(a)}(q)$.
Here and throughout the paper we let $\zeta_{N}=e^{2\pi i/N}$.

######  Lemma 2.1 (Hikami).

For coprime integers $M$ and $N$ with $N>0$ we have

$\widetilde{\Phi}_{m}^{(a)}(\zeta_{N}^{M})=m\sum_{n=0}^{mN}\chi_{2m}^{(a)}(n)\left(1-\frac{n}{mN}\right)\zeta_{N}^{Mn^{2}/4m}.$
(2.1)

Second, we need a result from the classical theory of Bailey pairs. Recall
that a pair of sequences $(\alpha_{n},\beta_{n})$ is called a Bailey pair
relative to $(a,q)$ if

$\beta_{n}=\sum_{k=0}^{n}\frac{\alpha_{k}}{(q)_{n-k}(aq)_{n+k}}.$ (2.2)

The Bailey lemma (see [1, Lemma 5.1]) says that if $(\alpha_{n},\beta_{n})$ is
a Bailey pair relative to $(a,q)$ then so is
$(\alpha_{n}^{\prime},\beta_{n}^{\prime})$, where

$\alpha_{n}^{\prime}=\frac{(b)_{n}(c)_{n}}{(aq/b)_{n}(aq/c)_{n}}(aq/bc)^{n}\alpha_{n}$
(2.3)

and

$\beta_{n}^{\prime}=\frac{(aq/bc)_{n}}{(q)_{n}(aq/b)_{n}(aq/c)_{n}}\sum_{k=0}^{n}\frac{(b)_{k}(c)_{k}(q^{-n})_{k}q^{k}}{(bcq^{-n}/a)_{k}}\beta_{k}.$
(2.4)

Putting this into (2.2) with $b,c\to\infty$ and using

$(q)_{n-k}=\frac{(q)_{n}}{(q^{-n})_{k}}(-1)^{k}q^{\binom{k}{2}-nk},$

we have the following.

######  Lemma 2.2.

If $(\alpha_{n},\beta_{n})$ is a Bailey pair relative to $(q,q)$, then

$(q^{2})_{n}\sum_{k=0}^{n}(q^{-n})_{k}(-1)^{k}q^{\binom{k+1}{2}+(n+1)k}\beta_{k}=\sum_{k=0}^{n}\frac{(q^{-n})_{k}}{(q^{2+n})_{k}}(-1)^{k}q^{\binom{k+1}{2}+(n+1)k}\alpha_{k}.$
(2.5)

With these two lemmas in hand, we are ready to prove Hikami’s conjecture.

###### Proof of Theorem 1.2.

We begin by recalling the Bailey pair relative to $(q,q)$ in [5, Prop. 4.1],

$\alpha_{n}=\frac{(1-q^{(a+1)(2n+1)})(-1)^{n}q^{\binom{n+1}{2}+(a+1)n^{2}+(m-a-1)(n^{2}+n)}}{1-q}$

and

$\beta_{n}=\beta_{n_{m}}=\sum_{n_{1},\dots,n_{m-1}\geq
0}\frac{q^{n_{1}^{2}+\cdots+n_{m-1}^{2}+n_{a+1}+\cdots+n_{m-1}}}{(q)_{n_{m}}}\prod_{i=1}^{m-1}\begin{bmatrix}n_{i+1}+\delta_{a,i}\\\
n_{i}\end{bmatrix}.$

Using this with $m=m-1$ and substituting into Lemma 2.2 with $n=N-1$ we obtain

$\displaystyle(q)_{N}\sum_{n_{1},\dots,n_{m-1}=0}^{N-1}$
$\displaystyle\frac{(q^{1-N})_{n_{m-1}}(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+Nn_{m-1}+n_{1}^{2}+\cdots+n_{m-2}^{2}+n_{a+1}+\cdots+n_{m-2}}}{(q)_{n_{m-1}}}$
(2.6)
$\displaystyle\times\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{a,i}\\\
n_{i}\end{bmatrix}$
$\displaystyle=\sum_{k=0}^{N-1}\frac{(q^{1-N})_{k}}{(q^{1+N})_{k}}(1-q^{(a+1)(2k+1)})q^{mk^{2}+(m-a-1)k+Nk}.$

Note that for $q$ a primitive $N$th root of unity the left hand side vanishes,
giving

$\displaystyle 0$
$\displaystyle=\sum_{k=0}^{n-1}(1-q^{(a+1)(2k+1)})q^{mk^{2}+(m-a-1)k}$
$\displaystyle=q^{\frac{-(m-a-1)^{2}}{4m}}\sum_{k=0}^{2mN}\chi_{2m}^{(a)}(k)q^{k^{2}/4m},$

where the second equality follows from a short computation involving
completing the square. Dividing both sides of (2.6) by $(q)_{N}$ and taking
$\lim_{q\to\zeta_{N}^{M}}$ we then have

$\displaystyle Y_{m,N}^{(a)}(\zeta_{N}^{M})$
$\displaystyle=\lim_{q\to\zeta_{N}^{M}}\frac{1}{(q)_{N}}\sum_{k=0}^{N-1}\frac{(q^{1-N})_{k}}{(q^{1+N})_{k}}(1-q^{(a+1)(2k+1)})q^{mk^{2}+(m-a-1)k+Nk}$
$\displaystyle=\frac{1}{N}\lim_{q\to\zeta_{N}^{M}}\frac{1}{1-q^{N}}q^{\frac{-(m-a-1)^{2}}{4m}}\sum_{k=0}^{2mN}\chi_{2m}^{(a)}(k)q^{k^{2}/4m}$
$\displaystyle=-\frac{1}{4mN^{2}}\zeta_{N}^{\frac{-M(m-a-1)^{2}}{4m}}\sum_{k=0}^{2mN}k^{2}\chi_{2m}^{(a)}(k)\zeta_{N}^{Mk^{2}/4m}.$
(2.7)

In the above we have used the fact that for $q$ a primitive $N$th root of
unity,

$\prod_{i=1}^{N-1}(1-q^{i}x)=\frac{1-x^{N}}{1-x},$

which gives

$(q;q)_{N-1}=N.$

Now, to finish the proof we compute

$\displaystyle\zeta_{N}^{\frac{M(m-a-1)^{2}}{4m}}Y_{m,N}^{(a)}(\zeta_{N}^{M})$
$\displaystyle=-\frac{1}{4mN^{2}}\sum_{k=0}^{2mN}k^{2}\chi_{2m}^{(a)}(k)\zeta_{N}^{Mk^{2}/4m}$
$\displaystyle=-\frac{1}{4mN^{2}}\Bigg{(}\sum_{k=0}^{mN}k^{2}\chi_{2m}^{(a)}(k)\zeta_{N}^{Mk^{2}/4m}+\sum_{k=mN}^{2mN}k^{2}\chi_{2m}^{(a)}(k)\zeta_{n}^{Mk^{2}/4m}\Bigg{)}$
$\displaystyle=-\frac{1}{4mN^{2}}\Bigg{(}\sum_{k=0}^{mN}k^{2}\chi_{2m}^{(a)}(k)\zeta_{N}^{Mk^{2}/4m}$
$\displaystyle\hskip
72.26999pt+\sum_{k=0}^{mN}(2mN-k)^{2}\chi_{2m}^{(a)}(2mN-k)\zeta_{N}^{M(2mN-k)^{2}/4m}\Bigg{)}$
$\displaystyle=-\frac{1}{4mN^{2}}\Bigg{(}\sum_{k=0}^{mN}k^{2}\chi_{2m}^{(a)}(k)\zeta_{N}^{Mk^{2}/4m}-\sum_{k=0}^{mN}(2mN-k)^{2}\chi_{2m}^{(a)}(k)\zeta_{N}^{Mk^{2}/4m}\Bigg{)}$
$\displaystyle=\sum_{k=0}^{mN}\chi_{2m}^{(a)}(k)\left(m-\frac{k}{N}\right)\zeta_{N}^{Mk^{2}/4m}$
$\displaystyle=\widetilde{\Phi}_{m}^{(a)}(\zeta_{N}^{M}),$

the last equality following from (2.1). This establishes Theorem 1.2.

∎

##  3\. Further examples

###  3.1. Preliminaries

In this section we find further examples of families of $q$-hypergeometric
false theta functions whose radial limits are given by evaluating a truncated
version of the $q$-series. There are certainly more, but we limit ourselves to
applications of some of the Bailey pairs in [5]. Here we will need a
specialization of the Bailey lemma different from Lemma 2.2. Namely, if we
take $a=b=q$ and $c\to\infty$ in (2.3) and (2.4) and then use the definition
(2.2) with $n\to\infty$ we have the following, which is well-known.

######  Lemma 3.1.

If $(\alpha_{n},\beta_{n})$ is a Bailey pair relative to $(q,q)$, then

$\sum_{n\geq 0}(q)_{n}(-1)^{n}q^{\binom{n+1}{2}}\beta_{n}=(1-q)\sum_{n\geq
0}(-1)^{n}q^{\binom{n+1}{2}}\alpha_{n}.$ (3.1)

We also require a formula which allows us to compute the radial limits of
false theta functions as we approach a root of unity. We cite this more or
less verbatim from Hikami [3, Proposition 9].

######  Lemma 3.2.

Let $C_{f}(n)$ be a periodic function with mean value $0$ and modulus $f$.
Then as $t\searrow 0$ we have the asymptotic expansion

$\sum_{n\geq 1}C_{f}(n)e^{-n^{2}t}\sim\sum_{k\geq
0}L(-2k,C_{f})\frac{(-t)^{k}}{k!},$ (3.2)

where

$L(-k,C_{f})=-\frac{f^{k}}{k+1}\sum_{n=1}^{f}C_{f}(n)B_{k+1}\left(\frac{n}{f}\right),$
(3.3)

with $B_{k}(x)$ being the $k$th Bernouilli polynomial.

In each of the following subsections we first use a Bailey pair from [5]
together with Lemma 3.1 to produce a family of $q$-hypergeometric false theta
functions. These are not all necessarily new, but we include the derivations
for completeness. Next we use Lemma 3.2 to compute the radial limits of the
false theta functions. Finally, we use the same Bailey pair in Lemma 2.2 to
produce a truncated version of the $q$-series whose values at roots of unity
coincide with the radial limits of the infinite series.

###  3.2. Example 1

For $m\geq 2$ define

$\displaystyle\widetilde{\Psi}_{m}(q)=\frac{2m-1}{2}q^{\frac{(2m-3)^{2}}{8(2m-1)}}\sum_{n_{1},n_{2},\dots,n_{m-1}\geq
0}$
$\displaystyle\frac{(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+n_{1}^{2}+\cdots+n_{m-2}^{2}+n_{1}+\cdots+n_{m-2}}}{(-q)_{n_{1}}}$
(3.4) $\displaystyle\times\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}$

and its truncated version

$Z_{m,N}(q)=\sum_{n_{1},\dots,n_{m-1}=0}^{N-1}\frac{(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+n_{1}^{2}+\cdots+n_{m-2}^{2}+n_{1}+\cdots+n_{m-2}}}{(-q)_{n_{1}}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}.$ (3.5)

Our goal is to prove that $\widetilde{\Psi}_{m}(q)$ is well-defined at odd
roots of unity and that the values are essentially given by $Z_{m,N}(q)$. We
restrict to odd roots of unity to avoid poles arising from the term
$(-q)_{n_{1}}$ in the denominator of $Z_{m,N}(q)$.

######  Theorem 3.3.

Let $q=e^{2\pi iM/N}$ be a primitive odd $N$th root of unity. Then

$\widetilde{\Psi}_{m}(q)=q^{\frac{(2m-3)^{2}}{8(2m-1)}}Z_{m,N}(q).$ (3.6)

This result will follow from the next three propositions. We first prove that
$\widetilde{\Psi}_{m}(q)$ is a false theta function.

######  Proposition 3.4.

We have the identity

$\widetilde{\Psi}_{m}(q)=\frac{2m-1}{2}\sum\limits_{k=0}^{\infty}\chi_{4m-2}(k)\,q^{\frac{k^{2}}{8(2m-1)}},$
(3.7)

where $\chi_{4m-2}(k)$ is the odd periodic function defined by

$\chi_{4m-2}(k)=\begin{cases}1&\mbox{if $k\equiv 2m-3\pmod{4m-2}$},\\\
-1&\mbox{if $k\equiv 2m+1\pmod{4m-2}$},\\\ 0,&\mbox{otherwise}.\end{cases}$
(3.8)

###### Proof.

We begin with the Bailey pair relative to $(q,q)$ [5, Proposition 5.1],

$\alpha_{n}=\frac{1-q^{2n+1}}{1-q}(-1)^{n}q^{mn^{2}+(m-1)n}$ (3.9)

and

$\beta_{n}=\beta_{n_{m}}=\sum_{n_{1},n_{2},\dots,n_{m-1}\geq
0}\frac{q^{n_{1}^{2}+\cdots+n_{m-1}^{2}+n_{1}+\cdots+n_{m-1}}}{(q)_{n_{m}}(-q)_{n_{1}}}\prod_{i=1}^{m-1}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}.$ (3.10)

Using this with $m=m-1$ in Lemma 3.1 we obtain

$\displaystyle\widetilde{\Psi}_{m}(q)$
$\displaystyle=\frac{2m-1}{2}q^{\frac{(2m-3)^{2}}{8(2m-1)}}\sum_{n\geq
0}(1-q^{2n+1})q^{\binom{n+1}{2}+(m-1)n^{2}+(m-2)n}$

and the result then follows after completing the square on the right-hand
side. ∎

Next we employ Lemma 3.2 to calculate the radial limits of
$\widetilde{\Psi}_{m}(q)$ as $q$ approaches a root of unity.

######  Proposition 3.5.

For coprime integers $M$ and $N$ with $N$ odd and positive we have

$\widetilde{\Psi}_{m}(\zeta_{N}^{M})=\dfrac{-1}{8(2m-1)N^{2}}\sum_{k=0}^{(4m-2)N}k^{2}\chi_{4m-2}(k)\zeta_{N}^{\frac{k^{2}}{8(2m-1)}M}$
(3.11)

###### Proof.

Regarding $\widetilde{\Psi}_{m}$ as a function of $\tau$ with $q=e^{2\pi
i\tau}$, we have

$\displaystyle\widetilde{\Psi}_{m}\left(\frac{M}{N}+i\frac{t}{2\pi}\right)$
$\displaystyle=\frac{2m-1}{2}\sum_{k=0}^{\infty}\chi_{4m-2}(k)e^{2\pi
i\left(\frac{M}{N}+i\frac{t}{2\pi}\right)\frac{k^{2}}{8(2m-1)}}$
$\displaystyle=\frac{2m-1}{2}\sum_{k=0}^{\infty}C_{(4m-2)N}(k)e^{\frac{-k^{2}}{8(2m-1)}t},$

where

$C_{(4m-2)N}(k)=\chi_{4m-2}(k)e^{\frac{Mk^{2}}{2(4m-2)N}\pi i}.$

We have

$\displaystyle C_{(4m-2)N}(k+(4m-2)N)$
$\displaystyle=\chi_{4m-2}(k+(4m-2)N)e^{\frac{M\pi
i(k^{2}+2kN(4m-2)+N^{2}(4m-2)^{2})}{2(4m-2)N}}$
$\displaystyle=\chi_{4m-2}(k)e^{\frac{M\pi ik^{2}}{2(4m-2)N}}e^{M\pi
i(k+N(2m-1))},$

and using the fact that $N$ is odd together with the definition of
$\chi_{4m-2}(k)$ this gives

$C_{(4m-2)N}(k+(4m-2)N)=C_{(4m-2)N}(k).$ (3.12)

Similarly, we have

$C_{(4m-2)N}((4m-2)N-k)=-C_{(4m-2)N}(k).$ (3.13)

Note that by (3.13) we have

$\sum_{k=0}^{(4m-2)N}KC_{(4m-2)N}(k)=0$ (3.14)

for any constant $K$.

Now using Lemma 3.2 we have the asymptotic expansion as $t\searrow 0$,

$\widetilde{\Psi}_{m}\left(\frac{M}{N}+i\frac{t}{2\pi}\right)\sim\frac{2m-1}{2}\sum_{k=0}^{\infty}\dfrac{L(-2k,C_{(4m-2)N})}{k!}\left(-\dfrac{t}{8(2m-1)}\right)^{k}.$

Using this together with $B_{1}(x)=x-\frac{1}{2}$ we compute the limiting
value as follows,

$\displaystyle\widetilde{\Psi}_{m}\left(\frac{M}{N}\right)$
$\displaystyle=\frac{2m-1}{2}L(0,C_{(4m-2)N})$
$\displaystyle=-\frac{2m-1}{2}\sum_{k=1}^{(4m-2)N}C_{(4m-2)N}(k)B_{1}\left(\dfrac{k}{(4m-2)N}\right)$
$\displaystyle=\frac{2m-1}{2}\sum_{k=1}^{(4m-2)N}C_{(4m-2)N}(k)\left(1-\frac{k}{(4m-2)N}\right)\
\ \ \text{(by \eqref{Cprop3ex1})}$
$\displaystyle=\sum_{k=0}^{(4m-2)N}C_{(4m-2)N}(k)\left(\frac{2m-1}{2}-\dfrac{k}{4N}\right)$
$\displaystyle=\sum_{k=0}^{(2m-1)N}C_{(4m-2)N}(k)\left(\frac{2m-1}{2}-\dfrac{k}{4N}\right)$
$\displaystyle\ \ \ \ \ \ \ \
+\sum_{k=0}^{(2m-1)N}C_{(4m-2)N}((4m-2)N-k)\left(\frac{2m-1}{2}-\dfrac{(4m-2)N-k}{4N}\right)$
$\displaystyle=\sum_{k=0}^{(2m-1)N}C_{(4m-2)N}(k)\left(\frac{2m-1}{2}-\dfrac{k}{2N}\right)\
\ \ \text{(by \eqref{Cprop2ex1})}.$

Using the fact that

$\frac{2m-1}{2}-\dfrac{k}{2N}=\frac{-1}{8(2m-1)N^{2}}\left(k^{2}-((4m-2)N-k)^{2}\right)$

we then obtain

$\displaystyle\widetilde{\Psi}_{m}\left(\frac{M}{N}\right)=\dfrac{-1}{8(2m-1)N^{2}}\Bigg{(}$
$\displaystyle\sum_{k=0}^{(2m-1)N}k^{2}C_{(4m-2)N}(k)$
$\displaystyle-\sum_{k=0}^{(2m-1)N}((4m-2)N-k)^{2}C_{(4m-2)N}(k)\Bigg{)}$
$\displaystyle=\dfrac{-1}{8(2m-1)N^{2}}\Bigg{(}$
$\displaystyle\sum_{k=0}^{(2m-1)N}k^{2}C_{(4m-2)N}(k)$
$\displaystyle+\sum_{k=0}^{(2m-1)N}((4m-2)N-k)^{2}C_{(4m-2)N}((4m-2)N-k)\Bigg{)}$
$\displaystyle=\dfrac{-1}{8(2m-1)N^{2}}\Bigg{(}$
$\displaystyle\sum_{k=0}^{(2m-1)N}k^{2}C_{(4m-2)N}(k)+\sum_{k=(2m-1)N}^{(4m-2)N}k^{2}C_{(4m-2)N}(k)\Bigg{)}$
$\displaystyle=\dfrac{-1}{8(2m-1)N^{2}}$
$\displaystyle\sum_{k=0}^{(4m-2)N}k^{2}\chi_{4m-2}(k)e^{\frac{k^{2}}{2(4m-2)N}M\pi
i},$

which gives the result. ∎

Now we determine the values of $Z_{m,N}(q)$ at primitive odd $N$th roots of
unity.

######  Proposition 3.6.

For $q=\zeta_{N}^{M}$, a primitive odd $N$-th root of unity, we have

$Z_{m,N}(q)=\frac{-1}{8(2m-1)N^{2}}q^{\frac{-(2m-3)^{2}}{8(2m-1)}}\sum\limits_{k=0}^{(4m-2)N}k^{2}\chi_{4m-2}(k)\,q^{\frac{k^{2}}{8(2m-1)}}.$
(3.15)

###### Proof.

Inserting the Bailey pair from (3.9) and (3.10) into Lemma 2.2 with $m=m-1$
and $n=N-1$, we obtain

$\displaystyle\sum_{n_{1},\dots,n_{m-1}=0}^{N-1}$
$\displaystyle\frac{(q^{1-N})_{n_{m-1}}(-1)^{n_{m-1}}q^{Nn_{m-1}+\binom{n_{m-1}+1}{2}+n_{1}^{2}+\cdots+n_{m-2}^{2}+n_{1}+\cdots+n_{m-2}}}{(q)_{n_{m-1}}(-q)_{n_{1}}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}$
$\displaystyle=\sum_{k=0}^{N-1}\frac{(q^{1-N})_{k}(1-q^{2k+1})q^{-\binom{k}{2}+(N-1)k+mk^{2}+(m-1)k}}{(q)_{N+k}}.$

Then taking $q$ to be a primitive odd $N$-th root of unity $\zeta_{N}^{M}$, we
get,

$\displaystyle Z_{m,N}(q)$
$\displaystyle=\lim_{q\rightarrow\zeta_{N}^{M}}\sum_{k=0}^{N-1}\frac{(q^{1-N})_{k}(1-q^{2k+1})\,q^{-\binom{k}{2}+(N-1)k+mk^{2}+(m-1)k}}{(q)_{N-1}(1-q^{N})(q^{N+1})_{k}}$
$\displaystyle=\frac{1}{N}\lim_{q\rightarrow\zeta_{N}^{M}}\frac{\sum\limits_{k=0}^{N-1}(1-q^{2k+1})\,q^{-\binom{k}{2}+mk^{2}+mk-2k}}{(1-q^{N})}$
$\displaystyle=\frac{1}{N}\lim_{q\rightarrow\zeta_{N}^{M}}\frac{q^{\frac{-(2m-3)^{2}}{8(2m-1)}}\sum\limits_{k=0}^{(4m-2)N}\chi_{4m-2}(k)\,q^{\frac{k^{2}}{8(2m-1)}}}{(1-q^{N})}$
$\displaystyle=\frac{-1}{8(2m-1)N^{2}}\zeta_{N}^{\frac{-M(2m-3)^{2}}{8(2m-1)}}\sum\limits_{k=0}^{(4m-2)N}k^{2}\chi_{4m-2}(k)\,\zeta_{N}^{\frac{Mk^{2}}{8(2m-1)}}.$

This completes the proof. ∎

Comparing Propositions 3.5 and 3.6 gives Theorem 3.3.

###  3.3. Example 2

We follow the same steps as in the previous example, this time with the
function

$\displaystyle\widetilde{\Gamma}_{m}(q)=(m-1)q^{\frac{(2m-3)^{2}}{8(m-1)}}\sum_{n_{1},\dots,n_{m-1}\geq
0}$
$\displaystyle\frac{(-1)^{n_{m-1}}q^{2\binom{n_{m-1}+1}{2}+2n_{1}^{2}+2n_{1}+\cdots+2n_{m-2}^{2}+2n_{m-2}}(q;q^{2})_{n_{1}}}{(-q)_{2n_{1}+1}}$
(3.16) $\displaystyle\times\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}_{q^{2}}$

and the truncated version

$U_{m,N}(q)=\sum_{n_{1},\dots,n_{m-1}=0}^{N-1}\frac{(-1)^{n_{m-1}}q^{2\binom{n_{m-1}+1}{2}+2n_{1}^{2}+2n_{1}+\cdots+2n_{m-2}^{2}+2n_{m-2}}(q;q^{2})_{n_{1}}}{(-q)_{2n_{1}+1}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}_{q^{2}}.$ (3.17)

As usual, these are valid for $m\geq 2$. We will show that
$\widetilde{\Gamma}_{m}(q)$ is a false theta function and that its values at
odd roots of unity are computed using $U_{m,N}(q)$.

######  Theorem 3.7.

Let $q=e^{2\pi iM/N}$ be a primitive odd $N$th root of unity. Then

$\widetilde{\Gamma}_{m}(q)=q^{\frac{(2m-3)^{2}}{8(m-1)}}U_{m,N}(q).$ (3.18)

This theorem will follow from the next three propositions. We first show that
$\widetilde{\Gamma}(q)$ is a false theta function.

######  Proposition 3.8.

We have the identity

$\widetilde{\Gamma}_{m}(q)=(m-1)\sum\limits_{k=0}^{\infty}\chi_{4(m-1)}(k)\,q^{\frac{k^{2}}{8(m-1)}},$
(3.19)

where

$\chi_{4(m-1)}(k)=\begin{cases}1&\mbox{if $k\equiv 2m-3\pmod{4(m-1)}$},\\\
-1&\mbox{if $k\equiv 2m-1\pmod{4(m-1)}$},\\\ 0,&\mbox{otherwise}.\end{cases}$

###### Proof.

We use the Bailey pair relative to $(q^{2},q^{2})$ [5, Proposition 5.3],

$\alpha_{n}=\frac{(-1)^{n}q^{(2m-1)n^{2}+(2m-2)n}(1-q^{2n+1})}{1-q^{2}}$
(3.20)

and

$\beta_{n}=\beta_{n_{m}}=\sum_{n_{1},\dots,n_{m-1}\geq
0}\frac{q^{2n_{1}^{2}+2n_{1}+\cdots+2n_{m-1}^{2}+2n_{m-1}}(q;q^{2})_{n_{1}}}{(q^{2};q^{2})_{n_{m}}(-q)_{2n_{1}+1}}\prod_{i=1}^{m-1}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}_{q^{2}}.$ (3.21)

Inserting this with $m=m-1$ into Lemma 3.1 (remembering to replace $q$ by
$q^{2}$ throughout) we obtain

$\widetilde{\Gamma}_{m}(q)=(m-1)q^{\frac{(2m-3)^{2}}{8(m-1)}}\sum_{n\geq
0}q^{(2m-2)n^{2}+(2m-3)n}(1-q^{2n+1}),$

and the result then follows after completing the square on the right-hand
side. ∎

Next we use Lemma 3.2 to compute $\widetilde{\Gamma}(q)$ at odd roots of
unity.

######  Proposition 3.9.

For coprime integers $M$ and $N$ with $N$ odd and positive we have

$\widetilde{\Gamma}_{m}(\zeta_{N}^{M})=\dfrac{-1}{16(m-1)N^{2}}\sum_{k=0}^{(4m-4)N}k^{2}\chi_{4(m-1)}(k)\zeta_{N}^{\frac{k^{2}}{8(m-1)}M}.$
(3.22)

###### Proof.

Regarding $\widetilde{\Gamma}$ as a function of $\tau$ with $q=e^{2\i i\tau}$
we have

$\displaystyle\widetilde{\Gamma}_{m}\left(\frac{M}{N}+i\frac{t}{2\pi}\right)$
$\displaystyle=(m-1)\sum_{k=0}^{\infty}\chi_{4(m-1)}(k)e^{2\pi
i\left(\frac{M}{N}+i\frac{t}{2\pi}\right)\frac{k^{2}}{8(m-1)}}$
$\displaystyle=(m-1)\sum_{k=0}^{\infty}C_{4(m-1)N}(k)e^{\frac{-k^{2}}{8(m-1)}t}$

where

$C_{4(m-1)N}(k)=\chi_{4(m-1)}(k)e^{\frac{Mk^{2}}{4(m-1)N}\pi i}.$

We note that

$C_{4(m-1)N}(k+4(m-1)N)=C_{4(m-1)N}(k)$ (3.23)

and

$C_{4(m-1)N}(4(m-1)N-k)=-C_{4(m-1)N}(k).$ (3.24)

Then using Lemma 3.2 we have, asymptotically as $t\searrow 0$,

$\widetilde{\Gamma}_{m}\left(\frac{M}{N}+i\frac{t}{2\pi}\right)\sim(m-1)\sum_{k=0}^{\infty}\dfrac{L(-2k,C_{4(m-1)N})}{k!}\left(-\dfrac{t}{8(m-1)}\right)^{k}.$

Using this and keeping in mind (3.24) we compute the limiting value

$\displaystyle\widetilde{\Gamma}_{m}\left(\frac{M}{N}\right)$
$\displaystyle=(m-1)L(0,C_{4(m-1)N})$
$\displaystyle=-(m-1)\sum_{k=1}^{4(m-1)N}C_{4(m-1)N}(k)B_{1}\left(\dfrac{k}{4(m-1)N}\right)$
$\displaystyle=(m-1)\sum_{k=1}^{4(m-1)N}C_{4(m-1)N}(k)\left(\frac{1}{2}-\dfrac{k}{4(m-1)N}\right)$
$\displaystyle=(m-1)\sum_{k=0}^{2(m-1)N}C_{4(m-1)N}(k)\left(1-\dfrac{k}{2(m-1)N}\right)$
$\displaystyle=\sum_{k=0}^{2(m-1)N}C_{4(m-1)N}(k)\left((m-1)-\dfrac{k}{2N}\right)$
$\displaystyle=\dfrac{-1}{16(m-1)N^{2}}\left(\sum_{k=0}^{2(m-1)N}k^{2}C_{4(m-1)N}(k)-\sum_{k=0}^{2(m-1)N}(4(m-1)N-k)^{2}C_{4(m-1)N}(k)\right)$
$\displaystyle=\dfrac{-1}{16(m-1)N^{2}}\Bigg{(}\sum_{k=0}^{2(m-1)N}k^{2}C_{4(m-1)N}(k)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\sum_{k=0}^{2(m-1)N}(4(m-1)N-k)^{2}C_{4(m-1)N}(4(m-1)N-k)\Bigg{)}$
$\displaystyle=\dfrac{-1}{16(m-1)N^{2}}\left(\sum_{k=0}^{2(m-1)N}k^{2}C_{4(m-1)N}(k)+\sum_{k=2(m-1)}^{4(m-1)N}k^{2}C_{4(m-1)N}(k)\right)$
$\displaystyle=\dfrac{-1}{16(m-1)N^{2}}\sum_{k=0}^{(4m-4)N}k^{2}\chi_{4(m-1)}(k)e^{\frac{k^{2}}{4(m-1)N}M\pi
i},$

and the result follows. ∎

Finally, we compute the values of the rational function $U_{m,N}(q)$ when $q$
is a primitive odd $N$th root of unity.

######  Proposition 3.10.

For $q=\zeta_{N}^{M}$, a primitive odd $N$-th root of unity, we have

$U_{m,N}(q)=\frac{-1}{16(m-1)N^{2}}q^{\frac{-(2m-3)^{2}}{8(m-1)}}\sum\limits_{k=0}^{(4m-4)N}k^{2}\chi_{4(m-1)}(k)\,q^{\frac{k^{2}}{8(m-1)}}.$
(3.25)

###### Proof.

Inserting the Bailey pair (3.20) and (3.21) into Lemma 2.2 with $m=m-1$ and
$n=N-1$ we find

$\displaystyle U_{m,N}(\zeta_{N}^{M})$
$\displaystyle=\lim_{q\rightarrow\zeta_{N}^{M}}\sum_{k=0}^{N-1}\frac{(q^{2-2N};q^{2})_{k}(1-q^{2k+1})\,q^{2(m-1)k^{2}+(2m-3)k+2Nk}}{(q^{2};q^{2})_{N-1}(1-q^{2N})(q^{2N+2};q^{2})_{k}}$
$\displaystyle=\frac{1}{N}\lim_{q\rightarrow\zeta_{N}^{M}}\frac{\sum\limits_{k=0}^{N-1}(1-q^{2k+1})\,q^{2(m-1)k^{2}+(2m-3)k}}{1-q^{2N}}$
$\displaystyle=\frac{1}{N}\lim_{q\rightarrow\zeta_{N}^{M}}\frac{q^{\frac{-(2m-3)^{2}}{8(m-1)}}\sum\limits_{k=0}^{(4m-4)N}\chi_{4(m-1)}(k)\,q^{\frac{k^{2}}{8(m-1)}}}{1-q^{2N}}$
$\displaystyle=\frac{-1}{16(m-1)N^{2}}\zeta_{N}^{\frac{-M(2m-3)^{2}}{8(m-1)}}\sum\limits_{k=0}^{(4m-4)N}k^{2}\chi_{4(m-1)}(k)\,\zeta_{N}^{\frac{Mk^{2}}{8(m-1)}}$

∎

Combining Propositions 3.9 and 3.10 gives Theorem 3.7.

###  3.4. Example 3

For $m\geq 2$ and $0\leq a\leq m-2$ define $\widetilde{\Lambda}_{m}^{(a)}(q)$
by

$\displaystyle\widetilde{\Lambda}_{m}^{(a)}(q)=\frac{2m-1}{2}q^{\frac{(m-a-1)^{2}}{(2m-1)}}\sum_{n_{1},\dots,n_{m-1}\geq
0}$
$\displaystyle\frac{(-1)^{n_{m-1}}q^{2\binom{n_{m-1}+1}{2}+2n_{1}^{2}+\cdots+2n_{m-2}^{2}+2n_{a+1}+\cdots+2n_{m-2}}}{(-q;q^{2})_{n_{1}+\delta_{a,0}}}$
(3.26)
$\displaystyle\times\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\\\
n_{i}\end{bmatrix}_{q^{2}}$

along with its truncated counterpart

$V_{m,N}^{(a)}(q)=\sum_{n_{1},\dots,n_{m-1}=0}^{N-1}\frac{(-1)^{n_{m-1}}q^{2\binom{n_{m-1}+1}{2}+2n_{1}^{2}+\cdots+2n_{m-2}^{2}+2n_{a+1}+\cdots+2n_{m-2}}}{(-q;q^{2})_{n_{1}+\delta_{a,0}}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\\\
n_{i}\end{bmatrix}_{q^{2}}.$ (3.27)

We will show the following.

######  Theorem 3.11.

Let $q=e^{2\pi iM/N}$ be a primitive odd $N$th root of unity. Then

$\widetilde{\Lambda}_{m}^{(a)}(q)=q^{\frac{(m-a-1)^{2}}{(2m-1)}}V_{m,N}^{(a)}(q).$
(3.28)

The steps should be familiar. We prove a series of three propositions,
beginning with the fact that $\widetilde{\Lambda}_{m}^{(a)}(q)$ is a false
theta function.

######  Proposition 3.12.

We have the identity

$\widetilde{\Lambda}_{m}^{(a)}(q)=\frac{2m-1}{2}\sum\limits_{k=0}^{\infty}\chi_{2(2m-1)}^{(a)}(k)\,q^{\frac{k^{2}}{4(2m-1)}},$
(3.29)

where

$\chi_{2(2m-1)}^{(a)}(k)=\begin{cases}1&\mbox{if $k\equiv
2(m-a-1)\pmod{2(2m-1)}$},\\\ -1&\mbox{if $k\equiv 2(m+a)\pmod{2(2m-1)}$},\\\
0&\mbox{otherwise}.\end{cases}$

###### Proof.

We begin with the Bailey pair relative to $(q^{2},q^{2})$ [5, Proposition
5.5]222We note that there is a typo in Proposition 5.5 of [5]. The term
$(1+x^{2a+1}q^{(2a+1)(2n+1)})$ should be $(1-x^{2a+1}q^{(2a+1)(2n+1)})$.,

$\alpha_{n}=\frac{1}{1-q^{2}}(-1)^{n}q^{2(m-a-1)(n^{2}+n)+2(a+1)n^{2}+n}(1-q^{(2a+1)(2n+1)})$
(3.30)

and

$\beta_{n}=\beta_{n_{m}}=\sum_{n_{1},\dots,n_{m-1}\geq
0}\frac{q^{2n_{1}^{2}+\cdots+2n_{m-2}^{2}+2n_{a+1}+\cdots+2n_{m-2}}}{(q^{2};q^{2})_{n_{m}}(-q;q^{2})_{n_{1}+\delta_{a,0}}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\\\
n_{i}\end{bmatrix}_{q^{2}}.$ (3.31)

Using the case $m=m-1$ of this Bailey pair in Lemma 3.1 (with $q=q^{2}$) we
have

$\widetilde{\Lambda}_{m}^{(a)}(q)=\frac{2m-1}{2}q^{\frac{(m-a-1)^{2}}{(2m-1)}}\sum_{n=0}^{\infty}(1-q^{(2a+1)(2n+1)})\,q^{(2m-1)n^{2}+2(m-a-1)n},$
(3.32)

and this gives the result after rewriting the right-hand side. ∎

Now we compute the limiting values $\Lambda_{m}^{(a)}(q)$ at roots of unity.

######  Proposition 3.13.

For coprime integers $M$ and $N$ with $N$ positive we have

$\widetilde{\Lambda}_{m}^{(a)}(\zeta_{N}^{M})=\frac{-1}{8(2m-1)N^{2}}\sum\limits_{k=0}^{(4m-2)N}k^{2}\chi_{2(2m-1)}^{(a)}(k)\,\zeta_{N}^{\frac{Mk^{2}}{4(2m-1)N}}.$
(3.33)

###### Proof.

Regarding $\widetilde{\Lambda}_{m}^{(a)}$ as a function of $\tau$ with
$q=e^{2\pi i\tau}$, we have

$\displaystyle\widetilde{\Phi}_{m}^{(a)}\left(\frac{M}{N}+i\frac{t}{2\pi}\right)$
$\displaystyle=\frac{2m-1}{2}\sum_{k=0}^{\infty}\chi_{2(2m-1)}^{(a)}(k)e^{2\pi
i\left(\frac{M}{N}+i\frac{t}{2\pi}\right)\frac{k^{2}}{4(2m-1)}}$
$\displaystyle=\frac{2m-1}{2}\sum_{k=0}^{\infty}C_{2(2m-1)N}(k)e^{\frac{-k^{2}}{4(2m-1)}t}$

where

$C_{2(2m-1)N}(k)=\chi_{2(2m-1)}^{(a)}(k)e^{\frac{Mk^{2}}{2(2m-1)N}\pi i}.$

We note that

$C_{2(2m-1)N}(k+2(2m-1)N)=C_{2(2m-1)N}(k)$ (3.34)

and

$C_{2(2m-1)N}(2(2m-1)N-k)=-C_{2(2m-1)N}(k).$ (3.35)

Then using Lemma 3.2 we have the asymptotic expansion as $t\searrow 0$,

$\widetilde{\Phi}_{m}^{(a)}\left(\frac{M}{N}+i\frac{t}{2\pi}\right)=\sum_{k=0}^{\infty}\dfrac{L(-2k,C_{2(2m-1)N})}{k!}\left(-\dfrac{t}{4(2m-1)}\right)^{k}.$

Using this and keeping in mind (3.35) we compute the limiting value

$\displaystyle\widetilde{\Lambda}_{m}^{(a)}\left(\frac{M}{N}\right)$
$\displaystyle=\frac{2m-1}{2}L(0,C_{2(2m-1)N})$
$\displaystyle=-\frac{2m-1}{2}\sum_{k=1}^{2(2m-1)N}C_{2(2m-1)N}(k)B_{1}\left(\dfrac{k}{2(2m-1)N}\right)$
$\displaystyle=\frac{2m-1}{2}\sum_{k=1}^{2(2m-1)N}C_{2(2m-1)N}(k)\left(\frac{1}{2}-\dfrac{k}{2(2m-1)N}\right)$
$\displaystyle=\frac{2m-1}{2}\sum_{k=0}^{(2m-1)N}C_{2(2m-1)N}(k)\left(1-\dfrac{k}{(2m-1)N}\right)$
$\displaystyle=\sum_{k=0}^{(2m-1)N}C_{2(2m-1)N}(k)\left(\frac{2m-1}{2}-\dfrac{k}{2N}\right)$
$\displaystyle=\dfrac{-1}{8(2m-1)N^{2}}\Bigg{(}\sum_{k=0}^{(2m-1)N}k^{2}C_{2(2m-1)N}(k)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\sum_{k=0}^{(2m-1)N}(2(2m-1)N-k)^{2}C_{2(2m-1)N}(k)\Bigg{)}$
$\displaystyle=\dfrac{-1}{8(2m-1)N^{2}}\Bigg{(}\sum_{k=0}^{(2m-1)N}k^{2}C_{2(2m-1)N}(k)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\sum_{k=0}^{(2m-1)N}(2(2m-1)N-k)^{2}C_{2(2m-1)N}(2(2m-1)N-k)\Bigg{)}$
$\displaystyle=\dfrac{-1}{8(2m-1)N^{2}}\left(\sum_{k=0}^{(2m-1)N}k^{2}C_{2(2m-1)N}(k)+\sum_{k=(2m-1)}^{2(2m-1)N}k^{2}C_{2(2m-1)N}(k)\right)$
$\displaystyle=\frac{-1}{8(2m-1)N^{2}}\sum\limits_{k=0}^{(4m-2)n}k^{2}\chi_{2(2m-1)}^{(a)}(k)\,e^{\frac{k^{2}}{2(2m-1)N}M\pi
i},$

which gives the result. ∎

######  Proposition 3.14.

For $q=\zeta_{N}^{M}$, a primitive odd $N$-th root of unity, we have

$V_{m,N}^{(a)}=\frac{-1}{8(2m-1)n^{2}}q^{\frac{-(m-a-1)^{2}}{(2m-1)}}\sum\limits_{k=0}^{(4m-2)N}k^{2}\chi_{2(2m-1)}^{(a)}(k)\,q^{\frac{k^{2}}{4(2m-1)}}.$
(3.36)

###### Proof.

Inserting the Bailey pair from (3.30) and (3.31) with $m=m-1$ into Lemma 2.2
with $N=N-1$, we obtain

$\displaystyle V_{m,N}^{(a)}$ $\displaystyle(\zeta_{N}^{M})$
$\displaystyle=\lim_{q\to\zeta_{M}^{N}}\sum_{k=0}^{N-1}\frac{(q^{2-2N};q^{2})_{k}(1-q^{(2a+1)(2k+1)})q^{2(a+1)k^{2}+k+2(m-a-1)(k^{2}+k)+2Nk-2k-k(k-1)}}{(q^{2};q^{2})_{N-1}(1-q^{2N})(q^{2N+2};q^{2})_{k}}$
$\displaystyle=\frac{1}{N}\lim_{q\rightarrow\zeta_{N}^{M}}\frac{\sum\limits_{k=0}^{N-1}(1-q^{(2a+1)(2k+1)})\,q^{(2m-1)k^{2}+2(m-a-1)k}}{1-q^{2N}}$
$\displaystyle=\frac{1}{N}\lim_{q\rightarrow\zeta_{N}^{M}}\frac{q^{\frac{-(m-a-1)^{2}}{(2m-1)}}\sum\limits_{k=0}^{(4m-2)N}\chi_{2(2m-1)}^{(a)}(k)\,q^{\frac{k^{2}}{4(2m-1)}}}{1-q^{2N}}$
$\displaystyle=\frac{-1}{8(2m-1)N^{2}}\zeta_{N}^{\frac{-M(m-a-1)^{2}}{(2m-1)}}\sum\limits_{k=0}^{(4m-2)N}k^{2}\chi_{2(2m-1)}^{(a)}(k)\,\zeta_{N}^{\frac{Mk^{2}}{4(2m-1)}}.$

∎

Theorem 3.11 now follows upon comparing Propositions 3.13 and 3.14.

##  4\. Concluding Remarks

In closing we observe that the evaluations in (2.7), (3.15), (3.25), and
(3.36) can be used to obtain $q$-series identities that hold at roots of unity
but not inside the unit disk. This is in the spirit of quantum $q$-series
identities described in [4, 5]. For instance, if we take $m=2m-1$ and $a=2a$
in (2.7) and compare with (3.36), we have that if $q$ is a primitive odd $N$th
root of unity, then

$\displaystyle\sum_{n_{1},\dots,n_{2m-2}=0}^{N-1}(-1)^{n_{2m-2}}q^{\binom{n_{2m-2}+1}{2}+n_{1}^{2}+\cdots+n_{2m-3}^{2}+n_{2a+1}+\cdots+n_{2m-3}}\prod_{i=1}^{2m-3}\begin{bmatrix}n_{i+1}+\delta_{i,2a}\\\
n_{i}\end{bmatrix}$ (4.1)
$\displaystyle=2\sum_{n_{1},\dots,n_{m-1}=0}^{N-1}\frac{(-1)^{n_{m-1}}q^{2\binom{n_{m-1}+1}{2}+2n_{1}^{2}+\cdots+2n_{m-2}^{2}+2n_{a+1}+\cdots+2n_{m-2}}}{(-q;q^{2})_{n_{1}+\delta_{a,0}}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\\\
n_{i}\end{bmatrix}_{q^{2}}.$

For another example, comparing the case $m=2m-2$ and $a=0$ of (2.7) with
(3.25), we have that if $q$ is a primitive odd $N$th root of unity, then

$\displaystyle\sum_{n_{1},\dots,n_{2m-3}=0}^{N-1}(-1)^{n_{2m-3}}q^{\binom{n_{2m-3}+1}{2}+n_{1}^{2}+\cdots+n_{2m-4}^{2}}\prod_{i=1}^{2m-4}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}$ (4.2)
$\displaystyle=2\sum_{n_{1},\dots,n_{m-1}=0}^{N-1}\frac{(-1)^{n_{m-1}}q^{2\binom{n_{m-1}+1}{2}+2n_{1}^{2}+\cdots+2n_{m-2}^{2}+2n_{1}+\cdots+2n_{m-2}}(q;q^{2})_{n_{1}}}{(-q)_{2n_{1}+1}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}\\\
n_{i}\end{bmatrix}_{q^{2}}.$

Note that the left-hand sides of the above two equations are essentially
Kashaev invariants for certain torus links, as shown by Hikami [3], and so the
right-hand sides provide alternative expressions for these invariants.

##  5\. Acknowledgements

The authors would like to thank both the Unversité Paris Cité for hosting the
second author’s research visit and the French Embassy in the United States for
making this visit possible through a Chateaubriand Fellowship.

## References

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