# Rotating Spacetimes generalizing Lifshitz Black Holes

Alfredo Herrera-Aguilar<EMAIL_ADDRESS>Jhony A. Herrera-Mendoza
<EMAIL_ADDRESS>Daniel F. Higuita-Borja<EMAIL_ADDRESS>Instituto de Física, Benemérita Universidad Autónoma de Puebla. Apdo. Postal
J-48, C.P. 72570, Puebla, México

###### Abstract

We present a spinning black hole solution in $d$ dimensions with a maximal
number of rotation parameters in the context of the Einstein-Maxwell-Dilaton
theory. An interesting feature of such a solution is that it accommodates
Lifshitz black holes when the rotation parameters are set to zero. We verify
the rotating nature of the black hole solution by performing the quasi-local
analysis of conserved charges and defining the corresponding angular momenta.
In addition, we perform the thermodynamical analysis of the black hole
configuration, show that the first law of thermodynamics is completely
consistent, and obtain a Smarr-like formula. We further study the
thermodynamic stability of the constructed solution from a local viewpoint, by
computing the associated specific heats, and from a global perspective, by
using the so-called new thermodynamic geometry. We finally make some comments
related to a pathology found in the causal structure of the obtained rotating
black hole spacetime and compute some of its curvature invariants.

##  I Introduction

During last decades, the AdS/CFT correspondence Maldacena (1999) has become a
milestone in modern physics. The idea of connecting, through a holographic
dictionary, gravitational aspects of a theory in $d$ dimensions with quantum
aspects of a field theory in $d-1$ dimensions seems very appealing. The
overwhelming success of this approach to different problems in physics has
motivated the quest to modify or even extend the correspondence from his
original and mathematical well-defined setup Maldacena (1999) to new
frontiers. It is the case of Condensed Matter Physics where systems are
usually represented by non-relativistic field theories, which are anisotropic
and allow characteristic directions. The strongly interacting regime of such
theories is usually difficult to solve and constitutes a rich ground for a
Gauge/Gravity duality since the gravitational dual falls into the weakly
interacting limit and it can be easily solved. To establish the correspondence
it is crucial to have objects sharing symmetries and physical properties to be
connected through the duality. In this work we will focus on anisotropic
gravitational configurations, being an important case the Lifshitz spacetime
which can be understood as a gravitational dual to non-Lorentz invariant
quantum field theories Kachru  _et al._ (2008)

$ds^{2}=-\left(\dfrac{r}{\ell}\right)^{2z}dt^{2}+\dfrac{dr^{2}}{\left(\dfrac{r}{\ell}\right)^{2}}+\left(\dfrac{r}{\ell}\right)^{2}d\vec{x}{}^{2}.$
(1)

The metric (1) is invariant under the anisotropic scale transformations

$D_{z}:\,\,\,\,t\mapsto\lambda^{z}t,\hskip
14.22636ptr\mapsto\dfrac{r}{\lambda},\hskip
14.22636pt\vec{x}\mapsto\lambda\vec{x},$ (2)

with $z$ known as the dynamical critical exponent that represents the degree
of anisotropy of the spacetime (the isotropic relativistic case $z=1$
corresponds to AdS). It is also invariant under space and time translations
and spatial rotations:

$\displaystyle H:$ $\displaystyle\,\,\,t\longrightarrow\,\,\,t^{\prime}=t+a;$
$\displaystyle P^{i}:$ $\displaystyle x^{i}\longrightarrow
x^{{}^{\prime}i}=x^{i}+a^{i};$ (3) $\displaystyle L^{ij}:$ $\displaystyle
x^{i}\longrightarrow x^{{}^{\prime}i}=L^{i}_{j}x^{j}.$

To include finite temperature effects in the holographic correspondence it is
necessary to introduce black hole configurations, being the first
asymptotically Lifshitz black hole the one reported in Taylor (2008). Since
then, many static asymptotically Lifshitz Black Holes have been found in
several gravitational theories Ayon-Beato  _et al._ (2009) and different
dimensions Ayon-Beato  _et al._ (2010), failing consistently to have a
rotating version. The only published advance in this direction is given in the
context of New Massive Gravity with $z=3$ through a boost Sarioğlu (2019),
with a detailed thermodynamical study in Kolev  _et al._ (2019). We will
extend these works to arbitrary dimensions and construct a rotating
configuration that accommodates Lifshitz black holes in certain limit. To
achieve this aim we will organize the paper as follows: In Sec. II we will
present the Einstein-Maxwell-Dilaton theory and our black hole metric ansatz
with the maximal number of independent rotation parameters in a
$d$-dimensional spacetime. Once the field equations are obtained and a
rotating black hole solution is found, the thermodynamical study has to be
performed. Then we shall need to compute global conserved charges related to
the Killing vector fields of our ansatz, different methodologies can be
applied to achieve this goal. However, we will use the off-shell quasi-local
formalism Kim  _et al._ (2013a); Hyun  _et al._ (2014a, b); Gim  _et al._
(2014a) since it is straightforward and well-defined for different kinds of
asymptotic behavior of the fields. Thus, Sec. III will be devoted to briefly
review the quasi-local formalism and to compute the relevant thermodynamical
quantities, unveiling the rotating nature of the black hole configuration by
calculating its angular momenta. We introduce a generalization of the Hawking
temperature definition that is compatible with the third law of
Thermodynamics, in the sense that avoids having degenerate states with finite
entropy and vanishing temperature, and allows for a reasonable description of
phenomenological condensed matter systems that attain a phase transition at
finite temperature Hartnoll  _et al._ (2016); Sachdev (2012). In Sec. IV we
use these thermodynamical quantities to construct a geometrization of the
thermodynamic space on the basis of the Gibbs free energy and analyze possible
phase transitions based on the specific heat and curvature invariants. We then
engage with the spacetime causal structure in Sec. V. We will end with some
conclusions and remarks regarding the scope of our work in Sec. VII.

##  II The Gravitational Configuration

The first Lifshitz black hole solution Taylor (2008)

$ds^{2}=-\left(\dfrac{r}{\ell}\right)^{2z}f(r)dt^{2}+\dfrac{dr^{2}}{\left(\dfrac{r}{\ell}\right)^{2}f(r)}+\left(\dfrac{r}{\ell}\right)^{2}d\vec{x}{}^{2},$
(4)

was engineered in the context of the Einstein-Maxwell-Dilaton (EMD) theory
defined by the following action

$S[g^{\mu\nu},\phi,A_{\mu}]=\int{d^{d}x\sqrt{-g}\left(\dfrac{R-2\lambda}{2\kappa}-\dfrac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi-\dfrac{1}{4}e^{-b\phi}F_{\mu\nu}F^{\mu\nu}\right)},$
(5)

with the following field equations

$\displaystyle\delta_{g}S$ $\displaystyle=0\Longrightarrow G_{\mu\nu}+\lambda
g_{\mu\nu}=\kappa T_{\mu\nu},$ (6a) $\displaystyle\delta_{A}S$
$\displaystyle=0\Longrightarrow\partial_{\mu}\left(\sqrt{-g}e^{-b\phi}F^{\mu\nu}\right)=0,$
(6b) $\displaystyle\delta_{\phi}S$
$\displaystyle=0\Longrightarrow\Box\phi=-\dfrac{b}{4}e^{-b\phi}F_{\mu\nu}F^{\mu\nu},$
(6c)

where

$T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\dfrac{1}{2}g_{\mu\nu}\nabla_{\alpha}\phi\nabla^{\alpha}\phi+e^{-b\phi}\left(F_{\mu\alpha}{F_{\nu}}^{\alpha}-\dfrac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right).$
(7)

It would be interesting to find a rotating version of this family of Lifshitz
black holes. In order to accomplish this aim, inspired by the (A)dS rotating
solutions found in Awad (2003), we will make use of the following metric
ansatz

$\displaystyle ds^{2}=-\left(\dfrac{r}{\ell}\right)^{2z}f(r)\left(\Xi
dt-\sum_{i=1}^{n}a_{i}d\phi_{i}\right)^{2}$
$\displaystyle+\dfrac{r^{2}}{\ell^{4}}\sum_{i=1}^{n}\left(a_{i}dt-\Xi\ell^{2}d\phi_{i}\right)^{2}+\dfrac{dr^{2}}{\left(\dfrac{r}{\ell}\right)^{2}f(r)}$
$\displaystyle-\dfrac{r^{2}}{\ell^{2}}\sum_{i<j}^{n}\left(a_{i}d\phi_{j}-a_{j}d\phi_{i}\right)^{2}+\left(\dfrac{r}{\ell}\right)^{2}d\vec{y}{}^{2},$
(8)

where $n=[(d-1)/2]$ is the maximal number of independent rotation planes in
$d$ dimensions, each one related with a rotation parameter $a_{i}$,
$\Xi=\sqrt{1+\sum_{i=1}^{n}a_{i}^{2}/\ell^{2}}$, and
$d\vec{y}{}^{2}=dy^{k}dy^{k}$ is the Euclidean metric on a
$(d-2-n)$-dimensional submanifold. We will prove later on that such a
configuration has indeed non-trivial angular momenta. In addition we shall
assume that matter fields share the same spacetime symmetries, taking the
following scalar field and vector potential ansätze Awad (2003)

$\phi=\phi(r),\qquad A=-\Xi h(r)dt+\sum_{i=1}^{n}a_{i}h(r)d\phi_{i}.$ (9)

By collecting all this information into the field equations (6) we arrive to a
first integral that solves the Maxwell equations

$\displaystyle\partial_{r}\left(\sqrt{-g}\,e^{-b\phi}F^{r\phi_{i}}\right)\propto\partial_{r}\left(\sqrt{-g}\,e^{-b\phi}F^{rt}\right)=0\quad\Rightarrow\quad
h^{\prime}(r)=\frac{Q}{e^{-b\phi}}\left(\frac{\ell}{r}\right)^{d-1-z},$ (10)

where $Q$ is an arbitrary integration constant.

In the 4-dimensional case we consider a general vector potential compatible
with the spacetime symmetries, $A=A_{t}(r)dt+A_{\phi}(r)d\phi$. Thus, from
Maxwell equations we are able to get two first integrals with their respective
integration constants. However, the evaluation of the complete set of
equations (6) restricts these constants to be equal and we arrive to the same
result coming from the ansatz (9), after a constant redefinition. We expect
something similar to happen in higher dimensions, being the reason to assume
the vector potential proposed in (9). Note that along with the electric charge
there are multiple magnetic charges in our configuration as it can be seen
from the Maxwell equations in (10). Since all of these charges obey the same
differential equation as a result of the assumed ansatz (9), we denote them by
the constant Q introduced in Eq. (10).

A combination of the Einstein equations that decouple the gravitational
potential is

$\mathcal{E}^{r}_{r}+\mathcal{E}^{y_{k}}_{y_{k}}=\dfrac{1}{2\ell^{2}}\left[r^{2}f^{\prime\prime}+\left(3(z+d)-7\right)rf^{\prime}+2(z+d-2)(z+d-3)f+4\lambda\ell^{2}\right]=0,$
(11)

which corresponds to a second order inhomogeneous Euler equation for $f$ with
the following general solution

$f(r)=\frac{k_{1}}{r^{z+d-2}}+\frac{k_{2}}{r^{2(z+d-3)}}-\dfrac{2\ell^{2}\lambda}{(z+d-2)(z+d-3)},$
(12)

where $k_{1}$ and $k_{2}$ are new independent integration constants. By
imposing the Lifshitz boundary condition,
$\displaystyle{\lim_{r\to\infty}f(r)=1}$, we get a negative definite
cosmological constant $\lambda=-\dfrac{(z+d-2)(z+d-3)}{2\ell^{2}}$.

The dilaton field can be found from the following Einstein equations

$\displaystyle\mathcal{E}_{t}^{\phi_{i}}$
$\displaystyle=\dfrac{1}{2}a_{i}\Xi\quantity{-2\kappa
Q^{2}e^{b\phi}\quantity(\dfrac{\ell}{r})^{2(d-2)}+\dfrac{1}{\ell^{2}}\quantity[r^{2}f^{\prime\prime}+(3z+d-3)rf^{\prime}+2(z-1)(z+d-2)f]}$
$\displaystyle=0,$ (13)

by taking the gravitational potential (12) as a source, obtaining

$e^{b\phi}=\dfrac{1}{\kappa
Q^{2}\ell^{2}}\quantity[(d-2)(z+d-4)\dfrac{k_{2}}{r^{2(z+d-3)}}+(z-1)(z+d-2)]\quantity(\dfrac{r}{\ell})^{2(d-2)}.$
(14)

The remaining Einstein equations are not identically fulfilled but establish
the constraints $b=2\sqrt{\dfrac{\kappa(d-2)}{z-1}}$ and $k_{2}=0$. Gathering
all this information and renaming the constant $k_{1}=-M$, the solution will
be given by the spacetime defined in (II) with the following gravitational
potential and matter fields

$\displaystyle f(r)$ $\displaystyle=1-\dfrac{M}{r^{z+d-2}},$ (15a)
$\displaystyle e^{-b\phi}$
$\displaystyle=\kappa\dfrac{Q^{2}\ell^{2}}{(z-1)(z+d-2)}\left(\dfrac{\ell}{r}\right)^{2(d-2)},$
(15b) $\displaystyle A_{t}$ $\displaystyle=-\Xi\dfrac{(z-1)}{\kappa
Q\ell}\left(\dfrac{r}{\ell}\right)^{d+z-2},$ (15c) $\displaystyle
A_{\phi_{i}}$ $\displaystyle=a_{i}\dfrac{(z-1)}{\kappa
Q\ell}\left(\dfrac{r}{\ell}\right)^{d+z-2},$ (15d) $\displaystyle\lambda$
$\displaystyle=-\dfrac{(z+d-2)(z+d-3)}{2l^{2}},$ (15e) $\displaystyle b$
$\displaystyle=2\sqrt{\dfrac{\kappa(d-2)}{z-1}},$ (15f)

where the integration constants of $A_{t}$ and $A_{\phi_{i}}$ were chosen to
vanish.

We would like to mention here that the limit $z\rightarrow 1$ for the scalar
field is smooth. In fact, the apparent singularity of (15b) is due to the way
in which we present the solution.

On the other hand, the limit $Q\rightarrow 0$ is not smooth. From Eqs. (9) and
(10) one realizes that such a limit turns off the interaction between the
scalar field and the Maxwell potential. This case effectively reduces the
action (5) to the Einstein-Hilbert action with a cosmological constant plus
the kinetic term for the scalar field, however, by solving the field equations
one realizes there is no Lifshitz black hole configuration supported by this
action. Therefore, it is necessary to appropriately modify this scalar-tensor
theory (with curvature corrections –as in New Massive Gravity- or with other
matter fields) to support our rotating Lifshitz black hole configurations.

Finally, the configuration (15) corresponds indeed to the Taylor static black
hole solution quoted in Taylor (2008) when all the rotation parameters $a_{i}$
are set to zero.

##  III Conserved Charges and Black Hole Thermodynamics

###  III.1 Conserved charges

In order to find the energy and angular momenta associated to our rotating
black hole configuration we shall make use of an off-shell formalism called
generalized ADT quasi-local method, foremost introduced in Kim  _et al._
(2013b). This method has been implemented and proved to be well suited in the
construction of conserved charges in higher order gravity theories (see for
instance Ayón-Beato  _et al._ (2015); Bravo Gaete  _et al._ (2017); Ayón-Beato
_et al._ (2019)), as well as in EMD theories (Gim  _et al._ (2014b); Herrera-
Aguilar  _et al._ (2021)). In this method, the conserved charge corresponding
to a Killing vector field $\xi$ is given by

$\mathcal{Q}(\xi)=\int d^{d-2}x_{\mu\nu}\left(\Delta
K^{\mu\nu}(\xi)-2\xi^{[\mu}\int_{0}^{1}ds\Theta^{\nu]}(\xi,s)\right),$ (16)

where
$d^{d-2}x_{\mu\nu}=\frac{1}{2}\frac{1}{(d-2)!}\epsilon_{\mu\nu\mu_{1}\mu_{2}\ldots\mu_{d-2}}dx^{\mu_{1}}\wedge\ldots\wedge
dx^{\mu_{d-2}}$ with $\epsilon_{\mu\nu\mu_{1}\mu_{2}\ldots\mu_{d-2}}$ is the
totally antisymmetric Levi-Civita symbol, whereas $s$ stands for a parameter
allowing to interpolate the black hole configuration between the solution of
interest ($s=1$) and the asymptotic one ($s=0$). Furthermore, $\Delta
K^{\mu\nu}(\xi)=K^{\mu\nu}_{s=1}(\xi)-K^{\mu\nu}_{s=0}(\xi)$ stands for the
total difference of the Noether potentials between the two end points of the
path, $s=1$ and $s=0$. Finally, $\Theta^{\nu}$ is the surface term obtained
after the variation of the corresponding action.

The Noether potential and the surface terms associated to our model (5) are
respectively given by

$K^{\mu\nu}(\xi)=2\sqrt{-g}\left[\frac{\nabla^{[\mu}\xi^{\nu]}}{2\kappa}-\frac{1}{2}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}A_{\nu})}\xi^{\sigma}A_{\sigma}\right],$
(17) $\Theta^{\mu}(\delta g,\delta\phi,\delta
A)=2\sqrt{-g}\left(\frac{g^{\alpha[\mu}\nabla^{\beta]}\delta
g_{\alpha\beta}}{2\kappa}+\frac{1}{2}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}A_{\nu})}\delta
A_{\nu}+\frac{1}{2}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\delta\phi\right).$
(18)

In order to derive the corresponding mass and angular momenta related to our
model (5) and black hole configuration (II) we shall consider a timelike and a
set of rotational Killing vector fields, respectively.

For the case of the mass, the Killing vector takes the form
$\eta^{\mu}=-\partial_{t}$. This vector field, together with (16), (17), and
(18) eventually gives rise to the following expression for the mass

$\mathcal{M}=\frac{V_{d-2}}{2\kappa\ell^{z+[d/2]}}\left[(d-z)\Xi^{2}+(z-2)\right]r_{H}^{z+d-2},$
(19)

where $V_{d-2}$ represents the volume of the transverse spatial dimensions and
the exponent $[x]$ denotes the integer part of $x$.

Furthermore, for the computation of the angular momenta we consider a set of
rotational Killing vector fields that take the form
$\zeta_{i}^{\mu}=\partial_{\phi_{i}}$, allowing us to find that the angular
momentum corresponding to the $i$-th rotation plane can be expressed as

$\mathcal{J}_{i}=\frac{a_{i}\Xi(d-z)V_{d-2}r_{H}^{z+d-2}}{2\kappa\ell^{z+\left[d/2\right]}}.$
(20)

On the basis of this fact we can assure that the black hole field
configuration rotates and the $a_{i}$ are indeed rotation parameters.

The electromagnetic charge can be computed through a Gaussian integral over a
spatial hypersurface $\Sigma$ at asymptotic infinity Bravo-Gaete and Juárez-
Aubry (2020)

$Q_{em}=\int_{\Sigma}d^{D-2}x\sqrt{|\gamma|}n^{\mu}u^{\nu}e^{-b\phi}F_{\mu\nu}=\Xi
QV_{d-2},$ (21)

where $\gamma$ is the induced metric on $\Sigma$, with $u$ and $n$ its
timelike and spacelike normal unit vectors

$u=\dfrac{\sqrt{f(r)}\quantity(\dfrac{r}{\ell})^{z+1}}{\sqrt{\quantity(\dfrac{r}{\ell})^{2z}f(r)\quantity(1-\Xi^{2})+\quantity(\dfrac{r}{\ell})^{2}\Xi^{2}}}\,dt,\qquad\quad
n=\dfrac{1}{\sqrt{f(r)}}\,\quantity(\dfrac{\ell}{r})\,dr.$ (22)

The electromagnetic charge is a conserved quantity derived from the formula
(21) and accounts for both the electric and the magnetic components of the
field, which are coupled in the case under study. In fact, one can drop the
magnetic part (as well as the rotating sector of the configuration) by setting
the rotation parameters to zero, but the electric part is not allowed to
vanish because it supports the whole field configuration.

###  III.2 Gravitational Thermodynamics

A formal relation between black holes and thermodynamics was systematically
presented by Bardeen, Carter and Hawking in Bardeen  _et al._ (1973), inspired
by early works of Smarr Smarr (1973) and Bekenstein. In Bardeen  _et al._
(1973) it is emphasized that the four laws of black hole mechanics represent
an analogy more than a physical reality. The analogy was pushed even further
by Bekenstein himself Bekenstein (1973) suggesting that the area of the event
horizon and the surface gravity were indeed related to the entropy and the
temperature of the black hole, respectively. Hawking further justified part of
this statement assuming particle creation near the event horizon Hawking
(1975) due to quantum mechanical effects over a curved spacetime. Part of such
particles tunnels through the event horizon and part of them escape to null
infinity producing a black body radiation with Hawking temperature. This
temperature is defined in terms of the surface gravity $\tilde{\kappa}$ by the
relation Hawking (1975)

$T_{H}\equiv\frac{\tilde{\kappa}}{2\pi},$ (23)

where the surface gravity can be expressed in terms of the null generator of
the event horizon $\chi^{\mu}$, according to

$\tilde{\kappa}^{2}=-\frac{1}{2}\left(\nabla_{\mu}\chi_{\nu}\right)\left(\nabla^{\mu}\chi^{\nu}\right).$
(24)

For our rotating black hole geometry this corresponds to the Killing vector
field

$\chi^{\mu}=\partial_{t}+\sum_{i=1}^{n}\Omega_{i}\partial_{\phi_{i}},$ (25)

with $\Omega_{i}$ the angular velocity linked to the $i$-th rotation plane
that can be obtained by considering the null character of the Killing vector
field (25) at the horizon. By doing this, one can easily find that the $i$-th
angular velocity related to the $i$-th rotating axis is given by

$\Omega_{i}=\frac{a_{i}}{\ell^{2}\Xi},$ (26)

from which we can express $\Xi$ in terms of $\Omega$ as follows

$\Xi=\frac{1}{\sqrt{1-\ell^{2}\Omega^{2}}},$ (27)

subject to the restriction
$\Omega^{2}\equiv\sum_{i}\Omega_{i}^{2}<1/\ell^{2}$.

All relations in black hole Thermodynamics are about geometrical quantities
Bardeen  _et al._ (1973); Hawking (1975); Wald (1993); Iyer and Wald (1994,
1995). The holographic duality between gravity and condensed matter systems
establishes a relation between the black hole Hawking temperature and the
temperature of a condensed matter system in a given thermodynamic phase. This
motivation allows us to realize that the Hawking temperature associated with
the configuration (II) reads

$T_{H}=T-T_{c}\equiv\frac{\tilde{\kappa}}{2\pi},$ (28)

where $T_{c}\geq 0$ is certain constant critical value of the temperature,
since there is no restriction to map $T_{H}\rightarrow T-T_{c}$ as far as this
quantity remains positive for a given thermodynamic phase. This difference
allows us to consider processes approaching the critical temperature from
above where the critical point is reached at the extremal case
$\ell^{2}\Omega^{2}=1$, evidencing the emergence of a possible phase
transition of the condensed matter system at finite temperature (see equations
(27) and (30)). Moreover, this definition allows us to give a consistent
physical interpretation of the low temperature limit of the Reissner-
Nordström-AdS black hole geometry, avoiding the issue of having a state with
finite entropy and zero temperature that contradicts the third law of
Thermodynamics Hartnoll  _et al._ (2016); Sachdev (2012).

Alternatively, in order to take into account thermodynamic processes
approaching the critical temperature from below and reaching a phase
transition we can use the following definition

$T_{H}=T_{c}-T\equiv\frac{\tilde{\kappa}}{2\pi},$ (29)

where now $T_{c}-T$ is a positive definite quantity.

Thus, this definition allows us to compute the Hawking temperature (28)
associated with the rotating black hole field configuration (II):

$T_{H}=T-T_{c}=\frac{1}{4\pi}\frac{(z+d-2)r_{H}^{z}}{\ell^{z+1}\Xi},$ (30)

whereas the entropy, giving in terms of the event horizon area $\mathcal{A}$,
adopts the form

$\mathcal{S}\equiv\frac{\mathcal{A}}{4G_{d}}=\frac{\Xi
V_{d-2}}{4G_{d}\ell^{[d/2]-1}}r_{H}^{d-2},$ (31)

where $G_{d}=\kappa/8\pi$ is the $d$-dimensional gravitational constant.

Finally, by making use of the expressions (19), (20), (26), (30) and (31), it
is easy to check that the first law of thermodynamics is reasonably satisfied

$d\mathcal{M}=\left(T-T_{c}\right)d\mathcal{S}+\sum_{i=1}^{n}\Omega_{i}d\mathcal{J}_{i}+\Phi_{em}dQ_{em},$
(32)

with the electromagnetic potential

$\Phi_{em}=-\dfrac{(z-1)}{\kappa Q\ell^{z+[d/2]}}\dfrac{r_{h}^{z+d-2}}{\Xi},$
(33)

obtained as an appropriate linear combination of $A_{t}$ and $A_{\phi_{i}}$ at
the event horizon. The variations in the first law of black hole
Thermodynamics (32) are taken with respect to two sets of independent
variables, for instance the temperature $T$ and the angular velocities
$\Omega_{i}$. In fact, the electromagnetic charge contributes to the
variations only with respect to the angular velocities.
The Smarr-like formula adopts the following form

$\mathcal{M}=\left[\dfrac{d-2+2\,c\,(z-1)}{z+d-2}\right]\left(T-T_{c}\right)\mathcal{S}+\sum_{i=1}^{n}\Omega_{i}\mathcal{J}_{i}+c\,\Phi_{em}Q_{em},$
(34)

where $c$ is an arbitrary constant.

These remarkable relations provide striking evidence that the Thermodynamics
of our rotating black hole configuration is physically consistent.

##  IV A look for the thermodynamic stability

Here we explore the thermodynamic stability of the constructed solution from a
local and a global perspective. We study local stability by computing the
associated specific heats, while the global one is explored using the so-
called new thermodynamic geometry (NTG) formalism as appears in Mansoori and
Mirza (2019). It is worth mentioning that the NTG formalism corresponds to one
of the most recent attempts for describing the thermodynamics of black holes
using Riemannian geometry. Although there are several earlier developments
Weinhold (1975); Ruppeiner (1995); Quevedo (2007); Bravetti  _et al._ (2013);
Quevedo  _et al._ (2020), this formalism has proved to be the most suitable
one since, besides being Legendre invariant, it also solves some of the
inconsistencies presented in the aforementioned developments.

Before we proceed, we remark that it is convenient to work in the Gibbs free
energy representation since this potential allows us to construct a positive
definite metric Kolev  _et al._ (2019). In this sense, the Gibbs free energy
can be obtained from a Legendre transformation of the energy (34), resulting
in

$\displaystyle\mathcal{G}(T,\Omega_{1},\ldots,\Omega_{n})$
$\displaystyle=\mathcal{M}-\quantity(T-T_{c})\mathcal{S}-\sum_{i=1}^{n}\Omega_{i}\mathcal{J}_{i},$
$\displaystyle=-\frac{zV_{d-2}}{2\kappa\ell^{z+[d/2]}}\quantity[\frac{4\pi\quantity(T-T_{c})\ell^{z+1}\Xi}{z+d-2}]^{\frac{z+d-2}{z}},$
(35)

in differential form this relation reads

$\displaystyle d\mathcal{G}(T,\Omega_{1},\ldots,\Omega_{n})$
$\displaystyle=-\mathcal{S}dT-\sum_{i=1}^{n}\mathcal{J}_{i}d\Omega_{i}+\Phi_{em}dQ_{em},$
$\displaystyle=-\mathcal{S}dT-\sum_{i=1}^{n}\left(\mathcal{J}_{i}-\Phi_{em}\dfrac{\partial
Q_{em}}{\partial\Omega_{i}}\right)d\Omega_{i},$ (36)

where in the last equality we have taken into account the dependence of the
electromagnetic charge on the angular velocities (21).

The angular momenta and entropy written in terms of the Gibbs free energy
natural variables $(T,\Omega_{1},\ldots,\Omega_{n})$, are given respectively
by

$\mathcal{J}_{i}=\frac{\alpha
z(d-z)}{z+d-2}\ell^{2}\Omega_{i}\Xi^{\frac{3z+d-2}{z}}\left(T-T_{c}\right)^{\frac{z+d-2}{z}},\qquad\mathcal{S}=\alpha
z\Xi^{\frac{z+d-2}{z}}\left(T-T_{c}\right)^{\frac{d-2}{z}}$ (37)

where for simplicity we have defined
$\alpha=\frac{(z+d-2)}{z}\frac{V_{d-2}}{2\kappa\ell^{z+[d/2]}}\quantity(\frac{4\pi\ell^{z+1}}{z+d-2})^{\frac{z+d-2}{z}}$.
Additionally, in these expressions we have made use of Eq. (27).

As it was previously stated, we shall make use of the specific heats in order
to study the local thermodynamic stability and phase transitions of the
system. In the $(T,\Omega_{1}\ldots,\Omega_{n})$ space of the Gibbs free
energy, we define the specific heats of the black hole following Mansoori  _et
al._ (2015) as follows

$\begin{split}C_{\Omega_{1},\ldots,\Omega_{n}}&=\left(T-T_{c}\right)\quantity(\partialderivative{\mathcal{S}}{T})_{\Omega_{1},\ldots,\Omega_{n}}=\left(T-T_{c}\right)\frac{\\{\mathcal{S},\Omega_{1},\ldots,\Omega_{n}\\}_{T,\Omega_{1},\ldots,\Omega_{n}}}{\\{T,\Omega_{1},\ldots,\Omega_{n}\\}_{T,\Omega_{1},\ldots,\Omega_{n}}}\\\
&=\alpha(d-2)\Xi^{\frac{d+z-2}{z}}\left(T-T_{c}\right)^{\frac{d-2}{z}},\end{split}$
(38)

and

$\begin{split}C_{\mathcal{J}_{1},\ldots,\mathcal{J}_{n}}&=\left(T-T_{c}\right)\quantity(\partialderivative{\mathcal{S}}{T})_{\mathcal{J}_{1},\ldots,\mathcal{J}_{n}}=\left(T-T_{c}\right)\frac{\\{\mathcal{S},\mathcal{J}_{1},\ldots,\mathcal{J}_{n}\\}_{T,\Omega_{1},\ldots,\Omega_{n}}}{\\{T,\mathcal{J}_{1},\ldots,\mathcal{J}_{n}\\}_{T,\Omega_{1},\ldots,\Omega_{n}}}\\\
&=-\frac{\alpha
z\Bigl{[}z\ell^{2}\Omega^{2}-(d-2)\Bigr{]}}{\ell^{2}\Omega^{2}(d+2z-2)+z}\Xi^{\frac{d+z-2}{z}}\left(T-T_{c}\right)^{\frac{d-2}{z}}.\end{split}$
(39)

From these relations we observe that both specific heats are regular
everywhere except when $\Omega^{2}=1/\ell^{2}$, where $\Xi$ is singular,
evidencing the emergence of a possible phase transition in the thermodynamic
system. This extremal point corresponds to one in which the metric is not
well-defined. Moreover, we also identify a change of sign occurring in
$C_{\mathcal{J}_{1},\ldots,\mathcal{J}_{n}}$ when

$\ell^{2}\Omega^{2}=\frac{d-2}{z},$ (40)

pointing to the existence of a thermodynamic instability.

It is well-known that in order to have local thermodynamic stability in a
system, one must require both specific heats to be positive definite. This
fact is fulfilled by imposing the condition

$\ell^{2}\Omega^{2}<\frac{d-2}{z}.$ (41)

On the other hand, we can also consider a weaker stability condition such that
$\ell^{2}\Omega^{2}<1$, implying the black hole being locally stable
concerning $C_{\Omega_{1},\ldots,\Omega_{n}}$, but allowing the system to
suffer a phase transition with respect to
$C_{\mathcal{J}_{1},\ldots,\mathcal{J}_{n}}$ from a stable into an unstable
one due to a change of sign.

An alternative way for exploring the stability but now from a global
perspective is through the NTG formalism, as we remarked earlier. In this
formalism, the thermodynamic metric in the Gibbs free energy representation is
defined according to the relation

$g_{ab}=-\frac{1}{\left(T-T_{c}\right)}\partial_{a}\partial_{b}\mathcal{G}(T,\Omega_{i}),$
(42)

with the indices $(a,b)$ taking the values
$a,b=T,\Omega_{1},\ldots,\Omega_{n}$. The explicit form of the metric
components read

$\displaystyle g_{TT}$
$\displaystyle=\alpha(d-2)\Xi^{\frac{d+z-2}{z}}\left(T-T_{c}\right)^{\frac{d-2z-2}{z}},$
(43a) $\displaystyle g_{T\Omega_{j}}$
$\displaystyle=\alpha(z+d-2)\ell^{2}\Xi^{\frac{d+3z-2}{z}}\left(T-T_{c}\right)^{\frac{d-z-2}{z}}\Omega_{j},$
(43b) $\displaystyle g_{\Omega_{i}\Omega_{j}}$
$\displaystyle=\alpha\ell^{2}\left\\{(3z+d-2)\ell^{2}\Omega_{i}\Omega_{j}\Xi^{3}+z\delta_{ij}\right\\}\Xi^{\frac{d+3z-2}{z}}\left(T-T_{c}\right)^{\frac{d-2}{z}}.$
(43c)

In addition we can write the metric in a differential form as follows

$ds_{\mathcal{G}}^{2}=g_{TT}dT^{2}+\sum_{ij}g_{\Omega_{i}\Omega_{j}}d\Omega_{i}d\Omega_{j}+2\sum_{i}g_{T\Omega_{i}}dTd\Omega_{i},$
(44)

from which we find that the curvature scalar is given by

$\begin{split}R_{\mathcal{G}}=-\dfrac{\quantity(1-\ell^{2}\Omega^{2})^{\frac{z+d-2}{2z}}(n-1)}{\alpha
z^{2}\quantity[z\ell^{2}\Omega^{2}-(d-2)]^{2}\left(T-T_{c}\right)^{\frac{d-2}{z}}}&\Biggl{\\{}(n-2)z^{3}\left[\ell^{2}\Omega^{2}-1\right]\ell^{2}\Omega^{2}+\\\
&\frac{n(d-2)}{4}\Biggl{[}(z\ell^{2}\Omega^{2}-2z-d+2)^{2}-4z^{2}\ell^{2}\Omega^{2}\Biggr{]}\Biggr{\\}}.\end{split}$
(45)

From this expression, it is evident that it vanishes when $d=4\to n=1$ and,
for $d>4$, we observe that the scalar is negative definite. Besides, we
identify the appearance of two singularities: one takes place when the
temperature adopts some critical value $T=T_{c}$ and another one when the
angular velocities are restricted by $\ell^{2}\Omega^{2}=(d-2)/z<1$. It is
interesting to remark that the singularity at $T=T_{c}$ does not appear in the
Gibbs potential or the specific heats, but it is present in the second
derivative of the Gibbs potential and the first derivative of the specific
heats, provided $(d-2)/z<1$. Therefore, we can say that a second-order phase
transition takes place due to thermal fluctuations.

On the other hand, the singularity at $\ell^{2}\Omega^{2}=(d-2)/z<1$ does not
appear in either the Gibbs potential or the specific heats, but it does
coincide with the change of sign in
$C_{\mathcal{J}_{1},\ldots,\mathcal{J}_{n}}$. As we have remarked earlier, a
transition from a stable into an unstable black hole occurs at this
singularity.

It is interesting to note that the metric singularity at the extremal point
$\ell^{2}\Omega^{2}=1$ is not present in the scalar curvature, which vanishes
at this point, clarifying that it is in fact a coordinate singularity. On the
other hand, the specific heats diverge at this point as mentioned earlier. A
similar situation takes place in the thermodynamic system studied in Kolev
_et al._ (2019), where by analyzing the specific heats in the entropic
representation one sees that this divergence does not appear, confirming the
coordinate nature of the aforementioned singularity in these physical
quantities.

Finally, we remark that these results are in perfect accordance with the
prescription of thermodynamic geometry, in the sense that singularities of the
curvature scalar for the thermodynamic metric coincide with the phase
transitions observed in the specific heats Mansoori and Mirza (2014).

##  V Causal Structure

The anisotropic nature of the metric (II) makes the spacetime structure more
involved with respect to the (A)dS rotating spacetimes Awad (2003). For
example, in this black hole field configuration the _ergosphere_ is defined as
the region where the norm of the timelike Killing field $\eta=\partial_{t}$
becomes positive

$\displaystyle\eta^{2}$ $\displaystyle=\eta^{\mu}g_{\mu\nu}\eta^{\nu}\equiv
g_{tt},$
$\displaystyle=\quantity[-\quantity(\dfrac{r}{\ell})^{2z}f(r)+\quantity(\dfrac{r}{\ell})^{2}]\Xi^{2}-\quantity(\dfrac{r}{\ell})^{2}$
$\displaystyle=-\quantity(\dfrac{r}{\ell})^{2z}f(r)+\quantity(\dfrac{r}{\ell})^{2}\sum_{i=1}^{n}\quantity(\dfrac{a_{i}}{\ell})^{2}>0.$
(46)

Below the radius of the event horizon $f(r)<0$ and the inequality (V) is
satisfied. However, a change of sign takes place at some point outside the
event horizon as it will be exemplified in Figure 1 for some parameters
election. Hence, there is a region between $r_{H}$ and the largest root of
$\eta^{2}=0$ where an observer cannot remain stationary. The spacetime drags
the observer even when he is outside the black hole’s event horizon.

Another issue that arises within this metric is related to the rotation
Killing fields $m_{i}$. There are $n$ independent rotation planes and over
each plane there is a rotation Killing vector field
$m_{i}=\partial_{\phi_{i}}\equiv m_{i}^{\mu}\partial_{\mu}$, $i=1\ldots n$,
which is cyclic – $\phi_{i}\in[0,2\pi]$ –. Their norm reads

$\displaystyle m_{i}^{2}$
$\displaystyle=m_{i}^{\mu}g_{\mu\nu}m_{i}^{\nu}\equiv g_{\phi_{i}\phi_{i}},$
$\displaystyle=\quantity[-\quantity(\dfrac{r}{\ell})^{2z}f(r)+\quantity(\dfrac{r}{\ell})^{2}]a_{i}^{2}+r^{2},$
(47)

that could be null, spacelike or timelike depending on the function $f(r)$.
Let us explore where the vector is timelike, $m_{i}^{2}<0$. From (V), by
taking into account that $a_{i}\neq 0$, we obtain

$f(r)>\quantity(\dfrac{\ell}{r})^{2(z-1)}\dfrac{\quantity(a_{i}^{2}+\ell^{2})}{a_{i}^{2}}.$
(48)

It will be timelike if $f(r)$ is greater than certain positive number and the
limit $r\rightarrow\infty$ implies $f(r)>0$. In the case of our solution
(15a), the function $f(r)$ is positive outside the event horizon and we will
end up with closed timelike curves accessible from null infinity. This
constitutes a pathology of the spacetime.

Figure 1: The metric components $g_{tt}$ and $g_{\phi_{i}\phi_{i}}$ with
parameters $\ell=1$, $M=0.5$, $a_{i}=1$, $z=2$ and $d=5$. The vertical axis
signals the event horizon radius $r_{H}$. From the event horizon to the
intersection of the _red_ curve with the r-axis we have the _ergosphere_. From
the intersection of the _blue_ curve with the r-axis to infinity we have
_closed timelike_ curves.

We have already proved the spinning black hole configuration given by (II) and
(15a) as a proper solution to the Einstein-Maxwell-Dilaton Theory (5) with a
well-defined Thermodynamics provided in Sec. III. This solution does not seem
to capture this pathological behavior. It would be interesting to explore
whether this pathology has any consequence in the dual field theory since so
far we are not aware of any restriction of this kind on it.

##  VI Curvature invariants

Here we compute the curvature invariants of our rotating black hole spacetime
in order to study its behavior. Thus, we proceed to display the curvature
Ricci scalar

$\displaystyle
R=-\frac{1}{\ell^{2}}\quantity{2\quantity[z^{2}+(d-2)z+\frac{1}{2}(d^{2}-3d+2)]+\frac{(d-2)(z-1)M}{r^{z+d-2}}},$
(49)

the square contraction of the Ricci tensor $\mathcal{R}=R_{\mu\nu}R^{\mu\nu}$

$\displaystyle\mathcal{R}=$
$\displaystyle\frac{1}{\ell^{4}}\Biggl{\\{}2(z^{2}+d-2)\quantity[z^{2}+(d-2)z+\frac{1}{2}(d^{2}-3d+2)]+\frac{2(d-2)(z-1)(z^{2}+d-2)}{r^{z+d-2}}$
$\displaystyle+\frac{(z-1)^{2}(d-2)^{2}M^{2}}{r^{2(z+d-2})}\Biggr{\\}},$ (50)

as well as the Kretschmann invariant — defined as the square contraction of
the Riemann tensor $\mathcal{K}=R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$ —

$\displaystyle\mathcal{K}=$
$\displaystyle\dfrac{d-2}{\ell^{4}}\Biggl{\\{}[d^{3}-2(z+2)d^{2}+(z^{2}+8z+2)d-6(2z-1)]\quantity(\dfrac{M}{r^{z+d-2}})^{2}$
$\displaystyle-4(z-1)\quantity[zd-(z+1)^{2}]\quantity(\dfrac{M}{r^{z+d-2}})+\dfrac{2d\quantity(d+2z^{2}-3)+4(z^{2}-1)^{2}}{d-2}\Biggl{\\}}.$
(51)

All of these invariants are singular at $r=0$ and regular at the black hole
horizon, $f(r_{H})=0$, for all $z\geq 1$ and $d>2$. Moreover, these scalars
tend to a constant at spatial infinity.

Here some comments are in order regarding the structure of these curvature
invariants. Their structure is the same for both static and rotating black
holes due to the metric ansatz (II) and the flat topology of the event horizon
that have been employed to construct our stationary black hole configuration.
In fact we have a rotating planar brane, therefore rotation is compatible with
its planar symmetry. Precisely the same situation takes place for the famous
AdS3 BTZ black hole Banados  _et al._ (1992) as well as for the Lemos black
hole Lemos (1995) and its multi-dimensional version given by the Awad metric
Awad (2003) (all these black holes are rotating and possess non-trivial
angular momenta according to the computation of the corresponding conserved
global charges). When computing these curvature invariants, a peculiar
mathematical fact arises indicating that they coincide for both static and
rotating black holes as it can be straightforwardly appreciated from their
expressions: they do not depend on the rotation parameters of the black hole
solution or the transverse coordinates of the line-element since the latter
are cyclic due to the flat topology they represent (this is in contrast to the
spherical or hyperbolic symmetries where these coordinates are not cyclic
anymore).

At least for the case of multi-dimensional ($d\geq 4$) rotating black hole
spacetimes, either in AdS or Lifshitz backgrounds, which are very similar
since both possess negative curvature, one would expect more complex metrics
to exist in the sense that the blackening factor entering the line-element can
have a functional dependence on the angular momenta of the metric, rendering
curvature invariants that explicitly depend on the rotating parameters. This
is an interesting issue which is currently under research within our group.

##  VII Conclusions

We have presented a rotating black hole exact solution, given by Eqs. (II) and
(15a), within the context of Einstein-Maxwell-Dilaton theory. We were able to
prove that the configuration rotates with angular momenta given by Eq. (20),
where the $a_{i}$-constants can be identified as rotation parameters. Once the
rotation parameters are set to zero, an asymptotically Lifshitz black hole is
recovered. In addition, we implemented the off-shell quasi-local analysis for
computing the mass of the black hole configuration (19) and compute the
entropy (31) to verify that the first law of black hole mechanics (32) is
plausible satisfied. In order to achieve this goal we introduced a
generalization of the Hawking temperature notion (28), where the critical
temperature can attain any positive definite or vanishing value, allowing us
to easily identify the emergence of phase transitions at finite temperature
and entropy, in accordance with the third law of Thermodynamics and the
phenomenology of many quantum matter systems showing superconductivity or
superfluidity, for instance. Moreover, a Smarr-like formula for quasi-
homogeneous functions has been constructed to give the relation (34). By
making use of the thermodynamic quantities of our spinning black hole
configuration we constructed a consistent geometrization on the basis of the
Gibbs free energy. We revealed the existence of critical points where phase
transitions arise in the specific heats, while singularities emerge in the
curvature invariants of the thermodynamic metric. We find a remarkable
correspondence between them, in complete concordance with the prescriptions of
thermodynamic geometry. We also explored a pathological behavior of the
spacetimes defined by (II) and (15a), that is not captured by the
Thermodynamics of the constructed metric. It would be interesting to explore
whether there is any sensible consequence of this pathology in the dual field
theory within the framework of the holographic correspondence. We finally
computed and analyzed the structure of the curvature invariants of our
rotating black hole configuration.

###### Acknowledgements.

The authors would like to thank Eloy Ayón-Beato, Mokhtar Hassaïne, Héctor I.
Zapata Rodríguez and Manuel de la Cruz López for multiple enlightening and
fruitful discussions. We are also indebted to an anonymous referee for
constructive reports that helped improving our work. A.H.-A. and D.F.H.-B.
thank the Sistema Nacional de Investigadores (SNI) for support. The authors
acknowledge financial support from a CONACYT grant No. A1-S-38041. D.F.H.-B is
also grateful to CONACYT for a Postdoc for Mexico grant No. 372516. J.A. H.-M.
also acknowledges support from CONACYT through a PhD grant No. 750974.

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# On a matrix element representation of special functions associated with
toric varieties111 Contribution to Proceedings of the Nankai Symposium on
Mathematical Dialogue 2021.

A.A. Gerasimov, D.R. Lebedev and S.V. Oblezin

Abstract. We develop representation theory approach to the study of special
functions associated with toric varieties. In particular we show that the
corresponding special functions are given by matrix elements of certain non-
reductive Lie algebras.

With toric manifolds and their duals one can naturally associate generalized
hypergeometric functions introduced by Gelfand-Kapranov-Zelevinski [GKZ].
These functions arise in description of quantum cohomology of toric manifolds
and its mirror dual (see e.g. [Ba]). In this note we consider a special
instance of these functions associated with toric varieties obtained by
Hamiltonian reduction of $\mathbb{C}^{N}$ (see e.g. [Au]). Precisely let us
fix the standard symplectic structure on $\mathbb{C}^{N}$

$\displaystyle\begin{array}[]{cc}\Omega=\frac{\imath}{2}\sum_{i=1}^{N}dz_{i}\wedge
d\bar{z}_{i}=\frac{1}{2}\sum_{i=1}^{N}d|z_{i}|^{2}\wedge d{\mathop{\rm
Arg}}(z_{i}),\end{array}$ (2)

and consider the Hamiltonian action of the compact torus $T^{n}=(S^{1})^{n}$
on $\mathbb{C}^{N}$. The corresponding toric manifold
$Z=\mathbb{C}^{N}//T^{n}$ is the Hamiltonian reduction under action of
$T^{n}$. The torus $T^{N}=(S^{1})^{N}$ and hence its subtorus $T^{n}\subset
T^{N}$ act on $Z$ with the following momentum maps

$\displaystyle\begin{array}[]{cc}\mathbb{C}^{N}\longrightarrow{\rm
Lie}(T^{N})^{*},\qquad\mathbb{C}^{N}\longrightarrow{\rm
Lie}(T^{n})^{*}.\end{array}$ (4)

Define a natural projection

$\displaystyle\begin{array}[]{cc}m\,:\quad{\rm
Lie}(T^{N})^{*}\longrightarrow{\rm Lie}(T^{n})^{*}\,,\end{array}$ (6)

which is dual to the inclusion $T^{n}\subset T^{N}$. Let $\\{E_{i},\,i\in
I\\}$ indexed by $I=\\{1,\ldots,N\\}$ be a basis in ${\rm Lie}(T^{N})^{*}$ and
let $\\{e_{\alpha},\,{\alpha}\in A\\}$ indexed by $A=\\{1,\ldots,n\\}$ be a
basis in ${\rm Lie}(T^{n})^{*}$. Then we have explicitly

$\displaystyle\begin{array}[]{cc}m(E_{j})=\sum_{{\alpha}=1}^{n}e_{\alpha}m_{j}^{\alpha},\qquad
j\in I\,.\end{array}$ (8)

Let $\\{v_{k},\,k\in\overline{A}\\}$ indexed by
$\overline{A}=\\{1,\ldots,N-n\\}$ be a basis in the kernel ${\rm Ker}(m)$ of
(6):

$\displaystyle\begin{array}[]{cc}v_{k}=\sum\limits_{i=1}^{N}v_{k}^{i}E_{i}\in{\rm
Lie}(T^{N})^{*},\qquad\sum_{i=1}^{N}m^{\alpha}_{i}v_{k}^{i}\,=\,0,\qquad{\alpha}\in
A,\quad k\in\overline{A}\,.\end{array}$ (10)

Let $Z_{1},\ldots,Z_{N}$ be cohomology classes dual to the coordinate
hyperplane divisors in $Z$ given by intersects of the coordinate hyperplanes
in $\mathbb{C}^{N}$ with $Z$. Then in the quantum cohomology $QH^{*}(Z,q)$ the
following relations hold (see [Ba]):

$\displaystyle\begin{array}[]{cc}\prod_{j=1}^{N}\,Z_{j}^{m_{j}^{\alpha}}=q_{\alpha},\qquad{\alpha}\in
A,\end{array}$ (12)

where $q_{\alpha}=e^{x^{\alpha}}$ are quantum deformation parameters. These
relations are deformations of the classical relations for $q_{{\alpha}}=0$
corresponding to the sets of hyperplanes with zero intersections.

The $T^{N}$-equivariant mirror dual to the toric manifold $Z$ is defined by
the data $(Y_{q},\mathcal{F},\omega_{q})$ where $Y_{q}$ is an affine space
given by the set of solutions of (12), $\mathcal{F}$ is given by

$\displaystyle\begin{array}[]{cc}\mathcal{F}(t,\lambda)=\sum_{i=1}^{N}(-t_{i}+\imath\lambda_{i}\ln
t_{i}),\qquad\omega_{q}=\prod_{{\alpha}=1}^{n}\delta(f_{\alpha})\,\frac{dt_{1}\wedge\cdots\wedge
dt_{N}}{t_{1}\cdots t_{N}},\end{array}$ (14)

where

$\displaystyle\begin{array}[]{cc}f_{\alpha}=e^{-x^{\alpha}}\prod_{j=1}^{N}\,t_{j}^{m_{j}^{\alpha}}-1,\qquad\qquad{\alpha}\in
A.\end{array}$ (16)

The special function associated with these data is then given by the following
period

$\displaystyle\begin{array}[]{cc}\Psi_{\lambda_{1},\ldots,\lambda_{N}}(q_{1},\cdots,q_{n})=\int\limits_{\mathbb{R}^{N}_{+}}\\!\\!\omega_{q}\,\,\,\prod_{i=1}^{N}t_{i}^{\imath\lambda_{i}}e^{-t_{i}}\,.\end{array}$
(18)

The integral representation (18) allows interpretation as a matrix element of
a suitable Lie algebra. Namely, consider the semi-direct sum $\mathcal{L}_{N}$
of abelian Lie algebra $(\mathfrak{gl}_{1})^{N}$ and the Heisenberg algebra of
dimension $2N+1$ defined by the set generators
$\\{\mathcal{E}_{i},\mathcal{F}_{i},\mathcal{H}_{i},\,i\in I\\}$, central
element $\mathcal{C}$ and relations

$\displaystyle\begin{array}[]{cc}[\mathcal{E}_{i},\mathcal{H}_{j}]=\delta_{ij}\mathcal{E}_{i},\qquad[\mathcal{F}_{i},\mathcal{H}_{j}]=-\delta_{ij}\mathcal{F}_{i},\qquad[\mathcal{E}_{i},\mathcal{F}_{j}]=\mathcal{C}\delta_{ij},\qquad
i,j\in I.\end{array}$ (20)

We fix a generic irreducible representation $\pi_{\lambda,c}$ in a suitable
space of functions in $t=(t_{1},\cdots,t_{N})\in\mathbb{R}_{+}^{N}$ with
images of the generators in this representation given by

$\displaystyle\begin{array}[]{cc}E_{i}=c\partial_{t_{i}},\qquad
F_{i}=t_{i},\qquad H_{i}=t_{i}\partial_{t_{i}}+\imath\lambda_{i},\qquad
C=c,\end{array}$ (22)

where $\lambda_{i}\in\mathbb{R},\,\,c\in\mathbb{R}_{+}$. In the following for
convenience we take $c=1$. Define left and right vectors by the following
conditions

$\displaystyle\begin{array}[]{cc}E_{i}|\psi_{R}\rangle=-|\psi_{R}\rangle,\qquad
i\in I\,;\\\
\sum_{i=1}^{N}v_{k}^{i}H_{i}|\psi_{L}\rangle=0,\,\,k\in\overline{A},\qquad\prod_{j=1}^{N}F_{j}^{m_{j}^{\alpha}}|\psi_{L}\rangle=|\psi_{L}\rangle,\,\,{\alpha}\in
A,\end{array}$ (25)

where $\\{v_{k},k\in\overline{A}\\}$ are given by (10). Now introduce the
following matrix element

$\displaystyle\begin{array}[]{cc}\Psi_{\lambda}(e^{y}):=\Psi_{\lambda_{1},\ldots,\lambda_{N}}(e^{y^{1}},\ldots,e^{y^{N}})=\langle\psi_{L}|e^{\sum\limits_{j=1}^{N}y^{j}H_{j}}|\psi_{R}\rangle,\end{array}$
(27)

with the non-degenerate pairing $\langle\,,\,\rangle$ given by

$\displaystyle\begin{array}[]{cc}\langle
a\,,\,b\rangle\,:=\,\int\limits_{\mathbb{R}_{+}^{N}}\prod_{i=1}^{N}d^{\times}t_{i}\,\,\overline{a(t)}\,b(t),\qquad
d^{\times}t_{i}=\frac{dt_{i}}{t_{i}}\,.\end{array}$ (29)

###### Theorem 1

The matrix element (27) allows the following integral representation

$\displaystyle\begin{array}[]{cc}\Psi_{\lambda}(e^{y})=\int\limits_{\mathbb{R}_{+}^{N}}\prod_{i=1}^{N}d^{\times}t_{i}\,\prod_{{\alpha}=1}^{n}\delta\Big{(}e^{-\sum\limits_{j=1}^{N}y^{j}m_{j}^{\alpha}}\,\prod_{j=1}^{N}t_{j}^{m_{j}^{\alpha}}-1\Big{)}\,\,\prod_{j=1}^{N}t_{j}^{\imath\lambda_{j}}\,e^{-t_{j}}\,\,,\end{array}$
(31)

and satisfies the equations:

$\displaystyle\begin{array}[]{cc}\Big{\\{}\prod_{j=1}^{N}\prod_{k=0}^{m_{j}^{{\alpha}}-1}(\imath\lambda_{j}+k-\partial_{y^{j}})-e^{\sum\limits_{j=1}^{N}y^{j}m_{j}^{\alpha}}\Big{\\}}\,\Psi_{\lambda}(e^{y})=0,\qquad{\alpha}\in
A,\end{array}$ (33)
$\displaystyle\begin{array}[]{cc}\sum_{j=1}^{N}v_{k}^{j}\partial_{y^{j}}\cdot\Psi_{\lambda}(e^{y})=0,\qquad
k\in\overline{A}\,,\end{array}$ (35)

with $v_{k},\,k\in\overline{A}$ defined in (10).

Proof. Explicitly for the vectors $\psi_{L,R}$ we have

$\displaystyle\begin{array}[]{cc}|\psi_{R}\rangle=e^{-\sum\limits_{i=1}^{N}t_{i}},\qquad|\psi_{L}\rangle=\prod_{i=1}^{N}t_{i}^{-\imath\lambda_{i}}\,\,\prod_{{\alpha}=1}^{n}\delta\Big{(}\prod_{j=1}^{N}t_{j}^{m_{j}^{\alpha}}-1\Big{)}.\end{array}$
(37)

Substitution of (37) into (27) after a simple change of variables leads to
(31). For the second assertion, consider the following polynomials of
generators acting trivially in representation $\pi_{\lambda,c}$ defined by
(22)

$\displaystyle\begin{array}[]{cc}\mathcal{C}_{\alpha}=\prod_{j=1}^{N}(F_{j}E_{j})^{m_{j}^{\alpha}}-\prod_{j=1}^{N}C^{m_{j}^{{\alpha}}}(H_{j}-\imath\lambda_{j})^{m_{j}^{\alpha}},\qquad{\alpha}\in
A.\end{array}$ (39)

Insertion of these expressions into the matrix element (27) then leads to the
first set of equations (33). In addition we have the following set of
equations (35) obtained by inserting the linear combinations
$\sum\limits_{j=1}^{N}v_{k}^{j}H_{j}$ into the matrix element (27) and taking
into account the defining relation (25). $\Box$

###### Corollary 1

The matrix element (27) effectively depends on a smaller number of variables
$x^{\alpha}=\sum_{i=1}^{N}y^{i}m_{i}^{\alpha}$, $\alpha\in A$ and allows the
following representation

$\displaystyle\begin{array}[]{cc}\Psi_{\lambda_{1},\ldots,\lambda_{N}}(e^{x_{1}},\ldots,e^{x_{n}})=\int\limits_{\mathbb{R}_{+}^{N}}\prod_{i=1}^{N}d^{\times}t_{i}\,\prod_{{\alpha}=1}^{n}\delta\Big{(}e^{-x^{\alpha}}\,\prod_{j=1}^{N}t_{j}^{m_{j}^{\alpha}}-1\Big{)}\,\,\,\prod_{j=1}^{N}t_{j}^{\imath\lambda_{j}}\,e^{-t_{j}}\,.\end{array}$
(41)

This function coincides with the mirror period (18) and solves the set of
equations for ${\alpha}\in A$

$\displaystyle\begin{array}[]{cc}\Big{\\{}\prod_{j=1}^{N}\prod_{k=0}^{m_{j}^{{\alpha}}-1}(\imath\lambda_{j}+k-\sum_{{\beta}=1}^{n}m_{j}^{\beta}\partial_{x^{\beta}})-e^{x^{\alpha}}\Big{\\}}\,\Psi_{\lambda_{1},\ldots,\lambda_{N}}(e^{x_{1}},\ldots,e^{x_{n}})=0.\end{array}$
(43)

A simple example of the proposed construction is provided by mirror
correspondence for complex projective space $Z=\mathbb{P}^{\ell}$. Projective
vector space allows the standard Hamiltonian reduction representation
$\mathbb{P}^{\ell}=\mathbb{C}^{\ell+1}//S^{1}$, where the action is defined as

$\displaystyle\begin{array}[]{cc}S^{1}\,:\quad\mathbb{C}^{\ell+1}\longrightarrow\mathbb{C}^{\ell+1},\qquad
e^{\imath{\theta}}:\,(z_{1},\ldots,z_{\ell+1})\longmapsto(z_{1}\,e^{\imath{\theta}},\ldots,z_{\ell+1}\,e^{\imath{\theta}}).\end{array}$
(45)

Therefore we have $N=\ell+1$, $n=1$ and the relation matrix is given by
$||m_{j}^{\alpha}||=(1,\ldots,1)$. The resulting representation for the mirror
period (18) reads

$\displaystyle\begin{array}[]{cc}\Psi_{\lambda_{1},\ldots,\lambda_{\ell+1}}(e^{x})=\int\limits_{\mathbb{R}_{+}^{N}}\prod_{i=1}^{\ell+1}d^{\times}t_{i}\,\,\delta\Big{(}e^{-x}\prod_{j=1}^{\ell+1}t_{j}-1\Big{)}\,\,\prod_{j=1}^{\ell+1}t_{j}^{\imath\lambda_{j}}\,e^{-t_{j}}.\end{array}$
(47)

By Theorem 1, (47) coincides with the following matrix element of the Lie
algebra
$\mathcal{L}_{\ell+1}=(\mathfrak{gl}_{1})^{\ell+1}\uplus\mathcal{H}_{2(\ell+1)}^{c}$
in the generic representation (22):

$\displaystyle\begin{array}[]{cc}\Psi_{\lambda_{1},\ldots,\lambda_{\ell+1}}(e^{x})=\langle\psi_{L}|e^{x\sum\limits_{i=1}^{\ell+1}a_{i}H_{i}}|\psi_{R}\rangle,\qquad\sum_{i}a_{i}=1,\end{array}$
(49)

where the left and right vectors are defined by

$\displaystyle\begin{array}[]{cc}E_{i}|\psi_{R}\rangle=-|\psi_{R}\rangle,\qquad
i\in\\{1,\ldots,\ell+1\\}\,;\\\ (H_{k}-H_{\ell+1})|\psi_{L}\rangle=0,\quad
k\in\\{1,\ldots,\ell\\},\qquad F_{1}\cdots
F_{\ell+1}|\psi_{L}\rangle=|\psi_{L}\rangle.\end{array}$ (52)

Particular choice of $\\{a_{i},\,i\in I\\}$ is irrelevant due to (52). The
defining equations for (47) are obtained by using the following operator
acting by zero in the representation (22):

$\displaystyle\begin{array}[]{cc}\mathcal{C}_{\ell+1}=\prod_{i=1}^{\ell+1}F_{i}E_{i}-C^{\ell+1}\prod_{i=1}^{\ell+1}(H_{i}-\imath\lambda_{i}).\end{array}$
(54)

The resulting equation

$\displaystyle\begin{array}[]{cc}\Big{\\{}\prod_{i=1}^{\ell+1}(\imath\lambda_{i}-\partial_{x})-e^{x}\Big{\\}}\,\Psi_{\lambda}(e^{x})=0,\end{array}$
(56)

is in agreement with the proper equivariant counting of holomorphic maps of
$\mathbb{P}^{1}$ to $\mathbb{P}^{\ell}$ (see e.g. [GLO]).

Acknowledgements: The research of the second author was supported by RSF grant
16-11-10075. The work of the third author was partially supported by the EPSRC
grant EP/L000865/1.

## References

  * [Au] M. Audin, Torus Actions on Symplectic Manifolds, 2nd Revised Ed., Progr. in Math. 93 , Birkhäuser-Verlag-Basel-Berlin, 2004.
  * [Ba] V. Batyrev, Quantum cohomology rings of toric manifolds, in Journées de géometrie algébrique d’Orsay, Juillet 1992, Astérisque 218 (1993).
  * [GKZ] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Hypergeometric functions and toric varieties, Funct. Anal. Appl. 23:2 (1989) 12–26.
  * [GLO] A. Gerasimov, D. Lebedev, S. Oblezin, Parabolic Whittaker functions and topological field theories I, Commun. Number Theory Phys. 5:2 (2011) 135–201.

A.A.G. Laboratory for Quantum Field Theory and Information,
xxxx Institute for Information Transmission Problems, RAS, 127994, Moscow,
Russia;
xxxx Interdisciplinary Scientific Center J.-V. Poncelet (CNRS UMI 2615),
Moscow, Russia;
xxxx E-mail address<EMAIL_ADDRESS>
D.R.L. Laboratory for Quantum Field Theory and Information,
xxxx Institute for Information Transmission Problems, RAS, 127994, Moscow,
Russia;
xxxx Moscow Center for Continuous Mathematical Education,
xxxx 119002, Bol. Vlasyevsky per. 11, Moscow, Russia;
xxxx E-mail address<EMAIL_ADDRESS>
S.V.O. School of Mathematical Sciences, University of Nottingham ,
xxxx University Park, NG7 2RD, Nottingham, United Kingdom;
xxxx Institute for Theoretical and Experimental Physics, 117259, Moscow,
Russia;
xxxx E-mail address<EMAIL_ADDRESS>


# Particle motion during the compaction of granular matter

Steven Slotterback<EMAIL_ADDRESS>Masahiro Toiya  Leonard Goff  Department of
Physics, and IREAP, University of Maryland, College Park, Maryland, 20742
Jack F. Douglas  Polymers Division and Center for Theoretical and
Computational Materials Science, National Institute for Standards and
Technology, Gaithersburg, Maryland  Wolfgang Losert<EMAIL_ADDRESS>Department of Physics, IPST and IREAP, University of Maryland, College Park,
Maryland, 20742

###### Abstract

We track particle motions in a granular material subjected to compaction using
a laser scattering based imaging method where compaction is achieved through
thermal cycling. Particle displacements in this jammed fluid correlate
strongly with rearrangments of the Voronoi cells defining the local spatial
partitioning about the particles, similar to previous observations of Rahman
on cooled liquids. Our observations provide further evidence of commonalities
between particle dynamics in granular matter close to jamming and supercooled
liquids.

###### pacs:

45.70.Cc,81.05.Rm,45.70.-n

Granular materials are strongly interacting particle systems that admit an
admixture of collective particle motions, as in crystals, and random particle
motions, as in simple fluids. Under flow or deformation conditions these
materials pass intermittently between a solid “jammed” state and a state of
flow. When a granular material jams, the individual particles are in stable
mechanical equilibrium with their local neighbors O’Hern et al. (2003). Small
perturbations such as tapping Knight et al. (1995) or shearing Toiya et al.
(2004) can move the particles, and lead to the evolution from one jammed state
to another, generally more compact state. The increase in particle density
with the number of perturbations is similar to the slow thermally driven
dynamics of fluids close to their glass transition. Observations of phenomena
of this kind has led to the development of a general framework for the jamming
transition, with temperature, density, and external forcing as the three main
variables characterizing this dynamical transition O’Hern et al. (2003); Liu
and Nagel (1998).

Although there have been a number of recent studies of particle trajectories
in glass-forming colloidal fluids Weeks and Weitz (2002); Marcus et al. (1999)
and other strongly interacting particle systems, little is known about the
nature of particle trajectories in granular materials. While recent studies
have successfully observed particle positions in a 3D jammed granular material
Richard et al. (2003); Aste et al. (2007), particle trajectories were not
observed. It seems plausible that the particle motion has features in common
with glass-forming liquids, such as the emergence of string-like collective
motion Douglas et al. (2006), and based on this hypothesis we examine whether
the shape fluctuations in the Voronoi cells environments about the particles
in compactified granular media correlate with the direction of particle
displacements, as noticed by Rahman Rahman (1966) in his pioneering studies of
collective particle motions in cooled liquids. On the other hand, the non-
equilibrium nature of these driven particle systems leads us to expect
differences from thermalized liquids that require a better understanding. In
this paper, we provide the first direct measurements of 3D particle
trajectories during compaction, with a focus on the statistics of particle
motion.

In particular, we study particle rearrangements in a column of granular
materials during the course of thermal cycling. We use a laser sheet scanning
method to reconstruct the configurations of the system at the end of each
cycle. The motion of individual particles is tracked over 10 cycles, and we
analyze how the local geometry affects local particle dynamics.

Thermal cycling provides a method of granular compaction without exciting
strong motions of particles. By thermal cycling we meanalternately heating and
cooling the material and container, with one heating and cooling cycle
corresponding to one perturbation. This technique leads to exponential
compaction of the material with two characteristic timescales Chen et al.
(2006) if the granular material and container have different thermal expansion
coefficients. While most methods of forcing rearrangements of granular matter,
such as tapping of the boundaries or air fluidization agitate individual
particles significantly, i.e. provide particles kinetic energy relative to
their neighbors, thermal cycling provides a quasi-stationary method of
evolving the structure towards a jammed state.

Experimental Setup: The granular material used is soda-lime glass beads
(diameter $D=3{\rm\>mm}\>\pm\>0.3{\rm\>mm}$), poured into a transparent
polymethylpentene (PMP) cylinder of diameter 5cm (16.6 D) to a height of
$19.5{\rm\>cm}\>\pm\>0.5{\rm\>cm}$ (63.3 D to 66.6 D). This is the same
materials as used in previous work on thermal cycling Chen et al. (2006). The
thermal expansion coefficient for PMP is $1.17\times 10^{-4}{\rm\>K^{-1}}$,
about one order of magnitude larger than that of the beads ($9\times
10^{-6}{\rm\>K^{-1}}$) Chen et al. (2006). The beads were immersed in index-
matching oil (Cargille Labs Type DF 111Identification of a commercial product
is made only to facilitate reproducibility and to adequately describe
procedure. In no case does it imply endorsement by NIST or imply that it is
necessarily the best product for the procedure.) which contains $2.08\>\mu{\rm
g/mL}$ laser dye (Nile Blue 690 perchlorate). A transparent weight of
$145.5{\rm\>g}$ was placed on top of the particles to monitor the filling
height and to apply a controlled vertical force.

The cylinder is placed inside a transparent water tank for thermal cycling at
a rate of $16^{o}$ C per hour. As the PMP container was heated/cooled the
cross-sectional area expanded/contracted by $0.46{\rm\>mm^{2}}$ for every
degree heated/cooled ($6.5\>\%$ of a single particle cross-section).

While bead expansion was negligible, the sample and oil are index-matched at
room temperature. Since the index of refraction changes with temperature
differently for glass and oil, we can only image our sample at the end of each
thermal cycle.

Figure 1: Experimental setup: Glass beads immersed in index matching oil in a
cylinder are subjected to thermal cycling via a water bath. The fluorescently
dyed fluid is imaged via a laser sheet that is moved across the sample. One of
600 cross-sectional images shown.

Three dimensional images of the system at each cycle were obtained using a
laser sheet scanning method, first described in Toiya et al. (2007). Briefly,
a laser sheet is sent through the specimen at the excitation frequency of the
laser dye to produce a cross-sectional image of the granular material in which
the particles appear as dark circles within the dyed fluid (see Fig. 1).
Images are taken with a high sensitivity cooled CCD camera (Sensicam, PCO). To
obtain a three dimensional image, the laser is translated in $100\>\mu{\rm m}$
increments along the axis of the cylinder for a distance of 60 mm (20 D),
starting from the top of the pile. The camera is not translated, but the
changing object distance is accounted for in the analysis.

A 3D bandpass filter is used to reduce noise and to smooth out images of the
beads. A particle tracking algorithm by Crocker Crocker and Grier (1996) and
Weeks Weeks (2007) detects the centers of $>98\>\%$ of all beads to within
$100\>\mu{\rm m}$, which is smaller than the uncertainty in the bead
diameters. The remaining $<2\>\%$ of bead positions are detected manually with
comparable resolution using an interactive 3D visualization of images with an
overlay of extracted particles. This allows us to generate complete maps of 3D
jammed states. Particle rearrangmenets are followed through multiple thermal
cycles. The particles move by less than 0.5 D in each thermal cycle, a scale
small enough to allow for reliable particle tracking.

Figure 2: Packing fraction vs. cycle number for various temperature
differentials

Results: We apply ten thermal cycles for three different temperature
differentials: $25{\rm{}^{o}\>C}$, $40{\rm{}^{o}\>C}$, and $60{\rm{}^{o}\>C}$.
The average packing fraction is determined from the vertical position of the
flat glass weight on top of the pile in the 3D image. The initial conditions
are generated by letting particles settle into the oil filled cylinder. After
the particles settle, the glass top is allowed to sink onto the surface of the
fluid immersed particles. Filling in oil first is necessary to eliminate air
bubbles from the sample, but leads to significant variability in the initial
packing fraction. Nevertheless, the increase in packing fraction during
thermal cycling can be fit with an exponential
$\rho(t)=\rho_{0}-A\>e^{t/\tau{}}$ as shown in Fig. 2 with $\rho_{0}=0.619\pm
0.0008$, $A=0.028\pm 0.0007$, and $\tau=4.87\>{\rm cycles}\pm 0.31\>{\rm
cycles}$. The compaction rate is slower than the faster of the two time
scales, 2.72 cycles, reported by Chen et al. (2006), consistent with a weaker
forcing of the particles due to buoyancy forces.

We find that the pair correlation function $g(r)$ does not evolve
significantly during compaction, as shown in Fig. 3. This is not surprising
since the particles only need to move closer by about $0.001\>D$ on average to
achieve an increase in packing fraction of $1.5\>\%$. The overall shape of
$g(r)$ is consistent with a random arrangement of particles in the presence of
some ordering at the boundaries.

Figure 3: Pair correlation function $g(r)$ averaged over 10 thermal cycles.

The Voronoi construction defines a unique partitioning of space where each
particle is assigned a local neighborhood, the Voronoi cell, corresponding to
the region of space closest to a given particle. The Voronoi cell structures
were determined using the QHULL algorithm Barber et al. (1996). To avoid
boundary artifacts, Voronoi cells were computed for all particles, but only
the volumes of particles that are not on the boundaries we used in the
analysis. The distributions of Voronoi volumes is shown in Fig. 4. The
distribution is non-gaussian with a sharp cutoff at small volumes and a broad
range of large volumes, similar to distributions observed in simulations in
glass-forming liquids Starr et al. (2002) and in recent experiments of
granular materials Aste et al. (2007). One possible functional form of the
distribution of voronoi volumes proposed by Aste et al. Aste et al. (2007)
contains only one fitting parameter $k$. Fitting our Voronoi cell volume
distributions to this function yields $k=24$, appreciably larger than the $k$
value, $k=12$, suggested by Aste et al. Aste et al. (2007), however. The
larger polydispersity in bead diameters in the experiments presented here
($10\>\%$ as opposed to $2\>\%\>{\rm to}\>5\>\%$), could explain the wider
distribution of Voronoi volumes. We find that the distribution of Voronoi
volumes does not change significantly as the system is compacted.

Figure 4: Histogram of Voronoi volumes

Following Rahman’s investigation of cooled liquids, we analyze particle motion
and its relation to the Voronoi cell shape. We define the vector
$\vec{v}_{i,j}$ to be the displacement of the center of particle $i$ from
cycle $j$ to $j+1$ (see Fig. 5). Next, we define a vector describing the
Voronoi cell shape relative to the particle center. From the perspective of
the particle, the vertices of the vornonoi construction correspond to the
direction in which a particle “sees” a void between three neighbors. The mean
position of all vertices therefore indicates where a particle would see more
local void spaces. The vector $\vec{u}_{i,j}$ defines the displacement from
the center of particle $i$ at cycle $j$ to the mean position of its
corresponding Voronoi vertices, as shown in Fig. 5. The averages for
$|\vec{u}|$ and $|\vec{v}|$ are approximately 1 mm (0.33 D) larger than the
accuracy with which we detect particle positions.

Figure 5: Definition of particle displacement vector v and u vector for a
cross-section of the 3D Voronoi volume. Since this is a cross-section of the
3D Voronoi cell, the particle need not be in contact with any of the edges of
the Voronoi volume.

To determine the correlation between particle displacement and Voronoi shapes,
we then consider the alignment of $\vec{u}_{i,j}$ and $\vec{v}_{i,j}$.
Specifically, we calculate $\cos(\theta)$ for each image, where $\theta$ is
the angle between $\vec{u}_{i,j}$ and $\vec{v}_{i,j}$ for each particle. The
probability distribution of $\cos{\theta}$ averaged over all cycles is shown
in Fig. 6. In addition, we show the distribution for the first and the last
cycle, and a control for two 3-D images without thermal cycling between the
images. We find that the distribution peaks at $\cos{\theta}=1$, indicating
that the particle moves toward the center of the vertices of its Voronoi cell.
Particle motion and Voronoi shape are indeed correlated.

Figure 6: a) Preferred Direction PDFs for system with $\Delta
T=40{\rm{}^{o}\>C}$ for each thermal cycle. b) Preferred Direction PDFs for
system with $\Delta T=25{\rm{}^{o}\>C}$ for each thermal cycle. c) Preferred
Direction PDFs for system at constant temperature d) 3D reconstruction of
system from the perspective of a given particle facing five vertices of its
Voronoi cell. Notice that the vertex is located in the void between particles.

Discussion:

By using the laser sheet scanning method, we visualize the internal structure
and dynamics of a jammed granular material. Under quasi-static forcing via
thermal cycling flows are slow enough (less than $\approx 0.5\>{\rm D}$
between frames), so we can observe the microscopic dynamics of this strongly
interacting particle system. While the differences in local structures are too
small to be observable in local distributions such as $g(r)$ or the Voronoi
volume distribution, the Voronoi volume appears to hold important clues for
the future dynamical evolution of the particles. Voronoi reconstruction
indicates that particles tend to move toward the centers of Voronoi vertices
as long ago observed in Rahman’s pioneering simulations of particle motion in
cooled liquids Rahman (1966). This correlated displacement in our granular
fluid is independent of the size of the Voronoi volume, the velocity of
particles, and the position of particles within the granular column.

Rahman Rahman (1966) and others following him Zwanzig and Bishop (1974)
inferred from the correlations in the tracer particle displacement and the
Voronoi cell shape that the particles were moving in tube-like environments,
thus anticipating the string-like motion in cooled liquids that has recently
become widely appreciated Donati et al. (1998); Douglas et al. (2006). This
raises questions about whether such collective interparticle displacements
also characterize particle motion in compactified granular materials. The
absence of equilibrium in granular fluids, however, complicates the definition
of “mobile” and collectively moving particles in granular materials, since
there is no model such as Brownian motion to provide a “baseline” for this
comparison. A recent study of a quasi-2D air-driven granular material Keys et
al. (2007) indicates the presence of collective string-like motion as in
cooled liquids so the general presence of such collective motions in driven
granular media seem highly plausible. However, we expect important differences
to emerge in systems such as ours where the volume of the material is evolving
in time with compaction. We plan to study the evolutionary or aging dynamics
of our granular fluid in the future.

###### Acknowledgements.

We thank Krisztian Ronaszegi for assistance in data analysis. This work was
supported by NSF grants CTS0457431 and CTS0625890. Official contribution of
the National Institute of Standards and Technology - Not subject to copyright
in the United States.

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###### Abstract

We give a reformulation of the Lehmer conjecture about algebraic integers in
terms of a simple counting problem modulo $p$.

title = On the Lehmer Conjecture and Counting in Finite Fields, author =
Emmanuel Breuillard and Péter P. Varjú, plaintextauthor = Emmanuel Breuillard
and Peter P. Varju, keywords = finite fields, Lehmer conjecture, Mahler
measure, year=2019, number=5, received=1 November 2018, published=21 May 2019,
doi=10.19086/da.8306,

[classification=text]

Let $\mathcal{S}_{d}$ be the set of all polynomials of degree at most $d$ with
coefficients in $\\{0,1\\}$. Recall that if $\alpha$ is an algebraic number,
with minimal polynomial $\pi_{\alpha}(X)=a_{0}\prod_{1}^{n}(X-\alpha_{i})$ in
$\mathbb{Z}[X]$, then the Mahler measure of $\alpha$ is the quantity

$M(\alpha)=|a_{0}|\prod_{1}^{n}\max\\{1,|\alpha_{i}|\\}.$

In [BV16] we showed that the Mahler measure is related to the growth rate of
the cardinality of the set

$\mathcal{S}_{d}(\alpha):=\\{P(\alpha);P\in\mathcal{S}_{d}\\}$

as follows.

######  Theorem 1 ([BV16, Thm 5, Lem 16]).

There is a numerical constant $c>0$ such that for every algebraic number
$\alpha$,

$\min\\{2,M(\alpha)\\}^{c}\leq\lim_{d\to+\infty}|\mathcal{S}_{d}(\alpha)|^{\frac{1}{d}}\leq\min\\{2,M(\alpha)\\}.$

One can take $c=0.44$. The limit above always exists by sub-multiplicativity
of $d\mapsto|\mathcal{S}_{d}(\alpha)|$. The celebrated Lehmer conjecture
posits the existence of a numerical constant $c_{0}>0$ such that every
algebraic number $\alpha$ that is not a root of unity must have

$M(\alpha)>1+c_{0}.$

In particular the following reformulation of the Lehmer conjecture follows
immediately from Theorem 1.

######  Corollary 2.

The following are equivalent.

  1. 1.

There exists $c_{1}>0$ such that

$\lim_{d\to+\infty}|\mathcal{S}_{d}(\alpha)|^{\frac{1}{d}}>1+c_{1}$

for every complex number $\alpha$ that is not a root of unity.

  2. 2.

The Lehmer conjecture is true.

Note that if $\alpha$ is transcendental, or simply not a root of a polynomial
of the form $P-Q$, $P,Q$ in $\mathcal{S}_{d}$ for some $d$, then
$|\mathcal{S}_{d}(\alpha)|=2^{d+1}$ for all $d$. Note further that if $\alpha$
is a root of unity, then $|\mathcal{S}_{d}(\alpha)|$ grows at most
polynomially in $d$.

The set $\mathcal{S}_{d}(\alpha)$ is the support of the random process
$\sum_{0}^{d}\epsilon_{i}\alpha^{i}$ where the $\epsilon_{i}$ are Bernoulli
random variables equal to $0$ or $1$ with probability $\frac{1}{2}$. The proof
of the lower bound in Theorem 1 consists in establishing a lower bound on the
entropy of the above process.

The purpose of this short note is to give a mod-$p$ version of the above
equivalence, showing that Lehmer’s conjecture is equivalent to an easily
formulated assertion about finite fields.

We first propose the following reformulation. We write $\log$ for the base $2$
logarithm. Given $C\geq 1$, we say that a prime $p$ is $C$-wild if there is
$x\in\mathbb{F}_{p}^{\times}$ of multiplicative order at least $(\log p)^{2}$
such that not every element of $\mathbb{F}_{p}$ is a sum of at most $(\log
p)^{C}$ elements from the geometric progression
$H:=\\{x,x^{2},\ldots,x^{[C\log p]}\\}$, where $[y]$ denotes the integer part
of $y$. (Here we allow choosing the elements of $H$ multiple times.)

######  Theorem 3.

The following are equivalent.

  1. $(a)$

The Lehmer conjecture is true.

  2. $(b)$

For some $C>1$, almost no prime is $C$-wild. That is, as $X\to+\infty$

$|\\{p\leq X\textnormal{ and }p\textnormal{ is
}C\textnormal{-wild}\\}|=o(|\\{p\leq X\\}|).$

The implication $(b)\Rightarrow(a)$ is the easier one, so this result should
not be regarded as making any progress towards a positive answer to the Lehmer
conjecture. The main point of this note is to show the converse implication. A
crucial ingredient in the proof is the lower bound for
$|\mathcal{S}_{d}(\alpha)|$ in Theorem 1.

As it turns out the method of proof actually yields a more precise result: the
validity of Lehmer’s conjecture implies a stronger statement than $(b)$, and
vice versa a weaker version of $(b)$ already implies Lehmer’s conjecture. We
spell this out in Theorem 4 below and to this aim we first introduce some
notation.

Let $\delta,\epsilon>0$. We will say that a prime number $p$ is _$\delta$
-bad_ if there is a non-zero residue class $x$ modulo $p$ with multiplicative
order at least $\log p\log\log\log p$ such that $|\mathcal{S}_{d}(x)|\leq
p^{\delta}$ for all $d\leq\log p$, where

$\mathcal{S}_{d}(x):=\\{P(x);P\in\mathcal{S}_{d}\\}.$

For $K\in\mathbb{N}$, we let $\mathcal{S}^{K}_{d}$ be the family of
polynomials of degree at most $d$ with integer coefficients in the interval
$[-K,K]$. Clearly

$\mathcal{S}_{d}\subset\mathcal{S}^{K}_{d}\subset\mathcal{S}_{d}\pm\ldots\pm\mathcal{S}_{d}\subset\mathcal{S}^{2K}_{d},$
(1)

where there are $2K$ summands in the sumset.

We will say that $p$ is _$(\delta,\epsilon)$ -very bad_ if there is a non-zero
residue class $x$ modulo $p$ with multiplicative order at least
$\sqrt{\frac{p}{\log p}}$ such that $|\mathcal{S}^{K}_{d}(x)|\leq p^{\delta}$
for all $d\leq\frac{1}{\delta}\log p$ and all $K\leq 2^{\epsilon d}$.

We can now state our second reformulation.

######  Theorem 4.

The following are equivalent.

  1. $(a)$

The Lehmer conjecture is true.

  2. $(b)$

There is $\delta>0$ such that for all $\epsilon>0$ as $X\to+\infty$,

$|\\{p\leq X\textnormal{ and }p\textnormal{ is
}(\delta,\epsilon)\textnormal{-very bad}\\}|=o(|\\{p\leq X\\}|).$ (2)

  3. $(c)$

For every $\varepsilon>0$ there is $\delta>0$ such that as $X\to+\infty$,

$|\\{p\leq X;p\textnormal{ is
}\delta\textnormal{-bad}\\}|\ll_{\varepsilon}X^{\varepsilon}.$ (3)

We have used the Vinogradov notation $\ll_{\epsilon}$ to mean that the
inequality holds up to a multiplicative constant depending only on $\epsilon$.

It is clear that $(c)$ implies $(b)$ in the above theorem, because for
$\delta,\epsilon\in(0,\frac{1}{2})$ every large enough
$(\delta,\epsilon)$-very bad prime must be $\delta$-bad.

The relevance of the Lehmer conjecture to the study of the family of
polynomials $\mathcal{S}_{d}$ is well-known and appears for example in
Konyagin’s work [K92, K99].

As we mentioned above, the growth of $|\mathcal{S}_{d}(\alpha)|$ is intimately
related to the behavior of the random process
$\sum_{0}^{d}\epsilon_{i}\alpha^{i}$ where the $\epsilon_{i}$ are Bernoulli
random variables equal to $0$ or $1$ with probability $\frac{1}{2}$. These
processes have been studied by Chung, Diaconis and Graham [CDG87] and
Hildebrand [Hil90]. Very recently they have been used to study irreducibility
of random polynomials in [BV18].

In the remainder of this paper we prove the above theorems. We first handle
Theorem 4 and then deduce Theorem 3.

##  1 Proof that $(b)$ implies $(a)$ in Theorem 4

We recall the following fairly well-known lemma (see Lemma 16 in [BV16] and §8
in [B07]).

######  Lemma 5.

Let $\alpha$ be an algebraic number. Then there is a constant $C_{\alpha}>0$
such that for every $d,K\geq 1$,

$|\mathcal{S}_{d}^{K}(\alpha)|\leq C_{\alpha}(Kd)^{C_{\alpha}}M(\alpha)^{d}.$

If Lehmer’s conjecture fails, then for each $\delta>0$ there is an algebraic
unit $\alpha$ such that $M(\alpha)<1+\delta^{2}/2$. By the lemma above we
conclude that if we choose $\epsilon$ in the interval
$(0,C_{\alpha}^{-1}\delta^{2}/4)$, then for all large enough $d$ we have:

$|\mathcal{S}^{2^{\epsilon d}}_{d}(\alpha)|\leq 2^{\delta^{2}d}.$

Let $F$ be the Galois closure of $\mathbb{Q}(\alpha)$. Let $p$ be a prime
number which splits completely in $F$ and let $\mathfrak{p}$ be a prime ideal
in $F$ above $p$. Denoting by $\bar{\alpha}$ the residue class of $\alpha$
modulo $\mathfrak{p}$ we must have:

$|\mathcal{S}^{2^{\epsilon d}}_{d}(\bar{\alpha})|\leq 2^{\delta^{2}d},$

for all sufficiently large $d\in\mathbb{N}$. In particular if
$d\leq\frac{1}{\delta}\log p$, we get

$|\mathcal{S}^{2^{\epsilon d}}_{d}(\bar{\alpha})|\leq p^{\delta}.$

In particular if the multiplicative order of $\bar{\alpha}$ is at least
$\sqrt{\frac{p}{\log p}}$, then $p$ is $(\delta,\epsilon)$-very bad.

On the other hand, the Frobenius-Chebotarev density theorem tells us that
there is a positive proportion of primes $p$ which split completely in $F$.
Therefore in order to complete the proof, it is enough to show that there is
only a vanishing proportion of primes $p$ for which $\bar{\alpha}$ has
multiplicative order at most $\sqrt{\frac{p}{\log p}}$.

######  Lemma 6.

Let $F|\mathbb{Q}$ be a finite Galois number field. Let $\mathcal{O}_{F}$ be
its ring of integers and let $\alpha\in\mathcal{O}_{F}\setminus\\{0\\}$. Let
$\mathcal{P}_{\alpha}$ be the set of primes $p\in\mathbb{N}$ such that there
is a prime ideal $\mathfrak{p}$ in $F$ above $p$ such that
$|\mathcal{O}_{F}/\mathfrak{p}|=p$ and $\alpha\mod\mathfrak{p}$ is non-zero
and of multiplicative order in $\mathbb{F}_{p}^{\times}$ at most
$\sqrt{\frac{p}{\log p}}$. Then as $X\to+\infty$,

$|\\{p\leq X;p\in\mathcal{P}_{\alpha}\\}|=o(|\\{p\leq X\\}|).$

###### Proof.

Note that if $\alpha^{n}-1\in\mathfrak{p}$, then
$p=N_{F|\mathbb{Q}}(\mathfrak{p})$ divides $N_{F|\mathbb{Q}}(\alpha^{n}-1)$.
In particular

$\prod_{p\in\mathcal{P}_{\alpha};\sqrt{X}\leq p\leq X}p\quad\textnormal{
divides }\quad\prod_{n\leq(X/\log X)^{1/2}}N_{F|\mathbb{Q}}(\alpha^{n}-1).$

On the other hand,

$|N_{F|\mathbb{Q}}(\alpha^{n}-1)|\leq\prod_{\sigma\in
Gal(F|\mathbb{Q})}(1+|\sigma(\alpha)|^{n})\leq\prod_{\sigma\in
Gal(F|\mathbb{Q})}2\max\\{1,|\sigma(\alpha)|\\}^{n}\leq
2^{[F:\mathbb{Q}]}M(\alpha)^{n}$

We consider this inequality for $n=1,\ldots,(X/\log X)^{1/2}$ and get

$|\mathcal{P}_{\alpha}\cap[\sqrt{X},X]|\log X\ll(\frac{X}{\log
X})^{1/2}[F:\mathbb{Q}]+\frac{X}{\log X}\log M(\alpha).$

The conclusion follows immediately given that $|\\{p\leq X\\}|\gg X/\log X$. ∎

######  Remark 7.

Clearly we can replace $(p/\log p)^{1/2}$ by $p^{1/2}\epsilon(p)$ for any
function $\epsilon(p)$ tending to $0$ as $p\to+\infty$ with the same proof.
Erdős and Murty improve this even further to $p\epsilon(p)$ in [EM99] in the
special case $F=\mathbb{Q}$ assuming the Riemann hypothesis for certain
Dedekind zeta functions.

##  2 Proof that $(a)$ implies $(c)$ in Theorem 4

The proof of this implication uses the harder part (the lower bound) in
Theorem 1.

Let $\mathcal{P}_{d}$ be the set of polynomials with degree at most $d$ and
coefficients in $\\{-1,0,1\\}$. Note that
$\mathcal{P}_{d}=\mathcal{S}_{d}-\mathcal{S}_{d}$. Let $\mathcal{I}_{d}$ be
the set of monic $\mathbb{Q}$-irreducible polynomials dividing at least one
non-zero polynomial in $\mathcal{P}_{d}$. We will say that a prime $p$ is _$d$
-exceptional_ if it divides the resultant $Res(D_{1},D_{2})$ of some pair of
distinct irreducible polynomials $D_{1},D_{2}$ in $\mathcal{I}_{d}$.

Recall that given a field $F$ every field element $\alpha\in F$ determines a
partition $\pi_{\alpha,d}$ of $\mathcal{S}_{d}$ made of the preimages of the
map $P\mapsto P(\alpha)$ from $\mathcal{S}_{d}$ to $F$. Clearly
$|\mathcal{S}_{d}(\alpha)|$ is the number of parts of $\pi_{\alpha,d}$.

_Claim 1_. If $p$ is not $d$-exceptional, then for every
$\alpha\in\mathbb{F}_{p}$ there is at most one $D\in\mathcal{I}_{d}$ such that
$D(\alpha)=0$ and moreover $\pi_{\alpha,d}=\pi_{\beta,d}$ if
$\beta\in\mathbb{Q}$ is a root of $D$.

###### Proof.

If $p$ is not $d$-exceptional, then the reductions mod $p$ of the polynomials
in $\mathcal{I}_{d}$ are pairwise relatively prime. In particular they cannot
have a common root in $\mathbb{F}_{p}$. So each $\alpha\in\mathbb{F}_{p}$ can
be the root of at most one $D\in\mathcal{I}_{d}$. If there is such a $D$, it
is unique, and for any $P,Q\in\mathcal{S}_{d}$ we have the equivalences :
$P(\alpha)=Q(\alpha)$ if and only if $D$ divides $P-Q$ and if and only if
$P(\beta)=Q(\beta)$, where $\beta$ is chosen to be a complex root of $D$. This
ends the proof. ∎

Note that if $\pi_{\alpha,d}=\pi_{\beta,d}$, then
$|\mathcal{S}_{n}(\alpha)|=|\mathcal{S}_{n}(\beta)|$ for all $n\leq d$.
Similarly for any $n\leq d$ we have $\alpha^{n}=1$ in $\mathbb{F}_{p}$ if and
only if $\beta^{n}=1$ in $\overline{\mathbb{Q}}$. Here we used that the $n$’th
cyclotomic polynomial is in $\mathcal{I}_{d}$ for $n\leq d$.

_Claim 2_. For $d$ large enough, there are at most $10^{d}$ $d$-exceptional
primes.

###### Proof.

If $D_{1},D_{2}$ are two polynomials in $\mathcal{I}_{d}$, their resultant can
be bounded above by Hadamard’s inequality:

$|Res(D_{1},D_{2})|\leq\|D_{1}\|_{2}^{\deg D_{2}}\|D_{2}\|_{2}^{\deg D_{1}},$

where $\|P\|_{2}$ denotes the $\ell^{2}$-norm of the coefficients of a
polynomial $P$. Recall that for every $P\in\mathbb{C}[X]$ we have
$\|P\|_{1}\leq 2^{\deg P}M(P)$. This is easily seen by expressing the
coefficients of $P$ as sums of products of roots of $P$. Recall also that
$M(P)\leq\|P\|_{1}$ as can be seen using the Jensen formula for $M(P)$. Here
$\|P\|_{1}$ denotes the $\ell^{1}$-norm of the coefficients of a polynomial
$P$. However if $D\in\mathcal{I}_{d}$, then $D$ divides some
$P\in\mathcal{P}_{d}$ and hence $M(D)\leq M(P)\leq\|P\|_{1}\leq d+1$, and thus

$\|D\|_{2}\leq\|D\|_{1}\leq 2^{d}(d+1).$

It follows that

$|Res(D_{1},D_{2})|\leq 2^{2d^{2}}(d+1)^{2d}\leq 2^{4d^{2}}.$

The number of distinct prime factors of $|Res(D_{1},D_{2})|$ is thus at most
$4d^{2}$ when $d$ is large enough. Since there are at most $d$ irreducible
factors of $P$ for any $P\in\mathcal{P}_{d}$, $|\mathcal{I}_{d}|\leq
d3^{d+1}$. Hence there can only be at most $4d^{2}|\mathcal{I}_{d}|^{2}\leq
4d^{4}9^{d+1}$ $d$-exceptional primes. ∎

We are now ready to conclude the proof of the implication $(c)\Rightarrow(a)$
in Theorem 4. We assume that Lehmer’s conjecture holds, and therefore that
there exists $\delta_{0}>0$ such that $M(\beta)>e^{\delta_{0}}$ for every
algebraic number $\beta$ that is not a root of unity. Let $\delta>0$ and
$X\geq 1$. Assume that $\delta<(c\delta_{0}/2)^{2}$, where $c$ is the constant
from Theorem 1. Let $d$ be the integer part of $\sqrt{\delta}\log X$.

_Claim 3._ Let $p\in[\sqrt{X},X]$ be a prime number. If $p$ is $\delta$-bad,
then $p$ is $d$-exceptional.

###### Proof.

If $p$ is $\delta$-bad, there is $\alpha\in\mathbb{F}_{p}\setminus\\{0\\}$ of
multiplicative order at least $\log p\log\log\log p$ such that
$|\mathcal{S}_{n}(\alpha)|\leq p^{\delta}$ for all $n\leq\log p$. In
particular

$|\mathcal{S}_{[\frac{1}{2}\log X]}(\alpha)|\leq X^{\delta}.$

On the other hand if $p$ is not $d$-exceptional, by Claim 1 above, there is
$\beta\in\overline{\mathbb{Q}}$ of degree at most $d$ such that
$|\mathcal{S}_{n}(\alpha)|=|\mathcal{S}_{n}(\beta)|$ for all $n\leq d$. In
particular if say $\delta<\frac{1}{4}$, we have $d\leq\frac{1}{2}\log X$ and
thus

$|\mathcal{S}_{d}(\beta)|\leq X^{\delta}\leq e^{\sqrt{\delta}(d+1)}\leq
e^{2\sqrt{\delta}d}.$

However according to Theorem 1,
$\max\\{2,M(\beta)\\}^{c}\leq|\mathcal{S}_{d}(\beta)|^{1/d}$ (this holds for
all $d$ by sub-multiplicativity of $d\mapsto|\mathcal{S}_{d}(\beta)|$). Hence

$M(\beta)\leq e^{\frac{2}{c}\sqrt{\delta}}.$

Since we assume the validity of the Lehmer conjecture, it follows that $\beta$
must be a root of unity. However $\beta$ is a root of a polynomial $P$ in
$\mathcal{P}_{d}$, hence has degree at most $d$ over $\mathbb{Q}$. Its minimal
polynomial must be a cyclotomic polynomial $\Phi_{m}$ dividing $P$ hence in
$\mathcal{I}_{d}$. But $p$ is not $d$-exceptional, so by Claim 1 we know that
$\Phi_{m}(\alpha)=0$.

It is well known that there is an absolute numerical constant $C>0$ such that
if $x$ is a root of unity in $\overline{\mathbb{Q}}$ and has degree at most
$d$ as an algebraic number over $\mathbb{Q}$, then the order of $x$ is at most
$Cd\log\log d$. Indeed, the degree of a root of unity of order $n$ is given by
the Euler totient function, which satisfies the lower estimate: $\phi(n)\gg
n/\log\log n$ (see [HW79, Thm 328]).

It follows that $\alpha$ has multiplicative order at most $m\leq
C_{0}d\log\log d$. As this is less than $\log p\log\log\log p$ if $X$ is large
enough (and $\delta_{0}$ chosen small enough), a contradiction. ∎

Now combining Claims 2 and 3 we get:

$|\\{p\in[\sqrt{X},X];p\textnormal{ is }\delta\textnormal{-bad}\\}|\leq
10^{d}\leq X^{3\sqrt{\delta}},$

for all $X\geq 1$, hence

$|\\{p\leq X;p\textnormal{ is }\delta\textnormal{-bad}\\}|\leq
X^{\delta}+\sum_{k\geq 0;2^{k}\leq
1/\delta}X^{3\sqrt{\delta}/2^{k}}\ll_{\delta}X^{3\sqrt{\delta}}.$

This concludes the proof of Theorem 4.

##  3 Proof of Theorem 3

The proof that $(b)$ implies $(a)$ is immediate from the corresponding
implication in Theorem 4. Indeed given $\epsilon\in(0,\frac{1}{2})$ and $C>1$
every large enough $(\frac{1}{C},\epsilon)$-very bad prime will be $C$-wild.

In the converse direction, in view of Theorem 4 it is enough to show that
assertion $(c)$ of Theorem 4 implies $(b)$ of Theorem 3. Let
$\epsilon=\frac{1}{2}$ and let $\delta$ be given by assertion $(c)$. It is
enough to find $C=C(\delta)>1$ such that every $C$-wild prime is $\delta$-bad.
We will show the contrapositive. So assume $p$ is not $\delta$-bad. Then for
every $x\in\mathbb{F}_{p}^{\times}$ with multiplicative order at least $(\log
p)^{2}$ we have

$|\mathcal{S}_{d}(x)|\geq p^{\delta}$

for some $d\leq\log p$. Now recall the sum-product theorem over
$\mathbb{F}_{p}$ :

######  Theorem 8 (Sum-product theorem, see 2.58 and 4.10 in [TV10]).

There is $\epsilon>0$ such that for all primes $p$ and all subsets
$A\subset\mathbb{F}_{p}$ at least one of three possibilities can occur: either
$AA+AA+AA=\mathbb{F}_{p}$, or $|A+A|>|A|^{1+\epsilon}$, or
$|AA|>|A|^{1+\epsilon}$.

Note further that
$\mathcal{S}_{d}^{K}(x)+\mathcal{S}_{d}^{K}(x)\subset\mathcal{S}_{d}^{2K}(x)$,
while
$\mathcal{S}_{d}^{K}(x).\mathcal{S}_{d}^{K}(x)\subset\mathcal{S}_{2d}^{dK^{2}}(x)$.
In particular, starting from $\mathcal{S}_{d}(x)$ and applying at most $n$
times either a sumset or a product set, we obtain a subset of
$\mathcal{S}_{2^{n}d}^{d^{2^{n}}}(x)$.

Applying the sum-product theorem, we see that after some $n=n(\delta)$ steps
$\mathcal{S}_{2^{n}[\log p]}^{[\log p]^{2^{n}}}(x)$ must be all of
$\mathbb{F}_{p}$. Setting $C=2^{n+1}$, we conclude (by $(\ref{sumset})$) that
$p$ is not $C$-wild, as desired. This concludes the proof of Theorem 3.

## Acknowledgments

PV wishes to thank Boris Bukh for interesting discussions on subjects related
to this note. In particular, PV first learned about a variant of the
implication $(b)\Rightarrow(a)$ in Theorem 3 from him.

We are very grateful to the anonymous referee for his or her careful reading
of our paper.

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[EB] Emmanuel Breuillard
DPMMS
University of Cambridge
United Kingdom
<EMAIL_ADDRESS>[PV] Péter P. Varjú
DPMMS
University of Cambridge
United Kingdom
<EMAIL_ADDRESS>