# Bit Error Rate Analysis for Reconfigurable Intelligent Surfaces with Phase
Errors

Imène Trigui, , Edouard Komi Agbogla, Mustapha Benjillali, ,
Wessam Ajib, , and Wei-Ping Zhu I. Trigui, E. K. Agbogla, and W. Ajib are with
the Département d’informatique, Université du Québec à Montréal, Montreal,
Canada. E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>Benjillali
is with INPT, Rabat, Morocco. Email<EMAIL_ADDRESS>Zhu is with the
Department of Electrical and Computer Engineering, Concordia University,
Montreal, Canada. E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we analyze the error probability of reconfigurable intelligent
surfaces (RIS)-enabled communication systems with quantized channel phase
compensation over Rayleigh fading channels. The probability density and
characteristic functions of the received signal amplitude are derived and used
to compute exact expressions for the bit error rate (BER). The resulting
expressions are general, as they hold for an arbitrary number of reflecting
elements $N$, and quantization levels, $L$. We introduce an exact asymptotic
analysis in the high signal-to-noise ratio (SNR) regime, from which we
demonstrate, in particular, that the diversity order is $N/2$ when $L=2$ and
$N$ when $L>2$. The theoretical frameworks and findings are validated with the
aid of Monte Carlo simulations.

###### Index Terms:

Reconfigurable intelligent surfaces, phase errors, BER analysis, Rayleigh
fading, diversity order.

##  I Introduction

Recently, a major design make-over of wireless communications enabled by
reconfigurable intelligent surfaces (RISs), has become an extremely active and
promising research topic [1, 2]. Thanks to their capability to easily
manipulate the incident wave proprieties (frequency, amplitude, phase) to
establish favourable channel responses [1, 2, 3] (and references therein), RIS
offer undoubtedly an additional degree of freedom to boost capacity and
enhance coverage with low cost and power-efficient infrastructure in 6G
networks. Nevertheless, several hurdles must be overcome to enable RIS-
assisted communications and make them work properly. One of the major
challenges facing the implementation of RIS is their vulnerability to phase
compensation errors at RIS elements [4]. In fact, while a suitable design of
the phase shifts of the reflecting elements is necessary to reap the
advantages of RIS-aided transmission [3, 4, 5, 6], it is, however, difficult
to implement these phases in practice due to hardware limitations. As a
consequence, phase quantization errors inevitably arise.

Motivated by these considerations, recent attempts for studying RIS-aided
systems in the presence of phase errors include the use of approximate
distributions and asymptotic analysis [7, 8, 10, 11, 9]. By using the central
limit theorem (CLT), the authors of [7] and [8] obtained approximate BER
expressions considering a large number of reconfigurable elements at the RIS.
It was recently shown, however, that the approximation error attributed to the
CLT can be significant for small number of elements with important
discrepancies in the high signal-to-noise ratio (SNR) regime [7, 9]. In [10],
the authors derived an approximate error performance of RIS in the presence of
phase errors by leveraging the moment-based Gamma approximation. The Gamma-
based framework appears, however, unsuitable for diversity analysis since it
fails to extract the full diversity order even in the absence of phase errors.
Recently, the authors of [9] and [11] identified sufficient conditions based
on upper and lower bounds for ensuring that RIS-assisted systems achieve the
full diversity order in the presence of phase noise. More specifically, it was
shown in [9] that if the absolute difference between pairs of phase errors is
less than $\pi/2$, RIS-assisted communications achieve full diversity.

Although the results from [7, 8, 10, 11, 9] are insightful, these works have
been successfully tractable due to approximate SNR distributions and bounds
yielding a diversity analysis in the presence of phase errors which is, so
far, steadily inaccurate [11, 9]. Moreover, to the best of our knowledge, no
exact error analysis for RIS with quantized phase shifts and arbitrary number
of reconfigurable elements has been reported in the literature.

In this letter, we investigate the impact of quantized phase shifts on the
performance of RIS-enabled communications. Analytical expressions for the bit
error rate (BER) of binary phase shift keying (BPSK) modulated signals over
Rayleigh fading channels, as a function of the number of phase quantization
levels $L$ (also known as phase resolution) and the number of reflecting
elements $N$, are derived based on exact expressions for the probability
density function (PDF) and the characteristic function (CHF) of the combined
signal amplitude. Approximate expressions for the BER are derived in the high-
SNR regime yielding simple closed-form expressions which can be used to
determine the coding gain, the diversity order, and the degradation in the
system performance. The obtained results unveil, in particular, that for two
level quantization, the diversity order is equal to half of the reconfigurable
elements number. For higher quantization levels, RIS-assisted communications
achieve full diversity order, despite the BER is not a linear function of the
SNR (in dB scale).

##  II System Model

We consider an RIS with $N$ reconfigurable elements which is deployed to
assist data transmission to a single antenna receiver by reflecting an
incident RF wave emitted by a single antenna transmitter. More specifically,
we assume that the direct transmission link between the transmitter and the
receiver is blocked, and, thus, the RIS is deployed to relay the scattered
signal and to leverage virtual line-of-sight (LOS) paths for enhancing the
strength of the received signal. The received signal of the considered system
is [3]

$r=\sum_{i=1}^{N}h_{i}g_{i}\rm{e}^{j(\theta_{i}-\theta_{i}^{{\cal
Q}})}\alpha+\widetilde{n},$ (1)

where $\alpha$ is the baseband transmitted symbol, $\widetilde{n}$ represents
a zero mean complex additive white Gaussian noise (AWGN) with variance
$N_{0}/2$, the channel amplitudes $h_{i}$ and $g_{i}$ are assumed to be
Rayleigh distributed with $\theta_{i}$ uniformly distributed between
$[-\pi,\pi]$, and $\theta_{i}^{{\cal Q}}$ is denoting the quantized phase
shift induced by the $i$-th reflecting element at the RIS. For an $L$-level
uniform quantizer the signal phase can be modeled as

$\theta_{i}^{{\cal Q}}=\theta_{i}-\theta_{i}^{{\cal E}},$ (2)

where $\theta_{i}^{{\cal E}}$ represents the phase error which is uniformly
distributed between $[-\pi/L,\pi/L]$.

In the rest of the work, we consider that $\alpha$ is chosen from a BPSK
signal constellation set $\alpha\in\\{-1,+1\\}$. Hence, only the real part of
the signal has an impact on the detection performance111Although the analysis
in this work is limited to BPSK modulation for tractability reasons, the
proposed method can be used to evaluate the BER of other modulation schemes as
well. For instance, for QPSK modulation, the term $v_{i}$ in (3) can be
replaced with $v_{i}=\sqrt{2}\cos(\theta_{i}^{\cal E}\pm\frac{\pi}{4})$ [13].
The detailed analysis of higher-order QAM modulations is more involved, and is
an important extension that we will address in future work.. This reduces (1)
to

$r=\sum_{i=1}^{N}h_{i}g_{i}v_{i}\alpha+{\rm Re}[\widetilde{n}],$ (3)

where $v_{i}\triangleq\cos(\theta_{i}^{{\cal E}})$, and $\rm{Re}[\cdot]$
denotes the real part of a complex quantity. The instantaneous SNR is defined
as

$\gamma=\rho\left(\sum_{i=1}^{N}h_{i}g_{i}v_{i}\right)^{2},$ (4)

with $\rho$ denoting the transmit SNR.

##  III Average BER With Phase Noise

The evaluation of the error probability usually involves the computation of
the PDF of $x=\sqrt{\gamma}$ which is formulated in terms of a linear
combination of the product of random variables. A common approach for
analyzing the distribution of $x$ is to leverage the CLT. However, this
approach is accurate only for a large number of reconfigurable elements [1, 7,
8] and the resulting analysis is usually not accurate in the high-SNR regime
and, therefore, for analyzing the attainable diversity order [9]. In general,
the calculation of the exact error probability of RIS-assisted communications
with phase noise is an open research issue, and is very intricate for
arbitrary values of $N$.

To tackle this issue, we proceed by writing the BER expression using the CHF-
based approach as [14]

${\cal P}_{e}=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\cal
G}(t)\phi_{x}^{*}(t)\textrm{d}t,$ (5)

where, under BPSK signalling, we have [14]

${\cal
G}(t)=\frac{1}{2t}\left(\frac{t}{\sqrt{\pi}}{}_{1}{\rm{F}_{1}}\left({1,\frac{3}{2};-\frac{t^{2}}{4}}\right)+j-j\rm{e}^{-\frac{t^{2}}{4}}\right),$
(6)

where $j=\sqrt{-1}$ and ${}_{1}{\rm{F}_{1}}(\cdot)$ represents the confluent
hypergeometric function[19]. Moreover, we denote $\phi_{x}^{*}(t)$ the complex
conjugate of the CHF of $x$ defined as

$\phi_{x}(t)=\prod_{i=1}^{N}\phi_{z_{i}}\left(\sqrt{\rho}t\right),$ (7)

where $z_{i}=h_{i}g_{i}v_{i}$, $i=1,\ldots,N$, and
$\phi_{z_{i}}\left(t\right)=\mathbb{E}_{z_{i}}\\{\rm{e}^{jtz_{i}}\\}$, with
$\mathbb{E}\\{\cdot\\}$ denoting the expectation operator. Thus, to evaluate
the error probability, we need first to determine the CHF of $z_{i}$, which is
a product of three random variables.

######  Proposition 1.

The PDF of $z_{i}$ can be expressed as

$f_{z_{i}}(z)=L{\rm{e}}^{-2z}-\frac{2L}{\pi}\int_{0}^{\pi/2}\rm{e}^{-2z\left(\sqrt{1+\frac{\tan\left(\pi/L\right)^{2}}{\sin(\psi)^{2}}}\right)}{\rm{d}}\psi.$
(8)

###### Proof.

As noted earlier, $\theta_{i}^{{\cal E}}$ is uniformly distributed on the
interval $[-\frac{\pi}{L},\frac{\pi}{L}]$. As a result, the PDF of $v_{i}$ is
given by

$f_{v_{i}}(v)=\frac{L}{\pi\sqrt{1-v^{2}}},\quad\text{for~{}}\cos(\pi/L)\leq
v\leq 1.$ (9)

Recall that under independent and identically distributed (i.i.d.) Rayleigh
fading $f_{y_{i}}(x)=2x{\rm{e}}^{-x^{2}}$, for $y\in\\{h,g\\}$, then we use
(9) to calculate the PDF of $z_{i}=h_{i}g_{i}v_{i}$ as

$f_{z_{i}}(z)=\int_{0}^{\infty}\frac{1}{x}f_{h_{i}}(x)f_{g_{i}v_{i}}\left(\frac{z}{x}\right)\textrm{d}x,$
(10)

where

$f_{g_{i}v_{i}}\left(x\right)=\frac{2L}{\pi}\int_{\cos({\pi/L})}^{1}\frac{\rm{e}^{-\frac{x^{2}}{z^{2}}}}{z^{2}\sqrt{1-z^{2}}}\textrm{d}z.$
(11)

By relabeling $z=\sin(\theta)$ in (11), and applying [19, Eq. (2.33.2)], we
obtain

$f_{g_{i}v_{i}}\left(x\right)=\frac{L}{\sqrt{\pi}}\rm{e}^{-x^{2}}{\rm
erf}\\!\left(x\tan\left(\frac{\pi}{L}\right)\right),$ (12)

where ${\rm erf}(\cdot)$ is the error function [19]. Substituting (12) into
(10) while resorting to the alternative form

${\rm
erf}(z)=1-\frac{2}{\pi}\int_{0}^{\pi/2}\rm{e}^{-\frac{z^{2}}{\sin(\psi)^{2}}}\textrm{d}\psi,$
(13)

we obtain the PDF in (8) with the aid of [19]. This completes the proof. ∎

######  Remark 1.

Based on Proposition 1, we evince that for 1-bit quantization based RIS
communications (i.e., when $L=2$), the PDF of $z_{i}$ reduces to

$f_{z_{i}}(z)=2\rm{e}^{-2z},\quad z\geq 0$ (14)

which is in agreement with [8, Eq. (17)].

Using (8), the CHF of $z_{i}$ follows from
$\phi_{z_{i}}(t)=\int_{0}^{\infty}f_{z_{i}}(z){\rm{e}}^{jtz}\textrm{d}z$ and
is obtained as

$\displaystyle\phi_{z_{i}}(t)=\frac{L}{jt+2}$
$\displaystyle\quad-\frac{2L}{\pi}\int_{0}^{\pi/2}\left(jt+2\left(\sqrt{1+\frac{\tan\left(\frac{\pi}{L}\right)^{2}}{\sin(\psi)^{2}}}\right)\right)^{-1}\textrm{d}\psi.$
(15)

Note that the CHF given in (15) can be efficiently estimated using the Gauss-
Chebyshev quadrature (GCQ) [20, Eq.(25.4.38)] as

$\displaystyle\\!\\!\\!\\!\\!\phi_{z_{i}}(t)\\!\\!\\!$ $\displaystyle\approx$
$\displaystyle\\!\\!\\!\frac{L}{jt+2}\\!-\\!\frac{L\pi}{2n}\sum_{k=0}^{n}\frac{\sqrt{1-a^{2}_{k}}}{jt+2\sqrt{1\\!+\\!\frac{\tan\left(\frac{\pi}{L}\right)^{2}}{\sin\left(\frac{\pi}{4}(a_{k}+1)\right)^{2}}}},$
(16)

where $a_{k}=\cos\left(\frac{\pi}{2n}(2k-1)\right)$. It is important to note
that the accuracy of the GCQ rule is extremely high; and a relative accuracy
of $10^{-15}$ is possible for all SNRs and RIS configurations (i.e., $N$,
$L$). Thus using (16) and (7) together with (5) can be considered as a
replacement for the closed-form solution to the BER. In particular, using GCQ
leads to significant computational advantages as compared to the case when (5)
is used with (15).

##  IV High-SNR BER-Diversity Analysis

In this section, we will analyze the communication robustness that can be
achieved with $L$-level quantization-based RIS by focusing on the BER at high
SNR. The objective is to quantify the diversity order and identify sufficient
conditions for achieving it. This is a fundamental open issue for designing
and optimizing RIS-aided systems. So far, by relying on the CLT, it was shown
in [15], for example, that the diversity order is
$\frac{N}{2}\frac{\pi^{2}}{16-\pi^{2}}$ in Rayleigh fading, which implies that
the full diversity order cannot be obtained even in the absence of phase
errors. By resorting to some bounds, however, the authors of [9] recently
showed that the full diversity order is achievable in Rayleigh fading if the
absolute difference between pairs of phase errors is less than $\pi/2$.

In what follows, building upon the exact high-SNR analysis of BER for
arbitrary $N$, we compute the exact diversity order and coding gain of RIS-
assisted systems. In order to evaluate the BER at high SNR, according to [16],
we are interested in analyzing the behavior of the PDF of $x$ around the
origin. Using (11) and resorting to the Mellin-Barnes integral representation
of the exponential and error functions [17], it follows that

$\displaystyle f_{z_{i}}(z)=\frac{4L}{\pi(2\pi j)^{2}}\int_{{\cal
C}_{1}}\int_{{\cal
C}_{2}}\frac{\Gamma(s_{1})\Gamma(s_{2}+\frac{1}{2})\Gamma(-s_{2})}{\Gamma(1-s_{2})}$
$\displaystyle\quad\int_{0}^{\infty}{\rm{e}}^{-x^{2}}\left(\frac{z}{x}\right)^{-2s_{1}}\left(\frac{z\tan\left(\frac{\pi}{L}\right)}{x}\right)^{-2s_{2}}\textrm{d}x\textrm{d}s_{1}\textrm{d}s_{2}.$
(17)

where ${\cal C}_{1}$ and ${\cal C}_{2}$ are two integral contours in the
complex domain. Then, with the help of
$\int_{0}^{\infty}x^{2b}\rm{e}^{-x^{2}}\textrm{d}x=\Gamma\left(\frac{1}{2}+b\right)/2$
and applying [17, Eq. (2.1)], we obtain

$\displaystyle f_{z_{i}}(z)$ $\displaystyle\overset{(a)}{=}\frac{2L}{\pi}$
$\displaystyle\hskip-22.76228pt\times{\rm
H}_{0,1:0,1;1,1}^{0,0:1,0;1,2}\left[\\!\\!\begin{array}[]{ccc}z^{2}\\\
z^{2}\tan\left(\frac{\pi}{L}\right)^{2}\end{array}\\!\\!\Bigg{|}\\!\\!\begin{array}[]{ccc}-:-;(1,1)\\\
(\frac{1}{2};1,1)\\!:\\!\left(0,1\right);\left(\frac{1}{2},1\right),(0,1)\end{array}\Bigg{.}\\!\\!\right]$
(22) $\displaystyle\underset{z\rightarrow
0}{\overset{(b)}{\approx}}\frac{4L}{\pi}z\ln\left(z\tan\left(\frac{\pi}{L}\right)\right)\tan\left(\frac{\pi}{L}\right),$
(23)

where $(a)$ follows from applying [17, Eq. (2.57)], and $H[\cdot|\cdot]$ is
the bivariate Fox’s H-function [17, Eq. (2.56)]. When $z\rightarrow 0$, $(b)$
follows by computing residues at left poles of the two corresponding
integrands [18, Theorem 1.3].
The Laplace Transform of the approximated $f_{z_{i}}(z)$ (i.e., when $z\to 0$)
is now given by

${\cal
L}_{z_{i}}(s)=\frac{2L\tan\left(\frac{\pi}{L}\right)}{\pi}\frac{\ln\left(s\right)}{s^{2}}.$
(24)

Accordingly, the Laplace transform of the approximated PDF of
$x=\sqrt{\rho}\sum_{i=1}^{N}z_{i}$ can be formulated as

${\cal
L}_{x}(s)=\left(\frac{2L\tan\left(\frac{\pi}{L}\right)}{\pi}\frac{\ln\left(\sqrt{\rho}s\right)}{\rho
s^{2}}\right)^{N}.$ (25)

The PDF of $x$ requires the computation of the inverse Laplace transform of
(25) which is involved due to the term $\ln(\sqrt{\rho}s)^{N}$. To proceed
with the analysis, we use the following identity

$\displaystyle\ln(\sqrt{\rho}s)^{N}=$ $\displaystyle\frac{1}{(2\pi
j)^{N}}\int_{{\cal C}_{1}}\ldots\int_{{\cal
C}_{N}}\frac{\prod_{i=1}^{N}\Gamma(1+t_{i})\Gamma(-t_{i})^{2}}{\prod_{i=1}^{N}\Gamma(1-t_{i})}$
$\displaystyle\quad\left(\sqrt{\rho}s-1\right)^{-\sum_{i=1}^{N}t_{i}}\textrm{d}t_{1}\ldots\textrm{d}t_{N}.$
(26)

Then we plug the above equality into (25) and we compute the inverse Laplace
transform with respect to $s$ by using [17, Eq. (2.21)] as

$\displaystyle\\!\\!\\!{\cal
L}^{-1}\left\\{\left(1\\!-as^{-1}\right)^{-c}s^{-c-2N},x\right\\}$
$\displaystyle\underset{x\rightarrow 0}{\overset{(b)}{=}}$
$\displaystyle\\!\\!\\!\frac{a^{-c}x^{c+2N-1}}{\Gamma(2N+c)},$ (27)

where $(b)$ holds since we are interested in values around zero. Hence, the
PDF of $x$ around zero is approximated using (25), (26) and (27), and can be
formulated as

$\displaystyle f_{x}(x)$
$\displaystyle=\frac{\left(\frac{2L\tan\left(\frac{\pi}{L}\right)}{\pi}\right)^{N}x^{2N-1}}{(2\pi
j)^{N}}$ $\displaystyle\times\int_{{\cal C}_{1}}\ldots\int_{{\cal
C}_{N}}\frac{\prod_{i=1}^{N}\Gamma(1+t_{i})\Gamma(-t_{i})^{2}}{\Gamma(2N+\sum_{i=1}^{N}t_{i})\prod_{i=1}^{N}\Gamma(1-t_{i})}$
$\displaystyle\times
x^{-\sum_{i=1}^{N}t_{i}}\textrm{d}t_{1}\ldots\textrm{d}t_{N}.$ (28)

which can be expressed in terms of the multivariate Fox’s H-function [17, A.1]
whose details are not provided due to space limitation. When $x\rightarrow 0$,
we apply the asymptotic expansion of the Mellin-Barnes integrals in (28) at
the double poles $t_{i}=0$ using [18, Eqs. (1.8.14), (1.4.6)], thereby
yielding

$f_{x}(x)\underset{x\rightarrow
0}{\approx}\left(\frac{2L\tan\left(\frac{\pi}{L}\right)}{\rho\pi}\right)^{N}\frac{x^{2N-1}\ln(\rho
x)^{N}}{\Gamma(2N)}.$ (29)

Using (29), we can now formulate the asymptotic BER as [14]

${\cal
P}_{\text{e}}=\frac{\left(\frac{2L\tan\left(\pi\right)}{\rho\pi}\right)^{N}}{2\Gamma(2N)}\int_{0}^{\infty}{\rm
erfc}(\sqrt{x})x^{N-1}\ln(\rho\sqrt{x})^{N}\textrm{d}x,$ (30)

where ${\rm erfc}(x)$ is the complementary error function. By rewriting the
integrands in (30) as Fox’s H-functions using [17, Eq. (1.43)] and applying
[17, Eq. (2.3)], we get (33) at the top of the page, where ${\rm H}[\cdot]$
stands for the Fox’s H-function [17, Eq. (1.2)]. The asymptotic expression of
the BER when $\rho\to\infty$ can be obtained by representing the expression in
(33) as a multiple Mellin-Barnes type integral and then computing the residues
of the integrands using [18, Eq. (1.5.12)].

$\displaystyle{\cal P}_{\text{e}}$ $\displaystyle=$
$\displaystyle\frac{\left(\frac{2L\tan\left(\frac{\pi}{L}\right)}{\rho\pi}\right)^{N}}{2\sqrt{\pi}\Gamma(2N)}\frac{1}{(2\pi
j)^{N}}\int_{{\cal C}_{1}}\ldots\int_{{\cal
C}_{N}}\\!\\!\\!\frac{\prod_{i=1}^{N}\Gamma(1+t_{i})\Gamma(-t_{i})^{2}}{\prod_{i=1}^{N}\Gamma(1-t_{i})}$
(33)
$\displaystyle\times~{}\frac{(-1)^{-\sum_{i=1}^{N}t_{i}}}{\Gamma(\sum_{i=1}^{N}t_{i})}H_{3,2}^{1,3}\Bigg{[}-\rho\left|\begin{array}[]{ccc}(1-N,\frac{1}{2}),(\frac{1}{2}-N,\frac{1}{2}),(1-\sum_{i=1}^{k}t_{i},1)\\\
(0,1),(\frac{1}{2},1)(-N,1)\end{array}\right.\Bigg{]}\textrm{d}t_{1}\ldots\textrm{d}t_{N}.$

Hence from (33), we can finally compute the asymptotic BER as

${\cal
P}_{\text{e}}=\left(\frac{2L\tan\left(\frac{\pi}{L}\right)}{\pi}\right)^{N}\frac{\ln(\rho)^{N}\Gamma(N+\frac{1}{2})}{2\sqrt{\pi}(N+1)\Gamma(2N)}\rho^{-N},$
(34)

from which the following important conclusions and performance trends are
unveiled.

  * •

Due to the presence of $\tan(\frac{\pi}{L})$, (34) holds for $L>2$.

  * •

(34) unveils a new scaling law of the BER which is $\ln(\rho)/\rho$ as
$\rho\to\infty$. The BER is not a linear function with respect to $\rho$ in
dB, while the slope of BER changes very slowly at high SNR. This new scaling
law generalizes the definitions of diversity order and coding gain typically
used in wireless communications [16].

  * •

(34) is, to the best of our knowledge, the first in the literature that yields
the exact asymptotic BER for arbitrary $N$. This is in contrast with the
recently reported expressions in [1, Eqs. (4), (7)], [7, Eq. (29)] and [10,
Eq. (39)], which are based on approximations (the CLT in [1], [7] and the
moment-based Gamma approximation in [10]).

In order to better quantify the diversity order of RIS-assisted systems in the
presence of quantized phase noise, it is important to study the case when
$L=2$. To this end, the asymptotic PDF of the normalized SNR $\gamma/\rho$
follows by resorting to the fact that ${\cal
L}_{\frac{x}{\sqrt{\rho}}}(s)=\left(2{\cal
L}\\{\rm{e}^{-2z},s\\}\right)^{N}=\left(\frac{s}{2}+1\right)^{-N}$, and hence

$f_{\frac{\gamma}{\rho}}(\gamma)\approx\frac{2^{\frac{N}{2}-1}\gamma^{\frac{N}{2}-1}}{\Gamma(N)}\rm{e}^{-2\sqrt{\gamma}},$
(35)

which implies, according to [16], that the asymptotic error probability is

${\cal
P}_{e}\underset{\rho\to\infty}{\approx}\frac{2^{N-1}\Gamma\left(\frac{N}{2}+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(N+1)}\rho^{-\frac{N}{2}}.$
(36)

Given (34) and (36), the diversity and coding gains of RIS-assisted
communications with BPSK signalling and quantized phase noise are obtained, as
stated in the following proposition.

######  Proposition 2.

The diversity order and coding gains of RIS-assisted communications in the
presence of phase noise are

${\cal G}_{d}=\left\\{\begin{array}[]{ll}N/2,&\hbox{ $L=2$;}\\\
N,&\hbox{$L>2$.}\end{array}\right.$ (37)

and

${\cal
G}_{c}=\begin{cases}\\!\\!\left(\dfrac{2^{N-1}\Gamma(\frac{N}{2}+\frac{1}{2})}{\sqrt{\pi}\Gamma(N+1)}\right)^{\\!\\!-2/N},&L=2;\\\
\\!\\!\left(\\!\\!\left(\dfrac{2L\tan\left(\frac{\pi}{L}\right)}{\pi}\\!\\!\right)^{\\!N}\\!\\!\dfrac{\ln(\rho)^{N}\Gamma(N+\frac{1}{2})}{2\sqrt{\pi}(N+1)\Gamma(2N)}\\!\\!\right)^{\\!\\!-1/N},&L>2.\end{cases}$
(38)

The diversity analysis above helps to discover the effects of an
$L$-level222Considering the fact that energy consumption at RIS increases
exponentially with the number of levels $L$ (also known as quantizer
resolution)[21], low-resolution quantizers, i.e., with small $L$ values, could
provide significant energy savings, notably, when the number of elements $N$
at RIS is large. quantizer based RIS on the BER system performance. In
particular, we conclude that

  * •

For RIS-aided systems with BPSK signalling over Rayleigh fading, full
diversity order can be ensured if at least three quantization levels (i.e.
$L>2$) are used. This result is in agreement with [11, 9] where it was proved
by using upper and lower bounds.

  * •

For $L=2$ the diversity order is only $N/2$. This is consistent with [11],
where it is proved that the diversity order cannot exceeds $(N+1)/2$ for
$L=2$. To the best of the authors knowledge, this difference in the diversity
gain between the scenarios $L=2$ and $L>2$ was never reported in the
literature.

  * •

The increase in the average BER as a quantization penalty is defined as

$\displaystyle\Psi(\rho,L)=10\log\\!\left(\frac{{\cal P}_{e}}{{\cal
P}^{\infty}_{e}}\right)$
$\displaystyle=\begin{cases}10\log\\!\left(\\!\dfrac{\rho^{\frac{N}{2}}}{\ln(\rho)^{N}}\dfrac{\Gamma(\frac{N}{2}+\frac{1}{2})(N+1)2^{2N-1}}{N\sqrt{\pi}}\\!\right)\\!,&\\!\\!\\!L=2;\\\
10\log\\!\left(\\!\dfrac{L\tan(\pi/L)}{\pi}\\!\right)\\!,&\\!\\!\\!L>2,\end{cases}$
(39)

where ${\cal P}^{\infty}_{e}$ is the average SEP with infinite number of
quantization bits obtained form (34) by using the small-angle approximation
$\tan(x)\approx x$ as $x\to 0$.

##  V Numerical Results

In this section, we present analytical and simulated BER results for RIS-
assisted communications using an $L$-level quantized phase shifts. The CGQ-
based CHF in (16) is used to numerically evaluate the performance of BPSK
signalling over Rayleigh fading channels.

Fig. 1 shows the average BER as a function of the average transmit SNR (i.e.,
$\rho$) with $L=2,3,4$ and $N=5$. We observe that the analytical expression of
the BER using (16) and its corresponding high-SNR approximation in (34) and
(36) are in close agreement with Monte Carlo simulations. In particular, with
GCQ, a high level of accuracy is achieved with only $n=20$. We observe a
noteworthy improvement in the average BER when $L$ changes from 2 to 3. This
increase in the average BER performance is expected in the light of
Proposition 2, which states that the full diversity order can be achieved as
long as $L\geq 3$. When $L=2$, however, the quantization errors induce a
diversity gain loss.

Figure 1:  BER versus average SNR for $N=5$.

Fig. 2 shows a comparison between the exact and approximate BER expressions
for several values of $N$ and $L$. We observe, in particular, that the BER
scales with
${{\ln\left(\rho\right)}\mathord{\left/{\vphantom{{\ln\left(\rho\right)}\rho}}\right.\kern-1.2pt}\rho}$,
as predicted by (33) and unveiled in [9] over Rayleigh fading channels and in
the absence of phase noise. As expected, the BER decreases significantly as
the number $N$ of reconfigurable elements of the RIS increases.

Figure 2: BER versus average SNR, for different values of $L$ and $N$.

##  VI Conclusion

This letter investigates the effect of low resolution quantization based-RIS
on the BER system performance. By conducting asymptotic analysis, this letter
concisely unveils the diversity order, coding gain and the degradation in
system performance with respect to quantization levels. It was shown, in
particular, that for very low resolution quantizers (i.e. $L=2$), RIS-assisted
communications achieve a diversity order equal to half of the reconfigurable
elements number. For higher values of $L$, however, a full diversity is
extracted, despite the BER slowly changing slope in high SNRs.

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Dualities of Adjoint QCD3 from Branes

Adi Armoni

Department of Physics, Faculty of Science and Engineering
Swansea University, SA2 8PP, UK

<EMAIL_ADDRESS>

Abstract

We consider an ’electric’ $U(N)$ level $k$ QCD3 theory with one adjoint
Majorana fermion. Inspired by brane dynamics, we suggest that for $k\geq N/2$
the massive $m<0$ theory, in the vicinity of the supersymmetric point, admits
a $U(k-\frac{N}{2})_{-(\frac{1}{2}k+\frac{3}{4}N),-(k+\frac{N}{2})}$
’magnetic’ dual with one adjoint Majorana fermion. The magnetic theory flows
in the IR to a topological $U(k-\frac{N}{2})_{-N,-(k+\frac{N}{2})}$ pure
Chern-Simons theory in agreement with the dynamics of the electric theory.
When $k<N/2$ the magnetic dual is
$U(\frac{N}{2}-k)_{\frac{1}{2}k+\frac{3}{4}N,N}$ with one adjoint Majorana
fermion. Depending on the sign of the fermion mass, the magnetic theory flows
to either $U(\frac{N}{2}-k)_{N,N}$ or $U(\frac{N}{2}-k)_{\frac{1}{2}N+k,N}$
TQFT. A second magnetic theory, $U(N/2+k)_{\frac{1}{2}k-\frac{3}{4}N,-N}$,
flows to either $U(\frac{N}{2}+k)_{-N,-N}$ or
$U(\frac{N}{2}+k)_{-(\frac{1}{2}N-k),-N}$ TQFT. Dualities for $SO$ and $USp$
theories with one adjoint fermion are also discussed.

Contents

toc

##  1 Introduction

Quantum chromodynamics in three spacetime dimensions (QCD3) attracted in
recent years a lot of attention (see [1] and references within). The vacuum
structure of the theory is rich as it depends on the number of colours, the
number of flavours, the Chern-Simons (CS) level, the representation of the
quarks and their masses.

One particular interesting variant of the theory is QCD with a single Majorana
adjoint fermion. When the mass is large it can be integrated out, resulting in
a shift of the Chern-Simons level as follows

$k\rightarrow k+{\rm sgn}(m)\frac{1}{2}h(G)\,,$ (1.1)

where $h(G)$ is the dual Coexter number of the group. Due to the level shift
it follows that the theory admits at least one phases transition as the mass
of the fermion is varied. The phases of the theory were recently studied in
[2], where it was argued that for $k\geq\frac{1}{2}h(G)$ the theory admits two
phases, while surprisingly for $k<\frac{1}{2}h(G)$ the theory admits three
phases: apart from the large positive mass phase and the large negative mass
phase, it admits an intermediate quantum phase. The physics in the vicinity of
the two phase transitions is described by two dual theories. Follow-up papers
which are relevant to the current work include the generalisations to two
adjoint flavours [3] and to other two-index representations [4].

One of the striking results of [2] is that $SU(N)_{0}$111Throughout this paper
we use the notation $G_{k}$ for a level $k$ Chern-Simons theory based on gauge
group $G$. with one adjoint fermion is not a confining theory. Instead, the IR
of the theory is described by $U(\frac{N}{2})_{\frac{N}{2},N}$ TQFT. As a
result the Wilson loop in the fundamental representation shows a perimeter
law. A convincing argument in favour of this result is anomaly matching: the
UV theory cannot flow to a trivial theory in order to match the anomalous
$1$-form symmetry, see [5] for a recent discussion.

In this note we use branes to re-derive the dualities of [2] as well as new
dualities. We will use a 3d version of Seiberg duality, closely related to the
Giveon-Kutasov duality [6]. The electric theory is $U(N)_{k}$222Here we
specify the level of the $SU(N)$ part. The $U(1)$ level is rather subtle and
will be discussed later. (or $USp(2N)_{2k}$, $SO(2N)_{2k}$, $SO(2N+1)_{2k}$)
with one adjoint fermion. The Seiberg dual for $k\geq\frac{1}{2}h(G)$ is
supersymmetric for a certain value of the fermion mass. The duality is valid
in the vicinity of this value. As we shall see both electric and magnetic
theories admit the same (Witten) index. Moreover, they both flow to the same
IR theory.

When $k<\frac{1}{2}h(G)$ (in analogy with $N_{f}<N_{c}$ in 4d SQCD)
supersymmetry is broken. Yet, we are able to recover two magnetic duals whose
matter content was conjectured in [2].

Therefore, string theory beautifully recovers known dualities for
$k<\frac{1}{2}h(G)$ and, moreover, predicts new dualities for
$k\geq\frac{1}{2}h(G)$ that can be verified by using several consistency
checks such as anomaly matching, agreement of an index and the coincidence of
the low energy dynamics at the weak coupling regime of large fermion mass.

Throughout the paper we use $h(U(N))=N$, $h(USp(2N))=2N+2$, $h(SO(2N))=2N-2$.
We use the convention $USp(2)\sim SU(2)$.

##  2 Brane dynamics

We derive the gauge theory dualities from a duality in string theory. Our
setup is similar to that of refs.[7] and [8]. We consider ’electric’ and
’magnetic’ branes configurations as depicted in figure (1). The configurations
consist of $N$ (or $\tilde{N}$) D3 branes suspended between NS fivebrane and a
tilted $(1,k^{\prime})$ fivebrane. The D3 branes span the $012$ directions and
a finite segment of the $6$ direction. The NS fivebrane spans the $012345$
directions. The $(1,k^{\prime})$ fivebrane spans the $01238$ directions and it
is tilted in the $(59)$ plane with an angle $-\frac{\pi}{2}-\theta$. In order
to preserve supersymmetry we choose $\tan\theta=g_{s}k^{\prime}$. The content
of the theory admits 3d ${\cal N}=2$ supersymmetry and it can be obtained by
dimensional reduction of 4d ${\cal N}=1$ SYM. The CS interaction reduces the
supersymmetrey to 3d ${\cal N}=1$. The Lagrangian of the theory may be written
in terms of ${\cal N}=1$ supersymmetry as a level $k^{\prime}$ vector
multiplet $V=(A_{\mu},\chi)$ coupled to an adjoint scalar multiplet
$\Phi=(\phi,\psi)$. The action is given by333The same brane configuration was
used in [9] to construct the Acharya-Vafa theory [10] of ${\cal N}=1$ SYM
domain walls.

$\displaystyle S=\frac{1}{4g^{2}}\int\text{d}^{3}x~{}{\rm
tr}\left((D\phi)^{2}-F^{2}+i\bar{\chi}\displaystyle{\not}D\chi+i\bar{\psi}\displaystyle{\not}D\psi+2i\bar{\chi}[\phi,\psi]\right)+$
(2.1) $\displaystyle+\frac{k^{\prime}}{4\pi}\int{\rm
tr}\left(A\wedge\text{d}A+\tfrac{2}{3}A\wedge A\wedge
A\right)-\frac{k^{\prime}}{4\pi}\int\text{d}^{3}x~{}\bar{\chi}\chi~{}.$

Let us begin with the electric theory. In order to obtain a low energy theory
with a gauge group $U(N)$ and a light adjoint fermion $\chi$, we choose
$k^{\prime}=k+N/2$. We then give a large negative mass to $\psi$. A mass term
for the fermion that lives on the branes can be achieved by tilting the
$(1,k^{\prime})$ fivebrane further, or by turning on background fluxes [11].
In particular in ref.[11] the authors discussed in detail how a 3-form
background flux in type IIB can be used to give mass to matter that lives on
D3 branes.

The level of the $SU(N)$ part is shifted by $-N/2$ due to the integration over
the massive adjoint fermion, $\psi$.444It is also possible to begin with
$-k^{\prime}=-k+N/2$ and to give a positive mass to $\psi$. When we do so, the
resulting dual magnetic theory covers a different range of masses of the
electric theory. We will later refer to that theory as magnetic’. Note that
the $U(1)$ level is not shifted because the adjoint fermion does not carry a
$U(1)$ charge. The resulting electric theory is a 3d ${\cal N}=1$
$U(N)_{k,k^{\prime}}$ YM-CS theory.

We follow the prescription of Elitzur-Giveon-Kutasov [12] and swap the
fivebranes in order to obtain the Seiberg dual, see fig.(1). We start with
$k\geq N/2$ in order to preserve supersymmetry. The resulting magnetic dual is
a ${\cal N}=1$ supersymmetric $U(\tilde{N})_{-\tilde{k},-k^{\prime}}$ YM-CS
theory555The levels of the magnetic theory are negative because the
fundamental string ends on the left fivebrane, as opposed to the right
fivebrane in the electric theory case. It is as if we integrated out an
infinitely heavy massive flavour, with a negative mass.. As we swap the branes
the number of D3 branes in the magnetic side becomes (similar to $N_{f}-N_{c}$
D4 branes in 4d SQCD [12])

$\tilde{N}=k^{\prime}-N=k-\frac{1}{2}N\,.$ (2.2)

The level of the $SU(\tilde{N})$ part in the magnetic theory, $\tilde{k}$, is
given by $k^{\prime}$ shifted due to the massive fermion in the scalar
multiplet

$\tilde{k}=k^{\prime}-\frac{1}{2}\tilde{N}=k+\frac{1}{2}N-\frac{1}{2}(k-\frac{1}{2}N)=\frac{1}{2}k+\frac{3}{4}N\,.$
(2.3)

We therefore propose the following Seiberg duality between two ${\cal N}=1$
supersymmetric YM-CS theories:

$U(N)_{k,k+\frac{N}{2}}\iff
U(k-N/2)_{-(\frac{1}{2}k+\frac{3}{4}N),-(k+\frac{N}{2})}\,.$ (2.4)

The above duality (2.4) is the main result of this note.

The first check of the duality (2.4) is the Witten index. The electric ${\cal
N}=1$ $U(N)_{k,k+\frac{N}{2}}$ theory as well as the
$U(k-N/2)_{-(\frac{1}{2}k+\frac{3}{4}N),-(k+\frac{N}{2})}$ admit the same
Witten index [13, 14, 15, 16]

$I_{W}=\frac{(k+\tfrac{1}{2}N)!}{(k-\tfrac{1}{2}N)!N!}\,.$ (2.5)

The Witten index can be derived from the brane configuration by using the
s-rule and counting all possible ways the $N$ (or $\tilde{N}$) D3 branes can
end on the $k+N/2$ constituent fivebranes [18, 17]. Both the electric and
magnetic descriptions lead to (2.5), in agreement with field theory.

Fig. 1: Seiberg Duality from branes. The electric theory (left) is a
$U(N)_{k^{\prime},k^{\prime}}$ gauge theory with one massive $m^{2}\gg g^{2}$
adjoint fermion and an additional light adjoint fermion. Similarly, the
magnetic theory (right) is $U(\tilde{N})_{-k^{\prime},-k^{\prime}}$ gauge
theory with one massive $m^{2}\gg g^{2}$ adjoint fermion and an additional
light adjoint fermion. $k^{\prime}=k+N/2$.

Let us briefly discuss the case with $k<N/2$. When we swap the fivebranes we
obtain $\tilde{N}=k-\frac{1}{2}N$ anti D3 branes suspended between the
fivebranes. The brane configuration breaks supersymmetry, in agreement with
the expectation from the gauge dynamics of the electric theory when $k<N/2$.
Typically a non-supersymmetric brane configuration is unstable and cannot be
trusted, however in the current configuration the D3 branes are locked between
the fivebranes and therefore the 3d field theory does not suffer from an
instability. An analogoues scenario in 4d SQCD leads to $N_{c}-N_{f}$
antibranes suspended between orthogonal NS fivebranes, however the situation
in 4d is more complex because there are mesons and thus one may explain the
Affleck-Dine-Seiberg runaway superpotential by a potential between
branes.666This argument is due to Shigeki Sugimoto. The 3d field theory that
lives on the branes is a non-supersymmetric
$U(N/2-k)_{\frac{1}{2}k+\frac{3}{4}N}$ YM-CS theory with a Majorana adjoint
fermion. Supersymmetry breaking occurs due to the interaction between the anti
D3 branes and the fivebranes. Our results in section (4) for $k<N/2$ are in
full agreement with the dualities proposed in [2] and reveals their string
theory origin.

Our picture and the conjectured dualities are supported by the gravity dual,
proposed in [19]. The dual of $SU(N)_{k}$ with ${\cal N}=1$ supersymmetry is
described by a confining geometry when $k=N/2$. When $k$ is increased
additional branes are added to the geometry in a SUSY preserving manner. When
$k$ is decreased antibranes are added and supersymmetry is broken. The
emerging picture from our setup is essentially the same. The advantage of the
current dualities over the holographic dual is that it applies to finite $N$
as well as for a wide range of masses, hence it is useful for exploring phase
transitions.

In the following sections we will study the field theory dynamics of the
electric and magnetic theories in more detail.

##  3 Gauge dynamics when $k\geq N/2$

The electric $U(N)_{k,k+N/2}$ theory is supersymmetric when the bare fermion
mass is $m=-g^{2}k$. It is conjectured that the theory admits two phases. The
low energy dynamics is characterised by two TQFT’s. At large positive mass it
is a $U(N)_{k+N/2,k+N/2}$ TQFT and at large negative mass it is a
$U(N)_{k-N/2,k+N/2}$ TQFT.

Fig. 2: The phase diagram of adjoint QCD with $k\geq N/2$. There are two
phases, separated by a phase transition denoted by the blue dot. The
supersymmetric point lies at $m=-g^{2}k$. The dual gauge theory (shaded
ellipse) describes the phase with negative mass in the vicinity of the
supersymmetric point. When $k=N/2$ the $SU(N)$ IR theory is trivial (gapped).

The proposed magnetic dual describes the supersymmetric point and its
vicinity. Starting with $U(k-N/2)_{-(\frac{1}{2}k+\frac{3}{4}N),-(k+N/2)}$ let
us integrate out the massive adjont fermion. The resulting theory is
$U(k-N/2)_{-N,-(k+N/2)}$ which is dual to $U(N)_{k-N/2,k+N/2}$ [20]. We
obtained the IR TQFT in the vicinity of the supersymmetric electric theory by
using the dual magnetic theory, see fig.(2). In addition, at the
supersymmetric point there is an additional adjoint fermion, as expected from
the electric side. As we already discussed, the Witten index of the dual pair
matches.

At $k=N/2$ the dual magnetic theory is trivial, apart from a free massless
$U(1)$. This is not surprising: the $SU(N)$ part of the electric theory is
expected to confine and hence the IR is gapped, except a free $U(1)$.

##  4 Gauge dynamics when $k<N/2$

When $k<N/2$ the electric theory admits three phases. In addition to the large
positive mass and large negative masses, there is an intermediate quantum
phase characterised by a TQFT [2], see fig.(3)

$U(N/2-k)_{N/2+k,N}\sim U(N/2+k)_{-N/2+k,-N}\,.$ (4.1)

Within this phase there exists a point with a massless Goldstino, due to the
spontaneous breaking of supersymmetry.

Fig. 3: The phase diagram of adjoint QCD with $k<N/2$. There are three
phases, separated by two phase transitions denoted by blue dots. The
supersymmetric point lies at $m=-g^{2}k$. The magnetic theory (shaded ellipse)
describes the vicinity of the (broken-)supersymmetric point, at masses both
below and above the phase transition. The magnetic’ theory covers regions
above and below the positive mass phase transition.

Naively, the proposed dual YM-CS is
$U(N/2-k)_{\frac{1}{2}k+\frac{3}{4}N,k+N/2}$ with an adjoint fermion. However,
such naive assignment of the $U(1)$ level is not compatible with the dynamics
of the electric theory. Since we have two magnetic duals we need to make sure
that their $U(1)$ levels reproduce correctly the intermediate phase. In order
to have a consistent picture we need to assign a level $N$ to the $U(1)$ of
the magnetic theory (and level $-N$ to the magnetic’ theory). The shift of the
$U(1)$ level from its naive value of $k+N/2$ to the ‘correct’ $N$ may be
achieved by introducing a RR 0-form flux along the $6$ direction, or perhaps a
localised flux at the intersection of the D3 antibranes and the fivebrane

$S=\int dx^{6}\,\int d^{3}x\,C_{0}F\wedge F=-\int dx^{6}\,dC_{0}\int
d^{3}x\,A\wedge F\,.$ (4.2)

This is a subtle matter and a better understanding of non-supersymmetric brane
configuration which include antibranes is needed.

Let us integrate out the adjoint fermion. Positive and negative masses results
in two TQFT. The negative mass results in $U(N/2-k)_{(k+N/2),N}$ TQFT and the
positive mass results in $U(N/2-k)_{N,N}$ (which is dual to
$SU(N)_{k-N/2,-N}$). Thus the dual gauge theory describes two regimes of the
three phases of the electric theory: the negative mass phase and the
intermediate phase, see fig.(3). In addition at some value of the mass there
is a massless fermion which corresponds to the Goldstino of the electric
theory. Similar to the magnetic theory, there exists a magnetic’ theory that
covers a regime of the positive mass phase and the intermediate mass phase.
The details are as follows: the magnetic’ theory is
$U(N/2+k)_{\frac{1}{2}k-\frac{3}{4}N,-N}$ with an adjoint fermion. After
integrating out the massive fermion we end up with either
$U(N/2+k)_{-(N/2-k),-N}$ TQFT or $U(k+\frac{1}{2}N)_{-N,-N}$ TQFT. They
describe (by virtue of level-rank duality) the $U(N/2-k)_{N/2+k,N}$ and
$SU(N)_{k+\frac{1}{2}N}$ phases of the electric theory, see fig.(3).

Our results in this section are in agreement with ref.[2].

##  5 SO and USp theories

The generalization to SO and USp theories is straightforward, yet we will
encounter new phenomena. We use the same brane configuration that led to the
$U(N)$ dualities, with minor modifications.

The USp theory is realized by adding to the brane configuration an $O3^{+}$
plane along the 0126 directions. The orientifold plane does not tolerate a
fractional brane, therefore we need to place $N$ D3 branes and their mirrors,
namely an even number of branes. The resulting theory is $USp(2N)$. Similarly,
$SO(2N)$ and $SO(2N+1)$ theories are realized by placing either even or odd
number of D3 branes on top of an $O3^{-}$ plane. In both USp/So cases we will
use a $(1,2k^{\prime})$ tilted fivebrane.

###  5.1 The $USp(2N)$ theory

We begin with the electric theory. Let $k^{\prime}=k+\frac{1}{2}(N+1)$. The
adjoint fermion with large negative mass shifts the level such that the
resulting electric theory that lives on the brane configuration is
$USp(2N)_{2k}$.

As we swap the fivebranes an additional D3 antibrane (and its mirror) is
created between the fivebranes due to the presesnce of the orientifold plane.
Therefore $\tilde{N}=k^{\prime}-1-N=k-\frac{1}{2}(N+1)$ and
$\tilde{k}=k^{\prime}-\frac{1}{2}(\tilde{N}+1)=\frac{1}{2}k+\frac{3}{4}N+\frac{1}{4}$.
The resulting magnetic theory is ${\cal N}=1$ supersymmetric
$USp(2(k-\frac{1}{2}N-\frac{1}{2}))_{-2(\frac{1}{2}k+\frac{3}{4}N+\frac{1}{4})}$
YM-CS theory, hence we propose the following Seiberg duality

$USp(2N)_{2k}\iff
USp(2(k-\frac{1}{2}N-\frac{1}{2}))_{-2(\frac{1}{2}k+\frac{3}{4}N+\frac{1}{4})}$
(5.1)

The supersymmetric magnetic dual can be trusted when
$\tilde{N}=k-\frac{1}{2}(N+1)\geq 0$. By using the assignment $N\rightarrow
N+1,k\rightarrow k+1$ with respect to the $U(N)_{k}$ case we propose an
invariant index777In USp/SO theories the index obtained from the branes is not
the Witten index, but rather a bosonic index[16]. I thank Jaume Gomis and
Zohar Komargodski for discussions on this matter.

$I=\frac{(k+\tfrac{1}{2}N+\tfrac{3}{2})!}{(k-\tfrac{1}{2}N+\tfrac{1}{2})!(N+1)!}\,.$
(5.2)

Both the electric and magnetic branes configurations yields the same result
(5.2).

The phase diagram of the $USp(2N)$ theory is similar to $U(N)$ theory. For a
negative fermion mass, where the duality can be trusted, the IR theory is
$USp(2N)_{2(k-\frac{1}{2}(N+1))}$ TQFT. The magnetic theory, upon integrating
out the massive fermion, is $USp(2(k-\frac{1}{2}N-\frac{1}{2}))_{-2N}$. The
two descriptions agree thanks to level-rank duality

$USp(2N)_{2(k-\frac{1}{2}(N+1))}\sim USp(2(k-\frac{1}{2}N-\frac{1}{2}))_{-2N}$
(5.3)

When $\tilde{N}<0$ the magnetic brane configuration consists of anti D3
branes. It describes the electric theory when supersymmetry is dynamically
broken. When anti D3 branes are placed on top of $O3^{+}$ plane the gauge
group is $USp$, but the fermion transforms in the two-index antisymmetric
representation [21], in agreement with the duality conjecture of [2].

The magnetic dual is therefore

$USp(2(\frac{1}{2}N+\frac{1}{2}-k))_{2(\frac{1}{2}k+\frac{3}{4}N+\frac{1}{4})}\,+\,{\rm
antisymmetric}\,\,{\rm fermion}$ (5.4)

Let us integrate out the fermion in order to obtain the IR theory. The level
is shifted as follows:

$\frac{1}{2}k+\frac{3}{4}N+\frac{1}{4}\rightarrow\frac{1}{2}k+\frac{3}{4}N+\frac{1}{4}\pm\frac{1}{2}(\tilde{N}-1)$
(5.5)

We therefore obtain for the negative mass phase a TQFT as follows

$USp(2(\frac{1}{2}N+\frac{1}{2}-k))_{2N}\sim
USp(2N)_{(2(k-\frac{1}{2}N-\frac{1}{2}))}\,.$ (5.6)

The quantum intermediate phase is described by a TQFT

$USp(2(\frac{1}{2}N+\frac{1}{2}-k))_{2(k+\frac{1}{2}N+\frac{1}{2})}\sim
USp(2(k+\frac{1}{2}N+\frac{1}{2}))_{(2(k-\frac{1}{2}N-\frac{1}{2}))}\,.$ (5.7)

Similar to the $U(N)$ case we can use the brane configuration to introduce a
magnetic’ theory

$USp(2(\frac{1}{2}N+\frac{1}{2}+k))_{2(\frac{1}{2}k-\frac{3}{4}N-\frac{1}{4})}\,+\,{\rm
antisymmetric}\,\,{\rm fermion}$ (5.8)

that covers the positive mass and the intermediate phase of the electric
theory.

The low energy of the magnetic and magnetic’ theories matches the low energy
of the electric theory.

###  5.2 $SO(2N)$ and $SO(2N+1)$ theories

Similar to the $USp(2N)$ theory, we can study SO theories by placing $N$ D3
branes and their mirrors on top of an $O3^{-}$ plane. The $O3^{-}$ tolarates a
fractional D3 brane on it, hence we can use brane configurations to study both
$SO(2N)$ theories and $SO(2N+1)$ theories. In our discussion we use the
notation $SO(2N)$ with $N$ either integer or half integer. We start with
$k^{\prime}=k+\frac{1}{2}(N-1)$. The discussion is almost identical to the
discussion about $USp(2N)$ theories. The main difference is that when the
fivebranes are exchanged a D3 brane is created and hence the dual gauge group
is $SO(2\tilde{N})$, with $\tilde{N}=k^{\prime}+1-N=k-\frac{1}{2}(N-1)$ and
$\tilde{k}=k^{\prime}-\frac{1}{2}(\tilde{N}-1)=\frac{1}{2}k+\frac{3}{4}N-\frac{1}{4}$.
Therefore for $\tilde{N}>0$, when SUSY is not broken the duality is

$SO(2N)_{2k}\iff
SO(2(k-\frac{1}{2}N+\frac{1}{2}))_{-2(\frac{1}{2}k+\frac{3}{4}N-\frac{1}{4})}$
(5.9)

with an index 888The index calculated using branes is not the Witten index. It
is rather a bosonic index, see [16] for a comprehensive discussion.

$I=\frac{(k+\tfrac{1}{2}N-\tfrac{3}{2})!}{(k-\tfrac{1}{2}N-\tfrac{1}{2})!(N-1)!}\,.$
(5.10)

The SUSY duality agrees about the low energy TQFT when the fermion mass is
negative

$SO(2N)_{2(k-\frac{1}{2}(N-1))}\sim SO(2(k-\frac{1}{2}N+\frac{1}{2}))_{-2N}$
(5.11)

When $\tilde{N}$ is negative the dual magnetic theory lives on a collection of
anti D3-branes placed on an $O3^{-}$ plane. It is
$SO(2(\frac{1}{2}N-\frac{1}{2}-k))_{2(\frac{1}{2}k+\frac{3}{4}N-\frac{1}{4})}$
with a two-index symmetric fermion. The magnetic’ theory is
$SO(2(\frac{1}{2}N-\frac{1}{2}+k))_{2(\frac{1}{2}k-\frac{3}{4}N-\frac{1}{4})}$
with a two-index symmetric fermion. Upon integration over the symmetric
fermion the magnetic and magnetic’ theories describe the vacua of the electric
theory in its three phases.

##  6 Conclusions

In this note we used string theory to derive new dualities as well as to
recover known dualities of adjoint 3d QCD. Branes turned out to be a very
useful tool to uncover a broad picture, where known dualities can be explained
in terms of 3d Seiberg duality (or Giveon-Kutasov duality).

For $k\geq\frac{1}{2}h(G)$ we found new supersymmetric dualities that hold
even when a mass is given to the adjoint fermion, indicating that brane
dynamics can be trusted even when supersymmetry is softly broken. For
$k<\frac{1}{2}h(G)$ we found an interesting phenomenon: Seiberg duality can be
extended to the regime where supersymmetry is dynamically broken. It suggests
than in 4d SQCD the Affleck-Dine-Seiberg superpotential may be computed using
a dual magnetic theory. While it is long known that string theory can be used
to compute the ADS superpotential [22], the computation was carried out in the
electric variables.

It would interesting to generalise the results of this note to other
representations (using other branes setups) and more flavours of adjoint
fermions (by tilting the branes differently while preserving more
supersymmetry). Another interesting direction of future work is to relate the
brane dynamics to ${\cal N}=1$ supersymetric theories with flavour [23] .

### Acknowledgements

I would like to thank professors Zohar Komargodski and Vasilis Niarchos for
discussions. I also thank the Simons centre for geometry and physics where
this work started and to Zohar Komargodski for an inspiring discussion at the
Simons centre that led to the duality idea. I am indebted to Jaume Gomis and
Zohar Komargodski for a critical reading of a draft of this paper.

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