Welcome to Humor Nation. Henry Cejudo is an exceptional athlete, not only a great martial artist, but also a gold medalist. Yeah, you read it right? He won a gold medal in the sport of freestyle wrestling at the 2008 Summer Olympics happened in Beijing. He became the youngest American ever to win a gold medal in freestyle wrestling. A three time Pan American Championships gold medalist and a Pan American Games one time winner.
Also Read: Take A Look At How Much IPL Cheerleaders Earn! More Than An Engineer!!
Henry Cejudo is also the first flyweight in the world to be able to ever defeat the mighty mouse Demetrious Johnson at UFC 227. Johnson who has been defeated only three times in his career, lost to Dominick Cruz in 2011, and to Brad Pickett in 2010. Henry managed to stop the Demetrious Johnson's streak of 11 consecutive successful championship defenses.
After winning the UFC Flyweight championship, Henry Cejudo successfully defended his title against the former UFC Bantamweight Champion TJ Dillashaw. Cejudo TKO'd Dillashaw in only 32 seconds to retain his championship.
The list of Henry Cejudo's achievements goes on and on, winning the gold medal at Olympics, defeating Demetrious Johnson, TKOing TJ Dillashaw, and now going for the Bantamweight championship.
So Let's Take A Look At Henry Cejudo's Olympic Gold Medalist Memes That Will Make You Go ROFL!
He never mentions that! Such a humble guy, but he should mention it pretty often.
That's how an Olympic Gold Medalist introduces himself.
So what do you think about our Olympic Champion? Do you think he could surprise the world again by stopping Marlon Moraes at UFC 238, and become the fourth double champ champ. Don't forget to share this interesting article with your friends!


Low volumes of pediatric patients and lack of experience make sick children some of the most anxiety-provoking patients for EMS responders. This course provides an overview of pediatric respiratory anatomy and physiology, emphasizes the importance of Basic Life Support (BLS) airway support and management, demonstrates how to differentiate among the most common airway problems, and uses clinical experience to explain how common pediatric respiratory issues can be better assessed in the field. In addition, the course provides tools for EMS responders to properly assess even the most uncooperative and agitated young child.
Our course subject matter expert on this important topic is Morgen Bernius, MD MS, FACEP, FAAP, of Franklin Square Hospital in Baltimore, Maryland.


Amlodipine, known under the trademark Norvasc to name a few, is a meicine recommended to deal with high blood pressure as well as coronary artery disease. Amlodipine comes from a set of medications referred to as calcium network blockers. It works by unwinding blood vessels so blood can move extra easily. Norvasc (Amlodipine) is likewise used to stop specific types of chest pain (angina). Take Amlodipine exactly as suggested by your medical professional. You may take Norvasc regardless of taking food. Take the medicine at the same time every day. Your high blood pressure will certainly require to be examined often. Your breast pain may come to be even worse when you first begin taking Amlodipine or when your dosage is boosted. Tell your physician if your upper body discomfort is extreme or recurring. Obtain urgent medical assistance in case you experience a severe side result including: seeming like you might pass out; swelling in your hands, ankles, or feet; battering heartbeats or trembling in your breast; or chest pain or heavy sensation, pain spreading to the arm or shoulder, nausea, sweating, basic sick sensation; tumefaction of your lips, face, tongue, or throat; hives; difficulty breathing.
Buying Norvasc online at our website is convenient and very fast. At the same time you ought to take your therapy with a great care. Aim to talk with your doctor before deciding to order Norvasc. Even if you are sure that the information above suffices for you to decide, medical talk is a must to avoid severe unfavorable impacts.


London, UK / Seoul, Republic of Korea – 27 September 2017: Ergomed plc (LSE: ERGO) ('Ergomed' or 'the Company'), a specialised pharmaceutical services and drug development company, and Boryung Pharmaceutical Co. Ltd (Korea Stock Exchange, KRX: 003850) ("Boryung"), a pharmaceutical business with R&D, manufacturing and marketing capabilities, are pleased to announce an agreement to commercialise Ergomed's haemostat products PeproStat™ and ReadyFlow™ in South Korea.
Under the terms of the collaboration, Ergomed will receive an upfront and a series of milestone payments as well as a double-digit share of all future product sales in this territory. Today's agreement is the first in a series of anticipated deals that will support global sales of the products, which were obtained through Ergomed's acquisition of Haemostatix in 2016.
PeproStat is an innovative haemostat for the control of bleeding during surgical procedures. The product is based on a peptide that binds to fibrinogen in blood to promote localized clotting in a wound. Current therapies are based on the enzyme thrombin which require multiple preparation steps and are manufactured from blood donations with the related risks of infection and a complex supply chain. In contrast PeproStat is synthetic, fast acting and ready-to-use. PeproStat has completed recruitment in a Phase IIb study in five European countries with data expected in Q4 2017, and it is anticipated that the product will reach the market in 2020 in Europe and 2021 in Korea. ReadyFlow is a flowable formulation that is particularly suited to less accessible bleeds. The product is in pre-clinical development. ReadyFlow was awarded first prize in the UK's Royal Society of Chemistry Emerging Technology Competition.
The global haemostat market is forecast to grow to $2.76billion by 2020[i]. Ergomed estimates that PeproStat and ReadyFlow together have peak sales potential of more than $500 million[ii].
[i] Hemostats Market by Product Type (Thrombin, Oxidised Regenerated Cellulose, Combination, Gelatin, Collagen), and Geography (Americas, Europe, APAC, MEA) - Global Forecast to 2020. MarketsandMarkets, February 2016.
Ergomed provides specialist services to the pharmaceutical industry and develops drugs both wholly-owned and through partnerships. Ergomed's fast-growing, profitable service offering spans all phases of clinical development and post-approval pharmacovigilance and medical information. Drawing on more than 20 years of expertise in drug development, Ergomed is also building a growing portfolio of drug development partnerships and programmes, including wholly-owned proprietary products for the treatment of surgical bleeding. For further information, visit: http://ergomedplc.com.
Boryung Pharmaceutical Co., Ltd is a leading pharmaceutical company in Korea fully integrated from R&D through manufacturing, marketing and sales with an established presence in Korea. The company has focused on the development and commercialization of new pharmaceutical products. Its main therapeutic categories are oncology, cardiovascular, anti-infective and respiratory system.


text:
SUPPORTING TALK OUT LOUD MENTAL HEALTH
Local schools received certificates at the Morecambe game
Schools from across Northamptonshire, who have demonstrated their commitment to positive mental health and wellbeing, were at Northampton Town Football Club on Saturday January 18th to receive Targeted Mental Health in Schools (TaMHS) awards.
Leader of Northamptonshire County Council Cllr Matt Golby, Northampton Town Footballers Harry Smith & Shaun McWilliams, Mike Simons, from the Northamptonshire Targeted Mental Health in Schools (TaMHS) Programme & Educational Psychology Service and Victoria Leitner from Public Health Northamptonshire joined schools to celebrate just some of the great work going on across the county.
The following schools were presented with awards: Avenue Infant (Bronze); Earls Barton Primary School (Bronze); Rockingham Primary (Bronze); Billingbrook School (Silver); Brackley Junior School (Silver), Grange Primary Academy, Kettering (Silver); Headlands Primary (Silver); Helmdon Primary (Silver); Kettering Buccleuch Academy (Silver); Newbottle & Charlton Primary School (Silver); Pattishall CE Primary (Silver); South End Junior (Silver); Woodford Halse Primary (Silver Renewal); and Kingsthorpe Village Primary (Gold).
The presentation also provided an opportunity for the Talk out Loud programme to launch a countdown to Northamptonshire Mental Health Awareness Day, an annual County-wide event to help raise awareness amongst young people about mental health and mental health stigma. It is set to take place on Friday 14 February 2020 and is part of the Public Health Northamptonshire Talk Out Loud Programme; Northamptonshire schools will be taking part in mental health awareness day by getting involved in activities to raise awareness of mental health stigma in lots of interesting, fun and inclusive ways.
Cllr Matt Golby said "We are delighted to present TaMHS awards to the Northamptonshire schools here today, and recognise their commitment to positive mental health and well-being. We would like to say thank you to Northampton Town Football Club for their continued commitment to encouraging the conversation about mental health."
Victoria Leitner, Programme Manager at Public Health Northamptonshire, said "Northamptonshire remains strongly committed to ensuring that young people have the confidence and understanding to acknowledge they have a mental health need, without fear of stigma. It is also really important that they know where to go for help and support. Our Talk out Loud Programme aims to get across to young people that we all need to look after ourselves to help achieve positive mental health and wellbeing. Sometimes we need help and support with our mental health and that's ok – initiatives such as Northamptonshire Mental Health Awareness Day help to start some of these conversations."
Kelvin Thomas, Chairman of Northampton Town Football Club added, "We are proud to support the Talk Out Loud Programme. Hopefully our involvement helps to spread the message that it is OK to ask for help and support when you need it. This is a fantastic campaign that we are pleased to continue to support."

redpajama_set_name: RedPajamaCommonCrawl


text:
Adulrach Namkul (; born 2 November 1997) is a Thai badminton player. He settled for the bronze medal at the 2015 World Junior Championships in the boys' singles event. He also a part of the Thailand national team that won the bronze medal at the 2017 Sudirman Cup.

Achievements

BWF World Junior Championships
Boys' singles

BWF International Challenge/Series (1 runner-up)
Men's singles

  BWF International Challenge tournament
  BWF International Series tournament
  BWF Future Series tournament

References

External links

1997 births
Living people
Adulrach Namkul
Adulrach Namkul
Competitors at the 2017 Southeast Asian Games
Adulrach Namkul
Southeast Asian Games medalists in badminton
Adulrach Namkul

redpajama_set_name: RedPajamaWikipedia


The Wellness Center will put on fitness and meal workshops throughout the semester to help students stay committed to their New Year's resolutions.
According to the Wellness Center webpage, the purpose of the workshops is to provide resources that will allow people to improve physically, emotionally, spiritually and socially.
"These workshops are open to any student who is interested and for those students who have made goals or New Year's resolutions to live a healthier lifestyle," said Joshua Guthrie, a Wellness Center staff member and a senior studying public health.
Guthrie said some of the most noteworthy programs include fitness classes put on by Wellness Center trainers. Some of these programs include how to stretch properly, how to set up your own workout plan and proper techniques for certain lifts.
The Meal Prep workshop is put on by the wellness coaches and is specifically geared toward any students willing to make changes in their diets.
Guthrie said the workshop teaches how students should go about meal prepping, the types of foods that are appropriate for a healthy diet and those that should be avoided. The workshop also encourages students to get involved, even if they are not knowledgeable in nutrition because they will learn how to be self-sufficient in prepping healthy meals.
Guthrie said at the end of each workshop session there are food samples, raffle prizes and refreshments.
The Wellness Center also promotes workshops geared toward helping students with dating and marriage.
Guthrie said the biggest thing they hope to accomplish is having students take what they learn in these workshops and help them apply it during their time at BYU-Idaho and after they graduate.
All fitness and meal workshops are from 6 to 7 p.m. weekly throughout the semester. Meal workshops will take place in the Taylor Cultural Hall. Fitness workshop locations are to be determined.
Students can find specific dates and more information about these workshops at the Wellness Center located in Hart 174.


In this chapter, you will be given a crash course introduction (or refresher!) to Spring, the core container, as well as some of the globally available facilities provided by the container. You will also become acquainted with the Spring XML configuration format as well as the annotation-driven support.


Everest or Upper Khumbu region is the second most popular trekking area in Nepal. It is the home of the world's highest mountains that include Mt.Everest (8850m), Cho Oyu (8153m), Makalu (8470m), Ama Dablam(6856m) and many other countless peaks. The region attracts thousands of adventure seekers to explore and experience high mountain valleys, high passes, Buddhist culture and stunning Sherpa villages on such high altitudes. Trekking to the region makes the travelers feel satisfied being at the base of Mt.Everest, the highest mountain in the world. The highlights of this region include: visit to Namche Bazaar, the famous monastery at Tengboche and excursion to Kala Patthar on the way to the Everest Base Camp. The region is also abundant with diverse alpine flora and fauna.


text:
News > Latest Poker News > 888Poker to Return as Lead Sponsor to 2016 World Series of Poker
February 29, 2016 Latest Poker News
888Poker to Return as Lead Sponsor to 2016 World Series of Poker
888poker will have its logo and presence stamped all over the Rio in Las Vegas this summer, as it returns as a key sponsor of the 2016 World Series of Poker.
The WSOP has announced 888poker will be its official sponsor in 2016. (Image: pokercommunitysites.com)
Around this time last year, the WSOP announced its partnership with the popular online poker site. But DraftKings remained the main sponsor at the Series last year. DraftKings had its logo on every WSOP table over the summer but, after Nevada deemed daily fantasy sports (DFS) to be a form of gambling, the Series decided to cut ties with the company before the November Nine.
With DraftKings off the market, the WSOP has turned to one of its other top sponsors, 888poker, to fill that void as main sponsor for the 2016 World Series of Poker. The poker site will bank on the added exposure the WSOP provides to expand its brand worldwide.
888poker, a partner of Caesars Entertainment Corporation, the company that owns the WSOP and WSOP.com, will reap the benefits of being the lead sponsor of the world's most popular poker series.
The world's second largest poker site received increased exposure at the 2015 WSOP Main Event when one of its own pros, Joe McKeehen, won the bracelet. Two other November Nine participants, Neil Blumenfield and Thomas Cannuli, were also sponsored by 888poker.
This year, 888 will have a more prominent sponsorship role at the Series, starting with an $888 buy-in bracelet tournament. Event #54, dubbed "Crazy Eights", is an 8-handed hold'em tournament with $888,888 guaranteed for the first place winner. Yup, we get the theme.
Last year's lead sponsor, DraftKings, had its name stamped on event #55, the DraftKings 50/50, a unique no-limit hold'em tournament where half the field was paid. That event, obviously, won't be on the 2016 schedule, which was (finally) released last week.
888 will have its logo all over the Rio, on featured and live stream tables, and on ESPN's broadcast of the 2016 WSOP. The price tag on becoming the WSOP's main sponsor has not been released publicly, but it's likely not cheap.
Another perk of the agreement for 888 is to be able to host WSOP satellite events for players outside of Nevada. Poker players in New Jersey and even outside the United States can enter online satellite tournaments at 888poker for the Main Event and Crazy Eights tournament.
Match Made in Heaven?
The World Series of Poker and 888poker's relationship spans many years, with many poker players having worn an 888 patch during past WSOP events.
In recent years, WSOP.com and 888poker have worked together to grow online poker in the US. 888 built the software for WSOP.com in both Nevada and New Jersey. The poker sites share liquidity in New Jersey.
Now, as the interchange between live and online poker heats up slowly once again in America, 888 is banking on what seems like a solid partnership.
viktorkre wrote...
March 2, 2016-5:17 am
888very good poker
888 Poker Reviews: English
888 Mobile


text:
\section{Introduction}
\if0
Asymptotically-free gauge theories exhibit various phases depending on gauge groups, matter contents, space-time dimensions and so on. It is generally difficult to analytically study the low-energy dynamics because those theories are strongly-coupled and non-perturbative.
Duality is a very powerful tool for studying the dynamics which is not accessible by a perturbation theory \cite{Seiberg:1994pq}. Duality gives us other descriptions of the theory, which are more tractable than the original theory. Supersymmetry is also very powerful and allows us to exactly obtain the non-perturbative information \cite{Seiberg:1994bz}.

Recently, the understanding of the three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories was deepened very well. In particular, the 3d Seiberg dualities were derived from the corresponding 4d dualities \cite{Aharony:2013dha, Aharony:2013kma}. In this derivation, it was important to correctly understand the Coulomb branch (monopole) operators and the non-perturbative superpotential coming from a circle compactification of 4d theories. The analysis of the Coulomb branch was developed also for theories with various matters and various gauge groups \cite{Csaki:2014cwa,Amariti:2015kha,Nii:2016jzi,Nii:2018erm,Nii:2017npz,Nii:2018tnd}.

However, the quantum structure of the Coulomb branch for the $SU(N)$ ``chiral'' gauge theory with $F$ fundamental and $\bar{F} (< F)$ anti-fundamental matters\footnote{We simply denote this theory as $SU(N)$ with $(F,\bar{F})$ (anti-)fundamentals.} has been less understood until now. In \cite{Intriligator:2013lca}, various chiral theories were extensively studied for the $U(1)$ and $SU(2)$ cases with and without a Chern-Simons coupling. Although the ``chiral'' $SU(N)$ theories were also studied in \cite{Intriligator:2013lca}, its analysis was semi-classical and discussed only a very limited class of the Coulomb branch, which would miss some important quantum aspects of the ``chiral'' theories. For instance, \cite{Intriligator:2013lca} claims that there is no s-confining phase for the ``chiral'' $SU(N)$ theories and that the supersymmetry will be broken for $N > \bar{F}$ without mentioning the allowed region of $F$. They claimed these two statements only from the semi-classical analysis of the Higgs branch and one particular Coulomb branch.
By studying various Coulomb branches, we will claim that the first statement is incorrect and that the second statement should be refined into $N>F>\bar{F}$. In addition, we will propose a new duality for the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F, \bar{F})$ (anti-)fundamental matters. By focusing on the particular matter contents, we will also find the duality between the chiral $SU(N)$ and $USp(2 \tilde{N})$ theories. Although the $U(N)$ ``chiral'' duality was discussed in \cite{Benini:2011mf}, the ``chiral'' $SU(N)$ duality gives rise to an additional problem of how to match baryon operators under the duality. We will show that the anti-baryonic operator can be mapped to one of the Coulomb branch operators.

In this paper, we will consider the three-dimensional $\mathcal{N}=2$ $SU(N)$ gauge theory with $F$ fundamental and $\bar{F}$ anti-fundamental matters with $F > \bar{F}$. For $F>N$, we will propose the duality and the confinement phases. The low-energy dynamics of this theory is less understood. In \cite{Aharony:2013dha}, only the completely chiral theory with $(F,\bar{F}=0)$ was studied. In \cite{Intriligator:2013lca}, the structure of the single Coulomb branch was studied only semi-classically, which is different from the full quantum picture of the moduli space. We claim that the quantum moduli space is more richer because one can generally construct so-called dressed monopole operators. By correctly finding the Coulomb branch operators, we can see the validity of our duality. By introducing the complex masses, we can flow to the theory with $N>F>\bar{F}$ which exhibits the spontaneous supersymmetry breaking.

The rest of the paper is organized as follows. In Section 2, we will introduce a set of the Coulomb branch coordinates which appear in the 3d $\mathcal{N}=2$ ``chiral'' $SU(N)$ theory. We divide the argument into $SU(2N)$ and $SU(2N+1)$ cases.
In Section 3, we propose the dualities of the ``chiral'' $SU(N)$ theory. Since the precise structure of the Coulomb branch depends on $F, \bar{F}$ and $N$, we will show dualities case-by-case in the subsequent subsections. The first two cases lead to the novel dualities between $SU(N)$ and $USp(2 \tilde{N})$ theories.
In Section 4, we study the confinement phases in the 3d ``chiral'' $SU(N)$ theories. By deforming the confinement phases, we can find supersymmetry-breaking phases and quantum-deformed moduli spaces for particular matter contents.
In Section 5, we will summarize our findings and discuss the future directions.
\fi

Asymptotically-free gauge theories exhibit various phases depending on gauge groups, matter contents, space-time dimensions and so on. It is generally difficult to analytically study the low-energy dynamics because those theories are often strongly-coupled and non-perturbative.
Duality is a very powerful tool for investigating the dynamics which is not accessible by a perturbation theory \cite{Seiberg:1994pq}. Duality gives us other descriptions of the theories, which are more tractable than the original theories. Supersymmetry is also very powerful and allows us to exactly obtain the non-perturbative information \cite{Seiberg:1994bz}.

Recently, a deeper understanding of the three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories was gained a lot. In particular, the 3d Seiberg dualities were derived from the corresponding 4d dualities \cite{Aharony:2013dha, Aharony:2013kma}. In this derivation, it was important to correctly understand the Coulomb branch (monopole) operators and the non-perturbative superpotential coming from a circle compactification of the 4d theories. The analysis of the Coulomb branch was developed also for theories with various matters and various gauge groups \cite{Csaki:2014cwa,Amariti:2015kha,Nii:2016jzi,Nii:2018erm,Nii:2017npz,Nii:2018tnd}.
For 3d ``chiral'' theories with unequal numbers of (anti-)fundamental matters, the dualities were discussed in the $U(N)$ cases \cite{Benini:2011mf}. For the $SU(N)$ cases, a completely chiral theory with no anti-fundamental matter was discussed in \cite{Aharony:2013dha}. In \cite{Intriligator:2013lca}, various chiral theories were extensively studied for the $U(1)$ and $SU(2)$\footnote{\cite{Intriligator:2013lca} considered the 3d $\mathcal{N}=2$ $SU(2)$ Chern-Simons-Matter theory with odd number of fundamentals.} cases with and without a Chern-Simons coupling.

However, the quantum structure of the Coulomb branch for the $SU(N)$ ``chiral'' gauge theory with $F$ fundamental and $\bar{F} (< F)$ anti-fundamental matters\footnote{We simply denote this theory as $SU(N)$ with $(F,\bar{F})$ (anti-)fundamentals.} has been less understood until now. Although the ``chiral'' $SU(N)$ theories were also studied in \cite{Intriligator:2013lca}, its analysis was semi-classical and discussed only a very limited class of the Coulomb branch which corresponds to the breaking $SU(N) \rightarrow SU(N-2) \times U(1) \times U(1)$. Since there are so many other classical Coulomb branches, this analysis \cite{Intriligator:2013lca} would be incomplete and miss some important quantum aspects of the ``chiral'' theories.
For instance, \cite{Intriligator:2013lca} claims that there is no s-confining phase for the 3d ``chiral'' $SU(N)$ theories since we cannot contract the flavor indices of the meson operator and then a confining superpotential is not available.  \cite{Intriligator:2013lca} also claims that the supersymmetry will be broken for $N > \bar{F}$ without mentioning the allowed region of $F$. They concluded these two statements only from the semi-classical treatment of the Higgs branch and one particular Coulomb branch. By carefully studying various Coulomb branches, we will find that the first statement is incorrect and the second statement should be refined into $N>F>\bar{F}$.

In this paper, we will consider the three-dimensional $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})$ (anti-)fundamentals with $F > \bar{F}$. We will claim that the very many classical Coulomb branches can survive quantum corrections and remain exactly massless in the ``chiral'' $SU(N)$ theories. This situation is very different from the vector-like theory with $F=\bar{F}$, where only a single Coulomb branch remains flat. Naively, one might consider that many Coulomb branches are not gauge invariant and they cannot be moduli coordinates due to the ``chirality''. However, we can construct the so-called ``dressed'' monopole operators by appropriately combining the bare Coulomb branch and matter operators, which are gauge-invariant and quantum-mechanically flat.
By taking into account these ``dressed'' operators, we will present various phases: For $F>N+1$, we will propose a dual description in terms of the $SU(F-N)$ gauge theory. For $F=N+1$, we will find a confinement phase where the low-energy degrees of freedom are governed by a cubic superpotential. For $N>F>\bar{F}$, the theory exhibits a spontaneous supersymmetry breaking.
In contrast to the ``chiral'' $U(N)$ duality, the ``chiral'' $SU(N)$ duality gives rise to an additional problem of matching the baryon operators under the duality.
We will demonstrate that the anti-baryonic operator is mapped to one of the dressed Coulomb branch operators.
By focusing on the duality with the particular matter contents, we will also propose a novel duality between the chiral $SU(N)$ and $USp(2 \tilde{N})$ theories.

The rest of the paper is organized as follows. In Section 2, we will introduce a set of the Coulomb branch coordinates which appear in the 3d $\mathcal{N}=2$ ``chiral'' $SU(N)$ theory. We divide the argument into $SU(2N)$ and $SU(2N+1)$ cases.
In Section 3, we propose the dualities of the ``chiral'' $SU(N)$ theory. Since the precise structure of the Coulomb branch depends on $F, \bar{F}$ and $N$, we will show dualities case-by-case in the subsequent subsections. The first two cases lead to the novel dualities between $SU(N)$ and $USp(2 \tilde{N})$ theories.
In Section 4, we study the confinement phases in the 3d ``chiral'' $SU(N)$ theories. By deforming the confinement phases, we can find supersymmetry-breaking phases and quantum-deformed moduli spaces for particular matter contents.
In Section 5, we will summarize our findings and discuss the future directions.
In Appendix, we will show an example of the various chiral and non-chiral dualities for the 3d $SU(2)$ gauge theory with $2F$ fundamental matters, especially focusing on the $F=3$ case.

\section{Coulomb branch in ``chiral'' $SU(N)$ theories}

In this section, we will introduce a ``typical'' set of the Coulomb branch coordinates for the 3d $\mathcal{N}=2$ $SU(N)$ gauge theories with chiral matter contents. By carefully studying the gauge invariance of these coordinates and by taking into account induced Chern-Simons couplings for unbroken gauge groups, we can find which coordinates remain exactly massless at a quantum level. Note that in this section we will only consider the ``typical'' Coulomb branch in the 3d chiral $SU(N)$ theories. Then, there might be additional Coulomb brach operators which could appear in case of particular matter contents and the ranks of the gauge group. In some cases, the Coulomb branch operators which will be introduced below are not minimal-monopole-creating operators and we must take $n$-th root of these operators in those cases. However, the understanding of these ``typical'' Coulomb branches will be sufficient and strong enough to understand various confinement phases and dualities.

\subsection{$SU(2N)$ with $(F,\bar{F})$}

We start with the Coulomb branch of the 3d $\mathcal{N}=2$ $SU(2N)$ gauge theories with $(F,\bar{F})$ (anti-)fundamentals. The matter contents and their representations are summarized in Table \ref{MCSU2N} below. Notice that, due to a parity anomaly, $F \pm \bar{F}$ should be even since we do not introduce a tree-level Chern-Simons term to the gauge interaction.

\begin{table}[H]\caption{Matter contents of $SU(2N)$ with $(F,\bar{F})$}
\begin{center}
\scalebox{1}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
  &$SU(2N)$&$SU(F)$&$SU(\bar{F})$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1& ${\tiny \yng(1)}$&$0$&1& $\bar{r}$ \\  \hline
  \end{tabular}}
  \end{center}\label{MCSU2N}
\end{table}
The Coulomb branch (or Coulomb moduli space) is a flat direction of the adjoint scalar fields which come from the $SU(2N)$ vector multiplet.  Typically, we will consider the following Coulomb branch which breaks the gauge group as
 %
\begin{align}
SU(2N)  &\rightarrow SU(N-a) \times SU(N-a) \times SU(2a) \times U(1)_1  \times U(1)_2 \\
U(1)_1 &\sim \mathrm{diag}(1,\cdots,1, 0,\cdots,0,-1,\cdots,-1) \\
U(1)_2 &\sim \mathrm{diag} \left( 1,\cdots,1 , -\frac{N-a}{a} ,\cdots,-\frac{N-a}{a},1,\cdots,1  \right) \\
{\tiny \yng(1)} & \rightarrow ({\tiny \yng(1)},\cdot,\cdot)_{1,1}+(\cdot,{\tiny \yng(1)},\cdot)_{-1,1}+(\cdot,\cdot,{\tiny \yng(1)})_{0,-\frac{N-a}{a}} \\
\overline{{\tiny \yng(1)}} & \rightarrow (\overline{{\tiny \yng(1)}},\cdot,\cdot)_{-1,-1}+(\cdot, \overline{{\tiny \yng(1)}},\cdot)_{1,-1}+(\cdot,\cdot, \overline{{\tiny \yng(1)}})_{0,\frac{N-a}{a}},
\end{align}
where $a=1,\cdots,N-1$. For $a=0$, the breaking pattern is slightly modified to
\begin{align}
SU(2N)  &\rightarrow SU(N) \times SU(N)  \times U(1)_1 \\
{\tiny \yng(1)} & \rightarrow ({\tiny \yng(1)},\cdot)_{1} +(\cdot,{\tiny \yng(1)})_{-1}\\
\overline{{\tiny \yng(1)}} & \rightarrow ( \overline{{\tiny \yng(1)}},\cdot)_{-1} +(\cdot, \overline{{\tiny \yng(1)}})_{1}.
\end{align}
There are $N$ types of the Coulomb branches which are related with $U(1)_1$. The corresponding operators are denoted as $Y_a^{bare}$. For the theory with $(F,\bar{F})$ (anti-)fundamentals, along these directions, the effective Chern-Simons terms are introduced. The (anti-)fundamental fields are massive and integrated out, which results in an $SU(N-a)_{\frac{F-\bar{F}}{2}} \times SU(N-a)_{-\frac{F-\bar{F}}{2}} $ pure Chern-Simons theory.
If we have an $SU(M)_k$ CS theory with $0\le |k|<M$, there is no SUSY vacuum \cite{Intriligator:2013lca}. Therefore, we require $N-a \le \frac{F-\bar{F}}{2}$, which means $a$ must satisfy
\begin{align}
\frac{2N-(F-\bar{F})}{2}\le a.
\end{align}
Notice that the effective Chern-Simons term is not generated for $U(1)_1$ since the low-energy $U(1)_1$ theory is vector-like. That is why these Coulomb branches can be flat directions.

These Coulomb branch operators are generally not gauge invariant and charged under the $U(1)_2$ since the theory is ``chiral''. Therefore, we have to multiply it by the chiral superfields $(\cdot,\cdot, \overline{{\tiny \yng(1)}})_{0,\frac{N-a}{a}} \in \tilde{Q}$ which belong to an $SU(2a)$ anti-fundamental representation. Then, we have to construct the baryonic operators from them in order to make an $SU(2a)$ singlet, which requires
\begin{align}
2a \le \bar{F} ~~~\mbox{for}~a \ne 0.
\end{align}
By collecting all the conditions for $a$, we find
 %
\begin{align}
 \frac{2N-(F-\bar{F})}{2} \le a  \le \frac{\bar{F}}{2},~~~0 \le a \le N-1
\end{align}
and the Coulomb branch operators exist for this range of $a$.
By computing the effective Chern-Simons term between $U(1)_1$ and $U(1)_2$, one can compute the $U(1)_2$ charge of the bare monopole operators $Y_a^{bare}$. By properly dressing the bare operators with $\tilde{Q}^{2a}$, the dressed Coulomb branches become
\begin{align}
Y_a^{dressed} \sim Y_a^{bare} (\tilde{Q}^{2a})^{\frac{F-\bar{F}}{2}}.
\end{align}
The charges of these operators can be computed from the mixed Chern-Simons terms as in Table \ref{CBSU2n} below.

\begin{table}[H]\caption{Charges of the monopole operators}
\begin{center}
\scalebox{1}{
  \begin{tabular}{|c||c||c|c|c| } \hline
  &$U(1)_2$&$U(1)$&$U(1)$&$U(1)_R$  \\  \hline
  $Y_{0}^{bare}$&0&$-FN$&$-\bar{F}N$& \scriptsize  $-FN(r-1)-\bar{F}N(\bar{r}-1) -2N^2$ \\[3pt]
$Y_{a \neq 0}^{bare}$ &$-(F-\bar{F})(N-a)$ &$-F(N-a)$&$-\bar{F}(N-a)$&\scriptsize  \begin{tabular}{c} $-F(N-a)(r-1)-\bar{F}(N-a)(\bar{r}-1)$ \\ $-2(N-a)(N+a)$ \end{tabular}\\[7pt]
$Y_{a}^{dressed}$&0&$-F(N-a)$&$-\bar{F} N +aF$  &\scriptsize \begin{tabular}{c} $-F(N-a)r-(\bar{F} N -aF)\bar{r} $\\ $+(N-a)(F+\bar{F} -2(N+a))$ \end{tabular}  \\[7pt] \hline
  \end{tabular}}
  \end{center}\label{CBSU2n}
\end{table}

Finally, we must be careful of minimal monopole operators. Here we constructed the naive monopole operators and there might be more smaller operators whose magnetic charges are smaller than here. Since the $SU(N-a) \times SU(N-a)$ gauge symmetry is unbroken in addition to $U(1)_1$, the minimal $U(1)_1$ magnetic charge could be more smaller. In some cases, the corresponding (minimal) magnetic charge becomes $\frac{1}{N-a}$ and the minimal bare operators are the $(N-a)$-th root of the naive ones. However, for those minimal operators, we cannot make them gauge-invariant by using $\tilde{Q}^{2a}$ in many cases. If this is possible, we have to construct more minimal operators from the naive ones by taking a root. Thus, our analysis below will become case-by-case depending on the matter contents and the rank of the gauge group.

\subsection{$SU(2N+1)$ with $(F,\bar{F})$}
For completeness of our paper, we also study the ``typical'' Coulomb branch of the 3d $\mathcal{N}=2$ $SU(2N+1)$ theory with $(F,\bar{F})$ (anti-)fundamental matters, which needs a small modification of the previous analysis. The matter contents and the global symmetries are the same as the $SU(2N)$ case (see Table \ref{MCSUodd} below).

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(2N+1)$ with $(F,\bar{F})$ (anti-)fundamentals}
\begin{center}
\scalebox{1}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
  &$SU(2N+1)$&$SU(F)$&$SU(\bar{F})$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1& ${\tiny \yng(1)}$&$0$&1& $\bar{r}$ \\  \hline
  \end{tabular}}
  \end{center}\label{MCSUodd}
\end{table}

We consider the Coulomb moduli space whose expectation value breaks the gauge group as
\begin{align}
SU(2N+1)  &\rightarrow SU(N-a) \times SU(N-a) \times SU(2a+1) \times U(1)_1  \times U(1)_2,
\end{align}
which is labeled by $Y_a^{bare}$ $ (a=0,\cdots,N-1)$. The $SU(2a+1)$ factor is absent for $a=0$. Notice that there are two $U(1)$ factors even for $a=0$. This is different from the $SU(2N)$ case. $Y_0^{bare}$ is not gauge invariant now and would need ``dressing''.
The (anti-)fundamental representations are decomposed into
\begin{align}
{\tiny \yng(1)} & \rightarrow ({\tiny \yng(1)},\cdot,\cdot)_{1,1}+(\cdot,{\tiny \yng(1)},\cdot)_{-1,1}+(\cdot,\cdot,{\tiny \yng(1)})_{0,-\frac{2(N-a)}{2a+1}} \\
\overline{{\tiny \yng(1)}} & \rightarrow (\overline{{\tiny \yng(1)}},\cdot,\cdot)_{-1,-1}+(\cdot, \overline{{\tiny \yng(1)}},\cdot)_{1,-1}+(\cdot,\cdot, \overline{{\tiny \yng(1)}})_{0,\frac{2(N-a)}{2a+1}},
\end{align}
where $(2a+1)$-dimensional representations become singlets for $a=0$ and are charged only under the $U(1)_2$. The effective Chern-Simons term between $U(1)_1$ and $U(1)_2$ is generated as
\begin{align}
k_{eff}^{U(1)_1 U(1)_2} =(F-\bar{F}) (N-a)
\end{align}
and the Coulomb branch operator $Y_a^{bare}$ is not gauge invariant (even for $a=0$). We can construct a gauge invariant composite by using $ (\cdot,\cdot, \overline{{\tiny \yng(1)}})_{0,\frac{2(N-a)}{2a+1}} \in \tilde{Q} $. The candidate of the dressed monopole operator is
\begin{align}
Y_a^{dressed} \sim Y_{a}^{bare} (\tilde{Q}^{2a+1})^{\frac{F-\bar{F}}{2}}.
\end{align}
The charges of these monopole operators are summarized in Table \ref{CBSUodd} below.
\begin{table}[H]\caption{Charges of monopole operators}
\begin{center}
\scalebox{1}{
  \begin{tabular}{|c||c||c|c|c| } \hline
  &$U(1)_2$&$U(1)$&$U(1)$&$U(1)_R$  \\  \hline
$Y_{a}^{bare}$ & \footnotesize $-(F-\bar{F})(N-a)$ &$-F(N-a)$&$-\bar{F}(N-a)$&\tiny \begin{tabular}{c} $-F(N-a)(r-1)-\bar{F}(N-a)(\bar{r}-1)$ \\ $-2(N-a)(N+a+1)$ \end{tabular}\\
$Y_{a}^{dressed}$&1&$-F(N-a)$& \footnotesize  $-\bar{F} N +\frac{F-\bar{F}}{2}+aF$  &\tiny \begin{tabular}{c} $-F(N-a)r-(\bar{F} N -\frac{F-\bar{F}}{2}-aF)\bar{r} $\\ $+(N-a)(F+\bar{F} -2(N+a+1))$ \end{tabular}  \\ \hline
  \end{tabular}}
  \end{center}\label{CBSUodd}
\end{table}

Along the Coulomb branch, the matter fields are massive and integrated out. This introduces the Chern-Simons terms for the two $SU(N-a)$ groups. The resulting low-energy theory is an $SU(N-a)_{\frac{F-\bar{F}}{2}} \times SU(N-a)_{-\frac{F-\bar{F}}{2}} $ Chern-Simons theory. Since this theory must have a supersymmetric vacuum, it requires \cite{Intriligator:2013lca}
\begin{align}
N-a \le \frac{F -\bar{F}}{2}.
\end{align}
Furthermore, since we need to construct a baryonic operator $\tilde{Q}^{2a+1}$ in order to dress the bare monopoles, there is a further constraint, $2a+1 \le \bar{F}$. By collecting all the constraints for $a$, we find
\begin{align}
0\le a \le N-1,~~~\frac{2N-(F-\bar{F})}{2} \le a \le \frac{\bar{F}-1}{2}.
\end{align}
The Coulomb branch $Y_a^{dressed}$ exists only for this range of $a$.

In this subsection, we focused on the general structure of the Coulomb branch. When we will study a specific example in the next section, there are more-smaller monopole operators available in some cases. For instance, we can consider a monopole operator
\begin{align}
Y_{a}^{\substack{minimal \\ bare}} \sim (Y_{a}^{bare})^{\frac{1}{N-a}},
\end{align}
which has a smaller magnetic charge and is still not gauge invariant. In many cases, we cannot construct a gauge invariant composite from $Y_{a}^{\substack{minimal \\ bare}} $ and $\tilde{Q}^{2a+1}$. However, in some cases (special choices of $N,F,\bar{F}$ and $a$), we can do it. Because the correct choice of the bare monopoles depend on the matter contents, we have to consider the dualities case-by-case.

\section{Dualities}

Here, we propose a new duality for the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $F$ fundamental matters $Q$ and $\bar{F}$ anti-fundamental matters $\tilde{Q}$. The following discussion assumes $F > \bar{F}$ without loss of generality. The proposed dual description is a 3d  $\mathcal{N}=2$ $SU(F-N)$ gauge theories with $F$ fundamental (dual) matters $q$, $\bar{F}$ anti-fundamental (dual) matters $\tilde{q}$ and a meson singlet $M$ with a following superpotential
\begin{align}
W=M \tilde{q}q. \label{super}
\end{align}

In order to illustrate how the duality works, we restrict ourself to the case where the anti-baryonic operators cannot be constructed on the electric side. Furthermore, in order to explain a generic structure of the duality, let us focus on the 3d $\mathcal{N}=2$ $SU(2N)$ with $(2F,2\bar{F})$ (anti-)fundamentals. One can easily generalize our analysis to more generic cases. The proposed dual description is
a 3d $\mathcal{N}=2$ $SU(2F-2N)$ with $(2F,2\bar{F})$ (anti-)fundamentals and a singlet $M$ with a superpotential \eqref{super}.
The matter contents and their quantum numbers are summarized in Table \ref{generic}.

\begin{table}[H]\caption{$SU(2N)$ with $(2F,2\bar{F})$ and its dual}
\begin{center}
\scalebox{0.96}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
  &$SU(2N)$&$SU(2F)$&$SU(2\bar{F})$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1& ${\tiny \yng(1)}$&$0$&1& $\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$& ${\tiny \yng(1)}$&1&1&$r+\bar{r}$\\
$B:=Q^{2N}$&1&\scriptsize \begin{tabular}{c}$2N$-th \\ antisymmetric\end{tabular}&1&$2N$&$0$&$2Nr$ \\ \hline \hline
&$SU(2F-2N)$&$SU(2F)$&$SU(2 \bar{F})$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$q$& ${\tiny \yng(1)}$& ${\tiny \overline{\yng(1)}}$&1&$\frac{N}{F-N}$&$0$& $\frac{N}{F-N}r$ \\
$\tilde{q}$&${\tiny \overline{\yng(1)}}$&1 & $\overline{{\tiny \yng(1)}}$&$-\frac{F}{F-N}$&$-1$&$2-\frac{F}{F-N}r-\bar{r}$ \\
$M$&1&${\tiny \yng(1)}$ &${\tiny \yng(1)}$ &1&1&$r+\bar{r}$ \\\hline
$N:=\tilde{q}q$&1&${\tiny \overline{\yng(1)}}$ & $\overline{{\tiny \yng(1)}}$ &$-1$&$-1$&$2-r-\bar{r}$ \\
$b:=q^{2F-2N} \sim B$&1&\scriptsize \begin{tabular}{c}$2N$-th \\ antisymmetric\end{tabular}&1&$2N$&$0$&$2Nr$ \\ \hline
  \end{tabular}}
  \end{center}\label{generic}
\end{table}

The charge assignment of the fields in the dual description can be fixed from the superpotential and the matching of the baryon operator.
The mesonic branch labeled by $M: = \tilde{Q} Q$ is mapped to a singlet $M$. The baryonic operator $B:=Q^{2N}$ is mapped to the baryon $b:=q^{2F-2N}$ constructed from the dual quarks $q$. When $\bar{F} \ge F-N $, we can construct the anti-baryonic operator on the magnetic side. One might consider that there is no counterpart of the anti-baryon on the electric side. However, as we will see in the next subsection, the anti-baryon corresponds to one of the Coulomb branch operators on the electric side.

The electric and magnetic theories are both UV-complete because the duality does not include a monopole superpotential. In addition, the proposed duality is very similar to the conventional 4d Seiberg duality \cite{Seiberg:1994pq}. Therefore, this 3d duality can straightforwardly pass several simple tests. By applying the duality twice, we can go back to the electric theory. The complex mass deformation on the electric side is mapped to the Higgsing of the dual gauge group and the duality is correctly preserved with reduction of $F$ and $\bar{F}$. The parity anomaly matching is also satisfied. The generic real mass deformation is now complicated and let us focus on one particular simple case. By weakly gauging the global $U(1)$ symmetries, all the electric matters can obtain the positive real masses and the electric theory flows to the $SU(2N)_{F+\bar{F}}$ SUSY CS theory. This flow is only possible for $F+\bar{F}  \ge 2N$ otherwise the supersymmetry is broken and we have to find a more complicated flow. On the magnetic side, the fundamental matters obtain positive real masses while the anti-fundamental matters obtain negative masses. The theory flows to $SU(2F-2N)_{F-\bar{F}}$ SUSY CS theory. This flow is possible for $F-\bar{F} \ge 2F-2N$. Therefore, only for $\bar{F}=2N-F$, these flows are simultaneously realized. In this case, the electric side becomes the $SU(2N)_{2N} $ CS theory while the magnetic side becomes the $SU(2F-2N)_{2F-2N}$ CS theory. The both sides have the same Witten index \cite{Witten:1999ds,Intriligator:2013lca} which is one. The most non-trivial part of this duality is the matching of the moduli operators and hence we focus on it in what follows.

Since the correct Coulomb branch operators will change case-by-case depending on $N,F,\bar{F}$ and $a$, we here only give a general rule of matching the Coulomb branch operators on both sides. The more precise matching (including the anti-baryon) will be shown in the following subsections. As we explained in the previous section, the Coulomb branch of the ``chiral'' theory is labeled by $Y_a$. The charges of these operator on both sides are summarized in Table \ref{CBmapping} below. Notice that the dressed operators are generally charged under the non-abelian flavor symmetry, but we omitted it for simplicity.

\begin{table}[H]\caption{Coulomb branch operators}
\begin{center}
\scalebox{0.63}{
  \begin{tabular}{|c||c||c|c|c| } \hline
  &$U(1)_2$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$Y_a^{bare}$&\footnotesize  $-(2F-2\bar{F})(N-a) $& \footnotesize $-2F(N-a)$&\footnotesize $-2\bar{F}(N-a)$&\tiny \begin{tabular}{c}
$-2F(N-a)r -2\bar{F}(N-a) \bar{r} $\\$+2F(N-a)+2\bar{F}(N-a)-2(N^2-a^2)$
\end{tabular}\\
\footnotesize $Y_a^{dressed}:=Y_a^{bare} \tilde{Q}^{2a \frac{2F-2\bar{F}}{2}}$&0& \footnotesize $-2F(N-a)$& \footnotesize $-2\bar{F}N+2aF$&\tiny \begin{tabular}{c}
$-2F(N-a)r -2\bar{F}(N-a) \bar{r} $\\$+2F(N-a)+2\bar{F}(N-a)-2(N^2-a^2)+a(2F-2\bar{F})\bar{r}$
\end{tabular} \\ \hline \hline
&$U(1)_2$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$\tilde{Y}_a^{bare}$& \footnotesize $-(2F-2\bar{F})(F-N-a)$& \footnotesize $\frac{2F(F-N-a)(\bar{F}-N)}{F-N}$&\footnotesize $2 \bar{F}(F-N-a)$&\tiny \begin{tabular}{c}
$\frac{2F(F-N-a)(2\bar{F}-N)}{F-N} r +2 \bar{F}(F-N-a) \bar{r} $\\$+2F(F-N-a)-2\bar{F}(F-N-a)-2((F-N)^2-a^2)$
\end{tabular} \\
\footnotesize $\tilde{Y}_a^{dressed}:=\tilde{Y}_a^{bare} \tilde{q}^{2a\frac{2F-2\bar{F}}{2}}$&0& \footnotesize $-2F(N-\bar{F}+a)$& \footnotesize $-2\bar{F}N+2F(\bar{F}-a)$&\tiny \begin{tabular}{c}
$-2F(N-(\bar{F}-a))r -2\bar{F}(N-(\bar{F}-a)) \bar{r} $\\$+2F(N-(\bar{F}-a))+2\bar{F}(N-(\bar{F}-a))-2(N^2-(\bar{F}-a)^2)+(\bar{F}-a)(2F-2\bar{F})\bar{r}$
\end{tabular}  \\ \hline
  \end{tabular}}
  \end{center}\label{CBmapping}
\end{table}

Let us consider the matching of the dressed monopole operators.
From Table \ref{CBmapping} above, the identification becomes
\begin{align}
Y_{a_e}^{dressed} \leftrightarrow \tilde{Y}_{a_m=\bar{F}-a_e}^{dressed}.
\end{align}
The index $a_e$ on the electric side runs over
\begin{align}
 N-(F-\bar{F}) \le a_e  \le \bar{F},~~~0 \le a_e \le N-1.
\end{align}
while the magnetic $a_m$ runs over
\begin{align}
\bar{F}-N \le a_m  \le \bar{F},~~~0 \le a_m \le F-N-1.
\end{align}
Generally, the numbers of the Coulomb branch operators are different for the electric and magnetic sides.
The missing operators appear from the anti-baryonic operators. For $F>N>\bar{F} \cap F-N < \bar{F}$, there are $F-N+1$ electric Coulomb branch operators while there are $F-N$ Coulomb branch operators and one anti-baryon in a magnetic theory. For $F>N>\bar{F} \cap F-N > \bar{F}$, there are $\bar{F}+1$ Coulomb branch operators and the baryons on both sides. For $F>\bar{F}>N \cap F-N < \bar{F}$, there are $F-\bar{F}$ Coulomb branch operators on both sides and (anti-)baryons are available also on both side. Finally, for $F>\bar{F}>N \cap F-N > \bar{F}$, there are $N$ Coulomb branch operators and (anti-)baryons constructible on the electric side, while there are $N+1$ Coulomb branch operators and only a single baryon is possible. In this way, the total numbers of the Coulomb branches and (anti-)baryons are identical and the proposed duality works well.

The precise matching of the Coulomb branch operators (including baryonic ones) is involved and depends on the combination $(N,F,\bar{F})$. Furthermore, depending on $(N,F,\bar{F},a)$, the Coulomb branch operators which have more smaller magnetic charges are available. Then, in what follows, we will consider the specific cases of the duality for the $SU(N)$ ``chiral'' theories.

\subsection{$SU(2N)$ with $(2N+2,2\bar{F})$}
As a first example of our duality, let us start with the case where the dual gauge group becomes $SU(2)$. This case is simplest since the bare monopole operator of the magnetic $SU(2)$ theory is gauge invariant without ``dressing''. The electric theory is a 3d $\mathcal{N}=2$ $SU(2N)$ gauge theory with $(2N+2,2\bar{F})$ (anti-)fundamental matters. The magnetic description is a 3d $\mathcal{N}=2$ $SU(2)$ gauge theory with $2N+2 +2\bar{F}$ fundamentals $(q,\tilde{q})$ and a singlet $M$. The magnetic theory contains the superpotential $W=M q \tilde{q}$. Table \ref{SU2nSU2dual} shows the quantum numbers of the matter contents.

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(2N)$ with $(2N+2,2\bar{F})$ (anti-)fundamentals}
\begin{center}
\scalebox{0.88}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
 &$SU(2N)$&$SU(2N+2)$&$SU(2 \bar{F})$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1& ${\tiny \yng(1)}$&$0$&1&$\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$& ${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$B:=Q^{2N}$&1&${\tiny \overline{\yng(1,1)}}$&1&$2N$&$0$&$2Nr$ \\ \hline
$Y_{\bar{F}-1}^{\substack{minimal \\ dressed}}$&1&1&${\tiny \overline{\yng(1,1)}}$&$-2(N+1)$&$-2$&$4-2(N+1)r -2\bar{r}$ \\
$Y_{\bar{F}}^{dressed} $&1&1&1 & {\scriptsize $-2(N+1)(N-\bar{F})$}&$2\bar{F}$& {\scriptsize $ -2(N+1)(N-\bar{F})r+2\bar{F} \bar{r}+ 2(N-\bar{F}) $ } \\ \hline \hline
&$SU(2)$&$SU(2N+2)$&$SU(2\bar{F})$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$q$& ${\tiny \yng(1)}$& ${\tiny \overline{\yng(1)}}$&1&$N$&$0$& $ N r$ \\
$\tilde{q}$&${\tiny \yng(1)}$&1 & ${\tiny \overline{ \yng(1)}}$&$-(N+1)$&$-1$&$2-(N+1)r-\bar{r}$ \\
$M$&1&${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\\hline
$N:=\tilde{q}q$&1&${\tiny \overline{\yng(1)}}$ & ${\tiny \overline{\yng(1)}}$&$-1$&$-1$&$2-r-\bar{r}$ \\
$B:=q^{2} $&1&${\tiny \overline{\yng(1,1)}}$&1&$2N$&$0$&$(2N+1)r$ \\
$\bar{B} := \tilde{q}^2 $ &1&1&${\tiny \overline{\yng(1,1)}}$&$-2(N+1)$&$-2$&$4-2(N+1)r -2\bar{r}$ \\ \hline
$\tilde{Y}_{SU(2)}$&1&1&1& {\scriptsize $-2(N+1)(N-\bar{F})$}&$2\bar{F}$& {\scriptsize  $-2(N+1)(N-\bar{F})r+2\bar{F} \bar{r}+ 2(N-\bar{F}) $ }\\ \hline
  \end{tabular}}
  \end{center}\label{SU2nSU2dual}
\end{table}

The bare monopole operators $Y_a^{bare}$ are allowed for $a =\bar{F}, \bar{F}-1$. These two operators are not gauge invariant and require ``dressing'' by $\tilde{Q}^{2a}$. For $a= \bar{F}-1$, the gauge invariant combination can be constructed from $\tilde{Q}^{2a}$ and the minimal bare monopole
\begin{align}
Y_{a=\bar{F}-1}^{\substack{ minimal \\ bare }}&:=( Y_{\bar{F}-1}^{ bare } )^{\frac{1}{N-\bar{F}+1}},
\end{align}
instead of the naive one $Y_{\bar{F}-1}^{ bare }$. Then, the dressed Coulomb branch will be described by this smaller monopole. For $a= \bar{F}$, the minimal monopole cannot be made gauge invariant by $\tilde{Q}^{2a}$ and we must use $Y_{a=\bar{F}-1}^{bare}$ for constructing the gauge invariant operators. The dressed coordinates become
\begin{align}
Y_{\bar{F}-1}^{\substack{minimal \\ dressed}}&:= Y_{\bar{F}-1}^{\substack{minimal \\ bare}} \tilde{Q}^{2(\bar{F}-1)} \\
Y_{\bar{F}}^{dressed} &:=Y_{\bar{F}}^{bare}  (\tilde{Q}^{2\bar{F}} )^{N+1-\bar{F}}.
\end{align}
In Table \ref{bareCBcharges}, the $U(1)_2$ charges of the Coulomb branch operators are summarized. We can see that $Y_{\bar{F}}^{\substack{minimal \\ bare}}:= (Y_{\bar{F}}^{bare})^{\frac{1}{N-\bar{F}}}$ cannot be dressed by $\tilde{Q}^{2a}$.

\begin{table}[H]\caption{Monopole and dressing operators on the electric side}
\begin{center}
\scalebox{1}{
  \begin{tabular}{|c||c| }  \hline
  & $U(1)_2$ \\ \hline
  $Y_{\bar{F}-1}^{\substack{minimal \\ bare}}$ & $-(2N+2-2\bar{F})$\\
  $Y_{\bar{F}-1}^{bare}$& $-(2N+2-2\bar{F}) (N-\bar{F}+1)$ \\
  $Q^{2(\bar{F}-1)} |_{a=\bar{F}-1}$ & $2(N-\bar{F}+1)$ \\ \hline
  $Y_{\bar{F}}^{\substack{minimal \\ bare}}$ & $-(2N+2-2\bar{F}) $\\
  $Y_{\bar{F}}^{bare}$& $-(2N+2-2\bar{F}) (N-\bar{F})$ \\
  $Q^{2\bar{F}}|_{a=\bar{F}} $ & $2(N-\bar{F})$\\ \hline
    \end{tabular}}
  \end{center}\label{bareCBcharges}
\end{table}

The operator mapping is manifest from Table \ref{SU2nSU2dual}. Notice that the $SU(2)$ magnetic theory allows only one Coulomb branch operator \cite{Aharony:1997bx} and an additional Coulomb branch comes from the anti-baryonic operator $\bar{B} := \tilde{q}^2 $. The identification becomes
\begin{gather}
Y_{\bar{F}-1}^{dressed} \sim \bar{B},~~~ Y_{\bar{F}}^{dressed} \sim \tilde{Y}_{SU(2)}.
\end{gather}
For $\bar{F}=N$, the anti-baryon operator $\bar{b}:= \tilde{Q}^{2N}$ is available while the Coulomb branch operator $Y_{\bar{F}=N}$ is not defined. The anti-baryon $\bar{b}$ is identified with the $SU(2)$ monopole operator $\tilde{Y}_{SU(2)}$ in this case. In this way, the duality works even for $\bar{F}=N$.

\subsubsection*{$USp(2(N+\bar{F} -1))$ dual}
Since the dual gauge group $SU(2)$ can be regarded as a member of the $USp(2n)$ series, one can apply the Aharony duality \cite{Aharony:1997gp} to the magnetic side. The Aharony dual is given by a 3d $\mathcal{N}=2$ $USp(2(N+\bar{F} -1))$ gauge theory. The matter contents are summarized in Table \ref{USpdual1}. Since the $USp(2(N+\bar{F} -1))$ theory has only the fundamental matters, the corresponding Coulomb branch is simple and one-dimensional, which is denoted by $Y_{USp}$.

\begin{table}[H]\caption{$USp(2(N+\bar{F} -1))$ dual of $SU(2N)$ with $(2N+2,2\bar{F})$}
\begin{center}
\scalebox{0.87}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
 &\scriptsize $USp(2(N+\bar{F}-1))$&\scriptsize $SU(2N+2)$&\scriptsize $SU(2 \bar{F})$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $b$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&$-N$&0&$1-Nr$ \\
$\tilde{b}$  &${\tiny \yng(1)}$&1& ${\tiny \yng(1)}$&$N+1$&1&$-1+(N+1)r+\bar{r}$ \\
 $M $&1&${\tiny \yng(1)}$& ${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$N\sim q \tilde{q}$&1&${\tiny \overline{\yng(1)}}$&${\tiny \overline{\yng(1)}}$&$-1$&$-1$&$2-r -\bar{r}$ \\
$B \sim q^{2}$&1&${\tiny \overline{\yng(1,1)}}$&1&$2N$&$0$&$2Nr$ \\
$\bar{B} \sim \tilde{q}^2$&1&1&${\tiny \overline{\yng(1,1)}}$&$-2(N+1)$&$-2$&$4-2(N+1)r-2\bar{r}$ \\
$\tilde{Y}_{SU(2)}$&1&1&1&\scriptsize $-2(N+1)(N-\bar{F})$&$2\bar{F}$&\scriptsize $2(N-\bar{F}) -2(N+1)(N-\bar{F})r+2\bar{F} \bar{r}$ \\ \hline
$b \tilde{b} \sim M$&1&${\tiny \yng(1)}$&${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$Y_{USp}$&1&1&1&\scriptsize $2(N+1)(N-\bar{F})$&$-2\bar{F}$&\scriptsize $2(\bar{F}-N+1) +2(N+1)(N-\bar{F})r-2\bar{F} \bar{r}$  \\ \hline
  \end{tabular}}
  \end{center}\label{USpdual1}
\end{table}

The superpotential takes
\begin{align}
W &=MN+N b \tilde{b}  +B b^2 +\bar{B}\tilde{b}^2 + \tilde{Y}_{SU(2)}Y_{USp}\\
&=B b^2 +\bar{B}\tilde{b}^2 + \tilde{Y}_{SU(2)}Y_{USp}
\end{align}
Since the mesonic fields $M$ and $N$ are massive, they can be integrated out. The F-term conditions lead to $N=0$ and $M\sim b \tilde{b}$. The matching of the flat directions with the $SU(2N)$ theory is manifest from Table \ref{SU2nSU2dual} and Table \ref{USpdual1}. The unnecessary flat directions are all lifted by the above superpotential.

\subsection{$SU(2N+1)$ with $(2N+3,2\bar{F}+1)$}
For completeness, we study the $SU(2N+1)$ duality whose dual is again given by an $SU(2)$ gauge theory.
The electric theory is a 3d $\mathcal{N}=2$ $SU(2N+1)$ gauge theory with $2N+3$ fundamentals and $2\bar{F}+1$ anti-fundamentals without superpotential. We assume $\bar{F} \le N$ in what follows.

The Coulomb branch operators $Y_a^{bare}$ are available only for $a=\bar{F}$ and $\bar{F}-1$. These are not gauge invariant and need ``dressing''. For $a=\bar{F}-1$, the more minimal operator $(Y_{a=\bar{F}-1}^{bare})^{1/(N-\bar{F}+1)}$ can be dressed by $\tilde{Q}^{2a+1}$ while the $U(1)_2$ charge of $(Y_{a=\bar{F}}^{bare})^{1/(N-\bar{F})}$ cannot be dressed by $\tilde{Q}^{2a+1}$. Therefore, we need to introduce the following Coulomb branch operators
\begin{align}
Y_{\bar{F}-1}^{\substack{minimal \\ dressed}} &:= (Y_{\bar{F}-1}^{bare})^{\frac{1}{N-\bar{F}+1}} \tilde{Q}^{2\bar{F}-1}, \\
Y_{\bar{F}}^{dressed} &:= Y_{\bar{F}}^{bare} (\tilde{Q}^{2\bar{F}+1})^{N-\bar{F}+1},
\end{align}
where the flavor index of $\tilde{Q}^{2a+1}$ is anti-symmetric as it should be. The matter contents and their charges are summarized in Table \ref{SUoddSU2dual} below.

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(2N+1)$ with $(2N+3,2\bar{F}+1)$ (anti-)fundamentals}
\begin{center}
\scalebox{0.83}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
 &\scriptsize$SU(2N+1)$&\scriptsize$SU(2N+3)$&\scriptsize$SU(2 \bar{F}+1)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1& ${\tiny \yng(1)}$&$0$&1&$\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$& ${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$B:=Q^{2N}$&1&${\tiny \overline{\yng(1,1)}}$&1&$2N+1$&$0$&$(2N+1)r$ \\ \hline
$Y_{\bar{F}-1}^{\substack{minimal\\dressed}}$&1&1&${\tiny \overline{\yng(1,1)}}$&\scriptsize$-(2N+3)$& $-2$&$4-(2N+3)r -2\bar{r}$ \\
$Y_{\bar{F}}^{dressed} $&1&1&1 & {\scriptsize $-(2N+3)(N-\bar{F})$}& \scriptsize$2\bar{F}+1$& {\scriptsize $ -(2N+3)(N-\bar{F})r+(2\bar{F} +1)\bar{r}+ 2(N-\bar{F}) $ } \\ \hline \hline
&\scriptsize$SU(2)$&\scriptsize$SU(2N+3)$&\scriptsize$SU(2\bar{F}+1)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$q$& ${\tiny \yng(1)}$& ${\tiny \overline{\yng(1)}}$&1&$N+\frac{1}{2}$&$0$& $ \left(N +\frac{1}{2} \right) r$ \\
$\tilde{q}$&${\tiny \yng(1)}$&1 & ${\tiny \overline{ \yng(1)}}$&$- \left(N+ \frac{3}{2} \right)$&$-1$&$2-\left(N+ \frac{3}{2} \right)r-\bar{r}$ \\
$M$&1&${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\\hline
$N:=\tilde{q}q$&1&${\tiny \overline{\yng(1)}}$ & ${\tiny \overline{\yng(1)}}$&$-1$&$-1$&$2-r-\bar{r}$ \\
$B:=q^{2} $&1&${\tiny \overline{\yng(1,1)}}$&1&$2N+1$&$0$&$(2N+1)r$ \\
$\bar{B} := \tilde{q}^2 $ &1&1&${\tiny \overline{\yng(1,1)}}$&$-(2N+3)$&$-2$&$4-(2N+3)r -2\bar{r}$ \\ \hline
$\tilde{Y}_{SU(2)}$&1&1&1& {\scriptsize $-(2N+3)(N-\bar{F})$}&\scriptsize $2\bar{F}+1$& {\scriptsize  $-(2N+3)(N-\bar{F})r+(2\bar{F}+1) \bar{r}+ 2(N-\bar{F}) $ }\\ \hline
  \end{tabular}}
  \end{center}\label{SUoddSU2dual}
\end{table}

On the dual side, the gauge group is $SU(2)$ and there are $2N+2\bar{F}+4$ fundamental matters (no difference between $\mathbf{2}$ and $\bar{\mathbf{2}}$). The dual theory has a superpotential
\begin{align}
W= M \tilde{q}q.
\end{align}
We need not introduce any dressed operators and there is a  single Coulomb branch operator $\tilde{Y}_{SU(2)}$ \cite{Aharony:1997bx}. The operator matching again shows that one of the Coulomb branch operators $Y_{\bar{F}-1}^{\substack{minimal \\ dressed}} $ is transformed into the anti-baryonic operator $\bar{B}$ constructed from the dual quarks $\tilde{q}$.

\subsubsection*{$USp(2(N+\bar{F}))$ dual}
Since the dual gauge group is an $SU(2)$, we can obtain additional dual description by adopting the Aharony duality \cite{Aharony:1997gp}. The $USp(2(N+\bar{F}))$ dual theory contains $2N+2\bar{F}+4$ fundamental matters and singlets $M,N,B,\bar{B}$ and $Y_{USp}$. Table \ref{USpdual2} shows their quantum numbers.

\begin{table}[H]\caption{$USp(2(N+\bar{F} ))$ dual of $SU(2N+1)$ with $(2N+3,2\bar{F}+1)$}
\begin{center}
\scalebox{0.825}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
 &\scriptsize $USp(2(N+\bar{F}))$&\scriptsize $SU(2N+3)$&\scriptsize $SU(2 \bar{F}+1)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $b$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&$- \left( N+\frac{1}{2}\right)$&0&$1-\left( N+\frac{1}{2}\right)r$ \\
$\tilde{b}$  &${\tiny \yng(1)}$&1& ${\tiny \yng(1)}$&$N+\frac{3}{2}$&1&$-1+ \left(N+\frac{3}{2}\right)r+\bar{r}$ \\
 $M $&1&${\tiny \yng(1)}$& ${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$N\sim q \tilde{q}$&1&${\tiny \overline{\yng(1)}}$&${\tiny \overline{\yng(1)}}$&$-1$&$-1$&$2-r -\bar{r}$ \\
$B \sim q^{2}$&1&${\tiny \overline{\yng(1,1)}}$&1&$2N+1$&$0$&$(2N+1)r$ \\
$\bar{B} \sim \tilde{q}^2$&1&1&${\tiny \overline{\yng(1,1)}}$&$-(2N+3)$&$-2$&$4-(2N+3)r-2\bar{r}$ \\
$\tilde{Y}_{SU(2)}$&1&1&1&\scriptsize $-(2N+3)(N-\bar{F})$&\scriptsize $2\bar{F}+1$&\scriptsize $2(N-\bar{F}) -(2N+3)(N-\bar{F})r+(2\bar{F} +1)\bar{r}$ \\ \hline
$b \tilde{b} \sim M$&1&${\tiny \yng(1)}$&${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$Y_{USp}$&1&1&1&\scriptsize $(2N+3)(N-\bar{F})$&\scriptsize $-(2\bar{F}+1)$&\scriptsize $2(\bar{F}-N+1) +(2N+3)(N-\bar{F})r-(2\bar{F}+1) \bar{r}$  \\ \hline
  \end{tabular}}
  \end{center}\label{USpdual2}
\end{table}

The superpotential takes
\begin{align}
W&=MN+N b \tilde{b}  +B b^2 +\bar{B}\tilde{b}^2 + \tilde{Y}_{SU(2)}Y_{USp}\\
&=B b^2 +\bar{B}\tilde{b}^2 + \tilde{Y}_{SU(2)}Y_{USp},
\end{align}
where the mesonic fields $M$ and $N$ are massive and can be integrated out. The F-term conditions lead to $N=0$ and $M\sim b \tilde{b}$. The unnecessary flat directions $b^2$, $\tilde{b}^2$ and $Y_{USp}$ are all lifted by the above superpotential.

\subsection{$SU(2N)$ with $(2F+1,1)$}
Let us consider the 3d $\mathcal{N}=2$ $SU(2N)$ gauge theory with $(2F+1,1)$ (anti-)fundamentals. In this case, the Coulomb branch is highly simplified and only $Y_{a=0}^{bare}$ can survive because we cannot anti-symmetrize the anti-fundamental matters. Table \ref{SUevenodd} shows the matter contents and their quantum numbers. The Coulomb brach $Y_{a=0}^{bare}$ is gauge invariant and does not need ``dressing''.

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(2N)$ with $(2F+1,1 )$ (anti-)fundamentals}
\begin{center}
\scalebox{0.75}{
  \begin{tabular}{|c||c||c|c|c|c| } \hline
 &$SU(2N)$&$SU(2F+1)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1&$0$&1&$\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$B:=Q^{2N}$&1&\scriptsize \begin{tabular}{c}$2N$-th \\ antisymmetric\end{tabular}&$2N$&$0$&$2Nr$ \\ \hline
$Y_0^{bare}$&1&1&$-(2F+1)N$&$-N$&\scriptsize $-(2F+1)N r -N\bar{r} +2N(F+1-N)$ \\ \hline \hline
&$SU(2F-2N+1)$&$SU(2F+1)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$q$& ${\tiny \yng(1)}$& ${\tiny \overline{\yng(1)}}$&$\frac{2N}{2F-2N+1}$&$0$& $ \frac{2N}{2F-2N+1}r$ \\
$\tilde{q}$&${\tiny \overline{\yng(1)}}$&1 &$-\frac{2F+1}{2F-2N+1}$&$-1$&$2-\frac{2F+1}{2F-2N+1}r-\bar{r}$ \\
$M$&1&${\tiny \yng(1)}$ &1&1&$r+\bar{r}$ \\\hline
$N:=\tilde{q}q$&1&${\tiny \overline{\yng(1)}}$&$-1$&$-1$&$2-r-\bar{r}$ \\
$B:=q^{2F-2N+1} $&1&\scriptsize \begin{tabular}{c}$2N$-th \\ antisymmetric\end{tabular}&$2N$&$0$&$2Nr$ \\  \hline
$\tilde{Y}_0^{bare}$&\scriptsize \begin{tabular}{c} $-2F(F-N)$ \\ under $U(1)_2$\end{tabular}&1&$-\frac{(F-N)(2F+1)(2N-1)}{2F-2N+1}$&$F-N$& \tiny \begin{tabular}{c} $-\frac{(F-N)(2F+1)(2N-1)}{2F-2N+1}r+(F-N)\bar{r} $ \\ $+2(F-N)(N-1)$  \end{tabular}  \\
$\tilde{Y}_0^{dressed}:= \tilde{Y}_0^{bare} \tilde{q}^{F}$&1&1& $-(2F+1)N$&$-N$&\scriptsize $-(2F+1)N r -N\bar{r} +2N(F+1-N)$   \\ \hline
  \end{tabular}}
  \end{center}\label{SUevenodd}
\end{table}

The dual description is a 3d $\mathcal{N}=2$ $SU(2F-2N+1)$ gauge theory with $(2F+1,1)$ (anti-)fundamental (dual) matters and a singlet $M$ with the superpotential
\begin{align}
W= M \tilde{q}q
\end{align}
The charges of the dual fields can be fixed from the superpotential and from the matching of the baryon operator as Table \ref{SUevenodd}. The magnetic Coulomb branch is also allowed only for $\tilde{Y}_{a=0}$. Since the dual gauge group is $SU(2F-2N+1)$, $\tilde{Y}_{a=0}$ need ``dressing'' in contrast to the electric side. The matching of the moduli fields are transparent from Table \ref{SUevenodd}.

\subsection{$SU(2N)$ with $(2F,2)$}
The next simple example is a 3d $\mathcal{N}=2$ $SU(2N)$ gauge theory with $(2F,2)$ (anti-)fundamental matters. The dual description is given by a 3d $\mathcal{N}=2$ $SU(2F-2N)$ with $(2F,2)$ (anti-)fundamentals and a singlet $M$. The dual theory includes the superpotential
\begin{align}
W=M \tilde{q}q.
\end{align}
The matter contents and their quantum numbers are summarized in Table \ref{SUeveneven} below.

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(2N)$ with $(2F,2)$ (anti-)fundamentals}
\begin{center}
\scalebox{0.7}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
  &$SU(2N)$&$SU(2F)$&$SU(2)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1& ${\tiny \yng(1)}$&$0$&1& $\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$& ${\tiny \yng(1)}$&1&1&$r+\bar{r}$\\
$B:=Q^{2N}$&1&\tiny \begin{tabular}{c}$2N$-th \\ antisymmetric\end{tabular}&1&$2N$&$0$&$2Nr$ \\ \hline
\footnotesize $Y_0^{bare}$&1&1&1&$-2FN$&$-2N$&\scriptsize $-2FNr -2N\bar{r}+2N(F+1-N)$ \\
\footnotesize $Y_1^{dressed}:=Y_1^{bare} (\tilde{Q}^2)^{F-1} $&1&1&1&$-2F(N-1)$&$2(F-N)$&\scriptsize $-2F(N-1)r+2(F-N)\bar{r}+2(N-1)(F-N)$ \\ \hline \hline
&$SU(2F-2N)$&$SU(2F)$&$SU(2)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$q$& ${\tiny \yng(1)}$& ${\tiny \overline{\yng(1)}}$&1&$\frac{N}{F-N}$&$0$& $\frac{N}{F-N}r$ \\
$\tilde{q}$&${\tiny \overline{\yng(1)}}$&1 & ${\tiny \yng(1)}$&$-\frac{F}{F-N}$&$-1$&$2-\frac{F}{F-N}r-\bar{r}$ \\
$M$&1&${\tiny \yng(1)}$ &${\tiny \yng(1)}$ &1&1&$r+\bar{r}$ \\\hline
$N:=\tilde{q}q$&1&${\tiny \overline{\yng(1)}}$ & ${\tiny \yng(1)}$ &$-1$&$-1$&$2-r-\bar{r}$ \\
$b:=q^{2F-2N} \sim B$&1&\tiny \begin{tabular}{c}$2N$-th \\ antisymmetric\end{tabular}&1&$2N+1$&$0$&$(2N+1)r$ \\ \hline
\footnotesize $\tilde{Y}_0^{bare}$&1&1&1&$-2F(N-1)$&$2(F-N)$&\scriptsize $-2F(N-1)r +2(F-N)\bar{r} +2(N-1)(F-N)$ \\
\footnotesize $\tilde{Y}_1^{dressed}:=\tilde{Y}_1^{bare} (\tilde{q}^2 )^{F-1}$&1&1&1&$-2FN$&$-2N$&\scriptsize $-2FNr-2N\bar{r} +2N(F+1-N)$ \\ \hline
  \end{tabular}}
  \end{center}\label{SUeveneven}
\end{table}

On the electric side, the Coulomb branch is allowed for $Y_{a}^{bare} \,(a=0,1)$. Since the $U(1)_2$ is absent for $a=0$, $Y_{a=0}^{bare}$ is gauge invariant. For $a=1$, $Y_{a=1}^{bare}$ is not gauge invariant and needs ``dressing''. The indices of the dressing factor $\tilde{Q}^2$ is anti-symmetrized and the dressed operator $Y_1^{dressed}:=Y_1^{bare} (\tilde{Q}^2)^{F-1}$ is a flavor singlet. On the dual side, there are also two Coulomb branches $\tilde{Y}_a^{bare}\,(a=0,1)$. $\tilde{Y}_0^{bare}$ is gauge invariant and corresponds to $Y_1^{dressed}$. $\tilde{Y}_1^{bare}$ is not gauge invariant and must be dressed by $\tilde{q}^2$. The dressed operator $\tilde{Y}_1^{dressed}$ can be identified with $Y_0^{bare}$.

For $F=N+1$, the dual gauge group is $SU(2)$ and only  one monopole operator $\tilde{Y}^{bare}_0\sim \tilde{Y}_{SU(2)}$ is available. The remaining Coulomb branch comes from the anti-baryonic operator defined as $\bar{b}:=\tilde{q}^2$. The dual Coulomb branch operator $\tilde{Y}_{SU(2)} $ is identified with $Y_1^{dressed}$. The identification between $Y_0^{bare}$ and $\bar{b}:=\tilde{q}^2$ needs a special care. On the electric side, we have to take the $N$-th root of $Y_0^{bare}$. Then, we find the matching $Y_0^{\substack{ minimal \\ bare }}:= (Y_0^{bare})^{\frac{1}{N}} \sim \bar{b}$.

\subsection{$SU(2N+1)$ with $(2F+1,1)$}
Let us move on to the duality of the 3d $\mathcal{N}=2$ $SU(2N+1)$ gauge theory with $2F+1$ fundamental matters and a single anti-fundamental matter. The dual description is a 3d $\mathcal{N}=2$ $SU(2F-2N)$ gauge theory with $2F+1$ fundamental (dual) matters, a single anti-fundamental (dual) matter and a singlet $M$ with a superpotential
 %
\begin{align}
W=M \tilde{q}q.
\end{align}
The matter contents and their representations are summarized in Table \ref{SUoddeven}.

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(2N+1)$ with $(2F+1,1 )$ (anti-)fundamentals}
\begin{center}
\scalebox{0.83}{
  \begin{tabular}{|c||c||c|c|c|c| } \hline
 &$SU(2N+1)$&$SU(2F+1)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1&$0$&1&$\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$B:=Q^{2N}$&1&\scriptsize \begin{tabular}{c}$(2N+1)$-th \\ antisymmetric\end{tabular}&$2N+1$&$0$&$(2N+1)r$ \\ \hline
\footnotesize $Y_0^{dressed}:=Y_0^{bare}\tilde{Q}^{F}$&1&1& $-(2F+1)N$&$F-N$&\scriptsize $-(2F+1)Nr +(F-N)\bar{r} +2N(F-N)$ \\ \hline  \hline
&$SU(2F-2N)$&$SU(2F+1)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$q$& ${\tiny \yng(1)}$& ${\tiny \overline{\yng(1)}}$&$\frac{2N+1}{2F-2N}$&$0$& $ \frac{2N+1}{2F-2N}r$ \\
$\tilde{q}$&${\tiny \overline{\yng(1)}}$&1 &$-\frac{2F+1}{2F-2N}$&$-1$&$2-\frac{2F+1}{2F-2N}r-\bar{r}$ \\
$M$&1&${\tiny \yng(1)}$ &1&1&$r+\bar{r}$ \\\hline
$N:=\tilde{q}q$&1&${\tiny \overline{\yng(1)}}$&$-1$&$-1$&$2-r-\bar{r}$ \\
$B:=q^{2F-2N} $&1&\scriptsize \begin{tabular}{c}$(2N+1)$-th \\ antisymmetric\end{tabular}&$2N+1$&$0$&$(2N+1)r$ \\  \hline
$\tilde{Y}_0^{bare}$&1&1&$-(2F+1)N$&$F-N$& \scriptsize$-(2F+1)Nr -(F-N)\bar{r}+2N(F-N) $  \\ \hline
  \end{tabular}}
  \end{center}\label{SUoddeven}
\end{table}

The Coulomb branch operators are allowed only for $a=0$ on both sides. The electric Coulomb branch operator $Y_0^{bare}$ is not gauge invariant and must be dressed by $\tilde{Q}$. On the other hand, the magnetic Coulomb branch $\tilde{Y}_0^{bare}$ is gauge invariant and can be used for a moduli coordinate. From Table \ref{SUoddeven}, we can see the exact matching these two operator under the duality.

\subsection{$SU(2N+1)$ with $(2F+2,2)$}
The final example is a 3d $\mathcal{N}=2$ $SU(2N+1)$ gauge theory with $2F+2$ fundamental and $2$ anti-fundamental matters. The proposed dual description is given by a 3d $\mathcal{N}=2$ $SU(2F-2N+1)$ gauge theory with $2F+2$ fundamental and $2$ anti-fundamental (dual) matters in addition to a singlet $M$. The dual theory has a cubic interaction
\begin{align}
W=M \tilde{q}q.
\end{align}
The matter contents and their quantum numbers are summarized in Table \ref{SUodd2}. Since we consider the region with $F>N$, anti-baryon operators are not available on both sides.

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(2N+1)$ with $(2F+2,2)$ (anti-)fundamentals}
\begin{center}
\scalebox{0.7}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
  &$SU(2N+1)$&$SU(2F+2)$&$SU(2)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1& ${\tiny \yng(1)}$&$0$&1& $\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$& ${\tiny \yng(1)}$&1&1&$r+\bar{r}$\\
$B:=Q^{2N}$&1&\tiny \begin{tabular}{c}$(2N+1)$-th \\ antisymmetric\end{tabular}&1&$2N+1$&$0$&$(2N+1)r$ \\ \hline
\footnotesize $Y_0^{dressed}:=Y_0^{bare}\tilde{Q}^{F}$&1&1& \tiny \begin{tabular}{c}$F$-th \\ symmetric\end{tabular}&$-2(F+1)N$&$F-2N$&\scriptsize $-2(F+1)Nr+(F-2N)\bar{r} +2N(F-N+1)$ \\  \hline \hline
&$SU(2F-2N+1)$&$SU(2F+2)$&$SU(2)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
$q$& ${\tiny \yng(1)}$& ${\tiny \overline{\yng(1)}}$&1&$\frac{2N+1}{2F-2N+1}$&$0$& $\frac{2N+1}{2F-2N+1}r$ \\
$\tilde{q}$&${\tiny \overline{\yng(1)}}$&1 & ${\tiny \yng(1)}$&$-\frac{2F+2}{2F-2N+1}$&$-1$&$2-\frac{2F+2}{2F-2N+1}r-\bar{r}$ \\
$M$&1&${\tiny \yng(1)}$ &${\tiny \yng(1)}$ &1&1&$r+\bar{r}$ \\\hline
$N:=\tilde{q}q$&1&${\tiny \overline{\yng(1)}}$ & ${\tiny \yng(1)}$ &$-1$&$-1$&$2-r-\bar{r}$ \\
$b:=q^{2F-2N+1} \sim B$&1&\tiny \begin{tabular}{c}$(2N+1)$-th \\ antisymmetric\end{tabular}&1&$2N+1$&$0$&$(2N+1)r$ \\ \hline
\footnotesize $\tilde{Y}_0^{dressed}:=\tilde{Y}_0^{bare} (\tilde{q} )^{F}$&1&1&\tiny \begin{tabular}{c}$F$-th \\ symmetric\end{tabular}&$-2(F+1)N$&$F-2N$&\scriptsize $-2(F+1)Nr+(F-2N)\bar{r} +2N(F-N+1)$ \\ \hline
  \end{tabular}}
  \end{center}\label{SUodd2}
\end{table}

The electric Coulomb branch is allowed only for $Y^{bare}_{a=0}$, which corresponds to the breaking
\begin{align}
SU(2N+1) \rightarrow SU(N) \times SU(N) \times U(1)_1 \times U(1)_2.
\end{align}
The $U(1)_2$ charge of the bare operator is $-2FN$. This charge should be canceled by $1_{0,2N} \in \tilde{Q}$. Then the dressed operator becomes $Y_0^{dressed}:=Y^{bare}_{a=0} \tilde{Q}^{F}$. On the magnetic side, the Coulomb branch is allowed only for $a=0$ and the dressed operator becomes $\tilde{Y}_0^{dressed}:=\tilde{Y}_0^{bare} (\tilde{q} )^{F}$. From the charges of these two operators, it is found that these are identified with each other. Notice that, in this dressing procedure, the Coulomb branch operators become $F$-th symmetric representations of the $SU(2)$ flavor symmetry.

\section{Confinement and SUSY breaking}

In this section, we will consider the confinement phases of the 3d $\mathcal{N}=2$ ``chiral'' $SU(N)$ gauge theory with $(F,\bar{F})$ (anti-)fundamentals. (We again assume $F> \bar{F}$.) The appearance of the confinement phase can be regarded as a special limit of our duality proposed in the previous section.
We will claim that (s-)confinement phases are more ubiquitous than expected from a naive semi-classical analysis of the moduli space of vacua since the dressed monopole operators would not be flavor singlets and then a confining superpotential is available by properly contracting the flavor indices of the meson, baryon and monopole operators. In earlier works \cite{Intriligator:2013lca}, these phases were overlooked due to lack of understanding of the quantum Coulomb branches.

\subsection{$SU(N)$ with $(F,\bar{F})=(N+1,N-1)$}
In \cite{Aharony:1997bx}, it was discussed that the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})=(N,N)$ vector-like matters exhibits s-confinement. Then one can naively guess that the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})=(N+1,N-1)$ also shows a similar confinement since this theory differs very little from the former theory and since the total number of the (anti-)fundamental matters does not change. We here claim that this guess is indeed correct.

Let us consider the Coulomb branch labeled by $Y=Y_1\cdots Y_{N-1}$, where $Y_i$ corresponds to the fundamental monopoles. The expectation value for $Y$ breaks the gauge group to $SU(N-2) \times U(1)_1 \times U(1)_2$ and $Y$ is related to a $U(1)_1$ subgroup. Due to the ``chiral'' nature of the theory, the effective Chern-Simons level is induced for $k_{eff}^{U(1)_1 U(1)_2}$ along the RG flow. Therefore, $Y$ is no longer gauge-invariant and charged under the $U(1)_2$ symmetry. The effective CS term is calculated as
\begin{align}
k_{eff}^{U(1)_1 U(1)_2} =F-\bar{F}
\end{align}
and then the $U(1)_2$ charge of $Y$ is $-2$ in the current case. In order to construct the gauge invariant Coulomb branch, we can appropriately combine $Y$ and $\tilde{Q}$ into $Y_d :=Y\tilde{Q}^{N-2}$. As a result, the Coulomb branch is not a flavor singlet. The quantum numbers of the moduli fields are summarized in Table \ref{Scon1}.

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(N)$ with $N+1$ fundamentals and $N-1$ anti-fundamentals}
\begin{center}
\scalebox{0.82}{
  \begin{tabular}{|c||c|c|c|c|c|c| } \hline
  &$SU(N)$&$SU(N+1)$&$SU(N-1)$&$U(1)_B$&$U(1)_A$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&1&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1&${\tiny \yng(1)}$&$-1$&1&$\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$&${\tiny \yng(1)}$&0&2&$r+\bar{r}$ \\
$B:=Q^N$&1&${\tiny \overline{\yng(1)}}$&1&$N$&$N$&$Nr$ \\ \hline
$Y:=Y_1 \cdots Y_{N-1}$&$-2$ under $U(1)_2$&1&1&$-2$&$-2N$&$2-(N+1)r-(N-1)\bar{r}$ \\
$Y_d:=Y \tilde{Q}^{N-2}$&1&1&${\tiny \overline{\yng(1)}}$&$-N$&$-N-2$&$2-(N+1)r -\bar{r}$ \\ \hline
  \end{tabular}}
  \end{center}\label{Scon1}
\end{table}

From Table \ref{Scon1}, we see that the low-energy effective degrees of freedom are described by three singlets $M, B$ and $Y_d$ and that the effective superpotential becomes
\begin{align}
W= Y_d M B. \label{sconSUN}
\end{align}
One can check that the parity anomaly matching is satisfied by this confining description. The another check of this phase is that we can derive the same IR description by using the duality discussed in a previous section. Since the dual gauge group is absent, the dual (anti-)quarks $q$ and $\tilde{q}$ are gauge invariant, which are identified with $B$ and $Y_d$ respectively. The dual superpotential takes the same form as \eqref{sconSUN}. This superpotential can be also derived from the 4d s-confinement phase. Let us consider the 4d $\mathcal{N}=1$ $SU(N)$ gauge theory with $N+1$ flavors \cite{Seiberg:1994bz}, which is s-confining and described by the superpotential
\begin{align}
W= -\det M + B^i M_{ia} \bar{B}^a.
\end{align}
By putting the theory on a circle and introducing the real masses for the $SU(2)$ subgroup of the second $SU(N+1)$ flavor symmetry, the electric theory flows to the ``chiral'' $SU(N)$ theory in Table \ref{Scon1}. The effects of the twisted instanton are turned off in this deformation \cite{Aharony:2013dha}.
On the confined side, the real masses are introduced to the meson and the anti-baryons. By identifying the anti-baryon as the dressed Coulomb branch and integrating out the massive components, the superpotential is correctly reduced to \eqref{sconSUN}.

In what follows, we will deform this confining phase by introducing the complex masses and flow to the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})=(N-a,N-a-2)$, where $a \ge 0$.

\subsubsection*{$SU(N)$ with $(F,\bar{F})=(N,N-2)$}
By introducing a complex mass to one pair of (anti-)fundamental matters, we can flow to a 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})=(N,N-2)$. The quantum numbers of the moduli coordinates are summarized in Table \ref{QDM}. In this case, the flavor indices of $\tilde{Q}^{N-2}$ inside the dressed monopole $Y_d$ is completely anti-symmetrized and $Y_d$ is now a flavor singlet.

\begin{table}[H]\caption{3d $\mathcal{N}=2$ $SU(N)$ with $N$ fundamentals and $N-2$ anti-fundamentals}
\begin{center}
\scalebox{0.89}{
  \begin{tabular}{|c||c|c|c|c|c|c| } \hline
  &$SU(N)$&$SU(N)$&$SU(N-2)$&$U(1)_B$&$U(1)_A$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&1&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1&${\tiny \yng(1)}$&$-1$&1&$\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$&${\tiny \yng(1)}$&0&2&$r+\bar{r}$ \\
$B:=Q^N$&1&1&1&$N$&$N$&$Nr$ \\ \hline
$Y:=Y_1 \cdots Y_{N-1}$&$-2$ under $U(1)_2$&1&1&$-2$&$-2N+2$&$-Nr-(N-2)\bar{r}$ \\
$Y_d:=Y \tilde{Q}^{N-2}$&1&1&1&$-N$&$-N$&$-Nr $ \\ \hline
  \end{tabular}}
  \end{center}\label{QDM}
\end{table}

From Table \ref{QDM}, we can introduce a quantum-deformed constraint between $B$ and $Y_d$
 \begin{align}
B Y_d =1, \label{QC}
\end{align}
while $M$ has no constraint and will become a free field. This phase can be also derived from the previous subsection via a complex mass deformation $\Delta W= m M_{N+1,N-1}$.

\subsubsection*{$SU(N)$ with $(F,\bar{F})=(N-1,N-3)$}
In this case, there is no Coulomb branch operator since we cannot construct the baryonic operator $\tilde{Q}^{N-2}$. By introducing the complex masses for the previous case, one finds no solution for the F-flatness condition of $M$. Therefore, the supersymmetry would be spontaneously broken for a 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})=(N-a,N-a-2),~a \ge 1$.

\subsection{$SU(2N)$ with $(F,\bar{F})=(2N+1,2\bar{F}+1)$}
We can generalize the confinement phase observed in the previous subsection. The confinement phases appear by taking a special case of the dualities in Section 3. Let us start with the $SU(2N)$ cases. The theory is a 3d $\mathcal{N}=2$ $SU(2N)$ gauge theory with $2N+1$ fundamentals and $2\bar{F}+1$ anti-fundamentals, where we assume $\bar{F} < N$.

In this case, almost all the Coulomb branches are lifted and only $Y_{a=\bar{F}}$, corresponding to the breaking
\begin{align}
SU(2N) \rightarrow SU(N-\bar{F}) \times SU(N-\bar{F}) \times SU(2\bar{F}) \times U(1)_1 \times U(1)_2,
\end{align}
can survive and remain exactly massless. The bare monopole $Y^{bare}_{\bar{F}}$ is not gauge invariant and should be dressed by $\tilde{Q}^{2\bar{F}}$. Now, we have to be careful about the choice of the correct monopole operator since the $U(1)_2$ charge of the minimal monopole
\begin{align}
Y_{\bar{F}}^{\small \substack{minimal \\bare}} := ( Y^{bare}_{\bar{F}})^{\frac{1}{N-\bar{F}}}
\end{align}
is $-(2N-2\bar{F})$ and this can be dressed by $\tilde{Q}^{2a=2\bar{F}}$. Therefore, the Coulomb branch will be described by the minimal dressed monopole instead of the naive one.
\begin{align}
Y_{\bar{F}}^{\small \substack{minimal \\dressed}} :=( Y^{bare}_{\bar{F}})^{\frac{1}{N-\bar{F}}} \tilde{Q}^{2\bar{F}}
\end{align}
The matter contents and the quantum numbers of the moduli coordinates are summarized in Table \ref{SCONeven}. From a dual theory point of view, $M:=\tilde{Q}Q$ is regarded as a singlet $M$. The baryon operator $B:=Q^{2N}$ is identified with a dual quark $q$ which is now a gauge-singlet. The Coulomb branch operator $Y_{\bar{F}}^{\substack{minimal \\ dressed}}$ corresponds to the dual anti-quark $\tilde{q}$.

\begin{table}[H]\caption{$SU(2N)$ with $(2N+1,2\bar{F}+1)$ (anti-)fundamentals}
\begin{center}
\scalebox{0.83}{
  \begin{tabular}{|c||c|c|c|c|c|c| } \hline
  &$SU(2N)$&$SU(2N+1)$&$SU(2\bar{F}+1)$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1&${\tiny \yng(1)}$&$0$&1&$\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$&${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$B:=Q^{2N}$&1&${\tiny \overline{\yng(1)}}$&1&$2N$&$0$&$2Nr$ \\ \hline
\footnotesize $Y_{\bar{F}}^{\substack{minimal \\ dressed}} :=(Y_{\bar{F}}^{bare} )^{\frac{1}{N-\bar{F}}} \tilde{Q}^{2\bar{F}} $&1&1&${\tiny \overline{\yng(1)}}$&$-(2N+1)$&$-1$&$2-(2N+1)r -\bar{r}$  \\  \hline
  \end{tabular}}
  \end{center}\label{SCONeven}
\end{table}

The theory is dual to a non-gauge theory with three chiral superfields $M, B$ and $Y_{\bar{F}}^{\substack{minimal \\ dressed}}$ with a cubic superpotential
\begin{align}
W=BM Y_{\bar{F}}^{\substack{minimal \\ dressed}} =M \tilde{q}q.
\end{align}
This s-confining phase generalizes the result of the previous subsection. By introducing a complex mass, it  is shown that a 3d $\mathcal{N}=2$ $SU(2N)$ gauge theory with $(2N,2\bar{F})$ has a single quantum constraint between the baryon and monopole operators $BY_{\bar{F}}^{\substack{minimal \\ dressed}} =1$ and there is a free meson $M$, which is the same as \eqref{QC}. By further intoducing the complex masses to more flavors, we find that a 3d $\mathcal{N}=2$ $SU(2N)$ gauge theory with $(2N-a,2\bar{F}-a)$ (anti-)fundamentals spontaneously breaks the supersymmetry.

\subsection{$SU(2N+1)$ with $(2N+2,2\bar{F})$}
Finally, we consider the confinement phase for the $SU(2N+1)$ chiral gauge theory. This case also comes from the $SU(2N+1)$ duality by taking the special limit where the dual gauge group becomes null. The electric theory is a 3d $\mathcal{N}=2$ $SU(2N+1)$ gauge theory with $2N+2$ fundamental and $2\bar{F}$ anti-fundamental matters. Since we are interested in a ``chiral'' theory, we require $\bar{F} \le N$ without loss of generality and the following argument is applicable only to this region.

The Coulomb branch $Y_{a=\bar{F}-1} $, corresponding to the breaking
\begin{align}
SU(2N+1) \rightarrow SU(N-\bar{F}+1) \times SU(N-\bar{F}+1) \times SU(2\bar{F}-1) \times U(1)_1, \times U(1)_2
\end{align}
can remain flat after including the quantum corrections. Since $Y_{a=\bar{F}-1}^{bare}$ is not gauge invariant, we have to dress the bare monopole operator by $\tilde{Q}^{2a+1}=\tilde{Q}^{2\bar{F}-1}$.
In this particular matter contents, we can dress the more smaller monopole by using $\tilde{Q}^{2a+1}$. The minimal monopole corresponds to
\begin{align}
Y_{\bar{F}-1}^{ \substack{minimal \\ bare}} :=( Y_{\bar{F}-1}^{bare})^{\frac{1}{N-\bar{F}+1}}
\end{align}
and the dressed minimal operator becomes
\begin{align}
Y_{\bar{F}-1}^{\substack{minimal \\ dressed}} :=Y_{\bar{F}-1}^{ \substack{minimal \\ bare}}\tilde{Q}^{2\bar{F}-1},
\end{align}
which takes an anti-fundamental representation of the $SU(2\bar{F})$ global symmetry. The matter contents and their charges are calculated in Table \ref{SCONodd} below.

\begin{table}[H]\caption{$SU(2N+1)$ with $(2N+2,2\bar{F})$ (anti-)fundamentals}
\begin{center}
\scalebox{0.81}{
  \begin{tabular}{|c||c||c|c|c|c|c| } \hline
  &$SU(2N+1)$&$SU(2N+2)$&$SU(2\bar{F})$&$U(1)$&$U(1)$&$U(1)_R$  \\ \hline
 $Q$ & ${\tiny \yng(1)}$ & ${\tiny \yng(1)}$&1&1&0&$r$ \\
$\tilde{Q}$  &${\tiny \overline{\yng(1)}}$&1&${\tiny \yng(1)}$&$0$&1&$\bar{r}$ \\  \hline
$M:=\tilde{Q}Q$&1&${\tiny \yng(1)}$&${\tiny \yng(1)}$&1&1&$r+\bar{r}$ \\
$B:=Q^{2N+1}$&1&${\tiny \overline{\yng(1)}}$&1&$2N+1$&$0$&$(2N+1)r$ \\ \hline
\footnotesize $Y_{\bar{F}-1}^{\substack{minimal \\ dressed}} :=(Y_{\bar{F}-1}^{bare} )^{\frac{1}{N-\bar{F}+1}} \tilde{Q}^{2\bar{F}-1} $&1&1&${\tiny \overline{\yng(1)}}$&$-(2N+2)$&$-1$&$2-(2N+2)r -\bar{r}$  \\  \hline
  \end{tabular}}
  \end{center}\label{SCONodd}
\end{table}

The low-energy dynamics is described by $M, B$ and $Y_{\bar{F}-1}^{\substack{minimal \\ dressed}}$ with a cubic superpotential
\begin{align}
W=BM Y_{\bar{F}-1}^{\substack{minimal \\ dressed}}.
\end{align}
One can easily check that the parity anomaly matching is satisfied by this dual description. The another check is that one can flow to the $SU(2N)$ s-confinement phase by giving an expectation value to $\braket{M}$.

By introducing the complex masses to this s-confinement phase, we find that a
 3d $\mathcal{N}=2$ $SU(2N+1)$ gauge theory with $(2N+1,2\bar{F}-1)$ (anti-)fundamentals has a quantum-deformed moduli space of the baryon and monopole operators $B Y_{\bar{F}-1}^{\substack{minimal \\ dressed}} =1$ with a single free field $M$. It is also found that the supersymmetry is spontaneously broken for an $SU(2N+1)$ gauge theory with $(2N+1-a,2\bar{F}-1-a)$ (anti-)fundamentals, where $a \ge 1$.
By combining the results of the $SU(2N)$ and $SU(2N+1)$ cases, we conclude that the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})$ (anti-)fundamentals exhibits a spontaneous supersymmetry breaking for $N>F>\bar{F}$.

\section{Summary and Discussion}

In this paper, we investigated the quantum structure of the Coulomb moduli space of vacua in the 3d $\mathcal{N}=2$ $SU(N)$ ``chiral'' gauge theories with $F$ fundamental and $\bar{F}$ anti-fundamental matters. In case of a vector-like theory with $F=\bar{F}$, almost all the flat directions of the classical Coulomb branches are lifted and only a single direction ($Y_{N-1}^{bare}$ in our notation) remains exactly massless. On the other hand, in the ``chiral'' $SU(N)$ theories, the very many Coulomb branch directions could generally remain flat, which were described by $Y_a^{bare} (a=0,\cdots,N-1)$. These bare operators are not gauge invariant due to the mixed Chern-Simons term. Then, we constructed the correct gauge-invariant operators by dressing them with the anti-baryonic operators $\tilde{Q}$. Due to this dressing procedure, the Coulomb branch operators are no longer flavor singlets. The previous research \cite{Intriligator:2013lca} was missing this possibility.

By using the ``dressed'' Coulomb branch operators, we proposed the duality for the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})$ (anti-)fundamentals. The dual description is given by a 3d $\mathcal{N}=2$ $SU(F-N)$ gauge theory with $(F,\bar{F})$ (anti-)fundamental matters and a meson $M$ with a superpotential $W=M \tilde{q}q$. Naively, one might think that the baryonic operators do not match on both sides of the duality. However, the dressed monopole operators remedy this and the duality works well. The total number of the (anti-)baryonic and Coulomb branch operators is identical on both sides and one can find the correct matching between the Coulomb branch and anti-baryonic operators.
By focusing on the duality where the dual gauge group is $SU(2)$ and by applying the Aharony duality \cite{Aharony:1997gp}, we found a novel duality between the $SU(N)$ and $USp(2 \tilde{N})$ theories.
For the 3d $\mathcal{N}=2$ $SU(2)$ gauge theory with $2F$ fundamentals, we can construct the various ``chiral'' dualities.

We also found the novel confinement phases of the ``chiral'' $SU(N)$ gauge theory. The confinement phases appear for $F=N+1$ and $F>\bar{F}$, where the dual gauge group becomes null. The confined degrees of freedom are the meson, the baryon and the dressed Coulomb branch operator. The low-energy dynamics is governed by the cubic interaction of these three fields whose flavor indices are naturally contracted. By introducing the complex masses to these confining phases, we found that the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(N,N-a)$ (anti-)fundamentals have the free meson $M$ and the quantum deformed moduli space, where $a$ must be even. By further introducing the complex masses, the 3d $\mathcal{N}=2$ $SU(N)$ gauge theory with $(F,\bar{F})$ (anti-)fundamentals exhibits a spontaneous supersymmetry breaking when $N>F>\bar{F}$ and $F+\bar{F}$ is even.

In this paper, we introduced the important class of the Coulomb branch operators $Y_a^{bare}$ which typically appear in the ``chiral'' theories. However, we do not exhaust all the Coulomb branch operators. Generally, there could be other operators. For example, we did not consider the possibility where the dressed monopoles are constructed from the bare monopoles, anti-quarks and gauginos. Typically, they will take the following form
\begin{align}
Y_a^{bare} \tilde{Q}^b \lambda^c, \label{missing}
\end{align}
where $b$ and $c$ are chosen as the above combination becomes gauge invariant.
These operators were studied in case of the $U(N)$ theory \cite{Aharony:2015pla}. One might consider that the matching of these additional operators ruins the proposed duality. However, the matching of the proposed Coulomb branch operators would be highly strong and non-trivial enough to test the validity of the duality. It is worth studying when these additional operators appear or disappear on both sides of the duality. This would be an additional check of our duality.

It would be very important to test our duality and confinement phases from the superconformal index computation \cite{Bhattacharya:2008bja, Kim:2009wb, Imamura:2011su, Kapustin:2011jm}. We have partially done this in Appendix only for the $SU(2)$ case. The additional Coulomb branch operators \eqref{missing} which could appear for more higher rank gauge groups can be studied from the index. Probably, in some cases, the lowest Coulomb branch operator which has a minimal conformal weight could be different from the operators which we defined in this paper. In such cases, \eqref{missing} will give the lowest contribution.
It is also worth understanding our duality and confinement phases from the four-dimensional Seiberg duality point of view by following the same path as \cite{Aharony:2013dha}. This could be achieved by introducing the real masses only for the anti-fundamental matters and flowing to the 3d limit.
The duality of the ``chiral'' $SU(N)$ Chern-Simons matter theories was discussed in \cite{Aharony:2014uya}. The $U(N)$ CS duality was studied in \cite{Giveon:2008zn, Benini:2011mf}. It is curious to find the RG flow from our duality to those Chern-Simons dualities.

\section*{Acknowledgments}
I would like to thank Ofer Aharony and Antonio Amariti for valuable comments and discussions.
This work is supported by the Swiss National Science Foundation (SNF) under grant number PP00P2\_157571/1 and by ``The Mathematics of Physics'' (SwissMAP) under grant number NCCR 51NF40-141869.

redpajama_set_name: RedPajamaArXiv


The following local residents have earned awards or honors from their respective colleges or universities.
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