text:
Prudence Island Light – ipl's Lighthouses: A Photographic Journey
Engineering, North America, North America Travel and Tourism, Quotations, Travel and Tourism, Travel News
Prudence Island Light – ipl2's Lighthouses: A Photographic Journey
"This 30-foot tall octagonal granite tower was originally located on the southern end of Newport's Goat Island. In 1851 it was dismantled and then reassembled at Sandy Point on Prudence Island. It was relighted in January of 1852." It was built in 1823.
AskOxford.com: Free online dictionary resources from Oxford University Press
Dictionaries, Quotations, Reference, Reference, Reference Resources for Teachers and Parents
Provides essays, definitions, resources, FAQs, and games about the English language, usage, history, grammar, and related topics. If your questions are not answered on the website, you can send them to the Oxford Word and Language service team.
Subject: Reference–Quotations; Reference–Dictionaries
The Columbia World of Quotations
Quotations, Reference, Reference, Reference Resources for Teachers and Parents
"The 65,000 essential quotations that constitute this authoritative collection represent the research of 154 experts. Entries from more than 5,000 authors and speakers are multiply classified into 6,500 subjects." Search full text, by author or category, or browse an alphabetical author index.
Subject: Reference–Quotations
Simpson's Contemporary Quotations
A quotation guide, convering 1950-1988. "Nearly 10,000 notable quotes from 4,000 quoatable notables. Covers all of modern American life–including politics and government, architecture, art, music and dance, theater, radio and television, and religion." Use either the author index or the subject index, or search for a quotation.
Biographies, Book Lists and Awards, Calculation and Conversion Tools, Calendars, Census Data and Demographics, Chemical Sciences (Chemistry), Dictionaries, Engineering, Finance, Genealogy, Geography, Language and Linguistics, Literature, Mathematics, Movies, Music, Personal Finance, Physics, Quotations, Time and Weather
Wolfram Alpha quickly locates facts, graphs, definitions, tables, numbers, formulas, and other data. Use the "Examples" page to browse and see all the different topics covered.
Subject: Arts & Humanities–Literature–Book Lists & Awards; Health & Medical Sciences; Reference–Geography; Reference–Genealogy; Science & Technology–Earth Sciences; Arts & Humanities–Language & Linguistics; Science & Technology–Chemical Sciences (Chemistry); Computers & Internet; Science & Technology–Life Sciences; Science & Technology–Astronomy; Reference–Dictionaries; Arts & Humanities–Literature; Arts & Humanities–History; Reference–Time & Weather; Reference–Biographies; Science & Technology–Engineering; Business & Economics–Finance; Science & Technology–Physics; Science & Technology–Mathematics; Regional & Country Information; Entertainment & Leisure–Music; Reference–Quotations; Reference–Thesauruses Or Thesauri; Reference–Calendars; Reference–Census Data & Demographics; Reference–Calculation & Conversion Tools; Entertainment & Leisure–Movies; School And Homework Help–Homework Help–Subject Area Help–Math; Clubs And Organizations; School And Homework Help–Homework Help–Subject Area Help–Geography; School And Homework Help–Homework Help–Subject Area Help–Biographies; Money And Work–Personal Finance–Saving And Investment; School And Homework Help–Homework Help–Subject Area Help–Music; School And Homework Help–Homework Help–Subject Area Help—-English And Literature; Sports, Entertainment, And Arts–Sports; School And Homework Help–Homework Help—-Reference; School And Homework Help–College; School And Homework Help–Homework Help–Subject Area Help–History
Quotations and Proverbs Search
Drama and Performance (Theater and Stagecraft), Fine Arts, Performing Arts, Quotations, Reference, Reference, Reference Resources for Teachers and Parents, Theater
This is a comprehensive directory of quotation and proverb resources available online. If you are looking for a variety of resources, this is a good place to start!
The Movie Quotes Site
Movies, Quotations, Screenwriting
This site has a huge collection of quotes from major motion pictures. It includes a "guess the quote" game and links to minutia and mistakes from each movie.
Subject: Reference–Quotations; Entertainment & Leisure–Movies
Confucian Analects : the Lun Yu
Online Texts, Original Contemporary Fiction, Philosophy, Quotations, Religion and Theology
This site contains the Lun Yu (499 sayings of Confucius) in more than twenty languages, as well as a varying amount of other Confucius related contents for each language.
Subject: Arts And Humanities–Online Texts–Classical Chinese Texts; Arts & Humanities–Religion & Theology–Faiths & Denominations–Other Faiths–Other Faiths Religious & Sacred Texts; Reference–Quotations; Arts & Humanities–Philosophy–Eastern–Chinese
The Quotations and Sayings Database
"A search portal for finding literary quotations and sayings from all the best databases on the internet."


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Rezső Patkoló, also known in Poland as Rudolf Patkolo (13 October 1922 – 1 September 1992) was a Hungarian-born footballer who played international football for both Hungary and Poland. He played as a striker for Gamma Budapest, Újpest, ŁKS Łódź, Polonia Bydgoszcz, Wisła Kraków, Stal Stalowa Wola and Kujawiak Włocławek.

References

1922 births
1992 deaths
Footballers from Budapest
Hungarian footballers
Hungary international footballers
Citizens of Poland through descent
Polish people of Hungarian descent
Hungarian people of Polish descent
Polish footballers
Poland international footballers
Dual internationalists (football)
Újpest FC players
ŁKS Łódź players
Wisła Kraków players
Stal Stalowa Wola players
Kujawiak Włocławek players
Association football forwards

redpajama_set_name: RedPajamaWikipedia


Freeside's east gate is a location in the Mojave Wasteland. Located just north-east of the Gun Runners, it is one of two direct routes that lead into Freeside; the other being Freeside's north gate.
Freeside's east gate opens to the portion of Freeside that contains Mick & Ralph's, as well as the Old Mormon Fort. These are the only two openly inhabited buildings in this portion of Freeside. The ghoul Rotface is usually leaning against a wall near Mick & Ralph's. Max and Stacey can be found chasing each other just inside the gate near a few bodyguards for hire.
Heading into the ruins and going left when entering Freeside (across Mick & Ralph's), one can find a dead body simply referred to as "man" in the western corner of the ruin . Within the man's inventory is a destroyed collar and pre-War casual wear. The collar is similar to the slave collar, and the one given by the Brotherhood of Steel at the beginning of the quest Still in the Dark. Lying next to the man is a wrench, detailing the man's failed attempt to take off the collar successfully.
Freeside's east gate appears only in Fallout: New Vegas.
The faction associated with this area is Freeside Gate Safe Faction, or as the game files say, DontTazeMeBroFaction.


Developers this is Haverhill's best kept secret – this 14 acre parcel is naturally beautiful, with mature trees, scenic space and an enormous flat area. It's so well hidden from the road you would have no idea it exists in a residential neighborhood. The Gile street side is sloping with a single family home. Abutting a townhome community & upscale, recently built neighborhood. Super convenient to town yet totally tucked away from traffic. The majority of this land is natural. One adorable ranch on front of the lot on Gile Street. All research and due diligence to be at buyer's expense. Potential uses will require investigation by buyer, ideas have been brought up including cluster development, planned unit development, townhome community, a cluster of luxury homes with oversized lots, solar farm, cell tower, large vegetable farm. Do not access land without an appointment under any circumstances. Offers due by 5/15. Video with drone footage of land, house & overhead photos to be posted.


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The Trident prototypes were first built in France but in August 1919 it was announced that the Trident would be manufactured in England. Production in the UK is thought to have never started with planned models of a one passenger taxi-cab and delivery van. The Trident was powered by an 8hp 2-cylinder engine that was mounted on, and driving, the single front wheel. The engine was mounted on the near side whilst the gearbox, clutch and magneto were mounted on the off side. The vehicle was a two-seater that sat two people in tandem with a steel body that was suspended by four laterally mounted and interconnected cantilever springs. Production ceased in 1920.
The 1919 Trident. (My thanks to Kerry Croxton for sending in this picture)
Google: Search for Trident

redpajama_set_name: RedPajamaCommonCrawl


text:
Black Indian
An Object of Art
Sean Drakes
Ashraph with a ring of red coral set in sterling silver. At right, his Black Indian adorned with a pendant made of smoky quartz.
Interview and photography by Sean Drakes
Ashraph is not a jeweller! His creative expression is not intended to be tethered to boundaries. As he develops one mixed media project, ideas nurture concepts for the prize-winning small Carnival band he co-created with friends. In the months between the season of mas, he channels his creative curiosity into gallery exhibits that are often fuelled by mas, politics or calypso, and incorporate wearable objects.
Most days he sells custom frames to avid collectors of Caribbean art from a Woodbrook shop. After a recent renovation, The Frame Shop also doubles as an exhibit space for select fine artists—that includes jewellers.
His third collection of objects of art in three years caters to a niche market that welcomes talent with bold ideas. Ashraph, 50, constructs the statement of his Black Indian collection with cedar, lace, sterling silver, smoky quartz and assertive feathers. To help you relate with why he does not claim to be a jeweller, he explains that the elegant element of his collection that is wearable, is conceived to complete the narrative of his objects. On their own they are statement pieces, but not the central character of his story.
To produce a catalog of striking images for his elegant objects, Ashraph toyed with hiring a model's manicured hands, then he considered the weathered fingers of a labourer. An ah-ha moment led him to douse his own skin in black paint. But that isn't what inspired his exhibition's title.
"Black Indian is a line from a David Rudder song that I like," explains Ashraph, "you know me, I'm into my Carnival."
"Taking photos of traditional mas, the fancy Indian and other variations you see, they wear a lot of paint and black make-up. I think it has more personality and character than what frames the face. So I started working with black stones, then I changed up because it's hard to find black stones.
Last year, I did a silver ring without a stone that looks like a dreamcatcher or the wire frame of a mas. That ring is why I decided to deconstruct the familiar Black Indian mas and not focus on the feathers in their mas. The main focus is the textural qualities of the face of the Indian mas and the reflection of light, not too much feathers."
It's the week before election day in T&T, Ashraph (also known as Richard Ramsaran) is coating his objects with black emulsion one more time, while curator Yasmin Hadeed inspects the collection and visualises their display in her gallery. Before he released his work to the world, he entertained a few questions to offer a glimpse into the backstory of his objects that will be admired on and off the wall.
Q: Is there a common thread in the narrative of your three collections of objects of art?
A: I normally work on a body of artwork to be exhibited, not just two or three paintings to put in an exhibit. I try to have the same discipline with my approach to jewelry design. However, in 2013, I took part in a group show though I didn't produce a full collection. Stuff sold, but it's not about that. I like my work to have a story—that show didn't. I would rather not do that again. Black Indian is my first solo show; this show started a year ago in my head.
What distinguishes your approach to producing wearable art?
I'm not creating just a ring, it's about creating an object that captures the story that I am trying to tell. My first collection was inspired by a Minshall band—Callallo. I saw a circle and ran with it. I created a mas band using calabash and ivory, and the jewelry was made of aluminum, ebony and silver. For the third group show, they gave me the name of the exhibition, and I created a story around light with fireflies and bugs who are the keepers of light. I had a friend write a story and I designed the pieces.
Explain the function of the bejewelled element to your objects of art.
The rings are part of the adornment of the object of art. People may see the jewelry as the main attraction, but that is not how I approach it. I did these rings because I wanted to connect them to the Black Indian heads I designed. I had two prototype heads made in pitch pine, then I started seeing the jewelry. The process is like paint going on a canvas. It's like putting another layer on a costume. It is not a typical run-of-the-mill jewelry. Some are not practical rings.
Some know you have worked extensively with mas legend Peter Minshall. When you meditate on your finished collection do you see hints of Minshall before you?
Minshall used to say, pay attention to the details, and less is more. In my collections of objects of art, the jewelry is the detail of the mas. This collection is more about paying attention to the details.
You don't have formal training in designing accessories, how did you discover accessory design?
I was encouraged by Jasmine Thomas Girvan and Rachel Ross, they're both jewellers. Jasmine also creates fabulous objects. I told them I'm not a jeweller, and they said a lot of people design stuff and get someone else to construct them. So that's how I decided to design jewelry to be part of objects of art. I work with two jewellers to create, I do the drawings, and we sit and resolve the balance and figure out how to make the ring or pendant work. From the first show, I learned that the jewelry needed to balance on the object.
What materials did you recruit to construct your Black Indian collection?
Cedar, fabric, aluminum. The rings are made with silver, bone, ebony, cow horn, red coral, smoky quartz, feathers, lava and druzy.
Do you contain yourself in temporary exile in order to produce your art?
My discipline is about focus, I don't disconnect from anything. It's putting in the work at night; often they are very long nights just looking at the object for hours. Then, before I go to sleep, I figure out the story.
Visit Ashraph's 'Black Indian' show September 21 thru October 3 at Y Gallery 26 Taylor Street, Woodbrook.
Film Festival starts Tuesday
Charlieville man turns 103


Store at Improvements Catalog until 06/02/2017 and get $25 Off Each $125 You Spend at Improvements with the code. By way of my private state of affairs, I had by no means missed a credit score payment or even made a late cost- I simply took out too many credit score accounts. I have personally went from a low to mid-manner credit rating to the absolute best by following these steps.
In the event you do your whole buying by way of a home improvement catalog, you also must be aware that you're going to be charged a number of transport prices. CPSC does not control this external website or its privateness coverage and can't attest to the accuracy of the information it accommodates.
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Guidelines for submitting work to the Arch and Bruce Brown Foundation's 2017 Playwriting Competition have been posted. Please read the guidelines carefully before submitting, as the rules have changed. The 2017 competition will be the foundation's final playwriting contest.


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New Visitor Centre
Green light for new visitor centre at Penllergare Valley Woods
The Penllergare Trust is celebrating after receiving planning permission from the local authority to build a new visitor centre at its historic site.
The new building will have a low carbon footprint and use green technology such as solar power and utilise the Trust's Archimedes screw water pump to generate energy.
It will be clad with timber shingles sourced from the woods and be located in a spot that will enhance the sensitive Grade II registered picturesque landscape of Penllergare Valley Woods.
There will also be extra toilets in the new centre and there will be more parking spaces available for local visitors and tourists from farther afield.
Work on the visitor centre – which will tell the incredible story of the Penllergare estate and its 19th Century owner John Dillwyn Llewelyn, will start soon.
The Penllergare Trust has been managing and restoring the historic estate since 2000 and the hard work of its army of over 100 volunteers has helped attract up to 120,000 visitors annually to the site.
Funding provided by The National Lottery Heritage Fund in Wales – along with visitors' donations, has helped pay for much of the infrastructure work, the new visitor centre and restoration of the walled gardens.
With attractions including a waterfall and lakes, spring flowers and rhododendron, Penllergare Valley Woods has been an important and valued green space for local people during the pandemic.
Paul Baker Chairman of the Penllergare Trust said
" We're delighted that we've received the support of Swansea City Council for the latest phase in the development of Penllergare Valley Woods. We're indebted to the National Lottery Heritage Fund for its grant funding and of course all the people who play the National Lottery. Penllergare Valley Woods is a special place and we have a very special team of volunteers and staff who work tirelessly to improve our visitors' experiences. To date we have lacked a building in which to tell its amazing story and international importance. The new building will allow us to host visitors, schools and communities in an environmentally-friendly, pleasant atmosphere and enable them to learn about everything that Penllergare Valley Woods has to offer, its history, ecology and biodiversity. An added bonus will be the incorporation of indoor seating within our extended coffee shop."
Andrew White, director of The National Lottery Heritage Fund in Wales said:
"The granting of planning permission for the new visitor centre is a landmark moment for the Penllergare Trust during its restoration of this important heritage site.
"This announcement will help ensure that Penllergare Valley Woods can flourish and be enjoyed by current and future generations and help encourage tourism to boost the economy of North Swansea."


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Q: How can I get GoLand to build everything? Is it possible to do the equivalent of go build ./... in GoLand, so I can see all the errors in the IDE in one go? I have modified some widely used structs and I want to find all the places I need to change.
I have tried a package build at the top of my tree but that "succeeds" very quickly whereas build ./... in the same directory fails as expected.
I also tried a directory build with ./... as a tool argument but that gives an error
can't load package: package -o: cannot find package "-o" in any of:

which I assume is because whatever GoLand is putting around the configuration is not compatible with that argument.

A: As @nestor-sokil, mentioned above, if you perform the refactoring operations via the IDE tools, Refactor | Refactor This..., then the IDE manages all the changes automatically for you. If you do find places where this doesn't happen or a refactoring tool is missing, you can open an issue on the issue tracker.
That being said, you can follow the instructions below to configure the IDE to run go build ./... for you. The caveat is that it may show problems that are not normally encountered in operations as the command is unaware of build tags, multiple main() function containing files in the same directory, and so on.
You can create an external tool via Settings/Preferences | Tools | External Tools and configure like this:

You can then go to Settings/Preferences | Keymap and search for the name of the tool and assign it a shortcut to execute it more conveniently.

redpajama_set_name: RedPajamaStackExchange


text:
Kevin Churko
Master Producer | Engineer | Songwriter
Awards + Press
In This Moment's Maria Brink Says Having Rob Halford Guest On 'Ritual' Is 'An Honor' – Blabbermouth.net Copy
by Kevin Churko	on May 7, 2019 May 7, 2019
IN THIS MOMENT singer Maria Brink says that having Rob Halford guest on the band's new album, "Ritual", is "an honor."
The JUDAS PRIEST frontman duets with Brink on the song "Black Wedding" — and Maria says that Halford added his vocals after all the other parts were already laid down.
"We actually recorded the song first, we had all of our stuff on it, and then he came in afterward and put his parts into it," she told the "DOMKcast" (hear audio below). "But we had already kind of shown him [how we wanted it to sound]."
According to Brink, "it's pretty common nowadays" for musicians to collaborate on records without actually being in the studio together, "because artists are always doing something — some of them are on tour" or in different parts of the world. "When I did our song on our last album with Brent [Smith] from SHINEDOWN, it was the same thing — we weren't in the studio at the same time," she said. "I wish we could be in the studio at the same time — it feels like it could be more connecting or something like that — but it went down this way. But it doesn't matter."
Maria added that she is "really excited" for people to hear "Black Wedding". "I think that's definitely gonna be a single," she said. "And I think that's gonna be a favorite, for sure, on this album. And we're honored to have [Rob] be a part of it. He's a legend. So we're so excited. We became friends with him, and what an honor to have him be a part of it."
IN THIS MOMENT guitarist Chris Howorth told Metal Wani in a recent interview that Halford was a natural choice for the "Black Wedding" track.
"Maria was doing the lyrics — and it's almost like a duet, but she's singing to herself with the mother and the priest and we're, like, 'Rob Halford is the priest! He would be the coolest dude ever to have on this song.'
"A couple of years ago, he actually expressed interest in our band and came to see us. It blew our minds completely. We became friends with him from that point.
"He's been super influential to us and when we were doing the song, Maria was, like, 'I'm going to see if Rob's interested,' and he jumped right on it. It's so cool to have him on this song with us."
"Ritual" was released on July 21 via Atlantic Records in partnership with Roadrunner Records.
Listen to "DOMKcast with Maria Brink From In This Moment" on Spreaker.
http://www.blabbermouth.net/news/in-this-moments-maria-brink-says-having-rob-halford-guest-on-ritual-is-an-honor/
Prev Post: In This Moment's Maria Brink Says Having Rob Halford Guest On 'Ritual' Is 'An Honor' – Blabbermouth.net
Copyright © 2019 Kevinchurko.com


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Black Jack 100: On board the light airs supermaxi chasing Sydney Hobart glory
"I always told Peter you never want to buy a 100-footer and eventually he ignored me. But now we're all pretty happy that we're here!" The words of Mark Bradford,…
Third time lucky as Sydney Hobart Race record falls to supermaxi Perpetual Loyal
A new record for Rolex Sydney Hobart Race record has been set by the supermaxi Perpetual Loyal. Owner Anthony Bell and his crew of 18 arrived in Hobart in the…
20 weird and wonderful facts about the famous Rolex Sydney Hobart Race
10 Key facts from the 69 year history of the Sydney Hobart Race
How to follow the Rolex Sydney Hobart Yacht Race 2016 – an easy spectator guide
As the start of the famous Sydney Hobart draws close, we take a look at the best ways to keep up-to-date with the action… Race start time: 1pm AEST on Boxing…
Find out about the favourites for the Rolex Sydney Hobart Yacht Race 2016
Some 90 yachts are entered for this year's Rolex Sydney Hobart Yacht Race. The 72nd edition of the 628-mile classic, which starts on Boxing Day, is one of the most important…
Stunning picture highlights from the Rolex Sydney Hobart Race 2015
Take a tour of supermaxi Comanche, a yacht so beamy she's called 'the aircraft carrier'
"The design office were told specifically by me that if this boat wasn't the worst rated boat in history they have failed," Dr Jim Clark said about his new raceboat.…
Wild Oats XI retires with sail damage as brutal squalls hammer Rolex Sydney Hobart Race
Gale force southerly headwinds and rough seas have been pummelling the favourites for the Rolex Sydney Hobart Race 2015 fleet and caused havoc among the leading super maxis. Eight times line…
Watch how Wild Oats XI was cut in half and rebuilt for the Rolex Sydney Hobart Race 2015
This video by Crosbie Lorimer shows Bob Oatley's now legendary Australian supermaxi Wild Oats XI in action earlier this month after her modifications at McConaghy Boats. They were done in preparation…
Rolex Sydney Hobart Race 2014: Wild Oats XI takes 8th line honours victory but not before the battle with Comanche had closed up once again
As Bob Oatley's Wild Oats XI crossed the finishing line to take the overall win in the Rolex Sydney Hobart Race 2014, she made yet another entry in the famous…
Gallery: Action from the Rolex Sydney Hobart Race 2014 – a race of confounding contrasts
The Rolex Sydney Hobart Race 2014 will go down as one of confounding contrasts that favoured the all-rounders, such as now 8th time line honours winner Wild Oats XI. Our…
Rolex Sydney Hobart: Perpetual Loyal suffers bow delamination as Wild Oats takes the overall lead from Comanche – was this the race defining moment?
Twenty four hours after the start of the Rolex Sydney Hobart, Wild Oats IX took the overall lead from Comanche.
Rolex Sydney Hobart: Christmas Eve fleet briefing points to 20-30 knots against the tide on Friday for the dead beat south
A punchy upwind start followed by lighter conditions for the big boats crossing the Bass Strait, before the breeze swings to the north and slingshots the smaller boats back into…
New 100ft supermaxi Comanche arrives in Australia to prepare for the Sydney-Hobart Race
The superfast 100ft maxi designed as a record breaker for Netscape founder Jim Clark and his wife, Kristy, has arrived in Australia by ship in preparation for the Sydney-Hobart Race.…


Scale your career by scaling our Agile practices.
The Lean Enterprise ensures the adoption of lean-agile principles and practices to improve our ability to deliver value to our customers, eliminate waste, and grow a culture of continuous improvement. We're looking for Lean Leads (Scrum Masters), Release Train Engineers, Lead-Agile Coaches, and people with experience in Lean-Agile implementation practices.


take a look at Xcode playgrounds.
I'm gonna open up Xcode from the dock here.
And the first thing we see is the welcome screen.
So, you get three main ways to start coding in Xcode.
The first is getting started with a playground.
And I'll go over this in a minute.
The second one is to create a new Xcode project.
So, you would select this if you're actually starting a new app.
And then, the third is check out an existing repository.
online remote source code repository.
For example, you could use this to check out code directly from GitHub.
we are mainly going to use playgrounds to practice coding in Swift.
we'll start by creating a new playground.
The first thing it wants is a name.
You can just choose any name here.
And then, the second option is to choose a platform.
platform APIs and frameworks for either iOS or OS X.
But we want to make sure to use iOS for this course.
somewhere where you wanna keep your project files for this course.
And it'll create a single file, a bundle of files, really, for this playground.
So, here is the basic Playground screen.
a results window on the right.
just type in a few lines here, what it will do as I'm typing it in.
It's going to evaluate the code live while I'm typing it in.
And then, on the right-hand side it shows you the results of each line of code.
and it'll bring a pop-up if I click on the, on the variable name.
example, the inferred type of this variable.
inferred types later on in this course with Swift.
I can hold down the Option key, click on it.
And it will give me actually a lot of information about that class.
into the code that you're typing.
that wasn't quite complete, it just gave me an error message there.
you can see that it's also live checking my code to look for errors.
And maybe sometimes will suggest a resolution for each error as well.
So, we'll just get rid of that error message.
So, this is a pretty minimalist interface.
You can actually add more stuff onto this screen.
There's more features to playgrounds than you can see initially.
you right click on the title bar at the top and go to Show Toolbar.
hide other panels in this that will give us more functionality in this screen.
So, first thing I'll do is look at the navigation panel.
From here, we can add resources and we can create additional swift files.
So, you don't have to keep all of your code in your playground in one file.
You can actually create different sources and add additional sources.
Maybe separate different classes into different files.
that will be included in your project and can be referenced from your code.
You can also drag in some, you can drag in resources.
that you can directly reference it in your code.
type in some code to load the image, go.
So, it can load that image that I've just included in this project.
And you'll see right now it's actually evaluating it live again.
This time it prints the size of the image by default as the result.
how it appears once it's been loaded.
actually show the results of this line directly in line with the rest of my code.
Okay. So, that is the navigation panel.
There's another panel that we can open.
have some more panels on the screen here.
And you can see there's a quick help option here on this right-hand panel.
And it's called the Utilities panel.
And it has two tabs.
It has the file inspector and the quick help tab.
it will look up help in the documentation and show it to me right away.
it's easy to check at a glance.
And then, finally, you have this tab here.
It says the meta data, set of a file meta data tab.
And it lists some details about the current file that you have loaded.
in your project, it's an easy way to find that.
So, that is the basics of playgrounds.
start typing in some Swift code and see what happens.
And in the next video we'll look at actual projects and how they work in X code.


Worth the Wait (The Harvest Ridge Series Book 1) by K.L. Ramsey. Price: Free. Genre: Steamy Romance Suspense, Contemporary. Rated: 4.8 stars on 7 Reviews. 176 pages. ASIN: B07HFDSJSH.
Step Bride (Repeat) by B. B. Hamel. Price: Free. Genre: Contemporary Steamy Romance, Military, New Adult & College. Rated: 4.5 stars on 117 Reviews. 446 pages. ASIN: B01BIE99CQ.


Although the FDA does not require expiration dates on supplements, Thrivous supplements include a "best used by" date, which is printed on the bottom of the bottle during the manufacturing process. The "best used by" date is usually two years from the time of manufacturing.
Individual nutrients in supplements may have an expiration or retest date set by the supplier, as disclosed in the certificate of analysis that Thrivous publishes. These dates indicate the recommended date by which the material should be tested again to ensure quality. They do NOT mean that the nutrient is no longer safe or effective after the date.
FDA has performed extensive stability testing of numerous pharmaceutical products. In 2006, the Journal of Pharmaceutical Sciences published an analysis of the FDA's stability test data. The analysis found that 88% of products maintained stability at least 1 year beyond their original expiration date for an average extension of over 5 years. And the analysis concluded that "the stability and quality of extended drug products can only be assured by periodic testing and systematic evaluation".
Accordingly, at time of manufacturing, Thrivous and third-party labs re-test all of our supplement nutrients and their combination to ensure quality. Tests generally include at least identity, potency, microbial, and heavy metals. Finished products are only shipped if they pass all tests.


To celebrate the female form and the African American culture.
Pamela is a sculptor working in bronze, marble, alabaster, clay and sawdust. She's worked with at risk youth in the community, and in residential treatment facilities and traveled with statewide teaching sculpture in small communities, at Joslyn Art Museum, and to death row inmates. She has a Bachelors of Arts in Studio Art, a Masters in Organizational Leadership, and a Masters of Arts in Teaching. She's on the Education Committee for Joslyn Kent Bellows Mentor Program and is an 8th grade Reading Teacher at Monroe Middle School.
Her passion for the natural physical beauty of women and her continuous exploration of herself as an African American female artist, has taken her on some amazing artistic journeys. She's used a variety of mediums to create artwork, but the themes of celebrating the female form and the African American culture remains consistent.


text:
Problem Cards
Effective date: September 18, 2018
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Stuff our lawyers would tell us to say if we were smart enough to hire some:


*To make Chinese 5-spice powder, toast 1 teaspoon peppercorns in a skillet until fragrant. In a spice grinder, grind the roasted peppercorns with 2 whole star anise, 1/4 teaspoon cloves, 1 teaspoon of fennel seeds. Stir in 3/4 teaspoons ground cinnamon.
1 Season the pork with 5-spice powder: Rub the 5 spice powder into the pork chunks, all over, and let sit while you prepare the sauce in the next step.
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3 Place pork and sauce in slow cooker: Place the seasoned pork shoulder pieces in a slow cooker, cover with the sauce.
4 Slow cook for 4 hours on high or 8 hours on low, until the meat easily pulls apart. About halfway through the cooking, turn the meat over in the pot.
5 Ladle off excess fat: Use a ladle or a large metal spoon to remove excess fat that has risen to the surface while the meat has been cooking. Don't discard it yet, you may want to add some of it back.
6 Shred the pork: Use two forks to pull apart the pork into shreds. If the pork still seems dry, add back some of the fat.
7 Serve over sweet Hawaiian dinner buns topped with sliced green onion or cilantro.


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text:
package com.github.gwtc3.api;

import com.google.gwt.core.client.JavaScriptObject;

/**
 *
 */
public interface ObjectHandler<T extends JavaScriptObject> {
	String handle(T element);
}

redpajama_set_name: RedPajamaGithub


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text:
\section{Introduction and Summary}\label{introduction}

Consideration of multiple $p$-branes suspended between other branes is
an effective way to study the dynamics of $p$-dimensional supersymmetric
field theories \cite{Hanany:1996ie,Witten:1997sc}.
In this paper, we  analyze the dynamics of multiple
M2-branes suspended between two M5-branes in the following setup
\cite{Hanany:1997vm}.
We use the coordinates $x^{0,\cdots,10}$, with a compactified $x^{10}$ direction.
Let us introduce complex combinations $v=x^4+ix^5$ and $t=\exp(x^7+ix^{10})$.
Then we have, as summarized in Figure \ref{config}, \begin{itemize}
\item an M5-brane  extending along $x^{0,1,2,3}$ and on the complex
one-dimensional curve $t=t(v)$,  at a fixed position
$(x^6,x^8,x^9)=(L,0,0)$,
\item an M5-brane (which we call the M5$'$-brane) extending along
directions $x^{0,1,8,9}$ and $v$, at a fixed potition
$(x^2,x^3,x^6,t)=(0,0,0,1)$, and
\item $k$ M2-branes extending along $x^{0,1}$ and suspended between
the M5 and the M5$'$ along the $x^6$ direction.
\end{itemize}
We are interested in the infrared limit of the theory on $k$ M2-branes.

\begin{figure}
\centering
\[
\begin{array}{l||c|c|c|c|c|c|c|c|c|c}
&x^0&x^1&x^2&x^3&v&x^6&t&x^8&x^9 \\
\hline
\text{M5}& - & - & - & - & - & L & t(v) & 0 & 0  \\
\text{M5$'$}& - & - & 0 & 0 & - & 0 & 1 & - & - \\
\text{M2}& - & - & 0 & 0 & \sigma & - & 1 & 0 & 0
\end{array}\qquad
\vcenter{\hbox{\includegraphics[width=.4\textwidth]{figure_M_brane_2D_N_2_2}}}
\]

\caption{Configuration of branes. $v=x^4+ix^5$ and $t=\exp(x^7+ix^{10})$.
\label{config}}
\end{figure}

If we reduce the theory along the $x^{10}$ direction,
we have a system of $k$ D2-branes suspended between one NS5-brane at $x^6=0$
and some configuration of branes at $x^6=L$.
This gives a 3d $\mathrm{U}(k)$ gauge theory formulated on an interval
with some boundary conditions at the two ends,
which reduces to a 2d theory with $\mathcal{N}{=}(2,2)$ supersymmetry
at distances longer than the length $L$ of the interval.
We first assume that all solutions to the equation $t(v)=1$
are non-degenerate. That is, if $\{v_i\}_{i\in I}$ denotes
the set of solutions, then $t'(v_i)\ne 0$ for each $i\in I$.
Then, from the M-theory description, the supersymmetric
vacua are described by $k$ M2-branes, separated along $v$ directions,
each at a fixed value of $v$ being one of $\{v_i\}_{i\in I}$.
The s-rule \cite{Hanany:1996ie, Hanany:1997vm} forbids that more than one
M2-brane have the same value of $v$.
This vacuum structure would arize if the low energy theory is
the theory on the Coulomb branch with the twisted superpotential
\begin{equation}
	\mathcal{W}_{\rm eff}=\mathop{\mathrm{tr}} P(\Sigma_T),\label{twistedsuperpotential}
\end{equation}
for the fieldstrength superfield
$\Sigma_T={\rm diag}(\Sigma_1,\ldots,\Sigma_k)$ for the maximal torus
$T\cong\mathrm{U}(1)^k$,
where the holomorphic function $P(v)$ is given by
\begin{equation}
\exp(P'(v))=t(v).\label{definitionofP}
\end{equation}
Indeed, the vacuum equation is $t(\sigma_a)=1$ for $a=1,\ldots,k$
and we expect that no supersymmetric vacuum is supported at the solutions with
$\sigma_a=\sigma_b$ for $a\neq b$.
Also, permutations of $\sigma_a$'s are gauge symmetry.

When $t(v)$ is a rational function, the M5 reduces to a number of
D4-branes ending on an NS5-brane, and the 2d theory can be interpreted as
a $\mathrm{U}(k)$ gauge theory with a number of fundamental and antifundamental
chiral multiplets, possibly with twisted masses \cite{Hanany:1997vm}.
There are only finitely many solutions to $t(v)=1$
and hence the number of supersymmetric vacua is finite.
When $t(v)$ is such that $P'(v)$ in (\ref{definitionofP})
is a polynomial, the 2d theory has a different type of interpretation:
It is the $\mathrm{U}(k)$ gauge theory without matter field and with the tree level
twisted superpotential
\begin{equation}
\mathcal{W}=\mathop{\mathrm{tr}} P(\Sigma)+\pi i(k+1)\mathop{\mathrm{tr}}\Sigma,
\label{Wtree}
\end{equation}
where $\Sigma$ is now the fieldstrength for the full $\mathrm{U}(k)$ vector
mutiplet. The second term is the theta term with $\theta=\pi (k+1)$.
It is non-trivial if and only if $k$ is even since
$\theta$ is a periodic parameter of period $2\pi$.
This is needed in order to have (\ref{twistedsuperpotential})
as the effective twisted superpotential on the Coulomb branch \cite{Hori:2011pd}.
The equation $t(v)=1$ has an infinitely many solutions, and correspondingly,
there are infinitely many supersymmetric vacua in this gauge system.

Each vacuum has a mass gap when, as assumed above, all the solutions to
$t(v)=1$ are non-degenerate.
Things would be more interesting if $t(v)$ is fine tuned
so that some of the solutions coincide, or equivalently,
some of the solutions are degenerate.
$N$ solutions coincide, say at $v=0$, when
\begin{equation}
t(v)=1+v^N+\cdots\quad\mbox{or}\quad
P(v)=v^{N+1}+\cdots,
\label{Pv}
\end{equation}
where the ellipses stand for possible terms of higher order in $v$.
In such a case, we expect to have a non-trivial conformal field theory
in the infra-red limit. In fact, for the case $k=1$, it is argued
in \cite{Tong:2006pa} that the vacuum at $v=0$ is the same as
the infra-red limit of the Landau-Ginzburg model with superpotential
$W=X^{N+1}$, which is believed to be equivalent to
the $\mathcal{N}{=}(2,2)$ superconformal minimal model of type $A_{N-1}$.
For $k>1$, we may have vacua where multiple M2 branes are at $v=0$.
We expect that all $k$ of them can sit there as long as $N\geq k$.
We would like to ask: What is the infra-red limit of such a theory?

We will argue that the theory under question is equivalent
to the infra-red limit of
the Landau-Ginzburg model of $k$ variables $X_1,\ldots, X_k$,
where the superpotential $W(X_1,\ldots,X_k)$ is $\mathop{\mathrm{tr}} \Sigma_T^{N+1}$
written in terms of the elementary symmetric functions of
$\Sigma_1,\ldots,\Sigma_k$;
\begin{eqnarray}
\sum_{a=1}^k\sigma_a^{N+1}&=&W(x_1,\ldots,x_k),
\label{WP}\\
x_b&=&\sum_{a_1<\cdots<a_b}\sigma_{a_1}\cdots\sigma_{a_b},
\qquad b=1,\ldots,k
\end{eqnarray}
This model is believed to flow to the $\mathcal{N}{=}(2,2)$ superconformal
Kazama-Suzuki model \cite{Kazama:1988qp,Kazama:1988uz}
of the coset type
\begin{equation} \frac{\mathrm{SU}(N)_1}{\mathrm{S}[\mathrm{U}(k)\times
    \mathrm{U}(N-k)]}.\label{KS}
\end{equation}
Thus, we claim that the answer to the question is this Kazama-Suzuki
model.

The behaviour (\ref{Pv}) is realized simply by
$t(v)=1+v^N$ or $P(v)=v^{N+1}$.
In the former case, the 2d theory is the $\mathrm{U}(k)$ SQCD
with $N$ fundamental matter fields with fine tuned twisted masses
\cite{Hanany:1997vm,Tong:2006pa}.
In the latter case, the 2d theory is the pure $\mathrm{U}(k)$ gauge theory with
the tree level twisted superpotential
\begin{equation}
\mathcal{W}=\mathop{\mathrm{tr}} \Sigma^{N+1}+\pi i(k+1)\mathop{\mathrm{tr}}\Sigma.
\label{pureW}
\end{equation}
For $1\leq k\leq N$, we shall argue that the theory has, among infinitely
many others, a set of ground states supported at $\Sigma=0$,
and this ``$\Sigma=0$ sector" flows to the superconformal
field theory under question.

Thus, we have a purely field theoretical duality statement:
for $1\leq k\leq N$,
\begin{itemize}
\item the $\mathrm{U}(k)$ SQCD with $N$ fundamentals having fine tuned twisted masses,
\item the $\Sigma=0$ sector of the pure $\mathrm{U}(k)$ gauge theory with
superpotential (\ref{pureW}),
and
\item  the Landau-Ginzburg model with superpotential (\ref{WP})
\end{itemize}
 all
flow to the infra-red fixed point given by the Kazama-Suzuki model  (\ref{KS}).

The aim of this paper is to give evidence of the claims above.
We compute the number of supersymmetric ground states and
the chiral ring in the respective systems,
and show that they agree.
We also study the BPS spectrum of the brane system and compare
it with the known field theoretical results.  Superconformal points
themselves are hard to analyze, and therefore we often make mass
deformations.  We also study the $S^2$ partition functions by using the
recently-developed technique of exact computations
\cite{Benini:2012ui,Doroud:2012xw,Gomis:2012wy,Honda:2013uca,Hori:2013ika}
and show that they indeed agree.

The rest of the paper is organized as follows.
In Section~\ref{sec:vac} and \ref{sec:chiralring},
we test our claim by studying supersymmetric ground states
 and the chiral ring.
In Section \ref{sec:BPS}, we study BPS solitions of
the M2-brane system and compare the structure with the BPS
spectrum of the Landau-Ginzburg theory \cite{Lerche:1991re}.
If we take a certain limit, the BPS spectrum can be determined
by using the technique of spectral networks \cite{Gaiotto:2012rg}.
In Section \ref{sec:pf}, we calculate the $S^2$
partition functions of the 2d theories on both sides of the claimed
equivalence and show that they agree.
The sections \ref{sec:BPS} and \ref{sec:pf} can be read independently.
In Appendix~\ref{KSreview}, we
briefly review the basic facts of Kazama-Suzuki models and their
correspondence with Landau-Ginzburg theories.
In Appendix~\ref{Morse}, we give a proof of some algebraic statement
needed for the study of chiral ring.
In Appendix~\ref{convergence}, we discuss the convergence of the integral
appearing in the $S^2$ partition function.

\section{Supersymmetric Vacua}\label{sec:vac}

As the first check, we look at the supersymmetric vacua of the respective
systems or compute the Witten index \cite{Witten:1982df},
and see if the results are consistent with the claimed duality.

\subsection{Brane System}

Let us first look at the brane system. As in the introduction,
we denote the set of solutions to $t(v)=1$ by $\{v_i\}_{i\in I}$
where we initially assume that each solution is non-degenerate $t'(v_i)=0$.
Supersymmetry requires each M2-brane to have a fixed position in $(t,v)$.
The boundary at M5$'$ fixes $t$ to be $1$ and allowes $v$ to be arbitrary,
while the boundary at M5 requires the relation $t=t(v)$.
Thus, each M2 must be at $t=1$ and has $v=v_i$ for some $i\in I$.
The s-rule requires different M2 branes to have different values of
$v$. Therefore, a supersymmetric vacuum is specified by picking $k$
distinct elements from this set:
\begin{equation} V\subset
  \{v_i\}_{i\in I}, \quad |V|=k.\label{vacua}
\end{equation}
When $t(v)$ is a polynomial of order $M$, the equation $t(v)=1$ has
$M$ roots. Generically, they are distinct and non-degenerate.
Then, the number of supersymmetric vacua is $0$ if $k>M$ and ${M\choose k}$
if $k\leq M$. We may also consider a special polynomial where some of the
solutions coincide. In this situation we do not know how to identify the
supersymmetric vacua. However, the Witten index \cite{Witten:1982df},
which does not change under continuous deformation, remains the same as
in the non-degenerate case.
When some number, say $N$, of the solutions are close to each other
while others are far away, then, we may consider the ``subsector'' in which
all $k$ M2 branes are at one of these $N$ solutions.
In particular,
when $N\geq k$ of them are at the same point,
we expect to have a {\it single} infra-red theory whose Witten index is
${N\choose k}$.
This discussion on subsectors and their Witten indices
is applicable even when $t(v)$ is not a polynomial and the equation
$t(v)=1$ have infinitely many solutions.

\subsection{Gauge Theory}

Let us next consider the $\mathrm{U}(k)$ gauge theory with the tree level twisted
superpotential (\ref{Wtree}) determined by a polynomial $P(\sigma)$.
The classical scalar potential takes the form
\begin{equation}
U={1\over 4g^2}\mathop{\mathrm{tr}}[\sigma,\sigma^{\dag}]^2
+{g^2\over 2}\mathop{\mathrm{tr}} ({\rm Re}P'(\sigma))^2.
\label{Uclass}
\end{equation}
Vanishing of the first term requires $\sigma$ to be diagonalizable,
\begin{equation}
\sigma=\mathop{\mathrm{diag}}(\sigma_1,\ldots,\sigma_k).
\label{Coulomb}
\end{equation}
When all the eigenvalues are well separated, the value of $\sigma$
breaks the gauge group $\mathrm{U}(k)$ to its diagonal subgroup $T\cong \mathrm{U}(1)^k$.
In this Coulomb branch, we may integrate out the off-diagonal components of
the vector multiplet. This induces a correction to the twisted superpotential.
As explained in \cite{Hori:2011pd} following \cite{Witten:1993xi},
the correction is given by $\pi i$
times the sum of positive roots,\footnote{Incidentally, this settles
a problem concerning the relation  observed in \cite{Hori:2006dk}
between the tree level theta angle
of the $\mathrm{U}(k)$ linear sigma model and the B-field of the corresponding
non-linear sigma model on a complete intersection
in the Grassmannian $G(k,N)$.
The shift (\ref{shifttheta}) was missed in \cite{Hori:2006dk}
and the relation is mistakenly stated as
``$B=\theta+(N+k+1)\pi$''. This must be corrected to $B=\theta+N\pi$.
Now it is understandable as result from integrating
out the ``$P$-fields'' \cite{Morrison:1994fr}.}
\begin{equation}
\Delta \mathcal{W}=\pi i\sum_{a<b}(\Sigma_a-\Sigma_b)
\equiv \pi i (k+1)\sum_{a=1}^k\Sigma_a.
\label{shifttheta}
\end{equation}
In the second equality, we used the periodicity $\theta_a\equiv\theta_a+2\pi$
of the theta angle for the group $T$.
This cancels the tree level theta term in (\ref{Wtree})
and hence the effective twisted superpotential is
\begin{equation}
\mathcal{W}_{\rm eff}=\mathcal{W}|_{T}+\Delta\mathcal{W}=\sum_{a=1}^kP(\Sigma_a).
\end{equation}
We denote the effective gauge coupling constant by $e^2_{ab}(\sigma)$. We know
that it approaches $g^2\delta_{ab}$ in the limit where all $\sigma_a$ are
infinitely separated.
We assume that it is positive definite in the region of $\sigma$
we are looking at,
and defines inner products, $\Vert y\Vert^2_{e^{-2}}=(e^{-2})^{ab}y_ay_b$ and
  $\Vert x\Vert^2_{e^2}=e^2_{ab}x^ax^b$, on the Lie agbera of $T$ and its dual.
The effective potential is given by
\begin{equation}
U_{\rm eff}={1\over 2}\left\Vert
{\rm Re}\mathcal{W}'_{\rm eff}(\sigma)\right\Vert^2_{e^2}
+{1\over 2}\left\Vert v_{01}\right\Vert^2_{e^{-2}}.
\label{Ueff}
\end{equation}
The first term, where
${\rm Re}(\mathcal{W}'_{\rm eff}(\sigma))^a
={\rm Re}P'(\sigma_a)$,  is the remnant of
the classical potential (\ref{Uclass}).
The second term is the electro-static energy
\cite{Coleman:1976uz,Witten:1993yc}.
In the Hamiltonian formulation, see e.g.~\cite{Hori:2003ic},
$(e^{-2})^{ab}v_{b01}+{\rm Im}(\mathcal{W}'_{\rm eff}(\sigma))^a$ are regarded as
the conjugate momenta for the holonomy of $T$, each of which has period $1$,
and hence have eigenvalues in $2\pi\ZZ$. In other words,
\begin{equation}
v_{a01}=\sum_{b=1}^k
e^2_{ab}(\sigma)\left(2\pi n^b-{\rm Im}P'(\sigma_b)\right)
\label{vW}
\end{equation}
where $n^a\in \ZZ$. In the sector with definite $n^a$'s,
the effective potential is
\begin{equation}
U_{\rm eff}
=\sum_{a,b=1}^k{e^2_{ab}(\sigma)\over 2}(P'(\sigma_a)-2\pi i n^a)
\overline{(P'(\sigma_b)-2\pi i n^b)}.
\label{Ueff2}
\end{equation}
Supersymmetric ground states must be at the zeroes of this potential.
That is, each $(\sigma_a,n^a)$ must satisfy
\begin{equation}
P'(\sigma)=2\pi i n,\qquad n\in \ZZ.
\label{vacueq}
\end{equation}
The above analysis is valid only when $\sigma_1,\ldots,\sigma_k$
are separated. We do not know how to analyze the region
near the diagonals where some of $\sigma_a$'s coincide.
In many examples, however, it is found that no supersymmetric ground state is
supported near the diagonals as long as the critical points of
the effective twisted superpotential are all non-degenerate.
See for example \cite{Hori:2006dk}. Here we assume that this
applies to our system. Note also that solutions related by permutations of
$(\sigma_a,n^a)$'s are related by the residual gauge transformations and must
be identified. Thus, when $P''(\sigma)\ne 0$ at each solution
to (\ref{vacueq}), a supersymmetric vacuum is specified by a choice of
$k$ unordered solutions $\{(\sigma_a,n^a)\}$ to (\ref{vacueq})
such that $\sigma_a\ne \sigma_b$ for $a\ne b$.
We see that there are infinitely many supersymmetric vacua.

The equation (\ref{vacueq}) may be written simply as
$\exp(P'(v))=1$. Then we see that the problem of finding supersymmetric
vacua in this system is identical to that in the M2 brane system where
the function $t(v)$ defining the M5 curve is given by (\ref{definitionofP}).

Let us write
\begin{equation}
P_u(\sigma)={1\over N+1}\sigma^{N+1}+\sum_{j=1}^N{u_j\over N+1-j}\sigma^{N+1-j}.
\label{Psu}
\end{equation}
for which the equation (\ref{vacueq}) reads
\begin{equation}
\sigma^N+\sum_{j=1}^Nu_j\sigma^{N-j}=2\pi i n,\qquad n\in\ZZ.
\label{releq}
\end{equation}
For a small but generic $u = (u_1,\ldots, u_N)$, the equation with
$n=0$ has $N$ distinct solutions close to $\sigma=0$,
while the equation with $n\ne 0$ has $N$ separated solutions at
$|v|\sim (2\pi n)^{1/N}$.
Our main interest will be the sector with $n^1=\cdots=n^k=0$.
The supersymmetric vacua must have $\sigma_a$ values from the
$N$ solutions near $0$.
The number of such vacua
is zero when $k>N$ and ${N\choose k}$ when $1\leq k\leq N$.
When we turn off $u$, the $N$ solutions all go to $\sigma=0$.
If $1\leq k\leq N$, we expect to have a single infra-red theory
from the $n^1=\cdots=n^k=0$ sector. Its Witten index is ${N\choose k}$.

\subsection{Landau-Ginzburg Model}

Finally, we consider the Landau-Ginzburg model.
Let $W_u(X)=W_u(X_1,\ldots,X_k)$ be the superpotential corresponding to
$P_u(\sigma)$ of (\ref{Psu}), that is,
$\sum_{a=1}^kP_u(\Sigma_a)$ written in terms of the elementary symmetric
functions of $\Sigma_1,\ldots,\Sigma_k$.

When we turn off $u$, the superpotential $W_0(X)$ is the one
(\ref{WP}) given in the introduction and is a quasi-homogeneous
polynomial. When $N\geq k$, it has an isolated critical point
at $X=0$ and the Landau-Ginzburg model
is believed to flow to a non-trivial superconformal
field theory of central charge $c=3k(N-k)/(N+1)$.
In fact the conformal field theory has been claimed to be equivalent to
the Kazama-Sukuki supercoset of the type (\ref{KS}).
See Appendix~\ref{KSreview}. The space of
supersymmetric ground states of the model is naturally identified with
the representation $\wedge^k\CC^N$ of $\mathrm{SU}(N)$ \cite{Lerche:1989uy}.
Its dimension ${N\choose k}$ matches the Witten index
of the M2 and the gauge systems.

The model with $u_j\ne 0$ can be regarded as a perturbation of
this superconformal field theory by the chiral primary fields $\phi_j(X)$
corresponding to $\sum_{a=1}\sigma_a^{N+1-j}$. These have R-charges
$2(N+1-j)/(N+1)$ and conformal weigths $(N+1-j)/(N+1)<1$ and hence
the perturbation is relevant.
In particular, the number of supersymmetric ground states
remains the same, ${N\choose k}$.
Moreover, for the particular deformation
where all $u_j$ but $u_N$ vanish, the ground states are labelled by
 the weights of the representation $\wedge^k\CC^N$ of $\mathrm{SU}(N)$
mentioned above \cite{Lerche:1991re}.
This picture matches with the one for the M2 and the gauge systems
if we regard the roots of $\sigma^N+u_N=0$
as the weights of the representation $\CC^N$.
This observation will be important when we compare the spectra of
BPS solitons.

For a generic choice of $u$, the correspondence of the ground states
with those of the gauge system can be seen more explicitly.
The map $\sigma\mapsto x(\sigma)$, defined by
the elementary symmetric functions
$x_1(\sigma),\ldots,x_k(\sigma)$ of $\sigma_1,\ldots,\sigma_k$,
is regular away from the diagonals, since the Jacobi matrix has determinant
\begin{equation}
\det\left({\partial x_b\over \partial\sigma_a}\right)_{1\leq a,b\leq k}
=\prod_{1\leq a<b\leq k}(\sigma_a-\sigma_b).\label{jac}
\end{equation}
The singular values, i.e., the image of the diagonals, shall be called
{\it the discriminant}.
Let us write $f_u(\sigma_1,\ldots,\sigma_k)=\sum_{a=1}^kP_u(\sigma_a)$.
Then, we have
\begin{equation}
f_u(\sigma)=W_u(x(\sigma)).
\end{equation}
Taking the first derivatives, we obtain
\begin{equation}
{\partial f_u\over\partial \sigma_a}(\sigma)=\sum_{b=1}^k
{\partial x_b\over\partial\sigma_a}(\sigma)
{\partial W_u\over\partial x_b}(x(\sigma)).
\label{1stder}
\end{equation}
This means that ``off the diagonals'' critical points of
$f_u(\sigma)$ modulo permutations of $\sigma_a$'s
are in one-to-one correspondence with ``off the discriminant''
critical points of $W_u(x)$.
Taking one more $\sigma$ derivative
and computing the determinant, one sees that
the Hessian of $f_u(\sigma)$ vanishes if $x(\sigma)$ is
a critical point of $W_u(x)$ on the discriminant.
Therefore, if all the critical points of
$f_u(\sigma)$ are non-degenerate, then, all the critical points of
$W_u(x)$, if there exist, are off the discriminant
and also non-degenerate.
(Note however that $f_u(\sigma)$ may have a non-degenerate
critical point on the diagonal that does not correspond to a critical point
of $W_u(x)$.)
This establishes a one-to-one correspondence between the
supersymmetric ground states of the $n^1=\cdots =n^k=0$ sector of
the gauge system and those of the Landau-Ginzburg model, for a generic
$u$ so that $f_u(\sigma)$ is a Morse function.
In particular, this is one way to see that
the number of critical points of $W_u(X)$ is zero for $N<k$
and ${N\choose k}$ for $N\geq k$.

\section{Chiral Rings}\label{sec:chiralring}

In this section, we shall study the chiral ring of
the gauge system and compare the result with
that of the Landau-Ginzburg model.
We consider the $\mathrm{U}(k)$ gauge theory with tree level twisted superpotential
\begin{equation}
\mathcal{W}=f(\Sigma)+\pi i(k+1)\mathop{\mathrm{tr}}\Sigma
\end{equation}
where $f(\Sigma)$ is an adjoint invariant polynomal of $\Sigma$.
The effective twisted superpotential on the Coulomb branch is
$\mathcal{W}_{\rm eff}=f(\Sigma_T)$.
We shall use the same notation $f(\sigma)=f(\sigma_1,\ldots,\sigma_k)$
for that symmetric polynomial,
and denote simply by $W(X)$ the corresponding superpotential,
$f(\sigma)=W(x(\sigma))$. Just as in (\ref{1stder}), we have
\begin{equation}
{\partial f\over\partial \sigma_a}(\sigma)=\sum_{b=1}^k
{\partial x_b\over\partial\sigma_a}(\sigma)
{\partial W\over\partial x_b}(x(\sigma)).
\label{1st}
\end{equation}
We assume that $f(\sigma)$ is a Morse function.
Then, $W(x)$ is also Morse, and
supersymmetric ground states of the $n^1=\cdots n^k=0$ sector of
the gauge system are in one-to-one correspondence with
those of the Landau-Ginzburg model.

The chiral ring of the Landau-Ginzburg model is generated by the
chiral variables $x_1,\ldots, x_k$ and the relations are
generated by
\begin{equation}
0\equiv \left\{{\bf Q}_B,g^{a\bar b}(\overline{\psi}_{b-}-\overline{\psi}_{b+})\right\}
=\partial_{x_a}W(x),\qquad a=1,\ldots,k.
\end{equation}
Here ${\bf Q}_B$ is the relevant supercharge,
$g^{a\bar b}$ is the K\"ahler metric that appears in the kinetic term,
and $\overline{\psi}_{b\pm}$ are the fermionic components of the antichiral multiplet
$\overline{X}_b$.
Hence the chiral ring is isomorphic to the Jacobi ring,
\begin{equation}
{\rm Jac}(W)=\CC[x_1,\ldots, x_k]/(\partial_{x_1}W(x),\ldots,
\partial_{x_k}W(x)).
\end{equation}

The twisted chiral ring of the gauge system is generated by
gauge invariant polynomials of $\sigma$. In the low energy description
on the Coulomb branch, they reduce to symmetric functions of
$\sigma_1,\ldots, \sigma_k$. To find the relations,
we note that
\begin{equation}
\left\{{\bf Q}_A,(e^{-2})^{ab}(\overline{\lambda}_{b-}-\lambda_{b+})
\right\}=(e^{-2})^{ab}(D_b+iv_{b01})
\label{QAL}
\end{equation}
where ${\bf Q}_A$ is the relevant supercharge while
$\overline{\lambda}_{b-}$, $\lambda_{b+}$ and $D_b$ are fermionic and auxiliary
components of the twisted antichiral multiplet $\overline{\Sigma}_b$.
The auxiliary fields $D_b$ are constrained to be
\begin{equation}
(e^{-2})^{ab}D_b=-{\rm Re}\,\partial_{\sigma_a}f(\sigma).
\label{eD}
\end{equation}
We also have equations like (\ref{vW}):
\begin{equation}
(e^{-2})^{ab}v_{b01}=-{\rm Im}\,\partial_{\sigma_a}f(\sigma)+2\pi n^a,
\label{ev}
\end{equation}
where $n^a$ are integers labeling the momenta of the holonomy variables.
Therefore the relations are
$\partial_{\sigma_a}f(\sigma)\equiv 2\pi in^a$.\footnote{In
Eqns (\ref{QAL}), (\ref{eD}), (\ref{ev})
 some fermion bilinear terms are ignored to simplify the expression.
However, these final relations are exact.}
Our main interest is the $n^1=\cdots=n^k=0$ sector. The relations are
\begin{equation}
\partial_{\sigma_a}f(\sigma)\equiv 0,\qquad a=1,\ldots, k.
\end{equation}
We shall also accept relations of the form
\begin{equation}
\sum_{a=1}^k{F_a(\sigma)\over \Delta(\sigma)^{\ell}}
\,\partial_{\sigma_a}f(\sigma)\equiv 0
\label{Ifgen}
\end{equation}
where $F_a(\sigma)$ are polynomials and $\Delta(\sigma)$ is
the Vandermond determinant
\begin{equation}
\Delta(\sigma):=\prod_{1\leq a<b\leq k}(\sigma_a-\sigma_b).
\end{equation}
We allow division by $\Delta(\sigma)$
because $\sigma_a$'s are assumed to be separated from each other
in the Coulomb branch. Let $I_f$ be the ideal
of the ring $\CC[\sigma_1,\ldots,\sigma_k]^{\mathfrak{S}_k}$ of
symmetric polynomials consisting of polynomials that can be
written in the form on
 the left hand side of (\ref{Ifgen}).
Then, the twisted chiral ring is
\begin{equation}
\CC[\sigma_1,\ldots,\sigma_k]^{\mathfrak{S}_k}/I_f.
\end{equation}
When $f(\sigma)$ is generic so that $W(x)$ has only isolated and
non-degenerate critical points (i.e. $W(x)$ is Morse),
one can show that this is isomorphic to the Jacobi ring ${\rm Jac}(W)$.

The proof goes as follows. First, we have an isomorphism
$\CC[x_1,\ldots,x_k]\cong\CC[\sigma_1,\ldots,\sigma_k]^{\mathfrak{S}_k}$
given by $\phi(x)\mapsto \phi(x(\sigma))$. It is enough to show that the
ideal $I_W=(\partial_{x_1}W,\ldots,\partial_{x_k}W)$ is mapped precisely
to $I_f$ under this isomorphism.
That $I_W$ is mapped into $I_f$ is obvious in view of (\ref{1st})
and the definition of $I_f$.
To show that the map
$I_{W}\rightarrow I_f$ is surjective, let $\phi(x)$ be a polynomial so that
$\phi(x(\sigma))$ belongs to $I_f$. Then,
$\phi(x(\sigma))$ vanishes on ``off the diagonals'' critical points of
$f(\sigma)$. Here we recall from the previous section
that $\sigma\mapsto x(\sigma)$ gives one-to-one correspondence between
``off the diagonals'' critical points of $f(\sigma)$ modulo permutations
and critical points of $W(x)$.
Therefore, $\phi(x)$ vanishes on the critical points of $W(x)$.
Since $W(x)$ is a Morse function, this means that $\phi(x)$ belongs to $I_W$.
See Appendix~\ref{Morse} for the proof of the last statement.

\section{BPS Solitons}\label{sec:BPS}

In this section we analyze the spectrum of the BPS states from
M2-branes,
building on \cite{Hanany:1997vm,Fayyazuddin:1997by,
Henningson:1997hy,Mikhailov:1997jv,Dorey:1999zk},
and compare the results with the spectrum of BPS solitons
in the Landau-Ginzburg model \cite{Fendley:1990zj,Lerche:1991re}.

In what follows, we are interested in M2-branes whose $(t,v)$ values
are confined into a small neighborhood of
$v=0$ and $t=1$. Therefore, we write $t=e^z$ and regard $z$ as a coordinate on a neighborhood of the origin of
of a complex plane $\CC$.
M5$'$ is at $z=0$ and
we consider the M5-brane wrapped on the curve
\begin{equation}
z= v^N +  u_1  v^{N-1} + \cdots +  u_N.\label{curve}
\end{equation}
Recall (\ref{vacua})
 that a supersymmetric ground state is specified for a choice
of $k$ distinct elements from the set $\{v_i\}_{i=1}^N$
of solutions to $v^N +  u_1  v^{N-1} + \cdots +  u_N=0$.
We are interested in solitonic M2-brane configurations that
interpolate two different ground states.

\subsection{A single M2-brane}

Let us recall the basics of BPS solitons arising from a single
M2-brane stretched between  two M5-branes. This setup was originally studied in
\cite{Hanany:1997vm} and later in \cite{Dorey:1999zk}.
The system may be regarded as an $\mathcal{N}{=}(2,2)$ supersymmetric field theory on
$\RR^2=\{(x^0,x^1)\}$ with a chiral multiplet taking values in the space of paths
$\phi:x^6\in [0,L]\mapsto (z(x^6),v(x^6))\in \CC^2$
from the M5$'$ at $z=0$ to the M5 at (\ref{curve}). It has
the superpotential
\cite{Witten:1997ep}
\begin{equation}
\mathcal{W}[\phi]=\int_{C_{\phi,\phi_*}}\Omega,\qquad\Omega:={\rm d} z\wedge {\rm d} v,
\end{equation}
where $C_{\phi,\phi_*}$ is a configuration that interpolates a reference
path $\phi_*$ and $\phi$.
Note that
\begin{equation}
{\delta\mathcal{W}\over \delta z(x^6)}=\partial_6v(x^6),\qquad
{\delta\mathcal{W}\over \delta v(x^6)}=-\partial_6 z(x^6).
\end{equation}
In particular, the action includes the usual kinetic term of a
 theory on three dimensions $(x^0,x^1,x^6)$.
A soliton is a configuration that approaches two vacua, say
$\phi_j\equiv (0,v_j)$ and $\phi_i\equiv (0,v_i)$,
as $x^1\rightarrow-\infty$ and $x^1\rightarrow +\infty$ respectively.
The central charge of such a solitonic sector is
\begin{equation}
Z_{ij}=\mathcal{W}[\phi_i]-\mathcal{W}[\phi_j]
=\int_{C_{\phi_i,\phi_j}}\Omega.
\end{equation}
A soliton preserves a half of the supersymmetry
if the configuration satisfies the BPS equation,
$\partial_1\phi=\zeta_{ij}\overline{\delta\mathcal{W}/\delta\phi}$
with $\zeta_{ij}:=Z_{ij}/|Z_{ij}|$,
i.e.,
\begin{equation}
\partial_1z=\zeta_{ij}\partial_6\overline{v},\qquad
\partial_1v=-\zeta_{ij}\partial_6\overline{z}.
\label{BPS}
\end{equation}
It follows that
\begin{equation}
\phi^*\omega=0,\qquad
\overline{\zeta}_{ij}\phi^*\Omega
={\rm d} x^1\wedge{\rm d} x^6\cdot(\mbox{real positive}),
\end{equation}
where $\omega:={i\over 2}{\rm d} z\wedge {\rm d} \overline{z}+{i\over 2}{\rm d} v\wedge
{\rm d} \overline{v}$ is the K\"ahler form.
This is equivalent \cite{Henningson:1997hy} to the condition
that the image of $\phi:\RR\times [0,L]\rightarrow \CC^2$
is a special Lagrangian submanifold.

One may also look at the usual
supersymmetry condition \cite{Becker:1995kb,Fayyazuddin:1997by,Dorey:1999zk}.
Let $\eta$ be the eleven-dimensional spinor obeying
$\eta=\Gamma_{012\ldots 9,10}\eta$.
Presence of the M5-branes imposes the condition
$\eta=\Gamma_{014589}\eta=\Gamma_{012345}\eta=\Gamma_{01789,10}\eta$ from which
we also have $\eta=\Gamma_{016}\eta$.
Then, the BPS equation (\ref{BPS}) is equivalent to
the existence of a spinor $\eta$ obeying
\begin{equation}
\eta={1\over 2}
\epsilon^{\alpha\beta}\Gamma_{0IJ}
\partial_{\alpha}x^I\partial_{\beta}x^J\eta
\end{equation}
(the summation over $I,J=1,4,5,6,7,10$ and $\alpha,\beta=1,6$ is assumed),
in the limit $|\partial_{1,6}x^{4,5,7,10}|\ll 1$ where
the eleven dimensional Planck length is set equal to one.
The preserved supersymmetry is
$(\zeta_{ij}\Gamma_{vz}+\overline{\zeta}_{ij}\Gamma_{\bar v\bar z})\eta
=\Gamma_{16}\eta$.

If we change the complex structure of $\CC^2$ so that
$\omega-i{\rm Im}(\overline{\zeta}_{ij}\Omega)$ is a holomorphic two form
and ${\rm Re}(\overline{\zeta}_{ij}\Omega)$ is a K\"ahler form,
the BPS equation (\ref{BPS}) has a different interpretation:
It is a Cauchy-Riemann equation with respect to the holomorphic coordinate
$x^1+ix^6$ of the domain $\RR\times [0,L]$.
That is, $\phi:\RR\times [0,L]\rightarrow \CC^2$ can be regarded as
a holomorphic map. With respect to the new K\"ahler form, the two M5 curves,
$z=0$ and (\ref{curve}), are Lagrangian submanifolds of $\CC^2$
which intersects at the $N$ points $\{(0,v_i)\}_{i=1}^N$. Therefore,
a BPS solition from $(0,v_j)$ to $(0,v_i)$ can be identified as a term of the
Floer differential of the pair of Lagrangian submanifolds, in the
intersection Floer theory \cite{FOOO}.

\subsection{$k$ M2-branes}

We now consider the case of general $k$.
Let us take two ground states specified by subsets $V$ and $V'$ of
$\{v_i\}_{i=1}^N$ of order $k$.
A soliton that interpolates $V$ and $V'$ is the superposition of
$k$ single M2-brane solitions, each of which approach
 $v_{i_a}\in V$ and $v_{i'_a}\in V'$ as $x^1\rightarrow -\infty$ and
$x^1\rightarrow +\infty$ respectively.
The central charge of such a solitonic sector is
the sum $\sum_{a=1}^kZ_{i'_a,i_a}$
while the mass is bounded below by
$\sum_{a=1}^k|Z_{i_a,i'_a}|$.
When $ u_j$ are generic, $Z_{ij}$ have different phases for
different pairs $(i,j)$.
Therefore, it saturates the BPS bound only when just one of the $k$ M2-branes
is a non-trivial soliton while the remaining $k-1$ stay fixed at the vacua.
This is possible only when $|V\cap V'|=k-1$.

In the picture where $\{v_i\}_{i=1}^N$ is regarded as the set of
weights of the fundamental representation $\mathbb{C}^N$ of $\mathrm{SU}(N)$,
a BPS state for the $k$ M2-brane system
exists only if $V$ and $V'$, which are regarded as weights of
the representation $\wedge^k\mathbb{C}^N$, are connected by
a root of $\mathrm{SU}(N)$.
This matches with the structure of the BPS spectrum of the
Landau-Ginzburg model:
In \cite{Fendley:1990zj,Lerche:1991re}, it was proposed
that there is exactly one BPS solition for each pair of vacua labelled by
weights of $\wedge^k\mathbb{C}^N$ that differ by a root of $\mathrm{SU}(N)$.
Therefore, we would like to  see that
there is exactly one BPS soliton for any pair of $v_i$ and $v_j$
in the single M2-brane system.

Showing this seems to be a difficult problem to the authors.
Instead of trying to find BPS configurations directly,
we shall take a certain limit  \cite{Mikhailov:1997jv}
that reduces the problem of finding BPS membranes to the
problem of finding BPS geodesics \cite{Klemm:1996bj},
and then use the technique of
spectral networks.

\subsection{BPS states via spectral networks}

So far, we have been using the metric
${\rm d} s^2=|{\rm d} z|^2+|{\rm d} v|^2$ in the $x^{4,5,7,10}$ directions.
We now change it to
\begin{equation}
{\rm d} s^2=|{\rm d} z|^2+\beta^2|{\rm d} v|^2
\end{equation}
and take a small $\beta$ limit. We also have
\begin{equation}
\omega={i\over 2}{\rm d} z\wedge{\rm d}\overline{z}
+{i\over 2}\beta^2{\rm d} v\wedge{\rm d}\overline{v},\quad\,\,
\Omega=\beta{\rm d} z\wedge {\rm d} v.
\end{equation}
The argument of \cite{Mikhailov:1997jv} shows that, in the limit $\beta\rightarrow 0$,
the projection of a BPS configuration $C_{ij}=C_{\phi_i,\phi_j}$
onto the $z$-plane is a real one-dimensional graph $\gamma_{ij}$,
and the tangent directions $\Delta z$ and $\Delta v$ obey the constraint
$\Delta z\cdot\Delta v=\zeta_{ij}\cdot$ (real number).
Over a generic point $z$ on the graph $\gamma_{ij}$,
$C_{ij}$ is a line segment from one solution $v_l$ to another $v_k$
of (\ref{curve}).
Of course, $(k,l)=(i,j)$ near $z=0$, but that may not be the case if
$z$ is far from $z=0$. See Figure~\ref{fig:M2_soliton}.
\begin{figure}[htb]
	\centering
	\begin{subfigure}{.4\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_M2_soliton_A_1.pdf}
		\caption{}
		\label{fig:M2soliton_A_1}
	\end{subfigure}
	\begin{subfigure}{.5\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_M2_soliton_A_2.pdf}
		\caption{}
		\label{fig:M2soliton_A_2}
	\end{subfigure}
	\caption{M2-brane solitons in the $\beta\rightarrow 0$ limit. Blue curves are parts of M5, and red lines are parts of M5$'$. The $(z,v)$ images of the M2-brane solitons are shaded.}
	\label{fig:M2_soliton}
\end{figure}

In a neighborhood of such a point, the graph $\gamma_{ij}$ is a curve
determined by the differential equation
\begin{align}
	\lambda_{kl} (z) \frac{\partial z}{\partial \tau}
= \exp(i \vartheta_{ij}) = \frac{Z_{ij}}{|Z_{ij}|}, \label{eq:BPS M2 boundary}
\end{align}
where $\lambda_{kl}{\rm d} z = (v_k(z) - v_l(z)){\rm d} z$ is
the difference of $\lambda=v{\rm d} z$ at the $k$-th sheet and at the $l$-th
sheet of M5-branes, and $\tau$ is a real parameter along the curve
$\gamma_{ij}$.
Such a $\gamma_{ij}$ is called a finite open web of BPS
strings \cite{Gaiotto:2012rg}.

We would like to find a solution to (\ref{eq:BPS M2 boundary})
which starts from a branch point of the covering $v(z)\mapsto z$ and call it
$\mathcal{S}_{kl}$.
For a generic value of $\vartheta=\vartheta_{ij}$,
it does not pass the endpoint of the ground-state M2-branes at
$x=0$ but goes to infinity,
meaning that it does not correspond to any of the BPS states.
These paths are called $\mathcal{S}$-walls.  When two
$\mathcal{S}$-walls $\mathcal{S}_{ik}$ and $\mathcal{S}_{kj}$ cross, another
$\mathcal{S}$-wall, $\mathcal{S}_{ij}$, can emerge, when there is a
supersymmetric junction of three M2-branes that satisfy $\lambda_{ik}
+ \lambda_{kj} = \lambda_{ij}$
\cite{Gaiotto:2009hg,Gaiotto:2012rg},
like in Figure \ref{fig:M2soliton_A_2}. The collection of $\mathcal{S}$-walls is called a spectral network
\cite{Gaiotto:2012rg}.  When there is an $\mathcal{S}$-wall $\mathcal{S}_{ij}$ that
passes $z=0$, then this gives us a BPS object with a finite central
charge.

\subsubsection{Deformation by \texorpdfstring{$ u_N$}{u\_N}}\label{symmetric}

Let us consider a particular deformation where the curve is
\begin{equation}
	 z =  v^N +  u_N.
\end{equation}
The $z$-coordinate is zero when
\begin{align}
	v_j = (- u_N)^{1/N} \omega^j,\quad
	\text{where}\quad \omega=e^{2\pi i/N}.
\end{align}
Hence, the vacua are depicted by the vertex of a regular polygon on the $v$-plane.

 The curve has a branch point of ramification index $N$ at $z =
  u_N$, and the differential equation that governs the behavior of
 each $\mathcal{S}_{ij}$ on the $z$-plane is
\begin{align}
	\alpha_{ij} (z -  u_N)^{1/N} \frac{\partial z}{\partial \tau} = \exp(i \vartheta), \label{eq:differential equation for ij-string of M'(N,k,mu_N)}
\end{align}
where $\alpha_{ij} = \omega^i - \omega^j$.
The solution is
\begin{align}
	z_{ij}(\tau) =  u_N + c\left(\frac{N+1}{N} \frac{\tau}{\alpha_{ij}} \right)^{N/N+1} \exp \left( \frac{N}{N+1} i \vartheta \right) \label{eq:solution for an S_ij}
\end{align}
where $c$ is an $(N+1)$-st root of unity.  This is a straight line
starting at the branch point $z= u_N$.  When
$\alpha_{ij}=\alpha_{i'j'}$ two S-walls can be on top of each other.
As an example, Figure \ref{fig:IRMN4k_S_Wall} shows the spectral network when $N=4$
for $\vartheta = 0$.  As can be seen there, $\mathcal{S}_{12}$ and $\mathcal{S}_{34}$
are coincident.
\begin{figure}[ht]
	\centering
	\includegraphics[scale=.5]{figure_IRMN4k_SWall_theta_zero.pdf}
	\caption{A spectral network around a branch point of ramification index $N=4$.}
	\label{fig:IRMN4k_S_Wall}
\end{figure}

When we change $\vartheta$ from $0$ to $2\pi$, the whole spectral
network rotates by $2\pi N/(N+1)$, and the endpoint of the M2-brane
meets $N(N-1)$ $\mathcal{S}$-walls in the process, implying there are
in total $N(N-1)$ BPS states in the BPS spectrum of this
theory. Therefore, for each distinct $i$ and $j$, there is one BPS
state in the sector with the right boundary set to the vacuum $i$ and
the left boundary set to the vacuum $j$.  It is easy to identify the
value $\vartheta$ when an $\mathcal{S}_{ij}$ wall hits $x=0$. There is one
value of $\vartheta$ for each $\mathcal{S}_{ij}$.

\begin{figure}[ht]
	\centering
	\begin{subfigure}{.4\textwidth}
	\centering
		\vspace{23pt}
		\includegraphics[width=\textwidth]{figure_IRMN4k1_diagram.pdf}
		\vspace{10pt}
		\caption{on the $v$-plane}
		\label{fig:IRMN4k1_diagram_v_plane}
	\end{subfigure}
	\hspace{.1\textwidth}
	\begin{subfigure}{.4\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_A3_fundamental_weight.pdf}
		\caption{in the weight lattice of $\mathrm{SU}(4)$}
		\label{fig:IRMN4k1_diagram_weight_lattice}
	\end{subfigure}
	\caption{Vacua and solitons, $k=1$.}
	\label{fig:IRMN4k1_diagram}
\end{figure}

On the $v$-plane, we can introduce a soliton of the $k=1$ theory by a
line connecting $v_i$ and $v_j$. Let us illustrate the case $N=4$.
Figure \ref{fig:IRMN4k1_diagram_v_plane} represents the four ground
states and twelve solitons on the $v$-plane.  We clearly see that
$Z[\gamma_{12}]$ and $Z[\gamma_{34}]$ has the same phase, as was also
reflected in the spectral network shown in
Figure~\ref{fig:IRMN4k_S_Wall}. Note that Figure
\ref{fig:IRMN4k1_diagram_v_plane} can be understood as obtained from
the projection of the weights of the fundamental representation of
$\mathrm{SU}(4)$ and the roots connecting the weights, representing the ground
states and the solitons respectively, as shown in Figure
\ref{fig:IRMN4k1_diagram_weight_lattice}. This structure of BPS
solitons is the same as that of the corresponding Landau-Ginzburg
model with a single chiral field, which has as its IR fixed point the
$\mathcal{N}{=}2$ $A_3$ minimal model \cite{Fendley:1990zj}.

\begin{figure}[ht]
	\centering
	\begin{subfigure}{.45\textwidth}
		\includegraphics[width=\textwidth]{figure_IRMN4k2_soliton_diagram_01.pdf}
		\caption{$[14] \rightarrow [12]$ soliton for $k=2$}
		\label{fig:IRMN4k2_soliton_diagram_01}
	\end{subfigure}
	\hspace{10pt}
	\begin{subfigure}{.45\textwidth}
		\includegraphics[width=\textwidth]{figure_IRMN4k2_soliton_diagram_02.pdf}
		\caption{$[13] \rightarrow [12]$ soliton for $k=2$}
		\label{fig:IRMN4k2_soliton_diagram_02}
	\end{subfigure}
	\begin{subfigure}{.45\textwidth}
	\includegraphics[width=\textwidth]{figure_IRMN4k3_soliton_diagram.pdf}
	\caption{$[123] \rightarrow [412]$ soliton for $k=3$}
	\label{fig:IRMN4k3_soliton_diagram}
	\end{subfigure}
	\caption{Examples of $k > 1$ solitons.}
	\label{fig:IRMN4k}
\end{figure}

So far we discussed the case when there is just one M2-brane,
$k=1$. For general $k$, we need to choose $k$ vertices out of $N$, and
a soliton is obtained by moving one of the $k$ vertices. In Figure
\ref{fig:IRMN4k}, some representative examples of the solitons with
$k=2$ and $k=3$ are shown.
For $k=2$, we see from Figures \ref{fig:IRMN4k2_soliton_diagram_01}
and \ref{fig:IRMN4k2_soliton_diagram_02} that a $k=1$ solitonic
configuration can connect two $k=2$ ground states. From this
consideration we can represent $k=2$ ground states and solitons as
shown in Figure \ref{fig:IRMN4k2_diagram_v_plane}. Again, we can
understand this as obtained from the projection of the weights of the
2nd antisymmetric power of the fundamental representation of $\mathrm{SU}(4)$
and the roots connecting the weights, as shown in Figure
\ref{fig:IRMN4k1_diagram_weight_lattice}. The same structure of BPS
solitons of the corresponding Landau-Ginzburg model is observed in
\cite{Lerche:1991re}, which is expected to flow in the IR to the
Kazama-Suzuki model based on $\mathrm{SU}(4)_1/{\mathrm{S}[\mathrm{U}(2)\times
  \mathrm{U}(2)]}$.

\begin{figure}[ht]
	\centering
	\begin{subfigure}{.4\textwidth}
	\centering
		\vspace{20pt}
		\includegraphics[width=\textwidth]{figure_IRMN4k2_diagram.pdf}
		\vspace{4pt}
		\caption{on the $v$-plane}
		\label{fig:IRMN4k2_diagram_v_plane}
	\end{subfigure}
	\hspace{.1\textwidth}
	\begin{subfigure}{.4\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_A3_k_2_weight.pdf}
		\caption{in the weight lattice of $\mathrm{SU}(4)$}
		\label{fig:IRMN4k2_diagram_weight_lattice}
	\end{subfigure}
	\caption{Vacua and solitons, $k=2$.}
	\label{fig:IRMN4k2_diagram}
\end{figure}

For $k=3$, because choosing $k$ ground states among $N$
indistinguishable ones is the same as choosing $N-k$ ground state, the
ground states and the solitons are represented by the same diagram as
Figure \ref{fig:IRMN4k1_diagram_v_plane}, thus we see the $k
\leftrightarrow N-k$ duality.

\subsubsection{General deformations}
Now let us consider how the spectral networks look when the
deformation parameters $ u_j$ are general.

\subsubsection*{BPS spectrum with $z=v^3$ }

Now we consider the case where we have three M5-branes ramified over
the $z$-plane:
\begin{align}
	 z =   v^3 +  u_2 v +  u_3.
\end{align}
For general $ u_2$ and $ u_3$, we have two branch points of
ramification index 2 on the $z$-plane as shown in Figure
\ref{fig:IRMN3k_SWall_after_joint}. In the figure, we chose
$ u_{2,3}$ so that a (12)-branch cut, a blue wavy line, comes out
from the upper branch point, and (13)-branch cut, a green wavy line,
from the lower branch point.  From the (12)-branch point we have three
$\mathcal{S}$-walls: two $\mathcal{S}_{21}$ with solid blue line and
one $\mathcal{S}_{12}$ with a dashed blue line.  Similarly, from the
(13)-branch point, we have two $\mathcal{S}_{13}$ with solid green
line and one $\mathcal{S}_{31}$ with dashed green line.  We can see
that one $\mathcal{S}_{21}$ and one $\mathcal{S}_{13}$ meet at a
point, from which another $\mathcal{S}$-wall, $\mathcal{S}_{23}$,
emerges.

\begin{figure}[ht]
	\centering
		\includegraphics[width=.3\textwidth]{figure_IRMN3k_SWall_after_joint.pdf}
		\caption{A spectral network with general $u_2$ and $u_3$.}
		\label{fig:IRMN3k_SWall_after_joint}
\end{figure}

\begin{figure}[ht]
	\centering
	\begin{subfigure}{.22\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN3k_SWall_0.pdf}
		\caption{$\vartheta \approx 0$}
		\label{fig:IRMN3k_SWall_0}
	\end{subfigure}
\hspace{5pt}
	\begin{subfigure}{.22\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN3k_SWall_13.pdf}
		\caption{$\vartheta \approx \vartheta_{13}$}
		\label{fig:IRMN3k_SWall_13}
	\end{subfigure}
\hspace{5pt}
	\begin{subfigure}{.22\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN3k_SWall_23.pdf}
		\caption{$\vartheta \approx \vartheta_{23}$}
		\label{fig:IRMN3k_SWall_23}
	\end{subfigure}
\hspace{5pt}
	\begin{subfigure}{.22\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN3k_SWall_21.pdf}
		\caption{$\vartheta \approx \vartheta_{21}$}
		\label{fig:IRMN3k_SWall_21}
	\end{subfigure}

	\caption{Rotation of a spectral network with general $u_2$ and $u_3$.}
	\label{fig:IRMN3k_SWall}
\end{figure}

As we now have the full spectral network, let us rotate it by changing
$\vartheta$ from $0$ to $2\pi$.  Figure \ref{fig:IRMN3k_SWall} shows
spectral networks at various values of $\vartheta$, $0 <
\vartheta_{13} < \vartheta_{23} < \vartheta_{21} < \pi$. We see that
there are $\gamma_{13}$, $\gamma_{23}$, and $\gamma_{21}$ at
$\vartheta_{13}$, $\vartheta_{23}$, and $\vartheta_{21}$,
respectively, between the branch point and the M2-brane
endpoint. Therefore there are corresponding three BPS states for $0 <
\vartheta < \pi$. There are another three BPS states for $\pi <
\vartheta < 2\pi$, each of which has the central charge
$Z[\gamma_{ji}] = - Z[\gamma_{ij}]$.

\begin{figure}[ht]
	\centering
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN3k_SWall_mu2_large.pdf}
		\caption{$ u_2 \approx 1$}
		\label{fig:IRMN3k_SWall_mu2_large}
	\end{subfigure}
\hspace{5pt}
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN3k_SWall_mu2_small.pdf}
		\caption{$ u_2 \ll 1$}
		\label{fig:IRMN3k_SWall_mu2_small}
	\end{subfigure}
\hspace{5pt}
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN3k_SWall_mu2_zero.pdf}
		\caption{$ u_2 = 0$}
		\label{fig:IRMN3k_SWall_mu2_zero}
	\end{subfigure}

	\caption{Evolution of the spectral network under the limit of $u_2 \rightarrow 0$.}
	\label{fig:IRMN3k_SWall_mu2_vanishing}
\end{figure}

Now let us take the limit $ u_2 \rightarrow 0$ so that the two branch points
collide, see Figure \ref{fig:IRMN3k_SWall_mu2_vanishing}.
Figure \ref{fig:IRMN3k_SWall_mu2_zero} shows the spectral network when
$ u_2=0$.  There is only a single (123)-branch point and a single
(123)-branch cut.\footnote{The notation is that around the
  $(n_1n_2\cdots n_k)$ branch cut the sheets are exchanged in the
  order $n_1\rightarrow n_2\rightarrow \cdots \rightarrow n_k \rightarrow n_1$.}  The whole spectral
network rotates by $3\pi/4$ when we change $\vartheta$ from $0$ to
$\pi$ continuously, and in the process we find three BPS strings
connecting the branch point and the endpoint of the M2-brane,
corresponding to three BPS states in $0 < \arg(Z) < \pi$.

\subsubsection*{BPS spectrum with $z=v^4$}

Let us now consider the case $N=4$, for more illustration.

\begin{figure}[ht]
	\centering
	\begin{subfigure}{.4\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_after_joint.pdf}
		\label{fig:IRMN4k_SWall_after_joint}
	\end{subfigure}
	\begin{subfigure}{.1\textwidth}
		\includegraphics[width=.75\textwidth]{figure_IRMN4k_SWall_legend.pdf}
		\vspace{20pt}
	\end{subfigure}
	\caption{A spectral network with $u_{2,3,4} \neq 0$.}
	\label{fig:IRMN4k_SWall_joint}
\end{figure}

Figure \ref{fig:IRMN4k_SWall_joint} shows the spectral network with
$ u_{2,3,4}$ chosen so that there is a $(124)$-branch point and a
$(34)$-brach point. See the legend for the nature of walls represented
by the colors and the styles.

\begin{figure}[ht]
	\centering
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_01.pdf}
		\caption{$\vartheta \approx \vartheta_{13}$}
		\label{fig:IRMN4k_SWall_01}
	\end{subfigure}
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_02.pdf}
		\caption{$\vartheta \approx \vartheta_{43}$}
		\label{fig:IRMN4k_SWall_02}
	\end{subfigure}
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_03.pdf}
		\caption{$\vartheta \approx \vartheta_{12}$}
		\label{fig:IRMN4k_SWall_03}
	\end{subfigure}

\vspace{10pt}

	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_04.pdf}
		\caption{$\vartheta \approx \vartheta_{42}$}
		\label{fig:IRMN4k_SWall_04}
	\end{subfigure}
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_05.pdf}
		\caption{$\vartheta \approx \vartheta_{41}$}
		\label{fig:IRMN4k_SWall_05}
	\end{subfigure}
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_06.pdf}
		\caption{$\vartheta \approx \vartheta_{34}$}
		\label{fig:IRMN4k_SWall_06}
	\end{subfigure}
	\caption{Rotation of the spectral network with $u_{2,3,4} \neq 0$.}
	\label{fig:IRMN4k_SWall_various_theta}
\end{figure}

Figure \ref{fig:IRMN4k_SWall_various_theta} shows the spectral network
at various values of $\vartheta$. At $\vartheta_{ij}$ there is
$\gamma_{ij}$ between one of the branch points and the endpoint of the
M2-brane, and Figure \ref{fig:IRMN4k_SWall_various_theta} is arranged
such that
\begin{align}
	0 < \vartheta_{13} < \vartheta_{43} < \vartheta_{12} < \vartheta_{42} < \vartheta_{41} < \vartheta_{34} < \pi.
\end{align}
That is, we can imagine the whole spectral network rotating
anti-clockwise as we increase $\vartheta$ from $0$ to $\pi$ and in the
course of the rotation we encounter six finite BPS strings. Therefore,
we can expect this theory to have twelve BPS states in total.  There
are four vacua, so there is one BPS state for each boundary condition
at the left and the right spatial infinity.

\begin{figure}[ht]
	\centering
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_a_large.pdf}
		\caption{$| u_2| \sim 1$}
		\label{fig:IRMN4k_SWall_a_large}
	\end{subfigure}
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_a_small.pdf}
		\caption{$| u_2| \ll 1$}
		\label{fig:IRMN4k_SWall_a_small}
	\end{subfigure}
	\begin{subfigure}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{figure_IRMN4k_SWall_a_zero.pdf}
		\caption{$ u_2 = 0$}
		\label{fig:IRMN4k_SWall_a_zero}
	\end{subfigure}

	\caption{Evolution of the spectral network under the limit of $u_{2,3} \rightarrow 0$.}
	\label{fig:IRMN4k_SWall_limit}
\end{figure}

Let us consider what happens when we have just one branch point of
ramification index 4. This limit corresponds to $ u_2,  u_3 \rightarrow 0$,
and the evolution of the spectral network under the limit is depicted
in Figure \ref{fig:IRMN4k_SWall_limit}.  Thus we see that the spectrum
of the BPS states at general $ u_{2,3,4}\neq 0$ is smoothly connected
to the more symmetric situation analyzed in Sec.~\ref{symmetric} with
$ u_{2,3}=0$ and $ u_4\neq 0$.

\section{\texorpdfstring{$S^2$}{S \textasciicircum 2} partition functions}\label{sec:pf}
As a final check of our proposal, we show in this section that the
partition function on $S^2$ of the 2d $\mathcal{N}{=}(2,2)$ $\mathrm{U}(k)$ gauge
theory with twisted superpotential
$\mathcal{W}=\mathop{\mathrm{tr}} P(\Sigma)+\pi i (k+1){\rm tr}(\Sigma)$ in the infrared
limit agrees with that of the Landau-Ginzburg model with chiral fields
$X_1,\ldots, X_k$ with appropriately chosen superpotential
$W=W(X_1,\ldots,X_k)$.  We employ the localization methods recently
developed in \cite{Benini:2012ui,Doroud:2012xw,Gomis:2012wy}.
The derivation can be easily generalized to arbitary gauge
group, and the quasihomogeneity of $P$ and $W$ is not required,
either.  The integrals below are only conditionally convergent. In
this section we perform the comparison of the partition functions
rather naively. The convergence issues will be explained in
Appendix~\ref{convergence}. It will be then clear that the
manipulations can be readily justified.

The partition function of the Landau-Ginzburg model of
$k$ variables $X_1,\ldots, X_k$ with the superpotential
$W(X_1,\ldots,X_k)$ is given by \cite{Gomis:2012wy} \begin{equation}
  Z_\text{LG}=(r\Lambda)^k
\int\limits_{\,\,\mathbb{C}^k\!\!\!\!} \prod_a dX_a d\overline{X}_a
e^{-ir[W(X)+\overline{W}(\overline{X})]}.\label{LG-Z}
\end{equation}
where $r$ is the radius of the sphere.
The factor in front, $(r\Lambda)^k$,
with $\Lambda$ being a renormalization scale, was not explicitly in
\cite{Benini:2012ui,Doroud:2012xw,Gomis:2012wy} but its presence is
mentioned in a footnote of \cite{Doroud:2012xw} and the computation
was done by the authors of these papers \cite{Jaume,Sungjay}.
The same applies to $(r\Lambda)^{-k^2}$ in (\ref{Zgaugegen}) below.
See \cite{Hori:2013ika} for a detailed explanation in a related context.
When  $W$ is quasi-homogeneous, a rescaling of fields
can absorb the $r$ in the integrand and yields the expected behaviour
$Z_{\rm LG}\sim r^{\hat{c}}$ with $\hat{c}$ being
the expected central charge of the infra-red fixed point
of the model \cite{Martinec:1988zu,Vafa:1988uu}.

The partition function of the $\mathcal{N}{=}(2,2)$ supersymmetric gauge theory
was first computed in \cite{Benini:2012ui,Doroud:2012xw}
up to a sign factor which was later corrected in \cite{Honda:2013uca,Hori:2013ika}.
The one for the theory with
gauge group $\mathrm{U}(k)$ and with the twisted
superpotential $\mathcal{W}(\Sigma)$ is given by
 \begin{multline}
Z_\text{gauge}=(r\Lambda)^{-k^2}\sum_{m\in \mathbb{Z}^k}
\int\limits_{\,\,\mathbb{R}^k\!\!\!\!} \prod_a d(r\tau_a)
\prod_{a<b}\left(r^2(\tau_a-\tau_b)^2 + {(m_a-m_b)^2\over 4}\right) \\
\times (-1)^{(k+1)\sum_{a}m_a}
e^{-i r[\mathcal{W}(\Sigma) + \overline{\mathcal{W}}(\overline{\Sigma})]}
\label{Zgaugegen}
\end{multline}
where \begin{equation}
  \Sigma=\mathop{\mathrm{diag}}(\tau_1,\ldots,\tau_k)+\frac{i}{2r}\mathop{\mathrm{diag}}(m_1,\ldots, m_k)
\end{equation}
in the exponent.\footnote{The sign factor $(-1)^{(k+1)\sum_{a}m_a}$ was not in
\cite{Benini:2012ui,Doroud:2012xw}.
Its presence only changes the weight of
the sum over the topological type of the $\mathrm{U}(k)$ gauge bundle, only when
$k$ is even. Therefore such a factor is rather sutble.
The presence  is demanded
 for the factorization of the sphere partition function into
two hemispheres \cite{Honda:2013uca,Hori:2013ika}.
As we will see, its presence is also needed for the match with the partition
function of the proposed Landau-Ginzburg model.}
For the twisted superpotential
$\mathcal{W}(\Sigma)={\rm tr}P(\Sigma)+\pi i (k+1){\rm tr}(\Sigma)$,
the formula (\ref{Zgaugegen}) reads
\begin{equation}
Z_\text{gauge}=
\Lambda^{-k^2}\sum_{m\in \mathbb{Z}^k}
\int\limits_{\,\,\mathbb{R}^k\!\!\!\!} \prod_a d\tau_a
\prod_{a<b}\left((\tau_a-\tau_b)^2
+ \left({m_a-m_b\over 2r}\right)^2\right)
e^{-i r[{\rm tr}P(\Sigma) + {\rm tr}\overline{P}(\overline{\Sigma})]}
\label{gauge-Z}
\end{equation}

Now, look at the infra-red regime $r\Lambda\gg 1$.
The sum $\sum_{m\in \mathbb{Z}^k}$ in \eqref{gauge-Z} turns into an
integral $(2r)^k\int_{\mathbb{R}^k} \prod_a d\upsilon_a$ for $\upsilon_a={m_a\over 2r}$,
and we have
\begin{equation}
Z_\text{gauge}\stackrel{r\Lambda\gg 1}{\longrightarrow}
\Lambda^{-k^2}r^k\int\limits_{\,\,\mathbb{C}^k\!\!\!\!}
\prod_a d\sigma_a d\overline{\sigma}_a \prod_{a<b} {|\sigma_a-\sigma_b|^2}
e^{-ir[\mathop{\mathrm{tr}} P(\Sigma)+\mathop{\mathrm{tr}} \overline{P}(\overline{\Sigma})]}\label{gauge-Z-IR}
\end{equation}  where $\sigma_a=\tau_a+i\upsilon_a$ and
\begin{equation}
\Sigma=\mathop{\mathrm{diag}}(\sigma_1,\ldots,\sigma_k).
\end{equation}

Let us introduce variables $X_a$ as the elementary symmetric polynomials of $\sigma_a$; equivalently, let us take
\begin{equation}
\det(z-\Sigma)=\sum_a X_a z^{k-a}. \label{sym}
\end{equation}
where $z$ is a dummy variable. Then the Jacobian between the variables
$\sigma_a$ and the variables $X_a$ are given as in \eqref{jac}, \begin{equation}
 \det\left(\frac{\partial X_b}{\partial \sigma_a}\right)_{1\le a,b\le k}= \prod_{1\le a<b\le k} {(\sigma_a-\sigma_b)}.
\end{equation}
Therefore, we see that the gauge partition function in the infrared,
\eqref{gauge-Z-IR}, agrees with the Landau-Ginzburg partition function
\eqref{LG-Z}, under the identification
\begin{equation}
P(\Sigma)=W(X_1,\ldots,X_k).
\end{equation}

We now have the equality of $S^2$ partition functions of the
$\mathrm{U}(k)$ theory with the twisted superpotential in the infrared limit
and those of the Landau-Ginzburg theory.
Two-point functions of BPS operators can be
dealt with in the completely same way, by just inserting the operators
in the integral.
It is well-known that the resulting integral expressions suffer from
subtleties: apparently spurious operators do not decouple
and the choice of representatives of the (anti)chiral
ring elements matters \cite{Cecotti:1991me}.
The agreement holds provided that the operators
in the gauge system are identified with those in the Landau-Ginzburg model
precisely via the isomorphism
$\mathbb{C}[\sigma_1,\ldots,\sigma_k]^{\mathfrak{S}_k}\,\cong\,
\mathbb{C}[X_1,\ldots,X_k]$.

\acknowledgments

It is a pleasure for the authors to thank helpful discussions with Keshav Dasgupta, Nick Dorey,
Sasha Getmanenko,
Jaume Gomis, Sangmin Lee, Sungjay Lee, Todor Milanov, Andy Neitzke,
Kyoji Saito,
John H.\ Schwarz,
Jaewon Song, Edward Witten, and Piljin Yi. C.~Y.~P.~would like to
thank Kavli IPMU for hospitality and support while this work was in the
initial and final stages. C.~Y.~P.~would also like to thank the
organizers of ``$\mathcal{N}{=}2$ JAAZ'' Workshop at McGill University and the
10th Simons Summer Workshop in Mathematics and Physics for hospitality
and support where part of this work has been done. The work of
C.~Y.~P. is supported in part by Samsung Scholarship. The work of
K.~H. and
Y.~T. is supported in part by World Premier International Research
Center Initiative (WPI Initiative), MEXT, Japan through the Institute
for the Physics and Mathematics of the Universe, the University of
Tokyo. The work of K.~H. is also supported in part by
JSPS Grant-in-Aid for Scientific Research No. 21340109,
and the work of Y.~T. is supported in part by JSPS Grant-in-Aid for Scientific Research No. 25870159.

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