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Puzo Michael J Sells 895 Shares of Home Depot Inc (NYSE:HD)
Posted by Chris Copeland on Jan 16th, 2020
Puzo Michael J lessened its position in Home Depot Inc (NYSE:HD) by 3.6% in the fourth quarter, according to the company in its most recent filing with the Securities & Exchange Commission. The firm owned 24,052 shares of the home improvement retailer's stock after selling 895 shares during the quarter. Home Depot makes up about 1.9% of Puzo Michael J's investment portfolio, making the stock its 22nd largest holding. Puzo Michael J's holdings in Home Depot were worth $5,253,000 at the end of the most recent reporting period.
Several other institutional investors and hedge funds have also recently added to or reduced their stakes in the company. Enterprise Trust & Investment Co bought a new stake in Home Depot in the third quarter worth $27,000. Belmont Capital LLC purchased a new stake in shares of Home Depot in the fourth quarter worth $28,000. Julex Capital Management LLC bought a new position in shares of Home Depot during the third quarter valued at $29,000. Proffitt & Goodson Inc. increased its holdings in shares of Home Depot by 106.2% during the third quarter. Proffitt & Goodson Inc. now owns 134 shares of the home improvement retailer's stock valued at $31,000 after acquiring an additional 69 shares in the last quarter. Finally, Financial Advantage Inc. purchased a new position in shares of Home Depot in the 3rd quarter worth about $37,000. Institutional investors own 69.87% of the company's stock.
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In other Home Depot news, EVP Teresa Wynn Roseborough sold 12,974 shares of the firm's stock in a transaction dated Friday, December 13th. The stock was sold at an average price of $213.70, for a total value of $2,772,543.80. Following the transaction, the executive vice president now directly owns 26,335 shares of the company's stock, valued at $5,627,789.50. The sale was disclosed in a legal filing with the Securities & Exchange Commission, which is available at the SEC website. Also, CEO Craig A. Menear sold 113,468 shares of the business's stock in a transaction dated Friday, December 13th. The shares were sold at an average price of $211.24, for a total value of $23,968,980.32. Following the sale, the chief executive officer now owns 277,604 shares of the company's stock, valued at $58,641,068.96. The disclosure for this sale can be found here. In the last three months, insiders sold 126,552 shares of company stock worth $26,765,281. 0.27% of the stock is owned by corporate insiders.
Shares of HD opened at $224.59 on Thursday. The firm has a market cap of $243.05 billion, a PE ratio of 22.71, a P/E/G ratio of 1.97 and a beta of 1.00. The firm has a fifty day simple moving average of $218.89 and a two-hundred day simple moving average of $222.23. Home Depot Inc has a 1-year low of $172.00 and a 1-year high of $239.31.
Home Depot (NYSE:HD) last released its quarterly earnings results on Tuesday, November 19th. The home improvement retailer reported $2.53 earnings per share (EPS) for the quarter, hitting the Zacks' consensus estimate of $2.53. Home Depot had a negative return on equity of 721.00% and a net margin of 10.01%. The firm had revenue of $27.22 billion for the quarter, compared to the consensus estimate of $27.51 billion. During the same period last year, the business posted $2.51 EPS. The company's revenue for the quarter was up 3.5% on a year-over-year basis. Research analysts expect that Home Depot Inc will post 10.08 earnings per share for the current fiscal year.
The company also recently disclosed a quarterly dividend, which was paid on Thursday, December 19th. Investors of record on Thursday, December 5th were given a dividend of $1.36 per share. This represents a $5.44 dividend on an annualized basis and a yield of 2.42%. The ex-dividend date was Wednesday, December 4th. Home Depot's dividend payout ratio (DPR) is presently 55.01%.
Several brokerages recently commented on HD. Evercore ISI reissued a "buy" rating on shares of Home Depot in a research report on Thursday, November 21st. Nomura reduced their price objective on shares of Home Depot from $237.00 to $224.00 and set a "neutral" rating for the company in a research note on Thursday, December 12th. Longbow Research reaffirmed a "buy" rating and issued a $255.00 target price on shares of Home Depot in a report on Wednesday, November 20th. ValuEngine upgraded shares of Home Depot from a "sell" rating to a "hold" rating in a research note on Thursday, December 26th. Finally, Telsey Advisory Group cut their price target on shares of Home Depot from $220.00 to $210.00 and set a "market perform" rating on the stock in a report on Thursday, December 12th. One equities research analyst has rated the stock with a sell rating, fourteen have issued a hold rating and seventeen have issued a buy rating to the company's stock. The stock has an average rating of "Buy" and a consensus target price of $237.31.
Home Depot Profile
The Home Depot, Inc operates as a home improvement retailer. It operates The Home Depot stores that sell various building materials, home improvement products, lawn and garden products, and décor products, as well as provide installation, home maintenance, and professional service programs to do-it-yourself and professional customers.
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Rena Owen (Bay of Islands, 22 de julio de 1962) es una actriz neozelandesa, más conocida por haber interpretado a Macy Fields en la serie Medivac.

Biografía
Rena es hija de un maorí y de Cynthia, una pakeha, y tiene ocho hermanos.

Estudió enfermería y trabajó cuatro años como enfermera; sin embargo, lo dejó después de que se mudara a Inglaterra. Se graduó del European Film Business School en 2006.

En 2002 se casó con el norteamericano Olo Alailima, un consultor de seguridad; sin embargo, el matrimonio terminó en 2007.

Carrera
En 2002 apareció en la película Star Wars: Episode II - Attack of the Clones, donde prestó su voz para el personaje de Taun We. En 2003 apareció en las películas Pledge of Allegiance y en Nemesis Game, donde dio vida a Emily Gray. En 2005 apareció en la película Star Wars: Episode III - Revenge of the Sith, donde interpretó a Nee Alavar.

En 2011 apareció como invitada en varios episodios de la serie Shortland Street, donde dio vida a Hine Ryan. En 2012 se unió al elenco de la serie The Straits, donde interpretó a Kitty Montebello.

Filmografía
Series de televisión

Películas

Videojuegos

Productora y directora

Teatro

Premios y nominaciones

Referencias

Enlaces externos

 Rena Owen's Official Web Site
 Rena Owen | NZ On Screen
 Rena Owen - Zimbio

Actrices de televisión de Nueva Zelanda
Actrices de televisión de Australia

redpajama_set_name: RedPajamaWikipedia


These band-aid solutions don't work.
They work for the people they're intended to work for.
In Lebanon, real estate "investment" is nothing short of a dirty word. It was real estate investment that made prices soar to unrealistic levels and lead to the current stagnation in the market. The country is overcrowded with one of highest population density on the globe and the population can't afford houses for themselves, sometimes including couples who both work to make ends meat.
There is no reason whatsoever to encourage buying real estate for investment purposes. If someone owns a house they don't actively live in as an "investment", they should pay a respectable amount of extra taxes on that property, in order to discourage this practice. Especially for foreigners, but this should also applies to Lebanese real estate investors.
No matter how one tries to spin and provide justifications, the motives of passing this law are clear in who's interest.
Price drop without a high volume of sales = stagnation. It means there is few selling at market price and the rest speculatively holding.
There is no risk of collapse unless all Lebanese economy collapsed first.
Because most house owners will keep their houses for life.
What about 330K dollar apartments?
Eitherway, a huge influx of money will drive prices up again and will become unattainable by the average Lebanese especially that they can't get favourable loans anymore.
Unless you're in real estate, do you think you'll see a whiff of that money?
As for palestinians and syrians, the gulf states have been pushing for a long time for the resettlement in Lebanon.
Do you think they won't get the funds from there?
First, expats don't need residency.
Second, expats or foreigners with spare cash to buy $500,000 house and use it once a year on vacation or worse rent it to Lebanese living in Lebanon - while otherwise not interacting or offering intrinsic lasting gains to Lebanese economy - are useless.
This kind of investment should be actively taxed and discouraged, if not aggressively combated.
You are rich Lebanese expat who considers Lebanon a vacation zone? OK pay good taxes because your vacation hurts the Lebanese economy.
You are foreigner who want residency in Lebanon? Offer something to the country like opening a business that meaningfully contributes to the economy and local growth. Parking money in the form of a vacation home doesn't help the economy.
i am not in real estate no.
also 330 k in cash is no small amount by any means.
ther palestinians that have 330k IN CASH to spare can immigrate to the west easily .
Again, palestinians will get bankrolled by the gulf.
At least in Lebanon they'll get a home, western countries will hardly accept them.
And in Lebanon 330K is not high end anymore, it's middle.
The government is launching this initiative specifically to encourage investment and bring more money into the country.
Both the minister of economy and the prime minister stated that.
This means that they're expecting a large inflow of funds.
Why would they do that if they weren't?
Why did they vote for it to start with???
It would have been better to consider investment as starting a business and creating X number of jobs for Lebanese citizens. Such investors should get permanent residency.
no outside beirut its still high end, you cant get a housing loan for that value of an appartment, in fact it should be half of that , you expect the gulf to pay 3.3 billion dollars to buy high end luxury appartments for palestinians in lebanon ?


Creating the session ticket is the second stage of the user connection process in a double-hop DMZ deployment.
The Web Interface communicates with both the XML Service and the Secure Ticket Authority (STA) in the internal network to produce session tickets for each of the published applications the user is authorized to access. The session ticket contains an alias address for the computer running Citrix XenApp that hosts a published application.
The STA saves the IP addresses of the servers that host the published applications. The STA then sends the requested session tickets to the Web Interface. Each session ticket includes an alias that represents the IP address of the server that hosts the published application, but not the actual IP address.
The Web Interface generates an ICA file for each of the published applications. The ICA file contains the ticket issued by the STA. The Web Interface then creates and populates a web page with a list of links to the published applications and sends this web page to the web browser on the user device.


Todd Begg's interest in knives began young and was fostered by his time in the military. He has formal machinist training in the aerospace industry which lends well to his craft of knifemaking. He takes great pride in producing top quality designs with superior form, function, fit and finish that are meant to last generations. He strives to create knives with zero tool marks, to look as if they are born and not built. Todd's got an amazing knack for this and his designs are some of the best!


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ARNRBKM – Chapter 1
If you guys find some mistakes or anything, feel free to leave a comment down below. Suggestions to improve this translation are very much appreciated as well.
Translator: Empress || Editor: absolutenobody
Novel Raw: 1 || Manga: I'm a Villainous Daughter, so I'm going to keep the Last Boss || Manga Raw: 悪役令嬢なのでラスボスを飼ってみました
Black crows flew out towards the clear sky with loud caws, and black cats were crossing the narrow trail.
Everything on her path gave her nothing but ominous signs.
Irene walked straight into the late night forest by only relying on the light from the Cantella she brought to light up her path while holding on to a slender sword on her other hand.
Suddenly, a giant rat jumped out of the bush and stopped at her feet. Its one giant eyeball was looking at Irene gently. Its ears were abnormally large with its sharp teeth protruding from its mouth.
(……a monster. It's my first time seeing it. It was something that you will rarely see at the first layer of this forest)
When she shone the Cantella's light to the giant monster rat, it jumped back into the bush and disappeared from sight.
「Crows……?I wonder why do they bring skulls with them. Is it possible that they are also monsters?」
What's more, they were all lined up on above a dead tree while looking down on Irene. If they decided to attack Irene together all at once maybe, she'll be in more than a sorry state.
In the midst of cawing crows, she was trying to think of a way to save herself when she heard a voice.
(The numbers are indeed a problem……but if I were to turn back from here, I'm going to die)
But if she were to proceed, she might die as well.
『Go home』
A large-sized crow suddenly spoke. As she had guessed, they were monsters.
Then voices of crows echoed around Irene as if they wanted to trap her in it.
『Go home, human』
『The Demon King's castle lies ahead』
『The Demon King is reading』
Strangely, there's an everyday schedule report mixed in with the warnings, but in this situation where the crows were talking, she didn't really feel like laughing.
『What is your purpose of coming here, girl? Do you come here to die?』
『It's the girl who was abandoned by her betrothed yesterday. Is this for revenge purpose? Ridiculous』
At that, Irene raised an eyebrow. And the voices of crows continued to ridicule her.
『If you are persistent, you'll be dumped, this is common sense』
『You're, someone with a bad reputation, the worst』
『Selfish, arrogant, thinking that she was loved by the Crown Prince――』
「You guys sure are noisy, do you guys want me to turn you to a "yakitori"?」
She stared directly at the noisy crows while giving them her best glares. She then straightened up her back and smiled gracefully.
「I wonder if the rumors are being gathered under the order of the Demon King? It is quite surprising to learn that the Demon King has such a vulgar hobby」
Just like that, the eyes of the crows were all gathered on her. The Demon King was an existence loved by all monsters. Insulting the Demon King is nothing but purposely trying to pick a fight with all the monsters. Provoking them is a suicidal act.
( If he turns out to be such an unreasonable being, I'll turn back and leave this place)
Like the way the Heroes would when they want to challenge the Demon King, Irene nobly declared.
「I, Irene Loren D'Authrice have come here to meet Claude Jeanne Elmeier. If you don't have anything relevant to say, then be quiet. It doesn't matter if you're a monster or a human, if you only know how to pass judgment on someone based on the rumors alone, then you don't deserve to be my opponent, so leave immediately.」
After dropping down those words, Irene gracefully left. After that, for some reason, the crows stopped talking but instead, they were flying around as if they were escorting Irene along the gloomy forest trail. Suddenly, something inside the bushes was moving to follow Irene as well.
I guess they can't help but being a sore loser. Irene, who was used to the attention, ignored them and continued her journey.
Eventually, her field of vision was cleared.
Under the starless sky, a castle came in sight. Most places had collapsed, the tapestries were in tatters, and the broken pillars were tangled with ivy. The surrounding trees around the castle had all withered and the small pond had become black and muddy, making it look like a swamp instead.
It's the Demon King's castle.
Once again, she braced herself.
(He should have a kind personality……but what if it's only going to be limited to the heroine or something――that's possible)
Since having wishful thinking is dangerous, she had no choice but to prepare herself for any possibilities. Then, she took a deep breath and lifted up her face.
She proceeded to pass by the scattered seemingly human-bones stumps and arrived in front of a rusty iron door. She tried to push it with all of her might but it barely moved. No matter how many times she tried, it remained still, unperturbed.
Since it's impossible for her to give up just because of that, she extended her hands to try pushing it once again; the moment she did that, she heard a voice from behind her.
「My. Thank you very much for your kind――」
The iron door blasted away, with a loud roaring sound.
Irene, while still keeping an amiable smile on her face, saw the stranger blasted the iron door away by using only a finger.
Glossy black hair colored deeper than the darkness itself, fluttered as the damp night wind blows, exposing the facial features beneath it.
Everything about the stranger was like a perfect first-class painting; elegant nose with thin lips, both of his facial features and his body were too perfect. But what's more striking than those features were his pair of shimmering bloody scarlet eyes.
(Com- compared to the CGs, the real person's intensity is much more amazing……!)
But, I recognized that face.――Irene chose to keep that knowledge to herself.
"Irene" does not have much time. The engagement annulment was the event flag for Cedric capture route. At this point, if I don't do something, I'll die.
It's going to be like in my previous world all over again; not even love, I can't even enjoy all of the joy of youth.
「What business does a human has with me?」
A Demon King as well as the First Prince of the Elmeier Kingdom, Claude Jeanne Elmeier spoke expressionlessly.
But she was too afraid to notice the details. Irene kept her chin up, tucked her hair gracefully behind her ears, and smiled as usual.
「I have good news, of course. I, came here to propose marriage to you」
Just when she thought that he had no reaction, lightning fell down from the sky, and struck down a big withered tree causing it to burn and split into two.
As if the God going to send down his divine judgment on her.
「Who is going to, propose, a marriage, to whom?」
To the Irene who kept on smiling, the person politely asked her the question.
The flames behind him burned up as if it was dancing, illuminating the surrounding. Almost like an inferno.
Thanks to her willpower, she was able to keep on smiling. Otherwise…
「Like I said, I, came here to propose marriage to――!」
When the lightning struck down thrice around her, her willpower lost to her instinct.
Before she could listen to his answer, just like that, Irene fainted, falling backward.
Tags: akuyaku reijo, Claude Jeanne Elmeier, Irene, Otome Games, Past Life, Reincarnation, villainess, Villainess Daughter
Previous Akuyaku Reijo Nanode Rasubosu o Katte Mimashita
Next ARNRBKM – Chapter 2
Ahahaha!
Nice 1st meeting .
Thanks alot 4 the fast update n new chapter.
Empress says:
You are most welcome! ☺
nakakure says:
Interesting interesting hahahhahah thanks for chapter. Hmm gentleman or ladies?
Not sure what you're asking about.. but if you're asking about us, we are ladies. ( ◡‿◡ *)
kariageweb says:
Thanks for the chapter! 🙂
you're welcome 😁
Nocta says:
I like the spirit of this MC, I'm looking forward to this story
Thank you very much for the translation
she needs to have that spirit. 🤣 You are welcome. ☺
sensei to be says:
Novels really do move slower than the manga, don't they (•̪ o •̪)
yeah, I was surprised to see the that length of both chapters wasn't even close to the progression of the story in the manga. But I guess it had a lot more to tell and described, that's why it moves at a slower pace?
The details were really harsh on an amateur translator like me who only has elementary level Japanese. In addition to that English isn't even my first language. I'm rambling aren't I?😅😅 Anyway, thank you for reading this chapter as well. 🤗🤗🤗
I think your English is very good, I couldn't tell at all.
The one thing I have to say is that the font and background colors are hard on my eyes. I like the story though.
manin0va says:
This is something new. I wanna read more. Doesn't look like demon lord'd gonna be an easy catch.
Yuhuuu says:
😍😍😍😍😍
alexandriavar says:
Thanks for the chapter!!! ❤
Tridentfrost says:
its look like this demon king have a natural talent in attracting a heavenly tribulation lightning
Wow! It's already this interesting…just hope it continues like this!
Elmithian says:
Thank you very much for translating this. For the past months I have been hooked on everything isekai shoujou/otome/etc variance. For some reason I can't get enough of it… I guess I am a bit of an isekai trash eh. Not that I am apologising for it. ^_^,
Btw, from what is that avatar image of your from, Empress? It be from something else interesting that is worthy of reading?
Nuh says:
Mantul busku. Siuk gila ni sorita.

redpajama_set_name: RedPajamaCommonCrawl


International Game Technology PLC (NYSE:IGT) hit a new 52-week low on Wednesday . The company traded as low as $13.23 and last traded at $13.28, with a volume of 181259 shares changing hands. The stock had previously closed at $13.44.
IGT has been the topic of several research analyst reports. Susquehanna Bancshares initiated coverage on International Game Technology in a research note on Friday, December 7th. They set a "positive" rating and a $22.00 price target on the stock. ValuEngine raised International Game Technology from a "strong sell" rating to a "sell" rating in a research note on Monday, November 26th. SunTrust Banks raised International Game Technology from a "hold" rating to a "buy" rating in a research note on Wednesday, January 9th. Argus decreased their price target on International Game Technology from $28.00 to $20.00 and set a "buy" rating on the stock in a research note on Friday, November 16th. Finally, Jefferies Financial Group set a $29.00 price target on International Game Technology and gave the stock a "buy" rating in a research note on Thursday, December 20th. One analyst has rated the stock with a sell rating, one has issued a hold rating and eight have given a buy rating to the stock. The stock presently has an average rating of "Buy" and a consensus target price of $23.64.
The company has a market capitalization of $2.74 billion, a P/E ratio of 13.54, a P/E/G ratio of 1.10 and a beta of 1.36. The company has a debt-to-equity ratio of 2.90, a current ratio of 1.14 and a quick ratio of 1.00.
International Game Technology (NYSE:IGT) last announced its quarterly earnings results on Thursday, March 7th. The company reported $0.24 EPS for the quarter, missing the Thomson Reuters' consensus estimate of $0.35 by ($0.11). The firm had revenue of $1.27 billion for the quarter, compared to analysts' expectations of $1.27 billion. International Game Technology had a positive return on equity of 7.64% and a negative net margin of 0.44%. International Game Technology's revenue was down 5.9% compared to the same quarter last year. During the same period last year, the firm earned $0.02 EPS. Analysts anticipate that International Game Technology PLC will post 1.22 earnings per share for the current year.
The business also recently announced a quarterly dividend, which will be paid on Thursday, April 4th. Investors of record on Thursday, March 21st will be paid a $0.20 dividend. This represents a $0.80 dividend on an annualized basis and a yield of 6.03%. The ex-dividend date is Wednesday, March 20th. International Game Technology's payout ratio is currently 81.63%.
A number of hedge funds have recently bought and sold shares of IGT. Marshall Wace North America L.P. lifted its stake in International Game Technology by 41.3% in the 3rd quarter. Marshall Wace North America L.P. now owns 5,747,367 shares of the company's stock valued at $113,510,000 after buying an additional 1,680,947 shares in the last quarter. Rubric Capital Management LP lifted its stake in International Game Technology by 65.9% in the 3rd quarter. Rubric Capital Management LP now owns 3,850,000 shares of the company's stock valued at $76,038,000 after buying an additional 1,530,000 shares in the last quarter. Norges Bank purchased a new stake in International Game Technology in the 4th quarter valued at approximately $20,893,000. TIAA CREF Investment Management LLC lifted its stake in International Game Technology by 336.8% in the 3rd quarter. TIAA CREF Investment Management LLC now owns 1,564,264 shares of the company's stock valued at $30,894,000 after buying an additional 1,206,152 shares in the last quarter. Finally, Samlyn Capital LLC lifted its stake in International Game Technology by 29.0% in the 4th quarter. Samlyn Capital LLC now owns 5,348,778 shares of the company's stock valued at $78,252,000 after buying an additional 1,202,082 shares in the last quarter. 47.76% of the stock is currently owned by hedge funds and other institutional investors.
ILLEGAL ACTIVITY WARNING: This piece of content was first published by The Cerbat Gem and is the property of of The Cerbat Gem. If you are viewing this piece of content on another site, it was copied illegally and reposted in violation of US and international copyright laws. The legal version of this piece of content can be read at https://www.thecerbatgem.com/2019/03/15/international-game-technology-igt-sets-new-1-year-low-at-13-23.html.
International Game Technology PLC operates and provides technology products and services across lotteries, electronic gaming machines, sports betting, and interactive gaming markets worldwide. The company operates through: North America Gaming and Interactive, North America Lottery, International, and Italy segments.
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On Friday night, actress Ruby Rose was kicked out of a restaurant, and now she wants you to know exactly what went down.
The owner of the restaurant, Seung Hong, told an entirely different story to NOLA.com. Hong claimed that Rose "cursed out" the bartender.
Hong claims that the bartender thought the first initial French fry fling was an accident and approached her to make sure everything was okay. "And she picked up a handful of French fries and threw them at him," Hong said. After that, he told her to leave.
That was the first time he had ever thrown someone out of his establishment.
"I worked in hospitality for 5 years and I have huge love, respect and admiration for servers around the world and how hard they work. I also have mad love for New Orleans and I want to come back 100 times if you will have me. Next time, dinner at my place," Rose wrapped it up with a winky face.
She also got herself a few more fries when she stopped in Chicago on Saturday.


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Richmond-Area Farmer Speaks to Collegiate Students About Nutrition
In honor of nutrition month, Collegiate School invited Joel Salatin, a farmer whose family owns and runs Polyface Farm in Swoope, Virginia, to campus today to speak to Middle and Upper School students.
Polyface Farm, open to the public year-round, grows and sells pastured livestock and poultry to individuals and Richmond-area stores. The farm is committed to raising its livestock in ways that are best for the environment, best for the animals (free ranging) and best nutritionally for the consumer.
Mr. Salatin's unique farming methods and commitment to ideals other than profits have made Polyface Farm a leader in the area of responsibly raised livestock. Mr. Salatin speaks all over the world about how his family farms and why they choose to farm this way.
He spoke to Middle School students about the importance of respecting the bacteria that live within the soil and their connection to food, nutrition and people.
"Most of us eat without thinking," he said. "I want you to leave here and the next time someone says, 'Where are we going to eat?' think, OK, where did they respect life so I can respect my life. You can be a catalyst to make better food choice decisions."
Later in the morning, Mr. Salatin spoke with 6th Graders, who are reading Chew On This: Everything You Don't Want to Know About Fast Food. In the afternoon, two Upper School forums focusing on sustainability — Know Your Farmer, Know Your Food and Brand This — were held.
Middle School PE teacher Jenny Lindner facilitated Mr. Salatin's visit. She hopes his talks have informed students about nutrition, sustainability and ethics.
"Not everyone has access to fresh foods," Ms. Lindner said. "Joel is a pioneer in that he shows how you can create food and sell it to people for a fair price."
Other efforts at Collegiate taking place around nutrition this month include a video created by Middle School students, as well as signage and decorations in Centennial Hall and McFall Hall, the School's two cafeterias, to remind students to make healthy choices to fuel healthy bodies.


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Egonu in Sanremo: "I thought it was a joke and instead it's a dream come true"
"When Amadeus called me, I thought it was a joke… Instead it's a dream come true!". These are the words of the Italian volleyball champion Paola Egonu. On Thursday 9 February at the Ariston Theater in Sanremo, Egonu will be co-host of the Italian Song Festival, alongside the artistic director Amadeus on the most popular stage in our country. "I am very happy, honored by the invitation and aware of this very special occasion", added the opposite of Vakifbank and leader of the Italian national volleyball team, considered one of the strongest volleyball players in the world
Paola Egonu's participation in the Sanremo Festival was announced on Sunday 15 January on Tg1 by Amadeus. Reigning European volleyball champion with the Italian national team, winner of two Champions Leagues with Novara and Conegliano and the Club World Cup in 2021, today star of VakifBank, Paola Egonu, born in Cittadella on 18 December 1998, is a volleyball icon on and off the field, woman of the year D of Repubblica in 2019, voice of Sognaluna in the animated film Soul by Disney Pixar and former host of a television episode of Le Iene. In addition to Paola Egonu, alongside Amadeus and Gianni Morandi during the 73rd Festival of Italian Song, from 7 to 11 February at the Ariston Theater in Sanremo, there will be the digital entrepreneur Chiara Ferragni, the journalist Francesca Fagnani and the actress Clare Francini.
Sanremo 2022, the artists competing between music and cinema and TV series
All the guests of the Sanremo 2022 Festival, evening by evening. PHOTO
Sanremo 2022, the most beautiful dresses of the third evening, from Elisa to Michele Bravi. PHOTO
Paola Turci, career and private life: from the accident to the marriage with Francesca Pascale
Arrest Messina Denaro, Giorgia Meloni: January 16 is the day of the fight against the mafia
Fir and the Italian Army together on the road to the Rugby World Cup


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Quality Assurance & Enhancement
Griffith in the Community
Established in 1974, Griffith College is Ireland's largest independent third-level institution with locations in Dublin, Cork and Limerick.
The main Dublin campus is located within a mile of St. Stephen's Green on Dublin's South Circular Road. Situated on seven acres and offering a wide range of amenities, it is a historic site with original campus buildings dating back to 1813.
Griffith College enjoys a national and a growing international reputation for student success. The College has gained an enviable, award-winning reputation for providing students with first-class lectures and excellent study material. Griffith College offers internationally recognised postgraduate and undergraduate degree programmes, complemented by a wide range of professional, short term and corporate training educational solutions - in Griffith College, there is something for everyone. Today, there are over 7,000 students studying in the College, which is a designated educational institute of Quality and Qualifications Ireland (QQI).
The College will mark an important milestone in 2024 when it will celebrate the 50th anniversary of its founding. Over the past half-century, Griffith has made a significant contribution to third-level education in Ireland. Many former students have gone on to distinguished careers in Ireland and internationally, in fields as diverse as accountancy, business, computing, design, engineering, law, media, music and drama. Our current students continue this tradition of success.
Recipients of the College's Distinguished Fellowship Award conferred on individuals who have made an outstanding contribution to Ireland, include Mary McAleese, the 8th President of Ireland, and Nobel Laureates Seamus Heaney and John Hume.
If you are interested in studying at Griffith, we'd be delighted to meet you at one of our upcoming open days to discuss the course that best suits your needs.
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For the spring summer collection 2016, Veja introduced the V-10 model to celebrate the 10th anniversary. The V-10 model made out of ecological and sustainable materials stands for 10 ans d'amour. The style comes up a little large. If you are between two sizes, it is better to take a size down.
These sneakers have navy and bright red accents, use them in your outfit and it will be smashing. Goes nicely with all black too.
For the V-10 we advice to take a smaller size if you are between sizes.


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text:
Charing Cross Hospital
Illness proves a spur to creativity for Fulham artist Nicola
When Fulham artist Nicola McCartney discovered she had a neuromuscular disorder, she had to find a new way of working.
Fulham artist Nicola McCartney, who has a new exhibition called Talking-Walking at Riverside Studios in Hammersmith
The act of applying paint to a canvas is a skill most artists would take for granted.
But five years after graduating from the Slade School of Fine Art, Fulham painter Nicola McCartney began finding it hard to lift up brushes or direct them with her usual level of precision.
That inexplicable fatigue turned out to be a symptom of myasthenia gravis, a disease in which the immune system mistakenly attacks voluntary muscles and the nerves that control them.
Being diagnosed with the condition forced a change of approach for the 28-year-old, who had to find a way of continuing to create art despite new physical restrictions.
"It started with blurry vision and when I was eating one side of my lip wouldn't come down to the meet the other. I went to the doctors and they told me to rest," said Ms McCartney, who lives in Rainville Road, where she also has a studio.
"It was only a month later when my voice stopped working and I went to Charing Cross Hospital that an A&E doctor diagnosed me. I was put on medication and told I had a tumour on my thymus.
"I found it difficult to lift my arm up to the canvas, I couldn't control it and found more delicate bits of the paintings difficult, so I began working facing down on plates at the local pottery cafe."
The result is a series of vivid, vibrant and sometimes slightly sinister paintings of wildlife and animals on earthenware dining plates, which will feature in a new exhibition called Walking-Talking at Hammersmith's Riverside Studios next month.
Now on medication following an operation, Ms McCartney says she is back to 'fighting fit'.
The artist, who is also a patron of the Hammersmith and Fulham Community Law Centre and has worked with students at William Morris Sixth Form, is proud of all she has achieved, having now returned to painting oil on canvas and to her PhD at Birkbeck College, University of London.
Her work on plates provoke questions about whether painted ceramics should be considered as fine art or purely decorative.
The images, including paintings of a slow lorris, which is an endangered species, mouse, guinea pig, musk ox skull, tiger and bat, explore notions of consumerism, hunting, collecting, displaying and eating.
She said: "I think of animals as another portrait, a subject. I paint them with the same intensity as I paint humans.
"None of the paintings are twee images of animals, they all come with a twist. I'm interested in human interaction with animals, stuffed animals, eating animals and fairy tale stories like Little Red Riding Hood and Goldilocks and the Three Bears. The plates are a cross between a work of art and a domestic object."
Each plate will be on sale for £75.
The exhibition also features fellow Slade graduate Laura Nash and her exploration of the patterns we often overlook on London's streets.
Talking-Walking is at Riverside Studios from September 1 to 23.
Ms McCartney is also planning a sponsored walk to raise cash for the Myasthenia Gravis Association on September 23. Visit www.justgiving.com/NAMcCartney to sponsor her.
Only Fools and Horses quiz: How well do you know the Tea for Three episode?
Only Fools and HorsesDel Boy Trotter goes on a flight over the countryside!


Lyndon Johnson and Robert Kennedy loathed each other.
They were a study in contrast — destined, it seemed, to disagree. They ranted, obsessed, sulked and brooded about one another, making few big decisions without first considering the feud.
It was the most important rivalry of their times. It spawned political turf battles across the United States. It weakened the Democratic Party by forcing its members to choose between them. It captivated the newly powerful media, that portrayed every disagreement between LBJ and RFK as part of a prolonged battle for the presidency or a claim on the legacy of the fallen JFK. It helped propel one man to the Senate and drive the other from the White House.
It is impossible to fully understand either man — or the decade they dominated — without understanding their troubled relationship with one another. Mutual Contempt explores the personalities, politics, and issues that wrenched these men apart, as well as the impact of their feud on civil rights, the war on poverty, and the war in Vietnam. Like a Greek tragedy played out on a nation's center stage, this book provides a prism through which to view two men, their times, and the nature of power.


Antarctica, on average, is the coldest, driest, and windiest continent, and has the highest average elevation of all the continents. Antarctica is considered a desert, with annual precipitation of only 200 mm (8 inches) along the coast and far less inland. The temperature in Antarctica has reached -89oC (-129oF). There are no permanent human residents, but anywhere from 1,000 to 5,000 people reside throughout the year at the research stations scattered across the continent. Only cold-adapted organisms survive there, including many types of algae, animals (for example mites, nematodes, penguins, seals and tardigrades), bacteria, fungi, plants, and protista. Vegetation where it occurs is tundra.


text:
\section{Introduction}

\noindent In a series of papers \cite{GN1,GN2,Imm3D}, we have recently introduced and studied a model of three-dimensional gravity that has proven to be very useful to understand some of the fundamental features of canonical loop quantum gravity and spin foam models. This new formulation of gravity in three dimensions is particularly interesting because it admits a Barbero-Immirzi parameter, and can be written as a Holst or a Plebanski action. As such, it can be used to study the imposition of the simplicity constraints in the construction of spin foam models, the role of $\gamma$ and of the $\text{SU}(2)$ Ashtekar-Barbero formulation, and the relation with the self-dual variables. In \cite{Imm3D}, it has been shown that this Lorentzian three-dimensional model can be written at the Hamiltonian level in terms of $\text{SU}(2)$ canonical variables (just like in the four-dimensional case), which leads to discrete and $\gamma$-dependent geometrical operators. However, it turns out to be possible to simplify the Hamiltonian constraint (and write it as a flatness constraint) by working with complex canonical variables, provided that one imposes simplicity-like conditions ensuring that the corresponding connection be $\mathfrak{su}(1,1)$-valued. In terms of these new variables, the spectra of the geometric operators become continuous and $\gamma$-independent.

In four-dimensional loop quantum gravity, the discreteness of quantum geometry at the Planck scale, together with the existence of a non-zero minimal eigenvalue for the area operator, play a crucial role in the loop quantum cosmology scenario of singularity resolution \cite{LQC} and are essential for the recovery of the semi-classical thermodynamical properties of black holes \cite{BHentropy1,BHentropy2,BHentropy3,BHentropy4}. Independently of the question of wether or not the geometric operators will remain discrete and $\gamma$-dependent at the dynamical level, it has been argued that already at the kinematical level a different choice of connection could potentially lead to continuous and $\gamma$-independent spectra \cite{alexandrov5}. This observation has always remained at the formal level, since no Hilbert space is known for the Lorentz-covariant theory in terms of the shifted connection of Alexandrov. It is however known that this approach is closely related to the quantization in terms of the complex self-dual Ashtekar connection, a route which has been recently reconsidered in the light of results on black hole entropy \cite{FGNP,BST,pranzetti} and the asymptotics of spin foam models \cite{BN}. In the absence of a known way of constructing a full quantum theory with the self-dual variables, one is forced to investigate toy models and to focus on particular calculations such as the derivation of black hole entropy. The hope is that this will provide informations as to how the self-dual quantum theory should be built. In particular, a key open question concerns the role of the Barbero-Immirzi parameter and of the $\text{SU}(2)$ variables. Are these fundamental variables of quantum gravity with $\gamma$ playing a true physical role, or simply a necessary regularization procedure needed in order to deal with the non-compactness of the gauge group of Lorentzian gravity? It seems to us that the latter point of view should be adopted. In this short paper, we give an example of the above-mentioned duality between the nature of the geometric spectra (discrete or continuous) and the choice of connection. For this, we use the Holst formulation of three-dimensional gravity to compare the quantum theories built with the $\mathfrak{su}(2)$ Ashtebar-Barbero and the $\sl(2,\mathbb{C})$ self-dual connection.

More precisely, our starting point for this study is the observation that the three-dimensional model of interest can be obtained from a symmetry reduction of the four-dimensional Holst action, defined by imposing a symmetry along an arbitrary spatial direction. Fortunately, this reduction preserves the presence of $\gamma$, and the resulting theory shares most of the properties of its four-dimensional counterpart. In particular, in the time gauge, the classical dynamical variables are given by an $\text{SU}(2)$ Ashtekar-Barbero connection $A$ canonically conjugated to an $\mathfrak{su}(2)$-valued electric field $E$. Furthermore, the six first class constraints that generate the isometries of the locally flat space-time can be decomposed, as in four dimensions, in terms of a Gauss constraint, a vectorial constraint, and a scalar constraint. Since the theory is topological in three dimensions, the physical degrees of freedom are captured by the holonomies $h_\ell(A)$ of the connection along the links $\ell$ of a unique graph $\Gamma$ (sufficiently refined to resolve the topology of the spatial surface), and the ``one-dimensional fluxes'' of the electric field $X_\ell$ along links $\ell^*$ (dual to $\ell$) of the dual graph $\Gamma^*$. At the quantum level, these (non-local) variables are promoted to non-commutative operators which satisfy the three-dimensional analogue of the holonomy-flux algebra. As usual, kinematical states are $\text{SU}(2)$ spin networks associated to the graph $\Gamma$, and the geometric operators are constructed in terms of the flux operators $X_\ell$. The spectrum of the length operator is given by a sum of fundamental contributions $\ell(j)=\gammal_\text{Pl}\sqrt{j(j+1)}$, where $j$ labels unitary irreducible representations of $\text{SU}(2)$, and $l_\text{Pl}$ is the three-dimensional Planck length. Motivated by the fact that this $\text{SU}(2)$ Hamiltonian theory should describe Lorentzian three-dimensional gravity, i.e. $\text{SU}(1,1)$ BF theory, the Hamiltonian constraint can be recast into the form of a flatness constraint. The canonical connection then becomes complex and $\sl(2,\mathbb{C})$-valued, although it is defined with $\gamma\in\mathbb{R}$. On top of this complex formulation, it is possible to impose two types of linear simplicity conditions similar to that used in spin foam models. The first form of the constraint (which relates the $\mathfrak{su}(2)$ generators and their complement in $\sl(2,\mathbb{C})$) gives back the $\mathfrak{su}(2)$-valued Ashtekar-Barbero connection, while the second one (which relates the $\mathfrak{su}(1,1)$ generators and their complement in $\sl(2,\mathbb{C})$) leads to an $\mathfrak{su}(1,1)$ connection. Constructing the quantum theory with the latter, we observe that the length operator is given by a sum of fundamental contributions $\ell(s)=l_\text{Pl}\sqrt{s^2+c}$ (with $c=1/4$ or $c=0$ depending on the regularization of the length operator), where $s$ is a real parameter labeling the continuous series of representations of $\text{SU}(1,1)$. This spectrum is therefore continuous and $\gamma$-independent.

This paper is organized as follows. In the next section, we briefly review the classical formulation that serves as the starting point for this analysis. In section \ref{sec:3}, we present the two different choices of connection that can be made at the classical level, and discuss the resulting quantum theories. Section \ref{sec:4} is devoted to the introduction of the classical geometric operators, while their spectra are studied in section \ref{sec:5}.

\section{Classical theory}

\noindent Let us start with the four-dimensional Holst action
\begin{equation}
S_\text{4D}[e,\omega]=\int_{\mathcal{M}_4}\left(\frac{1}{2}\varepsilon_{IJKL}e^I\wedge e^J \wedge
F^{KL}+\frac{1}{\gamma}\delta_{IJKL}e^I\wedge e^J\wedge F^{KL}\right),
\end{equation}
and perform a spacetime symmetry reduction that keeps the internal gauge group $\text{SL}(2,\mathbb{C})$ unbroken. For this, we assume that the four-dimensional spacetime manifold has the topology $\mathcal{M}_4=\mathcal{M}_3\times\mathbb{S}^1$, where $\mathcal{M}_3$ is a three-dimensional spacetime manifold, and $\mathbb{S}^1$ is a spatial direction along which the theory is invariant. Denoting by $x_3$ the coordinate along $\mathbb{S}^1$, the invariance of the basic fields can be expressed as
\begin{equation}\label{symmetry conditions}
\partial_3=0,\qquad\qquad\omega_3^{IJ}=0.
\end{equation}
As a consequence, the resulting reduced action is given by
\begin{equation}\label{3D action}
S[e,x,\omega]=\int_{\mathcal{M}_3}\mathrm{d}^3x\,\varepsilon^{\mu\nu\rho}\left(\frac{1}{2}\varepsilon_{IJKL}x^Ie_\mu^JF_{\nu \rho}^{KL}+\frac{1}{\gamma}\delta_{IJKL}x^Ie_\mu^JF_{\nu \rho}^{KL}\right),
\end{equation}
where $\omega^{IJ}_\mu$ is an $\sl(2,\mathbb{C})$-valued connection over $\mathcal{M}_3$, $e^I_\mu$ is an $\sl(2,\mathbb{C})$-valued one form over $\mathcal{M}_3$, and $x^I$ is an $\sl(2,\mathbb{C})$-valued scalar on $\mathcal{M}_3$. As usual, $\varepsilon^{\mu\nu\rho}$ is the totally antisymmetric spacetime tensor, $\varepsilon_{IJKL}$ denotes the totally antisymmetric internal tensor, and $\delta_{IJKL}=(\eta_{IK}\eta_{JL}-\eta_{IL}\eta_{JK})/2$ is defined in term of the flat Minkowski metric
$\eta$. We assume that $\mathcal{M}_3=\Sigma_2\times\mathbb{R}$, where $\Sigma_2$ is a two-dimensional surface with no boundaries.

In order to mimic the construction of four-dimensional loop quantum gravity, we performed in \cite{Imm3D} the canonical analysis of the three-dimensional action (\ref{3D action}) in the time gauge. This gauge fixing is defined by the conditions $x^0=e^0_a=0$, and breaks the internal gauge group $\text{SL}(2,\mathbb{C})$ into its maximal compact subgroup $\text{SU}(2)$. Just like in the four-dimensional theory, the resulting phase space is parametrized by the canonical pair $(E^a_i(x),A_b^j(y))$, where $A^i_a$ is the three-dimensional analogue of the $\mathfrak{su}(2)$ Ashtekar-Barbero connection, and $E^a_i$ its conjugate electric field. These variables satisfy the Poisson bracket
\begin{equation}\label{Poisson bracket}
\{E^a_i(x),A_b^j(y)\}=\gamma\delta^a_b\delta^j_i\delta^2(x-y),
\end{equation}
and are subject to the following Gauss, vectorial, and scalar constraints:
\begin{equation}\label{constraints}
G=\partial_aE^a+A_a\time E^a,\qquad
H_a=\varepsilon_{ab}E^b\cdot F_{12},\qquad
H_0=x\cdot\left(F_{12}-(1+\gamma^{-2})K_1\time K_2\right),
\end{equation}
where $F_{12}=\partial_1A_2-\partial_2A_1+A_1\time A_2$ is the curvature of $A$. The variable $\gamma^{-1}K^i_a$ is the three-dimensional analogue of the extrinsic curvature, and can be written as $\gamma^{-1}K^i_a=\gamma^{-1}\big(A^i_a+\omega^i_a(E)\big)=\omega_a^{0i}$, where $\omega(E)$ is the reduced version of the four-dimensional Levi-Civita connection $\Gamma(E)$. Its explicit expression is given by
\begin{equation}
\omega_a(E)=u\time\partial_au+\varepsilon_{ab}\frac{E^b\cdot\partial_cE^c}{|E^1\time E^2|}u,
\end{equation}
where $u^i=x^i/\sqrt{x^2}$. To finish with this brief overview of the phase space structure, let us mention that $x$ is colinear to $E^1\time E^2$, which makes the above expression for $H_0$ strictly similar to the scalar constraint of the four-dimensional full theory. Furthermore, because the vectors $K_a$ are orthogonal to $u$ \cite{Imm3D}, the quantity $K_1\time K_2$ is in the direction of $u$, and $H_a$ and $H_0$ can be viewed as the component of the same vector $H$ defined by
\begin{equation}\label{F=0 in time gauge}
H={F}_{12}-(1+\gamma^{-2})K_1\times K_2.
\end{equation}
This property is in some sense responsible for the fact that this three-dimensional model is exactly soluble.

\section{Quantum theory: from $\boldsymbol{\text{SU}(2)}$ to $\boldsymbol{\text{SU}(1,1)}$ via $\boldsymbol{\text{SL}(2,\mathbb{C})}$}
\label{sec:3}

\subsection{$\boldsymbol{\text{SU}(2)}$ quantum theory}

\noindent We now have all the ingredients to perform the loop quantization of the theory. Kinematical states are cylindrical functions of the connection associated with graphs embedded in the spatial surface $\Sigma_2$. Because of the topological nature of three-dimensional gravity, a single graph $\Gamma$ is sufficient to capture all the physical content of the theory. On this graph, let us introduce the holonomies of the connection along the links $\ell\in\Gamma$, and the fluxes of the electric field along links $\ell^*\in\Gamma^*$ dual to $\ell$. These variables, i.e.
\begin{eqnarray}\label{holflux}
\text{SU}(2)\ni h_\ell(A)=\overrightarrow{\exp}\int_\ell A_a^i\mathrm{d} x^a\tau_i,\qquad\qquad
\mathfrak{su}(2)\ni X^i_{\ell^*}=\int_{\ell^*}\varepsilon_{ab}E^a_i\mathrm{d} x^b,
\end{eqnarray}
are the building blocks of the three-dimensional (classical and quantum) holonomy-flux algebra. The action of the flux operator on the holonomies evaluated in the spin-$j$ representation of $\mathfrak{su}(2)$ is given by
\begin{equation}\label{actionofL}
X^i_{\ell^*}\triangleright\mathbf{D}^{(j)}(h_{\ell'}(A))=\mathrm{i}\epsilon\gammal_\text{Pl}\delta_{\ell,\ell'}\mathbf{D}^{(j)}(h_{\ell<c}(A))J_i\mathbf{D}^{(j)}(h_{\ell>c}(A)),
\end{equation}
where $\epsilon\in\{-1,+1\}$ is the index between $\ell$ and $\ell^*$, the point $c$ denotes the intersection $\ell\cap\ell^*$, and $\mathbf{D}^{(j)}:\text{SU}(2) \longrightarrow\mathbb{V}^{(j)}$ denotes the spin-$j$ representation matrix associated to the $(2j+1)$-dimensional vector space $\mathbb{V}^{(j)}$. The elements $J_i$ generate the Lie algebra $\mathfrak{su}(2)$ and satisfy, by convention, the Lie algebra commutation relations $[J_i,J_j]=\varepsilon_{ij}^{~~k} J_k$, where $\varepsilon_{ijk}$ is the totally antisymmetric tensor with $\varepsilon_{123}=1$, and indices are lowered and raised with the flat Euclidean metric $\delta_{ij}$. As usual, cylindrical functions associated to the graph $\Gamma$ form the kinematical Hilbert space where the scalar product is constructed from the $\text{SU}(2)$ Haar measure.

\subsection{Obtaining the $\boldsymbol{\text{SL}(2,\mathbb{C})}$ connection}

\noindent In principle, physical states and observables should be constructed by solving the quantum constraint $H\simeq0$. In this set of constraints, only the imposition of the scalar constraint $H_0\simeq0$ is problematic since the vectorial constraint $H_a\simeq0$ imposing spatial diffeomorphism invariance has already been solved implicitly by fixing the graph $\Gamma$. The difficulty in solving $H_0\simeq0$ is essentially the same as in the four-dimensional theory, and requires an appropriate regularization of the term $K_1\time K_2$ when written in terms of the non-linear expression $\omega(E)$. This difficulty can however be bypassed by noticing that in this three-dimensional model the constraints $H\simeq0$ given by (\ref{F=0 in time gauge}) can be written in the form of the flatness constraint of a BF theory. More precisely, one can look for a connection $\mathbf{A}$ for which $H\simeq0$ is equivalent to  $\mathcal{F}_{12}\simeq 0$, where $\mathcal{F}$ is the curvature of $\mathbf{A}$. It was shown in \cite{Imm3D} that, up to gauge transformations, there are only two connections that satisfy this requirement. These are the self-dual and anti self-dual components of the initial $\text{SL}(2,\mathbb{C})$ connection $\omega^{IJ}_a$. The connection $\mathbf{A}$ is therefore complex and has to be interpreted as an $\sl(2,\mathbb{C})$-valued connection. Therefore, we have somehow traded the problem of dealing with the complicated Hamiltonian constraint (\ref{F=0 in time gauge}) for that of imposing the reality conditions. The same fact is obviously true in four dimensions, where one can take $\gamma=\mathrm{i}$ to obtain the self-dual scalar constraint, at the expense of working with the complex Ashtekar connection.

We arrive at the conclusion that the quantum dynamics can in principle be solved in terms of the connection $\mathbf{A}$, since with this variable the Hamiltonian constraint is manageable. This means that one should consider $\text{SL}(2,\mathbb{C})$ spin network states instead of the standard $\text{SU}(2)$ ones, which introduces numerous technical problems due to the non-compactness of the Lorentz group. A priori, such spin network states require a regularization in order to be well-defined. Fortunately, as we are about to see, this problem does drastically simplify due to the constraints satisfied by the complex connection $\mathbf{A}$.

\subsection{Simplicity constraints and the $\boldsymbol{\text{SU}(1,1)}$ connection}

\noindent As explained in \cite{Imm3D}, the connection $\mathbf{A}$ can be written in the form
\begin{equation}\label{decomposition of A}
{\mathbf{A}}=\left[-u\time\mathrm{d} u\cdot(J\pm\gamma^{-1}P)\right]+
\left[(A\cdot u)(u\cdot J)\mp\gamma^{-1}(A\time u)\cdot(P\time u)\right],
\end{equation}
where $J_i$ are the $\mathfrak{su}(2)$ generators introduces above, and $P_i$ are the boost generators satisfying the standard $\sl(2,\mathbb{C})$ commutation relations. One can see from this expression that $\mathbf{A}$ possesses two different parts, which are the two terms between the square brackets. Interestingly, while the first term has no direct algebraic interpretation, the second term defines an $\mathfrak{su}(1,1)$ component. Indeed, it can easily be shown that the elements $J\cdot u$ and the two independent components of $P\time u$ generate the Lie algebra $\mathfrak{su}(1,1)$.

Now, one can use the fact that there exists another legitimate choice of gauge for the action (\ref{3D action}). Indeed, in the gauge $x^I=(0,0,0,1)$, the action (\ref{3D action}) reduces to that of $\text{SU}(1,1)$ BF theory, which is in complete agreement with the fact that it describes Lorentzian three-dimensional gravity. For this reason, consistency of the theory in the time gauge requires that we impose that the connection $\mathbf{A}$ be $\mathfrak{su}(1,1)$-valued as well. This requirement can be met if the first term in (\ref{decomposition of A}) is vanishing, which leads to the constraint $(J\pm\gamma^{-1}P)\time u=0$. This simplicity-like constraint is equivalent to a reality condition, in the sense that it selects a real section of $\text{SL}(2,\mathbb{C})$ corresponding to its non-compact subgroup $\text{SU}(1,1)$. Once it is imposed at the classical level, the linear simplicity constraint reduces the connection (\ref{decomposition of A}) to the following $\mathfrak{su}(1,1)$ connection:
\begin{equation}\label{cal A}
\mathbf{A}=\mp\mathrm{i}\gamma^{-1}(A^1F_1+A^2F_2-\mathrm{i}A^3F_0),
\end{equation}
where $A^i$ are the components of the initial Ashtekar-Barbero connection $A$ expressed in the basis $(J_1,J_2,J_3)$, and the family $(F_0,F_1,F_2)$ generates $\mathfrak{su}(1,1)$ with the commutation relations
\begin{equation}
[F_1,F_2]=\mathrm{i}F_0,\qquad
[F_0,F_2]=\mathrm{i}F_1,\qquad
[F_0,F_1]=-\mathrm{i}F_2.
\end{equation}
These generators are related by $(F_0,F_1,F_2)=-(\mathrm{i}J_3,J_1,J_2)$.

Now that the $\mathfrak{su}(1,1)$ connection (\ref{cal A}) is defined, one can in principle construct the physical Hilbert space by imposing the quantum $\text{SU}(1,1)$ flatness constraint. Physical states are $\text{SU}(1,1)$ spin networks with support on the graph $\Gamma$. This defines the vector space structure of the physical Hilbert space, and it remains to define the scalar product between the states and eventually to eliminate the zero-norm states. In order to do so, it is much simpler to choose $\Gamma$ to be a minimal graph. When the spatial slice $\Sigma_2$ is a Riemann surface of genus $g$ with no boundaries, the minimal graph consists in only one vertex $v$ and $2g$ loops $(a_1,b_1,\dots,a_g,b_g)$ starting and ending at $v$. The loops $(a_i,b_i)$, with $i\in\llbracket1,g\rrbracket$, are the non-contractible loops around the handles of $\Sigma_2$, and they can be identified with the standard generators of the fundamental group $\pi^1(\Sigma_2)$ of the surface. For obvious reasons, such a minimal graph is called a flower graph. Physical states are then totally defined by complex-valued functions $\varphi$ on $\mathcal{G}_0=\text{SU}(1,1)^{\otimes2g}/\text{SU}(1,1)$, where the coset by $\text{SU}(1,1)$ traduces the gauge-invariance at the unique vertex $v$ of $\Gamma$, and the physical inner product between two such states is formally defined by
\begin{equation}\label{physical product}
\langle\varphi_1,\varphi_2\rangle=\int\mathrm{d}\mu(a,b)\overline{\varphi_1(a,b)}\delta([a,b])\varphi_2(a,b),
\end{equation}
where $\mathrm{d}\mu(a,b)$ is a ``regularized'' measure on $\mathcal{G}_0$, $\delta$ is the Dirac distribution on $\text{SU}(1,1)$, and the commutator is $[a,b]=\prod_{i=1}^g a_ib_ia_i^{-1}b_i^{-1}$. For simplicity, we have identified here the states with the corresponding funtions on $\mathcal{G}_0$. At this point, the physical scalar product and the physical Hilbert space are defined only at the formal level, but we will make this construction concrete and explicit later on in the case of the torus. To finish, let us recall that the physical scalar product (\ref{physical product}) should reproduce the Lorentzian Ponzano-Regge amplitudes \cite{Freidel}, just like it is the case for the Euclidean signature \cite{NP1,NP2}.

\section{Classical geometric operators}
\label{sec:4}

\noindent We now introduce the three-dimensional length and area operators. For this, we first define the four-dimensional area and volume operators, and then reduce them with the symmetry (\ref{symmetry conditions}). Recall that in four spacetime dimensions the Hamiltonian theory is written assuming that $\mathcal{M}_4=\Sigma_3\times\mathbb{R}$. The area $\alpha_4(S)$ and volume $\nu_4(R)$ operators are then defined on the spatial slice $\Sigma_3$, and measure the area and the volume of a surface $S\subset\Sigma_3$ and a region $R\subset\Sigma_3$, respectively. The classical expressions for the area $\alpha_4(S)$ of a surface $S$ of normal $n$, and the volume $\nu_4(R)$ of a bounded region $R$, depend only on the electric field $E^\mu=\varepsilon^{\mu\nu\rho}e_\nu\time e_\rho/2$, with $\mu\in\{1,2,3\}$, and are given by
\begin{equation}\label{A and V in 4D}
\alpha_4(S)=\int_S\mathrm{d}^2x\,\sqrt{E^\mu\cdot E^\nu n_\mu n_\nu},\qquad\qquad
\nu_4(R)=\int_R\mathrm{d}^3x\,\sqrt{\left|\frac{1}{3!}\varepsilon_{\mu\nu\rho}E^\mu\cdot E^\nu\time E^\rho\right|}.
\end{equation}
Notice that for $\mu=a\in\{1,2\}$, $E^\mu$ coincides with the electric field (\ref{Poisson bracket}) of the three-dimensional action (\ref{3D action}), and is given by $E^a=\varepsilon^{ab}e_b\time x$, whereas for $\mu=3$ we have $E^3=e_1\time e_2$.

The compactification that we have used to obtain the three-dimensional Holst action is of the type $\mathcal{M}_4=\mathcal{M}_3\times\mathbb{S}^1$. Since we have further assumed that $\mathcal{M}_3=\Sigma_2\times\mathbb{R}$ in order to perform the canonical analysis, we can write that
\begin{equation}
\mathcal{M}_4=\Sigma_3\times\mathbb{R}=\mathcal{M}_3\times\mathbb{S}^1=\Sigma_2\times\mathbb{R}\times\mathbb{S}^1.
\end{equation}
In other words, the spatial slice $\Sigma_3$ on which the operators (\ref{A and V in 4D}) are defined is given by $\Sigma_3=\Sigma_2\times\mathbb{S}^1$. Therefore, the surface $S\subset\Sigma_3$ measured by $\alpha_4(S)$ can be of two types, as represented in figure \ref{operators}. It can either be of the form $S=L\times I$, where $L$ is a path (i.e. a one-dimensional submanifold) in $\Sigma_2$ and $I$ a segment in $\mathbb{S}^1$, or it can be a two-dimensional submanifold of $\Sigma_2$.
\begin{figure}[h]
\includegraphics[scale=0.45]{operators.eps}
\caption{Decomposition $\Sigma_3=\Sigma_2\times\mathbb{S}^1$ and representation of the two types of surfaces that can be embedded in $\Sigma_3$. The surface $S\subset\Sigma_2$ is a two-dimensional submanifold of $\Sigma_2$, while the surface $S'$ is of the type $S'=L\times I$, where $L$ is a one-dimensional submanifold of $\Sigma_2$ and $I$ a segment in the direction $\mu=3$. The length operator on $\Sigma_2$ can be derived from the area operator $\alpha_4(S')$ on $\Sigma_3$, while the area operator on $\Sigma_2$ can be derived from the volume operator $\nu_4(R)$ on $\Sigma_3$.}
\label{operators}
\end{figure}
Let us start with the case $S=L\times I$, where $L$ is a path in $\Sigma_2$ and $I$ a segment of length $d$ in the third direction $\mu=3$. Then, the normal $n_\mu$ is orthogonal to the third direction and has only components $(n_1,n_2)$ in the directions $a=1$ and $a=2$ (i.e. in $\Sigma_2$). The area reduces to $\alpha_4(S)=d\lambda(L)$, where $\lambda(L)$ is the length of $L$ given by
\begin{equation}\label{3D length}
\lambda(L)=\frac{1}{d}\alpha_4(S)=\int_L\mathrm{d} x\,\sqrt{E^a\cdot E^bn_an_b}.
\end{equation}
In this way, we obtain immediately the length of any curve $L$ in $\Sigma_2$ in terms of the electric field and the unit vector $n$ normal to this curve.

When the surface $S$ is a two-dimensional submanifold of $\Sigma_2$, then it is immediate to see that $\alpha_4(S)$ gives the area of the surface $S$. In this case, $n$ is necessarily in the third direction, i.e. $n_1=n_2=0$ and $n_3=1$, and the expression $\alpha_4(S)$ for the area reduces to
\begin{equation}
\alpha(S)=\int_S\mathrm{d}^2x\,\sqrt{E^3\cdot E^3},
\end{equation}
which can be expressed in terms of the triad field since $E^3=e_1\time e_2$. To write this expression in terms of the electric field $E^a$ only, we use a symmetry of the action (\ref{3D action}) described in \cite{Imm3D}, which states that $x$ can be chosen to be of unit norm (${x^2}=1$), and that it is orthogonal to $e^a$ (i.e. $e_a\cdot x=0$). If we make this choice, then the area takes the simple form
\begin{equation}\label{3D area}
\alpha(S)=\int_S\mathrm{d}^2x\,\sqrt{|E^1\time E^2|^2}=\int_S\mathrm{d}^2x\,\sqrt{(E^1)^2 (E^2)^2 - (E^1 \cdot E^2)^2}.
\end{equation}
The operators $\lambda(L)$ and $\alpha(S)$ defined in (\ref{3D length}) and (\ref{3D area}) measure respectively the length of a curve $L\subset\Sigma_2$ and the area of a surface $S\subset\Sigma_2$ in our three-dimensional model.

Notice that we could have derived the expression for the area operator $\alpha(S)$ of the three-dimensional theory by considering the volume operator $\nu_4(R)$ defined for a region $R$ of the type $R=S\times I$, where $S$ is a surface in $\Sigma_2$ and $I$ a segment of length $d$ in $\mathbb{S}^1$ (see figure \ref{operators}). Indeed, for such a region we have $\nu_4(R)=d\alpha(S)$, and one can then directly obtain (\ref{3D area}).

Now that we have derived the classical expressions for the geometric operators of the three-dimensional theory (that are defined on the two-dimensional surface $\Sigma_2$), we can quantize them and study their action on the above-defined $\text{SU}(2)$ and $\text{SU}(1,1)$ states. As it is the case for the four-dimensional volume operator $\nu_4(R)$, the three-dimensional area operator $\alpha(S)$ requires a regularization in order to be well-defined in terms of the quantum flux operators $X_{\ell^*}$ \cite{rovelli-smolin-area-volume,ashtekar-lewandowski-volume,giesel-thiemann-vol1,giesel-thiemann-vol2,thiemann-brunnemann,brunnemann-rideout1,brunnemann-rideout2}. These regularizations have been studied and well-understood in the literature, and they will not affect our result.

\section{Quantum geometric operators}
\label{sec:5}

\noindent The length operator is associated to links $\ell^*$ of the dual graph $\Gamma^*$, and the length $\lambda(\ell^*)$ of the link $\ell^*$ dual to $\ell$ can be expressed in terms of the fluxes (\ref{holflux}) according to
\begin{equation}\label{length operator}
\lambda(\ell^*)=\sqrt{X_{\ell^*}^2}.
\end{equation}
The length $\lambda(L^*)$ of any path $L^*\in\Gamma^*$ defined by a composition $L^*=\ell_1^* \circ\dots\circ\ell_n^*$ of a sequence of $n$ elementary links\footnote{A physical observable is given by a closed loop $L$ around non-contractible handles of the surface $\Sigma_2$. An open link $L$ would not correspond to a physical (Dirac) observable apart if particles are coupled to gravity and located at the end points of $L$ .} is simply given by the sum $\sum_{i=1}^n \lambda({\ell_i^*})$. Therefore, computing the action of $\lambda(\ell^*)$ on the holonomy $h_{\ell'}(A)$ or $g_{\ell'}(\mathbf{A})$ allows to compute the action of any length operator $\lambda(L^*)$ on the states of the $\text{SU}(2)$ or $\text{SU}(1,1)$ quantum theories. This action is non-trivial only when $\ell=\ell'$, it is diagonal and given by
\begin{eqnarray}\label{lengthspectrum}
\lambda(\ell^*)\triangleright\mathbf{D}^{(j)}(h_\ell(A))&=&\gamma\sqrt{C(\mathfrak{su}(2))}\mathbf{D}^{(j)}(h_\ell(A)),\\
\lambda(\ell^*)\triangleright\mathbf{D}^{(s)}(g_\ell(\mathbf{A}))&=&\sqrt{C(\mathfrak{su}(1,1))}\mathbf{D}^{(s)}(g_\ell(\mathbf{A})),
\end{eqnarray}
where $C(\mathfrak{su}(2))=-J^2$ and $C(\mathfrak{su}(1,1))=F_1^2+F_2^2-F_0^2$ are the quadratic Casimir operators of the Lie algebras $\mathfrak{su}(2)$ and $\mathfrak{su}(1,1)$, respectively. We have used the notation $j$ to denote unitary irreducible representations of $\mathfrak{su}(2)$ and generically $s$ for those of $\mathfrak{su}(1,1)$ (whose class has not been fixed yet). While $C(\mathfrak{su}(2))=j(j+1)$ is always positive and discrete because $j$ is a non-negative half-integer, the sign of $C(\mathfrak{su}(1,1))$ depends on the class of $\mathfrak{su}(1,1)$ representations $s$ under consideration. In order to have a self-adjoint length operator (with real and positive eigenvalues), $s$ must belong to the continuous series. In this case, the spectrum $\sqrt{C(\mathfrak{su}(1,1))}=\sqrt{s^2+1/4}$ of the length operator is continuous. Before studying the area operator, let us make some important remarks.

\begin{enumerate}
\item First, we would like to emphasize that, in addition to being continuous, the length spectrum becomes also independent of the Barbero-Immirzi parameter $\gamma$. The reason is simple and comes from the overall factor of $\gamma^{-1}$ in the expression (\ref{cal A}) for $\mathbf{A}$, which cancels with the factor of $\gamma$ coming from the Poisson bracket (\ref{Poisson bracket}) when computing the action of $X_{\ell^*}$ on $g_\ell(\mathbf{A})$, i.e.\footnote{The basis $F_i$ as to be understood with $i\in\{0,1,2\}$, while for $J_i$ we have $i\in\{1,2,3\}$.}
\begin{equation}
X^i_{\ell^*}\triangleright\mathbf{D}^{(s)}(g_{\ell'}(\mathbf{A}))
=(-\mathrm{i}\gamma)(\mp\mathrm{i}\gamma^{-1})l_\text{Pl}\delta_{\ell,\ell'}\mathbf{D}^{(s)}(g_{\ell<c}(\mathbf{A}))(\sigma F_i)\mathbf{D}^{(s)}(g_{\ell>c}(\mathbf{A})),
\end{equation}
where $\sigma\in\{1,-\mathrm{i}\}$ is equal to $-\mathrm{i}$ when one acts with the third component $X^3_{\ell^*}$ of the flux.

\item The factor of $\sigma$ is responsible for the fact that the action of the $\mathfrak{su}(2)$-invariant operator $X^2_{\ell^*}=X^i_{\ell^*}\delta_{ij}X^j_{\ell^*}$ produces the $\mathfrak{su}(1,1)$ quadratic Casimir when it acts on the holonomies $g_\ell(\mathbf{A})$. Of course, one could think of writing the action of the length operator on the $\text{SU}(2)$ holonomies $h_\ell(A)$ in the basis $F_i$ instead of $J_i$, but then the spectrum would come with an incorrect minus sign.

\item Even if the spectrum (\ref{lengthspectrum}) is continuous when using the connection $\mathbf{A}$ instead of $A$, there exists a length gap between ``no length'' and the first non-trivial possible length. Another regularization, discussed in \cite{freidel-livine-rovelli}, would give $s$ instead of $\sqrt{s^2+1/4}$ for the action of the length operator on holonomies colored with a representation $s$, and no length gap would exist in this case. We refer the reader to \cite{freidel-livine-rovelli} for a discussion of this ambiguity.
\end{enumerate}

The area operator in three dimensions is the analogue of the volume operator in four dimensions, and it acts on the vertices of the spin networks. For obvious reasons, we will be interested only in its action on three-valent vertices. Denoting by $\ell_1$, $\ell_2$ and $\ell_3$ the three links meeting at the vertex $v$, we will use the notation $\alpha(\ell_1,\ell_2,\ell_3)$ for the area operator at $v$. It is straightforward to obtain the following expression for the regularized area operator acting on the three-valent vertex $v$ \cite{freidel-livine-rovelli}:
\begin{equation}
\alpha(\ell_1,\ell_2,\ell_3)^2=\frac{1}{6}\sum_{I<J}\left(X_{\ell^*_I}^2X_{\ell^*_J}^2-(X_{\ell^*_I}\cdot X_{\ell^*_J})^2 \right),
\end{equation}
where $1\leq I<J\leq3$ label the edges meeting at the vertex. One can then show that the action of the area operator is diagonal, with eigenvalues given by
\begin{equation}
\alpha(\ell_1,\ell_2,\ell_3)^2=\frac{1}{6}\sum_{(I,J)}\left(\lambda_I^2\lambda_J^2-(\xi_I\cdot\xi_J)^2\right),
\end{equation}
where the sum contains only the three terms $(1,2)$, $(2,3)$ and $(1,3)$. The Lie algebra elements $\xi$ denote the $\mathfrak{su}(2)$ or $\mathfrak{su}(1,1)$ generators depending on the states that we act on (i.e. the $\text{SU}(2)$ or $\text{SU}(1,1)$ spin networks), and the scalar product in the internal space is defined with the flat Euclidean metric $\delta_{ij}$ or the flat Minkowskian metric $\eta_{ij}$, depending on the choice of connection. Finally, $\lambda_I$ denotes the eigenvalue of the length operator $\lambda({\ell^*_I})$ (\ref{length operator}). Using the Gauss law at the vertex $v$ to write $X_{\ell^*_1}^i+X_{\ell^*_2}^i+X_{\ell^*_3}^i=0$, an immediate computation leads to
\begin{eqnarray}\label{areaproduct}
\alpha(\ell_1,\ell_2,\ell_3)^2&=&\frac{1}{6}\sum_{(I,J)}\left(\lambda_I^2\lambda_J^2
-\frac{1}{4}(\lambda_I^2+\lambda_J^2-\lambda_K^2)^2\right)\nonumber\\
&=&\frac{1}{16}(\lambda_1+\lambda_2+\lambda_3)(-\lambda_1+\lambda_2+\lambda_3)
(\lambda_1-\lambda_2+\lambda_3)(\lambda_1+\lambda_2-\lambda_3),
\end{eqnarray}
where $K \neq I,J$. We obtain the classical expression for the area of a triangle with edges of length $\lambda_I$, and the geometrical interpretation of this formula is therefore immediate. However, an extra condition has to be imposed in order for the area eigenvalues to be real. Indeed, the product (\ref{areaproduct}) is positive only if the three lengths $\lambda_I$ satisfy the triangular inequalities. Contrary to the $\text{SU}(2)$ kinematical case where the triangular inequalities are automatically satisfied due to the standard properties of tensor products between unitary irreducible representations, here this condition has to be imposed by hand because the representations are in the continuous series of $\text{SU}(1,1)$.

To finish, let us study more precisely the case in which $\Sigma_2$ is a torus. The torus can be represented by the flower graph with two petals as shown in figure \ref{torus}.
\begin{figure}[h]
\includegraphics[scale=0.45]{torus.eps}
\caption{Discretization of a torus by a minimal flower graph. The flower consists only in a single four-valent vertex $v$ and two loops, $\ell_1$ and $\ell_2$, corresponding to the non-contractible cycles of the torus. A spin network state associated to the flower graph is an assignment of unitary irreducible representations of the suitable Lie algebra to the two loops and to the vertex. The vertex being four-valent, it can be decomposed into two three-valent vertices, with the spin previously attached to $v$ now coloring the link $\ell_3$. As shown on the right, this graph with three links can be seen as dual to the rectangle with identified opposite edges that defines the torus. This rectangle can be triangulated into two triangles sharing the side of length $j_3$ and dual to the vertices $v_1$ and $v_2$. The spins $j_1$ and $j_2$ coloring the loops $\ell_1$ and $\ell_2$ represent the edge lengths of the rectangle.}
\label{torus}
\end{figure}
In this case, one can construct the kinematical spin network states of the $\text{SU}(2)$ theory based on the three-dimensional Ashtekar-Barbero connection $A^i_a$. These states, denoted by $\varphi_\text{kin}(j_1,j_2,j_3)$, are totally characterized by the three spin labels $(j_1,j_2,j_3)$ coloring the two loops $\ell_1$ and $\ell_2$ (for the first two labels) and the vertex $v$. Note that the choice of a ``shape'' for the intertwiner is not important (i.e. the choice of a triangulation for the rectangle in figure \ref{torus} is arbitrary). Our choice, which is depicted in figure \ref{torus}, corresponds to a triangulation of the parallelogram into two triangles. Since the graph is minimal, the kinematical scalar product is given by
\begin{equation}
\langle\varphi_\text{kin}(j_1,j_2,j_3),\varphi_\text{kin}(j'_1,j'_2,j'_3)\rangle=\prod_{i=1}^3\delta_{j_i,j_i'},
\end{equation}
where $\delta_{j,j'}$ denotes the Kronecker delta. The length and area operators of the $\text{SU}(2)$ kinematical theory are unitary with respect to this scalar product since their action is diagonal with positive eigenvalues. Furthermore, the geometric interpretation of the spin labels is immediate: they are associated with the length of the three edge of the (similar) triangles that discretize the parallelogram whose opposite sides are identified to construct the torus.

Now, the construction of the physical states would a priori require to solve the Ashtekar-Barbero Hamiltonian constraint (\ref{F=0 in time gauge}). Just like in four-dimensional loop quantum gravity, it is not known wether or not at the physical level the $\gamma$ dependency will disappear and the nature of the geometric operators change. However, in the present three-dimensional model, the route to constructing the physical states is known. One can go back to the complex connection (\ref{decomposition of A}), impose the simplicity constraint that turns it into an $\mathfrak{su}(1,1)$ connection, and then the Hamiltonian constraint becomes simply the flatness constraint of $\text{SU}(1,1)$ BF theory. In this case, the scalar product between physical states is given by
\begin{equation}
\langle\varphi_\text{phy}(s_1,s_2,s_3),\varphi_\text{phy}(s'_1,s'_2,s'_3)\rangle=\prod_{i=1}^3\delta(s_i-s_i'),
\end{equation}
where $\delta(s-s')$ is the Dirac delta distribution, and any other quantization scheme (like the $\text{SU}(2)$ formulation in the time gauge) should lead to the same result in order for it to be anomaly-free.

\section{Conclusion}

\noindent In this short paper, we have studied the loop quantization of three-dimensional gravity based on two different classical formulations of the Hamiltonian theory. Using the results of \cite{Imm3D}, we have first written the canonical theory in terms of $\text{SU}(2)$ Ashtekar-Barbero variables with a real Barbero-Immirzi parameter. Then, we have argued that there is a simpler Hamiltonian formulation (in terms of a flatness constraint), in which the connection becomes complex and has to be supplemented by appropriate reality conditions. When these are implemented classically in the form of a linear simplicity constraint, one recovers an $\mathfrak{su}(1,1)$ connection which does still depend on real values of the Barbero-Immirzi parameter $\gamma$. However, as we have shown, this connection leads to a quantum theory with $\gamma$-independent and continuous spectra for the geometric operators. It is important to stress out that what we did was not to compare the new $\text{SU}(2)$ Ashtekar-Barbero formulation of three-dimensional gravity with usual first order Lorentzian gravity (i.e. $\text{SU}(1,1)$ BF theory). Instead, starting from the $\text{SU}(2)$ theory, we have naturally ended up with a new choice of connection that enables to work out the Hamiltonian constraint (just like the choice $\gamma=\pm\mathrm{i}$ in four dimensions), with the additional requirement that this connection should satisfy a reality condition similar to the spin foam linear simplicity constraint.

Which lessons can be learned for four-dimensional loop quantum gravity? The present model shows that already at the kinematical level one can have two very different quantum theories depending on the choice of connection. The same observation seems to be true in four dimensions, although it has never been made rigorous because of the absence of a kinematical arena for the self-dual connection of Ashtekar or the non-commutative shifted connection of Alexandrov. Moreover, it is clear in this three-dimensional model that physical states are $\text{SU}(1,1)$ spin networks, therefore raising the question of wether the implementation of the Ashtekar-Barbero Hamiltonian constraint (\ref{F=0 in time gauge}) will also lead to these physical states, and to the disappearance of the Barbero-Immirzi parameter. Even if we have not addressed this question here, our model could be taken as a starting point to investigate this direction.

To summarize, if one tries to support the point of view that $\gamma$ should not play a role at the physical level, there are essentially three possible scenarios.
\begin{enumerate}
\item The first one is a quantum theory based on the $\text{SU}(2)$ kinematics, and in which the Barbero-Immirzi parameter disappears in the quantum theory at the level of physical states (with possibly the spectra becoming continuous). From what is known today about the quantum scalar constraint, this does not seem to be the case. Moreover, the EPRL and FK${}_\gamma$ spin foam models are designed in such a way that $\gamma$ plays a central role in the definition of the dynamics of the theory.
\item Another possibility would be to go back to the self-dual connection already at the classical level, and to find a way of dealing with the reality conditions. This is what has been done in the three-dimensional model of \cite{Imm3D} and in the present paper. As we have seen, the reason for which this is possible in three-dimensions is that there is a natural notion of reality for the complex connection, which corresponds to considering its non-compact subgroup $\text{SU}(1,1)$ by means of the linear simplicity constraint. Then, the non-compactness can be handled due to the topological nature of the theory, and the resulting possibility of working on a fixed minimal graph.
\item It seems to us that the most promising direction would be to consider $\gamma$ as a regulator needed in order to deal with the non-compactness of the gauge group. Because the four-dimensional theory has local degrees of freedom, this non-compactness cannot be handled like in the three-dimensional case by fixing a minimal graph. The introduction of the Barbero-Immirzi parameter could therefore be seen as a generalized Wick rotation, which has to be unfolded at the end of the day in order to obtain physical results. This procedure of going through a compact gauge group in order to define the theory, and then using tools of analytic continuation to return to the non-compact case, was put forward in the computation of black hole entropy in four \cite{FGNP} and three dimensions \cite{BTZ} (although in this last reference the regulator is the Chern-Simons level rather than the Barbero-Immirzi parameter).
\end{enumerate}

If this last interpretation of the Barbero-Immirzi parameter turns out to be the correct one, then one should reconsider the results derived from the kinematical structure of the $\text{SU}(2)$ theory, since these rely heavily on the compactness of the gauge group and the value $\gamma\in\mathbb{R}$. The key open question is in fact that of knowing at which step to perform the analytic continuation back to the self-dual value $\gamma=\mathrm{i}$, if this can be given any well-defined meaning in the (canonical or covariant) full theory at all.

\section*{Aknowledgements}

\noindent MG is supported by the NSF Grant PHY-1205388 and the Eberly research funds of The Pennsylvania State University.

redpajama_set_name: RedPajamaArXiv


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at medium-high speed with a mixer until creamy, about 3 minutes.
3. Reduce mixer speed to low; add flour mixture to butter mixture. Add cream, and beat to combine.
4. Preheat oven to 350°. Line baking sheets with parchment paper; set aside.
baking sheets, at least 2 inches apart. Sprinkle with coarse sugar.
6. Bake for 8 to 10 minutes or until edges are lightly golden brown. Remove from oven, and let cool on pan for 5 minutes. Remove to wire racks to cool completely. Garnish with fresh lavender, if desired. Store, covered, at room temperature for up to 1 week.
1. In a medium saucepan, melt butter over medium heat. Cook until butter turns a medium brown and has a nutty aroma, about 10 minutes.
2. Remove from heat, and let cool to room temperature. Use any remainder for another use, if desired. Use immediately, or store, covered, in refrigerator for up to 1 week.
Lavender and almonds give this crumbly biscuit cookie a rich, delicate flavor.