text:
Legal & Politics
General Business News
Home » Lucky Cement records consolidated earnings of PKR 14.82 billion for fiscal year ended June 30, 2018
Lucky Cement records consolidated earnings of PKR 14.82 billion for fiscal year ended June 30, 2018
August 1, 2018	| General Business News | Share:
Karachi, August 01, 2018 (PPI-OT): On a consolidated basis, Lucky Cement Limited reported net profit after tax of PKR 14.82 billion after taking out PKR 1.35 billion attributable to non-controlling interests for the financial year ended June 30, 2018, which translates into earnings per share (EPS) of PKR 45.83 / share as compared to PKR 50.18 / share reported last year. The Board of Directors proposed a final cash dividend of PKR 8.00 per share subject to approval of shareholders in the upcoming Annual General Meeting scheduled to be held on September 28, 2018. This proposed final cash dividend is in addition to the interim cash dividend of PKR 5.00 per share, as previously recommended by the Board of Directors for the financial year ended June 30, 2018.
On a consolidated basis, the Company achieved gross turnover of PKR 124.68 billion which is 13.6%higher as compared to last year's turnover of PKR 109.80 billion.With regards to the Company's standalone performance, the gross sales revenue of the Company increased by 9.4% to PKR 67.38 billion compared to PKR 61.60 billion reported last year. The increase in revenue was mainly due to higher volumes and increase in Federal Excise Duty and Sales Tax. Furthermore, Lucky Cement recorded net profit after tax of PKR 12.20 billion which is 10.9% lower as compared to last year. Similarly, the standalone EPS of the Company is PKR 37.72 / share which is 10.9% lower as compared to last year's reported EPS of PKR 42.34 / share.
The local cement sales volume of the Company during the fiscal year registered a growth of 14.8%(North 18.1% and South 11.0%) to reach 6.63 million tons compared to 5.77 million tons reported last year, whereas, local clinker sales volumes declined by 80.1% to 0.06 million tons during the current financial year as compared to 0.30 million tons last year, resulting in the overall local sales growth of 10.2% to reach 6.69 million tons during the current financial year compared to 6.07 million tons last year; while export sales volume increased by 4.6% to 1.13 million tons compared to 1.08 million tons reported last year.
Lucky Cement also shared progress on its brownfield expansion for cement production of 2.6 million tons per annum at its Pezu Plant and the greenfield investment project for producing 1.2 million tons of clinker at Samawah, Iraq. In addition, the Company also reported progress on its investment projects of 1 X 660 MW supercritical coal based power project at Port Qasim and automobile manufacturing plant under license from Kia Motors Corporation.
Lucky Cement remains committed towards value-creation for the society in which it operates. In this regard, the Company extended numerous scholarships to deserving students for various leading universities in Pakistan and abroad. Keeping in view the importance and impact of womenempowerment in Pakistan, the Company in collaboration with Zindagi Trust continued its support for two leading Government girls' schools in Karachi.
The company also continued to donate generously towards health-based initiatives by supporting various welfare organizations. Lucky Cement always takes its responsibility towards environment seriously and for continuous efforts to curtail carbon emissions in the atmosphere, Lucky Cement has a comprehensive air quality measurement program that enables it to identify the limits of pollutionparameters in the ambient air in and around its cement plants and all mentioned parameters are closely monitored and kept well below the respective limits as specified by the National Environment Quality Standards (NEQS).
Lucky Cement
6-A Muhammad Ali Housing Society,
A. Aziz Hashim Tabba Street,
Karachi-75350, Pakistan
UAN: (+92-21) 111-786-555
Email: info@lucky-cement.com
Tags: Lucky Cement
Category: General Business News
« UNISAME urges Banks to develop products for SMEs
Board of directors meeting of Attock Petroleum Limited »
Subscribe for Newsletters in English
Latest General Update
Punjab Health Minister Dr. Yasmin Rashid has said that the intensity of the second wave of coronavirus is gradually reducing in the province. Addressing a press conference in Lahore today (Monday), she said that more than 2.7 million coronavirus tests have so far been conducted in Punjab. She said that 20 new state-of-the art laboratories […]
Punjab Governor Chaudhry Muhammad Sarwar will lay the foundation stone of the first project of the Punjab Aab-e-Pak Authority at Chak Jhumra near Faisalabad on Monday. The project costing over 161million rupees will provide clean drinking water to more than 57,000 families of 16 villages at their doorsteps. Talking about the project, Governor Sarwar said […]
Snowboarding event kicks off in Malam Jabba on Monday
Pakistan International Snowboarding Cup has kicked off in the scenic Malam Jabba area of Swat. The six days' event has been organized by Khyber Pakhtunkhwa department of Tourism in collaboration with a private business enterprise. Participants from across the country will take part in the snowboarding cup. Food and music festivals have also been arranged […]
RUDA sets up to build new city: Firdous
Punjab Chief Minister's Special Assistant for Information, Dr. Firdous Ashiq Awan has said that Ravi Urban Development Authority has been set up to build a new city and this project will create ample new job opportunities. Addressing a news conference in Lahore this evening, she said that all modern facilities will be provided in the […]
Firdous lauds services of late Munnu Bhai for welfare of ailing humanity
Punjab Chief Minister's Special Assistant for Information, Dr. Firdous Ashiq Awan has lauded services of late senior Columnist Munnu Bhai for welfare of ailing humanity. Addressing a ceremony of Sundas Foundation at Sialkot today, she said that Munnu bhai served ailing humanity and spread happiness among people throughout his life. She appreciated the contributions from […]
Pakistan felicitates Nepalese climbers on first winter K2 ascent
Pakistan has felicitated the team of Nepalese mountaineers for making a history by reaching the world's second highest mountain peak, K2, in winter. Located on Pakistan's side of Karakoram range, K2 is one of the world's 14 tallest mountains. In a tweet, Foreign Office spokesperson Zahid Hafeez Chaudhri termed the expedition as one of the […]
Latest Medical Updates
Dr Sania Nishtar inaugurates Ehsaas Delivery Unit in Islamabad
Special Assistant to Prime minister on Social Protection Dr Sania Nishtar has inaugurated the Ehsaas Delivery Unit in Islamabad. Speaking on the occasion in Islamabad on Tuesday, she said the main objective of this unit is to oversee implementation and assess progress of the Ehsaas programme. She said EDU will also track implementation of the […]
Sarwar lauds services of minority communities in war against COVID-19
Punjab Governor Chaudhry Muhammad Sarwar has lauded the services of minority communities living in Pakistan in the war against COVID-19. Addressing a press conference in Nankana Sahib today (Tuesday), he said that we distributed rations to two million families in the province with the help of philanthropists. He said that the real heroes in this […]
January 5, 2021	|
Special Assistant to Punjab Chief Minister on Information Dr. Firdous Ashiq Awan says the provincial government provided tax relief worth 56 billion rupees to people during Covid-19, despite financial difficulties. In a tweet, she said 42 percent increase has been witnessed in tax recoveries during the last six months and Current Account is also surplus. […]
Dr. Yasmin hopes COVID-19 vaccine may reach Pakistan by February
Punjab Health Minister Dr Yasmin Rashid has expressed the hope that coronavirus vaccine may reach Pakistan by February this year. She was addressing oath taking ceremony of new office-bearers of Pakistan Medical Association organized at the Cosmopolitan Club in Lahore. Dr Yasmin Rashid said the government has reserved Rs 14 billion to fight the pandemic. […]
Federal capital records 136 fresh corona cases in last 24 hours: NCOC
COVID-19 cases continued to surge in the federal capital as 136 new Corona cases were reported from the city during the last 24 hours, National Command and Operation Center (NCOC) said on Sunday. According to an official of NCOC, as many as 137 cases were reported on Saturday while 123 cases were reported on Friday. […]
December 27, 2020	|
Monthly Archives Select Month January 2021 December 2020 November 2020 October 2020 September 2020 August 2020 July 2020 June 2020 May 2020 April 2020 March 2020 February 2020 January 2020 December 2019 November 2019 October 2019 September 2019 August 2019 July 2019 June 2019 May 2019 April 2019 March 2019 February 2019 January 2019 December 2018 November 2018 October 2018 September 2018 August 2018 July 2018 June 2018 May 2018 April 2018 March 2018 February 2018 January 2018 December 2017 November 2017 October 2017 September 2017 August 2017 July 2017 June 2017 May 2017 April 2017 March 2017 February 2017 January 2017 December 2016 November 2016 October 2016 September 2016 August 2016 July 2016 June 2016 May 2016 April 2016 March 2016 February 2016 January 2016 December 2015 November 2015 October 2015 September 2015 August 2015 July 2015 June 2015 May 2015 April 2015 March 2015 February 2015 January 2015 December 2014 November 2014 October 2014 September 2014 August 2014 July 2014 June 2014 May 2014 April 2014 March 2014 February 2014 January 2014 December 2013 November 2013 October 2013 September 2013 August 2013 July 2013 June 2013 May 2013 April 2013 March 2013 February 2013 January 2013 December 2012 November 2012 October 2012 September 2012 August 2012 July 2012 June 2012 May 2012 April 2012 March 2012 February 2012 January 2012 December 2011 November 2011 October 2011 September 2011 August 2011 July 2011 June 2011 May 2011 April 2011 March 2011 February 2011 January 2011 December 2010 November 2010 October 2010 September 2010 August 2010 July 2010 June 2010 May 2010 April 2010 March 2010 February 2010 January 2010 December 1012
The Asia Net Pakistan is the website that gathers the news from every sector and publish it online. The reason or the basic concept behind this website is to keep our readers aware of every happening that can affect their lives.
Register Users
Categories Select Category 2020 Nansha International Talents Conference Advertising Africa Arabic Asean Automotive Board Meetings Bombardier Bonds Brokerage Building & Construction Business Business News Chambers of Commerce Changhong Children & Youth Science Center of CAST China International Contractors Association Chongqing Liangjiang New Area Administrative Committee Chongqing Municipality Company News CoreCard CS Global Partners CS Global Partners Attachments area DiaSorin District & City Governments Dividend Distribution DP World Education Elections / Appointments Embassies English Entertainment Equity & Non Equity Investment Instruments Fashion Fertilizers Financial Results Financial Services Food & Beverage Fund News Future Contracts General General Business News Government Institutions Guidepoint Hengtong Group HIMSS Household Goods http://asianetnews.net/view-attachment?attach-id=378202 Industries Inter Services Public Relations Ivanti Legal & Politics Material Information Medical MENA Metals & Mining Ministries Misc. Miscellaneous MobileIron Multiutilities National Network Communications Official News Oil and Gas OncologyEducation Other Language Pakistan Personal Goods President Secretariat Press Information Department Press Release Prime Minister Secretariat Real Estate SDLG CONMART SECP Announcements SHAREit Shenyang Municipal People's Government Shenyang Municipality South Africa Sphera Sports Sports Extra Stock Exchange Announcements Support Services Taxation TCL Electronics Telecommunication Textile & Garments The 7th Humen International E-commerce Festival The Government of Dominica the HPTN The Information Office of Dongguan Municipality The Information Office of Shandong Provincial People's Government The International Finance Forum 2020 Annual Meeting The Ministry of Natural Resources & Shandong Provincial People's Government The Organizing Committee of Asian Beach Games The People's Government of Hainan Province The Philippine Board of Investments (BOI) The Publicity Department of Baise Municipality The Publicity Department of Longnan Municipality The Publicity Department of Longyan Municipality The Publicity Department of Nansha District Committee The Publicity Department of Shenhe District The Publicity Department of Shenyang Municipality The Publicity Department of Wanzhou District Tobacco Trade & Shipping Travel & Leisure UnionPay International United Nations Urdu World Bank Xinhua Sports [INVNT GROUP]™ اطلاعاتی ادارہ برائے حکومت ڈونگوان ایشین بیچ گیمز کی آرگنائزنگ کمیٹی شنیانگ میونسپل پیپل گورنمنٹ شینڈونگ صوبائی عوامی حکومت کا دفتر اطلاعات شینیانگ بلدیہ کا محکمہ تشہیر ضلع وانژوچونگچن بلدیہ کا محکمہ تشہیر لونگیانگ بلدیہ کا تشہیری ادارہ ن الاقوامی فنانس فورم 2020 کی سالانہ میٹنگ نانشا کی ضلعی کمیٹی کا اشتہاری ادارہ نیشنل نیننگ اکنامک اینڈ ٹیکنولوجیکل ڈویلپمنٹ ایریا مینجمنٹ کمیٹی وزارت برائے قدرتی وسائل اور شینڈونگ پیپلز گورنمنٹ چونگ چنگ لیانگ جیانگ نیو ایریا انتظامی کمیٹی
Copyright © 2010 - 2021 AsiaNet-Pakistan. All Rights Reserved.


text:
Home > Archive > Archive > Around Ireland Tidal takeoff in Derry
Around Ireland Tidal takeoff in Derry
By Ray O'Hanlon
Ships leave and arrive with the tide. But planes? In Derry, the tides that surge up and down the River Foyle might indeed have to be taken into account by aircraft using the city's airport, which is currently under expansion.
The expansion plan was given the all clear two years ago and since then the airport's runway has been lengthened in the direction of the river.
Now, however, the City Council has concluded that the runway should have been built in a direction away from the river. The problem is, £5.3 million already spent on the runway project now stands to be washed away.
The runway plan u-turn, according to a report in the Derry Journal, is due to the fact that updated studies show that aircraft using the runway that runs toward the Foyle might have to deal with extra high tides or even mud if a combination of certain conditions prevail.
The airport has been increasingly busy in recent years and is seen vital to Derry's continued economic growth.
The chairman of the council's airport Committee, Andrew Davidson, said he was concerned that the council had been operating in a King Canute atmosphere.
"It is vital that we get to the bottom of this and sort it out as a matter of urgency," he told the Journal.
Hack lack
Dublin is not the only place in Ireland where hailing a taxi can be difficult. A Blarney, Co. Kerry, man recently attempted to hitch a ride home in a Garda squad car but was promptly told that the police did not provide a taxi service. According to the Southern Star, the man was directed by the Garda in the car to a nearby taxi stand.
Some time later, the Garda car drove past the taxi stand and the man was observed "roaring and shouting" by the officers. Apparently he was not roaring for a taxi. The man was arrested and charged with abusive behavior and refusing to obey an instruction delivered by a Garda officer.
University of Waterford?
Waterford does not have a university of its own and that is troubling to some local politicians. Progressive Democrat Councilor Oliver Cleary believes that the southeast of Ireland, and Waterford in particular, is losing out on multinational investment because it does not have a full-fledged university.
"Galway, Limerick, Cork and Dublin all have universities and Waterford is the only city which does not have one," Cleary said.
However, other local officials are not so sure that a new university is needed and instead want to see the city's Institute of Technology developed to its full potential. The debate is yet to graduate to the voting stage.
Some like this cot
Dublin hotel owner Barry Canny is offering guests the chance to sleep with a Hollywood legend — at least in their dreams.
Canny, who runs the exclusive Browne's Brasserie and Townhouse on St. Stephen's Green in Dublin, recently paid £15,000 at an auction in London for a bed once used by Marilyn Monroe in her Hollywood office.
Canny is charging £350 a night for the privilege of a nap in the "Some Like it Hot" star's onetime cot.
"It's an enormously comfortable bed. The suite is booked out for over a year," Canny told the Irish Independent.
The bed has even attracted some living celebrities. Actor Jack Nicholson spent a night in the bed. However, Canny has resisted the temptation to hike the price because of other star snoozers. Every penny of the overnighting price is due to the bed's original owner.
Monastic tribute
Four of County Clare's most historic early monastic settlements are to get big boosts for their collection plates.
The minister with responsibility for heritage matters, Síle de Valera, has announced that £4 will be spent on preservation work and upgrading of Ennis Friary, Quin Abbey, Kilfenora Cathedral and Scattery Island.
According to a report in the Clare Champion, de Valera believes that the importance of the monastic sites in national terms had not been matched by spending in recent years. She was now determined to set that to right.

redpajama_set_name: RedPajamaCommonCrawl


Its June! Winter has arrived and look what we're playing in!
Heres a lovely couple from Singapore on the Mountain Experience the other day, a you can see the sun is still shining! We get the best of both worlds at The Cardrona!
Think the mountains are beautiful in summer? The Cardrona winter will blow your mind!
We've had some really cold autumn temperatures recently which have left us with a bit of snow on the tops to remind us that winter is on the way.. And YES we WILL be riding!
The horses are getting nice and hairy in preparation for the cold and we've got our winter feed in the barn.
Pull on your wooly socks and get ready to saddle up in the snow with The Cardrona!
4TH BIRTHDAY PONY PARTY WITH ANDY & GINGER BEAR!
Trinitys 4th birthday party was an "amazing day could not have asked for much more" says mum Danielle… Many first time cowboys & cowgirls, Andy & Ginger Bear were the perfect gentlemen as always!
Overnight pack saddle trips to the huts way up on the stunning Pisa range, DOC consents underway – watch this space!
HAPPY EASTER FROM THE CARDRONA!
Look what we've been up to……..
Quad-horse combos to the pub for a beer! Attracts LOADS of attention in the hotel car park- fun for all ages & abilities- so if you don't want to ride a horse theres no excuse now!
A group of Australian Nuffield Scholars experiencing The Cardrona's full package- horses, quads and mid-ride treats in the mountains!


text:
Q: Update ArrayList of Specific Instance This has been confusing me for a while, picture this
public class someObject {

private ArrayList<Player> listOfPlayers = new ArrayList<Player>();
private String name;

public void addPlayer(Player player){
    listOfPlayers.add(player);
}
}

Multiple instances of this object can be created during the game, what i don't understand is how do you update the ArrayList of a specific instance, for example
There are 10 instances currently of the someObject class Player 1, 2 and so on. Player 5 wants to add a player to his listOfPlayers, how would i handle getting the arraylist of his specific instance where he is currently in the listOfPlayers.
What i have been doing is
for (int i = 0; i < objects.size(); i ++){ //ArrayList<someObject>
        someObject temp = someObject.get(i);
        if (temp.listOfPlayers.contains(player)){
            return true;
        }
}

Which works but it seems so wrong using for loops in every getter and setter of the someObjects class, is there a better way than this ?

A: A better solution/alternative would be using a Map.
Map<someObject, List<Players>> objPlayersMap=  new HashMap.....

Later, you can check your condition just like
objPlaersMap.get(someObject).contains(...

A: If you want, u can modify your "Player" class and add an id. With setter and getter u can handle these with this id. Its an idea, if u wanna prefer it more.

A: The best way is to use a map with the name of the player as a key. In that way you can get the list easily without a loop
Map<String, List<Players>> players =  new HashMap<String,List<Players>>();
players.put("player1", new SomeObject());
players.put("player2", new SomeObject());
....

after that you do that:
List<Players> player1 = players.get("player1");

redpajama_set_name: RedPajamaStackExchange


Bajaj Auto is on the highway today. The stock price spurted up over 4% to Rs.2702; it remains firmly in the green though it has come off the intra day high and is now at Rs.2670 levels.
The company reported very good sales number for October; in fact it was the highest ever monthly sales growth of both three as well as two wheelers.
Company's cumulative sale (export and domestic ) stood at 506,699 units last month, registering a growth of 32%.
Exports showed a good 38% jump while domestic sales rose 29%.
Motorcycle sales registered a jump of 33% at 4.32 lakh units while three wheeler sales rose308% at 73,714 units.


text:
Science Discovers Why A Lack Of Sleep Makes You Cranky
© Etienne Oliveau/Getty Images
Trending News: Here's Why Missing Sleep Makes You Act Like A Jerk
Ian Lang
Because sleep is bae.
Chinese scientists used brain scans to find out why chronic lack of sleep makes us so emotionally distressed. It turns out it literally rewires the part of the brain connected to emotion, destroying important sections of white brain matter.
Long Story
Anyone who works a demanding job or has kids knows that multiple nights with little or no sleep can leave you feeling like a zombie — detached, anxious and irritable. But consider the plight of insomniacs, who struggle to function under the crushing weight of a clinical inability to sleep. Now, scientists from Guangdong No.2 Provincial People's Hospital in China have discovered why missing out on sleep makes us feel so terrible — it literally rewires the brain.
The scientists used brain scanning technology to examine the brains of 23 people severe insomnia, who each reported having trouble sleeping for at least a month at a time. When they compared them to the brain scans of 30 healthy people, they found significant and obvious differences in the so-called "white matter" of the brain. While the insomniacs presented white matter degradation all over, the biggest changes were in the right side of the brain, which regulates emotion.
The scans measured the strength of electrical pulses throughout the brain, and they found that the culprit was the corpus callosum, the part that connects the two halves of the brain. It wasn't functioning properly, leading to the weakened responses in the right half. Making things worse, they also found damage in the thalamus, which is (among other things) responsible for regulating sleep and consciousness — by being unable to sleep, your body inadvertently makes it even harder to find relief.
The researchers didn't offer up any solutions to the problem, but it may offer a modicum of comfort to people who struggle to function when they lose sleep. It's not an issue of temperament or willpower — your brain may literally be broken.
Own The Conversation
Ask The Big Question
Is there a way to restore proper brain function?
Disrupt Your Feed
I'm useless if I stay up a little too late, so I can't imagine having insomnia.
Drop This Fact
Insomnia affects more than 3 million people in the U.S. every year.
Is Burnout Having a Negative Impact on Your Fitness Progress?Read More
Lack of Sleep Can Mess With Your Sex Life
Trending News: Americans Would Trade Raises for a Decent Night's Sleep
It Turns Out Daytime Snoozing Is Good for You
Trending News: Listen To This Before Bed To Improve Your Sleep (And Memory)
Trending News: Here's How Much Extra Sleep You'd Get By Going To Bed On Time


Smart Trade Technologies, a provider to the financial markets through its Liquidity Management System (LMS), has announced that the Smart Trade Trading Platform (STTP) meets MiFID (Markets in Financial Instruments Directive) requirements for best execution in cross-asset trading.
Smart Trade Technologies, a provider to the financial markets through its Liquidity Management System (LMS), has announced that the Smart Trade Trading Platform (STTP) meets MiFID (Markets in Financial Instruments Directive) requirements for best execution in cross-asset trading. By implementing the STTP, banks can smart routing clients' orders to different venues according to rules that enable the bank to define and deliver its best execution policy.
"We can simulate savings generated by market participants – buy-side, sell-side and broker – by running the smartTrade Trading Platform simulation locally in live conditions or through an ASP," says Harry Gozlan, CEO, Smart Trade. "By applying various conditions to incoming orders and hedging scenarios, banks gain an accurate picture of the risk-reward ratio of internalisation or crossing that can be achieved, measuring economic savings generated through internalising, including keeping the bid/ask internally and eliminating execution and clearing fees."
He continues, "This simulation can also profile categories of order flows – retail, institutional or professional – pinpointing what type of flow can match against other types. Another benefit gained through this fast-to-implement STTP simulation system is that it's very similar to the actual STTP production system, facilitating a fast, efficient move into production."
According to TABB Group in its December 2006 research note, Liquidity Management: Pushing Automated Trading beyond Agency Brokerage, liquidity management, the methodology surrounding how firms automate their trading desks, is set to become the next phase in the development of electronic trading as customer-flow and order-flow tools become increasingly widely used. Liquidity management includes rules around valuation and pricing of liquidity, handling of customer order-flow, matching of internal liquidity and automated rules and methodologies around provision of capital for internal market-making and proprietary trading.
A liquid market, says TABB Group, regardless of asset class, relies on three components to be successful: tightness in price (bid/ask spread), depth and resiliency. Although liquidity management is not a new concept to banks, until recently most banks have not fully automated this process. Algorithmic trading and market fragmentation have now changed this situation due to the sheer volumes and speed of the transactions. Reg NMS in the US and MIFID in Europe propel the need further, especially MIFID, whose directive is cross asset for best execution.
Regardless of size or location, banks face serious challenges in building the automation for liquidity management, primarily around multi-asset trading, because an internal market needs to be created in order to know your depth of the market. "Products cannot be easily combined to create this internal market as each one has their own unique characteristics and the model is not scalable, especially for synthetic products," explains Gozlan.
Smart Trade addressed this issue by building its STTP platform as an asset-agnostic platform. From a technical perspective, says David Vincent, CTO at Smart Trade, "the STTP has been developed using object-oriented methodology and built on a component approach. Each component can be extracted from the rest and replaced by another one via the Spring Framework."
The platform, he explains, has two engines: the execution engine, which enables multi-asset trading via its mathematical representation of the environment, and the routing engine whose function it is to decide the final destination or destinations of the orders, quotes and indicative prices.


text:
Flowering began on 25 May and took place evenly and quickly across the three varieties; it was complete by the beginning of June, some eight days earlier than the average for the last 30 years. The spring and summer of 1999 were relatively hot, with temperatures consistently higher than the seasonal average.
Mid-veraison was noted three to six days earlier than average, on 4 August for Merlot and 7 August for Cabernet Sauvignon. Temperature records at the end of August showed an increase of more than 1° C in comparison with the aggregate average for the last 30 years. Beneficial for the fruit, these conditions also encouraged constant growth of the vines, which were trimmed eight times rather than the customary five or six.
July and August were wetter than normal: 169 mm of rain fell during the two months, compared with an average of 107 mm. The rainfall, mainly due to storms on 27 July and 2, 6 and 18 August, did not have the same effect as steady rain, since the water ran off relatively quickly. Fine weather returned on 20 August, enabling the crop to ripen in good conditions. The grapes were of good quality, with high degrees of potential alcohol and a perfect balance between sugar and acidity. Anthocyan levels were high or very high, especially in some Merlot parcels, holding out the promise of a great vintage.
Crop thinning and leaf removal were carried out in the entire vineyard. Fine, very sunny days alternated with dull weather during the harvest period, making it possible to harvest a healthy crop of well-coloured and nicely tannic grapes.
The wine has a fine, rather vivid red colour with a ruby tint and a very open, complex nose on which fresh fruit (cherry, blackcurrant, redcurrant) and floral aromas (violet, lilac) mingle with denser notes of vanilla, tobacco leaf and cedarwood. The body builds rapidly from a substantial attack on round, harmonious tannins which unfold over elegant, well-integrated oak, developing a pleasant suggestion of liquorice, menthol and spice on the palate. The long, elegant finish displays highly refined tannins and lingering red fruit flavours.


text:
\section{Introduction}

String theory opens new possibilities in the study of fundamental problems
in black holes (BHs) physics, like string origin of BH entropy (for
a review, see \cite{P}). Moreover, string duality dictates various
relations between higher dimensional BHs and lower dimensional BHs,
as for example between 5D and 4D BHs (where D-branes methods have been
applied to calculate the entropy \cite{HS}) from one side and
3D and 2D BHs \cite{Hy,T} from another side. This fact indicates
that applying results of (quantum) calculations in 2D
one can learn about properties of higher dimensional BHs.

Another indication that such approach may be very useful comes from the
study of back-reaction of quantized matter to 4D BH when spherical
reduction is applied. Here, the resulting effective action turns out to
be the effective action of 2D dilatonic gravity with dilaton coupled matter.

Recently, some interesting effects have been discovered in this
direction. Working in large $N$ and $s$-wave approximation (quantum
minimal scalars) Bousso and Hawking \cite{BH2} demonstrated that
degenerate Schwarzchild-de Sitter (SdS) (Nariai) BH may not only
evaporate but also anti-evaporate. The effect of anti-evaporation
has been confirmed in refs.\cite{NO2} for conformal matter where
large $N$ approach and 4D anomaly induced effective action have been
used. It has also been shown in refs.\cite{NO2} that some of
anti-evaporating SdS BHs may be initially stable when Hartle-Hawking
no boundary condition \cite{HH} is used.

Further studying SdS BH perturbations due to quantum minimal scalars
Bousso\cite{B} found new effect: proliferation of de Sitter space,
which means disintegration of this space into an infinite number of copies
of itself (the instability of de Sitter space has been demonstrated
in ref.\cite{PGP}).

It could be really interesting to understand if all these developments
may be realized in other types of BHs. The natural candidate to think
about is charged BH. Working in large $N$ approach for dilaton
coupled quantum scalars we study here the quantum properties of 2D
charged BHs. The corresponding classical solutions have been found in
\cite{MNY} (see also ref.\cite{GP}). They may be considered as some
compactifications of Type II string solutions. Moreover, very naturally
they may be considered as 2D analog of Reissner-Nordstr\" om (RN) charged
4D BH. Note that some properties of 2D charged BHs \cite{MNY} have been
discussed in refs.\cite{NP,EO} (for the calculation of BH entropy in this
case, see\cite{GP,NP}).

On the other hand, Kaluza-Klein reduction of 3D gravity with minimal
scalars leads to 2D dilaton-Maxwell gravity with dilaton coupled scalars
and the rotating BH solution found by Ba\~nados, Teitelboim and Zanelli
(BTZ) can be also described by 2D charged dilatonic BH.
BTZ black hole (BH)\cite{BTZ} attracts a lot of attention due to various
reasons. In particulary, it is related via T-duality with a class
of asymptotically flat black strings \cite{H} and via U-duality
it is related \cite{Hy} with 4D and 5D stringy BHs \cite{HS}
which are asymptotically flat ones. Hence, microscopically computing
the entropy of BTZ BH \cite{SK} may be applied via duality relations
for the computing of entropy for higher dimensional BHs.
This fact is quite remarkable as above types of BHs look completely
different from topological, dimensional or space-time points of view.

BTZ BH being locally ${\rm AdS}^3$ without curvature singularity
may be considered as a prototype for general CFT/AdS correspondence
\cite{MW}. Indeed, 3D gravity has no local dynamics but BH horizon induces
an effective boundary (actually, 2D WZWN model). Hence,quantum studies
around BTZ BH may help in better understanding of above correspondence.

The paper is organised as follows. In the next section we consider
2D dilaton-Maxwell gravity which represents toy model for
4D or 5D Einstein-Maxwell theory and review its 2D charged BH
solutions discussed by Mc-Guigan, Nappi and Yost. The one-loop
effective action for dilaton coupled 2D scalar which is obtained by
spherical reduction from 4D or 5D minimal scalar is found. We
work in large $N$ approximation. Quantum dynamics of 2D charged BHs is
discussed in third section. In particulary, quantum corrected 2D
BH is constructed.
The analytical and numerical study of its perturbations is presented.
It shows the existence of stable (evaporating) and unstable
(anti-evaporating) modes. Calculation of quantum corrections to mass,
charge, temperature and BH entropy is done in section 4. Due to duality
with 5D stringy BH last result gives also quantum corrected BH
entropy for 5D BH. After that we study quantum dynamics of BTZ BH
due to 3D minimal scalars. We work in large $N$ and $s$-wave approach
where minimal scalars are described as 2D dilaton coupled scalars. The
quantum corrected version of BTZ BH is found. It is shown that it can
also evaporate or anti-evaporate. Higher modes
perturbations in the quantum spectrum for these two kinds of charged
BHs are also briefly discussed.

\section{One-loop effective action}

We start from the action which has been considered by McGuigan, Nappi
and Yost in ref.\cite{MNY}. As it has been shown in \cite{MNY}, this
action follows from compactification of heterotic string theory:
\begin{equation}
\label{cMNY}
S={1 \over 16\pi G}\int d^2x \sqrt{-g}{\rm e}^{-2\phi}
\left( R + 4 (\nabla \phi)^2 + 4\lambda^2 - 4 F_{\mu\nu}^2 \right)\ .
\end{equation}
where $\lambda^2$ is cosmological constant,
$F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$, $\phi$ is dilaton.
Note that above action has the form typical for 4D or 5D Einstein-Maxwell
theory spherically reduced to two dimensions. Hence, it can be considered
as toy model to describe 4D or 5D BH with spherical reduction.

It is remarkable that action (\ref{cMNY}) has the classical solutions which
correspond to the 2D charged black hole (2D analogue of
Reissner-Nordstr\" om black hole) with multiple horizon.
In the explicit form its metric and dilaton look like \cite{MNY,GP}
\begin{eqnarray}
\label{MNYmetric}
ds^2&=&-\left( 1-2m {\rm e}^{-2\lambda x} + q^2 {\rm e}^{-4\lambda x}\right)dt^2 \nonumber \\
&&+\left( 1-2m {\rm e}^{-2\lambda x} + q^2 {\rm e}^{-4\lambda x}\right)^{-1}dx^2 \nonumber \\
{\rm e}^{-2(\phi-\phi_0)}&=&{\rm e}^{2\lambda x}\ .
\end{eqnarray}
Here $q$ and $m$ are parameters related to the
charge and the mass of BH, respectively.
The extremal solution is given by putting $q^2=m^2$.
If we define new coordinates $r$ and $\tau$ by
\begin{eqnarray}
\label{r}
{\rm e}^{-2\lambda x}&=&{m+ \epsilon \tanh 2\lambda r \over q^2} \nonumber \\
t&=&{q \over \epsilon}\tau \nonumber \\
\epsilon^2&\equiv&m^2 - q^2\ ,
\end{eqnarray}
$r\rightarrow +\infty$ corresponds to outer horizon and
$r\rightarrow -\infty$ corresponds to inner horizon.
Taking the limit $\epsilon\rightarrow 0$, we obtain
\begin{eqnarray}
\label{exmetric}
ds^2&=&{1 \over \cosh^2 \lambda r}\left(d\tau^2 - dr^2\right) \nonumber \\
{\rm e}^{-2(\phi-\phi_0)}&=&{1 \over q}\ .
\end{eqnarray}
Note that the dilaton field becomes a constant in the limit.

We will discuss now the quantum corrections induced by $N$ free conformally
invariant dilaton coupled scalars $f_i$ (no background scalars $f_i$):
\begin{equation}
\label{sS}
S^f=-{1 \over 2}\int d^2x \sqrt{-g}{\rm e}^{-2\phi}
 \sum_{i=1}^N(\nabla f_i)^2\ .
\end{equation}
Note that the above action (\ref{sS}) appears as a result of
spherical reduction from 4D or 5D minimal scalar.

We present now the two-dimensional metric as the following:
\begin{equation}
\label{2dmetric}
ds^2={\rm e}^{2\sigma}\tilde g_{\mu\nu}d x^\mu d x^\nu
\end{equation}
where in conformal gauge $\tilde g_{\mu\nu}$ is the flat metric.
Then the total quantum action coming from quantum scalars
is given as (we work in large $N$ approximation what
justifies the neglecting of proper quantum gravitational corrections):
\begin{equation}
\label{Gamma}
\Gamma = W + \Gamma[1, \tilde g_{\mu\nu}]
\end{equation}
where $W$ is conformal anomaly induced effective action \cite{BH,NO}
and $\Gamma[1, \tilde g_{\mu\nu}]$ is one-loop effective action for scalars
(\ref{sS}) calculated on the metric (\ref{2dmetric}) with $\sigma=0$ (the
general covariance should be restored after all). Note that general
structure of conformal anomaly has been discussed in ref.\cite{DS} while
quantum conformal field theory in general dimensions (bigger than two) has
been discussed in ref.\cite{EHO} (for earlier work,see \cite{JLC}). It
could be interesting to generalize above works for the case when external
dilaton presents.

Following \cite{NO} (due to corresponding result for conformal anomaly of
(\ref{sS})\cite{NO,BH,A}), we get
\begin{equation}
\label{qc}
W=-{1 \over 2}\int d^2x \sqrt{-g} \left[
{N \over 48\pi}R{1 \over \Delta}R  -{N \over 4\pi} \nabla^\lambda \phi
\nabla_\lambda \phi {1 \over \Delta}R +{N \over 4\pi}\phi R \right]\ .
\end{equation}
We are left with the calculation of the conformally invariant part of the
effective action $\Gamma[1, \tilde g_{\mu\nu}]$. This term has to be
calculated on the flat space, after that the general covariance should be
restored. It is impossible to do such calculation in closed form. One has
to apply some kind of expansion. Using Schwinger-De Witt type expansion, we
get this effective action as local curvature expansion.

Using the results of ref.\cite{NO}, one can obtain:
\begin{equation}
\label{Gamma1}
\Gamma[1, \tilde g_{\mu\nu}]={N \over 24\pi}\int d^2x
\sqrt{-\tilde g}(-6\tilde \nabla_\mu \phi
\tilde \nabla^\mu \phi ) \ln \mu^2 + \cdots
\end{equation}
here $\mu$ is dimensional parameter.\footnote{Note that more exactly, one
should write $\ln {L^2 \over \mu^2}$ where $L^2$ is covariant cut-off
parameter.} Now, one has to generalize $\Gamma[1, \tilde g_{\mu\nu}]$
in order to present it in general covariant form:
\begin{equation}
\label{Gamma1b}
\Gamma[1, g_{\mu\nu}]=-{N \over 4\pi}\int d^2x
\sqrt{- g}\nabla_\mu \phi \nabla^\mu \phi \ln \mu^2 + \cdots \ .
\end{equation}
This action is conformally invariant as it should be in accordance with
Eq.(\ref{Gamma}) where $W$ gives scale-dependent part of total one-loop
effective action. Note that terms which are not written explicitly in
Eq.(\ref{Gamma1b}) are conformally invariant, higher derivatives terms on
$\phi$. Moreover, as it is easy to see for minimal scalars $\phi=0$,
$\Gamma[1, \tilde g_{\mu\nu}]=0$. That corresponds to general result that
in two dimensions, effective action (\ref{Gamma}) is defined completely by
only anomaly induced action $W$ (which is not the case in the presence of
dilaton).

It is also interesting to note that if one uses some form of non-local
expansion in the calculation of $\Gamma[1, \tilde g_{\mu\nu}]$, one would
get the leading terms of this expansion similar to some terms in
Eq.(\ref{qc}). In other words, such procedure would just decrease some of
coefficientes in Eq.(\ref{qc}) (apart from appearence of few local terms.)

\section{Quantum dynamics of 2D charged BHs}

Let us work in the conformal gauge
\begin{equation}
\label{cg}
g_{\pm\mp}=-{1 \over 2}{\rm e}^{2\rho}\ ,\ \ g_{\pm\pm}=0
\end{equation}
The equations of motion with account of quantum corrections are given by
variations of sum $S$(\ref{cMNY})$+W$(\ref{qc})$+\Gamma$(\ref{Gamma1b})
over $g^{\pm\pm}$, $\rho$, $\phi$ and $A_\mu$:
\begin{eqnarray}
\label{eqnpp}
0&=&{1 \over 8G}{\rm e}^{-2\phi}\left(4\partial_\pm \rho \partial_\pm\phi
- 2 \partial_\pm^2\phi\right) +{N \over 12}\left( \partial_\pm^2 \rho
- \partial_\pm\rho \partial_\pm\rho \right) \nonumber \\
&& +{N \over 2} \left\{\left( \partial_\pm \phi
\partial_\pm\phi \right)\rho+{1 \over 2}{\partial_\pm \over \partial_\mp}
\left( \partial_\pm\phi \partial_\mp\phi \right)\right\} \nonumber \\
&& +{N \over 4}\left\{ 2 \partial_\pm \rho \partial_\pm \phi
-\partial_\pm^2 \phi \right\} + t^\pm(x^\pm) \nonumber \\
&& +{N \over 2}\left(\partial_\pm \phi\partial_\pm\phi \right)\ln \mu^2 \\
\label{req}
0&=&{1 \over 8G}{\rm e}^{-2\phi}\left(2\partial_+
\partial_- \phi -4 \partial_+\phi\partial_-\phi
- \lambda^2 {\rm e}^{2\rho} + {\rm e}^{-2\rho}(F_{+-})^2\right) \nonumber \\
&& -{N \over 12}\partial_+\partial_- \rho
-{N \over 4}\partial_+ \phi \partial_- \phi
+{N \over 4}\partial_+\partial_-\phi \\
\label{eqtp}
0&=& {1 \over 8G}{\rm e}^{-2\phi}\left(-4\partial_+
\partial_- \phi +4 \partial_+\phi\partial_-\phi +2\partial_+\partial_- \rho
+ \lambda^2 {\rm e}^{2\rho}+ {\rm e}^{-2\rho}(F_{+-})^2\right)  \nonumber \\
&& +2N\left\{-{1 \over 8}\partial_+(\rho \partial_-\phi)
-{1 \over 8}\partial_-(\rho \partial_+\phi)
-{1 \over 8}\partial_+\partial_-\rho \right\} \nonumber \\
&& -{N \over 2} \partial_+ \partial_- \phi \ln \mu^2\\
\label{eqA}
0&=&\partial_\pm({\rm e}^{-2\phi-2\rho}F_{+-})\ .
\end{eqnarray}
Eq.(\ref{eqA}) can be integrated to give
\begin{equation}
\label{eqA2}
{\rm e}^{-2\phi-2\rho}F_{+-}=B\ (\mbox{constant})\ .
\end{equation}
We first consider the static case and replace $\partial_\pm$ by $\pm {1
\over 2}\partial_r$. Then, using (\ref{eqA2}), Eqs.(\ref{eqnpp}),
(\ref{req}) and (\ref{eqtp}) are rewritten as follows:
\begin{eqnarray}
\label{eqnpp2}
0&=&{1 \over 8G}{\rm e}^{-2\phi}\left(\partial_r \rho \partial_r \phi
-{1 \over 2} \partial_r^2\phi\right) +{N \over 48}\left( \partial_r^2 \rho
- \partial_r\rho \partial_r\rho \right) \nonumber \\
&& +{N \over 8} \left( \rho + {1 \over 2}\right)
\partial_r \phi \partial_r\phi
+{N \over 16}\left\{ 2 \partial_r \rho \partial_r \phi
-\partial_r^2 \phi \right\} + t_0 \nonumber \\
&& +{N \over 8} \partial_r \phi \partial_r \phi \ln \mu^2\\
\label{req2}
0&=&{1 \over 8G}{\rm e}^{-2\phi}\left(-{1 \over 2}\partial_r^2
\phi + \partial_r\phi\partial_r\phi
- \lambda^2 {\rm e}^{2\rho} + B^2{\rm e}^{2\rho+4\phi}\right) \nonumber \\
&& +{N \over 48}\partial_r^2 \rho
+{N \over 16}\partial_r \phi \partial_r \phi
-{N \over 16}\partial_r^2\phi \\
\label{eqtp2}
0&=& {1 \over 8G}{\rm e}^{-2\phi}\left(\partial_r^2\phi
- \partial_r\phi\partial_r\phi -{1 \over 2}\partial_r^2 \rho
+ \lambda^2 {\rm e}^{2\rho}+ B^2{\rm e}^{2\rho;4\phi}\right)  \nonumber \\
&& +{N \over 8}\partial_r(\rho \partial_r\phi)
+{N \over 16}\partial_r^2\rho + {N \over 8}\partial_r^2 \phi \ln \mu^2 \ .
\end{eqnarray}
Since the classical extremal solution can be characterized by constant
dilaton field $\phi$, we assume $\phi$ is constant even when we include
the quantum correction:
\begin{equation}
\label{constphi}
\phi=\phi_0\ (\mbox{constant})\ .
\end{equation}
Then (\ref{req2}) and (\ref{eqtp}) have the following forms:
\begin{eqnarray}
\label{req3}
0&=&{\rm e}^{-2\phi_0}\left(-\lambda^2 + B^2{\rm e}^{4\phi_0}\right)
{\rm e}^{2\rho}+{GN \over 6}\partial_r^2\rho \\
\label{eqtp3}
0&=&{\rm e}^{-2\phi_0}\left(\lambda^2 + B^2{\rm e}^{4\phi_0}\right)
{\rm e}^{2\rho}+\left(-{1 \over 2}{\rm e}^{-2\phi_0}
+{GN \over 2}\right)\partial_r^2\rho \ .
\end{eqnarray}
The condition that Eqs.(\ref{req3}) and (\ref{eqtp3}) are compatible with
each other gives
\begin{equation}
\label{A}
B^2={\lambda^2{\rm e}^{-4\phi_0}\left({\rm e}^{-2\phi_0}-{4GN \over 3}
\right) \over {\rm e}^{-2\phi_0}-{2GN \over 3} }
\end{equation}
and we find the scalar curvature $R$ is constant
\begin{equation}
\label{constR}
R=-2{\rm e}^{2\rho}\partial_r^2\rho=R_0
\equiv -{8\lambda^2 {\rm e}^{-2\phi_0} \over
{\rm e}^{-2\phi_0}-{2GN \over 3} }\ .
\end{equation}
We see that quantum corrections decrease the value of curvature if compare
with the classical case. (\ref{constR}) can be integrated to give
\begin{equation}
\label{rho0}
{\rm e}^{2\rho}={\rm e}^{2\rho_0}\equiv {2C \over R_0}
\cdot {1 \over \cosh^2 \left(r\sqrt{C} \right)}\ .
\end{equation}
Here $C$ is a constant of the integration. Finally (\ref{eqnpp2}) gives
\begin{equation}
\label{t_0}
t_0={GN \over 6}C\ .
\end{equation}

We now investigate the (in)stability of the above obtained extremal
solution by the perturbation
\begin{equation}
\label{pert}
\rho=\rho_0 + \epsilon R\ ,\ \ \ \phi=\phi_0 +\epsilon S\ .
\end{equation}
Here $\epsilon$ is an infinitesimally small constant. By following Bousso
and Hawking's argument \cite{BH2}, we neglect the second term in
(\ref{qc}). (Note that we keep similar term with $ln \mu^2$ as it could be
large). Then (\ref{req}) and (\ref{eqtp}) (and (\ref{eqA2})) give
the following equations:
\begin{eqnarray}
\label{reqpert}
0&=&{1 \over 8G}{\rm e}^{-2\phi_0}\left\{2\partial_+\partial_-S
- 2\lambda^2{\rm e}^{2\rho_0} (R-S) + 2B^2 {\rm e}^{2\rho_0+4\phi_0}(R+S)\right\} \nonumber \\
&& -{N \over 12}\partial_+\partial_- R+ {N \over 4}\partial_+\partial_-S \\
\label{eqtppert}
0&=&{1 \over 8G}{\rm e}^{-2\phi_0}\left\{-4\partial_+\partial_-S
+ 2\partial_+\partial_-R -4\partial_+\partial_-\rho_0 S
+ 2\lambda^2 {\rm e}^{2\rho_0} (R-S)\right.\nonumber \\
&& \left. + 2B^2 {\rm e}^{2\rho_0+4\phi_0}
(R+S)\right\} - {N \over 4}\partial_+\partial_- R
-{N \over 2}\ln \mu^2 \partial_+\partial_- S \ .
\end{eqnarray}
We are going to study (in)stability of black holes in the same way as in
four dimensions \cite{NO2}. As in the previous works, we assume $R$ and
$S$ have the following form
\begin{eqnarray}
\label{RS}
R(t,r)&=&P\cosh t\alpha\sqrt{C}\cosh^\alpha r\sqrt{C} \nonumber \\
S(t,r)&=&Q\cosh t\alpha\sqrt{C}\cosh^\alpha r\sqrt{C} \ .
\end{eqnarray}
Then we obtain the following equations
\begin{eqnarray}
\label{eigenRS}
\cosh^2\left(r\sqrt{C}\right)\partial_+\partial_- R &=& A R \nonumber \\
\cosh^2\left(r\sqrt{C}\right)\partial_+\partial_- S &=& A S \nonumber \\
A&\equiv& {\alpha (\alpha - 1) C \over 4}\ .
\end{eqnarray}
Note that there is one to one correspondence between $A$ and $\alpha$ if
we restrict $A>0$ and $\alpha<0$. Then (\ref{reqpert}) and
(\ref{eqtppert}) become algebraic equations:
\begin{eqnarray}
\label{alg1}
0&=&\left\{{\rm e}^{-2\phi_0}\left({R_0 \over C}A + 2\lambda^2
+ 2 B^2{\rm e}^{4\phi_0}\right) + {GNR_0 \over C}A\right\} Q \nonumber \\
&& + \left\{{\rm e}^{-2\phi_0}\left(- 2\lambda^2
+ 2 B^2{\rm e}^{4\phi_0}\right) - {GNR_0 \over 3C}A\right\} P \\
\label{alg2}
0&=&\left\{{\rm e}^{-2\phi_0}\left(-{2R_0 \over C}A -{R_0 \over 2}
- 2\lambda^2 + 2 B^2{\rm e}^{4\phi_0}\right)
-2GN (\ln\mu^2) {R_0 \over C}A\right\} Q \nonumber \\
&& + \left\{{\rm e}^{-2\phi_0}\left({R_0 \over C}A + 2\lambda^2
+ 2 B^2{\rm e}^{4\phi_0}\right) - {GR_0 \over C}A\right\}P\ .
\end{eqnarray}
In order that Eqs.(\ref{alg1}) and (\ref{alg2}) have a non-trivial solution
for $Q$ and $P$, we find
\begin{eqnarray}
\label{det}
F(A)&\equiv&\left\{{\rm e}^{-2\phi_0}\left({R_0 \over C}A + 2\lambda^2
+ 2 B^2{\rm e}^{4\phi_0}\right) + {GR_0 \over C}A\right\} \nonumber \\
&& \times \left\{{\rm e}^{-2\phi_0}\left({R_0 \over C}A + 2\lambda^2
+ 2 B^2{\rm e}^{4\phi_0}\right) - {GNR_0 \over C}A\right\} \nonumber \\
&& - \left\{{\rm e}^{-2\phi_0}\left(- 2\lambda^2
+ 2 B^2{\rm e}^{4\phi_0}\right) - {GNR_0 \over 3C}A\right\} \nonumber \\
&& \times \left\{{\rm e}^{-2\phi_0}\left(-{2R_0 \over C}A -{R_0 \over 2}
- 2\lambda^2 + 2 B^2{\rm e}^{4\phi_0}\right) \right. \nonumber \\
&& \left. -2GN (\ln\mu^2) {R_0 \over C}A\right\}=0\ .
\end{eqnarray}
Eq. (\ref{det}) should be solved with respect to $A$.

When we compare the model with the 4D one, ${\rm e}^{-\phi}$ is identified with
the radius coordinate. Since the (apparent) horizon is a null surface,
the horizon is given by the condition
\begin{equation}
\label{horizon}
\nabla\phi\cdot\nabla\phi=0\ .
\end{equation}
Substituting (\ref{RS}) into (\ref{horizon}), we find the horizon is given
by
\begin{equation}
\label{horizonb}
r=\pm\alpha t\ .
\end{equation}
Therefore on the horizon, we obtain
\begin{equation}
\label{Shrzn}
S(t,r(t))=Q\cosh^{1+\alpha} t\alpha\sqrt{C} \ .
\end{equation}
This tells that the system is unstable if there is a solution
$0>\alpha >-1$, i.e., $0<A<{C \over 2}$. On the other hand, the
perturbation becomes stable if there is a solution where $\alpha<-1$,
i.e., $A>{C \over 2}$.

The radius of the horizon $r_h$ is given by
\begin{equation}
\label{hrds}
r_h={\rm e}^{-\phi} ={\rm e}^{-\left(\phi_0 + \epsilon S(t,r(t))\right)}\ .
\end{equation}
Let the initial perturbation is negative $Q>0$. Then the radius shrinks
monotonically, i.e., the black hole evaporates in case of $0>\alpha>-1$.
On the other hand, the radius increases in time and approaches to the
extremal limit asymptotically
\begin{equation}
\label{Shrzn2}
S(t,r(t))\rightarrow Q{\rm e}^{(1+\alpha) t |\alpha|\sqrt{C}} \ .
\end{equation}
in case of $\alpha<-1$. The latter case corresponds to the anti-evaporation
of Nariai black hole observed by Bousso and Hawking \cite{BH2}.

Eq.(\ref{det}) can be solved with respect to ${A \over C}$:
\begin{eqnarray}
\label{Asol}
{A \over C}&=&{1 \over 2}\left(-2304 + 192 g + 36 g^2 + 24 g^2 \ln\mu^2
\right)^{-1} \nonumber \\
&& \times \left\{ -2304 + 144 g - g^2 - 6g^2 m \pm \left( 313344 g
+ 63744 g^2 \right.\right. \nonumber \\
&& -8640 g^3 + g^4 + 27648 g \ln\mu^2 + 53568 g^2 \ln\mu^2 \nonumber \\
&& \left.\left.
- 5556 g^3  \ln\mu^2 + 36 g^2 ( \ln\mu^2)^2 \right)^{1 \over 2}\right\} \\
&\sim&{1 \over 2} \pm {1 \over 8} g^{1 \over 2} + {\cal O}(g)
\nonumber \\
g&\equiv&8GN{\rm e}^{2\phi_0} \nonumber
\end{eqnarray}
In the classical limit $g=0$, $A={C \over 4}$,
what tells that there does not occur any kind of the radiation
in the solution.  Near the classical limit $g\sim 0$, there are solutions
corresponding to both of stable and instable ones.
It might be surprizing that there is an instable mode since the
extremal solution is usually believed to be stable.

The global behavior of $A$ as a function of $g\equiv 8GN{\rm e}^{2\phi_0}$ and
$\ln\mu^2$ is given in Figures. In Fig.1, the vertical line corresponds to
${A \over C}$ and the horizontal one to $g$ when $\ln\mu^2=1$. There is a
singularity near $g\sim 5$. In the wide range from $g=0$ to near the
singularity, there coexist the stable and instable modes. Near the
singlularity, both of the modes become stable ones. Beyond the singularity,
${A \over C}$ becomes negative and there is no consistent solution. In
Fig.2a and 2b, we show the global behavior of two modes as a function of
$g\equiv 8GN{\rm e}^{2\phi_0}$ and $\ln\mu^2$. Fig.2a corresponds to the
instable mode and 2b to the stable one. In Fig.2a, the singularity
appeared in Fig.1 disappears near $g=3$.

\section{ Quantum corrections to BH parameters}

We investigate now the quantum corrections to mass, charge, etc. by
considering the limit of $\phi\rightarrow -\infty$. In the large $N$
limit, the leading order is proportional to $N$. Since the leading order
is the contribution of the one-loop correction it is also proportional to
the effective gravitational constant ${\rm e}^{2\phi_0}$. Therefore the quantum
correction under consideration appears in the form of $N{\rm e}^{2\phi_0}$.
Since ${\rm e}^{-\phi_0}$ is the scale of the black hole radius in the
corresponding 4 dimensional model, the above expansion also corresponds to
the large radiusexpansion. Therefore we can assume that the quantum
corrections for $\rho$ and $\phi$ appear in the following form
\begin{equation}
\label{pert2}
\rho=\rho_{cl} + 8GN{\rm e}^{2\phi_0}\rho_1\ ,\ \ \
\phi=\phi_{cl} + 8GN{\rm e}^{2\phi_0}\phi_1
\end{equation}
neglecting the second and higher power of $N{\rm e}^{2\phi_0}$. Here
$\rho_{cl}$ and $\phi_{cl}$ are the classical parts of $\rho$ and $\phi$,
respectively, which are given by (see (\ref{MNYmetric}))
\begin{eqnarray}
\label{clmetric}
{\rm e}^{2\rho_{cl}}&=& 1-2m {\rm e}^{-2\lambda x}
+ q^2 {\rm e}^{-4\lambda x} \nonumber \\
{\rm e}^{-2(\phi_{cl}-\phi_0)}&=&{\rm e}^{2\lambda x}\ .
\end{eqnarray}
Since the limit of $\phi\rightarrow -\infty$ means the large radius of
black hole, we expand $\rho_{cl}$ and $\phi_{cl}$ as power series of
${\rm e}^{-2\lambda x}$. Using the Eqs.(\ref{eqnpp2}), (\ref{req2}) and
(\ref{eqtp2}) and the boundary conditions
\begin{eqnarray}
\label{bc}
\rho_1,\ \phi_1\rightarrow 0\ \ &&\mbox{when}\ \ x\rightarrow -\infty \\
(\rho\rightarrow \rho_{cl}\ ,\ \ \phi\rightarrow \phi_{cl})\ , \nonumber
\end{eqnarray}
we find
\begin{eqnarray}
\label{asybh}
\rho_1&=&-\left({7 \over 32} + {1 \over 4}\ln\mu^2\right)
{\rm e}^{-2\lambda x} - \left( {5 \over 12} + {1 \over 2}\ln\mu^2
\right){\rm e}^{-4\lambda x} + {\cal O}({\rm e}^{-6\lambda x}) \nonumber \\
\phi_1&=&-{1 \over 32}{\rm e}^{-2\lambda x}
+\left({13 \over 96}+{1 \over 16}\ln\mu^2\right)
m{\rm e}^{-4\lambda x}+ {\cal O}({\rm e}^{-6\lambda x}) \nonumber \\
t_0&=&-\lambda^2GN (1 + \ln \mu^2)\ .
\end{eqnarray}
Here we change the radial coordinate by
\begin{equation}
\label{chcoord}
dx ={\rm e}^{2\rho}dr\ ,
\end{equation}
which gives the metric of the form
\begin{equation}
\label{Gmetric}
ds^2=-G(x)dt^2 + {1 \over G_(x)}dx^2\ ,\ \ G(x)\equiv{\rm e}^{2\rho(r(x))}\ .
\end{equation}
Since the parameter $m$ related to the black hole mass is classically
given by
\begin{equation}
\label{rhocl}
\rho_{cl}=-m{\rm e}^{-2\lambda x}+{\cal O}({\rm e}^{-4\lambda x})\ ,
\end{equation}
we can find the quantum correction $\delta m$ to $m$ as
\begin{equation}
\label{dm}
\delta m=\left({7 \over 32}+{1 \over 4}\ln\mu^2
\right)8GN{\rm e}^{2\phi_0}\ .
\end{equation}
If we use the mass formula from \cite{MNY}
\begin{equation}
\label{mass}
M={8 \over \sqrt{\alpha'}}m{\rm e}^{-2\phi_0}\ ,
\end{equation}
$\delta m$ gives the quantum corection to mass,
\begin{equation}
\label{dM}
\delta M={8 \over \sqrt{\alpha'}}
\left({7 \over 32}+{1 \over 4}\ln\mu^2\right)8GN{\rm e}^{2\phi_0}\ .
\end{equation}
On the other hand, (\ref{eqA2}) and (\ref{asybh}) tell that the asymptotic
behavior of the electric field is not changed by the quantum correction.
This would restrict parameter $q$, which is related with $B$ by
\begin{equation}
\label{Bq}
B=\lambda q {\rm e}^{-2\phi_0}\ ,
\end{equation}
and the charge of black hole would not change.

Eq.(\ref{asybh}) tells
\begin{eqnarray}
\label{G}
G(x)&\equiv&{\rm e}^{2\rho} \nonumber \\
&=& 1+2\left\{-m - \left({7 \over 32} + {1 \over 4}\ln\mu^2\right)
8GN{\rm e}^{2\phi_0}\right\}{\rm e}^{-2\lambda x} \nonumber \\
&& + \left\{q^2 - \left( {5 \over 6} + {1 \over 2}\ln\mu^2
\right)8GN{\rm e}^{2\phi_0}
\right\}{\rm e}^{-4\lambda x} + {\cal O}({\rm e}^{-6\lambda x}) \ ,
\end{eqnarray}
what gives the shift of the position of the horizons
\begin{eqnarray}
\label{horizonshift}
{\rm e}^{2\lambda x^\pm}&=&m\pm\sqrt{m^2 - q^2} \\
&& +8GN{\rm e}^{2\phi_0}\left\{-\left({7 \over 32}
+ {1 \over 4}\ln\mu^2\right)\mp{m \over \sqrt{m^2 - q^2}}\left(
{1 \over 96}+{1 \over 4}\ln\mu^2 \right)\right\}\ .\nonumber
\end{eqnarray}
When the metric has the form of (\ref{Gmetric}) and $G(x)\rightarrow 1$
when $x\rightarrow +\infty$, the temperature $T$
of the black hole is given by,
\begin{equation}
\label{tempe}
T={1 \over 4\pi}\left.{\left|{d G \over dx}\right|}\right|_{x=x^+}\ .
\end{equation}
Using (\ref{G}), we find
\begin{eqnarray}
\label{tmpG}
4\pi T&=&{4\lambda \over q^2}\left\{-m^2+q^2+m\sqrt{m^2-q^2}\right\} \nonumber \\
&&+32\lambda GN {\rm e}^{2\phi_0}\left[-{7 \over 32}-{1 \over 4}\ln\mu^2
+ {m^2 \over q^2}\left({5 \over 6}+{1 \over 2}\ln\mu^2\right) \right. \\
&& \left. +{1 \over q^2\sqrt{m^2-q^2}}\left\{-\left({5 \over 4}
+{3 \over 2}\ln\mu^2\right)m^3
+\left({1 \over 48}+{1 \over 2}\ln\mu^2\right)q^2m\right\}\right]\ .
\nonumber
\end{eqnarray}

Recently Teo has found that the model of McGuigan, Nappi and Yost is
related with five dimensional extremal black hole in Type II superstring
theory \cite{T} by the duality. The five dimensional black hole can be
obtained as a solution of ten dimensional Type II supergravity when five
space coordinates $(x_5, x_7, \cdots x_9)$ are compactified. The metric has
the following form:
\begin{eqnarray}
\label{5dmetric}
ds^2&=& - (H_1K)^2 f dt^2 + H_1^{-1}K \left( dx_5 - ({K'}^{-1}
-1)dt\right)^2  \nonumber \\
&& + H_5\left( f^{-1}dr^2 + r^2 d\Omega_3^2\right) +\sum_{i=6}^9 dx_i^2 \ .
\end{eqnarray}
Here $d\Omega_3^2$ is the metric on the unit three-sphere and
\begin{eqnarray}
\label{HK}
&r^2=\sum_{i=1}^4 x_i^2\ ,\ & f=1-{r_0^2 \over r^2}\ , \nonumber \\
&H_1=1 + {r_0\sinh^2\alpha \over r^2}\ ,\ \
&H_1'=1 - {r_0\sinh\alpha\cosh\alpha \over r^2}H_1^{-1}\ ,\nonumber \\
&H_5=1 + {r_0\sinh^2\beta \over r^2}\ ,\ \
&H_5'=1 - {r_0\sinh\beta\cosh\beta \over r^2}H_5^{-1}\ ,\nonumber \\
&K=1 + {r_0\sinh^2\gamma \over r^2}\ ,\ \
&K'=1 - {r_0\sinh\gamma\cosh\gamma \over r^2}K^{-1}\ .
\end{eqnarray}
The entropy is, as usually, given by
\begin{equation}
\label{5entropy}
S={A_H \over 4G_5}={2\pi^2r_0^3\cosh\alpha\cosh\beta\cosh\gamma
\over 4G_5}\ .
\end{equation}
Here $A_H$ is the horizon area and $G_5$ is Newton constant in five
dimensions. The extremal limit is obtained by taking $r_0\rightarrow 0$
but keeping $r_0\sinh\alpha$, $r_0\sinh\beta$ and $r_0\sinh\gamma$ to be
finite. When taking the extremal limit and setting $\alpha=\gamma$, it was
shown \cite{T} that the metric (\ref{5dmetric}) is dual to the following
metric
\begin{equation}
\label{dualmetric}
ds^2=-\left(1+{r_1^2 \over r^2}\right)^{-2}dt^2+{r_5^2 \over r^2}dr^2
+ r_5^2 d\Omega_3^2 + + \sum_{i=5}^9dx_i^2 \ .
\end{equation}
Here
\begin{equation}
\label{r15}
r_1=r_0\sinh\alpha\ ,\ \ \ r_2=r_0\sinh\beta\ .
\end{equation}
The metric is direct product of 5 dimensional compact space corresponding
to $(x_5,x_6,\cdots,x_9)$, three sphere of the radius $r_5$ and
two-dimensional spacetime. Changing the coordinate
\begin{equation}
\label{Ccoord}
\sqrt{m^2-q^2} {\rm e}^{-\lambda x}={r_0^2 \over r^2 + r_1^2}\ ,\ \ \
r_0={1 \over \lambda}\ ,
\end{equation}
the metric of the two-dimensional spacetime becomes that of
(\ref{MNYmetric}). It was also shown that the entropy $S$ in
(\ref{5entropy}) is given in two dimensional language as follows
\begin{equation}
\label{2entropy}
S=4\pi{\rm e}^{-2\phi_0}{\rm e}^{\lambda x^+}\ .
\end{equation}
Using the quantum correction in (\ref{horizonshift}), we find the quantum
correction to the entropy
\begin{eqnarray}
\label{entropy}
S&=&4\pi{\rm e}^{-2\phi_0}\Bigl[m+\sqrt{m^2 - q^2} \nonumber \\
&& +8GN{\rm e}^{2\phi_0}\left\{-\left({7 \over 32}
+ {1 \over 4}\ln\mu^2\right)-{m \over \sqrt{m^2 - q^2}}\left(
{1 \over 96}+{1 \over 4}\ln\mu^2 \right)\right\}\Bigr]\ .
\end{eqnarray}
Hence, we demonstrated that two-dimensional considerations may be useful
to define quantum corrections not only to two-dimensional parameters
but also to five-dimensional ones (like entropy).

\section{Kaluza-Klein reduction of 3d gravity}

Let us apply above technique to
 BTZ BHs. Three dimensional Einstein gravity with the
cosmological term has the exact rotating black hole solution \cite{BTZ}
and the solution can be regarded as an exact solution of string theory
\cite{H}. In this section, we first consider the circle reduction of three
dimensional Einstein gravity to two dimensional one. Now we identify the
coordinates as $x^1=t$, $x^2=r$, $x^3=\phi$. Here $t$, $r$ and $\phi$ are
the coordinates of time, radius, and angle. We now assume that all the
fields do not depend on $x^3$. Then, under the following infinitesimal
coordinate transformation $x^3\rightarrow x^3 + \epsilon (x^1,x^2)$, the
metric tensors transform as follows;
\begin{equation}
\label{metrictrnsf}
\delta g^{(3)3\alpha}=\delta g^{(3)\alpha 3}=
g^{(3)\alpha\beta}\partial_\beta\epsilon\ ,\ \
\delta g^{(3)33}=2g^{(3)3\alpha}\partial_\alpha\epsilon\ ,\ \
\delta g^{(3)\alpha\beta}=0
\end{equation}
Here $\alpha, \beta=1,2$.
Eq.(\ref{metrictrnsf}) tells that we can identify the gauge vector field
$A_\alpha$ with $g^{(3)3\alpha}=g^{(3)\alpha\beta}A_\beta$.
Eq.(\ref{metrictrnsf}) also tells that the metric tensor $g^{(3)\mu\nu}$
($\mu,\nu=1,2,3$) can be parametrized a la Kaluza-Klein as
\begin{equation}
\label{3dmtrc}
g^{(3)\mu\nu}=\left(\begin{array}{cc}
g^{\alpha\beta} & g^{\alpha\gamma}A_\gamma \\
g^{\beta\gamma}A_\gamma & {\rm e}^{2\phi} +g^{\gamma\delta}A_\gamma A_\delta \\
\end{array}\right)\ .
\end{equation}
Then we find
\begin{eqnarray}
\label{g3}
g^{(3)}&\equiv&{1 \over {\rm det\,} g^{(3)\mu\nu}}=g{\rm e}^{-2\phi} \ \
(g\equiv {1 \over {\rm det\,} g^{\mu\nu}}) \\
\label{ginv}
g^{(3)}_{\mu\nu}&=&\left(\begin{array}{cc}
g_{\alpha\beta}+{\rm e}^{-2\phi}A_\mu A_\nu & -A_\gamma \\
-A_\gamma & {\rm e}^{-2\phi} \\
\end{array}\right)\ .
\end{eqnarray}
In the following, the quantities in three dimensions are denoted by the
suffix ``$(3)$'' and the quantities without the suffix are those in two
dimensions unless we mention. In the parametrization (\ref{3dmtrc}),
curvature has the following form
\begin{equation}
\label{cvsclr}
R^{(3)}=R+\Box \phi - \partial_\alpha\phi \partial^\alpha\phi - {1 \over 4}
{\rm e}^{-2\phi}F_{\alpha\beta}F^{\alpha\beta}\ .
\end{equation}
Here $F_{\alpha\beta}$ is the field strength: $F_{\alpha\beta}
=\partial_\alpha A_\beta - \partial_\beta A_\alpha$. Then the action $S$
of the gravity with cosmological term and with  $N$ free scalars $f_i$
($i=1,\cdots,N$) in three dimensions is reduced as
\begin{eqnarray}
\label{redS}
S&=&{1 \over 4\pi^2}
\int d^3x \sqrt{g^{(3)}}\left\{{1 \over 8G}\left( R^{(3)} + \Lambda \right)
+ {1 \over 2}\sum_{i=1}^N \partial_\mu f_i \partial^\mu f^i \right\}\nonumber \\
&\sim&{1 \over 2\pi} \int d^2x \sqrt{g}{\rm e}^{-\phi}\left\{
{1 \over 8G}\left(R - {1 \over 4} {\rm e}^{-2\phi}F_{\alpha\beta}F^{\alpha\beta}
+\Lambda \right)+ {1 \over 2}\sum_{i=1}^N \partial_\alpha f_i
\partial^\alpha f^i \right\}\ .
\end{eqnarray}
Note that the contribution from the second and third terms in
(\ref{cvsclr}) becomes the total derivative which maybe neglected.
One BH solution for the classical action is given by \cite{BTZ}
\begin{eqnarray}
\label{BTZclass}
&& ds^2=-{\rm e}^{2\rho_{cl}}dt^2 + {\rm e}^{-2\rho_{cl}}dx^2 \ ,\ \
{\rm e}^{2\rho_{cl}}=-M+{x^2 \over l^2} + {J^2 \over 4x^2} \nonumber \\
&& {\rm e}^{-\phi}={\rm e}^{-\phi_{cl}}={\rm e}^{-\phi_0}x
\end{eqnarray}
Here ${1 \over l^2}={\Lambda \over 2}$. $M$ is the parameter related to
the black hole mass and $J$ is that of the angular momentum in 3D model or
the electromagnetic charge in the 2D model. The extremal limit is given by
$J^2=l^2 M^2$. In order to consider this limit, we change the coordinates
as follows
\begin{equation}
\label{chgcoord}
x^2={l^2 M \left( 1 + \epsilon \tanh r \right) \over 2}\ ,\ \ \
t={l \over \epsilon\sqrt{2}}\tau\ .
\end{equation}
Then $r\rightarrow +\infty$ corresponds to outer horizon and
$r\rightarrow -\infty$ corresponds to inner horizon.  Taking the limit of
$\epsilon\rightarrow 0$, we obtain
\begin{equation}
\label{extsol}
ds^2={l^2 M \over 4\cosh^2 r}\left(d\tau^2 - dr^2\right)\ ,\ \ \
{\rm e}^{-\phi}={l{\rm e}^{-\phi_0}}{\sqrt{M \over 2}}\ .
\end{equation}
Note that $\phi$ becomes a constant in this limit.

Let us now discuss the quantum corrections induced by $N$ free conformally
invariant dilaton coupled scalars $f_i$:\footnote{Quantum corrections due
to matter near BTZ black hole have been also studied in \cite{BVZ}.}.
We use (\ref{qc}) and (\ref{Gamma1b}) by replacing $2\phi\rightarrow \phi$.
In the study of quantum corrected BH we work in the conformal gauge
(\ref{cg}). We are going to study (in)stability of black holes in the
same way as in four dimensions \cite{NO2}.

The equations of motion with account of quantum corrections are given by
\begin{eqnarray}
\label{eqnppBTZ}
0&=&{1 \over 8G}{\rm e}^{-\phi}\left(-\partial_\pm \rho \partial_\pm\phi
- {1 \over 2} \partial_\pm^2\phi + {1 \over 2} (\partial_\pm\phi)^2\right)
+{N \over 12}\left( \partial_\pm^2 \rho
- \partial_\pm\rho \partial_\pm\rho \right) \nonumber \\
&& +{N \over 8} \left\{\left( \partial_\pm \phi \partial_\pm\phi \right)
\rho+{1 \over 2}{\partial_\pm \over \partial_\mp}
\left( \partial_\pm\phi \partial_\mp\phi \right)\right\} \nonumber \\
&& +{N \over 8}\left\{ 2 \partial_\pm \rho \partial_\pm \phi
-\partial_\pm^2 \phi \right\} + t^\pm(x^\pm)
+{N \over 8} \left( \partial_\pm \phi\partial_\pm\phi \right)\ln \mu^2 \\
\label{reqBTZ}
0&=&{1 \over 8G}{\rm e}^{-\phi}\left(4\partial_+ \partial_- \phi
- 4 \partial_+\phi\partial_-\phi
- \Lambda {\rm e}^{2\rho} + 2{\rm e}^{-2\phi-2\rho}(F_{+-})^2\right) \nonumber \\
&& -{N \over 12}\partial_+\partial_- \rho
-{N \over 16}\partial_+ \phi \partial_- \phi
+{N \over 8}\partial_+\partial_-\phi \\
\label{eqtpBTZ}
0&=& {1 \over 8G}{\rm e}^{-\phi}\left(4\partial_+ \partial_- \rho
+ {\Lambda \over 2} {\rm e}^{2\rho}
+ 3{\rm e}^{-2\phi-2\rho}(F_{+-})^2\right)  \nonumber \\
&& -{N \over 16}\partial_+(\rho \partial_-\phi)
-{N \over 16}\partial_-(\rho \partial_+\phi)
-{N \over 8}\partial_+\partial_-\rho
-{N \over 4} \partial_+ \partial_- \phi \ln \mu^2\\
\label{eqABTZ}
0&=&\partial_\pm({\rm e}^{-3\phi-2\rho}F_{+-})\ .
\end{eqnarray}
Eq.(\ref{eqABTZ}) can be integrated to give
\begin{equation}
\label{eqA2BTZ}
{\rm e}^{-3\phi-2\rho}F_{+-}=B\ (\mbox{constant})\ .
\end{equation}
We now assume $\phi$ is constant as in the classical extremal solution
(\ref{extsol}) even when we include the quantum correction:
$\phi=\phi_0\ (\mbox{constant})$. Then we obtain
\begin{equation}
\label{constRBTZ}
B^2={\Lambda {\rm e}^{-4\phi_0} \over 2 } \ ,\ \
R=R_0
\equiv -{4\Lambda^2 {\rm e}^{-\phi_0} \over
{\rm e}^{-\phi_0}-{GN \over 4} }\ .
\end{equation}
Quantum corrected extremal (static) solution corresponding to the
classical one (\ref{extsol}) can be written as in (\ref{rho0})
($r={x^+ - x^- \over 2}$).

In order to investigate the (in)stability in the above solution, we
consider the perturbation around the extremal solution as in (\ref{pert}).
We neglect the second term in (\ref{qc}) again. Then (\ref{reqBTZ}) and
(\ref{eqtpBTZ}) (and (\ref{eqA2BTZ})) lead to the following equations:
\begin{eqnarray}
\label{reqpertBTZ}
0&=&{1 \over 8G}{\rm e}^{-\phi_0}\left\{4\partial_+\partial_-S
-\Lambda {\rm e}^{2\rho_0} (2R-S) + 2B^2 {\rm e}^{2\rho_0+4\phi_0}(2R+3S)\right\} \nonumber \\
&& -{N \over 12}\partial_+\partial_- R+ {N \over 8}\partial_+\partial_-S \\
\label{eqtppertBTZ}
0&=&{1 \over 8G}{\rm e}^{-\phi_0}\left\{4\partial_+\partial_-R
- 4\partial_+\partial_- \rho_0 S
+ {\Lambda \over 2} {\rm e}^{2\rho_0} (2R-S)\right.\nonumber \\
&& \left. + 3B^2 {\rm e}^{2\rho_0+4\phi_0}
(2R+3S)\right\} - {N \over 8}\partial_+\partial_- R
-{N \over 4}\ln \mu^2 \partial_+\partial_- S \ .
\end{eqnarray}
By assuming $R$ and $S$ have the form of (\ref{RS}), we can rewrite
(\ref{reqpertBTZ}) and (\ref{eqtppertBTZ}) as algebraic equations and the
condition that there is  non-trivial solution for $Q$ and $P$ is given by
\begin{eqnarray}
\label{detBTZ}
F^{BTZ}(A)&\equiv&{\rm e}^{-2\phi_0}\left({2R_0 \over C}A + 4\Lambda\right)^2
- \left({GNR_0 \over C}A\right)^2 \nonumber \\
&& + {GNR_0 \over 3C}A\left\{{\rm e}^{-\phi_0}\left(-{R_0 \over 2}
+ 4\Lambda\right) - GN (\ln\mu^2) {R_0 \over C}A\right\}=0\ .
\end{eqnarray}
Here $A={\alpha(\alpha -1 ) C \over 4}$. ${\rm e}^{-\phi}$ is identified with
the radius in 3D model, so horizon is given by the condition of
(\ref{horizon}).

Eq.(\ref{detBTZ}) can be solved with respect to $A$,
\begin{eqnarray}
\label{AC}
{A \over C}&=&\left[ -4096 - 2176 g + 48 g^2
\pm \left\{ 18939904 g + 4321280 g^2 \right.\right.\nonumber \\
&& - 208832 g^3 + 2304 g^4 + \left(786432 g - 49152 g^2 \right. \nonumber \\
&& \left.\left.\left. + 768 g^3 \right)\ln\mu^2
\right\}^{1 \over 2}\right]
 \times \left\{2\left( -4096 + 16 g^2 + 192 g^2 m \right)\right\}^{-1}\nonumber \\
&\sim&{1 \over 2} \pm {3 \sqrt{2} \over 4}g^{1 \over 2}
+ {\cal O}(g)
\end{eqnarray}
Here $g \equiv 8GN{\rm e}^{\phi_0}$. Note that there are two solutions near the
classical limit $g\rightarrow 0$, which correspond to stable and instable
modes, respectively. It might be surprizing that there is an instable mode
since the extremal solution is usually believed to be stable. The global
behavior of ${A \over C}$ is given in Figures. In Fig.3, the vertical line
corresponds to ${A \over C}$ and the horizontal one to $g$ when
$\ln\mu^2=1$. There is a singularity when $g\sim 0.4376$. The singularity
occurs when the denominator in (\ref{AC}) vanishes. In the range from
$g=0$ to near the singularity, there coexist the stable and instable
modes. In Fig.4a and 4b, we present the global behavior of two modes as a
function of $g\equiv 8GN{\rm e}^{g\phi_0}$ and $\ln\mu^2$. Fig.4a corresponds
to the stable mode and 2b to the instable one.

\section{Higher order perturbations and multiple BHs}

Recently Bousso\cite{B} has shown the possibility that de Sitter space
disintegrates into an infinite number of copies of de Sitter space.
In the scenario, Schwarzschild de Sitter black hole becomes an extremal
one which is known to be the Nariai solution. The topology of the Nariai
space is $S^1\times S^2$, therefore the solution can be regarded to
express the topology of a handle. If there is a perturbation, multiple
pair of the cosmological and black hole horizons are formed. After that
the black holes evaporate and the black hole horizons shrink to vanish.
Therefore the handle is separated into several pieces, which are copies
of the de Sitter spaces.

The metric (\ref{exmetric}) or (\ref{extsol}) in the extremal limit or
its quantum analogue (\ref{rho0}) has the same spacetime structure as the
Nariai solution except its signature. Therefore there is a possibility of
the multiple production of universes from the extremal soltion.

If we define new coordinate $\theta$ by
\begin{equation}
\label{theta}
\sin\theta = \tanh \left(r\sqrt{C}\right)
\end{equation}
the metric corresponding to (\ref{rho0}) becomes
\begin{equation}
\label{thetametric}
ds^2={2C \over R_0}\left(-dt^2 + {1 \over C}d\theta^2\right)\ .
\end{equation}
Since $\theta$ has the period $2\pi$, the topology of the space with
the metric (\ref{thetametric}) can be $S_1$. Note that Eq.(\ref{theta})
tells there is one to two correspondence between $\theta$ and $r$.

The operator
\begin{equation}
\label{Delta}
\Delta\equiv C\cosh^2\left(r\sqrt{C}\right)\partial_+\partial_-
\end{equation}
in Eq.(\ref{eigenRS}) can be regarded as the Laplacian on two-dimensional
Lorentzian hyperboloid, i.e., the Casimir operator of $SL(2,R)$. The
raizing and lowering operators $L_\pm$ of $SL(2,R)$ are given by
\begin{equation}
\label{rl}
L_\pm={\rm e}^{\pm t\sqrt{C}}\left(\sinh\left(r\sqrt{C}\right)
{\partial \over \partial t} \pm \cosh\left(r\sqrt{C}\right)
{\partial \over \partial r}\right)\ .
\end{equation}
The eigenfunction of $\Delta$ in $S$ and $R$ (\ref{RS}) is the sum of
highest and lowest weight representation
${\rm e}^{\pm t\alpha\sqrt{C}}\cosh^\alpha r\sqrt{C}$.
We now consider the following eigenfunction
\begin{eqnarray}
\label{S2}
S(t,r)&=&{Q \over 4} \left(L_+{\rm e}^{ t\alpha\sqrt{C}}\cosh^\alpha r\sqrt{C}
+ L_-{\rm e}^{- t\alpha\sqrt{C}}\cosh^\alpha r\sqrt{C}\right) \nonumber \\
&=&Q\sinh\left(t(\alpha+1)\sqrt{C}\right)\cosh^\alpha r\sqrt{C}
\sinh r\sqrt{C}\ .
\end{eqnarray}
Since the evaporation of the black hole in \cite{B} corresponds to the
instable mode, we restrict $\alpha$ to be $0>\alpha>-1$. Then the
condition of horizon (\ref{horizon}) gives
\begin{equation}
\label{horizonS2}
\tanh\left(r\sqrt{C}\right)=\pm{\alpha+1 -
\sqrt{(\alpha + 1)^2 - 4\alpha\tanh^2\left(t(\alpha+1)\sqrt{C}\right)}
\over 2\alpha\tanh\left(t(\alpha+1)\sqrt{C}\right)}\ .
\end{equation}
We find the asymptotic behaviors of $S$ are given by
\begin{eqnarray}
\label{Sasym}
S(t,r(t))&\stackrel{t\rightarrow 0}{\rightarrow}&\pm (\alpha +1)CQ
t^2 \nonumber \\
&\stackrel{t\rightarrow +\infty}{\rightarrow}&\pm{Q \over 2^{\alpha+2}}
\left({1-\alpha \over 1+\alpha }\right)^{\alpha + 1 \over 2}
{\rm e}^{t(\alpha + 1)(\alpha + 2)\sqrt{C}}
\end{eqnarray}
Since the radius of the horizon is given by (\ref{hrds}), the horizon
corresponding to $+$ sign in (\ref{Sasym}) grows up to infinity and that
to $-$ sign shrinks to vanish when $t\rightarrow +\infty$ although the
perturbation does not work when $S$ is large. The horizon of $+$ sign
would corespond to outer horizon and that of minus sign to inner horizon.
Since there is one to two correspondence between the radial coordinates
$r$ and $\theta$, there are two outer horizons and two inner ones. When we
regard the model as the model reduced from four (three) dimensions, the
handle with the topology of $S_1\times S_2$ ($S_1\times S_1$) is separated
to two pieces when the two inner horizons shrink to vanish. This result
would be generalized if we use higher order eigenfunctions as perturbation
\begin{equation}
\label{Sn}
S(t,r)={Q \over 4}\left((L_+)^n {\rm e}^{ t\alpha\sqrt{C}}\cosh^\alpha r\sqrt{C}
+ (L_-)^n{\rm e}^{- t\alpha\sqrt{C}}\cosh^\alpha r\sqrt{C}\right)\ .
\end{equation}
 From the coorespondence with Bousso's work \cite{B}, there would appear
$2n$ outer horizons and $2n$ inner horizons and the spacetime could be
disintegrated into $2n$ pieces when the inner horizons shrink to vanish.
Of course, the confirmation of this effect should be checked with some
other methods. It is also interesting to note that proliferation of de
Sitter space is somehow different from baby universes creation based also
on the topological changes (for a review,see ref.\cite{GWG}) as it occurs
at large scales \cite{B}.
Note that horizon equation for the higher mode is $2n$-th order
polynomial with respect to $\tanh r\sqrt{C}$, which is the
reason why it is expected that there can be $2\times 2n$
horizons. However, it is difficult to confirm that all solutions
are real solutions.

\section{Discussion}
We studied quantum properties of 2D charged BHs and BTZ BH. The quantum
effects of dilaton coupled scalars in large $N$ approximation have been
taken into account. Quantum corrected solution for 2D charged BH has been
found and quantum corrections to mass, charge, Hawking temperature and
BH entropy have been evaluated. As 2D dilaton-Maxwell gravity with
non-minimal scalars under discussion may be also considered as toy (or
spherically reduced) version for 4D or 5D Einstein-Maxwell-minimal scalar
theory then higher dimensional interpretation of results may be done. (It
is also possible due to duality of 2D BH with 5D BH). Quantum
(in)stabilities of 2D charged BH and BTZ BH are discussed and it is shown
the presence in the spectrum of the mode corresponding to anti-evaporation
like in the case of 4D SdS BH. Hence, the important qualitative result
of our work is that we show:quantum anti-evaporation of BHs
(even BTZ BHs) is quite
general effect which should be expected for different multiple BHs (and
not only for SdS BH). Numerical calculation shows that evaporation is
``more" stable effect, nevertheless. The interpretation of higher modes
perturbations as the reason of disintegration of the space itself to
copies is given. It is clear that in order to understand if the above
effects may really happen in early universe one should select boundary
conditions corresponding to BH formation and analyse above processes
subject to such boundary conditions. Depending on the choice of boundary
conditions they may be compatible with anti-evaporation as it happens for
Nariai anti-evaporating BH \cite{NO2}.

It is not difficult to generalize the results of this work to other higher
dimensional BHs where metric may be presented as the product of 2D charged
BH with D-2-dimensional sphere. Then large $N$ approximation and spherical
reduction may be applied as well
(as it was done in 3D case) and there are no principal problems to
study quantum properties of such BHs (at least in large $N$ approximation).
>From another point, duality may help in the study of connections between
low and higher dimensional BHs on quantum level. Number of new effects are
expected to be found in such studies.

It could be very interesting also to consider supersymmetric
generalization of above results. It seems to be possible to do as anomaly
induced effective action for dilaton coupled supersymmetric 2D or 4D
theories is known \cite{NOG}.

\noindent
{\bf Acknoweledgments.} We thank R. Bousso and S.Hawking
 for some discussion.
The work by SDO has been partially supported by RFBR, Project No.96-02-
16017.

redpajama_set_name: RedPajamaArXiv