Okay, Traders, welcome to this week's live analysis session with me, Patrick Munnally. If you can hear me and you can see the screen, if you type a Y in the chat box, that would be great. Good stuff. Okay, let's get going here. Okay, so before we jump into today's content, as always, we want to adhere to the risk disclaimer understanding that trading any financial instrument carries an inherent risk, and we could potentially lose more capital than we necessarily have on deposit. And more importantly, for today's session, any opinions expressed by me today are solely mine. They are not representative or indicative of those held by Tick Mill UK or Tick Mill Europe Limited. So for those of you joining us for the first time today, I'll just give you a brief introduction as to my background, I'm going to say my name is Patrick Munnally. After I graduated from King's College London, I joined a city PLC consulting firm. After a couple of years of learning the ropes, I left with some colleagues and went on to co-found and successfully exit a consulting startup post-emerger in 2004. And then moved on to explore my passion for markets, with some capital to play with and some time in my hands, I started day trading or more appropriately day gambling, the S&P 500. After some early beginners luck, I racked up some quite solid and then some quite significant gains. However, as is often the case, that beginner's luck ran out. And as the market phase changed, I began to average down into losing positions very quickly, giving back all the gains and then ultimately taking a six-figure hit on my personal capital. To say this was a gut-wrenching and sobering experiences and understatements, at this point I really had to stand back and figure out if it was really feasible for me to make a living from the markets. So I decided to get serious about trading and I sought out a mentor with an excellent trading track record. Working with my mentor for about 18 months or two years. It was a period during which I upped not just my technical game in terms of researching and developing a strategy that importantly suited my personality, I extensively back and forward tested and developed a rigorous risk management approach to underpin the strategy. But most importantly during the period of mentorship, I significantly developed my mental game and probably the most important development was the watershed shift really from being a highly goal-orientated and focused on financial gains to becoming a process-orientated individual. So what does that actually mean? Well, it means I stop focusing on what I could make from the markets and start focusing solely on managing my mindset to allow me to consistently execute my trading strategy, often times in the face of negative feedback from the markets in the form of losing trades. But once you make that shift and you become process-orientated and you have a professional trading mindset, understanding the true nature of trading insofar as that it really is a numbers game in which you're simply playing the probabilities, you lose the emotional investments and that hellish emotional roller coaster of living and dying by the outcome of individual trades. So I'm no longer concerned the outcome of individual trades or even strings of trades. My focus is on the next 100 trades because I know if I focus on excellence in execution, my edge will demonstrate itself over an extended series of outcomes. My multi-strategy approach has delivered profitable annual returns since 2008, since 2013, the performance figures you can see on the screen at the moment. I've also been managing investor capital through a managed account service, again delivering annual positive returns. I'm currently responsible for managing a multi-million dollar portfolio. Also since 2010, I've personally mentored over 100 private traders of all experience levels from complete novices to former CME floor traders in developing the technical and mental skills to reach consistent returns from the markets. I've also consulted to numerous brokers and trading education brands contributing written webinar and live presentation content on a range of topics from market analysis to trading psychology to strategy development and execution. In addition to my fund management and private mentoring, I'm also the resident market expert at Tickmill providing daily market analysis and a daily setup that I'm tracking in the markets. I'm currently registered through the blog to receive those updates via email to get a flavor for my style and approach to the markets. My other passion project really is as head of trading and trader education for fxcareerswap.com. We offer development and funding to retail trading talent and fxcareerswap. We don't just develop retail traders market and trading strategy knowledge, we work on mindset development through our structured program that culminates in managing the firm's capital at zero personal financial risk on a profit share basis. For those who are interested, you can reach out to me through LinkedIn or contact me via the website there fxcareerswap.com and I'm happy to give you more information with respect to that program. That gives you a flavor of where it is I'm coming from. What I want to do now is jump into some charts, but not the technical charts that we're going to look at in a minute. This is some sentiments and flow information that I track. We're going to start where we ended last week, which is with Bitcoin. Not an instrument that I actively trade, but we looked at this idea last week with respect to historical volatility and being drained out of the market. The past seven or eight times that's occurred has preceded some quite significant moves in the market. If we jump to the Bitcoin chart here now, we were looking last week at this descending trend line and we actually had this one as long as we were looking at here and we got the breakout just really after we finished the session last week. Some of that breakout was actually on the news that came out that Square, the financial payments business that is run by the founder of Twitter, has actually put a $50 million investment into Bitcoin and that helped pop Bitcoin this time last week and we've since traded higher. And whilst we hold now above this ascending trend line support, I think we, like I said last week, we've got a crack at the projected ascending trend line resistance, which is starting to tick up a bit now and we could be heading up towards $14,000 potentially. So that's one to keep an eye on and you can see there how that information, that flow information or volatility information feeds into the market movements there and that said, let's look at some foreign exchange flow data. This is from Credit Agricole that just enhanced their FX positioning and flow tracker data, which should deliver some pretty interesting signals. They've back tested this strategy over the past eight years and annualized delivered a 15% return and I'll share these signals with you on a weekly basis. Obviously, by the time you see these on the Thursday, they'll be a little bit stale because these signals are issued on Monday. I'll be showing them with the guys on the trading team at FX Careers, what though, but anyway, what we're looking at here is the flow data and you can see that last week, the euro was close to three standard deviations away from the mean in terms of long positioning in the market. We've seen a bit of a pullback this week, but we're still pretty stretched and so that suggests that we could see some corrected downsides in the euro and we'll look at some charts in a minute that would underpin that. We also see the Kiwi and the Aussie pretty stretched and the Swissie, less so obviously the dollar and to the downside. So what we're basically seeing here is if you take it in its rawest form, the euro sec, the Swedish Krona here would give an interesting short position as euro longs are stretching the upside and the biggest short positioning is in the Swedish Krona at the moment. So in its rawest sense, that will be the signal, but you can see here in terms of if you think about FX majors and the dollar, the dollar longs paired significantly and so the stretch played in terms of the FX majors is looking at a signal on the short side in the euro dollar and that's underpinned further. This is the JP Morgan flow data and we're seeing a bit of a tick up here now. We were very stretched on the downside in terms of US dollar shorts and we're seeing a bit of a pullback there. So we'll start to see now or we'll look now as we go into the charts how that might impact our technical views. So let's check in with the fractals. This is the fractal going back to 2017 that we've been tracking. There's been working as a decent market map for current price action. And we've put in this low or we're trying to hold the low here and tick higher in terms of the dollar. What we're actually looking for now is a dollar high to occur into the US elections. Now, that's kind of premised on the moment that we're currently looking at the betting in terms of the US elections that Biden looks the favorite at the moment or shoe in almost at this stage to get in. And if he does, we think that that will have implications for the dollar potentially weakened the dollar. And so we're looking for this November 3rd into that election high. If we can hold this current low, you can see how we're tracking almost to the day here. October 12th low and we have an October 16th low. So there are there about some terms of the fractal overlay. So we'll see now if this dollar can push higher. And so that obviously has implications for the euro, which we would look to trade lower. And we can see we're kind of in the ballpark there with the euro looking for looking for some weakness now to play out. And I'll show you on the execution charts how I'm set up for that. But that's what we're looking at with the euro cable. Obviously very, very choppy environments in terms of sterling at the moment. Headline driven and you can see that we're popping lower again now after yesterday's whipsaw. All this is based obviously on source comments coming out with respect to Brexit and the extension of negotiations. Yesterday morning we were at lows in the market on any cable cross or cable itself. We're trading down maybe a hundred bits or more. And we have one headline come out and within a second the Algos took it back up through the highs and we're now selling off again. So like I said last week what I expect is a messy, a messy low to occur in cable. And we probably see that into into the elections now. So we could see cable drift back down into the lows here, maybe a little bit higher just off the lows before we recover into a potential US election low. But the water is kind of muddied with cable a little bit obviously because we're always at the whim of the next Brexit headline. So I prefer to play it through the euro at this stage. The other one I'm playing at the moment is the Aussie highlighted the potential for a correction. A similar story again, almost to the tick here, October the 16th was the high in 2017 coming off the highs. And what do we have here? October the 12th was our high. So you know with this 2017 model is really tracking quite nicely at the moment and again, as with all these things, I'm not just trading the model, I'm looking at specific setups. But where you have this historical or seasonal implications for these pairs, it's certainly worth paying attention to and you can use it as a guide. So let's see how that plays out. Last but not least, let's check in with the S&P 500. And interesting, I looked at this on a line chart basis and went back to all the way back to 2009. You can see what we've done here is we took out this ascending trend line in that last high. We've since come back down through to retest it from below here and we're struggling to get back above it. We'll take a look a little bit closer at the S&P in a minute on the execution charts. But you can see the potential pattern here, this broadening pattern is in play at the moment. And we'll see what that looks like now in terms of setups and opportunities. So let's start as always with the dollar. The charts I've highlighted here with the AMBA tags are the ones I'm going to look at. If you have a chart that you'd like me to take a look at that I don't cover, I'm happy to do that at the end. Equally, if you have any questions, you can file those into the chat box at the end. I'll let you know once we're done with the review of these charts. So this is the dollar index broad basket. And what we're looking at now is for this current load to hold. I'm for us to get up into into November, the beginning of November here to test this resistance zone at the 95.86 versus this swing low. We have an equality objective as well at 95.93. So it's all nicely poised here for that test. We do need to clear this trend line, though. We've had three tests. We've held three like like the guys I work with know that when you get that fourth test of trend line, that's the trend line tends to give way on the fourth test. But more certainly the third test we always look for a whole. So we'll see if the dollar can get through here. If it can clear the trend line, then we look for a retest of the price cycle. Highest man 473 on route then to an ideal test of the 96 area heading into the into the elections. Equal way to dollar index. Let's remove these and bring in this. So in terms of the equal way to dollar index, we've got a similar story here. What we're looking for is that current load to hold and for this equidistant swing to play out. Take us a little bit higher here. Let's just zoom out a couple of layers. So we're back into that resistance area just ahead of 123. And that's versus this 111. 120 holding, sorry. 119 not 111, 119. And if that holds, then we're looking for 122.88 into the US election as the target there. Now this is the equal way to dollar index. So this is the dollar index that incorporates the Aussie, and it has an equal weight versus the Euro, Aussie, yen and sterling. So if this is ticking up, what that should mean is that we're seeing a weight in those in the in the counterpart. So let's let's take a look at this. First with the Euro. So the Euro has obviously been trending higher. We're looking to take out some trend line support here on the Euro on a closing basis today. If we do, then what I'm looking for is a test down to this 115 area. And if we bring in the equality objective versus the current pattern here. So I'd actually suggest slightly low, almost retesting this prior high, the 114.15. So if we can get through this trend line and closing basis, certainly look for a retested lows. And the next marker is going to be the 115. And through there, we look for the equality objective for setting up then for that, that leg higher into that November lows.  So what we look for is something like this. Let's see here. And then from there, we look for longs to reload. And then we have at least a 122 on the upside would be the initial objective. So that's the type of pattern I'd be looking to play out in the Euro. So the dollar basket signal is a short Euro, it's short cable. So let's have a look at that. So cable trading in this interim channel here. We can take out support now at yesterday's low, 12850. Then we look for cable to test down into the channel support. 12850 on, sorry, 125 would be the downside of the objective here. Let's just take a look at the symmetry swing objective. Sorry, the equality objective. Quality objective even lower, we could back down trading 123 there versus this initial decline. And if this is going to be the corrective high, we'd be able to see that. But certainly it means I'm looking a little weak now for cable. We weren't able to recapture the upside in terms of the trend line resistance that was broken. We've almost tested it from below and rolled over here. But again, all about the closes. These are daily time frames. And the confirmation only comes at the close of the candle. So that's cable. Let's take a look at the Aussie. This is the trade. I mean, at the moment, short the Aussie through the break of this ascending trend line support. We had the RBA governor out overnight suggesting that the economic crisis in Australia equal to that of the Great Depression. And certainly there are now implications that race adjustments are imminent in Australia. Employment data wasn't too hot either. So the initial target for this move now is going to be the 69 handle, which is the equality objective versus the last swing. Could go deeper than that. And it could test down into the 6830, which are these prior loads over here. So the first objective certainly will be 6930-ish versus the current move. So we're getting confirmation there. We've got the Euro, Sterling, and the Aussie. So let's see what the yen is doing. So the yen is kind of, I mentioned this last week, the yen is kind of trapped with US yields at the moment. So we're not seeing too much of a reaction here in the yen. But whilst we hold this 105 handle, and I still think we have potential to test this descending trend line resistance in this triangle, almost triangle within triangles here in terms of the yen, but certainly we could see that 106-78. So almost the yen isn't confirming the broader cell signal that we've got in terms of the dollar index. But like I say, kind of trapped by the yield story in the US. So let's take a look at Swissy. So Swissy, whilst we can hold this swing low here at the 9077 level, we can look for the Swissy to trade higher to the equality objective and this descending trend line resistance at 9350-9370 area before we potentially make the next leg to the downside in terms of the Swissy. The Looney, obviously with the Aussie weakening, we can look now for the Looney to strengthen and look for a test up into this 3526. That's versus this 131 low. So again, look for a strong close today in terms of the Looney and then we could play pullbacks into the London session tomorrow to get in on the long side and reduce your risk reward there. Euro yen, another trade that I've been in. And we're just coming down into the projected descending trend line support now. We took out the channel and we rolled over and I took some profits on this. But what I'm looking for now is if we can take out this 12290 area, then I'll be back in on the short side looking for the equality objective here, which should see us down into 121. We could go deeper if we play versus the swing high here. Could be looking at 12050. So between 12050 and 121, once we take out this 12290, that's the downside objective currently in the Euro yen. Sterling Swiss, this is a box trade or potentially a box setup. There are two targets here. If we break the box support at 11750, then we have an equality objective versus this structure, which would take us down to 11330. If we take the box to the top side looking less likely at the moment, if we did, then we'd have an equal objective at 12126. So watch for the box to break. And then you can get in on intraday time frames and look for a retest of the box support to act as resistance. And then you've got ample downsides to run if this thing is going to roll over as projected. So that's one to keep an eye on, probably heading into next week now or tomorrow. Sterling, we talked about. Sterling yen, I got whips all out of this yesterday. And lo and behold, it rolls over. So again, if this Sterling yen now takes out the channel support here yesterday's low, let's say 13550, we take that out, then there is a big target on the downside here, 12830. So again, you can go on to the intraday charts and look for a pullback to this support area at 13550 as resistance to play for the bigger targets on the downside in terms of Sterling yen. Sterling CAD, nothing for me to do there. Sterling Kiwi, I also had this trade running, took some more profits on this. Not really a clear set up for me at the moment there, those two should be blank. Sterling Aussie, trying to break to the top side here. This one has been tricky. Looks like we could potentially today even get another failure below the trend line. So again, not in the immediate action. I've covered the Aussie, the Aussie Kiwi, another one in at the moment. We took out the sending trend line support. So that gives us this channel to play for now. Certainly we pay attention to 7393, probably a little bit sticky there. We've got a wheat DS3, price swing lows. But once we consolidate and ultimately take that out, and we'll be looking for the descending projected descending trend line support, 7270. And then the equality objective versus this swing would have us down at 72 level on the downside. So those are the downside targets for that trade and I'm currently running. Aussie Kiwi, it's taken out a sending trend line support. So whilst we hold below the median point of this channel now, then I think the Aussie Kiwi can move lower, certainly to test down into the 106 area. The projected descending trend line support is the logical target there. Let's take a look at the Kiwi. So looking at the Kiwi for this test, into the potential reversal zone, which gives us this head and shoulder scenario. We just came shy of it yesterday and obviously we've rolled over hard today. So watching for a breach now of the ascending trend line support to set up or move to the downside. In terms of the technical target there, we'll measure the heads to the neck. So once we, if we can break through So if we take out the support here at 6489 and there's a 6189 downside target. So what I'd be looking for would be this type of scenario. So we move down into the neck, always a crowded area on that initial test. What I'd like to see there is we get a middle pull back into retest the ascending trend line support then as resistance. And that would be the area that I'd be looking to get in on the short side and get trying to get into that move there. So that's what I'm looking at with Kiwi. Kiwi yen, similar to the Aussie yen we've taken out the sports area. It wasn't a signal for me as such there but certainly it started to look weak and the downside objective could be into the lower, lower part of this channel now, 6780 area. Kiwi Swiss, I have this overnight I've taken the profits now. Watch for the test here of the ascending trend line support. 5998 through there and we've got some clear water certainly down to 59 and we could be heading lower than even to test this area as range support down to 57. But again watch how we trade at this ascending trend line support in terms of Kiwi Swiss. Kiwi CAD looks like it's going to give up its trend line now. So certainly if we get through 8660 in terms of Kiwi CAD then we've got downside to 8570 will be the objective on the downside. Swiss yen, I've got an order waiting to be filled in this one through these lows hasn't been triggered yet. But if we do trade, if we can get through the lows through 1490 and we've got a downside objective here at 11220 and I've got an order to take that trade if it plays out. S&P 500 just talked about this one. Pay attention in terms of the fact that the S&P is obviously driving risk sentiment. If we don't take out this trend line support and there is a chance that we're in a bear trap leading to a bull trap. So let me just draw this in for you. Yeah, this is a bit bigger. So we've got a potential five-way sequence here. So we have one, two, this will be three, this will be four and the fifth there I think would set up a margin on new high versus our current highs. And that would just get everybody excited on the upside. But what I'd be paying attention to on that margin on new high is divergence. And we could get a great divergence set up here.  But at this point, pay attention really to this 34, 30 area, 3400 if we get through there. Then I'd say the high is in and we can discount this count. And what we'd be looking for then would be that this is our A, B and we'd have a C point here, 31, 60 heading into the elections as people reposition basically ahead of the election and take cash off the table. NASDAQ came up into its quality objective and it's rolling over again. Watch the trend channel here. We take out the 11,500 area to suggest that this is the A, B and we've got a C target, which would be a C target down there at 10, 481. So we'll be looking at this type of pattern into an election low and then it probably takes off again to the upside. Gold, tricky, still trading in a very tight range. I think this is bound pretty much by the elections at this stage. Wasting to see what the shakeout is before gold's going to make its move. Crude oil again trapped in this range at this point. What I am interested in is copper here, copper.  We've traded up, retest the prior, we're sending trend line support as resistance. Looks like we're potentially rolling over here. The cycle indicator is already bearish and copper certainly leads those commodity currencies. If we see weakness in copper now, we can expect risk sentiments to meaningfully roll over. So those are the charts I'm watching, setups I'm watching, positions I'm in, Aussie and Euro mainly at this stage, watching some key potential trend line breaks to play pullbacks and then getting on some moves. Okay, so that wraps up the charts I wanted to discuss today. Are there any questions? Would anyone like me to take a look at the chart? I haven't covered. You can type it in the chat box and I'll do so. If you don't have a question, if everything's clear and then in the chat box, it's equally helpful. So I know we're all on the same page. Okay, to here. Hi to here. Hello. Hello. To here. Yeah, how are you doing? I'm doing well. Great, Mr. Patchwick. Nice having you by this time of the day. Very good, very good. Well, how can I help you? Yeah, concerning the trends you just predicted or you just forecasted, basically some investors are having a schedule ahead of the US elections. But what I need to clear up on is I don't consider it any fundamental or certain investor sentiments in your analysis. So I just got to notice that you only use technicals concerning price movements. So here to here. This is how I look at things. To me that the chart that the price, the current price contains all current information known by all market participants. And so the technical setups or the patterns that I'm looking at are a indication of a probable scenario. So one probability being higher than another. But what drives those patterns to complete is often fundamental or market supply or demand shocks. So obviously the US election is a major component to that at the moment as is Brexit. And so the patterns to my mind proceed or give us the edge for an opportunity that may precede new market news. Does that make sense? All right. Yes, that's true. So I'm fully cognizant and conversant in all the current market narratives and themes. And like I say, the guys I work with, you have to have that because having that information allows you to almost look at the charts and where you see a technical pattern the knowing what could be coming down the pike in terms of market narratives and dynamics allows you to see what could drive the price action. All right. Thank you. Actually, I have to say thank you for improving my trading strategy with your forecasters and all those stuffs you are giving. Thank you. You're very welcome. Thanks to here. Any other questions? Okay. If there aren't any other questions, I'm going to wrap this one up here and we will reconvene at the same time next week. Have a great weekend and a great trading week ahead. All the best. Thanks.


how important it is to learn history because in some way it dictates how it can help our behavior. It didn't help yours. No, no, no. I've lived, hey, I lived just a life just like a lot of people. It may have been a long time. Ben Hurd had a life like, remember we saw that movie Ben Hurd? You know what I mean? He was a wealthy man. He became a slave. He became a great conqueror as far as athletics were concerned. He became a star. He became a very, he became the best celebrity. And I believe wherever it was, Actavia, wherever, wherever city they took him to, he rescued that, he rescued the general of that ship. And after all that, he still went, he still couldn't save his family. They put him with the lepros. And his main objective, he went back and he got his family from the lepros. His sister and his mother. And then again, he came back. He may not have been famous again, but he got his family and that was a success. We look at success, me, myself, you know, at different levels of success. Some people's success is not using drugs, not drinking anymore. Some people's success is making a million dollars. Some other people's success is making a billion dollars. Some people's success is not going to prison no more.


Innovating products and services to be client-centric and meeting the needs and demands of our customers is essential for us at the UPU. And that is reflected in our new strategy for the upcoming cycle. Similarly, in postal financial inclusion and financial services, we are continuously looking at how posts can innovate their products and services to meet the needs and demands of our clients around the world and how people who are underbanked and unbanked can be served better by postal operators around the world. What we are looking at at the moment is a study that draws upon the views of experts in the sector, postal experiences and the voice of the customer to see how posts can innovate their products and services to meet the needs in the years to come. We invite you to read this study and we hope that you will benefit from it and be able to design your own products and services that are innovative, client-centric and can provide a wealth of benefit to the people of the world. Thank you very much.


Your coca-cola bottler presents Claudia based on the play of novels by Rose Franken Brought to you transcribed Monday through Friday by your friendly neighbor who bottles coca-cola Relax and while you're listening refresh yourself Have a Coke and now Claudia Hello. Hello. Oh blast it. I've disconnected myself again. Hello. Hello. Hello. Hello. Is there an architect in the house? Hello. Hello. There are two architects in the house and that's all that I've Roger. You're sitting at the switchboard again. I Certainly am. Where's Lottie Lottie is home taking care of her mother who has a bad cold in her chest. Oh Is David here David certainly is here. Hello David. I'm here to a fine thing Isn't the man's office sacred anymore? Well, if you're not glad to see me Roger and I will walk right out and leave you here with a switchboard on your lap. Who says I'm not glad to see you. Nobody says and I am delighted to see you Claudia.  What kind of a freight terminal do you want us to build for you now. Sit down. Have a cigar. Oh I don't smoke cigars gave it up last week. I thought you seemed different. Your face looks absolutely empty. That's a nice way to talk to a prospective customer. No wonder you've only got one freight terminal to build. One is more than enough. If only Carrington would come to New York. Our one would be underway immediately. Instead he only phones and things are endlessly delayed now. We're expecting to hear from him any minute. That's awful. What's awful about it. I was hoping you'd come and look at wallpapers with me. That's why I'm downtown. What do we want with wallpaper for the house in the country. We hardly have a wall. That is the trouble with architects. No imagination. No imagination. I suppose you've imagined something better than we have. I certainly have. Have you ever seen how hard it is to put wallpaper on a wall. My new system does away with all that. David we should listen to this very carefully. Occasionally good ideas come from highly improbable places. All right. Let's hear it. What is the Claudia Brown Norton system. I put up the wallpaper first. Then paste the wall on to the back of it. It is vision. What do you think of it Roger. It is probably the greatest idea since the invention of the sky hook for hanging roof. It's got striped paint beat a mile. You two sound like a Vogueville team. Darling can you come and look at the wallpapers with me. Well I'd love to but you see Roger has to go to the bank and we're expecting a call from Carrington and somebody has to stay here to answer the phone. If I leave right now you will be able to get away sooner. See you very soon. Goodbye. I'll only be a few minutes. Goodbye. Say David the wallpaper places right across the street. We'll go just as soon as Roger is back and in fact we can go just as soon as Carrington calls. I wish I could get him to come to New York. That's the only way we'll really get things ironed out. There he is now. Kelly and a Norton. Good morning. Give me that telephone. Mr. Norton. Who is calling. Hand over that telephone. Mr. Kelly. One moment please. Mr. Kelly. I'll see if Mr. Norton is in. You don't say what he wants. Hello. Hello Kelly what's on your mind. The bid. What bid. It's due today. Well I'll rush it right down there myself. Yeah thanks Kelly for reminding me. What did David we're bidding on a school job with Kelly and it now it looks like I'm going to have to rush it right down to the board would you like me to stay here I'll talk to Mr. Carrington well I hate to ask you to but I think I wish you would I don't mind darling it'll help you out besides which Mr. Carrington is a very nice man we always have lots to talk about. Certainly as lucky you showed up now where is that bid I think it's over here on this day look at all the mess on that desk honestly Lottie isn't here for one day in this place looks like well it looks like an office with two men in it I was sure I left it over here I'll help you find it what does a bid look like looks like an envelope that's what it is an envelope well there are dozen envelopes here you sure it isn't one of these it isn't I look David look at all the things you have on these debt newspapers and blueprints what are these things Oh specifications never mind them now and look at all these advertisements and calendars don't you people ever throw anything away Lottie is the admiral in charge of throwing things away around would you like me to clean this mess up while you've gone you can do anything you like when I'm gone only help me find that bid now you sure it isn't one of these envelopes I am positive what's not when you have in your hand it's one of the envelopes where did it come from it wasn't there a minute ago that's the bid darling fancy fancy now give it to me I haven't much time Mr. Find it all how long you'll be gone oh half hour at the most the board is right near here so long I'd better hurry goodbye oh David now what wait for the traffic lights hello oh hello mama how's your cold you sure it's better yes I'm in David's office and it's lucky I came because he and Roger both had to go out no I won't touch anything only I'm going to clean up the place while they're gone isn't it awful the way men leave an office when there are no women around oh all kinds of papers I'm going to put them in the safe and lock them up yeah I thought it was a sensible idea too I'll speak to you later goodbye mama think I'll take a newspaper and do a crossword puzzle one horizontal six letter word meaning South American tree oh for heaven's sakes one vertical siamese coin 97 horizontal turkish nobleman beginning with e four letters see now it's the most foreign puzzle hello hello roger you're back quickly only 20 minutes where's david did kerington call he hasn't called yet david had to take a bid someplace a bid yeah oh you must mean a school job bid oh I thought that wasn't due till tomorrow that's what david thought too but mr kelly called up and david had to go rushing out well it's a lucky thing you were here uh roger do you notice anything different different oh yes all those papers you clean them all up that was nice of you floria it certainly makes a difference doesn't it a wide expanse of desktop what's mr kerington going to call you about you never did tell me well we hope it's for a last minute checkup on the specifications oh he's just about ready to put the freight terminal out for bidding hello everybody did kerington call no not yet did you win win what win the bid of course we uh we don't know yet well if you don't know yet then what was the big hurry roger you you explain to her will you while we try to figure out why kerington hasn't called yet well what time is it let's see i have a quarter after one quarter after one well it's an hour earlier in chicago we mustn't forget that david you didn't even say thanks for how nice your office looks well it looks empty that all the thanks i guess oh that's kerington now do you want to talk to him or shall i no no no no you you go ahead you go ahead hello hello yes yes this is uh killian and not chicago calling yes yes i yes i'll take it myself it must be kerington now you talk to him david while i find specifications now they must be here somewhere fine fine fine no hello yes this is mr notton yes yes i'll speak to mr kerington oh oh hello mr kerington david do you have those papers uh it uh excuse me a moment mr kerington haven't you got the i think they're here on the table that's a lot of where did you put the specifications that were on the table i'm uh i'm sorry mr kerington i'll i'll be with you in just a moment i cleaned up everything and put everything in the safe oh no you locked it i don't know the combination do you dated me i thought you did i i i keep it written down but i keep it in the safe well try something try anything come over here and help me date uh claudia claudia here take the phone and start talking what about here you have to clean the office of one moment please i think i think it starts with 75 three times to the left to the left i i thought it was to the right that's the this way hello mr kerington this is claudia notton yes well you must come to new york mr kerington it doesn't open claudia keep talking i'm sure it goes to the left 75 yes mr kerington when did you pass it you got to pass the number that's uh well we'd love to see you again mr kerington we just bought a house in canada and we're going to have a carl well it's a real old sock box yes yes yes yes mr kerington with 78 acres and a brook three times to the left and twice to the right david i'm running out of things to say never happened before say anything it won't open david you try it you try it but of course i mean mr kerington we'd love to have you come to new york and then maybe you'll come off the eastbrook with us and look at the house and what am i saying three times to the right that's right twice to the left that's it that's it 48 is it coming oh yes oh claudia had to tidy up things i don't see why she couldn't have put the safe away and left the papers where they were i have been intending to throw this this thing away for many years yes mr kerington that's wonderful goodbye roger roger i think we've got it i i heard something click take it easy david be very careful it's the crankiest old safe in the world let's see you 47 that's it 48 that's right 49 there there oh there they are there they are right on top the specification you talk to him roger no no you go ahead david you talk to him but he's gone he's gone who's gone mr kerington he hung up oh no oh yes oh dear do you mean you ran out of things to say i mean he decided to come to new york next weekend person and you know the funniest thing he seemed to have forgotten why he called while he was talking to me that i can understand he's coming to new york well that's marvelous that's just what we were hoping naturally naturally naturally and now i am putting these specifications in this fine old safe and then i'm blocking it and then i'm taking my two partners out for months shift your packages to the other arm man when that bright red cooler hits your eye in the local food store stop for a few minutes treat yourself to ice cold coca cola and it'll be a lot pleasanter to go about your business stands the reason you shop more comfortably when you shop for refresh hello there mr king well if it isn't mr killian you had quite a time getting that safe open i see but uh i suppose you should feel lucky that the safe didn't have one of those time locks on it you'd still be waiting to get in a time lock and claudia the combination is too much for me she's a charming girl but i don't think she ever got anywhere on time in her life claudia and david are gonna have dinner with aunt louisa tomorrow night you know mr killian and uh i understand that aunt louisa runs her house as punctually as a railroad that you 20 cents as claudia says that she and david don't arrive at the right time if they don't well i think i'll be listening to hear the consequences and so will i mr king goodbye goodbye mr killian as i was about to say every day monday through friday claudia comes to you transcribe with the best wishes of your friendly neighbor who bottles coca cola so listen again tomorrow at the same time and now this is joe king saying oliv war and remember whoever you are whatever you do wherever you may be when you think of refreshment think of coca cola or coca cola makes any pause the pause that refreshes and ice cold coca cola is everywhere and now here's a word from your friendly neighbor who bottles coca cola


Svaj, je stejn. Ti stejn. I'm going to scoop your next paper. OK, we can start the afternoon session. We have Sameer Monti from KINX, who will join all what we have seen so far, localization, supergravity in so on. We talk about localization in supergravity. Wait. Thanks. So, let me begin by thanking the organizers for this invitation or maybe I should say push to give these lectures. This is not here yet. So there is. So my title is Quantum Black Hole Entropy and Localization in Super Gravity. And what I would like to discuss is a set of ideas about the quantum entropy of black holes. I'll explain what I mean by that as we go along. That have been developed in the last 10 or 15 years. And in parallel there's a set of techniques which have been developed to address such problems. And the main technique is localization about which we heard a lot applied to field theory. And here what we want to do is to apply this technique to supersymmetric theories of gravity. So that comes with its own set of challenges and problems. So I'll try to explain that as we go along. So my handwriting is not very good. So at some point of time I'll try to type up what I write and put it on the website. But for now at least I'll scan in my notes and do it after every lecture. So for now I think this should be already on the website. There's just a plan of the lectures that I have. So I have four lectures and I've divided it into four parts roughly corresponding to one part per lecture. I don't know why it came out like this. I had single pages but now I'm sure it cannot read it with my handwriting. So I'll just try to do it like this. Maybe this is better. So the first part is, which is today, which is, I'll just tell you about the idea of quantum entropy of black holes. So I'll just review some basic black hole thermodynamics, which I'm sure all of you know, just to put all of us on the same page. Then I'll talk about supersymmetric black holes and their classical entropy and then start to define what quantum entropy means. The second part, which is tomorrow's lecture will be giving a precise context to it and the context will be what Stefan discussed very nicely in his first two lectures, namely supersymmetric black holes in four-dimensional n equal to supergravity. This is in asymptotically flat space. So that's tomorrow. And then part three is the discussion of localization applied to this problem and the computation of exact quantum entropy in this class of black holes. And part four, so this has a whole bunch of stuff. So maybe a bit of this will already be tomorrow and some of this will be on Friday. And then part four is, so loose ends, but actually I replaced this. I call this advanced topics because this looks very casual. But based on a little bit of the kind of questions that have arisen here about one loop determinants and so on. So I just want to summarize our status of knowledge about that. I'll just try to go through this in some detail. But applying these two supergravities is actually, there are some serious issues there about which there's been some progress. Some of it you'll hear in the workshop by Intox of Bernaduid and also of Imtak Jeon, who's here, who's not here, I's here. And hopefully I'll come and at least give you an introduction to the topic so that you can then really take advantage of those talks. Okay, so today will be just essentially an introduction. Okay, very good. So, of course, please feel free to interrupt with questions. And also if this is not visible, just pick up. I just want to make sure that, so is this sort of readable in the last row? Okay, yeah. No, when I was sitting there, not all the lectures were readable, so just, and no one was picking up. So please tell me if I write something illegible or if I say something, which you don't understand. Okay, so let's start with some very basic notions. So start with black hole, so ideally I should be drawing some kind of space-time diagram, pendulum diagram, but it actually helps a lot to keep this very simple minded picture that there's a black hole with some even horizon, okay, with some radius r. Maybe I'll sometimes call this h. Okay, and the story starts with the fact that black holes have thermodynamics. So the first statement is that of Bekenstein, so I'm not being completely chronological, but there are lots of works of Bekenstein and Hawking and collaborators, but I think logically this is a good way to say it. So the statement of Bekenstein was that black holes carry thermodynamic entropy, and the argument for this is very simple. I'm sure all of you know it, but just to put everybody on the same page. In fact, here is the paper of Bekenstein and he says, let's start by considering this very simple thought experiment.  Just throw something which carries energy and entropy and throw it into the black hole. So something like this, okay, you have a body which carries a lot of entropy. And then what you want to do is that it goes and falls inside the black hole. So actually Bekenstein had Mexican roots, so I'm sure he will be happy with this. Sorry for the resilience, I know there are a few of you. Okay, sorry. So I hope Bekenstein's spirit forgives me for this joke. But anyway, based on this, so you argue the following. You argue that there is a generalized second law, and the argument is very simple. It says that consider the second law of thermodynamics, okay, and you know that the total change in entropy of the world of the universe should increase. So if you think of this inside and outside of a black hole. So you've lost some entropy from the outside world. So this is entropy of out, okay, and this is negative, right? But this should be positive, and the only way in which this is even possible is if you assign an intrinsic entropy to the black hole itself, okay. So this should be positive, okay, and that's the argument that the black hole must have entropy. So very simple minded argument, but a very profound one, and in fact you can sort of carry this further to say that a black hole must also be the thing that packs entropy in the most efficient way, because whatever you throw in, the total entropy has to increase, okay. So now the second statement is that, well, if black holes have entropy, then they must also, sorry, if the black hole has entropy, they must also have thermodynamics and temperature, and this was a hard calculation, a beautiful one by Hawking.  Then, so the horizon radius here is 2 gm, this R, and Hawking's beautiful calculation showed that the temperature of the black hole is h bar over 2 pi times the Boltzmann constant times the surface gravity. So this is gm over R square, and if you plug that in, pi and this is equal to h bar over 4 pi kb times 1 over R, okay. So that's the temperature. So now use the first law of thermodynamics, and it says that the infinitesimal change in the entropy is dm over t. Just plug that in, and you'll get kb over h bar times g times 2 pi R dr. So you can integrate this equation now to get famous formula, which is kb over g h bar c cubed. I've reinserted a c times area of the horizon divided by 4, and this is what we call area of the horizon divided by 4 in planke units, okay. So that's the famous area law of bacon, stun, and Hawking, all right. So it's a very beautiful equation with all the constants, most of all the fundamental constants of nature appearing. Now in any theory of quantum gravity we expect that we are the Boltzmann equation. This thermodynamic entropy should be equal to Boltzmann constant times the logarithm of the number of microstates of the black hole, okay. And this should be true in the thermodynamic limit when things simplify. So these dots means that things depend on the details of the system, which just gets washed away in the thermodynamic limit. So when the black hole becomes very large, we're just left with this, okay. And for reasons that will become clear soon, I'm gonna call this the classical entropy. I should really be saying semi-class, but it's easier to write class, okay. Now, these little, yeah, S is what? Well, I've made a KB in both cases, so it's, sorry, KB. Thank you. There's a KB here. Any other questions? Yeah, so L-plank square is this, of course. Now, this law of bacon, stun, and Hawking is universal in that it applies to any black hole in generativity, and so that's a very nice thing. Universal laws are good, but it also means that it's, that's a limitation of, it's a self-limiting thing. If you want to understand what is the fine structure of gravity, what's the ultraviolet properties of gravity, all those things get washed away in this thermodynamic limit, okay. So if you want to understand the quantum structure of gravity, you actually need to understand these corrections, and that's what the notion of quantum entropy is about. So these corrections, you can think of either as finite size effects, okay, and by size, I mean, size is the area, the horizon in plankerings, okay. So that dimensionless object, I'm going to call size, when the black hole is large, this is correct, and now there are going to be corrections when the black hole, when the curvature is not so small, okay.  So that's what we're interested in. So this, you want to ask, what are these effects? So this whole thing, okay, I'm going to call.  And from now on, I'm going to put kb to 1, so you don't see this issue. So what I'm asking is really some macroscopic question. So in Boltzmann's equation, this is micro, and this is macro, okay. And I'm just saying that in a theory of quantum gravity, there must be an equation like this, and the question I'm asking actually is just about the gravitational theory, okay. So let's be a little bit more precise. So let's ask ourselves some questions. So one question is, what is the definition of this thing? So Baconstein and Hawking gave us the definition of the classical entropy. What is the definition of this quantum entropy? Okay, so it must be something defined in the gravitational theory, which reduces to Baconstein Hawking when the area is very large, and otherwise agrees with the logarithm of the number of microstates of the system, okay. Yeah, so once you have the, yeah, so once, they have entropy because they have area. That's not how I set up the argument, so you should think of extremal black holes as zero temperature limits of non-extremal black holes. And in the limiting procedure, if you ask what is, so think of the entropy of the black hole as limit t goes to zero of entropy of the black hole of this, you get an area, yeah. So, yeah. Okay, other questions? Yeah, yes. If it's a quantum effect, so shouldn't one anticipate so UV sensitivity at some higher order? Absolutely, so that's actually, that was my point, so you said it in a better way than I did. Thanks. What I'm saying is that if you, these things are not universal, they depend on the ultraviolet completion of gravity. In that sense, if you manage to compute this, actually get some information about the UV theory, okay. What I'm saying is you can actually try to, so there are just looking ahead a little bit in string theory, we have sort of two notions of what a black hole is. One is that of some effective theory and a solution and so on with higher derivative and quantum interactions and the other notion is some microscopic interactions. So, the idea is that, sorry, microscopic excitations of some ensemble. The idea is you want to use the first effective type description and see how far you can push that. And in the supersymmetric case, it turns out that you can go very far with these ideas. Okay, thank you. Other questions, yes. Yeah, so in fact, you'll see that they can be positive or negative. They're small compared to this one, but you'll see actually that in the one example where I know how to actually compute this all the way to the end, sometimes they're negative. It's because they're fermions. Yes. Yeah. Yes. Vox. Yes. Gravity in 10. Indeed. Or just the ones from the black hole. I understand. So, the answer is the latter and I'll answer this in five minutes. It's a very good question. I'm going to come to that very soon. Other questions. Okay, so one question is what's the definition? The question is can we compute in any sensible model? And another question which I'm not going to discuss today is compare to microscopics. So, in string theory, as we all know, to the work of Sen and Stromingenwaffe, we know that a black hole is really some, at least in the supersymmetric situation, is ensemble of some kind of microscopic states which you can count and you get some integer here. So, the question is, is the exponentials, if I can calculate this, is the exponential of this equal to that integer? How far can you go? What I'll be talking about in this set of lectures is just these two topics. Joan Gomesh, who is studying tomorrow, will talk about the microscopic side of the story. And in my third lecture, and maybe in his third lecture, we'll try to somehow, at least in the future. So, just keep an eye out for his lecture and try to connect with this, with this, what I'm saying. Okay, now, whatever this, let me make one more comment. Is that this kind of, whatever this quantum m2p is, just based on mathematical analysis, it starts like this. It just starts with this and then there'll be some logarithmic and power law suppressed, right? And maybe some exponential suppressed corrections. Again, I'm being schematic. Here, I'm sort of roughly assuming that a h is the only small, sorry, large, one over h is the only small parameter in the problem. Sometimes you have more parameters, so it's a multi-dimensional expansion, but roughly speaking, something like this is true. So, I'm just putting this out to sort of, so that you know what to expect towards the end. Okay, so these are the questions I want to answer. So, let me first begin by asking why at all are there these corrections, the origins of, what is the origin of quantum corrections? Well, one obvious answer is the following. You have, so write down the gravitational action. So, it starts like this, okay? At low energies is this coupled to other fields, this Einstein-Hilbert coupled to other fields, okay? But at higher energies, so whatever the theory of quantum gravity is, I think there's a universal agreement that it's not just this, okay? Whether, whatever philosophical bias you might have from an effective field theory point of view, you know that there are operators of this kind that are going to correct the two derivative low energy action, okay? L-plank square, R-square here, I'm schematic, this could be any four derivative term and so on, okay? Now, in the, in such, with such an effective action, and it's not just GR, the Bacon-Stein-Hulking law does not obey the laws of thermodynamics. I forgot to say this, but I think all of you know that this work of Bacon-Stein-Hulking and Carter and I'm missing one more name, Bardin, all of, they showed that in fact this entropy and that temperature obey the laws of thermodynamics. So, the first law I already used, but also that in any physical process, the second law is true and so on. Okay, so the first, even the first law is no longer true for such an action. Now, this problem has been solved with some assumptions and oops, I shouldn't erase that. Yeah, and there's something called the Wald formula developed by Wald and Ier and their collaborators, which says that given some, suppose this is a local effective action, okay? Then, there's a formula, which says that the black hole entropy is, so let's say this is spherically symmetric again, it's something like this.  There's a set of rules which goes here that you might have some functions of the Riemann tensor or there can be other covariant derivatives and so on. There's a set of rules that, when you have anti-symmetrized combinations, you have to make it symmetric and all the anti-symmetric guys, you have to write in terms of the Riemann, then you think of Riemann as an independent object and you just vary the action with respect to this and compute this formula. I'm not going to go in this direction, but this is an extremely well understood formula and it's completely explicit and this obeys the first law of thermodynamics. It was rigged up, so as to obey the first law of thermodynamics, okay, in given some local effective action. Now, this is not the direction I want to take because there is another problem which is in a sense as interesting and more interesting, namely that in the quantum theory, the correct action of the theory, the correct means one PI action which respects all the symmetries of the theory, need not always be local effective action because massless particles run in loops, okay. So suppose you have some loop of this kind, I'm not telling you what observable this is, but something like this, this is going to give you some contribution of this type where epsilon is some IR cutoff, right, and this one versus log epsilon, okay, and that's certainly non-local. Think of this as some box, we'll have log box or something like this. It's a very non-local effect and once formalism is not enough to take care of that, so you want some formulation in which you want to think of being able to compute such effects, okay. So that's what I'm going to turn to next. So this, so the breakthrough came to work by Ashok Sen in 2008 in the context of supersymmetric black holes. Before I go on one technical comment, through all these lectures sometimes I'll talk about classical and quantum entropy. So far I said classical is area over four. Sometimes people also call this wall entropy the classical entropy just because if there is a local effective Lagrangian which you can minimize, that's like a classical problem. It might have higher derivatives, okay, and you talk about these loop effects as the quantum effects. So it's a question of semantics, but if I might switch between the two, if it's unclear, please ask me. So look at supersymmetric black holes. So these are extrema zero temperature black holes and as was already anticipated, the way we should think about it is to think of a very low temperature black hole, compute all physical quantities and then take the zero temperature limit. That's how we'll think of extrema black holes and this avoids all kinds of problems which people have invented when you directly take extrema limits, not invented, found, but I think this is the right way to do it. Okay, now the fact is in the limit these black holes are isolated quantum systems and that's a very important thing philosophically and this comes to your question. Sorry, I don't know your name, but the question. So usually you have a black hole which is interacting, which is environment and to ask what is the number of states of the black hole? Suppose I want to be ultra powerful and say I want to compute the full integer number of states associated with black hole. For a real non-zero temperature black hole, this question is not even well defined because there's Hawking radiation which goes in and out.  It's only the full ensemble, the box as you said, which has a well defined entropy, but there you might have a canonical ensemble. So those are the kind of things which one has to compute. But for extrema super symmetric black holes they have zero temperature, they're completely quiet objects and they're isolated quantum systems. So there is a sense in asking what is the number of quantum states associated to that black hole and that's the question I want to answer, okay? So that's an important point. So let's then do this in order to do this. So the first idea is that the entropy of a black hole is somehow stored near or behind the horizon of the black hole. The horizon is the reason why there is an entropy. So the first step is to sort of go near horizon and before going to the quantum entropy I want to do an exercise which was done by various people, some of whom may be here, namely to express this classical entropy, this wall entropy for the case of super symmetric black holes as a quantity which you can purely compute in the near horizon region, okay? So let me do that and in fact, so there are two advantages. Firstly you'll see very soon, it's a very elegant formulation, it's a very simple formulation and secondly it allows us to go beyond and define quantum entropy, okay? So I'm going to take some kind of near horizon limit and in order to do this I'll just be very specific just for now. Let me talk about this Einstein Maxwell theory in four dimensions, 16 pi. Okay, so there is, in fact, Stefan even wrote the Reissner-Aston solution, I'm going to repeat it in slightly different variables.  So that's your Reissner-Aston solution and if you take the extremal limit what you'll find is that, first I should tell you more about this, finish the solution, so there are two parameters, rho plus and rho minus, this is not how Stefan presented it, I think, but it's completely equal and I should also tell you what the fields are, electric fields, gauge fields are, so there's an f rho tau, so these are charged black holes and there's an electric charge q here and in fact you can also take a black hole which is dionic, so let's take that, that's the most general thing, p sin psi, okay?  So that's the Reissner-Aston solution in all detail and the extremal limit is a limit in which the two horizons coales, rho plus, both of them go to a common value which I'm going to call rho horizon, rho h and that turns out to be equal to square root of g times q square plus p square over 4 pi and m equals 1 over 4 pi g times q square plus p square, okay? So that's the extremal limit so the mass is completely determined by the charges and the two horizons coales, yes? Yes, so I haven't yet done that, that's exactly what I'm going to do next, let me just do it and then you can ask a question if it remains, so now what I want to do is, so far I haven't talked about so I want to take a near horizon limit but I want to take a simultaneous limit so this is the comment I made earlier so what I want to do is to go to some take a scaling limit where I go near the horizon but always keep the horizon at a finite distance so if I do this naively then the horizon goes up to infinity but I want to do it in a way that I keep it at a finite distance so you do, take a simultaneous extremal plus near horizon limit as follows so you do so plus minus rho minus is 2 lambda and I'm going to take lambda to 0 at the same time it take rho minus rho horizon equals lambda times r I'm going to define a new variable called r okay and tau, so if I do this and look at the metric but tau also has to be rescaled in the other way otherwise there's no sensible metric tau over rho h square t over lambda okay and then I take lambda goes to 0 okay in doing so what I've effectively done let me finish the analysis so what I get is that the metric is v times r square minus 1 dt square plus dr square over r square minus 1 plus d omega 2 square where v is rho h square okay so this side just freezes to the horizon here so this is an s2 and this is two-dimensional anti-decider space okay but just to answer your question I think there was another question about this it's not your someone else's that I've done this in a way so that effectively this also looks like there are two horizons to it although it's all in the near horizon limit and that's important okay so what you get is some near horizon limit to be ads2 times s2 now this fact ah sorry and what happens to the gauge field so f equals let me write it as e over 4 pi times dr vh dt plus p over 4 pi sin psi d psi vh d phi okay so what I get is s2 times s2 and I get constant electric and magnetic field okay so for those of you who have never seen it I recommend this exercise I've spelled out everything it'll take you less than 10 minutes but it's nice to do it where e equals where e equals q so it's just q over 4 pi and p over 4 pi so that's what you get in this resume question tau has disappeared from the story yeah thanks okay now this fact so it's very easy to do in this metric for this theory in fact this is much more general so let's stick to 4 dimensions and give ourselves an arbitrary theory of the metric coupled to some bunch of h fields and a bunch of scalar fields and maybe fermions this is the kind of theory we've been seeing in the last few days okay so suppose you have so this is very general so this is general suppose you give yourself s which is before x the square root of minus g times some Lagrangian of metric some bunch of scalar fields a mu i and some bunch of sorry bunch of gauge fields bunch of scalar fields phi a and maybe fermions this is still true you always get in the near horizon limit in ads 2 times s2 in this theory in asymptotically flat spaces in a second the changes are that let me take the question because I am going to take 2 minutes to explain the Reissner Nostrom this is just ads 2 times s2 this is the near horizon geometry of the black hole you've really gone to the horizon and taken that patch near the horizon and that's what this is I mean in the pendulum starting the horizon looks like that you really have some very small patch next to it that's this okay so what happens is that this is much more general so given any such Lagrangian you write a Reissner Nostrom type solution so there's some electric and magnetic charges now the near horizon region of the supersymmetric or extremal black hole always has ads 2 times s2 symmetry so this means there's an sl2 associated with this and su2 associated with this and if I can just write down directly what the near horizon geometry should be so this always looks like ads 2 times s2 with some value of v the field strengths become constant because those are the only things allowed by the symmetries and now where should I write it so comma I'll go on I wanted to fit it here comma so phi a is also constant and all the fermions are 0 okay so these are the this is the most general configuration consistent with the symmetries and this relation that the electric field was equal to the charge for one gauge field now gets changed that e ai is a function of all the qis okay yes so the near horizon is a very thin shell that's right so this is still true this is true now v is some v constant f i becomes e i plus p i e i is some function of the electric fields scalars are constant fermions are 0 at the same time yes yes that's correct that's correct okay and now in this limit sorry in this near horizon configuration this Lagrangian effectively becomes just a function in these fields now it just becomes a function of of course q1p qi p i semicolon now it's just the parameters v e i phi a I hope I didn't miss anything sorry is there a question yes that's the solution that's the solution in its own right so maybe I said this too you said it's the near horizon limit of a solution so yeah that's the first statement but then it's a solution in its own right meaning that that is a solution now so I showed you this sorry so I should have added one sentence so in the case of just Maxwell Einstein you start with Reissner-Norsturm take the limit you get this thing I should have added a sentence saying that itself solves the equation of motion and this fact is much more general I'm saying that there's a very easy way to do it you just assume the symmetries of the near horizon sl2 times su2 and that's the most general configuration I can write down consistent with the symmetries and the Lagrange I still haven't solved equation of motion just give me a second but there is you'll see that this is a solution the Lagrangian now becomes a function of these fields of these numbers of these real variables so what was a complicated what was gr now it just becomes multivariable calculus ok and now in fact this is very general it's not just general but very general in the sense that it's also true if you don't have four dimensions so what then changes is this part of course if you don't have four dimensions if you have five dimensions there will be something else so this correct statement is now you always have ads2 times a compact manifold and it's so you can be any dimension any d and any theory so including the type of theory that Stefan talked about in his second, sorry, in his third and fourth lectures need not have asymptotically flat space kind of thing that Alberto was talking about you can have ads2 times some Riemann surface you can have s3 whatever so the statement is that as long as there is a consistent limit to gr the near horizon region always has an ads2 factor ads2 times a compact factor constant electric fields and these magnetic fields now become some scalar fields think of it as a two-dimensional theory and constant scalar fields so what we're going to do is to think of this as some two-dimensional theory on ads2 just compactify the compact part think about it as a two-dimensional theory so yes so this is thus far a very classical statement about equation of model I'll finish the classical thing here quantum so it could be what, quantum what at the quantum point of view this could be a troncation that is not so I'll discuss that once I start here just give me 5 minutes yes so the charges are just given to you the parameters that vary are the size v yeah I'll come to that so I'm saying that the most general so maybe I should have done the whole thing very quick the most general most general configuration consistent with the near horizon symmetries is this where v is arbitrary e i are arbitrary and in this case p is and phi i are arbitrary but now you have to impose equations of motion and here's the statement so firstly I wanted to say that it's not just four-dimensional with spherical symmetry so it's even more general than that so in that case this L you should think of some two-dimensional Lagrangian which comes from compactifying this higher-dimensional Lagrangian in our case is just equal to v sin theta sorry sin psi d phi times L of the four dimensions and this is just 4 pi v L and S is just another integration of that and you get a v of this two-dimensional thing just look at the metric there's a v in front of the radius 2 so the square root of g of this two-dimensional part is also v and this part is also v and that explains all these factors of v that I've been pulling out I just do the integration over the two-dimensions I do further integration over the radius 2 I get this and now so is that clear was that too quick there ok so now comes the statement the statement is that statement is that the there's something called a classical entropy function which are defined as follows e is function of q i pi v so those are fixed v e i phi a which is defined to be 2 pi times e i times q i minus v times l2 it's just a Legendre transform of the action of the theory with respect to the electric fields ok and the two statements are then one implies it's just exactly the same as a variational principle delta e is equal to 0 and I said this is just multivariable calculus so that means what I mean is d e d v is 0 d e d e phi a is equal to 0 and d e d e i is equal to 0 so the statement is that if you just take your Lagrangian think of Einstein Hilbert plug in this ansatz what I'm saying is that the near horizon symmetries have completely fixed the tensor structures completely so the only things that are left to vary are the sizes so the shapes have been fixed by either supersymmetry or extremality as you want to think of the only things that are left are the sizes but if you take the motions of motion then just become this extremization problem ok so this is just extremization ok and notice in particular that this is the same oops sorry this one is the same as saying that q i is d v l 2 d e i which is the same as Gauss now and part 2 of the statement is that the classical entropy of the black hole is the value of e at this extreme ok so I'm not going to derive this for you you can think of so please take this as an exercise they're both sort of medium level exercises if you want to do a nice review paper by Ashok Sen in 2007 you can do this so Jaum Reis sitting there he's also worked on it you can bug him but anyway this is not a very difficult problem in my opinion ok so that's the classical entropy statement ok any other questions about this did I answer all questions or did I postpone something which I still haven't answered Leopoldo did you it's clear now quantum quantum but I'm going to come to that in a second so for the classical it's just you can prove it as I said all tensor structures are fixed and there's nothing else you can do it as a 2 dimensional problem or a 4 dimensional problem it doesn't matter because of the symmetry I mean here I'm assuming there's a lot of symmetry if not suppose this thing is some Riemann surface like Albert then you'll have more complicated problem ok so in this case let me be careful and say this ok the main point is this equation implies that that's always true and the entropy is equal to that under a large more or less the other way is also true but let me not commit right here we can discuss this later so before writing here before general and very general I said let's stick to 4 dimensional or simplatically flat space this statement is true even for when I break such assumptions so the kind of theories that Alberto was discussing Stefan was discussing all of that is true now of course I'm assuming supersymmetry so what I've assumed here is extremality actually so that was my assumption in looking at rational nostrum but that was because I'm interested in supersymmetric black holes which then become extremal so the technical assumption so far is extremality yes no so that's what I said so the first example I did was spherically symmetric then it remains spherically symmetric but the Lagrangian changes now I'm saying even that's not true any theory arbitrary alright of course the details will change if you have some other symmetry this is not a sphere it will be something ok now let's come to the quantum entropy no so far it's completely general what I said was completely general sorry I want to stress this try to say this I really mean any sphere so the only assumption I've made is extremality or soon I'll say supersymmetry but so far it's just extremality but now I'm going to go towards something more specific and then maybe at the end tell you about the generalities sorry who's the chair can I take five more minutes because I have lots of questions already I mean I have five minutes and that's great please continue like this but it's just that I want to do one little thing before I finish so I've restated so if you want that you can think of that as a definition of an extremal daco that in its near horizon that fact is true for you can prove that it's true for again an arbitrary theory of gravity coupled to matter any extremal black hole as long as it has a limit to Einstein's theory in the sense that the corrections that I put to Einstein's theory must be small if I within that set of assumptions you can prove this I forgot who did it it was kunduri, luciati, bunch of these people but otherwise for our purposes you can just take that to be the definition that's correct what if I add a total derivative total derivative no I think I've not made any assumptions of that sort so I think it's because these are constants I think any Lagrangian is okay can we just postpone this for a bit I think that's correct but let me just think it through okay so what I want to do now in the last x minutes is define the notion of quantum entropy so the classical entropy problem has been recast as an extremization problem in the near horizon geometry so since idea was that the quantum entropy for super symmetric black holes should be a functional integral so the classical problem is an extremization as was said there's an on shell problem so now you make it off integrate over all the fields so that's the sort of moral idea so extremization now is promoted to a path integral so what I want to tell you next is how to do it in the next 50 minutes so the first thing I want to do is to go Euclidean because otherwise the circle is not well defined so in order to do that I'm going to take t to i theta so what I'll have here is now the classical problem will become plus become theta square and here I'll get a minus i times EI so that's the only effect on these fields alright and now what is the idea so that's Euclidean ideas 2 so that looks like a disk it's a hyperbolic disk with some asymptotic boundary the idea is that so in the classical problem everything is constant as we just saw in the quantum problem I want to fix the boundary conditions according to the classical problem and then in the interior I want to let all the fields fluctuate integrate over these fields so I need some falloff conditions so as r goes to infinity I want to say that g mu nu is that classical one which is on the other blackboard times 1 plus o of 1 over r square a mu is a mu i classical which is there times 1 plus o of 1 over r phi a is phi a classical which is constant times 1 plus o of 1 over r now people have analyzed this that this is a sort of consistent boundary conditions so another way to say this is that the ds square in the ads 2 part will now become like this v times r square plus some o of 1 times d theta square plus dr square times 1 over r square plus o of 1 over r to the fourth that's the kind of fluctuation I am going to take and a i is minus i e i times r plus o of 1 o of 1 and phi a is constant phi a plus 1 over r constant plus 1 over r and now given this conditions I am going to define quantum entropy so now I think I can erase this more time define the quantum entropy so e to the s quantum entropy for black hole is the path integral z over ads 2 is a function of q i and p i which is defined as the expectation value in the quantum theory of the exponential of minus i q i times wilson line of all the gauge fields under which is charged and this wilson line is placed at the boundary of ads 2 as r goes to infinity and this is with asymptotic ads 2 boundary conditions of the type that I just said and there is a certain renormalization that you have to do which is what I call finite so that's the definition and I want to spend at least explaining how one should think of this path integral then next time I will pick up with that so this is euclidean ads 2 I am going to put a cutoff and r0 so there are many many issues with this with this path so the way what is the definition you think of the two dimensional theory this includes gravity and all the fields that you have all the whatever fields are in the theory and then there is this wilson line according to all the electric charges of the black hole the magnetic charges in two dimensional description becomes some scalar fields I give myself these boundary conditions and integrate over all fluctuations what are the issues the issues is well firstly how should I think of this integral I really think of it as an integral d over all the fields e to the minus exponential times this many issues with this one issue is that this action itself is divergent because the volume of ads 2 is divergent so this is an infrared problem from the point of view of gravity so let's first tackle that so think of some bulk action which is what our Lagrangian was so you have dr0 from some interior to sorry dr from something to r0 and d theta square root of g times l bulk now remember that the Lagrangian evaluated on the classical field started with a constant we already saw that and therefore this thing will start as as something like that so constant times the volume the volume is r0 it diverges as r0 so it's going to start like this then there's going to be some finite piece and then there's going to be some 1 over r piece ok this is going to be my definition of the finite part so in general in ads cft or in ads when you think of ads quantum gravity as we heard in Janis' lectures there's a whole procedure of holographic renormalization and that's what you should really apply here it actually just becomes very very simple and I'll make some comments about this because the only divergence is this linear divergence which you can essentially subtract I'll make some more comments about this but for now I can just take this as the definition just take this c1 to be the finite piece just one second now similarly for the Wilson line so think of now the Wilson line lives on the boundary the proper length of the boundary if you just integrate the length you'll get v square root of v times 2pi times r0 and therefore the Wilson line eI qI times integral ai also goes similar behavior there is some r0 piece plus some c1 prime piece plus order of 1 over r and again I'm going to take this and a better way to say this so I'm going to define something called renormalized action which is s bulk plus sorry minus i qI integral ai plus some boundary term boundary counter term that I have to add and you choose it so as to cancel this leading order divergence and I'll make comments about this after I finish and now with this definition z ads 2 I should really think of as really a path integral d of all the fields phi gravitational which this metric et cetera et cetera is s renormalized of these phi vector and now I'm going to say one more thing and then stop so suppose you want to take the classical limit that's the first thing to check so when the black hole is large that implies v goes to infinity remember v was the size of the horizon and if v goes to infinity remember l the Lagrangian started its life by with a term which is proportional to v times l2 in what I wrote therefore the action grows therefore you can do a saddle point approximation okay that's just saying that in the classical limit you have a saddle point because v there's a 1 over h bar in front of v and in that case the s bulk let's be more careful so r remember sorry r went from 1 to r0 d theta went from 0 to pi v times l2 so this is just 2 pi this is a constant in the classical limit r0 minus 1 times v l2 and minus i qi integral ai is just minus 2 pi times qi ei times r0 minus 1 and that implies that normalized is just 2 pi times this is the finite part so it's this one and that is this one I just drop the r just get qi ei minus v l2 and that's precisely what the classical entropy was so this is not a surprise of course I've started with the classical thing and went up to the fluctuation problem just to make sure that everything is consistent so if you do the saddle point evaluation of that so in this limit you get s e to the s quantum just reduces to e to the s classical so this is just s classical so I think I'll stop there next time what I'm going to do is to apply this to this n equal to 2 black holes and we'll take it from there thank you very much sorry I already postponed some questions