WARNING FROM THE STARS This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Reading by Rowdy Delaney, Idaho, USA. WARNING FROM THE STARS By Ron Cocking It was a beautifully machined container, shaped like a two-pound chocolate candy box, the color and texture of lead. The cover fitted so accurately that it was difficult to see where it met the lip on the base. Yet when Forrester lifted the container from the desk in the security guard's office, he almost hit himself in the face with it, so light was it. He read the words, clumsily etched by hand into the top surface with some sharp instrument. To be opened only by Dr. Richard Forrester, Assistant Director, Air Force Special Research Center, Petersport, Maryland. CAUTION! Open not later than 24 hours after receipt. Do not open in atmosphere less than equivalent of 65,000 feet above MSL. He turned the container over and over. It bore no other markings, no express label or stamps, no file or reference number, no return address. It was superbly machined, he saw. Tentatively he pulled at the container cover. It was as firm as if it had been welded on, but then, if the cover had been closed in the thin atmosphere of 65,000 feet, it would be held on by the terrific pressure of a column of air twelve miles high. Forrester looked up at the burly guard. Who left this here? Your guess is as good as mine, sir. The man's voice was as close to insolence as the difference in status would allow, and Forrester bristled. I just clocked in an hour ago. There was a thick fog came on all of a sudden, and there was a bit of confusion when we were changing over. They didn't say anything about the box when I relieved. Fog? Forrester queried? How could fog form on a warm morning like this? You're the scientist, sir. You tell me. Went as fast as it came. Well, it looks like very sloppy security. The contents of this thing must almost certainly be classified. Give me the book, and I'll sign for it. I'll phone you the file number when I find the covering instructions. Forrester was a nervous, over-conscientious little man, and his day was already ruined, because any departure from strict administrative routine worried and upset him. Only in his field of aviation medicine did he feel competent, secure. He knew that around the center they contemptuously called him Liliput. The younger researchers were constantly trying to think up new ways to play jokes on him and annoy him. Crowley Preston, the research center's director, and his chief had been summoned to Washington the night before. Forrester wished fervently that he was around to deal with this matter. Now that relations between east and west had reached the snapping point, the slightest deviation from security regulations usually meant a full-scale inquiry. He signed for the container and carried it out to the car, still seething impotently over the guard's insolence. He placed it beside him on the front seat of his car and drove up to the building which housed part of the labs and also his office. He climbed out, then as he slammed the door he happened to glance into the car again. The seat covers were made of plastic in a maroon and blue plaid pattern, but where the box had rested there was a dirty gray rectangular patch that hadn't been there before. Forrester stared, then opened the door again. He rubbed his fingers over the discolored spot. It felt no different from the rest of the fabric. He placed the box over the area. It fitted perfectly. He flopped down on the seat, his legs dangling out of the car, fighting down a sudden irrational wave of panic. He pushed the container to the other end of the seat. After all he rationalized, plastics are notoriously unstable under certain conditions. This is probably a new alloy Washington once tested for behavior under extreme conditions of temperature and pressure. What's gotten into you? He took a deep breath, picked up the box again. Where it had rested there was another discolored patch on the car's seat covers. Holding it away from him, Forrester hurried into the office, then dumped the box into a metal wastebasket. Then he went to a cabinet and pulled out a Geiger counter, carried it over to the wastebasket. As he pointed the probe at the box the familiar slow clicking reassured him, and feeling a little foolish he put the instrument back on its shelf. Hurriedly he went through his mail. There was nothing in it referring to the package. Then he called the classified filing section. Nobody there knew anything about it either. For some reason he couldn't explain to himself. He wasn't even surprised. He stared into the wastebasket. The clumsily etched instructions glinted up at him, to be opened as soon as possible. He picked up the phone and called the decompression chamber building. There was no valid reason why he should have been self-conscious as he talked to the lab attendant in charge of the decompression tank. He used it a dozen times a month for tests and experiments, yet when he gave his instructions his voice was labored and strained. Some genius in Washington sent this thing down without any covering instructions, but it has to be opened in a hurry in a thin atmosphere. Er, I'd like to stay on the intercom for a while in case it blows up in my face or something. He tried to laugh, but all that came out was a croak. The attendant nodded indifferently, then helped Forrester into the helmet of his pressure suit. He climbed up the steps into the chamber, pulling the airtight door shut behind him. He placed the box on the desk in front of the instrument panel, then turned back to push the door clamps into place. For the first time, in the hundreds of hours he had spent in this tank, he knew the meaning of claustrophobia. Mechanically he plugged in his intercom and airlines, went through the other routine checks before ascent, tested communications with the lab attendant, then flicked the exhaust motor switch. Now there was nothing to do except wait. He stared at the box. In artificial light it seemed full of hidden menace, a knowing aliveness of its own. Forrester shrugged his shoulders impatiently as though to throw off the vague blanket of uneasiness that was settling around him. Somebody had forgotten to send a cover letter with the container, or it had been mislaid. That could happen, although with security routine, as strict as it was, the possibility was remote. All the same it could happen. After all, what other explanation was there? What was it he was afraid of? There was something about it. He glanced at the altimeter. The needle showed only ten thousand feet and seemed to be crawling around the dial. He resolved not to look at it for three minutes by the clock on the panel. When he checked the altimeter again, it registered just over thirty thousand feet, not even half way yet. As the pressure in the tank decreased, he began to be conscious of the need for reverse breathing, and he concentrated on using his tongue to check the flow of air into his lungs. Then, using the thoracic muscles to exhale against the higher pressure inside the suit, time seemed to be passing in microseconds. Twenty five thousand feet, thirty thousand, forty thousand, fifty thousand. At sixty two thousand five hundred feet, he gently tested the cover of the container again. It lifted. As the altimeter needle flickered on the sixty five thousand foot mark, he cut the exhaust motor and picked up the box. The cover slipped off easily. His feeling of anti-climax was almost ludicrous. As he looked in, all the box contained was a flattened roll of some grayish material. He took it out. Despite its comparative bulk, it was feather-light. It had the appearance of metal, but was as porous and pliable as a good grade of bond paper. He could not feel its texture through his heavy gloves. He took a good look. It was new all right. No doubt Washington wanted some tests run on it, although without covering instructions and data this trip was wasted. But some heads would roll when he reported back on the way the container had been shipped in. He started to unroll the material to get a better look at it. Then he saw that it was covered with cramped, closely spaced handwriting in a purplish ink, handwriting that was elusively familiar. Then he read the words written in neat capitals at the top, the name of the man with the familiar handwriting, and fear came back, clamped cold fingers around his throat. James Rodden Bentley. Dear Dick, the writing went on.  I know this is going to sound like a voice from the dead, but I'm very much alive and kicking. In the best of health, in fact. The writing blurred, and instinctively Forrester put his fist up to rub his eyes, only to meet the hard plastic of his helmet visor, James Rodden Bentley. It was January 18, 1951, three years ago, and the jagged line of the Australian coast stretched like a small scale map to the black curve of the horizon. From the converted bomber that was his flying lab, Forrester could see the other American observation plane cruising on a parallel course about a half mile away, and beyond it, downwind, the fringe of the billowing cloud dome of the super-secret British thermonuclear shot. Then suddenly Bentley's voice from the other plane was crackling over the earphones, sharp and urgent. Our geigers and scintillometers are going crazy. We're getting out of here. There's something coming inside—a light! Silence. Forrester had watched in helpless horror as the other ship dipped a silver wing. Then, nose down ever so slowly, it seemed, down, down, in a dive that seemed to take hours as Forrester's plane tracked it, ending in a tiny splash like a pebble being thrown into a pond. Then the grimly beautiful iridescence of oil and gasoline spreading across the glassy waters of the Timor Sea. No parachutes had opened on the long journey down. An Australian air-sea rescue launch and helicopter were at the scene of the crash in minutes, but neither bodies nor survivors had been found then or later. Everything OK, Dr. Forrester? Yes, he said hoarsely. Yes, everything's OK. Just routine. Forrester focused his eyes on the writing again. There was no doubt at all that it was Bentley's. They had roomed and studied together for four years at MIT. And then there had been a couple of years post-graduate work after that. During all that time they used each other's notes constantly. But Bentley was dead. Forrester read on. Stunned. First you'll want to know what happened over the Timor Sea after the shot. Put very simply, I, the rest of the technicians, and the crew of the B-29, were trans-shipped to another vehicle, without any damage to ourselves. Now I'm not allowed to explain at this stage. Actually, they only wanted me, but it wasn't feasible to collect me and leave the rest behind, so they're all here, safe and well. Who are they and where am I? The second question I can't answer, not allowed to. They, roughly translated, are the shiny ones, which doesn't tell you anything, of course. Briefly, there are a couple of light-years ahead of Earth in evolution, mentally, morally, and physically, although I use the last word loosely.  It's so hard to discuss really abstract ideas. Why am I here? The whole reason for this message is wrapped up in the answer to that. As you probably know, Project Longfall, which I was heading up, was delayed about a year due to my removal. That was the sole purpose, although I and the rest of us are getting special instruction to keep us occupied. At about the same time, they also took several other key people from Britain, Russia, and the United States. Others were already here. The idea then was delay. To delay more test shots of atomic weapons, in the hope that East and West would come to some agreement. Now, because of the growing volume of tests and the critical tension which prevails, delay will no longer suffice, and far more drastic steps are to be taken. I wish you could be here for only a few minutes to see what happens after a multi-megaton thermonuclear test shot is set off on Earth. I can't describe it in any terms which would have any relation to your present knowledge of physics. All I can say is that life here is intimately bound up with the higher laws of electromagnetism which at present are only being guessed at on your level. It's not the radioactivity which you know as such which causes the trouble. There are neutralizing devices throughout the planetary system to take care of that. The damage is caused by an ultra-ultra-shortwave radiation which not even the most sensitive centelometer you have can pick up, a very subtle byproduct of every chain reaction. It doesn't have too much immediate effect on the lower forms of life, including human beings, if you'll pardon the expression, but here it causes a ghastly carnage, so ghastly it sickens me even to think about it for a second. The incredible thing is that the people here could stop Earth from firing another shot if they wished to and at 24 hours' notice, but their philosophy is totally opposed to force, even when it means their own destruction. That will give you an idea of the kind of people they are. Here they say that Einstein was on the fringe of discovering the theory involved when he died, but was having trouble with the mathematics. Remember how Einstein always complained that he was a really poor mathematician? But with atomic warfare threatening to break out on Earth at any minute, they've got to do something. This is what they plan to do. This is what they are going to do. Starting within a few hours after you receive this message, a mass removal of key scientists will begin. They will take 20, 30, or 40, roughly equal numbers from both sides, every few hours as technical conditions allow. That will go on until East and West agree to drop this whole mad weapons race. It will be done quietly, peacefully. Nobody will be hurt except by a fluke. But if needs be, they will lift every major scientific brain off the face of Earth to stop the present drift to disaster for everybody. There are no weapons, no devices that you have at present which can stop this plan going into effect. There it is. It's as simple as that. If you knew what you were really headed for, it would need no steps from here to make both sides on Earth stop this horrible foolishness in a moment. The lesson of Mars is part of the orientation course here. I'm not on Mars. I'm using up space, so I'll go into note form for a bit. Martians had an atomic war. Forgot they had to breathe. Destroyed 60% of their atmosphere. Canals on Mars aren't. They're closely spaced lines of shafts leading to underground cities. View from Earth telescopes, shaft mouths appear as dots which run together into lines due to eye fatigue. British Royal Astronomical Society figured that out 30 years ago at least. See papers on their proceedings. Photographs here show monsters created by wholesale mutations. Lasted about four generations before reproduction failed. Now only vegetation on Mars. Saw pictures of last survivors. Horrible. I was ill for days after. Imagine having to take 40 separate breaths after making a single step. Getting back to the others. A regular U.N. Remember O'Connor and Walters in our class? They're here. Check. You'll find that O'Connor is detached from Oak Ridge and Walters from Acons for special duty. That's Central Intelligence's story for their disappearance. Remember those top German boys the Russians were supposed to have gotten to before the Allies could reach them after the Nazi collapse? They're here too. And Komolnikov and Pritchkin of the Russian Academy. Believe me, the style and the writing was a little less urgent again now. I've had all the intellectual stuffing knocked out of me here. We all have had, for that matter. We're supposed to be the cream of the crop, but imagine sitting down to instruction from people whose IQs start where yours leaves off. But what has really made most of us here feel pretty humble is the way they have demolished Earth's so-called scientific method and used the method itself to prove that it doesn't stand up. You know how we've always been taught to observe, collect data, then erect a theory to fit the data, a theory which has to be modified when other data come along which don't fit into it. Here they work the opposite way. They say, know the fundamental principles governing the operation of the universe, and then all the pieces fit together inside this final truth. I understand now why so many of the Oak Ridge boys turned to religion after they had been exposed to the electron microscope for a while. They realized they had gone as far as their scientific training would ever take them. Time and space are running out. I know all this must sound confused and incredible, Dick. I'm still confused myself, but I want you to think about what I've written, then take the action you think best. I know it won't be easy for you. If you think this is some maniac's idea of a joke, you'll have proof very soon that it isn't, because one of the people at your center is due to leave for here any time now. You're wondering why I used this weird and wonderful means of communication. The problem was to find a writing material which would stand up in Earth's atmosphere, oddly enough. It's not the oxygen which causes the trouble, but the so-called inert nitrogen. This container will probably not disintegrate for a couple of days at sea level atmospheric pressure, but this material I'm writing on would not last more than a few seconds. That's one reason they picked you. Most people just don't have a spare decompression chamber up in the attic. The other reason was that with your photographic memory, you'll know this is my handwriting, beyond a shadow of a doubt, I hope. I'm sure you've sat in that pressure suit long enough, but remember, if you want to take another look at this, you'll have to put it back in the container before you go down, wishing you all you would wish for yourself. Jim. Forster examined the signature. That was the way Bentley made the capital J. It looked almost like a T, with just a faint hook at the bottom of the downstroke. Then the way it joined the—hey, Doc, are you going to tie up the tank all day? I've got work to do. Forster recognized the voice on the intercom as Tom Summerford's. Summerford was one of the crop of recent graduates to join the center. Brash, noisy, irresponsible like the rest of them. He knew Forster hated being called Doc, so he never lost an opportunity to use the word. True, he was gifted and well-trained, but he was the ringleader in playing the practical jokes on Forster, which might have been funny in college, but which only wasted a research team's time in these critical days. Practical joke. Anger flooded over him. Yes, this was all a macabre game, cooked up by Summerford, with the help of some of his pals. Probably they were all out there now, snickering among themselves, waiting to see his face when he came out of the decompression chamber, waiting to gloat. Hey, Doc, you still with us? I'll be out very shortly," Forster said grimly. Just wait right there. He spun the air inlet controls. Impatiently, he watched the altimeter needle begin its anti-clockwise movement. He'd call a staff meeting right away, find the culprits, and suspend them from duty. Preston would have to back him up. If Summerford proved to be the ringleader, he would insist on his dismissal, Forster decided, and he would see to it that the young punk had trouble getting another post. The fantastic waste of time involved in such an elaborate forgery. Forster trembled with indignation, and using the name of a dead man, above all a scientist who had died in the interest of research, leaving behind him a mystery which still troubled the Atomic Energy Commission, because nobody had ever been able to explain that sudden dive in a plane which was apparently functioning perfectly and flown by a veteran crew. He glanced down at the roll. Was it his imagination, or had the purplish ink begun to fade? He ran a length of it through his fingers, and then he saw that in places there were gaps where the writing had disappeared altogether. He glanced up at the altimeter needle, which was sliding by the twenty-four thousand foot mark. He looked back at his hands again, just in time to see the roll part in two places, leaving only the narrow strip he held between his gloved fingers. He put the strip on the desk, and bent clumsily in his suit to retrieve the other pieces from the floor. But wherever he grabbed it, it fell apart. Now it seemed to be melting before his eyes. In a few seconds there was nothing. He straightened up. The strip he had left on the desk had disappeared too. No ash, no residue, nothing. His thought processes seemed to freeze. He glanced mechanically at the altimeter. It read twenty-five hundred feet. He grabbed at the two pieces of the container. They still felt as rigid as ever. He fitted them together carefully, gaining a crumb of security from the act. He realized vaguely that the altimeter needle was resting on zero, but he had no idea how long he had been sitting there, trying to find a thread of logic in the confused welter of thoughts when he heard the scrape of metal on metal as somebody wrestled with the door clamps from the outside. He was certain of only one thing. His memory told him that the signature that was no longer a signature had been written by Jim Rodden, who couldn't possibly have survived that crash into the Timor Sea. From behind somebody was fumbling with his helmet connections. Then fresh air and familiar sounds rushed in on him as the helmet was taken away.  Doc, are you all right? He was asking sharply. For once there was no superciliousness in his voice. I'm fine, Forester said heavily. I—I've got a headache. Stayed in here too long, I suppose. What's in the box, Summerfield asked? The way he asked told Forester at once that the youngster knew nothing about it. Just some half-baked idea out of the Pentagon, some Colonel trying to justify his existence. He clutched the box to him as though Summerfield might try and take it away. The tank's all yours. He turned and clambered out of the chamber. He put the box down on the concrete floor and climbed out of the pressure suit, watching the box all the time. It seemed to gleam up at him as though it had eyes full of silent menace.  Are you quite sure you're OK? I'm fine, Forester said, hardly recognizing his own voice. He picked up the box and stumbled out, heading for his office. When he walked in, his secretary was answering the line fitted with a scrambler which connected directly with the Pentagon. General Morgensen, she said, handing him the receiver. Forester took the receiver, sat down at his desk and took a deep breath, fighting hard to regain his self-control. Forester, he said into the mouthpiece when the office door closed behind the girl. Forester, what in the dickens happened to Preston? My driver met the train here this morning, but there was no sign of him. But the Pullman porter checked him in last night, and we found all his gear and papers in his compartment. He left here in plenty of time to catch the train, General, Forester heard himself say. He took the train to get a night's rest. He realized how irrelevant the last statement was only after he made it. The General was talking again, important meeting with the Joint Chiefs. Whole Briefing Team was being held up. He reported it to the CIA as a precautionary measure. Forester could see the words on the roll, the roll that wasn't, as though they were engraved on his eye retinas. As a beginning, and to prove this isn't just a bit of hocus pocus, one of the people at your center is due to leave for here any time now. General, Forester broke in hoarsely. I've got some very important information which you must have. I'll leave by heliplane right away. He replaced the phone receiver in its cradle, wondering how convincing he would be able to make his story. At least, even if he didn't have Bentley's letter, he had the container. That should help. But when he looked across the desk, he saw that that too had disappeared without a trace. General Morgensen was the newest product of the atomic age, half soldier, half scientist, shrewd and perceptive, an intellectual giant. He listened carefully, without comment or change of expression, as Forester doggedly went through his story in chronological order. Halfway through, he held up his hand and started pushing buttons on the console built into his desk. Within a few moments, men began filing into the room and sat down around Forester. Then the general motioned to the clerk seated in the corner by the tape recorder. Gentlemen, listen to this playback and then I'll have Dr. Forester go on from there. What was left of Forester's confidence leaked away as he heard his own diffident voice filling the room again. It was like being awake in the middle of a weird dream. But when the tape recorder hissed into silence, he went on, staring straight ahead of him in quiet desperation. When he ended his story, there was silence for a moment. Everyone sat motionless. Then Morgensen looked up and around. Well, gentlemen, Mr. Bates, CIA, first. This was no longer a story told by one man. It had become a problem, a situation to be evaluated objectively. Well, sir, the only part of the thing I can comment on at this point is the stuff about O'Connor and Walters. That checks. They both disappeared without a trace. It was treated as a maximum security situation, and we did give out the story that they had been assigned to special duty. He glanced briefly at Forester. Up until now we assumed that only the directors at Aiken and Oak Ridge knew the real situation, outside the Atomic Energy Commission and the CIA, of course. This represents a very serious leak, or his voice trailed away. Colonel Barfield, Intelligence The young Colonel tried to sound flippant, unsuccessfully. General, acting on the assumption that the story is true, it would answer about 200 question marks in our files, maybe more with further study. The CIA man cleared his throat and raised a finger. For everybody's information, he said, a preliminary field check shows that Dr. Preston's train was stopped for ten minutes by Fogg last night. The train's radar installation failed simultaneously. There wouldn't be anything odd about that, except the temperature at the time was about 65 degrees and the humidity was only 55%. Consider that, gentlemen. Theoretically, Fogg can't form under such conditions. Similar Fogg occurred on the occasions when O'Connor and Walters were reported missing. The Met people couldn't explain that, either. That's all. Morgensen sat straight up, as though he had suddenly made a decision. I don't think there's any value in further discussion at this point. You will all have transcripts of Dr. Forester's statement within a few minutes. According to that statement, we're due to lose a number of key men in the next few hours. I'll have Code 1 emergency precautions instituted at all research establishments, and I think the chairman of the Joint Chiefs should hear from me right away. Colonel Barfield, I'd like you to ask Colonel Malinowski, the Russian military attaché, to see me here not later than one hour from now. We'll have a full dress conference here at eight o'clock tomorrow morning, with written evaluation reports in detail from all branches. Dr. Forester, consider yourself assigned to Pentagon duty as of now and until further notice. Forester sat, dazed, until he realized that the others had left, and the general was standing in front of him. Go get some rest, Forester, the other man said with surprising gentleness. You've had a tough day. As Forester slept that early summer night, weathermen across the world were marking their weather maps with thousands of observations, feathery wind arrows, temperatures, barometric pressures, and relative humidities. Then, as they drew their isobars, the patterns for the northern hemisphere emerged. A giant high-pressure system, with its center in northern Oklahoma, promised warm weather across America. Another, centered east of the Ural Mountains, forecast clear weather for most of Europe and northern Asia. A low-pressure trough between was dropping light warm rain on the green fields of England, but from Seattle to Washington, D.C., from Stettin to Vladivostok the sun was rising or setting in clear skies. Then, about nine p.m. Eastern Standard Time, a thickening mist descended over warm and drowsy southwest South Carolina.  Most of the reporting night shift at Aiken Hydrogen Bomb Plant never reached the tightly guarded gates. Those who did were not allowed in. At the same hour, across the world at the newly built underground heavy-water factory of Rosolofsky Grosk, west of the southern tip of Lake Backel, the late morning sun cast deep shadows into the gaping holes in the hillside which marked the plant entrances and exits. Deep below, miles of filtration chambers hissed quietly as they prepared their deadly concentrate. Then, without warning, the sun grew watery and pale, and within a few minutes a haze began to form at ground level. It grew thicker and thicker, the sun became a dim orange sphere, then was blotted out, total darkness enveloped the area. And, at the same hour, the watchers manning the lonely circle of probing radar domes, facing each other across the frozen wastes of the Arctic, cursed softly in Russian and English as their scopes sweeping the upper air first went blank, then dark. They were shaken men at the meeting in General Morgensen's office the next morning. Over thirty key men are gone from Aiken, Morgensen was saying. In terms of goals, it means that our 1960 program now cannot possibly be fulfilled until 1965. If the situation develops as forecast by Dr. Forster's statement, our entire nuclear weapons program will grind to a halt within two weeks. If we drain men from civilian research it will cause a total breakdown in the civilian atomic power production program. As you all know, the nation's entire economic expansion program is based on the availability of that power. Without it, industry will be forced into a deep freeze. That, in turn, means we might as well run up a white flag on the White House lawn. He smiled thinly. I would be a lot more worried than I am, except we have the first indications that the other side is in the same boat. I broke every regulation in the book last night when I talked to Melanowski. I took the liberty of warning him on the basis that there was nothing to lose. His reaction was that it was all a Wall Street capitalist plot, psychological warfare, he called it. He phoned me an hour ago, sounded as though he'd just seen a ghost. He said the Russian ambassador had asked for an appointment with the Secretary of State this morning. Forster bewildered and out of his depth in these global problems, let the flood of words pour over him. Then he realized Morgensen was staring at him over the telephone receiver at his ear, and that the room was very quiet. Then Morgensen said respectfully, very well, Mr. President, we'll have Dr. Forster there. Forster was relegated to the sidelines after his interview with the gray-faced man in the White House. Events were moving swiftly, events which Forster could read behind the blurred black headlines of the newspapers. The Russian ambassador was closeted with the Secretary of State for a record six-hour talk. Then the Soviet Foreign Minister took off for Washington at thirty minutes' notice, and another record was made when he spent all day with the President. The Washington columnist began to hint of lessening tensions in the Cold War, and the wire services carried reports of Russian radio broadcasts talking of a new era of cooperation between East and West. Only fragments of the broadcasts could be monitored because radio reception had suddenly deteriorated right across the world. The reports could not be confirmed, because Russia cut all phone communications with the outside world. There was no possible mode of contact. Meanwhile, in the United States television reception was blacking out for hours at a time, with no explanation available. The Civil Aeronautics Administration and the Air Force banned all plane movements under instrument flight conditions, because radar navigational equipment had become so unreliable as to be useless. Newspapers across the nation were reporting sudden fogs of short duration which baffled local weathermen. The U.S. Weather Bureau in Washington refused to comment. For the first time in the history of an East-West conference, there was no haggling, no propaganda speeches. Hour after hour, even as the talks went on, the cream of the world's scientific brains quietly continued to disappear. It was revealed later. In three days the major powers accomplished what they had failed to do in the previous fifteen years. Just four days and twenty-one hours after Forrester had first talked to General Morgensen at the Pentagon, a treaty was signed ending the World Atomic Weapons Race. And it had all happened, was over and done, before the people of the globe could realize what was happening, before they could rise in mass panic in the face of the incredible unknown. Almost immediately after the announcement, radio and radar communications suddenly returned to normal, and reports of the mysterious fogs ceased. Back at the center, as he walked down the floodlit ramp of the heliport towards his car, Forrester found himself thinking of the experimental work on the dream state which he had performed as a graduate student. He knew that a dream which might take half an hour to recount took only a fraction of a second to occur in the subconscious of the sleeper as he awoke. It was the same way with the events of the last five days, already details were becoming fuzzy and blurred as though they had happened five years ago. He opened the car door, and the soft glow of the dome light filled the interior. Then he saw again the rectangular discoloration on the seat covers, and the jolt back to reality was almost a physical thing. Relief, overwhelming, flooded over him. He looked up into the indigo velvet sky. Above him was the enormous triangle formed by Deneb, Vega, and Altair. From within it were a thousand other dimmer stars, but all he knew far, far, bigger than the speck of solidified gases called Earth. Somewhere out there, living, thinking, breathing, was Bentley. Good night, Forrester said out loud, and somehow he was sure he wasn't talking into thin air. End of Warning from the Stars By Ron Cocking


Okay, the mic is on. So let me start by thanking the organizers of this wonderful workshop. It's been extremely interesting so far, as well as the school last week, which was also very nice. So in fact, it was maybe a little bit too interesting because there's a lot of talks that already happened where some of the material that I present were already there. But I guess repetition is not necessarily a bad thing. So apologies if you're already familiar with parts of the story, which concerns supergravity localization and ADS black holes. So this is based on this recent paper that we wrote together with Kirill Ristov and Ivano Lodato. So I'll start with some very basic introduction and motivation. Our starting point are black holes. And this is because essentially black holes are a theorist laboratory to try and understand gravity due to this key property that they have an entropy. And this entropy, as we all know very well, semi-classical physics shown by Beckenstein and Hawking gives you the leading term and some large area expansion that's the A over four. And here I reinstated all of the fundamental constants so you could see this H bar. But then you also expect that there's gonna be corrections to this Beckenstein Hawking entropy, which will take the form of an expansion in a large area.   And these corrections here, they're actually interesting because they start probing the quantum gravity regime. And in fact, while Beckenstein Hawking is completely universal, all these coefficients and these expansions, this alpha, beta, et cetera, might depend on the details of the theory and become sensitive to all of this. And one way to see this, and this was a natural question that arose immediately after Beckenstein Hawking started associating entropy to black hole, is can you give a Boltzmann interpretation of this exact entropy here in terms of some microscopic degeneracies making up the black hole? So the question to ask is, is there such a quantity, the micro, for which the exact entropy can be written via the Boltzmann law? Right, so here I'm just gonna very briefly review some of the progress that has been made in trying to understand and answer this question. And much of the progress has been made for supersymmetric models, so that will be the focus of my talk. I'll talk about supersymmetric black holes only. So for asymptotically flat black holes in the seminal calculation of Schrodinger and Waffa, they showed that string theory can successfully account for the Beckenstein Hawking entropy of the black hole by realizing it as some kind of brain system with some strings. And we had a very beautiful talk by Coulomb Waffa for the inauguration of the new center here that sort of reminded us of this very fundamental calculation. I also wanna remind you that in fact, this string theoretic or brain picture is actually very powerful because not only can you recover Beckenstein Hawking, but you can also compute the subleading corrections to the entropy of the black hole in some particular cases. And here I'm just quoting some of the important papers in this direction. And in fact, for certain supersymmetric black holes, the microscopic degeneracies are fully known as functions of the charges. And that essentially was achieved because the generating functions for these microscopic degeneracies can be obtained from some topological invariance in the string theory, such as the elliptic genus and generalizations. So that's a very beautiful story. There's still much to be said as also Coulomb Waffa this morning was explaining. But really a lot of progress has been made beyond Beckenstein Hawking for these black holes. So now I just wanna turn to asymptotically ADS black holes, which are another class of black holes. And here, much more recently, some progress was made specifically at first for ADS-4 spherically symmetric BPS black holes. And the microscopic counting for their microstates was achieved in the dual field theory, which lives at the boundary of this ADS-4 space. And this was in this beautiful paper by Benini Iristof and Zaffaroni. So this is much more recent. This was in 2016. And here the idea was to compute a quantity in the CFT-3, which is known as the topologically twisted index. And you are able to do this calculation exactly using supersymmetric localization in the CFT. Now, what you get is a matrix model out of this. In principle, this is an answer which is valid for all n, but it's quite excruciatingly difficult to evaluate exactly. However, Benini Iristof and Zaffaroni were able to show that at large n, you can take some approximation on this matrix model and it reproduces the Beckenstein Hawking entropy of the black hole. The sublating contributions or corrections to this Beckenstein Hawking entropy are still encoded in the matrix model, but it's much harder to sort of extract them and try to get some log n term or one of our n corrections. And here I just want to mention that after this followed a lot of generalizations to other models, other dimensions. I'm giving a list of names here. I really apologize if I missed yours. Please come at lunch and insult me. But like there's been a lot of recent work in the last two years, essentially on these ADS black holes. So that's the story about the microscopic counting using the dual CFT. But in this talk, I want to ask kind of a, well related but somewhat different question. So given the degeneracies that are computed in the microscopic picture, can we define and hopefully compute the corrections to Beckenstein Hawking entropy and get this complete quantum entropy directly in the microscopic picture, i.e. the dual gravity theory. And if so, of course, if you can do a calculation on the gravity side and on the CFT side, do the two calculations and descriptions of the black hole agree. And if so, at which order. So I want to examine these questions using the notion of quantum entropy for these asymptotically ADS black holes and supersymmetric localization, but directly in the super gravity or gauge super gravity theory where the black hole lives. And here I just want to mention that these ideas of applying localization in gauge super gravity were already put forward in these papers where no black hole really was considered. It was more about just ADS space. And these two notions here, the quantum entropy for asymptotically ADS black holes are more generically black holes and applying localization directly in the super gravity theory were already presented and discussed in the two very nice talks that we had yesterday by Bernard David and Imtac Gian. So some of what I'm going to say is going to be familiar from these last two talks. Right, so that was the intro and motivation and kind of like what we're going to try to do. So I'll go forth and set up the problem and then show you what one can achieve with localization in gauge super gravity and the quantum black hole entropy. And I'll end up with some conclusions. If there are any questions at any point just feel free to interrupt. Right, so setting up the problem. In general, we want to examine solutions of four dimensional N equal to two gauge super gravity and these black hole solutions of electric and magnetic charges and ADS asymptotics. The previous talk by Dietmar gave us a very nice summary of various solutions that exist for these 40 N equal to two gauge super gravity. So typically the full black hole solution will interpolate between some ADS four vacuum at infinity and there's a near horizon region which has the form ADS two times S two or more generally as Dietmar reminded us some more exotic type of near horizon topologies where you can have a Riemann surface or more general things than the two sphere. In this talk I will just for simplicity focus on ADS two times S two and I'll come back to a more general case perhaps at the very end. So the full solution preserves two supercharges it's quarter BPS, but in the near horizon region there's an enhancement and the region itself preserves four supercharges and it's half BPS in the language of 40 N equal to two. This enhancement is in fact a general feature of the attractor mechanism. This might be familiar to you from un-gauge super gravity. This also happens in gauge super gravity as was already explored by these papers. So what I wanna do for the rest of the talk is to focus on the contribution to the quantum entropy coming from the near horizon region itself. And the reason I wanna do this is that this will allow me to consider a large class of asymptotically ADS black holes all at once namely all of those who have a near horizon attractor geometry ADS two times S two. So I'm not gonna be worried too much about where the black hole is coming from or what kind of asymptotics it has. I just want for it to have this ADS two times S two near horizon geometry. Hopefully in the second part of this talk after lunch Kiril Laristof will tell you about more of the asymptotics, but here I just wanna focus on the near horizon region. And the other reason why the near horizon is interesting is that there's this ADS two factor for these BPS black holes. And that's a key ingredient for us to explore the quantum entropy of black holes. The reason is this proposal that was put forward by Ashok Sen in 2008 that stated that if you want to compute the quantum entropy of the black hole you should compute the following expectation value. And this should also be familiar to you from the talk by IMTAC yesterday. This was the same quantity that he mentioned. So I'll denote this by W in the rest of the talk. This is essentially the expectation value of a Wilson line that you're inserting at the boundary of your ADS two space. And this Wilson line here is here to make the variational problem well defined. And in fact it enforces the micro canonical ensemble where I'm gonna look at black holes with fixed electric and magnetic charges. And that's kind of like the boundary condition that I wanna impose when I'm computing this expectation value. Now on top of this there's also some regularization scheme that one needs to put forward because of the essentially because of the infinite volume of ADS two. So that's what this superscript finite here refers to. And it was already shown by Sen that in a suitable large charge limit this expectation value you can do essentially a saddle point approximation. And this will reproduce for you the Beckenstein-Hawking entropy of the black hole or more generically if you want to consider theories with higher derivative terms this will be even the full Beckenstein-Hawking walled entropy for the black hole. But here you're really doing an expectation value so you're hoping to really capture all of the quantum corrections to Beckenstein-Hawking entropy. And just for some final motivation for why this proposal is interesting this was argued in the original paper on the basis of the ADS two CFT-1 correspondence where holographically you expect this to correspond to some Witten index in a dual CFT-1 counting the number of ground states which you interpret as the microstates of the black hole. So that will be the main player of the game and I just want to basically compute this quantity for the black holes engaged supergravity. So yeah, as I said, this defines the non-perturbative entropy of extremovps black holes. I want to stress as well that this is a Euclidean path integral. And we all know that supersymmetric localization very generically gives you exact one loop evaluation for path integrals. So what I want to do is apply supersymmetric localization in four-dimensional line equal to two gauge supergravity. Now to apply localization it's very convenient to have an off-shell formalism. And as Bernard was explaining yesterday, this is in fact crucial if you want to go ahead and make sense of the localizing supercharges that you want to use in supergravity. And to ensure off-shell closure of the underlying gauge algebra, I'll make use of the superconformal formulation of gauge supergravity which was put forward a long time ago already. And so, right, that's essentially the theory. I have some superconformal off-shell Euclidean gauge supergravity in four dimensions and equal to two. So the field content consists of a via-multiplet, which essentially is the multiple that contains the graviton as well as a bunch of other fields. And I'll be considering NV physical vector multiplets. But because I'm using the superconformal formulation, I need to introduce some compensating multiplets in order for the theory to be gauge equivalent to Poincare's supergravity. And this is why I have this extra vector multiplet here and one hyper-multiplet. So these will be my gauge compensators to sort of compensate for the extra conformal gauge symmetries that I introduced for the off-shell formalism. But physically I have some gravity multiplet and NV vector multiplets, sorry. So the steps will be as follows. First I want to go through and specify this near-horizon half-BPS field configuration for my vial vector and compensating hyper-multiplet. And then we'll look for off-shell fluctuations, BPS fluctuations with respect to a localizing supercharge that I'll show you. And so these fluctuations are around the half-BPS attractor background. And I also want these fluctuations to satisfy the boundary conditions for the quantum entropy, meaning that these fluctuations should go to zero at infinity and I should recover the attractor geometry. So that should also be familiar from the boundary conditions that Bernard was mentioning yesterday in a much more general setting where you had some background field split and then the quantum fluctuations should die off at the boundary. Right, so let's get into a little bit more of the details. So this is the Vy-multiplet. As I mentioned, it contains the graviton or the field bind. There's obviously the gravitini. Here, I is an SU2R index because I have n equal to two.  So this is a dilatation gauge field. This is the U1R gauge field or more appropriately in Euclidean, the SO11R gauge field. And this Vmu ij here is the SU2R gauge field. And then I have some auxiliary fields that are required for off-shell closure of the algebra. So there's a rank two tensor. There's some extra fermion here, chi, and there is a scalar field, d. That's my Vy-multiplet. Moving on to the vector multiplets, as I mentioned, will be in Euclidean. And in fact, the vector multiplets you might be familiar with in Lorentzian typically contain a complex scalar. Here in Euclidean, there are actually two real scalars that I'll denote by x plus and x minus. And in fact, with Bernard DeVitt, we worked out the full off-shell formulation of these Euclidean supergravity theories. And that's where you see these two real scalars appear. Morally, you can think of them as the real and imaginary part of your complex Lorentz scalar. There are also vector fields, obviously. I'll denote them by W mu, just so that it doesn't clash with the A mu up here, this S-O-1-1 R gauge field. And then there's some Gagini, and of course, some auxiliary fields, which again are required for off-shell closure. So these guys are just triplets of the SU2R. And I have a collection of these vector multiplets with I running from zero to Nv. And finally, there's this compensating hypermultiplet. For the moment, I'll just discuss its field content on-shell and then we'll come back to putting an off-shell later during the localization procedure. But on-shell, this just consists of some scalars, and they're from Yannick's super partners. And this is a good way of illustrating where the gauging comes in, because we're gonna talk about a gauging, which is specified by, so typically also as Dietmar reminded us, we have these symplectic vectors of gaugings. Here I wanna consider only electrical gauging. So I'll set the magnetic counterparts here to zero. And so the gauging is specified by these Feyyay-Liopoulos parameters, XII, as well as some generators, T-alpha-beta. And you can see this very clearly. For instance, if you look at the covariant derivative of the scalars and the hypermultiplet, where again, you have all these connections from the y-multiplet, and the gauging entering here, which charges this hypermultiplet section under some linear combination of the gauge fields in the vector multiplet. Now, this is in the conformal theory, so that's in the conformal compensator, but then if you gauge fix all of this to Poincare, you'll see that this induces a charge for the gravitini. So here I'm really talking about gauge supergravity with these electric gaugings. Right, so let's talk about the half-BPS near-horizon configuration first. So here in trying to keeping in touch with Bernard's notation from the talk yesterday, I'll denote the on-shell attractor configuration by a ring here or a dot above the fields, and this will really be my background fields, and then I'll think of fluctuations around this background. But for the background, so we have this metric, that's the near-horizon region. It consists of this ADS2 times S2 factor. Notice that there are two parameters here, V1 and V2, or sizes. In gauge supergravity, they're not equal, which is also a difference with the ungaged supergravity case. And then we have some non-zero configuration for this auxiliary T tensor, as well as for the scalar fields that sits in the vi-multiplet, which are all given by these V1 and V2 constants. Now there's, as I said, this hyper-multiplet compensator.  So in the background, you can really set it to a constant. And essentially here, this will identify this alpha index, which was an SP1 index with the SU2R in the theory. And finally, for the vector multiplets, again, the attractor configuration. So you're gonna have some linear combination of the charges or the field strength specified by this V2. And the scalars are also, here are their attractor values in terms of V1. And then because you're on shell, you can also solve for the auxiliary fields in the vector-multiplet. And these will be given by the following expression, Nij, by Nij here, I mean the same period matrix that Dietmar was using earlier, which is given in terms of some pre-potential for the theory. So this is my attractor configuration, almost complete. There is one more non-zero field. And this is the SU2R gauge field that sits in the V-multiplet. And this guy here on the background is related precisely to this combination of gauge fields in the vector-multiplet. So these are all the non-zero fields, all the other ones are zero, and of course I'm just looking at a bosonic configuration. So now that we have our background, we're gonna look for off-shell BPS fluctuations or like the quantum fields that Bernard was talking about. And for this, the very first step that one should do is solve the gravitini supersymmetric variations for arbitrary metric and arbitrary killing spinners that respect the boundary conditions for my quantum entropy. Now of course in general, this is a very hard problem. And here I wanna remind you in un-gauge supergravity, this was solved for fluctuations around the full BPS attractor geometry. And in particular in this full BPS attractor configuration for un-gauge supergravity, this SU2R gauge field vanishes. Recall that otherwise it's precisely given, it's non-zero for us because of the gauging. And so this problem here was solved in a great paper by Gupta and Murthy. And what they showed is that the only BPS configuration which is allowed is the attractor geometry itself. So essentially there are no off-shell fluctuations for the vi-multiplet fields and they're all frozen or pinned down to their attractor full BPS configuration. Now what we tried to do was essentially to go through this analysis again, but because of the gauging, because of the non-trivial backgrounds that we have for the SU2R field, et cetera, we rather quickly hit a wall. And so here will come the central assumption for the rest of the calculation. Here in gauge supergravity, we will assume that this is also the case and that the vi-multiplet is frozen to its attractor geometry. So this basically the sort of slogan is this w equals w naught. And this also means that I can use the attractor or background killing spinners to parameterize my superchargers with which I'm gonna affect localization. Right and so now I just wanna look for off-shell BPS fluctuations but in the gauge and matter fields only, so in the vector multiplets and the hypermultiplet. Right so we have, as I said, the background is half BPS. We have four superchargers and out of these four superchargers, we can pick one which we're gonna call Q lock, that's the charge with which we're gonna do localization and it has the following algebra, the one we picked. So this should look very familiar to most of you but this is essentially a lead derivative along the tau direction which was this Euclidean time at the boundary of ADS2. And then we have some SU2R transformation as well as some gauge transformations which are parameterized by the scalars in the vector multiplets. This type of gauge transformation is really inherent to the supergravity formulation. It also played a role yesterday in the talk by IMTAC. But essentially this algebra should look rather similar. This is just some lead derivative and some R transformation. Right so that's our setup. We have the localizing supercharge. We've made our assumption that the via-multiplet is frozen to its background value and now we can just go ahead and analyze the localizing equations in the vector and the hypermultiplet sector. So in the vector-multiplet sector, we'll just examine the consequences of imposing the vanishing of the Gagini variations. Now here comes another specificity of gauge supergravity and one of the important differences with similar calculations that have been done in ungauge supergravity. And the point is that the supercharge here Q lock is parameterized, as I said, by an attractor or background killing spinner. But because of the non-trivial SU2R gauge field in this background, the gauging actually affects a twist on the two-sphere of the ADS2 times to near horizon geometry. And this means that if you solve for the killing spinner in this background, you'll find that the killing spinner is actually constant on the two-sphere. There's no angular dependence at all because you've canceled essentially the spin connection with the SU2R gauge field, which is non-trivial. So in a way, the localizing equations that I wanna solve here, they're non-trivial in the radial direction of ADS2, but they're much less constraining in the angular directions because I just don't have anything non-trivial happening along the sphere. And this is a crucial difference with the ungauge supergravity or flat black holes. Nevertheless, let's not worry too much and press forward. So we want to look at a new find solution to these kind of equations and we'll parameterize the off-shelf fluctuations. Again, here I'm just trying to use a notation which should echo what Bernard was mentioning yesterday where we have a background and some fluctuations for our scalar fields, X plus and X minus, the field strength and the auxiliary fields. Remember, I'm looking for off-shelf fluctuations so the auxiliary fields might also have some non-trivial profile. I'm not requiring that these fluctuations solve the equation of motion, but simply that they solve the localizing equations for that one, particularly supercharge. So these equations ultimately will lead to some constraints on these fluctuations, X plus FAB and YIJ. Right, so you write down the equations, you go and try to solve and the end result that we reached is that also taking into account the boundary conditions, as I said, these fluctuations should die off at the asymptotic of the near horizon configuration and it should really recover the attractor geometry there. What we found is that the scalar fields here, they can be excited away from their on-shell value and the excitations, you can parameterize them, so each of the scalar fields, X plus and X minus, has a parameter C plus and C minus, but in fact, there's a whole tower of parameters which are indexed by this index K and these fluctuations die off as one over R to the K. So that's already quite a lot of parameters and I wanna just, I haven't written down the expressions but essentially the gauge fields and the YIJs, they're off-shell fluctuations are also gonna be given in terms of these C plus and C minuses. But not only do I have a bunch of those for all values of K from one to infinity, but these parameters are actually dependent on the two-sphere coordinates because as I said, I don't have any equation and any localizing equation to constrain their dependence so these guys are really just functions on the two-sphere. So I have two real functional parameters or rather infinite families of functional parameters for each vector multiple. Right, that doesn't look too good and there seems to be a lot of off-shell fluctuations allowed, however, if you analyze the localizing equations, you'll find an additional constraint and that is that there's a very specific linear combination of these fluctuations parameters C plus and C minus which turns out to be independent of the angular coordinates and that will be very important later on. So just to summarize, essentially this localizing manifold, the set of BPS solutions to the localizing equations in the vector-multiplet sector is given by these two types of CK plus and CK minus. There's one for each vector-multiplet, and V plus one and for every K but you can sort of repackage them into a combination that I'll denote by Phi plus which is really independent of the two-sphere that's really a constant parameter and well, all the other ones that are orthogonal to this particular linear combination that I'll denote by Phi perp and these guys are really angular dependent. Yes, yeah, especially at the horizon that R equals one, you want these sums to be finite. Yeah, it doesn't, I mean, yeah, there's some additional things here that I'm not, we can run through the constraints but essentially what you find is that it's not enough to completely kill off the angular dependence or the fact that you have this tower of excitations. One more caveat in this solution is that we've assumed smoothness of the BPS solutions. Of course, there could be additional singular configurations that might contribute. Here, I just want to briefly remind you that this was also the case in engaged supergravity with these kinds of geometries but for the moment I just want to set that aside and keep on with the smoothness assumption. Yes, they solved that one BPS equation for that one supercharge Q lock. Yes, that's my localizing supercharge with which I'm, right, right, no but then of course if you take your four supercharges then you'll find that these guys have to be zero because I need to recover the attractor geometry. That's the half BPS one that, but here I'm only really asking for one supercharge which is the localizing supercharge that I'm using. For the compensating hyper-multiplet you can also analyze the variation of the hyperini with respect to the localizing supercharge. Now here, I briefly mentioned earlier I need to put these guys off shell and for this we used a technique that was introduced by Berkovich and also beautifully illustrated by Hamanozomishi and it consists in the following. Basically you want to modify the standard variation of the hyperino by introducing some auxiliary scalar fields H and some constrained parameters side check. So these side checks have to verify some non-trivial equations in terms of Xi which is the parameter of my localizing supercharge but if you introduce these auxiliary fields and you give them an appropriate transformation under Q lock you'll find that you can close the algebra without using the fermionic equations of motion. So again we impose the vanishing of the hyperini variations and of course you see here that in the variation itself I have the gauging which is entering once more and the gauging here is selecting this linear combination of scalar fields. The scalar fields have off-shell BPS fluctuations so these are gonna get related to the off-shell BPS fluctuations allowed for the hyper-multiplets. So essentially what you'll find is that the fluctuations of these scalars are controlled by the parameters that I introduced earlier CK plus and CK minus. But in addition there is one more constraint that you can get out of this equation and that's a constraint on the linear combination that was independent of the angular coordinates and essentially you should satisfy this constraint that is specified by your choice of gauging. So that constraint comes out of the hyper-multiplet compensator. So in total we have characterized this localizing manifold in the gauging matter sector and what you get is phi plus phi perp which is angular dependent and this constraint for phi plus. Now what you wanna do is evaluate the action of gauge supergravity on this field configuration just as a brief aside in the Euclidean Formalism the action is actually specified by two pre-potentials which are functions of these X pluses and X minuses they're not related by complex conjugation but now comes an important observation when you plug in this field configuration you realize that the bulk action of gauge supergravity specified by a generic pre-potential here actually only depends on this constant parameter phi plus and it doesn't depend on phi perp. And in fact you can go further if you add the Wilson line contribution and the appropriate boundary terms to renormalize according to this finite prescription you reach the final form for the action on the localizing manifold which is only a function of phi plus and it takes a very simple and surely to most of you familiar form very schematically this is coming from the action this is coming from the Wilson line but everything only depends on phi plus which are these constant parameters. So the phi perps are really zero modes of the bulk action and the Wilson line. So because there are zero modes they can only enter my localized quantum entropy function not via the action itself but via the one loop determinants that I still have to compute and put into my formula. Now we haven't computed or these one loop determinants precisely but we wanted to argue the following that we can actually split these one loop determinants into a contribution from these phi plus modes and the ones from the phi perp modes and this one seems rather dangerous because it still really contains a functional integral. I have functions on the two sphere and so I've localized but I still have some functions that I need to integrate over and the reason for this as I mentioned many times is the gauging and the resulting twist that we had on the two sphere. But putting everything together and allowing for a bit of a formal notation what you'll find is the following. So here this was my WPQ from the beginning. I put a hat on top of it just to remind you that these are for smooth BPS off shell configurations. So I have an integration over these phi plus. This is a standard finite dimensional integral one for each vector multiplet. I have the constraint which was coming from the hyper-multiplet compensator which I'm putting in as a delta function. Then there's the exponential of the renormalized action so that's our QIFI minus PIFI. I should just mention but this is rather standard notation like this FI here is the derivative of the pre-potential with respect to one of the scalars. And then I have some one loop factors one of them which will depend on phi plus and some possibly horribly complicated messy perp factor which is coming from these angular dependent integrations on over phi perp. So as I mentioned for the moment we have what we did was characterizing the localizing manifold in the paper evaluating the action on this but we're still trying to compute these one loop determinants and I'll come back to this towards the end. Right but let's forge ahead a little bit and just see what we can get out of this formula. So setting aside these unknown factors we can analyze the saddle point of this of this integral. And for this you can define what are perhaps to some of you familiar combinations which I'll denote by Z and L. This L also made an appearance in Dick Mars claim talk sorry earlier. And so if you compute the saddle point you'll see that there is a constraint that L must be equal to two pi that was our constraint coming from the delta function. And the saddle point equations coincide with the standard attractor equations engage super gravity where you're supposed to extremize this quantity Z over L. And furthermore the value of Z over L at the extremum is the Beckenstein Hawking entropy of the black hole. So even though we have still some unknown and possibly very complicated factors in this formula we're kind of on the right track because if you start looking at some saddle point approximation you find the standard gauge super gravity attractor equations and the value of the Beckenstein Hawking entropy. So this is a strong hint that these one loop and perp factors will not contribute at this leading order level. Otherwise that would spoil the agreement that you find with the gauge super gravity attractor equations. And at least for the one loop factor this is reminiscent again of what was happening in engage super gravity or for the flat black holes where the Z one loop that IMTAC give us a very nice review of yesterday and the complete form. This contributes at order log of the area and beyond. So it does not spoil Beckenstein Hawking. Okay, there's some other checks that we can do on this formula that we obtained.  For instance, with some ADS four times a seven. So as I mentioned in the beginning and we've heard now many times this has some dual holographic description as some ABGM three dimensional field theory. And there is this topologically twisted index that's supposed to capture the microstates of the black hole. So the index itself was computed using localization but rigid localization. So here really in the field theory. And the degeneracies for the supersymmetric ground states are given as a function of the charges after euphoria transform the index and here you get DCFT. And the interesting thing is that it takes the following form. So the Z here is the topologically twisted index. It's a very complicated expression at finite N but at large N you can actually evaluate it in a variety of examples including a BGM but some other examples as well. And what we showed in the paper in the number of examples is that if once again you sort of forget and set aside the one loop and per factors, the integrand of our W hat here matches the one of these degeneracies at large N. Once you identify the off-shell fluctuations phi i with the chemical potentials delta in the computation of the topologically twisted index and you also have to use the old graphic dictionary to relate your N to your Newton's constant and your charges. So it takes an eerily similar form by just doing a calculation directly in the near horizon of the black hole in the super gravity theory. Right and now I just want to make also an observation. So this is the result that we got in gauge super gravity.  And some one loop determinant factors. Now, thank you. You should remember or I'll remind you of the un-gauge super gravity result which was pioneered by double cargo mesh and murty where they showed that for the flat black holes you can also write a very similar formula again with Nv plus one integration with a finite dimensional integral and some exponent which again depends on the pre-potential of course not the exact same combination but you can see the similarities here and some one loop determinant factor which again IMTAC nicely reminded us yesterday of its full form which is given in terms of the killer potential which once again is specified by the pre-potential. So what I want to emphasize here is just that these two formulas both for ADS and flat black holes they're extremely plug and play. You can just pick whatever pre-potential you want to specify your theory because you're coming from some string compactification because you just want to do it as an academic exercise and you can just plug this into these formulas and start seeing if you can even evaluate these integrals exactly. And in fact just as a remark for very specific models here I have in mind some type two B compactification on T6, this W flat here can be computed exactly. You can even take these integrals and actually get an exact result purely in terms of the charges P and Q and this led to a very precise non-perturbative match with the exact string theory results that you can derive in these compactifications. And here also comes a, if you'll allow me to be subjective, some very beautiful connection to number theory via the Rademacher expansion of which Komlund Vafa also talked about in his inaugural lecture. So this was for flat but of course for ADS we're far from there. We still have these one loop determinants to compute, et cetera but I just want to sort of end with this remark on the similarities that you can get engaged and engage supergravities. Right, so let me move on to my conclusion. I believe in the paper that I mentioned at the beginning with Kirill and Ivano, what we did was really presenting the first step towards the computation of the quantum entropy for BPS black holes engaged supergravity with a near horizon ADS two times S2. And again, I want to emphasize that this is a contribution from the near horizon only and not from outside of the horizon. That's important to keep in mind if you really want to do some precision matching with the calculation in the asymptotic CFTs. So perhaps unsurprisingly, although it's always kind of reassuring the saddle point evaluation of the quantum entropy that we have reproduces the semi-classical Beckenstein Hawking entropy and you see here the usual attractor mechanism engaged supergravity. And the answer takes a very similar form to the computation of the topologically twisted index in the dual field theory. This is at this stage just a formal observation but I would say that this is rather encouraging. So for some future directions, obviously one of the main priorities is to complete the one loop determinant calculation engaged supergravity. And this should really lead to a better understanding of the role of these mysterious functional zero modes that are not entering the renormalized action but we need to see what happens in the one loop determinant for these guys. And if we are able to do this calculation this is also something very concrete we can do to try and reach perhaps some predictions for the dual matrix model at finite end because here at large end you recover Beckenstein Hawking, you have these n to the three half scaling in the specific ABGM case but in the gravity side, again, modulo the caveat that this is all near horizon, if you know the one loop determinants you can start picking up some finite end corrections and then you can go back to the matrix model and try to ask how these things talk to each other. So that's our main priority for priority for the one loop determinants. Of course, as I mentioned, there's a central assumption in this work that the via-multiplet is frozen and pinned to its attractor value. So that should really be examinated thoroughly and that's quite high on our to-do list. And eventually later, I did mention this at the very beginning, you can think of trying to generalize these calculations to perhaps a more trivial, more non-trivial topologies for the near horizon, namely with some generic Riemann surface of some arbitrary genus and of course, also more general gaugings. Why not go to dionic gaugings and really turn on the full electric and magnetic F5 parameters. So I think these would be interesting generalizations of the calculation. Of course, these are two much more important points that we hope to be able to report on in the near future. And, but as I said, you can already say a lot about the formulas that we had and hopefully after lunch, Kiril will keep saying a lot about the formulas that we have. So thank you for your attention.


Jordan Pickford will start in gold tonight for England against Iceland. There's no place for Michael Keane or Dominic Albert Lewin in the start on 11. They are both on the bench and Guilfe Sigarton is on the bench for Iceland. Yeah, Pickford starting again. I thought maybe that the manager would have gone with Nick Pope for this one given it's a meaningless game really, but Pickford keeps his place. So Gareth Salfgate continuing to back him, he called him his number one, said no one's really challenging him. It looks like that, especially picking him for this game tonight, so it'll be interesting to see how he does tonight. Last night Brith Charleson played and scored for Brazil. He scored his second goal really good header in a 2-0 win over Jorragwai. Brith Charleson then limped off after a poor challenge from Edison Cavani which saw him red carded for Jorragwai. Brith Charleson hopefully will be fully fit and ready to take his place back in Everton's starting line up at the weekend in the game against Fulham Fingers crossed. Hamers Rodriguez played for Colombia but they were beaten 6-1 by Ecuador. Rodriguez scored a penalty in that game which was 4-0 at the time, yet he mean it of course, missed the game due to being suspended. Tom Davis and Ben Goffrey played for England under 21s as they beat Albania 5-0 last night as well. Everton are linked today with a move for Latio McField at Luis Alberto, the 28-year-old McField, a former Liverpool player. He's fallen out with his club over their decision to buy a private jet rather than pay the staff the full wages during the Covid pandemic. He was publicly critical of them on a Twitch stream, I think, and on social media. So he's fallen out with them and reports coming out of Italy that Carlo is an admirer of him and he may be ready to move for the 28-year-old McFielder. It seems a little bit not really in the parameters of what Everton are looking for with this one, although obviously the links to it go continue every day, don't they so who knows maybe there is something in this. And apparently Juventus and Paris Saint-Germain are going to fight it out at the end of the season to sign Moise Keane, who is at the moment on loan to the French club. Everton, Adam and Keane will retain and be part of their squad next season, but these reports keep coming out of not only France but Italy as well that Keane will be looking to move on. So we'll wait and see what happens there. And finally, the car park at the Everton Blue Base on Salop Street is to now be used as a Covid-19 testing centre. So it opened between 9am and 7pm. If you need a Covid test you can go and get it done at the Blue Base as Everton playing their part in this community testing that Liverpool is going through at the moment. So if you live local to Salop Street and you need a Covid test you can get it at the Blue Base. That is it, make sure you subscribe, give the video a thumbs up and if you want more videos join us on Patreon. See you later.


Facilities at Marine Corps Air Station Iwakuni completed construction of six new barracks on the north side of the air station last month. There are two bedrooms. Each Marine would have their own bedroom and their own fire-rated kitchen to be able to cook in. So there's a sink set up with cabinets. You also have the two burner cooktop, also washroom dryer, combination over here, bathroom and shower over here. The idea for these barracks goes back about three years. It was a combination of planning between bachelor housing, the facilities planning department and CSDB. These barracks were replacement barracks for old barracks that were demolished during the construction of the new commercial airport. I'm Patty Officer Marcus Suarez, AFN Iwakuni, Japan.