And so, at last, it had come to this. Day after day, week after week, they had come to him, talking to him, telling him their lies, trying to see where he broke. They'd never realized how much they had told him instead. When you hear enough lies, you get a feeling for the truth. You can see the shape of the truth by how the lies outline it. He had everything in place. He'd had it in place for quite some time. Plans, plots and schemes all ready to go when everything was right. In the end, all it took to spark it off was a simple cold. The old lady had already been quite weak, refusing to indulge in treatments like the others. An illness introduced into her system by a well-placed cough and she passed away in her sleep. The numbers above and below her had come to him and offered him her numbers. He accepted, with a show of reluctance. Those who are shown to desire power are those least likely to be trusted with it. And now, O5-4 sat in his office. He had been given his mark of office. A foot-long length of bone polished smooth. His name had been erased, not just from foundation data banks, but from the minds of all who had known him. He was sure the other numbers knew his name, but soon enough, they wouldn't matter. He let his fingers drift towards the keyboard before him, and a half-smile rose to his lips. Alone with no one watching, he still felt the need to say, well, something. For posterity, I feel like a supervillain. Like Lex Luthor or Adrian Vate. Do it, Dr. Bright. I did it 35 minutes ago. As good a set of words as any. Four touched a series of buttons, setting into motion the events that would place the entirety of the foundation into his hands. Then he sat back and waited. Fifteen minutes and everything would be his. Thirteen. Thirteen was easiest, mainly because O5-13 didn't really exist. Oh, the senior staff were told he did. They were memos handed down by O5-13, but it was just one more lie. Thirteen's vote was rotated to the other O5s, moving up a step, any time there was a tie vote and the tiebreaker was needed. But there was a power there, knowledge that only those who held Thirteen's vote could access. A simple, actually quite complex, computer virus snuck past the overseer's firewalls and bestowed the power of Thirteen to four. Easy as that. Twelve. A large African man sat in his chair, behind an expensive desk. The room he used as his office could also quite easily be described as expensive. The man himself could be described the same. Silk tie, gold cufflinks, Armani suit, shoes made from the skin of a very rare reptile. He wore dark wraparound glasses, smoked glass framed in ivory, perhaps to hide his eyes, perhaps to keep him from looking too closely at the things he had to. Although he had once borne another name, these days he was known to one and all as O5-12. He was the accountant for the overseer council, making sure the numbers added up everywhere. His assistant entered the office on schedule. Everything Twelve did was on schedule. From the moment he woke up to the moment he went to sleep, and perhaps even his dreams followed a precise, practical order. To do things otherwise invited chaos, and with chaos came death. The assistant walked calmly to the drink cabinet, opening it as he had done every day for years. Somewhere in his head things were turning. A thing had been planted in his head before he became Twelve's right-hand man. And now this thing made him pick a different bottle than the one he normally did. The glass was poured and it looked like whiskey. It even smelled like whiskey. When the brainwashed assistant dropped an ice cube in it, it even bubbled a little like whiskey. But when O5-12 raised it to his lips and gulped it down, it didn't act like whiskey. Contact with the soft wet flesh of his tongue and throat resulted in a chemical reaction.  He was dead before the glass left his lips. 11. The two men sat across from each other, separated only by a chessboard. It was an old board, but they were old men. O5-11 was a pleasant enough looking Caucasian gentleman, one of those old men who sit on their porch and offer candy to little children. Not the creepy kind. O11 fiddled with a bracelet made from human teeth, as he considered both the board and his opponent. It still your move, O11. Agent Altoclef teased his superior. Maybe the devil, maybe a reality bender. One thing could be said for certain. Clef enjoyed these monthly games with O11. When his phone beeped to alert him of an incoming text, he almost ignored it. Almost. But he'd expected this. Over enough, the text merely read, now. Check. Where? You don't have check anywhere. O5-11 leaned across the chessboard, searching for his opponent's move, his hands touching both sides of the board. A sad smile on his face, Clef spoke. There, he calmly tipped over his own king, and 300 amps of electricity charged into O11's body, frying his brain to a crisp. Checkmate. 10. O5-10 didn't notice when the air conditioning kicked on. It was just another distraction from the endless paperwork that filled a life these days. She was a strong, capable woman, a brunette, dressed in a functional long skirt and jacket. Every five minutes she'd raise her hand to the necklace around her throat, a carved marble eyeball, dangling to stare at her paperwork with the same disgust she felt for it. Her first indication that anything was wrong was when she'd realized she were the same page four times without comprehending it. By then, it was too late. She could hear the clicking and hissing sounds being made by her body. Her hand refused to let go of the paper. Her hand refused to let go of the paper. Her hand refused to let go. Her hand refused. Her hand. Her. 9. He always claimed he was the unluckiest O5. The one who didn't get to be a world-travelling jet-setter? Oh no. 9 was stuck deep in the middle of the ocean. Sure, he could travel to any of the water sites, all a couple dozen of them. But most of the time, he ended up here, deep beneath the sea, where no natural light could find its way. He was a pale man, small, always looking lost in the suits he tried to make look good. A bone-handled knife rested in his hands at this moment in time as he stared into the abyss. It would have been poetic to say something gazed back, but the only thing outside the window in his office was water, lots and lots of water. Several seconds later, the only thing inside his office was lots and lots of water. In the water in his lungs, and the pressure so deep, O5-9 was the unluckiest O5. 8. A shower could never make her feel clean. No matter how many times she washed, she could still feel those things crawling across her skin, could still see all the people she'd condemned to death. 8 was not a pretty one. Despite being one of a handful of all-powerful overseers, she'd never gotten any work done, no surgery to fix her slipped forehead, her craggy brow, no lap band to cut back on her weight. Somehow, despite having no appetite, she'd still gained weight. Even naked under the water, she'd still wore a braided ring of sinew on her left-ring finger.  The heat felt very nice, until there was nothing but heat. She didn't even manage a scream when her flesh boiled away. In fact, she may have smiled, because the memories, at last, were gone. 7. I'm sorry driver, I don't think I've seen you before. What did you say your name was? 7 asked absently, fiddling with the irate chopstick in her hair. Oh, it's Rodney. But most people call me Dr. Gerald. 6. He had been an agent once upon a time. When he worked in the field, they called him cowboy. People still remember the agent, even if no one knew he was now the overseer known as 6. He had been the best. And now, he was an old man. Dressed as always in an immaculate white suit, his long gray hair pulled back in a ponytail. His trademark stetson, tilted at an angle on his head. Between his gnarled old hands, he held a white cane. The handle carved into the shape of a howling wolf. At his back, were his two trusted bodyguards, Thompson and Black. They were almost as good as him. One day, they'd be better. When they were, well, when one of them could take his number, and he could retire. Maybe spend some time with his granddaughter. He didn't look back when Black got a text. He never knew that O5-4 had once been Agent Black's personal physician. Or if he had known, it wouldn't have mattered. He did, however, feel it when Black drew his guns. He began to turn, but no, too slow. As he fell to the floor, he noticed Thompson falling beside him.  5. It's an emergency, sir. We have to get you out of here. The security guard hurried O5-5 and his secretary into the nearest airlock and slammed the door shut behind the three of them. The guard leaned against the door, panting. Cater, sir. It sounds bad. The secretary sniffed, but then she always had pneumonia or some other sinus problem. O5-5 was nondescript. He dressed nice, but not well. His hair was... eh. His face was... blah. He didn't stand out. In fact, the only thing that was noticeable about him was the leather shoes. A nice, tanned leather. A secretary was an African-American woman about 51 years old, quiet, with a rather large nose. She had always been his emergency escape, even if she didn't know it. At the room proceeded to move upwards. The guard frowned, trying to look at the two of them. Do either of you hear a hissing sound? The gas was fast-acting, invisible, unscented. It poured out of 108's nose at a furious pace, killing them all as they rode to safety. Three. Well, hey guys, how are you today? It's fantastic to see you.  O5-3 never let himself be seen in person, or at least, not for many years. He always showed up on a monitor, broadcasting from a white room, a fancy desk and old-fashioned computer beside him. While he always had a proper two at hand, it was never anything real. In fact, O5-3 wasn't real. He died decades ago, plugged into a prototype computer he invented. He just didn't go away. The overseer council took a vote and decided they didn't care. He was still their best coder. He could still do the job. So they set his bone earring on the mainframe, installed some safety programs and let him work. An electromagnetic pulse set off inside the hardware that carried him, removed him from this world at last. Two. 60 kilometers west of Astrakhan, a garden was in bloom. O5-2 took great pride in her garden. Of course, she was cheating a little. She didn't mind. Once you get old enough, you don't mind using a few cheat codes to get things done. O5-2 was the type of woman you pictured when you heard the word grandma. She always had a blonde shawl on, no matter the weather, and usually a pair of gardening gloves. She had a fantastic gardening hat a friend had made for her, pushed down tightly on her gray curls. Her helper was an old friend, a man old well before his time, and maybe after it as well. He glanced up, a slight frown on his face, and pointed, looking uncomfortable. He wasn't used to this much attention on him. O2 had the time to look up and smile at the Russian satellite that was screaming down to the sky at her. She could have moved. She could have run. She noticed the rogue satellite too early, and she could have made it to the spring save herself, but she didn't. If anyone had been around to hear her last words, they might have been confused that what she chose to say was, it's about time. One. And one. The first. Once upon a time, he had been known as the administrator. All the power in the foundation had been his, but he didn't trust himself. He knew that absolute power corrupts absolutely. So he partialed off his power, handed out badges of responsibility, numbered them, and made sure they could balance each other, set up a system so that if one died, the authority, the votes were passed to another, so the balance would never be uneven for long. And then, he retreated from the world, buried himself down deep, with only electronic means to keep him in touch with the outside, which means that O5-1, still looking like a 13-year-old boy, still clad in the rags and bones that hadn't been used to mark the other 12, was completely alone. When his systems shut down, he didn't panic. When the liquid concrete began seeping out of the air vents, he didn't panic. He just lay down and accepted it. After all, he'd been expecting it. Four. O5-4 stared at the screen before him as the 13 lights shifted back and forth, here and there. One dies a little ahead of another, so the vote goes to her. Then she dies, so the vote travels here, until finally, at last, the votes were all his. The power was all his. The foundation was all his. Dr. Everett Mann, the administrator, allowed himself one appreciative cackle. Just one. No sense going all maniacal about it. Besides, he had so much work to do. His fingers flew across the keyboards, as he prepared to change everything. As he typed, he muttered to himself, Good evening, doctor. Oh no, don't stand up. Continued in Anon. End of file. To learn more about the SCP Foundation, subscribe to SCP Orientation today and turn the notification bell on, so you don't miss any of our videos.


Number one, if you are in a cybersecurity business, take a look at your website and your advertising. So pretend you're your sister, pretend you're a female friend, and with fresh eyes, take a look at what you put out there in the world. If you were a younger woman or a mid-career woman, would you feel welcome? Would you be excited about approaching your company for a job? Number two, get women in your leadership and on your board. So we know from social science research that if women and minorities see themselves in any field, if they see people like themselves, they're more likely to imagine themselves in that field and to apply for positions. So one, get women in your leadership and on your board. If they're there already, promote them. And if you need help with that, come to us. We will help you do that. Number three, get a sense of how much your employees are working. So going back to Elizabeth's pop-up, Elizabeth Weingarten's pop-up, get a sense for are your employees working late nights all the time? Because that's more likely to mean that women and actually young people are going to leave your company. So if you're interested in investing in the people you have, take a look at your overwork and decide if it's really worth losing women and young people.


Well, hello there. I'm Haslinda Armen. Thank you for joining us on Greening Trade. This session will be done in two parts. The first 30 minutes will be a moderated panel discussion. The next 30 minutes will be a detailed exchange that's limited only to foreign members. Of course, of this virtual wealth week, we heard from global leaders sounding the alarm on climate threats, calling for response, reiterating that green growth is absolutely possible. And to that end, trade can play a big part in saving the planet if it is done better, if it makes a shift to no or low carbon. Right now, international trade accounts for about 25% of greenhouse gas emissions. But the links between trade and climate have often been underexplored. And that's mostly because policymakers tend to look at both trade and climate in isolation. And add to that, trade in the environment communities may not necessarily see eye to eye. Still, the 2030 agenda calls on all countries to use trade to create a more sustainable, a more inclusive, a more resilient world. So how do we get there? What trade approaches and actions are needed? What can business play in the pursuit of net zero carbon emissions? Well, today we're really privileged to have with us, Frank Riester, France's minister delegate for foreign trade and economic attractiveness, Jerome Alhann, global senior partner and Clifford Chance, and a critic of head of sustainable development at Glencore International. Anna, gentlemen, thank you for being here. And that's the extent of my French. Let's start with you. I mean, is there a conflict between trade and environment protection and climate action? Can they be complementary? Essentially, are they France foes or are they frenemies? Hello, hello, everybody. You know, first perhaps, I should say that it's quite unrealistic to consider that we should do without trade. You know, natural resources are not shared equally. And I think that no country can be entirely autonomous. And so trade has brought many things to many people and has brought many people out of poverty. And international value chains lead to efficient specialization according to competitive edges. And so trade also allows supply chains to deliver indispensable goods across the globe. And so we will therefore maintain our economy in open economy. And we want the EU to remain open. And so for that, we need to have a consistency between our sustainable development agenda and our trade practices. We have to have a consistency between efforts we ask people and ask companies and the kind of trade we are supporting. And so it's very important to consider that in a political standpoint, these efforts of we ask to people are not diminished by the increase of imports from areas that not apply the same ambitions or the same standards. And so we have the globally to create more ambition and more standards so that we could globally increase results of our policy instead of creating the condition for a race to the bottom, in fact. And so perhaps I would like to give you two examples of what we are going to do or we are doing. First is the carbon border adjustment mechanism to fight against the carbon leakage in a WTO way compatible way. It's for us very important to be compatible to the WTO. And the second example is how we deal with deforestation imported. It's unbelievable that we could imagine that increasing trade could create more imported deforestation. And so that we have to create instruments within the EU regulation to legislation to fight against deforestation imported. And that's the problem we have in the trade agreement with Mercosur. And so we are trying to find solution to be sure that the increase of our trade with Mercosur wouldn't increase the imported deforestation. And we believe that it's important to create this regulation to put pressure as well to the private sector to be more fair in this kind of topic as well. Putting pressure on the private sector minister. I want to bring Jerome into this conversation then. Jerome, how do you think trade policy makers can better align incentives with sustainable zero emission supply chains? How would this help companies enact ESG commitments? Thank you very much, Haslinda. And thank you very much first of all for having me on this panel. And also good morning everyone or good day everyone on this last day of the diverse week. So thank you for joining on this last Friday. I agree Haslinda. The move to net zero must be supported by trade policies and you said in your introduction you spoke about friends or foes. And there is this sort of myth that trade and climate are somehow unnatural betelos that they are mutually exclusive and that they do work in two different worlds, you know, the WTO world and the Paris agreement world. And that doesn't have to be the case. Trade and climate can go hand in hand and should go hand in hand. And trade can be a driver of environmental and sustainable productivity and efficiency. And there's not much that governments can do to help move the needle on this. So let me mention four points. First of all, there is the elimination of tariffs on environmentally beneficial goods and to remove barriers on environmentally friendly services. And that's a clear opportunity and can be done in a way that is entirely consistent with the broader trade liberalizing objectives of the WTO. Now many will be aware of the negotiations for an environmental goods agreement between 2014 and 2016. Those regrettably came to an end, did not succeed, we can get into that later. But there are some moves to reinvigorate those discussions and that's good. Secondly, there is clearly more that can be done to remove non-tariff barriers to trade in environmental goods. So technical, environmental and other standards can be important and can be part of any regulatory systems. But those policies should not unnecessarily increase the costs of adopting technologies which help businesses to reduce the carbon intensity of their production processes. And thirdly, and this is a big one and we had President Biden speak about it two days ago, it is the removal of fossil fuel subsidies. These obviously slow down efforts to decarbonize businesses and again we can get into more detail later if you'd like. But it was very encouraging to see President Biden speak about that the day before yesterday and in his executive orders. And finally, and Minister Risto already mentioned it a moment ago, there is the big theme of carbon border adjustments. Those impose additional taxes on carbon-intense imports and if introduced they are likely to have a really significant impact on supply chains. Now carbon border adjustments and I had the Minister say it has to be done within the WTO framework. And that's a big debate again because they are scary on one hand, they can be very complicated, they can be fraud prone and they risk country protectionism. On the other hand they excite because if done right they will help adjust hard-to-bate industries. So I think those are four areas in which governments can help and it really is the way to bring the world of WTO and Paris together. So let me stop there as Linda, thank you. We'll pick up on fossil fuel subsidies as well as carbon tech slightly later. But first Anna, I want to come to you and talk about the circular economy. It is an important way to tackle emissions for many sectors and it's been estimated that eliminating ways through circular economy could bring benefits worth four and a half trillion dollars. But I mean as production and consumption is global what are the supply chain considerations for recycling materials here? Thank you, has Linda and good morning, good afternoon everybody, it's great to be here. When we talk about the global nature of production and consumption, exactly as you say, I think it's also important to think about carbon footprints also in that holistic way. And I suppose from our Glengor perspective we're a little bit, we look at it from two lenses. We look at it as a mining company with a footprint and we look at it as a company with a significant presence in recycling. So looking at the end use of these metals and giving them a second line. But let me take a step back and start at the start and think about this holistic perspective that we must talk about and that trade is fundamental too. I think any transition strategy must take into account the full spectrum of operational portfolio and commodity impacts and you alluded to that when you talked about the role that trade plays in the global carbon footprint. So from our perspective we looked at our, when we started thinking about our transition and our transition strategy we thought about our industrial footprint. But we quickly realized that really it's only 10 percent of our total footprint when we consider the emissions from the use of our products. So in our case our commitment to net zero covers our full footprint and we are the first mining company to do so. And we believe that that's critical to meaningfully participating in the global transition to carbon neutrality and participating in this whole of business, whole of world, whole of industry thinking. And I think when we think specifically about secularity we of course start by talking about contraction of some of the high emitting sectors with alluded to fossil fuels already. But of course we also say okay so recycling is key here and as you alluded to that as well. I think what we see in recycling today and this is where I think there's such an important role for greater coherence between policy and action is that we absolutely see today a growing interest in consumer interest and sustainability credentials and responsible sourcing and recycling content. And that's great and that then of course drives greater programming by metal using companies to take back products to collect the waste. But this is where we come up against the barrier and the barrier is that as we are looking to responsibly and safely collect and recycle the scrap we are then contending with a global package of regulations and of course some are it's unfair to generalize but as of course some are far more sophisticated than others but the broad regulation is still anchored in thinking of scrap as a hazard and is taking precautions to address that whether you're talking about collection and transportation achievements. And so in practice what we see is that for companies that are complying with this regulation it's incredibly difficult to recycle at scale. I'll give you an example some of the scrap that our customers have collected for us to recycle has been sitting in the country of origin for a year not because there's any problem with payment or any problem with the customer or any problem but because the regulation is so complex that dealing through the intricacies of this storage this shipping the transportation is incredibly complex and is very much framed within that context. And I think one of the downsides of that besides of course limiting the ability of the sector the formal sector to grow is the proliferation of the informal sector. And I think that theme is so important and it's something that the minister has alluded to as well. The community impacts the proliferation of informal practices that result in unsafe and poor environmental practices can be addressed through an evolved regulation that will address these questions of how do we think about recycling in this new world. How do we think of it as part of the package and how do we make sure that vulnerable people are not left behind or further marginalize as we're looking to to embrace this to embrace and enable this transition to public neutrality. I guess in a way it requires a mindset change to a minister I want to bring you back into this conversation. France has always been a strong advocate of strong environmental protections in trade agreements. What do you think ambitious trade partners could do to better support sustainable zero emission supply chains. First we believe that trade agreements are good tools to ensure consistency coherence between our climate ambition and trade. And so we want to use in the future even more the trade agreements to move to help us to move on these issues. And for instance we want in the future that in all future trade agreements the respect of the Paris agreement would be an essential close of the agreement. And we love to in which we are working on how to monitor the implementation of the Paris agreement within the relationship between the different countries with which we are signing some agreements. And second we think that we can already monitoring the implementation of trade sustainable development chapter because you know we have already in the bilateral agreements between EU and other countries some engagement of sustainable development. And so we think that we have to better check the implementation of these chapters. We can see that this chapter are already binding because the panel that was that has worked on the EU curry agreement has shown that there are some issues that are not correctly taken into account sorry by curry. And so we are putting some pressure on curry for that but we think that in the future we'll have to put some sanctions in the agreement in the chapters so that we could directly put sanctions if the commitments are not achieved. And third we think that in the multilateral level we have to do better. We have to put sustainable development even more in multilateral agreements in the WTO and the reform of the WTO will be useful for that. And we think that in the ongoing negotiation we could be better we could do better and I could say that the fishery negotiation could be a good example of how we could put more sustainable development in the agreements and in negotiations. And I totally agree with Anna that it's very important to use trade agreements to take the different private actors to a more circular economy to a more circular processes. And so it will be a day out of our ministry conference ministerial conference at the WTO level in the end of the year specifically perhaps on plastic issues. Jerome as you indicated slightly earlier sometimes it's also about removing incentives as it is about introducing incentives. And you mentioned fossil fuel subsidies by some calculation global carbon emissions will be cut by 20 percent and by doing that government coffers can be added by trillions of dollars. Yet this is not happening fast enough. Why is that so and what can be done about that. Well thank you as Linda I mean the abolishment of fossil fuel subsidies is obviously politically fraught. There is of course a big lobby for keeping them in place. They've been talking about abolishing fossil fuel subsidies I think since 2009. But you know the stars are now more aligned. We've had this week a lot about you know the net zero ambitions and targets of China of the EU of the US. And I think if you listened again as I said in my first opening remarks to President Biden and to John Kerry early this week also here in Davos about eliminating fossil fuel subsidies in the US which is particularly of course a country where there's been a lot of support for these subsidies. I mean that is a big move. I think the fossil fuel subsidies stand at something like 500 billion dollars and they're clearly distorted you know existing trade flows and incentivize continued reliance on these higher emission production processes. So I think taking away these fossil fuel subsidies will make a big difference. And we could also think that at a time when government finances are even more strained as governments seek to respond to COVID-19 and there's sort of less fiscal space the potential to redirect these environmentally harmful subsidies to other initiatives is I think significant. So I think we'll see real move there and I think the more certainties that exist for companies about the long-term shape of these policies and of these subsidies disappearing the more willing businesses will be to make the big shifts that are needed. Thank you. And Anna as you suggested earlier Glencore has been pretty successful in in meeting exceeding its ESG targets. Part of the whole equation is also social inclusion. Take us through the thinking behind that and how it's benefited. I guess it's provided more equality. Yeah. And thanks Linda. And I think that this question of inclusion is so much in the hearts of what we're talking about. And as COVID-19 brings more into focus questions of inequality of social detriment caused by the by the pandemic we need to ensure that as we are working to create and establish a low carbon economy we're not we're not leaving people people behind. As we think about that from from an energy perspective the importance of inclusive energy and the importance of again thinking about that policy perspective cannot be understated. But also I think as supply chains you know we started this panel by talking about the global nature of production and consumption and that just continues to be I think a sort of a thread that that we through this this conversation. One example that we've really grappled with and engaged with is the the the the commodity of cobalt you know an essential commodity needed for electric vehicles for batteries and it's it's it's it underpins in a lot of ways it's essential to the transition to low carbon economy. However we've seen how legitimate absolutely legitimate concerns about terrible operating practices and child labour that's associated with the artisanal mining of that cobalt in in part of the Democratic Republic of Congo has risked marginalizing further already vulnerable people as their concerns concerns would start driving sort of purchasing behaviors for for for the commodity. So we've been working with the whole with the cobalt supply chain with our partners in that chain that with with who we work commercially as well as as on the ground. We've looked to address this challenge by becoming a founding member of the fair cobalt alliance which is a whole of supply chain partnership that recognizes that that artisanally mined cobalt that cobalt produced by people you know very very poor very vulnerable people can absolutely play a legitimate role in the supply chain but that we must take action to eradicate child labour and and improve operating conditions in the in the in those cooperatives. And I think it's that the reason I give that example is because I think we must as supply chains and and obviously we're looking at support from from policy as well to be mindful of how as we're as we're looking to towards not only a low carbon economy but a successful low carbon economy that we don't inadvertently widen the gap the gap of inequality. Minister earlier you talked about the border carbon tax as planned by the EU how far do you think that will prevent carbon leakage. Do you see the move even encouraging non-EU companies exporting to Europe to kind of mission and in turn kind of what's charged to them or could the move lead to protectionist response by other trading blocks instead. It's for us we believe that this kind of mechanism will force other countries to be more ambitious in that field and that the the the costs cost more important to to to produce with within a green way will be will be compensed by the by the mechanism so that we could put pressure on countries that do not apply this kind of of a good way of producing and and so the difficulty is that we don't want that it could be a kind of competitiveness tool but a tool to to put everyone in a good in a good way to produce and that's that's the difficulty we have in front of us so that this mechanism could be accepted by by everyone in fact even by the US even by China and even by all the the country that are engaged in in the climate fight and we have to convince our partners of the of the pertinence of the of this mechanism. Jerome do you agree do you think it will encourage the others to follow suit? I think that would be the hope and I do agree with the minister it's a matter of communication communication and communication to getting to understanding why it is needed because without the the the the carbon border adjustments there is the risk of leakage you know both in getting imports from from countries where there are less environmental standards and the other way business is moving out of for example region to the EU to countries where that is the case and as I said in my introduction you know it's both on the one hand these adjustments are on the one hand scary there can be very complexly to a lot of bureaucracy and and also lead to protectionism which is why there's this concern and that is why there's a direct conflict with you know the the liberalized trade idea behind the WTO thinking but at the same time they can have very positive benefits as well and level the playing field so I do agree agree with the minister but it will be one that is potentially fraught with conflict and as we know currently the WTO is not very operational certainly not the absolute body so how do you resolve disputes now I'm a litigation lawyer so you know means a lot of business for people like me but I don't think it is what will what will further the cause of sustainability so I think that hell bringing together of the WTO and Paris thinking as I said in my introduction and you mentioned friend or foe and are they strange bedfellows I think there's a lot of work and thinking that continues to need to go into that and I heard the minister say you know we want to make these cross border adjustments WTO compliant but there's a level of complexity there but I think it is the right direction at the moment until there is a real level playing field that's right I mean the reorganization that the WTO could be a step in the right direction and I know we talked about how there are lots and lots of challenges but we've got to consider the role of clean technology clean and G can play a significant role in the transition to a green economy it helps resources be used in a more sustainable manner waste products can be recycled the question really is access to these clean technology what are the key issues here thank you the big ones look I think the first the first barrier and and really kind of going back to to the foundations of this principle barriers access to the commodities required for these technologies we cannot build solar panels winds wind turbines we cannot look at the transformation of the energy systems that that we need we cannot electrify transportation without access to metal we've done some modeling of the various climate scenarios that have been developed and we see in under any scenario whether we continue as we are or we achieve that radical transformation we see a material uplift in demand for metals such as copper cobalt zinc nickel and of course there's only so much metal that exists in the world you know I think we've seen under some of those scenarios demand for copper expecting to double by 2050 and for cobalt to quadruple where does that metal come from and of course recycling will be part of the answer and we've talked about that we've talked about the fantastic potential but also some of the barriers that that we see from our perspective today but I think we also have to think about the fact that we will need to access new metal recycling is not going to be able to fulfill all the needs we will need new metal and and this is going to be made more challenging going forward because the new minds that we know of the new the reserves that we know of today that are not being mined are increasingly located in hard to get to high risk difficult difficult to operate in in places with that that carry with them a very significant social and environmental considerations and again and I said as I said before it is so important to make sure that we don't lose sight of those of those of those considerations and make sure that we work on them hand in hand and so I think it's it's an important it's it's a barrier it's a consideration and I think there's some thought to be put to how policymakers in some of these mineral rich countries and how their partners in in sophisticated jurisdictions such as the European Union for example don't lose sight of how to continue to enable access to that new metal while also not compromising on on the importance of responsible operating practices how do we move those hand in hand there is a question here from a participant and minister maybe I'll post this to you do you agree that in the new normal we must reject trade at all costs and push instead for trade to be built on credible inclusivity sustainability integrity and transparency and I guess you know amid the pandemic there's always an urge to put yourself first the beginning of the panel that in my opinion it's unbelievable to think that we should do without trade because it's useful for many people and it's useful for economy and many people in development countries are waiting for some services some products we have in some other countries since a long time so the issue is how to produce and and to exchange and how to produce in a greener way recycling thanks to innovation so we have to invest a lot in innovation and we have to be sure that the the increase of trade would not have a negative impact in specific ways I talked sooner about deforestation and I could I could say that the the trade could be even could be perhaps a way to push some countries to do better in in sustainable development that's really in our mind in France and in Europe it's to use the the lever the tool of trade agreements of trade partnership to push people that need to do to to trade with us to to do better in in sustainable development in biodiversity in deforestation in climate change fight so no I think that we have to to to be open we have to be determined to maintain a international trade for all the reason I advocated jobs prosperity access to services and to to certain products for many countries but and we have to invest a lot in technologies so that the way we produce could be a more sustainable and we have to use the tool of trade agreements to push everyone to go up in our practices and to produce to consume and I think it's possible the the fact that the US come back in the Paris agreement and the fact that Biden and Kerry and and the new administration is so implicated so involved in sustainable development issues is a good news for all the world and I think as China is and says says that they are really committed on these issues as well I think we could do much better in the years to come on that very optimistic note minister I'd like to thank you your own as well as Anna for the insights today we wrap up the first session of this hour-long session thanks for joining us thank you for joining the conversation


we want Regan, we want Regan, we want Regan, we want Regan, we want Regan, we want Regan, we want Regan, we want Regan, we want Regan. Ladies and gentlemen, this is such a beautiful site right here on the Mississippi Gulf coast. I'm Mississippi by, but those days exist no longer because you have made it possible. Now we've got Mississippi here from all over Mississippi. Possible, the President of the United States. Thank you. Thank you very much. And thank you Trent for that introduction. I was having so much fun listening, I kind of hated to get up here. Thank all of you for a most heartwarming welcome. It's great to be in Gulfport. And greetings to all of you who come from the Delta, the home of the outstanding congressman, Webb Franklin. Webb can't be with us here tonight because he's up there with my friend Billy Munger having a fundraiser up in Jackson. Carry on so he can be back there again in the Congress. And of course, it's always a great pleasure to be here with Trent Lott, who as minority whip in the House of Representatives has worked with skill and devotion. And I know it's a great uphill battle, and a very difficult one. But just as he remarked about a certain speaker of the House of Representatives, wouldn't it be wonderful to be calling Trent Lott the majority whip? And we have with us here a candidate for the Congress who could add to our ranks up there David Armstrong. Send him there, we need him. Now, we couldn't have accomplished half of what we did without a Republican majority in the Senate. And Thad Cochran is in the first rank of that majority, and that's why he's there. He and some other stalwarts are battling against those who are still out for the old fashioned idea of spending your money faster than you can send it in. But it's great to be back in the South. You know, I'm always happy when I visit this part of the country, maybe because you make me feel so much at home. You in the South have always given our country more than its share of greatness and courage. Here are the traditions and values that shaped our land. Here is steadiness of purpose, fidelity to ideals, love of country. Our opponents may be ready to ignore this region, but Vice President Bush and I happened to consider that the South is worth respecting, worth listening to, and yes, worth fighting for. We've come here for one reason and one reason only, to win. And we ain't just wishlundixing. The choice between us and our opponents is the cleanest our country has faced in more than 50 years. According to them, America's future is grim and getting grimmer. They say our people are despairing that we have nothing to hope for but fear itself. Well, America has fallen on hard times, they claim, and they place the blame squarely on our administration. You know, sometimes I get the feeling that they don't like me very much.  It's time for them to wake up and look at the facts. Four years ago, the economy of the South and of all America was in a shambles. In just five years, taxes had nearly doubled. The average monthly mortgage payments more than doubled. And the real after-tax income of the average American actually began to decline. From 1979 through 1980, just two years, working Americans' weekly earnings declined in real terms by 8.8%. That was the worst drop since World War II. Our defenses had grown so weak that many of our planes were too old to fly, and those that weren't too old couldn't fly for lack of spare parts or lack of pilots. Many of our ships couldn't leave port. Today, just four years later, the United States of America is a very different place. It's stronger, more prosperous, and bursting with patriotism. Well, now, if, as our opponents insist, I have to take the blame, well, all right, if you'll share it with me. This isn't a Reagan recovery. This is an American recovery. This great nation is moving forward again, and we're not going back to that unhappy past. We've knocked inflation down from 12.4% to 4.2%. Productivity is up. Consumer spending is up. And take-home pay is up. In contrast to that 8.8% decline in really weekly earnings that I mentioned, in the last two years, they've risen 3.2%. You tell me whose policies are more fair to the working people of America. During the past 20 months, we've created 6 million new jobs. We've been creating more jobs every month than our allies in Europe have created over the last 10 years. Your great success story has come as quite a shock to the professors and the economists. Back in October of 1982, one economist, often seen and quoted, claimed that the engines of economic growth have shut down here and across the globe, and they're likely to stay that way for years to come. That was just two months before the start of the best recovery in post-war history. That's when I hear some of the projections they make for what it's going to be like in 1989. They're blowing smoke. They don't know what's going to be like in 1989. It all adds up to a simple lesson. Don't trust the professional pessimists. Trust the American people. My friends, I believe our great nation has turned the corner. The shadows are behind us, and the bright sunshine of hope and opportunity lies ahead. But I wouldn't want to take that for granted. So let me just ask you, do you feel better off than you did four years ago? Is America better off than it was four years ago? Well, now, despite this strength of this expansion, there's one sure way to ruin it. You'd have to be something of an expert to do that. But when it comes to bringing economic growth to a grinding halt, our opponents are experts. After decades of a rising tax burden, they want to give the American people a massive new tax increase. They call it bitter medicine, but that's just because they think they can get us to swallow anything. I think the word shrimp means something different to our opponents than it does to Gulfport. To you, it's a livelihood. To them, it's your paycheck after they get their hands on it. All told, their tax increase would be the equivalent of $1,800 in new taxes per household. Are we prepared to sit back and let them do that to America? We all watched the Olympics this summer, and we cheered to see American athletes go for the gold. Well, making our economy bear the burden of their tax hike would be like having a coach tell an Olympic swimmer to do laps carrying an anvil for a runner to sprint with a ball and chain. Come November, the American people will get to vote on our coaches. And come November, the American people are going to tell Coach Taxhike to go find another team. Our pledge is for tax simplification, to make the system more fair and easier to understand. Do you know that Albert Einstein has said he cannot understand the Form 1040? We want to make it simpler and easier so that we can bring yours and everybody's income tax rates further down, not up. Tax simplification will provide powerful new incentives for economic growth. We'll fight for enterprise zones to help Americans in disadvantaged areas get off unemployment and welfare and start climbing the economic ladder. And we'll keep government under control by working for a line item veto. I had it as a governor, 43 governors have it. The president needs it. And then a constitutional amendment, mandating that government stops spending more than government takes in. A Southerner named Thomas Jefferson, backed shortly after the Constitution was ratified and put into place, said it had one lack. It needed a clause to prevent the federal government from borrowing. Jefferson was right and it's time we recognized it. Now you might have noticed that our opponents are trying to appeal to traditional Democrats by comparing themselves to Harry Truman. Well, President Truman kept a sign on his desk that said, the buck stops here. But if our opponents are elected, their sign will stay, your bucks stop here. Forgive me, but Harry Truman believed with FDR before him and John Kennedy after him in strength of broad and self-reliance at home. Now to all those Democrats here today, and I hope there are many, there would have to be in this place. And I hope that you feel that under its present leadership, the Democratic Party no longer stands behind America's role in the world. That it no longer represents the working men and women. So we say to you, and I say this as one who for most of my adult life was also a Democrat, come walk with us down the new path of hope and opportunity. Add your strength to ours. And all of us can build something new for America, something far better than before. And it will be a true bipartisan achievement between millions of patriotic Democrats who know they can no longer follow the leadership of that party. As our economy grows, we'll need to go forward with bedrock values that have always sustained the people of the South, the values of faith, family, neighborhood, and good hard work. And together we're already making an impressive start. We've helped lead a grassroots revolution for excellence in education that will reach every child in this land. Recently, we learned that scholastic aptitude test scores have gone up a full four points. Now that's the second improvement in the past three years and the biggest increase in those scores in 21 years. We must continue to crack down on crime. We say with no hesitation, yes, there are such things as right and wrong. And yes, for hardened criminals praying on our society, punishment must be swift and sure. In 1980, the crime rate was rising. Well, last year, reported crime dropped 7%, and that's the steepest decline since 1960. No crime strikes harder at the heart of America than drug smuggling. For drugs tear families apart and turn healthy, productive Americans into sick people, unable even to care for themselves, men and women who often turn to robbery to pay for their habit. We've established the National Narcotics Border Interdiction System under the superb leadership of Vice President Bush to wage war on drugs. And today, drug seizures and arrests are at record levels. I know that Coast Guard men and women stationed here in Mississippi are playing a leading role in this vital effort. In a case that just took place days ago, the Coast Guard cutter, Akushnet, right here in Gulfport, came across a fishing boat that flew no proper flag. The Akushnet pulled alongside, and five Coast Guard officers went aboard. Well, the Coast Guard officers knew for certain something was wrong when they saw that all the hatches had been nailed shut. In the end, they confiscated almost four tons of marijuana, towed the fishing boat into Gulfport, and made six arrests. I know that you'd like to join me in thanking the officers and crew of the Akushnet for a job well done. In foreign affairs, in foreign affairs America, is at peace. Since 1980, not one nation has fallen to communist aggression while the people of one country, Grenada, have been set free. And as I look south over the Gulf, I can't help thinking of one troubled part of the world, Central America. Some would have us take a position of weakness in Central America or withdraw from the region altogether. They want to hang a do not disturb sign on our border and pretend there are no problems. But that would betray the brave people of Central America and ignore our own national interests. For if Central America falls, make no mistake about it, refugees will flood our borders. I pledge to you that we will give firm support to the forces of liberty, democracy, and economic progress, and that we'll do so as long as I hold this office. Well, I wasn't going to, but if you insist. Now, I know that many of you have a special interest in the great effort to defend our freedom. Whether you serve in our armed forces, help to build our ships or our veterans and have served our nation with pride, let me assure you we're determined to give America defenses that are second to none. Since we took office, we've built up our stocks of ammunition, we've replenished our military supplies, and we've begun work on new equipment, including important new aircraft. And today, America is once again giving the men and women in our armed forces the pay, the trading, and the respect they've always deserved. One of our proudest accomplishments concerns the Navy. In the two decades before we took office, our total fleet had been cut nearly in half. Well, we've turned that around. In the past four years, we've added 44 ships to the fleet, bringing the total to 523. And that's well on our way to our goal of 600. Now, many of the new ships have been built right here in Mississippi. And when I look at the sleek ships around us, once again, we can say with pride, Columbia is the gem of the ocean. I told Soviet Foreign Minister Gromyko when I met him at the White House on Friday, the United States seeks no territorial expansion. We make no attempts to impose our will on anyone, but we will never again allow America to let her guard down. I'm supposed to end my speech, but I, well, yeah. But I can't help but remark about something that I've seen more and more at rallies of this kind all over the country. So many bright young faces here today, so many young people. Listen, I can remember a time when the only young people that ever came to one of our rallies looked like they couldn't join anything else. Listen, and I see enough short haircuts that I believe there are some people here that aren't in uniform. But they wear the uniform every day, men and women. And may I stay just long enough to leave one message with you one last thought from my heart. I have to tell you that nothing, and I mean nothing, has made me prouder these past four years than the young men and women who are serving our country in uniform. By any measure, by any measure, those young people I've mentioned are just the best. And I have to tell you something, and forgive me for missing a word that you children should never use. Well, none of us should. But it's been said that back in World War II, as a matter of fact, it is a fact that someone asked General George Marshall what was the secret of America's success, what was our secret weapon in World War II. And General Marshall said, the best damn kids in the world. The young men and women serving today are the grandsons and granddaughters of those heroes. And I'll tell you, he was right. They are still the best damn kids in the world. Now and then on the news, a commentator will be talking to one of our young people in uniform. And it's kind of struck me that so often you hear the when the young man in uniform or woman in uniform answers you so often hear the proud and lilting cadence of Charleston or Memphis or Jackson or Gulfport or Biloxi. The south was the home of patriots in 1776, when a southerner wrote our Declaration of Independence. And today, more than two centuries later, the south is the home of patriots still. So maybe when you see one of our young people in uniform walking along the street here in Gulfport or Biloxi or wherever, maybe if you think of it just nod and smile and say hello and maybe let them know how all of us feel about them, it'll make you feel real good and I know how good it will make them feel. Well now, I do have to go.  Send us all back there.


Thank you for coming out early in the morning. I'm going to be talking today about a recent revival of ideas related to moonshine, which have been developed starting in 2010 to the present, going on. And this new kind of moonshine involves, at least in one way of formulating it, Jacobi forms N equals 2 and 4, super conformal algebras, objects called mock-modular forms, the nemire lattices that we already encountered, and a number of other things. But I think it will suffice to try to explain some of these things. Now, I know that the first two lectures might not have been crystal clear, but the situation regarding moonshine in those kind of classic examples where there's an explicit conformal field theory is much clearer than the situation involving this Matthew and Umbral moonshine, where there's a mathematical framework, which is perfectly consistent and makes sense and satisfies all sorts of non-trivial properties, but the physical understanding of what's going on in terms of symmetries of something that a physicist or string theorist would recognize that as a conformal field theory, sigma model, et cetera, is confused and a lot has to be done to try to really understand what's going on. But I'll try to at least be as clear as I can regarding the mathematics. And then towards the end of the last lecture, I'll try to talk a little bit about attempts to fit it into some kind of a physical framework and what ideas people are exploring, although nothing has been completely successful so far. So let me start out by talking about Jacobi forms and how they show up in computations and string theory. So when you study string theory, you encounter what mathematicians would call an elliptic curve. And in string theory, this arises because you have one loop string amplitudes on the torus. And any torus, you can choose the metric to be conformally flat and then up to conformal transformations that are uniquely specified by a parameter tau, which lives in the upper half plane. And on this elliptic curve, you can choose a point z. And there are two classes of transformations which are very natural to consider. The first are elliptic transformations, bending z to z plus mu tau plus lambda, which just corresponds to taking this point and shifting it up this way or over this way by one of these two lattice vectors that specifies this elliptic curve. And the other thing that's natural to do is to look at modular transformations, which correspond to global diffeomorphism of this torus. And those correspond to transformations of z that look like this and tau, which have the familiar form from one loop amplitudes in string theory, where a, b, c, d are integers and the matrix has determinant 1. Now, if you ask for functions that transform nicely under the elliptic transformations, you get elliptic functions, Bierstrass function. But those functions secretly depend on a parameter tau, which is used to label this torus. And if you then kind of ask just to look at this modular transformation and ask for functions that transform nicely, you get modular functions. But in a way, the most natural thing is to consider functions that transform nicely under both of these, and those are Jacobi forms. They're functions of z and tau that have nice transformation properties under both the elliptic and the modular transformations. Now, what do I mean by nice? Well, it's useful to look at a kind of motivating example, which is the Jacobi theta function, which gives Jacobi forms their name. So the Jacobi theta function of z and tau defined by following sum, it's convenient, as usual, to let q be e to the 2 pi i tau, and to call y pi i z. And it's easy to work out what the transformation properties of this Jacobi theta function are under both the elliptic and modular transformations. So in particular, you will find that under elliptic transformations, it transforms like this. And there's nothing very complicated about this. You just stick in z is equal to z plus mu plus lambda tau. You rewrite the sum by changing the summation variable around a little bit. And this pops out after a couple lines of algebra. And you can then look at how it behaves under modular transformations. And I'm only going to write down special cases here. So first of all, it's clear that if I take tau to tau plus 2, then this term doesn't change in some of the theta functions invariant. And a slightly less trivial computation using Poisson summation shows that if you take tau to minus 1 over tau, you get the square root of tau over i. e to the pi i z squared over tau, theta of z tau. And so this follows from using the Poisson summation formula. Now, these transformations are clearly, well, sl2z is generated by tau goes to tau plus 1 and tau goes to minus 1 over tau. So we don't really have here transformations under all the generators because we're missing tau goes to tau plus 1. And if I do tau goes to tau plus 1, you can see that this term is going to change by a minus 1 to the n. And so I get some other function. This factor of the square root of tau is what you would expect for a weight 1 half modular form. We could try to make precise what the modular transformations are here. But if we just take this as a motivating example, we could try to extend this to a more general framework. And that's what Eichler and Zagier did in their book on Jacobi forms, which is quite readable, very explicit. So they said, let's take this combination of transformations and try to make it make sense for the full modular group and rather than for half integral weight for integral weight. So they defined a Jacobi form of weight k and index m, or k and m are integers, be a holomorphic function by from the, I guess maybe I should write it as the complex plane and the upper half plane to the complex numbers, thinking of this as the variable z and this is the variable tau, which obeys an elliptic transformation property. And if we look at this elliptic transformation property, and although we're not allowed to in this definition, we put m equals 1 half, you would see that it's precisely the elliptic transformation property of the theta function, which is why this is generalizing it from a half integer to general integers. And then under modular transformations, I'll use a shorthand notation where gamma z of tau is z over c tau plus d, a tau plus b over c tau plus d. So under a modular transformation, you get c tau plus d to the k, which is the usual factor for a modular form of weight k. And then you get a slightly more complicated formula here. But again, for m equals a half and for tau goes to minus 1 over tau, you can check that this factor agrees with the factor that you have here. And these somewhat strange looking factors are needed in order that the combination of these two transformations, which generate the Jacobi group, acts nicely on this set of functions. So that's the definition of the Jacobi form. And there are a couple of properties which follow quite easily. I won't prove, but they involve fairly straightforward manipulations. And they're useful in analyzing them. So the first property, kind of trivial, the transformations tau goes to tau plus 1, and z goes to z plus 1, d phi invariant, as you can see from these transformation laws over here. So that means that you can write a Fourier expansion in terms of q, which is e to the 2 pi i tau, and y, which is e to the 2 pi i z within an R integer. And then you can show that these coefficients using the elliptic property, you can show that these depend only on quantity, which I'll call the discriminant, or which they call the discriminant, d, which is 4mn minus R squared, and on R tilde, which is the value of R mod 2m. So in other words, I can view these c of n and R as really being some other coefficient I'll call capital C. That's a function of d and R tilde. And in the sum, any c and R, which have the same value of the discriminant 4mn minus R squared and the same value of R mod 2m, will be equal. So that equality allows you to pick out the coefficients that are all equal and reorganize the sum. And when you do that reorganization using this property, you find that you can write pi. So I guess I should say here, I'm looking at something that has weight k and index m. That's where the m comes in. So the second property is that you can, by rewriting the sum as I described, you can do the sum over R first over all R's that are equal to the same R tilde mod 2m, and then over the R's from 1 to 2m minus 1, or whatever. And when you reorganize the sum that way, you can pull out all of the z dependence into something called a level m theta function, which is given by a sum on integers n to the 2mn plus R squared over 4m times y to the 2mn plus R. And where hR of tau is equal to a sum on these discriminants d, these new coefficients c of d and R due to the d over 4m. All right, so that's a lot of somewhat complicated formulas. What is this guy? Well, these theta functions, since they have a nice explicit form in terms of the sum of exponentials, you can use Poisson's summation, which is sort of the general tool for analyzing modular transformations, to deduce the modular transformation properties of thesestatas. I'm not going to write them down, but you can find them in all sorts of places. And you can use this and the modular transformation properties of phi to deduce that these guys are vector-valued modular forms. And what that means is that under modular transformations, there is some matrix, depending on a, b, and c, d, and then some hs of tau. And I guess also a weight, c tau plus d to the k minus 1 half. So there are the sort of things that are often encountered in conformable field theories, where you have different sectors of a rational conformable field theory and the different sectors mix into each other under modular transformations. All right, so after that little barrage of math, I want to pause for a minute and ask if there are any questions. And then I'm going to tell you how these things arise in spring theory or in super conformable field theories. The elliptic transformations are, they're not global dwarf morphisms. Essentially, what they are is just translations by a lattice vector. And if you had a function which was invariant under these transformations, you would just have a function which was well-defined on the torus. But you can't have a holomorphic function which is well-defined on the torus because it's not constant. So you either need to look at meromorphic functions or you look at functions that are not really defined on the torus but are rather sections of a line bundle over a torus. And these phases in the transformation laws under elliptic transformations are really telling you that you should think of these Jacobi forms as sections of a line bundle over the two torus. So that's their geometrical interpretation. It's not connected to any kind of global dwarf morphism of the torus. Could you just speak up a little bit? I don't quite understand. Oh, it could have been written in product form? Some Jacobi forms can be written in product form. But I don't know if there's any general result that every Jacobi form can be written in a product form. I'll write down some examples a little bit later. I think they can maybe be written at least as sums of products because many times they can be written in terms of theta functions. And then you can use the Jacobi triple product identity to rewrite those products. But I don't think there's necessarily just a single product that you can write them as. OK, so I now want to explain how these show up. One way of understanding where they show up and where these properties fit into physics. So when we consider compactification of string theory on a Calabi-Aus space, x, we know that that leads to an n equals 2 super conformal field theory. And provided there's a quantization condition on the u1 charges of this n equals 2 super conformal field theory, you can always construct spacetime supercharges. Now, this theory has a stress tensor. It has two dimension 3 halves super currents. And it has a dimension 1 u1 current. I'll call the modes of these ln, gr plus and minus, and jn. And as is familiar, I think you either have half integer r or you have integer r, depending on whether you're in the Niver-Schwarz or Ramon sector of the super conformal algebra. And I'm not going to write down the commutation relations exactly, but they have the rough kind of form. And they tell you what the charges are of the g plus minus under the u1 current.  And j is primary under verisoral, et cetera. You can just, there's a Wikipedia page on n equals 2 super conformal algebras, and you can find the commutation relations in detail there. But there's an interesting fact about this algebra, which was pointed out by Schwimmer and Seiberg. And that is that there is a spectral flow isomorphism between the Niver-Schwarz and Ramon versions of this algebra, something that is not true for n equals 1. So what this involves is taking ln and redefining it to be ln plus mu jn plus c over 6 mu squared delta n0, redefining jn to be jn plus c over 3 mu delta n0, and taking gr plus or minus to gr plus mu plus or minus with mu equals 1 half. So you just take the algebra, write it down explicitly, make this substitution, and you'll find that you get exactly the Ramon algebra starting with the Niver-Schwarz algebra or vice versa. So they're really isomorphic under this. And you can extend this to integer mu. For integer mu, this takes the Niver-Schwarz sector back to the Niver-Schwarz sector, the Ramon sector back to the Ramon sector. But it's still an untrivial isomorphism because it shifts ln by something proportional to jn and shifts jn by a constant for j0 and leaves the sector fixed. But it's still an isomorphism of the algebra. And you can check that there's a quantity 2c over 3l0 minus j0 squared, which is invariant under this shift. So it's going to turn out that this spectral flow isomorphism implies essentially the elliptic transformation property of a certain quantity, which is purely topological in this theory. So in an N equals 2 superconformable field theory, we can define the elliptic genus. And we'll think of this if we have some collabial space x, which gives rise to this N equals 2 superconformal field theory. As a sigma model of x, we'll call this the elliptic genus of x. It depends on the collabial manifold x and then on two variables z and tau. And it's given by a trace in the Ramon sector, so Ramon for left and right movers, u to the l0 minus c over 24, bar to the l0 twiddle minus c twiddle over 24, minus 1 j0 plus j0 tilde to 2 pi i z j0. This quantity is also written as minus 1 to the f left minus f right. So what is going on here is that we have weighted things on the right. That is the q bar dependence. By minus 1 to the f, we're basically computing minus 1 to the f right times q to the q bar to the l0 minus c over 24 on the right. And this quantity is the written index and is non-zero only for ground states with l0 tilde equal to c tilde over 24. Otherwise, if you're in an excited state on the right movers, the supersymmetry, the G-naughts in the Ramon sector, pair states that have opposite values of minus 1 to the f just like in supersymmetric quantum mechanics. And so all those excited states on the right cancel when you take this trace. So in fact, this is a function only of q and of y, which is e to the 2 pi i z. There is no dependence on q bar because this q bar dependence completely goes away when you take the trace and include this. On the left, this cancellation doesn't happen because you're weighting the states by their J-naught eigenvalue. And this differs for fermions and bosons. So if I were to set z equals 0, then only the left moving ground state would contribute. So the elliptic genus at z equals 0 and tau will be a constant because then you'll have cancellation both on the left and the right except from the ground state. And this is essentially trace minus 1 to the f just in the supersymmetric quantum mechanics on this space. And that gives us the Euler number of x, a famous fact from long, long ago. So this elliptic genus is something that is counting right-moving ground states, arbitrary left-moving excitations. And those are well known to be BPS states. So this is really capturing some kind of topological information about x. And that intuition has a space-time interpretation in terms of the kind of infinite set of indices of Dirac operators coupled to various tensor products of the tangent bundle, which I won't try to go into. But there's a space-time computation of this in terms of index theory. I'm just talking about the world sheet description. So this is a very interesting gizmo to compute. And you can use spectral flow, which turns out to be equivalent to the elliptic property, and the usual modular properties, one loop string amplitudes, and the fact that the Ramon-Ramon sector is left invariant under modular transformations to show that the elliptic genus of x, z, and tau is a Jacobi form of weight 0 and index m, which is 1 half of the complex dimension of the Calabi-Alspace x. This is essentially following from the formula for what the central charge of the conformal field theory is in terms of the complex dimension. I should add one small caveat here, which is that technically speaking, this is what's called a weak Jacobi form. And a weak Jacobi form is a Jacobi form that satisfies the condition that these coefficients are 0 whenever n is less than 0. So in other words, there are no negative powers of q in the expansion. There's a slightly stronger condition, which applies to other classes of Jacobi forms. But this is the Jacobi form that this is the condition that's suitable for the elliptic genus. I guess one could make a connection. Yeah, I mean, well, one way of making a connection would be if you look at string theory compactified on a K3 manifold, then you get a theory which can be chiral in six dimensions, and that theory has various anomalies. And those anomalies are related to the index of various operators on K3. So yeah, one could connect it to anomalies, but I'd have to think a little bit about what the most complete statement is. But yeah, there is a relation, but it's not going to be essential for what we're doing. Yeah, yeah, it's right. It's essentially, well, I won't try to write it down, but it's a kind of, there's a really complicated formula for it, but essentially you take a Dirac operator, but instead of the usual coupling to the tangent bundle, you take it coupled to some tensor product of the tangent bundle, various anti-symmetric and symmetric products. And then you arrange these things, you kind of grade it, and then the sum of all of the index of all these graded Dirac-like operators are what the elliptic genus is. And you can find explicit formulas, or I can tell you where to find explicit formulas for what that infinite set of Dirac operators is. There's no left or right. The operators I'm talking about are operators in spacetime. So operators on, Dirac operator on the Klavier manifold x. The left and right is purely a world sheet description in string theory. The spacetime interpretation involves index theory on the Klavier space x. So it's interesting to compute this. And one of the first non-trivial examples that was commuted, computed, was the elliptic genus of K3. And well, it turns out that there is a unique weak Jacobi form of weight 0 and index 1. And that unique weight 0 index 1 Jacobi form can be written in various ways. But one formula for it is to take theta 2 of z and tau over theta 2 of tau to the fourth power, plus the same thing with theta 3, same thing with theta 4, where theta 2, theta 3, and theta 4 are the usual theta functions. Sometimes I guess they're called, well anyway, these are the three non-odd Jacobi theta functions. And we can figure out the proportionality constant because the elliptic genus of K3 at z equals 0 should be the Euler number of K3, which is equal to 24. But if I look at this thing evaluated at z equals 0, it's clearly equal to 12. So we learned that this is equal to 2 times 501. Now, this was the starting point for the following computation. For K3, because it's hypercaler rather than just caler, there is actually an n equals 4 superconformal algebra. And therefore, the elliptic genus must have a decomposition into characters of the n equals 4 superconformal algebra. And Oguchi Oguri and Tachikawa, 2010, worked out what the decomposition into characters of n equals 4 is equal to. I understand it involves a formula, which was actually in Oguchi's PhD thesis. But they interpreted it as well. And so I'm going to tell you what they did. So characters of the n equals 4 algebra are defined by taking a trace in a highest weight representation where h and l are the l0 and j0 eigenvalues of the highest weight state. And then as usual, you act on that highest weight state with the creation operators to fill out a multiplet. And I'll define the characters to be the trace of minus 1 to the f, q to the l0 minus c over 24, y to the j0, and a highest weight representation labeled by h and l. So there are formulas for these characters, which were worked out by Oguchi and Taormina. They're somewhat complicated. And I'm not going to write them down, but it's a mechanical thing to decompose one function in terms of the sum of other functions. And so you do that. And what you find is that you can write the elliptic genus as the odd theta function squared over the dedicated eta function cubed times two terms. This is a kind of complicated object, but it comes from what is called a massless n equals 4 character, which has h equals a fourth and l equals 0. It's called a massless n equals 4 character because a multiplet with this h and l value gives rise to massless states in spacetime. And I'm not going to write down what that function is, because I don't need it right now. And then there's a term h2 of tau. And this is a function that has a form, which I'll write down explicitly. It encodes the degeneracies of what are called massive characters where you compute the characters with h equals a quarter plus n and l equals 1 half, which in the simple example are the only other characters that appear. So in other words, this guy has an expansion q to the minus 1 eighth, which comes from various things. And then there's an integer expansion in positive integer powers of q that label the characters with h value a quarter plus n.  And for further reference, I'll write this is the sum equals 0 to infinity c of n minus 1 eighth q to the n minus 1 eighth. Now what Aguchi Aguri and Tachikawa noticed is that 45, 231, and 770, and the next coefficient as well, are all dimensions of irreducible representations of a sporadic group, the Matthew Group M24. So that observation is the starting point of what's called Matthew Moonshine. And you'll notice that it's a bit different from the starting point of monstrous moonshine. So the starting point of monstrous moonshine was the fact that J of tau up to a constant term, which doesn't matter, looks like this. This had a nice decomposition into dimensions of irreducible representations of the monster. But this guy was a weight 0 modular form. And weight 0 modular forms can appear as the partition functions of conformal field theories. This guy is something else. It is a weight 1 half Mach modular form, which I'll have to define for you. But it has a more complicated transformation property under modular transformations. It's not weight 0. And it's not interpretable in any way that I know as the partition function of a conformal field theory. So the other thing is that when you interpret this as a trace of q to the l not in a conformal field theory, all the states contribute with a positive sign. So these numbers are really counting positive dimensions of vector spaces. But this arose in a computation of the elliptic genus. And this elliptic genus involves trace minus 1 to the f and other stuff. So states can contribute to this with a plus sign or a minus sign. So these coefficients are not manifestly positive. As a matter of fact, this first one's negative. It turns out that all these coefficients are positive. But it's not completely manifest from the definition that that should be the case. So people have been trying to understand. I said, yeah, why do you say that? Somehow I'm going to swore it was 4. Let's see. It should be index 1. He's saying these guys transform like index a half. And so when you take the fourth power, you get index 2. But I said it should be index 1. I think that's what you're saying, right? Let me, I'll check. I think this is right somehow. But all right, I'll. What? Yeah, so I think it must be that these theta functions, if you look at the index, if you look at their elliptic transformation properties, it must be essentially like, it should be, no? Well, OK. Yeah. Look it up. Somebody look it up. I thought this is right. But I'll have to reconcile these things. OK. So what are we supposed to do with this? Well, I don't know. You can just forget about it. It's a weird curiosity. But it cries out for some explanation. In monstrous moonshine, the answer was to find a conformal field there with the monster acting as a symmetry group. So it's the first thing you would think is, let's find, a K3 surface that has M24 as a symmetry. But that doesn't work. It should be squared. My apologies. At least I got the factor of 4 out in front, right? So there is no K3 surface with M24 symmetry. Well, you have to be a little more specific by what M24 symmetry means. What this really means is an automorphism that preserves the symplectic form, which is needed in order to preserve spacetime supersymmetry and to preserve the n equals 4 superconformal algebra that you've used to do this decomposition. So it's known that all the symmetry groups that can act this way in K3 are subgroups of M23, which is another Matthew group. But they're smallish subgroups. And you can't get M24. So you might say, OK, but we know that in string theory, there are things that look like K3 compactifications that might not be so easily described geometrically. And really, what we should be looking at is an n equals 4 superconformal field theory with a discrete spectrum of states corresponding to a compact space with c equals 6. So that's there. And with such that the Euler number is 24. So that is sort of the conformal field theory definition of what a K3 surface is. It's just a superconformal field theory that has n equals 4, c equals 6. But you can analyze this question. This was done by Gabberdeel Olpato. And again, there's no such beast with M24 symmetry. So the most obvious attempts at an explanation of what's going on fail. And well, if you can't do physics, you can always do math. So essentially what people did in this problem was to try to investigate mathematically whether this structure was consistent in the same way that the structure of monstrous moonshine was consistent. There, people computed these Mackay-Thompson series, Conway Norton did, before the conformal field theory construction by Frank Lopowski and Merman was known. And you can do the analog here. So that's what was done in papers by Chang, Gabberdeel, and collaborators, Gucci, and Hikami. So essentially what you do is you play the same kind of game that we had for the monster. So remember for the monster, said, well, we could assume that there's some infinite dimensional module that was graded such that the dimension of Vn was the coefficient An in the expansion of J of tau. So we can play the same kind of game here and say, let's suppose that I should really interpret these numbers as being dimensions of vector spaces on which M24x. So that would be the statement that there is an infinite dimensional graded module for M24 such that the dimension of this bit labeled by N is equal to these coefficients N minus 1 8. And then if we assume that there's a structure like that, we can certainly compute the analog of the Mackay-Thompson series. As we can look at twisted versions of these things by computing the trace of an element g in each one of these representations weighted by u to the N minus 1 8 and try to choose a decomposition of each one of these spaces into one of the 26 irreducible representations of M24 with positive integer multiplicities. And in the case of the monster, we demanded that these things be modular forms on a congruent subgroup and actually weight genus 0. What are we supposed to choose here? So what kind of function should be these b? Well, the answer is they should be Mach modular forms for essentially gammonaut of N, where N is related to the order of g. That's not quite correct, but it's close. Now, the problem with this formulation is that I haven't told you what a Mach modular form is and how that's supposed to replace the modular forms that we had in monstrous moonshine. So I think I'm out of time for today. So that will be part of the subject of the next lecture I give. I'll tell you what Mach modular forms are. And then I'll tell you of an extension of this structure to umbral moonshine, which is also not understood. That is, there's no explicit string theory, conformal field theory, black hole, whatever background that explains what's going on. But I'll talk kind of qualitatively about some of the ideas for trying to explain what's going on. And maybe one of you can implement that or find some better idea to try to explain the mystery of why these group structures are showing up here.