and I'm excited to journey with you as we learn how to take our health back. Streaming live from studios in Think Tech, Hawaii in downtown Honolulu. Today, we will hear from Paul Gates, who is the executive director of the River of Life mission and chocolate on a mission. So let's welcome Paul Gates. Aloha, Paul. Aloha, thanks for having me on. Thanks for having me. That's great to see you. Thanks for being here with us. Before we get started, let's learn a little bit about Paul Gates and your beautiful family. I love to start there. So tell us about them. Yeah, so, you know, I tell people all the time that I definitely married up in the food chain. And my wife, Shauna, is incredibly, just incredibly beautiful, not only beauty on the outside, but beauty on the inside. She is, I call my wife, Salt of the Earth. And we have an incredible story. We won't get that today, but God just redeemed both of our lives. We're actually a blended family.  But we don't really talk about it like that anymore. We have seven kids and all our kids love Jesus serving the Lord. And we're just really thankful for what God's done for us. But my wife, Shauna, is a huge part of that. Really, really thankful for her. Wow, that's beautiful. I love to hear when families get together and they blend so well. When you don't even know, yeah, as you said, you don't call them blended. But being the baby in the family, the little baby in pink, I understand her name is Sienna. Share a little bit about her. What are, what's so special about Sienna?   I don't mind diapers. I'm like, ah, get me in there. I'll change diapers, no problem. But Sienna, we just decided to have Sienna and there's a whole story to that as well.  And I'm not sure if I didn't have a vet take care of that for me, but anyways, we got Sienna out of it. And she is an absolute gem. And the kids just adore her. We have four kids that are back in California. They call Sienna all the time on FaceTime. It's so cute. She holds the phone and they're kissing each other. So we just love family. And Sienna honestly is a big part of that. Wow, precious. So I've been working a lot with the River of Life mission for a few years. And I know that our pastor Bob needed to retire and return to his other days of life and his other duties. And there was a call made out to you to become our new executive director for the River of Life mission. Please share with us a little bit about what really goes on at the River of Life mission. Yeah, so Bob was honestly, Wendy, as you know, really a legend, just an amazing man. So I'm not trying to fill any shoes here and wouldn't try, you know, but as you said, Bob was ready to retire and the van that I'm driving now shows that. Bob was a man that loved to laugh. And so Bob, as he was getting older and aged, swiped every side of the van that I still drive today. And so he was ready and, you know, my wife and I just felt like God was calling us here. I was born in Papua, New Guinea. We'll talk a little bit about that. My parents were Wycliffe Bible translators there. So I've been, you know, saying to a lot of my friends, say I was born on an island and back on an island. And my wife and I just, I'd always had a heart for the homeless, even in my early 20s. And five years ago, my wife and I came to Hawaii with a mentor couple. And we went off on her own and talked about five things on her own that we really felt like God may be stirring in our lives. Then we came back together and shared those five things. And one of those things that she felt the Lord leading and I felt the Lord leading five years ago is that we would move to Oahu. And my wife, Shauna reminded me the other day that actually three of the five have come to pass since we did that strategizing, dreaming with this mentor couple here in Hawaii and we're here and we love it. And River of Life is just, as you know, Wendy, I don't even have to say a lot other than I have absolutely fallen in love with this place. Falling in love with the team, falling in love with the volunteers, falling in love with the board, falling in love with what we do. And I don't think a lot of people understand what we really do here at River of Life mission, but in the month of August, we picked up giving out meals for another organization that had to shut down because of COVID. And we were basically preparing and giving out 1,300 meals a day. And I looked it up in the average restaurant in America, it gives out 300 meals a day. And we were doing 1,300 with one chef. And so this team worked so hard, but the whole reason we do it is to share the love of Jesus. That's our theme verse is first John 419, we love because Jesus first loved us. And that's why we love what we get to do is we love to love on others in the name of Jesus. Well, and I gotta give a shout out to your chef Hilda, who is a member of your staff at the mission and she has a great story. And I love one day to bring her back on to share her mission and her story and how God used her in miraculous ways and then brought her back to the mission as she fulfills her lifelong dream of being a chef right there for the people who need her the most. Fabulous and love her spirit, love her heart. You know, whenever I talk about the mission, everyone refers to it as, oh, the building with all the people standing in front on Hawaii street. So Paul, what are they waiting for while they're waiting in line there? They're waiting for some love. And Wendy, that's why you love River Life so much and why we love you at River Life is you're such a lover of people.   But even today as I've been forgiving Tuesday out interviewing guests, I've been saying, asking the guests, what is it about a meal? And they've all said it's not the meal. It's that somebody calls me by name at River Life. We have thousands of guests that come in and out of here on a rare daily basis and our staff knows their name and gives them love and just showers them with something that is not a part of their typical day. Just to put it lightly. If you can imagine, you know, I've really been impacted even today thinking about what it must be like to live on the streets and not knowing if you're gonna get beat up or robbed or worse. And to have a place where people love on you, that's why there's lines here. Right. People coming there, not just for the meals, but yes, to be called by name and just to be acknowledged as a human. And so that's what the mission really represents is that your doors are open and love exudes out. And that's what they know.  Sometimes they just knocking on the door because they just want that door to open and say, whoa, what do you need? Hey, Johnny, what do you need? So, oh, they called my name. I'm good, I'm good now, right? And so that goes on throughout the course of the day, I know. And the spirit of your workers or the staff members in the mission is amazing. And I've probably been with the mission, maybe at least a good 20 plus years and never faltering in either any which way. They may have a bad day at home but when they come to the mission, they really realize that somebody else is having it worse. And so their troubles are left behind and now they too put on smiles to help those who have less. So that's what they're looking for when they come in, they're standing on the street there for the door to open. So a blessing for you, Paul, to receive and then a blessing for the mission to receive you as the next leader to guide us through the next 20, 30, 40 years. So yeah. Keep on going. So you've got more than seven kids now. You've adopted many, many more. You honed the rest of us to be under your leadership and we so are grateful that you stepped to the plate. Truly, truly grateful. So thank you so much. You know. Yeah, and the only way I'm gonna make it to that 30, 40 years whatever you just said is to go boogie boardin' with you or surfin' with you, right? Hallelujah, yeah. You gotta take the time. That's why he made that beautiful blue ocean. It's for us to rest and relax. And it's all for free. I love it. I agree. Right, right on. So, you know, I have a question from the audience and they wanted to know, how did COVID impact the mission? Yeah, you know, so we've been seeing COVID as a blessing and we, you know, but it has caused a lot of organizations to adjust. And, you know, I'll give you just a little bit of the landscape when COVID hit, we lost almost all of our volunteers. So if you can imagine, you know, going from hundreds of volunteers a week down to almost none, that was a real hit for us. And then also we couldn't have our guests into the dining room and, you know, talkin' to Chevy and Tracy, some of the legends here at Riverlight Mission, talkin' about how that dining room experience was just an opportunity for some reprieve for our guests. And I've really taken that to heart because it was the one moment they could come in and not watch their back, especially for females living on the street. And so it has caused us to adjust, but here's what's amazing. Last year in 2020, in the pandemic, our meal distribution increased by 30% with hardly any volunteers and existing staff. So that shows you how hard our staff works. But one of the things I wanted to share with you, and the board and I really been talking about this, that we actually believe that our ministry opportunities have almost grown by handing meals out the door instead of having people in the dining room. And we really mourn the fact that we don't give our guests an opportunity for a reprieve to come in even under this air conditioning, you know, and to not have to watch their back. But because we're serving a meal out the door, we believe we're getting more one-on-one time with our guests. And so we're trying, we're telling our staff if, hey, if David or Angela gets engaged in a conversation, somebody else has to step in and fill their job and let them take that moment with that guest. And we've had story after story after story of people being reunited with their family. I can tell you about a guy, Larry, who are one of our staff guys, Eleni, got reunited with his family in North Carolina and he's back living in a family home in North Carolina. He sent us a letter saying, thank you. And now we found out he's starting to write music again, which we celebrate like crazy. And so, you know, I think our impact in COVID because of this one-on-one industry is even greater. And I could tell you more stories, but I'll stop there. Okay, I know we could go on and on because every day there's another story or 10. But what other services do you offer our friends that visit the mission? Yeah, so what we're really trying to do, Wendy, is one of the things we've gotten involved in is the HMIS system, which is, we're getting trained on that where we can see what services our guests have actually already accessed. And so we're trying to work with other government agencies and other ministries and NGOs on the island. And one of the things I've always said is you can't be a professional in every area. And so one thing we need to do is learn our limits and partner with other professionals in their area. So one thing, you know, we do some housing here at River of Life Mission, but we don't have a massive capacity for that. And so we've been working with Hope Center over in Kailua and getting men over to Hope Center to go through a one-year discipleship program or get them over to IHS. Connie and I at IHS have been spending a lot of time together. And how do we collaborate? And so I really believe that it's not one organization trying to do everything. It's all organizations doing everything together. Working together, yes. Well, you figured it out and early in the game. So that's amazing. I love that forward-minded thinking. So I know that we always are looking for donations. Yes, monetary donations are always welcome, but we also look for donations of all kinds, soap, socks, underwear. So how do you get these donations and how do we get them out to those in need? Yeah, so I mean River of Life's a great place to give your donations. In fact, we had a guy that's filming for our Giving Tuesday Live Day today, brought at 4.45 a.m. today a bag full of men's clothes. And then it was the coolest thing. We were up interviewing Noel, who does our clothing. And Noel said, yep, I already got your bag and I already distributed your clothes. So the one thing I want you to know, and I told Jordan now when he was donating the clothes, I go, we'll use them quick. And so just to let you know, one of the things we really need is actually men's clothing. We have a lot of women's clothing, but men's clothing is one of the biggest needs. And then with food, you know, we'll take, that's the beauty of, you talked about Hilda, we'll take any kind of food. We have sometimes people just pull up with fresh fish and then Hilda makes fresh fish dish.  I would have never guessed it, but salad is one of our, you know, and I'm not sure how the donation works with salad. You probably have some of them, but you know, we can accept donations pretty much from nine a.m. to five p.m. every day and just come up to the front door of Riverlight Mission and just knock on the door and say, I have a donation and we'll take them. And we've been here for so long. Everybody to do that. Or I'm also the drop off point, they drop it off to me, then I go knock on the door and say, hey, I got more bags. And I said, I don't need a receipt, which I know you were very loving to give us a tax receipt, but most times I don't even need that. I just want to drop off and get them on the people's back. So it's a great source for us to come to help in indirectly, as well as physically come to volunteer. And I know that you use all kinds of God's helpers in any which way that comes to your door. So please share with us. I know that you like to reach out to assist those in need. So you're always, always helping with people in need. Tell us more about that. Yeah, I mean, so we have talked a lot as a staff that there's three questions that we ask all of our guests. And one is if they don't know their name, first is call them by name. If they don't know their name, ask them, what's their name? And then secondly, how can we help you? And then the third question is, tell me about your family. And that's just a big priority for us to try and make. I interviewed this morning, Orlando, who's actually on the screen right there. And this is actually a screenshot of Orlando on a Zoom call with some of his children and grandchildren that he hadn't seen for 20 years. And Orlando in the, I'm looking at the screen in the bottom right hand corner as his mask on his chin. He was a world champion boxer. He fought with Muhammad Ali and he legitimately won a world title in boxing. He fought 47 fights and 30 knockouts. But he lives on the street. And it was because of a family rift that separated him from his families, from the Philippines. And he just left his homes and all his money. He said, give my money to my grandchildren for education. He educated 30 of his children and grandchildren. And until we got him on a Zoom call that you just saw a screenshot of, you didn't even know that he had educated his children and grandchildren. One of his sons is a police officer in Australia now. One of his daughters is an Uber successful business woman in Florida. One of them's a nurse in San Diego.  And he's just saying, I love you. I love you. And they're all crying. And I'm holding the computer and I'm crying. You know? We're still making strides to reunite Orlando with his family. But we just use, we believe that Jesus- Ideally it's gonna happen. It's gonna happen. Yeah, Jesus largely use questions. And so we just ask questions. And one of those is, how can we help you? Yes, wow. So another question coming in from the audience real quickly. How does one get to volunteer at the River of Life mission? Great question. We just redid our website. So if you just go on our website, which is riveroflifemission.org and right on our website, we just redesigned it to be an engage page. And so right when you go on on our home screen, there's two buttons that says I need help, which is anybody that needs help. And then underneath it says I want to help.  We've got a whole new system for it that's all virtually done. So there's no paperwork. It's awesome. It's really streamlined. I mean, obviously with COVID now and protocols, there's a little bit more to it. So we tell people just be patient with us. But basically just go to our website, riveroflifemission.org, click on the button I want to serve and we'll connect with you. It's as easy as that. Well, that's the key word. That's the key word, easy. You know, make it simple and they will come. And they will be coming even more. And usually during this time of the year, everybody has the servant's heart and they really want to come out. But we just want to encourage everyone to come out throughout the course of the year and serve every day because I tell you the rewards coming back is much greater than whatever hour or time you could give. You're going to receive even greater than what you give. Promise. I promise you. I believe it. I believe it. Now I know that there's a sweet side to your job. So that word, there's word on the street that you have a chocolate factory on the third floor. How many years has it been now that the mission has been making Hawaii smile with chocolates? It's, I think it's been about 11 years. Now I'm still claiming a little bit of new guy status because I don't know exact dates.  And I love it because I've been encouraging nonprofits, you know, all over the mainland even and here in Hawaii that you have to have a social enterprise. And that means to have, you know, some type of self-sustaining business that helps fund what you do. And chocolate's been a big part of that for River of Life mission. And we've just rebranded our chocolate and there's some of the chocolate right there. And I'm telling you, our chocolate is amazing. And let me tell you, let me tell you why it's amazing. A lot of people don't know this, but our chocolate was put in private label boxes for years and years. There was high-end private label that they thought was a certain brand, but it was actually River of Life chocolate. And so that when I heard that, it immediately told me our chocolate's good. And then I tasted it. Our chocolate is a blend of milk and dark chocolate. So people who love dark chocolate love it. People love milk chocolate love it. We have bars, we have barks, we have bombs that are coming out soon. So look out for that. We have a new retail space on Monicaia Street.  You can get chocolate for your corporation, for your office, for Christmas gifts. Get them in early because Christmas is a really busy season for us. But we're having great success with our chocolate right now. It is cranking. We've got a young team behind it. So get your chocolate now. It's chocolateonemission.com. Our website's brand new. Our packaging is colorful and beautiful Hawaii. And we say it's handmade and helping Hawaii. It doesn't get any better than that handmade and helping Hawaii. Wow. It's better than a Willy Wonka story. I know you have a picture here of one of your head chocolateeers.  I'm just gonna call her a Willy Wonka-like chocolateeer. And so I know she's very dedicated. She has many, many years of service in making chocolate. So shout out to Anna, who is doing a fabulous job. And previously you did see her, one of the other staff members, which has his name is Justin. And Justin is doing also a very fabulous job manning the fort of there and putting out Hawaii's best chocolate on a mission. So- Yeah, they've been here since day one and they're a mother-son combo, which I just think is awesome. Oh, I mean, really. Once a son hits normally 18, he wants nothing to do with mother, but this young man is just loving to be working there at the mission with the mission in his heart alongside his mother. And so you've got a great team there working on chocolate on a mission. And I just commend and continue to promote it because I tell everyone, if we look at the next slide, there's a picture of a gift box of chocolates and I wanted to throw that in for all the companies looking for gifts for Christmas time. Instead of buying a mainland company buy local, support local, support the mission. And it is the best and the best part about it is made fresh daily. I mean, I could almost say that sometimes when I pick up my chocolate, I know that it was just made sealed box and I'm waiting downstairs for the door to open for me to receive all my goodies.  It is the best, the best. Door open and then mouth open, right, Wendy? Yes, amen. You know, one of our best sellers. Everybody says, oh, but I cannot find parking in downtown. I'm like, no need to find parking. You call it in, you pay online. You drive up on Pawahi Street. You let them know you're there. They open the door and voila, all comes the best chocolate in all of Hawaii. So. Herb side. Herb side and with a smile. And you know what, if you want, you can talk story with the staff and hear their story as well. And you're going to be even more touched. Yeah, so it's a blessing. They, you bless and they bless us. So great partnerships. So now I really want to get, spend some time talking about Giving Tuesdays, which it is today.  So I know that you're going virtual today, all day from this morning till, I think it's 12 hours of going virtual, promoting the mission for that straight on 12 hours. But I wanted you to just take a break, 30 minutes and just share your heart with us here on Think Tech Hawaii and share the heart of the mission. So please tell us a little bit more about More Meals Before 2022, the campaign and what it is really all about. Yeah, thanks Wendy for having us on. I mean, this is a really big day and we just wanted to show a live event. We're calling it a day in the life of River Life Mission. And we just really wanted to show people what it's like on the streets, interviewing our guests, we interviewed staff members, we even interviewed a board member. It was awesome as a board member came up, he actually got down on his hands and knees and actually cared for a woman who had some injured, some injuries on her legs. And before I could even interview him, he was down on the sidewalk helping care for this, this lady who had some really significant injuries on her legs. And it's just been a really special, special day. And, you know, if you want to watch it, you can watch it even after the show you can get on live right now, it's riveroflifemission.org slash live. And it shows live there and you can donate towards meals there. But you know, I was reading on Giving Tuesday and it's incredible that 2.47 billion dollars was given in 2020 on Giving Tuesday. And I was like, that is awesome, the generosity of people to help others in need. And I was just like, I just want to do a live event on Giving Tuesday. So we just went for it and God has been so good. I mean, we've literally had some incredible interviews, God has just showed up, but you can still tune in even up until sundown, 6 p.m. at riveroflifemission.org slash live. Super simple, go there, you can help us raise more meals going into 2022.  And you know, Paul has invited everyone to go on to watch the balance of the live virtual feed there. But even after the 12 hour, the heart of the people of Hawaii and the people of the world will continue to shine. And please don't hesitate to continue to donate even after the 12 hour. We're always looking for donations, hearts, time, money, underwear, men's clothes, whatever you need to share with the rest who have less, we always welcome it at the riveroflifemission. So Paul, right now our time has come to an end. You've been watching Taking Your Health Back on Think Tech Hawaii. Mahalo to you, Paul, the executive director of the riveroflifemission and chocolate on a mission. Thank you for talking story with us and sharing the heart of this mission. Hawai'i's Mahalo's all to you and your team for doing this daily. I'm Wendy Lowe, we'll be back in two weeks with Auntie Melvin Lee as we close off for the season for 2021. Aloha and Mahalo to you, Paul and the riveroflifemission. Aloha.


Now I'm going to Metcalfe here. I'm going to Metcalfe this needs some extra time here.


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Hello everyone, this is Lixiang from Channelbuy. This time, I want to introduce one AI evolution with the OLAB case. Wiping to the ETSI AI which is implemented by Channelbuy with the cooperation of Intel, Cloud, and Hong Kong S3. Okay, let's begin. Firstly, with more and more infrastructure being involved into the network, that's the network becoming a lot more competitive. So how to make the operation more impenetrable and lead the network more tangent and smarter is a big challenge. Thanks to the OLAB, however, network and services design fantasy. What is OLAB? OLAB is a comprehensive platform for orchestration management and automation of network and edge computing services for network operators, cloud providers, and enterprise. At the same time, policy-driven orchestration and automation of physical and virtual networks functions enable repeat automation of new services and complete life cycle management critical for 5G and next-generation networks. What is the AI? Its printable network intelligence is defined as connectivity. Network management architectures using AI techniques and contains a wide policy to adjust offered service based on change in the user's needs, environmental conditions, and business goals. It therefore fully deface the 5G network with automated service-providing operation and assurance as well as optimized select management and resource orchestration. AI has also launched the protocol concepts aiming to demonstrate how AI techniques can be used to assess network operations include 5G. The user of AI techniques in the network were so proper of future networks deploying and operations. We introduced a case of AI combined with OLAB, which is an AI property. It is an intelligent telecom network energy optimization, which is mainly implemented by users of OLAB, approved in January 2020, not for one year. The team members include China Mobile, Intel, ConeCloud, and S3. The case is meant to showcase the network FAAO7 proposed and developed by the co-led network with a special attention to the AI ML aspects in the context defined by the AI. The case intends to test an automated AI-based approach that is proposed by the AI architecture of network self-organization and energy optimization. This case concerns two scenarios. The first scenario is to demonstrate that the AI-based techniques enable the network to be horizontal and a medical scanning network of PMF to be staffed and shut down. The second scenario shows that CPU frequency can be wise to turn off a turning down internet to CPU power saving. The case mainly can be split into four parts, real-time series telemetry data connector from the CPU and VF-EPC infrastructure by the platform connection agent. Using all of AI application technology and inference to be processed online or offline as required. VF-Scanning, PMF-Shutdown, and CPU-Turningoff-Turningdown commands send back to the EEPC infrastructure. In the following video, it will showcase how we achieved intelligent power management by Old Diamond. We have several scanning and adjusting CPU frequency based on the integration with OLAP and ML Drive data elastic platform. In this project, we work together and offer the highly-managed BNT, Easier Maintenance, and Adriven Management, orchestrating several achievable elastic and local server scanning and intelligent power management. Working something in 5G environment must be covered before to forfeit high-speed network connectivity and various kinds of server-level agreements. In this case, we developed an Adriven Management with Automatic Auschwitz to pre-configure and pre-segulate the servers for the future demands. Also CPU frequency is optimized to improve power efficiency. When the system determined to scan in, the system automatically slowed down the CPU frequency. Then, start to terminate two waves and mark the status penalty late. Show in the dashboard. In the same scenario, when the system determined to scale out, the system must be lab, CPU frequency meanwhile, the system will also launch two waves, and mark the status assigned to show in the dashboard to scale out the community of the servers. Okay, that's all. Thank you.


So remember we have on the one side string theory and ads three cross s three cross. And that we described in terms of an SL to our resume in a written model at level one. And equals one version say, I mean I'm a bit. I'm writing this again in the NSR formulation. I had the impression that the hybrid formalism didn't reach enormous level of enthusiasm so that's just stick with this and you can understand this as well and that's description as in the hybrid description. So this is a string theory on this. So this is the world sheet. And this is still to the symmetric orbit fold of T for which is the CFT. And what we want to do is we want to calculate a correlation functions in that theory. And the boundary of ADS three you should think of as being a cylinder and if you compactify it, it will be a sphere so these correlators will always live on the sphere.  And throughout today, the coordinates in this space will always be denoted by X. So X will always be a coordinate of the correlation functions of the symmetric orbit fold of T for. So this will be distinguished from the parameter on the world sheet. Now the world sheet will not always be a sphere, although for most of today I'll concentrate on the case where the world sheet is a sphere. So in general here I will have a world sheet of whatever genius is appropriate. But for most part I look at G equal to zero. And here I will denote by Z. So, so just to be clear, whenever you see a Z, it will be a coordinate on the world sheet. Whenever you see an X, it will be coordinate in the symmetric orbit fold theory. And I'm looking at sphere correlators of twisted sector states in the symmetric orbit fold theory so this will be something like, if you if you wish. So the simplest case would be a correlator of this sort of type, the three point function x one x two x three are three points on the sphere. And this is say the twisted sector ground state in the w one w two and w three cycle twisted sector. So that's the sort of thing we are going to calculate in the X space in the symmetric orbit fold space. And I first of all want to explain to you how you calculate this correlate correlator in the symmetric orbit fold, irrespective of whether it comes from ideas or not just I mean how would you calculate this. And then I want to relate it to a calculation in the world sheet theory. And I'll explain to you how this correlation function of X will be reproduced from the string theory perspective. The string theory perspective life is a bit complicated, because as you know in general, we'll have to integrate over the Z variables. The insertion points of the vertex operators on the world sheet are modular you have to integrate over, say for a four point function you would have to integrate over the cross ratio of the Zs. And what we should manage to achieve is that after we've done this integral, we should reproduce the correlator as we calculated from the dual CFT. That's the roadmap we have in mind. So how do you calculate this correlate. I mean there are different ways of doing it, but there's one smart way of doing it. That will fit very nicely with what we have in mind. So how do you calculate that. Now remember what a sigma w field looks like. Remember, this has the property that if I start. So let's take w to be the cycle from one to w. And then if I start remember in this simple case of a single boson, if we take the Xi around this field, we ended up with the Xi plus one. So that was the effect of this twist field right it was to make sure the ice one, the first one goes to the second the second one goes to the third, the third, the w one goes to the first one. So it's a w.  That's the structure of of this twist detector fields and now how what's the smart way of calculating it. So the smart way of calculating it is to change coordinates, and to think to go to the covering space. So we want to find the covering space locally. And what do I mean by the covering space. Well the covering space will be a map gamma, and I'll write it in a map of Z. So this said at this stage, it's not obvious that we will have anything to do with this. At this stage, it's just the parameter I call that to distinguish it from X, but I call it that because ultimately it will be connected to that set but that's far from clear stage. So gamma of that will be a map from some neighborhood of gamma to the minus one of X, who a neighborhood of X. So let's call this X zero, which is the point where the twist fields it. And this map should be such that that gets mapped for X zero plus a times X as that minus and let's call this point is at zero zero to the top. So what you're looking for is a map that sort of w fold covers the point X. So why is this a smart thing to do. Well think about in the coverings. So here's the covering space. This is the covering space in the covering space we have the point that zero. This gets mapped down in our real space. This gets mapped to X zero. And this map has the property that if I go once around up here. If I go, if I multiply that minus that zero by each of the two pie is that minus that zero, then I go w many times downstairs right because of the power of w. I circle once around here. I circle w many times around here. And you see after going around w many times, it's become single value. So this is the map that locally maps the individual coordinates the Xi single value.  You go from X one to X two to X three to X four up to X w back to X one X two, each of them gets mapped to itself so from the point of view of the covering space. This map will be single value in terms of the individual the Xi's. So in some sense you've removed the effect of this twist by going to this covering space. Now locally, it's clear how you do it. I mean you just write down something like that. What you do there is, if you calculate the correlator of that kind. What you should be doing is you should patch together the local covering maps of that type into a global covering map, where you have some remand surface on top. That has the property that near it needs near each of these insertion points. It's doing exactly that and it's therefore undoing the twisting that happened downstairs. So first you may think is this possible. Okay, so let's give an example. The simplest example is you take three points with w equals to three. So let's take w equals to three at X one w equals to three at X two, and w equals to three at X three. So let's say you set X one equal to one X two zero zero one and infinity. And let's choose also the sets, the pre images to sit at zero one and infinity. That is without loss of generality, because you see by the movie symmetry, I can always move three points to three points. So the corresponding map, the corresponding covering map in this case is of the form z to the four minus two times z to the three divided by one minus two z. Okay, I learned this by heart, I wouldn't be able to spot it as I'm not that good a calculator but what you can do you can, you can either be smart and seed in your head or you can type it into mathematical. So let's go back with this how this behaves how does this behaves near zero. Well near zero you see the first time goes like minus two times that to the cube. So this is exactly of that form that map zero to zero plus some coefficient which turns out to be minus two in that case, times the third power of that minus zero, and then what happens near one. So if I calculate gamma of the Z minus one, but let's write a gamma of Z is of the form if I expanded around one it goes like one minus two times that to the minus one to the power three for that near one. And then at infinity what I have to look at is gamma to the minus one of one over you. So, caught in a chart around infinity and you map it back to run zero, and then you can read off that it goes like minus to you cubed, which describes it for you equal to zero, which is the same as that goes to infinity. So this is an example of such a map. I mean it's holomorphic. It's in fact a map from the sphere to the sphere. It's fourfold. I mean every generic point has four pre images, because it's a fourth order polynomial divided by a fourth order polynomial, the fourth order polynomial in the denoted denominator has been degenerate but it's fourth order over fourth order. So it has, it has four pre images, and it maps zero to zero one to one infinity to infinity. And at each of the points, it has a threefold covering. So it will undo this correlator. And the idea is that instead of calculating this correlator, I applied this covering map, which is a holomorphic map correlation functions transform nicely under coordinate transformation and under holomorphic maps. So I lifted up to a map of the covering space, and then on the covering space, this map is this correlator is effectively trivial because you see once you've moved it up to the covering space, then everything is single valued, any trace of this funny sort of insertions where things got sort of a twisted have disappeared. And because I'm looking at this twisted sector ground state. The idea is that once I've lifted it up to the covering space. That's the vacuum correlator, every, every single trace of this twist field has disappeared. The only purpose of the twist field in life was to impose this twisting, but by this local coordinate transformation I've undone the twist things or nothing is left. So therefore this correlation function is just a conformal factor associated to this conformal transformation. So I just have to calculate the conformal anomaly that comes from this coordinate transformation and then I've calculated this correlation function. That's my idea. That's the smart way of defining of calculating the correlation functions of symmetric orbital theories. No question. Yes. Shouldn't we expect some remnant of the parameter capital N. Oh, sorry. What I'm saying here is is what, well, no. Sorry, yes and no. Yeah, I'll come to that in a second. Let, let, yeah, you're right, but let me explain one thing and then I'll explain. Okay, so, so the point is, so in this example, the covering map, I've written down is, is a sphere. So in this example, this example is a map from the sphere to the sphere, but in general, the covering surface need to be a sphere. So the covering surface. So the covering map. If you allow yourself a possible covering maps, and you allow yourself your covering space to have arbitrary topology, then you get different answer you get different covering maps there's not a unique covering map. There's a finite number of covering maps, and in general the covering space that covers say this configuration needn't be a sphere. In fact, there is the Riemann Hurwitz formula. The Riemann Hurwitz formula tells you exactly what's the, what the genus of the covering surface is in terms of the parameters w and the number of covering so the Riemann Hurwitz formula is I is equal to one to n. So if an endpoint function w i minus one over two plus one minus G, where G is the genus of the covering surface is equal to M and M is the number of pre images number of pre images at a generic point. For example, for this example, you see w is equal to three, so three minus one over two is equal to one. There are three points so this gives you three plus one is four is equal to four so the genus is equal to zero. So this is an example of a map that's from the sphere to the sphere a covering map. It's a fourfold cover, each point in the downstairs sphere has four pre images in the upstairs. But in fact, there's also a torus covering. There is in fact also a torus covering where you choose the number of pre images to be three, and then you find out that the genus has to be equal to one. There are a number of different coverings, in particular for this configuration is at least a sphere covering in the torus. Now, coming back to Francesca's questions what is the role of the different. What is the role of the genus of of this covering map, and what you discover is that a correlation functions in their large and expansion. The one over and dependence of the correlators is controlled by the genus of the covering space. So the way you should think about it is that if you calculate these correlators, what you have to do you have to sum over all possible covering maps. You get some conformal factor associated to the covering map gamma. And if you expand this in a one over and expansion. What you realize, and this was an observation of rastelli and rather much is that a correlation function, if you ask. So the diagram of for given gamma, it contributes as n to the one minus G minus and over two. And then is the end of the symmetric overflowed. So this and so I'm thinking of taking n large. I'm calculating this correlation function in a larger and expansion. And what I'm saying is that the prescription this is some of all possible covering maps, from the point of view of conformity theory that's summing over the different conformal blocks. Given covering map contributes, if you think about it in the one over and expansion in a way that depends on the genus of the covering. And so that's a fact that you can just read off from the structure of the symmetric or before current. I mean, there's a, there's a theorem that you can calculate them in this fashion and then if you look at each term there you can say each term goes as n to the one minus G minus and over two where G is the genus of the covering map of the covering surface that I'm considering. Now, remember that n is to be identified with a one over n is to be identified with the with the genus. Now, from the point of view of the dual string theory that the genus should go like the string coupling constant should go like one of a square root of n. So if I write this in terms of the genus this will go like the genus of the string, sorry that string coupling constant to the minus two G. Now, it'll, it'll suggest that the genus, but that is exactly how the genus expansion of a string theory should appear so what it suggests is that somehow the contribution that comes from a specific genus here, if I interpret it, why are the ADS CFT reality.  The contribution that's natural here that comes from just expanding out the one over X and expansion of the symmetric orbital correlators should let should match with the genus expansion as you write it down and conform a feed theory because the one over and expansion go behaves exactly like the G string expansion under the usual correspondence between the string coupling constant and one of a thing there's a question there. Okay, when we do an orbit for the usually today we, we get some symmetry, and that we usually call quantum seemingly corresponding to this orbit folder. And usually when we have a symmetry, the correlators will will fit in some, we will follow some selection rules or something related to the symmetry. In the year we can understand something of this, what we get from from this or. Well, first of all, I think the quantum symmetry is usually you can undo the orbital by taking the orbital with respect to the quantum symmetry that only works in the abelian case. You can't undo non abelian orbitals. But obviously you're still have you may so so the obvious quantum symmetry just work for a billion orbitals, because the twisted sector just you can characterize the twisted sector by some face. And then the faces give you selection rule and if you're all before by that to go back to the original theory from the non abelian orbital you can't do it. So it's this complication that the twisted sectors are now labeled by conjugacy classes rather than group elements so there's no easy way to reconstruct the full original theory from taking the orbital of the orbital. At least that's as far as I know that's not known. There is no the story about non invertible seamless in this case. So maybe there's something like that but certainly none of the conventional or befalls will do the job. Now maybe maybe there is some more general construction but I'm not aware of a general construction. And I actually I talked to you fun about this and I think I think that would be interesting to see what one could use non invertible symmetries here but that's certainly not not now obviously there are selection rules so for example, this can only be non zero. If you can find three, three cycle permutations that multiply to give you the identity. So, but that you can do. Right, I mean you could for example take three times the same one, but that would give you the tourist covering here you have to take one that involves for four different numbers. There are different ways of multiplying three cycle permutations together to give you the identity but beyond that I don't think there's any global symmetry for a non a billion or before like this metric or before. Thank you. There's another question over there. So the cover coordinate you called Z. Yes, and then the same Z as the word. So this is anticipating what I just explained. I mean, at a covering space at a moment I shouldn't have called that I should have called baby Z term because it does at this stage nothing to do with the Z. But over there I explained because of the one of our independence you will expect this Z to be the Z and the upshot of my talk today is that this that is this. So I'm I'm I'm I'm sort of suggesting to you already where I'm going, but I'm using a notation that will go there but at the moment this has nothing to do with the world. This is just some abstract covering space. So I want to be sure I understood this sum over gammas. So triple and function. It seems that there are possibly a unique. So are you referring to higher point function where you want to take. But three point function. I mean the question is how many channels do you have right. Yeah, so you have mine the four or higher point function I guess. I mean they know P where you have an infinite number of triple and function then an infinite number of gambas something like that. I mean for a fixed three point function there. There are different channels so I'm probably going to screw this up but so so there's one obvious channel which is so you pick the permutation 123 123 and 123 right. That would be one solution right. But remember in the symmetric or default, you have to pick representatives. So you. So when I say a w cycle to detect that what I mean you I take any w cycle. But I haven't specified that it's necessarily 123 right because I'm labeling the states by elements in the conjugacy class. So this will be one configuration that will contribute to the three point function, but there's also another. And if I can find my notes, I can tell you what it is. And maybe I can't tell you it is there is one. There's another one. 341324 and 123 that will also do the job. Right. So one goes to two, two goes to four, four goes to one, two goes to three, three goes to two, three goes to one, one goes to three, four goes to three, three goes to four. And that's also another solution. So there are two conformal blocks for the three point function of a three cycle twisted sector operator, because that is a different channel than that. Right. Because you see that involves three, three elements in the covering space that develops that involves a fourfold cover. So that's the genius one contribution and that's the genius zero contribution. And when you calculate this correlation function you have to sum over the conformal block coming from here and from here. And what I'm saying is in the large and limit this conformal block dominates over this conformal block, because this conformal block will be down by a power of one over N, because it has genius one relative to the zero. So if you look at more complicated correlators you get finite number of contributions from different genius, and they're one of our independence is controlled by the genius of the corresponding covering surface. Thanks. So another mild question is, there are other correlators right this is twist fields. Why are you focusing on the specific set of correlators matching and not. So first of all, I'm looking at a single cycle. I'm looking at a single particle states from the string theory perspective so I'm only going to look at states that live in a single cycle twisted sector. So they'll always live in a cycle in a cycle of one length, because otherwise they would be multi particle states and I wouldn't see in perturbative string theory. So I don't necessarily have to look at the ground state. I couldn't look at some excited state, and the punchline is, if you look at some excited state, then you don't get a trivial correlator on the covering surface but you get a correlation function of the seed theory on the covering surface. So you can relate it to something relatively simple and the interesting bits that I'm trying to explore to explain to you really hinges on which cycle it comes from rather than exactly which state you look at the covering map is always the same Okay, any further questions. Okay, good. So now, now I've written here some. Now, people know that when I write sums I don't necessarily mean sums maybe sometimes I mean integrals because I've been sloppy when I wrote down the spectrum of the SL2 hours we know written model. But the point is here I actually mean a sum covering surf covering maps are rare. And they don't exist covering maps are something that's, they're really a finite number of covering that says not a continuum of covering. And in order to explain that and that will be crucial for what we are going to see. Let's look at the case where you look at the genius zero covering. So let's look at the genius zero case. So let's lose use the remind who of its formula for the example of G equal to zero. So then this time isn't there. So, if you have on the sphere, what does the most general holomorphic map look like. Well my function gamma of that will be of the form minus PM M of Z divided B by P plus M of Z, where P minus plus of M of Z is a polynomial of degree and M. So this is the most general M to one map from a sphere to the sphere, right, I mean it's, it's just the ratio of polynomials. Now what is the condition for this to be a covering map. Well the condition has to be that gamma of Z should be equal to Xi plus something of order that minus that I to the w i. So I have in mind. I have an endpoint function. So I'm having endpoint function, and the point sit at coordinate Xi, and they have winding w i, and I want the, the covering the pre images to sit at position that I. So I'm asking. So what this means is that near that equal to set I it has to map to Xi, and then it has to be order that minus that I to the w i. That's the covering that condition. So you see, we can just plug this answer into this formula. So what does it say, but now maybe people can see anymore so maybe I'll continue over them. So what this means is that minus PMM of that divided by P plus M of that is equal to Xi plus order that minus that I to the w i. So we multiply by P plus M of that this is some standard generic polynomial, and then bring it is to the other side and what do we learn we learn we get the equation PM minus of that. Plus Xi P plus M of that must be of order that minus that I to the w i. So that means I have to satisfy in order for this to be a covering, right. I mean if I multiply this rule, this is a generic polynomial it will generically not have any zero or anything at z equal to z i. So therefore, you can just multiply through so Xi multiplies P plus, you bring P minus to the other side, and then has to be of order that minus that I to the w i. So let's count how many parameters we have in our ansatz and how many constraints we have to satisfy. Well how many parameters do we have well this is not polynomial of order M. So it has M plus one parameters. This is a polynomial of order M so it has another M plus one parameters so I have to M plus two free parameters, but one parameter drops out because I'm looking at a ratio. The overall scale drops out. So the number of parameters in my ansatz is equal to two m plus one. On the other hand, how many constraints do I have. So the number of constraints. I have to satisfy is equal to well at each point Xi. So at some for each I so some I is equal to one to N. I get how many constraint, I get w i constraints, because I have to make the order zero term vanish the order one term vanish up to the order W minus one So I get w i many constraints. But now let's look at the Riemann Hurwitz formula. You see the Riemann Hurwitz formula tells you that the sum over N of w i is equal to two M minus two. So this is equal to two M minus two plus and that's the number of constraints that you have to solve. If you want if you so what we are specifying here is we're specifying the coordinates Xi, we specify the W is and we've specified the Z is. So with all that information we have two M minus two plus and constraints, but we have only two M plus one, three parameters. So the number of parameters minus constraints is equal to two M minus plus one minus two M minus two plus N. So that this is equal to three minus N. So what does this tells you is that you see generically if N is bigger than three, you have more constraints than parameters, which is another way of saying that generically the covering map doesn't exist. And that will only exist if the Z is satisfied and minus three constraints. I mean that the Z is must must lie on a hypersurface of co dimension and minus three in order for you, at least generically to have a chance to find a covering map because the number of parameters is simply too small to satisfy all the constraints that you have to satisfy. So that's a very suggestive number. Right, because, you know, if we are doing world sheet theory on the sphere, if we're calculating a string diagram, we have to calculate, we have to integrate over the Zs. Three of that integrals are for free because of the furniture symmetry. So we always have to do n minus three Z integrals, we have to cross integrate and minus three cross ratios. So there are exactly n minus three constraints that allow us for the existence of a covering map. So this is the structure. So if you. So this suggests the following, the following way in which this wants to work. So suppose I can, I can write these constraints in terms of the variables is that for up to ZN, then I would expect that from the world sheet point of view. So, so, so far I've done symmetric orbital analysis now I make a leap of faith. Now I say how could I possibly get it from the string theory side. And there's a very, very natural way in which you can get it from the string theory side and the natural way is the following the vertex operators on the string Well, they're going to depend on Xi and that X one and that one, and up to WN X and and that and, but they should be a sum over all the covering maps, labeled by Jay, and it's for each covering map. So you get an n minus three delta functions, and the delta functions will be of the form, gamma j to the minus one of X for Xi minus z i times some conform the factor. So I propose, that is a natural way in which the world sheet theory should look like, and why is this natural. Well this is natural because once you do string theory. Remember in string theory we have to do the integral over DZ for up to the ZN of this correlation function. Right, because if you calculate an endpoint function on the sphere there and minus three more July we have to integrate over here I've for simplicity I've chosen that one the two and that three to be fixed. So then I have to integrate over that for up to ZN.  So this is exactly the configuration of z for which the covering map exists, and then I will end up with the sum of our covering maps with the conform the factor of the covering map, which is exactly what the symmetric or before the answer is. And if you stare at the symmetric or before answer given the fact that the covering map only exists on co dimension and minus three, what it suggests is that a world sheet correlators are actually delta function localized on the loci, loci, where the covering map exists and then the integral over the modular of my world sheet will automatically produce the structure of the correlation function that's calculated by the symmetric or before theory. It's a bit of a, I mean, probably takes a moment to get used to it I mean it's very unusual to think about a correlation function to be delta function localized that's not if you open a yellow book on a random page. That's not what CFT correlators look like CFT correlators are rational functions so this looks very far fetched. But if you think about what would the world sheet theory have to be like in order to reproduce the symmetric or before answer. That's the only thing that can really work right it has to.  At least, if this was true, then it would manifestly reproduce the symmetric or before the answer because the integrals over the free parameters in my string theory just kill the delta functions that appear here and I'm left with a some over covering maps as calculated from the symmetric or before the perspective. So that's our claim our claim is that this world sheet theory has exactly this property. Now this as I said this is a, is a bold claim because this is not what normal conformity is looked like. But our conformity there isn't entirely normal. It has this funny spectrally float sectors it has all this funny stuff, and unfortunately I don't have that much time left but what I want to do in the rest of my lecture is to convince you that we have actually proven that this is true for our world sheet theory our world sheet here has exactly that structure. And thereby it manifestly reproduces this metric over for the answer, and this does not only work a genus zero it also work at higher gene so if you do the corresponding analysis here you get a minus G term from here, counting for the, for the genius, the genus of the genius, and they, they are always exactly getting fixed by these constraints, and it, it reproduces exactly the structure that I was outlining here, namely that the world sheet gives you exactly the contributions, the world sheet of genius G gives you the contributions that come from the covering surface of genius G. That's exactly how it works. Now, this is a little bit. So, so what I want to spend the rest of my. So, unless there are questions, what I want to spend the rest of my lecture is to explain in how you can prove that this identity is true in our specific world sheets here. And obviously I won't give the full proof because I don't have enough time but I want to sketch how this works. So if you ask, ask me anything if this is unclear what I'm trying to do. I'm not sure whether that means that I've lost all of you or whether it's abundantly clear. But if you don't ask me there's nothing I can do. Okay, so, so how do we, how do we try to prove this from the symmetric from the world sheet perspective. So there's one key key insight, which in retrospect is a pretty obvious key insight, but maybe it's something that people didn't quite appreciate in its full significance. So in the first three lectures what I explained to you was how the states of the symmetric or before theory can be reduced from a physical state on the world sheet. So I established a map from a vector space of states here to the vector space of states here. Now, as you're probably familiar in two dimensional conformity in fact in higher dimension conformity as well. There's what's called the operator state correspondence to every state, you can construct and operate. But here we have to be a little bit careful, exactly what we mean by. So suppose I pick a state in the symmetric or before theory. This is a state now. And in the symmetric or before theory, I do the usual operator state correspondence which means I identified states with the corresponding vertex operator at x equal to zero. That's the usual way in which you do it and conform a feed theory at x equal to zero the vertex operator creates the state, but this is at x equal to zero because this is the symmetric or before T for that lives in X space. Now, what I explained to you is that this state corresponds to a state that which I call Phi hat in the world sheet theory. And in the world sheet theory, I also have the operator state correspondence. So to that state, I can associate it a vertex operator which I'll denote by Phi head of Z. So that's the vertex operator that's associated to the state. And again, I make the association as is conventional to say that the vertex operator at zero, that light on the vacuum produces for me the state, just like you always do into the CFT. Now, let me remind you how you can characterize the Z dependence of such a vertex operator. The vertex operators actually totally obvious, because you see we know what the translation operator is on the world sheet the translation operator is e to the Z times L minus one. And therefore, this vertex operator is really necessarily of the form e to the Z L minus one, the vertex operator associated to Z equal to zero, e to the minus Z L minus one right the the Z dependence is always trivial it just comes from with the with the translation operator and the translation operator on the world sheet is L minus one where L minus one is the real server generator on the world sheet. But it so far this describes the vertex operator that corresponds to the state being inserted at X equal to zero. So this is written here. This is the vertex operator corresponding to X equal to zero, and that, and this is the vertex operator associated to X equal to zero, and that is that equal to zero. Because you see I've identified it with a state and the state is really associated to the vertex operator in the dual CFT being at X equal to zero. So you can ask what is the vertex operator associated to this symmetric or before state if I inserted at an arbitrary position X, and an arbitrary position X in the dual CFT, and an arbitrary position that on my world sheet. Now, given that I've just explained to you how it works for the world sheet CFT. It's now obvious what I have to do. Also in space time, there is a translation operator, there's a mobius group, and therefore the X dependence of the space time vertex operator comes from conjugation by the translation operator in space time. What's the translation operator in space time but it's L minus one space time, but L minus one space time from the point of view of my world sheet theory is simply J plus zero. So obviously this is equal to e to the X, a plus zero e to the Z, L minus one, the vertex operator associated to the state X equal to zero at coordinate zero, and then it's just conjugated by the corresponding. And what I'm using here is that J plus zero is the same as L minus one in space time. The vertex operators on the world sheet will depend on X and on Z. And there is no ambiguity in how to define this once I've identified the states. It's clear how it will depend on X and Z independently because I know what the translation operator is on the world sheet. And I know what the translation operator is in space time. And note, there is no ordering ambiguity here because J plus zero commutes with L minus one. So there is no minus one and J plus zero commute so there is no, there's no funny ordering I have to choose whether I first translate in space time, first on the world sheet and then in space time or first on the world sheet and then space and doesn't matter. They commute with one another. And that's, that's how the vertex operator will depend on X and Z. And this is what I meant with this symbol. This is the vertex operator that will depend on where I insert a state in the dual CFTI where this crosses is an X space. And it depends on that, that is where the corresponding vertex operator is inserted on my world sheet on my world sheet surface, and the Z is the stuff I have to integrate over when I do a string theory, so that at the end of the day. I'm only left with the function of the exits. What I have to do is I have to calculate the correlation functions of these vertex operators. Okay, so now I at least know which problem I have to solve. And now I can go about trying to solve that problem. Now the, the, the key trick. So, so you may think, oh, what's the big deal about this funny exponential, but this funny exponential has a very significant impact. It has a significant impact of that exponential. Well, so I was planning to, it's most easily explained in terms of, if I can find my notes. It's most easily explained in terms of introducing the free field realization of SL2R. But, or should I do it, can I have maybe a few more minutes. Yes, yes, sure. Okay, so. So I have to, so at this stage I have to do introduce a fee for this is for the purpose of making this a little bit more pedagogical otherwise it becomes a little bit complicated. What I claim is that this has a free field realization in terms of what I like to call symplectic bosons. And string theory people tend to call this a beta gamma system. So these are bosonic coordinates, and they're characterized by the following commutation relations psi alpha beta beta s is equal to epsilon alpha beta delta r minus s. So beta take values in plus minus epsilon plus minus is equal to plus one and epsilon minus plus is equal to minus one and the other epsilon zero. So these are, this is like a beta gamma system. And then I claim this beta gamma system realizes the SL2R f and cut smoothie algebra at level one. So I write the plus the j plus and j minus generators that are bilinear of the form.  Okay, so I hope that this is not not scary for you I mean these are free fields this is like a beta gamma systems or this is not so different from stuff you've seen. And what I claim is if you look at these combinations, they all spin a half field so these guys has been one fields. And when you calculate the OPEs you realize SL2R at level one. That's what I claim. For example in string. Sorry, is it exactly the beta gamma system or for the boson ice to like in string theory we have this. Well, I mean it's it's bosonic field to have a simple pole OPE right that I thought is called beta gamma system right. I mean it's, I mean the corresponding field psi alpha eta beta of Z and W goes like epsilon alpha beta over Z minus w. That's the OPE right I thought that's what you call a beta gamma system. I mean that's what I mean, that's what it is. Right, and if I write it in terms of mode that's what it looks like. And then if I look at the normal order products that gives me SL2R at level one. Thanks. Okay, so that is that is. Now I've lost my notes again. Okay, so this is, so this is something you can check and it's not, it's not a rocket science to check it, and it's true. It requires maybe a little bit care but it's true. Okay, so let's take that. And then remember, in the world sheet theory don't have just the SL. So I'm looking at just the bosonic piece of it and just looking at the SL2R. I'm ignoring the fermions. So the SL2R bosonic algebra can be generated by these fields. And then the other thing we need to know is what spectral flow does. And I explained to you a spectral flow does to this fields, but actually this spectral flow comes from a natural spectral flow that you can formulate in terms of the, in terms of the three fields.  So this is how spectral flow acts on the three fields and that just generates the spectral flow as we've seen in the generator.  So the plus component gets shifted downwards by two minus a half, so it gets shifted down by minus one. So the minus component gets shifted moved up by two plus one, so it gets shifted up by plus one, and then the zero mode stays the same and a normal ordering term gives you the shift in the zero mode. Anyway, I mean it's, if you don't trust me on that then I probably have no chance to explain you the rest. This is not, this is not rocket science. This is something to do. So why, why is this funny formula, why am I so excited about this formula about the correct definition of the vertex operators. So why is this such an important formula. You see what we want to do in order to understand these correlators, they want to understand the world identities with respect to these three fields. What we're going to do is we're going to look at the correlators where we insert one of the symplectic bosons and see what the poles are of the symplectic bosons with respect to everybody else. So what we're interested in is the pole, the OPE of psi plus of zeta with a vertex operator of this kind. What's important is you see j plus is proportional to eta plus and psi plus, and eta plus and psi plus commutes both with psi plus, because the only commutator that's not trivial is between psi plus and eta minus, but there's no eta minus inside. So this psi plus doesn't care about the fact that there's an e to the xj plus zero standing here, because it just goes through. So it will behave exactly as it originally behaved. And because it gets shifted downwards, what you find is that this OPE you can calculate. So this goes like the sum of our of the mode numbers and it'll be so let's specify the spectrally flowed sector by this upper index. What you find is that this is going to be psi plus r minus w over two in this w fold spectrally flowed sector you shift this down by w times minus a half acting on fine hat, and then x and z.  That's just the usual way the OPE works and this is independent of the x variable because it commutes with the e to the xj plus zero term. And then if you look at it what's the first time that survives. If I had this the highest rate state, then this will survive is R is a is a less than w over two so the leading term will go like zeta minus z to the minus w plus one over two. And then it'll go like this term. But that's the structure of this OPE. So the XI plus OPE with this is this form. Okay, so that's how XI plus works. But now what about XI minus. So when you try to do the same calculation with XI minus you see previously we didn't have this factors of of x here after we didn't have this factor of e to the xj plus. So it's like the OPE we would have calculated previously, except we have to ask what happens when I move XI minus past e to the xj plus. And that you can easily calculate what is XI minus over zeta with e to the xj plus zero. But if you think about it, you can expand this out j plus zero has a term that matters.  And what you're finding is that this is equal to e to the xj plus zero times XI minus of zeta that's the term where nothing has happened, and then you get a correction term, the correction term is goes like minus X times XI plus of zeta. And that just comes from the fact that you are moving to see as you move this through, you pick up a commutator term of the XI minus with the j plus zero, and that produces you have XI plus term. And that's really algebraic right I mean there's no, there's no guesswork involved here this just follows from the commutation relations of these modes with one another. So, so white XI plus is invisible to the e to the xj plus the XI minus isn't. And therefore, when I calculate this OPE. If you pass the e to the xj plus zero, I have to calculate the OPE of minus x times the OPE I've just calculated, and then I have to calculate it so this will go like minus X times this OPE, except I sort of moved out the, but actually does there it doesn't matter because that commutes with X. It has a term that goes like plus the OPE I would get from XI minus, and the OPE I get from XI minus is actually much more regular than normal because under spectral flow XI minus modes get shifted the opposite way. So what they did what you find is, it goes like the correction term goes like plus zeta minus z to the power w minus one over two times the w of XI minus of zero on Phi X and Z. So, so the important fact is that XI plus and XI minus have a different OPE when you bring them near these vertex operators and more importantly, this OPE depends in an intricate way on the coordinate X, where you've inserted the vertex operator in the dual CFT, and algebraically it comes from this identity. This is the key identity. The key identity is that the translation operator in space time has an impact on XI minus geometrically what this means is you see you've picked the spectral flow direction. But under translation in space time the spectral flow direction isn't really invariant and you pick up correction terms, and the correction term you pick up is this correction term. Obviously when you said X equal to zero this goes away, but for X not equal to zero, you get this correction term. I mean it really comes all out of just doing things properly. I mean there was no ambiguity in how to define this operator. There's no ambiguity in calculating that. There's no ambiguity in calculating these OPEs. This is just dictated to you once I've defined the theory. There's no freedom here. Okay, so now, now there's one small additional fact I have to mention and then we'll, and you'll see what happens. Super string theory and super string theory correlators are awkward, because there are things like picture changing and stuff. So we have to incorporate picture changing, and the correct way of incorporating picture changing in this context is that you, is that you can use the real correlator as some additional picture changing operators inserted on and what this means is that the, the real correlator that you have to calculate is not just the product of all of these vertex operators, but the real real correlator that you have to calculate is the product that has alpha equals to n minus two of these sort of picture changing type operators. And then you have this product of I is equal to one to N, VWI of Phi hat I xi and so that's the physical correlate that's the physical correlate. So what we're discussing here the only thing you have to understand is the OPE of xi plus and minus with the W's and the OPE of xi plus and minus of zeta with W of u alpha goes like order of zeta minus u alpha to the power one. They're all, they're all regular. Okay, I've accumulated all the data that I need in order to explain to you the water identities. Yes. My question is about the correlators that do not necessarily preserve the winding number. So the spectral flow variable. So, they are totally singular. These are the vertex operators where you've moved the points here all into one position. These are non-zero, right, so you can compute them. But they're singular. They are not, they are not the right correlators to consider. These are the, so I mean if you ask about it from the point of view of the of the symmetric orbit fold, the winding number will only add if the two points sit on top of each other, then the winding number obviously adds. So then don't sit on top of each other it never adds. But what would be the string shear interpretation because from the string shear side we know that the winding numbers can be in principle violated in this case. So these are correlated so previously people have calculated these correlators but they didn't introduce the X variable. If you don't introduce the X variable what this means is that all your points are bunched up on one point. So then from the point of view of the dual CFT you're calculating not a correlation function, you're calculating the limit of all the points coming together then obviously winding number adds. So if you want, if you're interested in the honest correlator of the symmetric orbit fold then you have to evaluate them at different points, and then winding number will not add as we will see and it's all characterized in terms of covering. But that's the data that really controls the correlators of the symmetric orbit fold that controls the one over N dependence and so on. So, so the correlators that were looked at previously are sort of the funny singular limits. And it's not the correct I mean in some sense you must be able to retrieve them from there but they are very very singular that's the limit where the covering surface degenerates. So the regular correlators where they sit at different points winding number can never add and you get the correlate you get the, the, the structure of the fusion rules that are appropriate for the symmetric orbit fold. That's exactly how people calculate symmetric orbit fold correlators. Okay, so now our plan is that we're looking at these correlators, and we want to analyze the world identities and in order to analyze the world identities. What we have to do, we have to insert side plus and minus and now I've managed to mislead my crucial piece of paper. But maybe I find it here. Yes, I find it is the final one so the end is near. So, I propose to define the following functions. So, okay, let me let me just write it down it's a product of one to N of zeta minus Xi to the w i plus one over two divided by the product of alpha is equal to an n minus two of zeta minus you alpha. And then it's the correlation function where I insert zeta xi plus of zeta, and then I put the product of the w and the product of the piece. So it's basically this correlator.  Okay, so. So now what is what are these functions. Well, let's think about it. So let's ask, where, where do these functions have a pole. Well, originally you would think these functions have a pole and Xi plus and minus go close to one of the vertex operators. But if Xi plus comes close to a vertex operator the Polish of order minus w plus one over two. So Xi minus comes to occur to the vertex operator the Polish of the same order because of this term there's also a pole of that order, but in those types, I've exactly removed that pole by multiplying it by zeta. So this is zeta minus z i with a w i plus one over two. So this factor is designed to kill this pole. Right. I mean, Xi plus near each of these vertex operator has a pole of order w plus one over two, and I multiply it by a pre factor by hand that is just a zero of that corresponding order. So there's no poles at zeta equal to the z is and at zeta equal to w it has a simple zero. And I've cut divided by the simple zero so it still doesn't have a pole. So this function doesn't have a pole in zeta, and a function that doesn't have a pole in zeta and is holomorphic is called the polynomial. So P plus and minus of zeta are polynomials, because they don't have any zeros and they don't have any poles. So this is the degree of the polynomial. Well the degree of the polynomial is what, well I have to calculate the degree of the pre factor so I get the sum from is equal to one to n w i plus one over two. Then I get minus. The pre factor coming from here that obviously adds to the degree and then this subtracts to the degree. So then I have minus the sum of alpha is equal to one over and minus two times one. These are the things I divide out through. And what is the degree of this. So the degree of this you can determine by asking how does it behave when zeta goes to infinity, but you know how it behaves when zeta goes to infinity because zeta is a spin a half field. So it'll go like one over zeta. So there'll be a minus one from the behavior. That's the zeta behave the large zeta behavior of this correlation function which simply follows from the fact that zeta does xi plus does out at infinity. So now if I find my, my help sheet, then I will realize that this expression here is exactly equal to. So now I can't find my sheet anymore but this expression is exactly equal to M. So this goes exactly like, I can rewrite this this goes like this goes like the sum over is equal to one to N of w i minus one over two, because I can subtract these two these things out, and then plus one, because I'm only checking out N minus two ones and I have to add in another one so this is the degree of this map. And this is exactly M as coming of the Riemann for its formula that I described before. So these are polynomials of degree M. Right. I mean they have no, they have no poles. That's the degree and the degree is exactly the degree as predicted by the Riemann for its formula. So what about the characteristic identity that characterizes the covering map. Remember, what we have to look at is P minus of zeta minus x plus xi of P plus of zeta. But look at this, you see if you look at this combination, then see minus of the C minus of zeta plus x time P plus this kills exactly this term. So what's left behind is it goes like something to the order. So this goes like order z minus zeta minus z i to the power w i minus one over two. And remember there's an additional pre-factor standing outside that goes like zeta minus z to the power w i plus one over two plus w i plus one over two. So you see that this combination goes exactly like zeta minus z i to the w i because w i minus two plus w i plus one over two is w i. And therefore, these functions are polynomials of degree m that satisfy the characteristic equation to be defined in the covering map so I conclude that the covering map gamma of zeta is actually equal to P minus divided by P plus. So this is the minus sign for button and somebody remind me where I dropped my piece of paper. So this is my so this is minus. This is minus P minus of zeta divided by P plus of zeta. What this tells you is that the worksheet theory allows you to to recover the covering map. The covering map arises naturally out of the worksheet theory by virtue of looking at these correlation functions and just dressing them up so that they become polynomials of the right degree and the construction guarantees that these correlation functions of the worksheet theory produce for you exactly the covering map. But remember, the covering map doesn't generally exist. But if the correlators produce you the covering map, then something with this construction must go wrong for those configurations where the covering map doesn't exist, because we know the covering map generically doesn't exist. Now if you trace through this argument carefully what you see is that the thing that has to go wrong is that P plus has to go to zero. So the question must break down, and then you can show that if you plus goes to zero then the correlator without the insertion of psi plus must also go to zero. So you prove from that that because you can reconstruct the covering map from the correlators by inserting an additional field, and you know that the covering map will sometimes not exist. You can prove that whenever the covering map doesn't exist. You can prove that the correlators, even without the insertion of the xi fields must be equal to zero. I the correlators on the world sheet can only be non zero at the location where the covering map exists. And if you do a little bit more work you can show that it's actually a delta function type localization property, so that when you do the integral over the world sheet modular you really recover the sum of covering maps. So that's the reason why this world sheet theory knows about covering maps, and what it really tells you is, you see this identity. This identity tells you that psi plus and psi minus want to be the twister variables. And this is like the incidence relation that recovers the space time coordinate from the twister variables. What I strongly suggest is that the symplectic bosons played a role of the space time twisters, and the localization, the incidence relation of the space time theory is encoded by the world identities of this symplectic boson theories. And this is the idea that has then subsequently motivated us to try to imitate this also for ideas five and, but obviously I don't have any time anymore to sketch that but here in the ideas three case we can really nail this down. And this is exactly with what you would expect from the symmetric or default answer. So my time is truly up so I'll stop here and thank you for your attention. Thank you very much for this beautiful set of lectures.