The Orc is a stain upon our galaxy, make no mistake, a rampaging mycelial race, more than it would seem, ancient engineering and design, loose upon the void. The greatest environmental or biological or both disaster in galactic history, from a certain point of view, as elaborated upon in a previous record, identags to be found suspended to this one, the Orc is an almost perfectly engineered weapon machine, a facet of their existence not readily discernible to the casual observer of their horrifically degenerate and barbaric character. This Mien almost totally disguises their robust, biological, one hesitates to use the word sophistication when discussing these creatures, yet one has no special choice in this regard. Quite how their race accomplishes much of what they have been able to, despite the, on surface level, contradictory aspects of their existence, is a facet of the biology itself, and one that has deserved its own entry into my chronicles for the sheer ramifications it possesses. So then, that this is a record of the Orcoid race's Gestalt and genetic consciousness, the pervasive new sphere that appears to tie their entire race together from birth until death, the bluntly named waar effect. The Orcoid Gestalt, as I will refer to it from here on, in order not to debase Imperial Gothic with the grotesque renderings of their tongue, is, it would appear, a psychic phenomenon unique to the Orcoid genus, and one that permeates all levels of their species at a subconscious and apparently genetic level. As one since deemed heretical Mechanicus biologis adept posited, it is effectively a manifestation, species wide, of an animalistic pack instinct. This is an apt comparator, as the Gestalt has apparently little to no effect on solitary orcs. It does, however, scale exponentially, which is key to how it functions within the macro structure of this genus. Once orcs begin to gather, which they inevitably do, the field begins to form, and strengthen, as do the orcs themselves. As discussed in one's previous record, Orcoid biology is a curious hybrid of features humans have ascribed to both anomalia and to fungi, although the majority of the latter's characteristics are found within the organs and adrenal and musculature systems. Their brains are entirely animal, and larger than most would anticipate. One particularly overdeveloped region is the Paleopaleum, what humanity refers to as the Old Brain, the part of the cerebellum that in many species contains the root evolutionary elements derived from biological ancestors. It typically governs baser animal instincts, such as aggression and close social group predispositions. However, the orcs have no biological ancestors. This has been the conclusion of almost every majorist biologist worth their salt, who have studied them in depth, and whose capabilities can consider such an outcome that the orcs are an artificial, engineered species. Their lack of vestigial evolutionary throwbacks, such as the human appendix, for example, is evidence of a lack of process, as well as the sheer robust durability of their biology. Is the Old Brain of the orcs not a throwback? Based on the treatises of Magi, one has perused? No. It was simply selected at their creation for how useful it is. If one is to take the purpose of the orcs as a biological weapon system of sheer blunt force, the basic instincts of a predator species would naturally be the best place to begin. But if the brain is largely given over to the instincts that govern the individual, then where does the generative force of the Gestalt field lie? So far, one does not believe either the Mechanicus or the Ordosinos have been able to offer an explanation for this. The prevailing theory, such as it stands, is that its generation occurs at the cellular level, encoded within the very DNA of the species, inherent and untrammeled by the restrictions of sentience and independent thought. Indeed, this explanation certainly seems apt given how, initially, the Gestalt field affects the Xenos in question. Once enough quantities of the creatures have gone through the fungal spore reproduction cycle, they will instinctually seek each other out, both to form hunting bands and rudimentary tribal structures, and for competition. This latter aspect in these early stages has been theorized as one of the primary catalysts of the Gestalt, as if there is one thing orcs seem to adore above all else, it is combat. The purpose of these brutal early bouts is two-fold, both sociological and biological. In the case of the former, it allows the Xenos to establish a robust hierarchy, where the... Mightiest is Rightiest. The survival of the fittest method of establishing primitive dominance that sees the strongest or cleverest, or both, orc rise to a position of authority quite quickly, and with little in the way to challenge them, unless there is a stronger or cleverer opponent that emerges. As it happens, the first biological effects of the Gestalt begin to take hold. The individual aggression of the orcs are bolstered, pushing them into more frequent and more violent competition that, as noted, allows a tribal head to emerge, but also, thanks to the means by which their bodies function, causes them to grow, developing musculature and adrenal systems quite rapidly. Orc is also apparently affected, as at this point, the orcish society begins to propagate odd boys, as they refer to them. These variant orcs are effectively specialists in some form of information or skill that is not learned so much as simply known. The variants do not study their specialities, they are simply aware of them. The phenomenon has been marked by Mechanicus and Inquisition alike as further proof of the engineered nature of the orc. Once a smaller yield and mass of the creatures has been reached, the variants emerge to help their kin move further up the social and technological evolutionary ladder. Whether the knowledge is drawn from the Gestalt field, or it is embedded in their cellular structures to be merely activated, is currently up for debate by those who are aware of its mysteries, and one fully intends to elucidate upon it later in this record. Biologically, at least, Orcoid DNA possesses a unique third helical strand, one that is entirely fungoid in nature. This helix remains consistent in form and composition, no matter what subspecies of the Xenos is examined, be it a squig, a snotling, a grot, or an orc. All of them bear the same strand. It would certainly seem highly likely that this unique strand is the catalyst, which spurs on Orcoid development. At a certain point of population density for squigs, the minute Gestalt these creatures create triggers the growth of snotlings, and so on and so forth up the ladder until orcs eventually emerge. Given the strands' presence in the entire macro species, this has been theorized as being the access point also for the knowledge base, although precisely how is yet to be established. The Magi, parley to Esoterica of lost ages, have speculated that this is the result of some hitherto unknown Orcoid access to the Akashic record. This concept, one originally born of philosophy based on old earth in ancient times, was once pioneered by a near-herotechnical adept of the ancient Mechanicum, Coriel Zeth, in an attempt to gain access to a hypothesized claim of pure psychic thought, believed to contain some total of all universal knowledge. Every thought and every idea ever possessed by any sentient being at all points across linear chronology. Unfortunately for Zeth, and indeed humanity, her research ended with the destruction of the magma city upon Mars during the death of innocents at the very outbreak of the Horus heresy. What scraps of it remain have either been sealed behind fanatical adeptus Mechanicus dogma, or stolen away to reaches unknown by the dark pantheon-pledged adepts of the so-called true Mechanicum. Such theosophizing, where it pertains to the orc, has been largely suppressed, although one believes this has less to do with scientific inquiry, and more to do with the idea that any such primitive xenos able to access this plain denied to humanity is a heretical concept in the extreme, and not to mention terrifying unequal measure. One must, however, defy this suppression in order to explore the idea here in its fullest, for the implications therein are frankly staggering, before one even touches upon the potential ramifications of such knowledge that could be made for imperial defence and scientific progression. What we know of Coriel Zeth's research was lost in its near entirety during the death of innocents. But what we can establish is that the psychic energy needed to access the Akashic records was phenomenal, requiring not only a collective of specialised psychers, with a specially crafted chamber upon Mars, but a machine that was capable of harnessing the power of the astronomical itself for the briefest of instances, effectively to boost the capabilities of the humans empowering the reader, as Majos Zeth referred to it. Taken from a perspective of measuring pure psychic power, a tribe of orcs could not possibly generate enough energy with their meager gestalt to achieve access to what Coriel Zeth was seeking. However, that being said, the knowledge the orc variants access is usually only of the most basic nature, but crucially, one that tends to scale with the size of the tribe as it grows. Initially, technological knowledge is restricted to simple tools and weapons, usually how to construct both out of scavenged materials, negating the need for the Xenos to mine or process metals. Slavenger is of course completely off the table, although orcs are omnivorous, their initial dietary habits are as varied, predatory and scavenger based as their industrial development, not to mention their somewhat carnivorous inclinations with the skrots, snotlings, and squigs beneath them. As the benefits of the gestalt field become more pronounced, the system becomes self-perpetuating. The expanded technology brought by the orc variants allows the tribe to grow and prosper, to seize more territory and fight more enemies. As more orcs join the clan group, the information the variants are able to draw upon is expanded. The physician class becomes able to graft whole limbs and crude bionics onto injured orcs, where previously they had only served to augment the species already formidable healing capabilities. The shaman class reaches a degree of psychic ability that allows them to channel the gestalt field as a human combat psycher would, emboldening those around them into greater displays of violence, where previously it simply been spiritual leaders marked by their fellows as touched by strange Xenos god composites. The technician class evolves from spears and crude ballistics to an understanding of combustion engines and the principles of rocketry, as well as improvements to production lines and industrial optimization, however, scattershot and haphazard the latter may be. This is true galaxy wide. One may separate two tribes of orcs by a whole segmentum, placing them upon completely different worlds and observe that, in time, the species develops to an almost identical, but still idiosyncratic, level of technology and society, with the devices, social structures and traditions only shaped by local environments to a certain degree. All of this is literally instinctual. At a certain level of development, usually the one outlined above, all of these variants begin to develop beyond simple parameters laid down by instinctual orc knowledge and begin to experiment. For the medical cast, this often presents as obsessions with a certain aspect of orc biological augmentation, such as training orc muscular cultural development in one specific capacity or a hyper focus on perfecting the usage of certain bionics. In some cases, this has led to orc variants capable of pushing the boundaries of the species biology in disturbing ways, such as full cranial transplants to literally graft an orc head onto a brand new body. The technologist class experiments run with wild abandon, often plunging headlong into fields of weaponry that seem to defy the laws of physics. No orc technology is mass produced. All of us is simply assembled with gleeful abandon by those that the guest outfield activates for its purpose and by those they beat into submission to assemble the simpler aspects of technology. Encaptured by Adeptus Mechanicus' adepts for study, it appears to all intents and purposes to be little more than piles of junk that seemingly should not function, only upon closer inspection revealing a robust form of competency in its creation. Orc technology merely has to complete the task it is set to accomplishing, usually to go fast or kill as many sentient beings as possible, and in many cases both. The knowledge of the process by which the guest outfield activates orc variants and its expanding power concurrent to the amount of orcs in a local area has led to a common misconception surrounding its effects upon their technology. Numerous poorly educated imperial elements have become possessed by the conviction that orc technology works because they believe it does, that the power of the guest outfield can turn any piece of scrap that resembles a gun into a functional device capable of firing live ammunition, that it can transform a ramshackle buggy into a powerful land armored vehicle by virtue of the fact that the orc pilots believe it to be one. Such conclusions vastly overstate the effect of the guest out to the overwhelming majority of orc bioforms. While it is true that the field is not well understood and while there have been instances that lend some credence to the minor far more subtle effects the field may have to boost orc technology, it is not so powerful as to warp the fabric of reality. Ultimately the knowledge that the field bestows, the empowering effect it has upon the individual orc and their biology is where it lies. We would underestimate the cunning and tenacity of these creatures at our peril. Many, many humans have throughout history and none have lived to make the same mistake twice. The knowledge that the orc species simply actualizes into as a social group matures is this true power. It allows orcs separated by space, a whole planetary systems to ultimately rise to the point where spaceflight is possible and the correlation of massive Xenos forces equally so. At this point we have reached the infamous Horde phase of societal progress whose name, Wa, is naturally shared by the guest outfield that it has enabled. At this point the original infestation has developed into a planet system or sector threatening military force as the Imperium has encountered upon countless occasions. Multiple variants have emerged to catalyze various subsections of the Horde and these variants may often find their experiments with technology or biology push the boundaries of what even imperial science has considered possible. The war bosses, the leaders of the various clan groups inevitably submit to a single individual whose body and mind have grown glutted upon the powers of their own internal systems and the effects of the guest outfield. The devastation unleashed as the Horde careens along its warpath only pulls in more and more orcs into the fold whether drawn by rumor or biological compulsion to seek out the combat the Horde's war boss has promised his underlings. Please also bear in mind that orcs do not tire when engaged in combat as humans do for example if anything it actually provides them with energy. Since times long past the assassination or destruction of the lead war boss has become the standard imperial engagement protocol for dealing with such concentrations of the species. While this is of course a feat easier said than done it has typically led to an immediate disruption of the leadership structure as also Bordenant bosses begin vying for the prime role as instinct dictates them to do leading typically to significant infighting and ideally the eventual dissolution of the Horde as breakaway elements seek more fruitful volumes for combat and plundering. This is in effect the course their biology has dictated for them their reason for existence propagate expand conquer repeat. When taking into consideration the almost certainty that the orcoid is an engineered creature one cannot help but admit to the genius of their function. Every aspect of them however crew degenerate and unsophisticated is nevertheless finely tuned to achieve bellicosity endurance and a sustainable yet exponential expansion. They are a weapon system created to thrive under impossible privation orchestrated in their genetic coding to actively seek and desire the state of being all other life forms struggle through and in truth they appear to be a weapon whose masters have long since disappeared to however return to an earlier point where does this relate to the potential of the Akashic records. If the conclusions of the learned are to be believed the orcs are indeed a creation of the ancient progenitor species known simply as the old ones whose war with the Necron menace drove them to create biological weapons that would it appear one day devolve into the modern orc form. Since the disappearance of the progenitors from our galaxy long before humanity had even evolved it is a decent supposition to presume the degradation of the orcoid system has taken place to such a degree that they only possess fragments of their initial capabilities all save the most robust elements just as societies fracture and decay when they're more advanced technologies and structures are placed under undue privation or come to lack the skills necessary for their maintenance so too with the orc a technological system in and of itself just a hybrid animal fungoid one laps into decay following their abandonment by their creators it is a testament to the abilities of these ancient terrible xenos that their creations have outlasted their builders and risen to a position of if not primacy within the galaxy then consistent survival and prosperity a perfect self-replicating war machine however if their capabilities have indeed been degraded we must presume this would hold true for their ability to access the more advanced aspects of their knowledge base perhaps at one point during their war with the necrons the progenitors creations had access to a far larger suite of potential knowledge fueled by a more coherent and well-maintained gestalt than the rudimentary one they now possess the old ones were by a ldari and indeed necronteer mytho tracts supposedly superlative masters of the esoteric and one would imagine given their unrivaled dominance in this regard that if the akashic records do indeed exist their access to them would have been unquestionable that they in their desperation in the war of heaven would deign to create creatures who could do so similarly but at a primal and instinctual level as opposed to a learned one all in the name of becoming a better weapon system the thought is frightening in the extreme but not exactly beyond the realms of possibility given what we know of this titanic conflict there is of course another possibility and one that is in some ways quite a bit more esoteric fungoid organisms typically form biological structures referred to as mycelia networks of root structures below and above the loam or its local equivalent through which nutrition and even rudimentary information can be carried it is already a verifiable product of orcoid development that their spore pods that are used for gestation form extensive mycelial networks through which said development is regulated and expanded it has long been a theory of rather fringe elements of the major biologist divisions of the mechanic us the certain fungi given a density may form a discreet but robust connection beyond the boundaries of the known extending perhaps into the warp and potentially a substrat of the sea of souls hitherto unknown to humanity this galaxy spanning macro connection network in effect removed from the interference of the primordial annihilator and the dark pantheon would go a long way is to explaining many of the questions within this record as well as a general resilience the orc bears to the corruptive elements of the warp in order to lose you know elucidate is far too strong a word expand upon certain material here to forth discussed I am appending to this record a recording of a blood axe orc of an especially mercenarial variety during an exchange of materials with a fairly radical member of the hordozinos you came he's got all these big questions about how it all works we don't really bother with all that well sometimes we do I suppose can't be in a fight all the time after all when you're thinking meets not busy working out where it's best to put all the DACA starts asking questions like what is DACA where does it come from and if you've got too much thinking meet for your own good why is DACA mostly though even for the lads with a bit too much meat in their heads life simple when you're an orc mentally speaking Regan you lock make it too complicated in my insteemed opinion you got this obsession with the truth what is the truth where shall we find it how will we cut it out of them when we do and then what if it's the wrong shape and won't stick to the bits we've already got well I'll tell you something and I'll tell you for free there ain't no truth not to us we'll rather there's more of it and we know what to deal with so it don't really matter you know like grots why have we got so much to spare easy what an orc believes is the truth even if it's a rubbish orc even if it's a rubbish truth and if they have a bit of a think and end up with another more different truth well then that's the truth as well now you're looking a term for this actually I know these things you see hang on I even did some glyphs of it on one of these bits of metal so I can ever remember yeah we go cognorage dissonance it's a thing where you believe two things at the same time and it makes your brain hurt well imagine that only instead of hurting it makes your brain extra muscly now you're thinking like an orc maybe the reason it's so easy for us to fit all these new truths in our heads is because we're already used to ideas showing up in them what we ain't even thought of orcs just no things big things mega things I'll give you an example I saw a grot once get caught up in all the shattery wind blowing out the back of a Dacker jet engine he got blasted right off his claws right into the chest of a mech who was standing nearby totally boosted all the mechs ribs look my truck it was amazing but you know what that mech didn't even bother getting a new lung or nothing they just got this weird look in their eye scampered off to the scrapyard to do mech things next day they come out with this massive gun that shoots grots through the actual warp so they come out the other side or mental unbiety from all the devils and that mech came up with this thing overnight just after seeing a grot get chucked about by a load of angry wind and that's not all I've seen that same Zoggin gun get invented four other times by four other mechs since then because you can't just think something like that together all by yourself it's like the idea is already there in your blood maybe your guts or your earwax and you just need to you need to remember that you know it like well same way if someone ran up to you with a massive pipe or something and they wanted to brain you you wouldn't stand around panicking about what your hands were and what they were for you just batter them right well you probably call them a heretic or whatever as well but you you get my point having these ideas in you is just part of being an orc and it's funny because the more orc stuff you do meaning the more you fight the more other lads you have fighting with you and the more you win the more of those ideas you remember the more know what's a packed into every one of them you get enough orcs together doing enough fighting doing enough winning they'll make things you wouldn't believe so who hid all these ideas in our blood then I don't know you're a weird boy once told me the brain boys designed us that way so they could just let us get on with it without having to teach us loads of boring numbers and that and again same weird boys best mate was an old wheel and I saw him give a medal to a rock once so you know eat it with a load of salt as you look like to say but all this is doing my bouncing it's like I told you at the start really ain't stuff we feel the need to do loads of mega thinking about what matters is this all I has got to do to win at being me is be me be an orc batter stuff have a massive laugh battering it if I do a good job Gorka Mork the big lads themselves in the great green they'll let me know by making me bigger and fight here if I ain't done a good job I'll get burst I'll zog off back to the great green to wait for another turn it's a good saw a life always wondered why you lot don't tried it really the time you've taken to ask me all this and then for me to answer you could have had at least two and a half decent fights I reckon eating someone's ear for a laugh got a bullet wound with an interesting shape shout with the emperor's name into a bin I don't know whatever you chose to do would have been more fun than this so do yourself a favor yeah grab your shooter find a get to crump and crump them then keep doing that till you get to the get big enough to crump you nothing to ponder nothing to doubt just mashing stuff forever well good existing as surprising as it no doubt is to hear a member of this benighted species speak with such relative lucidity and eloquence we must endeavor to focus instead on the insights that can be gained from the content of this deposition much of what has been elucidated upon already has been corroborated and although of course from an orchoid point of view it does substantiate what imperial xenobiological intelligence has been able to establish about the species the references to the big green are troubling to say the least per ones suppositions previously laid out this could refer potentially to either the akashic records which the orcs may believe they have some claim to or the hypothetical substrate of the imiterium connected to the species through mycelial means beyond this however the real inside stretches to the psychological the orcs as expressed by this individual are utterly unconcerned with all existential questions whatever the brain boys as they refer to their progenitors encoded them to do they simply act out it is blunt but if this orc is to be believed it is also a liberating form of existence slaves to one's own biology yes but not too much beyond that say perhaps another orc that is bigger than you this is of course programming in and of itself orcs do not have to care about any of the larger questions their existence raises and this in turn leads to a society in a culture utterly stagnant completely locked into patterns expressed through instinct and not else interrogators have often found a difficulty making headway through orc debriefings simply because the species is capable of holding two completely opposing ideas or memories in their minds at the same time cognitive dissonance has literally been engineered out of them what possible use could such a thing provide if an orc remembers something then that thing effectively happened regardless of the objective empirical truth of the matter the merging of ego and in these creatures is as superlative as it is truly maddening but one cannot help but wonder as to what would happen if a member of their species was able to supersede this programming what happens when a weapon becomes self-aware what happens when a gun realizes it's a gun if a system like this should somehow evolve past its programming well would we not be in the same situation humanities ancestors were placed in so many millennia ago during the machine wars at the downfall of the first human galactic dominion the patterns and behaviors exhibited by the warlord gas called mag uruk Thrakka have already caused grave concern amongst members of the Inquisition and the biologists worried no doubt that such a crude awakening may already be underway in seemingly typical orcoid fashion and corroborating the story provided by the blood acts earlier the awakening of this orc was thanks to a massive traumatic injury in Gaskell's case a bolt around the skull should that wound have awoken some latent programming or destroyed it should Gaskell be the emergence of a form of orcoid not seen since old one warred with katana I shoulder to think what that could mean for humanity the mechanic has oft warns of the dangers of ancient technology lying dormant in the outer dark somehow being recovered and accidentally awakened rampant genocidal thinking machines devoid of all that is holy actualizing in a universe that thought them long dead well we may yet face a weapon system older still one wrought by intelligences beyond our comprehension and who are unaware of even their own potential we must pray not such a day never arrives of a imperator gloria in excelsis terra this video and this channel were made possible thanks to the very kind donations and support from my patreon subscribers if you'd like to help support the channel head on over to patreon.com slash oculus imperia if you'd like to receive more updates about the channel and any future videos you can contact me or follow me on twitter at oculus imperia otherwise please like subscribe comment let me know your feedback and as ever thank you very much for watching


What's going on everyone? Sam here with late in sports card, German 23, Penina legacy football hobby, six box break number six Random team style. There's a look again at your random results. Whoa As always you can find those on the website under the results tab as well as the link that's in chat Thank you all again for filling this up. We got flawless coming up guys Like I said, just three to go Three to go. Let's get that filled. We're almost there on flawless almost there Flawless basketball tiered random team style gets you one Team from the top tier value wise you are guaranteed a top 15 team in the product one team from the bottom tier value wise which Honestly, sometimes tend to hit really well if you have a logo man of a bottom tier team It's always awesome. And then you also get four promo spots into our national promo four chances to win a 2009 Mike trout pits That is a Nick Bosa under the lights Just upside down. Okay Darnell Washington that's here looking for as a stealer. I would say he's like the tight end.  We got Henry a Jalen Carter prized the prospects for the Eagles Got a full case of select draft picks coming up and then we're on to some Tony dorsett to 299 Zay flowers to 65 dare to tear oh, that's it for the Ravens There we go. Wow. That's awesome. Congrats Baltimore. That is terror Uh Cory G. Congrats, Cory. There you go, buddy. Nice day. It's a Brock Purdy under the lights Yes, there might be something cool in there. Who knows 75 Michael Meyer they used to do dude There was this set back in the last one past Tony dorsett back in like the 2000s It was like what was it called tops?  There you go Dallas wear and Parsons Does anybody else remember that the tops like I forget it was like minted or something cops minted was it called? The ages of Dion fries and put LT as a jet. I'm really shocked. I didn't put LT as a jet these Weird teams that they got going on here. That is a Tanner McKee 250 auto. I think he's an is he an eagle McKee. I think he might be an eagle Tanner McKee, there's a under lights Bijan Red DJ more to 299 a worn moon glass from the past Joe Green Arch Manning as a Viking still weird Still weird to me again guys three to go on flawless got to get that filled up three to go Go and drop that in Kyle's shirt. Will Levitt's red to 299 for the Titans that gate of dominance is a Michael Irvin George Kittle orange under lights did 149. We've got a Bruce Smith Legends Warren moon is a sea. Ah, that's so weird Clayton tune auto under lights believe he is a cardinal if I'm not mistaken James Harrison Surprise input James Harrison is a Bangalore Patriot laces out Dan. There's Peyton Manning to 100 Jordan Addison prize prospects for the Vikings Like he's a cardinal Whoops Brock birdie MJD Just chill on the back of the card That is a mini of Vita Vaya Kishan and Garrett Wilson And we'll do a dice roll on any of the the duals Had its mark Andrews to 199 Russell Wilson Thysman Erlocker base and base and one of those black and white CD lamb nice one there under the lights not numbered Jim Kelly blast in the past Zay flowers auto for the Ravens nice one that Ravens doing really well actually With the dare to tear now the Zay flowers auto very nice John Riggins decade of dominance Campbell and base For just three initials. I kind of like flowers is yeah, it's not bad Etn to 150 Ed Reed is a Raven. Thank God I've been so mad I would be right on the phone with them say what what are we doing here? No, I'm gonna change Darrell Washington I'm dead. No, you're not recalls Please send back all your Ed Reed decade of dominance It's a guy new version. He's had a new version dropping Rakeem Jarrett on the red 250 nice one there for the bucks a Rishi rice silver rookie Herschel Walker and a base Paris Johnson again there on to the next one Top one here. Good luck every body Poipole Jaren Hall to 10 to a 10 nice one there or Jaden Hall No, Jaren Hall Jaren Hall Alan Page cool one there. It was that about Bruce Smith. I think we've seen all the base I Don't know somebody somewhere. I just don't know where Shalen hurts gold ice to 25 10 of 25 under the lights for the Eagles Warren moon Viking fights See I'm drafted Zach Evans auto nice one there. Is he a bangle Chris Evans or Zach Evans? Zach Evans. No, he's not a bangle. He's a Nope, he's a ram I've been close There you go, LA Marcus Alan blast in the past What let's have Google do the yeah, honestly That is a Derek Johnson cool one there to 100 for the green for the Chiefs Matt Stafford decade of dominance. There's a Trenton Simpson rookie for the Ravens damn Rino Trevor Lawrence mini bronze to 75 for the Jags Eric Dickerson or in Deandre Hopkins to 199 Josh Downs prize prospects Vernon Davis Tana McKee got one of those black-and-white tendon hooker for the Lions. Nice. There you go, Detroit It's Saturday Warren moon USFL that actually be kind of cool Jeremy. I kind of be in on that I'd kind of be in on a Warren moon USFL card Hending Hooker to 149 another orange auto for the Lions. There you go, Detroit's Congrats on that one D town For the ages is Alex Smith And base Antonio Brown What did he own like an AFL team? Yeah, there's CJ Stroud to 150 for the Texans Swill levis we'd have to put owner in quotation marks Yeah question mark there you go Clayton Tune Herschel Walker legends Joe Namath and base There's a orange mini of Justin Fields to 125 Jalen Hyatt Favre dorset base and base I'm not gonna lie the USO the USFL is the hardest to watch football league really that there is in my opinion Like X XFL still have like good talent like they still good players true SFL is like watching like high school. It was so bad three to go everybody. Sorry could not watch Yeah, that's not great three to go everybody on flawless got to get it filled up cured random team style for promo spots available up for grabs per spot you purchase and flawless you will get Four promo spots into the national promo, so thank you all That is a Kendra Miller to 249 dual rookie Jersey auto for the Saints again. Congrats New Orleans go to Henry Dion a Archie Manning glass on the past Bryce young rookie there. There's a Cedric Tillman. Nice one there. He's a brown. I believe Grats on that Cleveland. I think so at least More and set up gonna have Donnie confirm that Chris Carter is a dolphin. It's so weird So like not right Miles Murphy for the bangles to 299 on the red Steve young Yeah, Tillman is a brown Tillman is a brown perfect. Yeah, so Cedric Tillman is in fact a Cleveland Brown We have another dare to tear here potentially. I think we do. Yes. We do 14 of 65 Tyree kill dare to tear for the dolphins very why I swung there buddy and That's how you know what it is tear it right there. See if there's a card inside Kshom beauty for the Patriots Do you get one of these did you tear it where you're not there depends who it is Flowers I would not tear I'd rather just have Brock birdie to 125 to save flowers prize prospects Tony Dorsett base base rookie Jarrett is well in there talk about a box That answers the question You Red Devin witherspoon to 299 last first rounder to sign his rookie deal yesterday the Seahawks Jimmy Garoppolo to 100 remember so Joey Bosa was like Held out for a while on his rookie deal. I remember and I remember someone made like a fake like Like conversation between the Chargers and Joey Bosa and like their notes app. Yeah Another dare to tear Paul Kraus 42 is 65 for the Vikings nice one there two dare to tears in one box very cool Yes, go into the Vikings Paul Kraus. It's got Minnesota. Yeah, right there Barry why again Barry's got two dare to tears Jaylen Carter under the lights for the Eagles to 100 Yeah, the all-time interceptions leader still really yeah Not a lot of people know that's why I think Paul Kraus is one of the in my opinion one of the best defenders of all time Devontae Smith to 50 on the red mini. Well, I didn't know who he was so base And they're like so to be like Chargers, how much do you want Joey Bosa? One billion and it was like it was like Chargers. Okay. Be serious 10 billion It was like Chargers. We can't pay you that much money goes 11 billion and it just goes Chargers sigh To go to the Broncos Marford babes.  I remember I was dying at it Because again, that was like 2016 or 2017. So I was like in high school That is a Josh Allen to 299 a under lights will levis blue to 35 nice.  We are almost there three spots remaining in flawless I know because they the legacy box. They're big. They don't really like break Mike Evans. That's Tanner McKee the Eagles Tim Brown legends and Mini Vaughan Miller the Bills and John Riggins Decade of Dominance I was I was in the E5 Sugar Bowl during The New Jersey generals.  Wow. Oh say flowers. Nice. That's just on the lights It's still cool. I do you get my my say flowers fixed today, though? Say flowers fixed today with the auto that we pulled in the dare to tear. Oh, you're fix. Okay. Thanks. Yeah, Ed Reid for the ages I think so you got it fixed. I'm like, how'd you get the idea to get fixed? Yeah Who does that thick one here? Yeah to 25 Aiden O'Connell Laundry tag for the Raiders. Congrats Vegas that'll be going to Joshua K Let's go Joshua. Nice one there, buddy. I see some game time Jimmy Gizzard. True. Very true. Joey Porter Jr. Rookie He almost passed Breeze's Purdue record Didn't really I did not know that he was he either passed it or tied it for most games with 350 passing yards. Wow. Bijan to 299 for the Falcons. That's kind of wild actually. Decade of dominance Jonathan Ogden I think Breeze had it at 12 and he either got either tied it or he passed it That wow dude another dare to tear To 65 go into the Niners. Congrats San Fran Jonathan s Jimmy the intern really loaded this case Jimmy the intern was looking out for us on that one apparently under the lights Dak Prescott This they put both Montana the night or uni. I know yeah True very true Deandre Hopkins under the lights to 75 last one to pass Joe Montana Matt Judon to 299 for the Patriots Garrett Wilson and Kishon again. We haven't had to do a dice roll for any of those yet It's still three to go on flawless everybody.  We're almost there almost there for the ages Russell Wilson Navarro Bowman Wow, that's awesome. I Need one of these Wait for a 50 Navarro Bowman He was so sick dude Jalen Hyatt. He was Like the year they made the Super Bowl he was like the linebacker next to Patrick Willis And he was just like he led the league in tackles I think two years in a row and then pours Achilles and was never right again God, he was like one of those like undrafted free agents that just turned into an absolute like stud dude Gotcha, there's Randall Cunningham six of 25. I remember I was in My eighth grade Spanish class. We all got to pick Spanish names in Navarro was one of the names you could pick So I picked Navarro and I spelled it just like them. I spelled it just like I did and a capital V Oro, that was my name in Spanish Navarro after Navarro Bowman Sounds like we're all the policy of Navarro from here. Yeah Navarro Navarro Pay attention No Be interesting definition of our name That's so sick dude, I need one of those cards Austin Eckler auto nice the 25 for the Chargers. There's some cool like vet autos in here, which I really like Randy Moss and Devonte Adams is Raiders. God Stop stop it Stop it That is a Rakeem Jarrett to 50 on the blue. He is a Tampa Bay Buccaneer. Yes Last from the past Drew Pearson, that's cool for the Cowboys mean Joe Green and Archie Manning Vikings Archie Manning Yeah, oh, yeah, rice played for the Raiders rice has like a decent amount of Raiders like autos and stuff CJ Stroud to 50 There we go Congrats Texans on that one Barry why? Like an impeccable I think yeah Jerry Rice is a raider, right? That Moss is a patriot and not what I believe this year. He pulled up. Yep Yeah, rice also played for the Seahawks and then had a very brief stint in Denver as well Matthew Stafford mini Warren moon Like when was it? Let me see when he was on the Broncos Yeah, it was like a practice squad guy in 2005 and his 20th year in the league. I Think yeah, it was like a preseason cot or something probably. I don't think we'll see a receiver play as many years No and be as dominant for as many years as he was right Orange Christian Gonzalez the 190 for the Patriots Jerry Rice that get dominance base He probably wasn't expecting to do any dice rolls. Yeah force did not take his dice with him.  We'll not stop like ready for it Clayton tune auto for the Cardinals again Congrats, Arizona James B CJ Stroud on the lights chrome Sanders Tim Brown Tim Brown played somewhere. I think playing like Hamplar or something. So I'm surprised they didn't put him as a buck in the air Henden Hooker prized prospects to at 249 Lawrence Taylor Strayhand Rice Let me make sure that he did Tim Brown really play. I don't know if it's Yeah, take play for one year in Tampa. Let's go dude. No bottle No, I can't roll my Rs. Dude Jameer Gibbs rookie Like whenever we had to do the alphabet and we've got to like that double R I always like sounded like I was choking because I couldn't roll my Rs. Dude There's Rakeem Jarrett orange to 125 CJ Stroud what's going on with Navarro?  That's what everybody was saying. Come on Tighten up. It was Sean. Sean game in for one day. Tighten up Matt Judon to 299 for the Patriots Don't take me on wicks chrome Paul Krauss Dickerson Paris Johnson and Henden Hooker Alright guys do a quick recap here pretty solid break overall. I would say we had no points either So not bad Clayton tune Austin Eckler Navarro bowman like the coolest card I've pulled dude. I'm love this thing Yeah, team high 143 tackles Yeah, he he made four all-pro teams Wow, that's Aiden O'Connell nice one there to 25 with a little bit of a laundry tag Very cool Cedric Tillman Kendra Miller Henden Hooker Zach Evans Zay flowers Clayton tune Tanner McKee and Kendra Miller All right guys, thanks again for the break or thanks again for the fill that'll do it for the break up next select draft picks full key


Today will be much less ranty and less physicsy, well, somewhat less physicsy. There'll be an increasing amount of geometry, hopefully. So last time I told you about what ADSEFT is, it's a conjectured correspondence or duality of physical equivalence between, on the one hand, certain quantum field theories, which are theories, as we discussed, that live on some fixed spacetime and a quantum mechanical. And on the other side of the correspondence, certain quantum theories of gravity we could call them. But really, they're quantum theories where space and time is dynamic. That's what one might most generally call a gravitational theory. And there are conjectured specific examples where you know what both the theories on both sides are. And there's a lot of evidence, in particular for this canonical example, this n equals 4 super Yang-Mills we talked about. There's a lot of evidence that this conjecture is true. It's still a conjecture. And as I said, it's very unlikely to be proven. Because to prove it, you would need to know what both sides look like. And understanding quantum field theories at a rigorous level of proof is already very hard for free field theories, let alone interacting theories. So I don't think it's reasonable to expect this to be proven. And as I said before, if you believe it's not true, I think by now you probably need to have some rather specific reason you believe it not to be true. However, that said, it's still important to stress. This isn't known to be fact yet. So I talked about relativistic quantum field theories, which maybe to many of you is very familiar and to many of you is a little bit alien, probably half and half. Today I'm going to tell you what a CFT is. And given what we've said before, I think we can go reasonably quickly. Conformal field theory is a huge subject. It's basically a statement. A conformal field theory is a quantum field theory with some additional symmetries beyond relativistic invariance of the Poincaré symmetry. It's a huge subject, particularly in low dimensions. And we won't be interested in low dimensions by which I mean one plus one theories where all sorts of beautiful things happen and will need very little from the general story. So I will only say things about what we'll actually need in order to define ADS CFT. So we'll start off by talking about CFTs. And hopefully we will also discuss ADS, which how many of you are familiar with ADS spacetime? How many aren't? OK, so hopefully we'll be able to discuss both of these things today. And then I'm hoping tomorrow I'll be able to state, actually, in detail how the correspondence works. We'll see. So let's just recap. In the usual coordinates on Minkowski, we have the Lorentz and translation symmetries of the Poincaré group, we can think of as being generated in this way. And we are going to now consider a theory that is not only invariant under those, but it also has a scale invariance. So we'll also require our QFT to have scale invariance. And scale invariance is an invariance of Minkowski spacetime, again as with the Poincaré group, where I just scale all my coordinates by a constant, time and space. So that's a scale transformation. And our fields that we build our theory from, if we want our theory to have a scale invariance, we should have that the action is scale invariant, and that we should have our fields have some definite transformation under this. And we require our fields to transform in the following way. So maybe there's other indices on the field that some tensor field, but let's suppress those. So the field transforms as an invariant as a scalar quantity up to some overall scaling, where delta is some constant associated to the field, and delta is called the scale dimension of phi. And remember, we've taken mass units. So we've taken h bar equals c equals 1, and we're thinking of all the units being given now in terms of mass. And this scale dimension, at least classically, is just the mass dimension of the field, if you have a little think about how this works. So classically, it's just the mass dimension. And the generator for this transformation, we call the dilatation or dilatation operator d. And it looks like this. So for an infinitesimal scale transformation, the infinitesimal shift in my field is governed by this acting on the field. This is how the field transforms under this transformation. So lambda is a constant. Yeah, I think we're butting up against physics versus maths in a very fundamental and deep way. And I'm not sure at this point, I'm not sure I can raise this to a mathematical level on the fly that's going to be useful and correct. So we can think, let's see, how do I think about this? So I can think of this as some diffeomorphism equivalently. And then this operator here under this diffeomorphism would be the Lie, I mean this diffeomorphism is generated by some vector field. And this is the lead derivative of this field contracted with that generating vector field. So that probably still wasn't high enough level. So then together, remember we already have our generators for the Poincaré group, which are the translation and Lorentz generators. And this new generator has some definite, I'm going to Well, it has some definite trans, we can now add it to our algebra, this is now our algebra. These are all the commutation relations of the Lie algebra together with the Poincaré algebra that we would require our generators for this scale invariance to satisfy. And just something to note, now M squared, which was a Casimir of the Poincaré group is no longer commutes with everything, so no longer a Casimir. And so in particular, you can't have a field with a mass as we're used to in relativistic field theory no longer is a good representation under this larger Poincaré plus scale invariance group. And in a scale invariant theory, all couplings in the action should be dimensionless, i.e. that the action should have no mass scales within it, because obviously those mass scales won't be scale invariant. So action should have scales. So a theory that's scale invariant is of this marginal type that's truly scale invariant is of this marginal type that we wrote down before. And in particular, roughly speaking, it will look the same at every scale. So it's a very special theory that's guaranteed to exist at high energies, at low energies, everywhere you like. So these are complete theories. These are good theories in the sense of if you want a fundamental theory that describes, that works up to arbitrarily high energy scales or short distance scales, these types of theories would be in that class. You might wonder whether there's a limit on what delta could be. Delta should be real and greater than some number that's greater than zero. And for those of you who know about quantum field theory, there are unitarity bounds given the, we might see this later, but depending on the spin of this field, there's some different number here that depends on the number of dimensions you're in in the spin of the field. But in particular, delta had to be greater than zero for unitarity reasons. Yeah, was there a question? Sorry, yes, I've suppressed indices, as I said. So these could be some spinner, tensor, fields. Yes, generally, generally. That's right. So indeed, for example, the Dirac field transforms in some analogous way. OK, let's have some examples then. So the free scalar that we've discussed, so I'm suppressing the Lorentz index there, this is a scale invariant, I'm sorry, of course, what am I doing? The free scalar, firstly, I can't have a mass term. I should have put a cross through that. I'm not allowed a mass term, because that will obviously break scale invariance. But if it's a massless free scalar, that is scale invariant provided my field has its usual dimension, delta is d minus 2 over 2. So provided delta is d minus 2 over 2, this action will be invariant under the scale transformation. It's certainly invariant under Poincare, as you all know, but it's invariant under scale transformation. The reason being that under a scale transformation, this will pick up d factors of lambda. The derivatives are inverse d by dx, so they will give me 1 over lambda squared. And so I'll require my field to transform with the appropriate numbers of powers of lambda to cancel all of that and leave the action invariant. Yes, that's at the quantum level, yeah, exactly. Yes, I mean, don't worry about this. That's for people who we might say more about it later, but it's not important. Here's another example. Classical, say, Phi to the 4 or Yang-Mills in four dimensions has a scale invariance. So if I write down a theory like this, let's have some coupling, let's call it C. Normally you call it lambda, but I realize I've used lambda, so I won't. So this thing, again, for the same, this is now in four space dimensions, for the same delta here, so delta would be 1 in four dimensions, this is also scale invariant. So this interaction term. Remember, this is a free theory. The action is quadratic. It's sort of trivial theory, but this is now an interacting theory. Classically, if you solve the wave equation sort of in modes, the modes would all couple together through this nonlinear interaction here. This term is obviously scale invariant because the four lambdas you would get by transforming this under the scale transformation are then canceled by the inverse four you would get from the transformation of phi itself. You may be wondering what this has got to do with Yang-Mills theory, but Yang-Mills theory structurally looks very similar. If I write it out, non-Avelian Yang-Mills in terms of its vector potential, roughly speaking, you'll get a kinetic term and you'll get some interaction terms, one of which looks like this and the other one will scale the same way. So this is true classically. Now we can wonder quantum mechanically, it looks like these actions scale as they should. One of the key points that I didn't, I can't remember if I said yesterday, but let me reemphasize it if I didn't, is that just because an action has a symmetry classically, doesn't mean when you put it into a path integral and think about your quantum theory that that quantum theory will still have the symmetry. It's a very subtle business and it's a very subtle business because in the path integral you have this horrific integral over all field configurations that you have to do and to make sense of it you nearly always have to introduce some way of getting rid of the infinite number of degrees of freedom in the field that you're integrating over. And in particular if that, when you regulate your theory by, for example, saying I'm only going to integrate over configurations of my field that have a say a frequency or momentum less than a certain amount, if that regulator breaks the symmetry, when you remove it, it's far from clear that your theory will still have a symmetry. It's far from clear your theory will even exist in a nice way, but if it does, it's far from clear it will have the symmetry. And in particular this scale symmetry is all about scale and the whole point of introducing these cutoff, the regulator and then renormalizing the theory by removing the regulator is to kill off very, very short distance behavior that is causing the problems with the path integral in the first place. And so it manifestly breaks this nice scale symmetry that our action may have. And it's then a very subtle question when you remove your regulator, whether you recover a theory that's scale invariant. And in fact in this case, for the free theories, it's sort of, it all works and it's fine, but in any interacting situation, it's very subtle and these theories are not scale invariant. Okay, sort of famously. So Yang-Mell's theory in four dimensions is famously not scale invariant. The coupling is dimensionless, that's true, but when you regulate the theory, remove the regulator, renormalize and so on, you secretly pick up some scale dependence in the coupling. And this is what's called the running coupling. So an example of this, a very physical example is that in our theory of sort of hadronic physics, QCD, the theory of quarks and gluons, the coupling in that is indeed dimensionless, but as you look at processes at different scales, the effective coupling changes. It's a very subtle effect. So scale symmetry is not something trivial in, the quantum level is not something trivial in the action, it's far more subtle. Nonetheless, this example of N equals four super Yang-Mills, again, looks like some sort of Yang-Mills theory with some scalars and so on. So it sort of structurally looks a little bit like this, it's classically scale invariant. That is a theory that is quantum mechanically scale invariant, but it's not a triviality that it's by any means that it's scale invariant at the quantum level. Anyway, so let me just say scale invariance is often of an action, is often broken, but not always UV regulation and the subsequent renormalization. However, as I said, there are certainly theories that retain this scale invariance, and then it's an interesting fact that in all cases we know of, unless there have been very recent developments I'm unaware of, which there may have been, but certainly up to recently, in all cases we know of, when you have a Poincaré invariant theory that is also scale invariant, in fact, you inherit an additional, slightly larger symmetry group which is the conformal group. So you land up with a conformal group or a conformal field theory. So to my knowledge, there's no proof of this. Well, there are proofs of it which require some assumptions about the theory. So I think that it's not clear that what the most general proof of this would be, but there aren't sort of relevant examples where you have Poincaré and scale, but not this slightly larger group. And what is the slightly larger group? There's an additional generator of this conformal algebra or conformal group, which is pretty odd and looks like this. It's called the special conformal transformation. I'm not sure why it's special, I should say, but maybe when you see it, you'll see. So again, using the usual Minkowski coordinates, the, it would be generated by a diffeomorphism that looks like this. Okay, so, okay, so where A is some D vector, so A squared is it contracted with itself and so on. So it's a rather strange action on Minkowski space. It is, if you're brave and plug this into the usual Minkowski metric, you can check that it leaves the metric invariant, the form of the metric invariant, and then you can look at its generator which I won't write down. It's not particularly illuminating, but once you have the generator, you can compute the algebra of its generator which we'll call K with the other generators we've written down and I won't write out it in gory detail. I'll just be schematic. This is the usual Minkowski metric and then there's some other terms. There's another term with a different index structure. And so there's some commutation relations with the other generators. Oh, I'm sorry, this should be, this should be mu. But anyway, I mean, I haven't, you know, it doesn't really matter the precise form here because I'm gonna write it more nice, well, more nicely in a nicer way. The algebra is actually, well, the group is SO2D. And it's not very obvious from the way I've written it down as it's been written down, you know, as we've sort of added generators, but if you repackage everything, it takes a much nicer form. So let's introduce indices AB, which are zero, one up to D plus one, where zero to D minus one were our usual spacetime mu nu indices. So we've got two more indices. And then if we write or make the following identifications, and you find that these are with, I should say J, so JAB is anti-symmetric, so the other components are determined. So these are now the generators of this SO2D in a much more familiar form. I'm sorry, I think I mean this. I'll leave it as an exercise for you to check the ordering of the signs here. So this looks like the usual rotation generators or Lorentz generators, but just with different numbers of minuses here. So that's why we have SO2D, we've got two minuses in this. So this is the group that leaves this invariant. So the conformal group, so if we have Poincaré and scale symmetry, we always land up with a theory that has a larger symmetry group, this conformal group, and it's SO2D. That's the important point. Now, obviously when a theory has symmetry is it has consequences, physical consequences. Let's just recap some of the physical consequences of the Poincaré symmetry for vacuum correlators. So we talked about how correlation functions can be computed from this path integral generating functional for them, but these are important. Quantities remember they look like, these are examples of correlation functions. They tell you in some sense about the behavior of your theory. This in some sense tells you what a field is doing at a given point. This object is telling you about how a field propagates from one position in space time to another. If I have more phi's, for example, that would tell me about how particles in this theory scatter, at least for Poincaré invariant theories. And Poincaré invariants already tells us that if we're looking in the, for the field theory in its usual vacuum state and we look at these correlation functions, this must be a constant. It can't depend on space and time and that just trivially comes from translation invariance of the vacuum. And this quantity here, well, whatever it is, it's only a function of the difference in the space time positions. And in fact, it's only a function, I should say, of the norm of the difference in the space time positions because of translation invariance and because of Lorentz. So already, that's quite constraining, but scale invariance now constrains things further or rather, well, so for a CFT and not just a relativistic field theory where these would be true, it constrains things a little further. This has to vanish. And that's basically because if the field has any scale and in vacuum, this was non-zero but constant, this would have some scale and it would break scale invariance explicitly. So the vacuum's not allowed to break scale invariance in a scale invariant theory and so this has to vanish. Note that if the dimension of this was zero, this wouldn't be true, but for reasons of unitarity that we said before, that's not allowed. There's a lower bound, deltas should be positive. Yeah, no, no, no, no, it's under the whole thing. This, yeah, the special transformations as well, but at least as of, I'm not quite sure what the state of the art is, but there aren't, I don't know of any examples of theories which are not, which are invariant on the scale, but not invariant under this extra symmetry. So, but in fact, some of what we're gonna say doesn't depend on the whole conformal group, it actually only just depends on the scale part. Anyway, the form of the two-point function, which is perhaps a little bit more interesting physically, is rather constrained. You can sort of understand intuitively why this roughly the scale of two phi's is, well, so this form here, which you relativistic symmetry basically just told you it was a function of the norm of X minus Y is now entirely constrained to a particular power of that norm. So I should say this is the norm of X minus Y to the two delta where C is some constant, which of course you can set to one by just redefining your field appropriately. And in fact, more generally, if I have, if you have fields, I'm now gonna put some indices on these, let's say A with scale dimension delta A and another field B with scale dimension delta B, the two-point function or rather, let me just say for some, I'm being silly for some A's. Let's assume I've got a few fields and I've labeled them with some index A. It's more sensible. And as a corresponding scale dimension delta A, then if I look at the two-point function of two of these things, it's relatively easy to see that it will vanish if the dimensions are not the same. And it will take a form like this if they are the same. So this is nice. So conformal field theories are nice because the propagation of particles is then completely fixed. Well, I should be more careful. Because these particles don't have a mass, it's very subtle whether one can't really interpret them. Conformal field theories, if they're interacting, always have very long range interactions and it's rather subtle. The normal manipulations you go through to convert a correlation function like this to a statement about propagation of particles don't really work anymore. But nonetheless, these correlators exist and they tell us roughly about how if you disturb the field in one place, information propagates physically through excitation of the whatever it is that however you want to think about what this field is doing. It's also true that the three-point function is constrained up to some constants. So it's functional form. So if I have some fields again like this, A, B and C, I look at the three-point function which morally speaking, although not for a conformal field theory, but morally speaking in a normal relativistic theory would tell us something about scattering. So this form is fixed in terms of X, Y and Z. I won't write it down because it won't be relevant to us. Higher point functions and I'm not constrained. They're in the sense that they have non-trivial dependence that you don't know a priori on the positions of where the fields are living. So they are constrained but they're not constrained enough to know explicitly all of this. Yeah, I suppose maybe it's, I mean, I'm not sure what it is. I don't know if I can give you a deep reason. Maybe think of it like this. If I think of the correlation function of two fields in relativistic theory with different spin and I think of the two-point function. If they've got the same spin, they can have a non-trivial two-point function. If they have different spin, they can't. Why? Well, it's because of the structure of the vacuum. Well, sorry, they, sorry, I should be more careful. They may be able to have a non-trivial correlator but they may not depending on the spins. I should be more careful there. But yeah, so, I mean, I could show you in more detail after how this works but it's not, you have to use properties of the vacuum being invariant under deletions and Poincaré. Let me see. Lastly, every CFT, well, if we, every quantum field theory, we expect to have a, we expect to be able to take our theory, put it on a curved space and in doing so, we can vary the metric and generate a stress tensor in the usual way. It's one way to think about the stress tensor for the theory which tells us then putting it back, putting the metric back to Minkowski that will give us our Minkowski stress tensor. And because of this structure, we see that the stress tensor itself must always have dimension D, okay? Because the action's obviously dimensionless, there's D, X is in the measure, the metric's dimensionless in my conventions here. And so the stress tensor has to have dimension D and that's always true. And the stress tensor can have some non-trivial two-point function which again takes a similar form. I should have said, what I should have said here is these really are scalar fields as I've written things. There are analogous formulae when you have indices say tensor indices, but now this object here will be, again determined by some constant but will also have some tensor structure, obviously. Let me give you an example of that here. The two-point function of the stress tensor always takes this form, but obviously we've got some index structure which has been suppressed there which comes out in some object here which I build from, I won't write it explicitly, it's not terribly illuminating, but I build from these objects which are scale invariant. So the metric and also the combination X mu X nu over X squared which is scale invariant. And so this is a slightly complicated expression in terms of these things but it's not difficult to write down. And this quantity here in a conformal field theory is usually, well it's often called the effective central charge, it doesn't matter in two dimensions, it is a central charge for the conformal algebra but in more than two dimensions it's by analogy it's called this but you can think of it as counting the degrees of freedom. This is roughly telling us how energy propagates from one place to another and roughly speaking the more fields you have around to propagate the more energy can propagate. But in particular let me just, because this may be relevant if we have time later, you can calculate what this is in this theory N equals four super Yang-Mills, this example we had before, and it's equal to some number which in fact is independent of the coupling interestingly although that's not usually the case. So it only depends on N, the rank of the gauge group which indeed counts the number of degrees of freedom in your theory, the number of fields or elementary excitations you could think of. But it's some number, yeah, was there a question? Yeah, well it doesn't matter. What is the dependence? It's something like, I'm not sure I can reproduce it on the fly, it's some combination of these objects. Well no, it's some, I'm not sure, there's a, this has got various symmetries, it's symmetric in these, it's symmetric in these. This is also a conserved tensor, one has to build in conservation, so imagine the most general set of terms you could write down using this sort of object and this sort of object that are compatible with symmetry under these two indices and the whole thing will be conserved if I act with a derivative of mu or nu or alpha or beta. And there's only one thing of that sort and I'll leave you to look it up or compute it yourself, it's not very hard, but it's not illuminating. Was there another question? No, okay. Good, so, and the last thing then I want to say about conformal field theories and then you're all experts or at least you know enough to carry on with what we need for ADS CFT is that in our path integral, remember we wrote down this generating functional for correlation functions, which if I functionally differentiate it will give me these correlation functions and schematically again, there may be many fields, I've just, I'm just calling them all if I'm suppressing indices, Lorentz indices and there may be many, obviously there are many local observables that I could write down. Let me just write one, but you could imagine lots of them and remember we call, so this is some local functional of the fields and in this we call a source for whatever operator it is and then we think of this as a functional of J and really because there are lots of different local operators, really this is a function of lots of different Js, one for each local operator I could write down and then if I functionally differentiate with respect to J appropriately I and then set the source to zero, I generate these sort of correlation functions we were talking about, but the only thing I want to say here is that in a conformal field theory because this will have some dimension, let's call it delta O, the source because the, when I add these sources the new action with sources should still be scale-imbarant otherwise I've explicitly broken scale-imbarant, the source should also have a transformation on the scale so that let's say the scale dimension of the source, the way it transforms will be correlated so it'll go like D minus the dimension of the operator so that this whole thing is scale-imbarant and that will again be important later and then the very last thing I want to say but I won't dwell on this is that I've really talked about a conformal field theory on a flat spacetime, on Minkowski spacetime just as with any, just as with a Poincaré invariant relativistic theory on Minkowski you can then put it on a curved spacetime but this, if you're on a curved spacetime you've manifestly broken Poincaré invariance but the action is still constrained by the Poincaré invariance although there are potentially new terms you can add to your action that will have vanished in Minkowski terms depending on curvatures, in the same sense you can promote a conformal field theory to a curved spacetime and then the theory usually will then have a symmetry under vial transformation or scale transformation of the metric where I change the metric by some but this is a bit like a scale if this was Minkowski this would look like a scale transformation but more generally for any space I can do a local scale transformation but like this, a vial transformation and if I take my CFT action and covariantize it appropriately and I can usually ensure that the theory will have some invariance or at least covariance under transformations like this it won't be important for what we have to say but it's just worth pointing out okay, so now that's enough so many of the things that I've written on this board or these boards we will see coming up later in very different contexts so do bear certain things in mind at least the form of these two-point functions for example, just keep this in your mind okay, that this goes like this we'll definitely see this within a lecture or so in a completely different context and it'll be very relevant, yep oh, I see, sorry, yes thank you good, I mean of course so now let's talk about the second part of ADS CFT the other bit of jargon which is ADS so many of you will know about ADS space time or have seen aspects of it so let me, but many of you won't so it's a peculiar space I'm gonna consider D plus one dimensional ADS so we were talking about D dimensional field theory this will be D plus one dimensional ADS and it embeds as a hyperboloid which I'm gonna take to have some radius let's say L in a sort of two-timed Minkowski space I'm not sure what to call it, R2D as so if I take these coordinates on this two-timed Minkowski space then ADS can be embedded so that the induced metric is the correct ADS metric in the following way as this hyper surface so just note that whilst the ambient space that we're gonna embed the metric in has two times because of the way the embedding works the induced metric will only, well, will be Lorenzian so note constant Xi are time-like closed curves in this embedding so if I sit at constant X I can still move essentially around this circle parameterized by U and V and if I ask what's the proper distance I move or the proper displacement I suppose I actually move a proper time it's a time-like direction so I'm moving, I mean a hyperboloid obviously looks something like this but I'm not sure I can imagine the two-timed directions in a nice way but anyway the point is you're moving around the hyperboloid but both the directions you're moving in a time-like so this is a closed time-like curve and so ADS is not this embedding alone it's the covering space of this so ADS, it can be embedded as that but ADS is the, I'll use the word universal cover but I'm not sophisticated enough as a geometer to really know what that means so if you know what that means that's great I'll be more pedestrian it's the universal cover of the hyperboloid as physicists we can be more explicit I parametrize the hyperboloid like this these thetas parametrize a D minus one sphere a unit and D minus one sphere you can verify that if you plug that into there using basic trigonometry it satisfies the embedding condition and furthermore covers all points on that hyperboloid and if we were just covering the hyperboloid tau would be an angle it would be the angle around this hyperboloid but because we actually want a space that doesn't have closed time-like curves ADS space time, for ADS space time we think of tau being a real, not an angle so as tau goes around I wind round and round my hyperboloid infinitely many times and the induced metric if you now substitute this into here to compute the induced metric so the metric and this is a global chart usually what's called the global chart of ADS looks like the following so it's got a simple form we can also write it in a schwarzschild-like form where this function f looks like this so you see the relation between rho and r is just some simple relation like this and you see that when rho or r is small it's an origin of spherical coordinates but when r becomes large this deviates from flat space this term becoming dominant over the usual one here that is all I would have in flat space reflecting the asymptotics being different now what are the isometries of this space the embedding we've used allows us to see the isometries immediately and they're SO2D they're obviously SO2D the isometries are just the equivalent of the Lorentz group for this two-time metric is SO2D not sure what you call it if there are two times but two, well, anyway SO2D and it also leaves this this is the inner product of two vectors if you like and that's left invariant again by this so it leaves the embedding invariant and therefore these will be this is the group of isometries of ADS it's not manifest, you know, this larger this group isn't manifest obviously in this particular chart which manifests only some rotations SOD-1 and an SO2 if you like or just translation symmetry in tau but from this we can see that there is a larger isometry group hiding now let's try and understand the asymptotics so as rho or R equivalently goes to infinity this metric is dominated by an exponential blow-up of the time and angular pieces like this so as rho becomes very large you can see the space is becoming very big in these directions and rho goes to infinity okay so this becomes obviously arbitrarily large now what what does that mean what sort of asymptotics is this well this is a space which is conformally compact so meaning there's an asymptotic region which can be compactified in the following sense so a conformally oh by the way the word conformal here is not the same conform I mean in geometry this is called a conformally compact space but it's not really related to conformal transformations well directly there is some relation but it's a slightly confusing terminology I mean or rather it's a fine terminology but don't be confused by the competing terminology as I should say so a conformally compact space can be written as in the following way so we have a we have a metric here a regular metric a manifold M with boundary that encloses M and Z is what's called the defining function and so Z is greater than zero inside M and vanishes on the boundary of M linearly let's say vanishes linearly i.e. DZ is not equal to zero on the boundary of M and such a space so G is a perfectly regular metric with a real boundary and the full space time and the full space time then has an asymptotic region due to this defining function one over it blowing up but it blows up in a sort of controlled way so the full space time has what's called a conformal boundary so and it lives let's say on M minus its boundary so the boundary now is some asymptotic regime region so the full space doesn't have a boundary anymore you can only think about its asymptotic behavior we say it's got a conformal boundary although there's no boundary at all it's a boundary only in the sense that you can conformally by multiplying appropriately you can then turn it into a real boundary of some other metric space space time and we say it has a conformal boundary metric which is equal to that induced on the boundary of this regular space by this regular metric okay so this regular space has some real boundary with a real induced metric on it and we say that's the conformal boundary metric of this full space really there's no boundary at some asymptotic region but there's some notion of it having a boundary you know having some metric that describes that that asymptotic region but and it's not going to be relevant for us but obviously it's inherited what I've said I have a freedom in how I can formally compactify a space time that has a conformal boundary if you give one positive smooth function on this space and recover a new defining function with it you know with a different regular metric so this whole construction is only defined up to multiplying this defining function by some positive now strictly positive function and therefore the conformal boundary metric is defined only up to a vial transform so again it won't play much role in what we're going to do but it's just worth noting and now we can see let me see I started at two fifteen right so ten minutes now we can see what this is okay this is a conformal boundary this is a conformally compact space and the conformal boundary metric at least a representative for it i.e. it's only defined up to vial transformation so we talk about conformal classes of the boundary metric as in the equivalent elements within the class being generated by vial transformations so this thing that the conformal boundary metric of this is actually just or a representative of it is just this it's just minus d tau squared plus d omega squared so let me do that more explicitly so for ADS over there if we take e to the rho e to the rho equals one over z then asymptotically just asymptotically we could we see the metric tends to this form and this I can write as one over let's call it big z squared this is our defining function it certainly goes to zero in a nice this is a regular metric I can say take z to be from zero to infinity and then or in fact I don't need sorry I don't even need to so I'm only doing an analysis locally anyway but z has a boundary at zero is what I want to say and extends some to some positive value here this defining function indeed goes to zero at the boundary and is positive away from it so this is a conformal boundary and so the conformal boundary metric let me write it as the the boundary of let's call it this here this space m and write it as the boundary of m even though it's not a boundary it's a conformal boundary let me just write it like that so this is what people would usually call the Einstein static universe but it's basically just time cross a sphere around sphere so that's the boundary of this ADS spacetime and we can then draw a nice picture of it in our picture we've conformally compactified it so we're sort of drawing a picture of it in this conformally compactified sense we should think of it as an infinite cylinder, solid cylinder where boundary of this cylinder so the ADS space is the interior the boundary of this cylinder if you like in this regular space is what will give you the conformal boundary of the full spacetime so this is the conformal boundary it's not a real boundary remember it's an asymptotic region the ADS spacetime lives in the interior and our tau coordinate goes up angular coordinates go around and if you like our radial coordinate as we've drawn it there's a sort of center of spherical symmetry and then our radial coordinate goes out and reaches infinity of this conformal boundary tends to infinity there and one of the key points that's very easy I mean it's a simple calculation one of the key points about ADS is that whilst this is an asymptotic region this conformal boundary is time like the Lorentzian manifold and in particular one can ask supposing I send a null ray in my space out towards the boundary does it reach it and in fact as I've drawn things this isn't really a conformal diagram I mean it's not a Penrose diagram but morally speaking light rays would travel at 45 degrees in this diagram so in particular light rays really do reach the boundary if I emit them from anywhere in the interior with my flashlight they travel out and they meet they hit the boundary at a finite time tau and that's a key physical point about ADS space it's actually not true if you take a space like curve and take it out to the boundary it's an infinite proper distance very easy to see that but a null a null geodesic travels out to the boundary and hits it in a finite time tau the key physical point there is that ADS acts like a box like a real physical box so even though this isn't a real boundary it's a conformal boundary it's not some asymptotic region actually for null rays they really get to the boundary in a finite proper time for observers in the interior so imagine someone sitting arbitrarily far out you know in a finite time in the interior you could send a message arbitrarily far out get your friend to send it back to you and you'll reach, you know, you'll recover it in a finite proper time so what that means physically is if I want to define dynamics in this space I must impose some sort of boundary conditions out here you know all my fields, all my dynamics high energy modes, short distance wave mode or you know high frequency wave modes will propagate out to the boundary and I'll have to deal with the boundary when I deal with dynamics so I can't ignore it and in some sense I will expect stuff to go out and with appropriate boundary conditions it may come back at me okay so it really acts like a box yeah there's a couple of questions, yeah if I've got a time like curve here could be a geodesic oh this is some null ray I just mean it will reach the boundary at some in the conformal boundary at some finite tau and then or if you like it will go what do I mean or sorry how do I answer the question shall I say so I think it is a I actually can't remember if it's a finite affine time along the curve you can do the calculation and tell me but it's not the important point and I think it's not a finite affine time along it's not a finite affine parameter along the curve and for the physicists that's because the affine parameter basically tells you about the blue shift or red shift of a photon if you like which becomes infinite so it isn't a finite affine parameter but nonetheless it will reach in a finite time tau and therefore someone arbitrarily far away can catch this before it escapes and send it back and for me in a finite time tau I will then see that see some information returned sorry was there another question okay oh yeah maybe I I'm not sure I'm not sure what you mean it's not a physical boundary no it's an asymptotic region it's an infinite proper distance spatially if I if I send a spatial curve out I mean you can just you know do the do the calculation for for the metric I gave you if you if you look at a curve that's on a constant time slice and ask how long is it it's got a proper an infinite proper distance length to the to infinity yeah so it's it's really not a not a not a boundary in the usual sense of boundary but the important point is it is a boundary for null rays um let me just finally in the one last one minute write down another uh chart which we will be using really or this is the way we'll think about ADS and it's the Poincaré chart and in terms of this embedding if I did everything right but you can go away and check there's a rather complicated way of parameterizing not all of the hyperboloid but half of it using a different parameterization than we had before so this is only covers half the hyperboloid and what half is it or well or rather um there's a there's a complicated map when you think about what portion of this space you're covering so you only cover half of the hyperboloid but remember ADS is the cover of the hyperboloid so what bit of this solid cylinder are we covering and basically we're taking a null wedges I think this is the way to think about it you take a null wedge that intersects the boundary or slices through this cylinder and then take another null wedge that just intersects the boundary here at the same point and basically the interior of this wedge is what's covered by this and so if we thought about a conformal diagram for what these what we're covering really there's a boundary or a conformal boundary and then we've got some Cauchy horizons here and here associated to the these null surfaces but the induced metric now takes a very simple form and just to be clear here A ran from one up to D minus one and now you know mu is our usual zero one up to D minus one okay so this is the metric of ADS in this Poincaré chart it doesn't cover it's not geodesically complete obviously I can extend it through these Cauchy horizons but note I'll say this again next time this manifests some interesting isometries in particular there's a Poincaré-like isometry which is the usual one associated to these coordinates this is the Minkowski metric so clearly if I do a Poincaré transformation on these X's I generate an isometry of this metric and there's also an interesting one here where I scale my Z coordinate and the X's in the same way okay that's obviously also an isometry it leaves this metric invariant but think about the conformal boundary now Z is a defining function in its own right now what's the conformal boundary it's at Z is zero it's just the Minkowski metric and these isometries therefore have an action on the conformal boundary and what's the action on the conformal boundary it's Poincaré and scale and in fact whilst this doesn't manifest this special conformal transformation that's also there it's just not manifest in these coordinates more complicated isometry in these coordinates is something nasty so as we'll say next you know as I'll just re-emphasize the beginning of next time we see that the isometries of ADS have an action on the boundary that looks like it's the conformal group action okay so sorry I ran over