## Counterterms in gravity and N = 8 Supergravity                  Video in TIB AV-Portal: Counterterms in gravity and N = 8 Supergravity

 Title Counterterms in gravity and N = 8 Supergravity Title of Series Vadim Knizhnik Memorial Conference Number of Parts 12 Author Contributors License CC Attribution 3.0 Unported:You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. Identifiers 10.5446/16412 (DOI) Publisher Release Date 2013 Language English

 Subject Area Mathematics Abstract I will discuss counterterms in gravity using the light-cone frame formulation and show that also in this fully gauge fixed formulation we do need a local symmetry to find the correct counter terms. Supergravity Group action Goodness of fit Dependent and independent variables Multiplication sign Lattice (order) Flow separation Rule of inference
Kinematics Positional notation Term (mathematics) Conjugacy class Multiplication sign Expression Booby trap Line (geometry) Gauge theory Group action Limit (category theory)
Kinematics Formal power series Group action Equations of motion Term (mathematics) Multiplication sign Kinematics Expression Order (biology) Mereology Nichtlineares Gleichungssystem Local ring
Point (geometry) Group action Momentum Euclidean vector Connectivity (graph theory) Direction (geometry) Parameter (computer programming) Mereology Theory Frequency Group representation Sign (mathematics) Term (mathematics) Operator (mathematics) Körper <Algebra> Physical system Area Focus (optics) Generating set of a group Reflection (mathematics) Expression Mereology Connected space Formal power series Radical (chemistry) Bounded variation
Point (geometry) Group representation Frequency Degrees of freedom (physics and chemistry) Multiplication sign Gravitation Water vapor Perturbation theory Theory
Greatest element Multiplication sign Binomial coefficient 3 (number) Maxima and minima Rule of inference Power (physics) Number Derivation (linguistics) Order (biology) Many-sorted logic Term (mathematics) Operator (mathematics) Moving average Modulform Cuboid Körper <Algebra> Linear map Area Generating set of a group Uniqueness quantification Closed set Point (geometry) Exponentiation Expression Commutator Grand Unified Theory Price index Incidence algebra Term (mathematics) Formal power series Factory (trading post) Calculation Order (biology) Gravitation Right angle Gastropod shell Object (grammar) Curve fitting
Derivation (linguistics) Term (mathematics) Well-formed formula INTEGRAL Correspondence (mathematics) Multiplication sign Buffer solution Expression Modulform Cuboid Nichtlineares Gleichungssystem
Derivation (linguistics) Term (mathematics) Expression Order (biology) 1 (number) Gastropod shell Energy level Gauge theory
Point (geometry) Slide rule Frequency Term (mathematics) Modulform Physical law Resultant
Dynamical system Group action 1 (number) Group theory Set (mathematics) Parameter (computer programming) Mereology Explosion Derivation (linguistics) Mathematics Invariant (mathematics) Conjugacy class Many-sorted logic Gravitation Körper <Algebra> Physical system Covering space Geometry Constraint (mathematics) Proper map Closed set Term (mathematics) Derivation (linguistics) Symmetry (physics) Order (biology) Right angle Energy level Bounded variation Point (geometry) Free group Transformation (genetics) Real number Sigma-algebra Autocovariance Modulform Infinity Theory Frequency Degrees of freedom (physics and chemistry) Flow separation Term (mathematics) Modulform Energy level Selectivity (electronic) Nichtlineares Gleichungssystem Associative property Loop (music) Condition number Supergravity Autocovariance Expression Theory Variance Incidence algebra Gauge theory Binary file Quadratic form Similarity (geometry) General relativity Invariant (mathematics) Algebraic closure Conjugacy class Combinatory logic Charge carrier Gravitation Local ring Maß <Mathematik>
Complex (psychology) State of matter Multiplication sign 1 (number) Insertion loss Parameter (computer programming) Mereology Dimensional analysis Derivation (linguistics) Supersymmetry Invariant (mathematics) Many-sorted logic Square number Körper <Algebra> Series (mathematics) Social class Theory of relativity Spinor Maxima and minima Price index Arithmetic mean Symmetry (physics) Duality (mathematics) Order (biology) Linearization Resultant Directed graph Geometry Transformation (genetics) Modulform Canonical ensemble Rule of inference Theory Number Goodness of fit Root Term (mathematics) Well-formed formula Natural number Symmetry (physics) Energy level Modulform Game theory Data transmission Dynamical system INTEGRAL Virasoro-Algebra Forcing (mathematics) Expression Mathematical analysis Division (mathematics) Transformation (genetics) Quadratic form Supersymmetry Formal power series Invariant (mathematics) Gravitation Musical ensemble Family Maß <Mathematik>
started that and then there was a essentially just a long considered that the like engage bridges that a plus while equipped would engage in isn't a 0 and I wanted do use notation run right it's plus-minus and I will use but explodes as the Times and then their minus is the conduit so that would mean also that no plus it's just a space derivative but OK and also that I will I will also use limitation that is so if I introduced this here and then the information into I can add and subtract and then what we find is that but I will get to that the bill plus or a minus a would be something like that and the Weishaar I cannot what understood there were no switch on the next line so you will see 2 I
will have this expression that is the I and then there is a term which is G if paid seeing 1 of their players a booby prize and they'll plus a city
please well I said we mention also voted to continue present a and cable and they here 1 Rydell plus which is so cause in all localities in it's another gathered along the like good but it's a it's nothing really dangerous about that so I will have lost expressions with people like that I if I just do this on introduce it back into the into the action I get an expression
like the 1 upstairs and you recognize In order kinetic term but you know we have to interpret as a said that passes the time this is a three-point capping an accomplice going a four-point company like that but so but you know this is the real year a formalism which is working on shall so what I will do it said was really look at equations of motion if you look at this equation emotions so miners are their mind this is really like P minus while on bad the miners on this but it was
this thing this is really the the momentum operator acting on that which means that this is 1 of the generators of the .period Algebra I. We will interpret this this is really as our right here's the variation with respect to this generator the season and if I if I write that a wonder where investors reflection on him using an expression like that said so this this generator I Is there a nonunion realized on the field this is a kinetic drum up there I'm up with the view .period cupping gives this chairman and I write code it's going to get there it's not the complex going at that but if you go back 1 step you see that here a term which has a a a and here's another 1 which has a BA AA when I take the the question motion respective they have vary with respect to a bar so this term is as we know it which means that I'm not going to bother about this term I I know I know what it is but 1 sign of this terminal that do not directly the goal the other Lagrangian so that I can go back you can find all the point Kerry generators acting on the field and you get to a completely the group theoretic approach and every generator that minus components will be dynamical and hence nonlinear this anomaly in the realization of the blank area all year for example the at this this is suspended the operator in the system minus 1 this is just tell you that it has been 1 the so that's the only the only thing you see from this being part when you're when you act here in this way and so what we did at this 30 years ago was that we summer abstracted out from this formalism on how to write so that any higher spin you just find this representations you find this generator and the focus will also have to look at the other generators the so I was right there is the use of the dynamic Alonso what I call
this and there's still stabbed complex applied also this is into the transverse direction but this administration this is the it's you will have they'll minus minus it's minus dealt last public all of us would express desire we do the argument expressing a desire to them that this history because it won't have to worry about this guy disconnect and this With the about the minus so the fine and if he is serious you massless future stock with you only have to find that these generators and we make sure that they stance correctly with the the deal Kennedy generators 1 In this where you can you know it's just celebrated area could you construct any of the known a few years and I don't think I
gave a talk when it .period Kerry was 150 years old I give a talk old you know a few serious representations of what the Piketty algebra so so this just representations of his algebra said looks in completely group theoretic and so this is a different way of looking at the theory and this is what I sort all the time that this was so completely gage fixed version and you can imagine that on on another planet in somewhere in the American way where he majored you have some creatures stated that this is might have been the way that the constructed the theory that the might of the water by Cajun variants in the just because what you're dealing with the their adjust the 5th this again degrees of freedom To Avendano Babatunde
Yusuf freedom and there was this was the starting point for when we did it was foreign and by doing and it was for like that to prove that it was fine acting in perturbation theory this is what I discussed in in 84 in In Moscow town and but then you can ask what happens when you go to gravity what what does gravity look like in this way understanding again what you would do if it were it would just have to find just start with finding this three-point .period coupling and in fact I was sure
at EU update this summary simple expression the three-point coupling this is really binomial coefficients innocence here and that's why it's very easy to to generalize this deterrent to higher spins because you know the number of derivatives it just tells you what he and essentially so this is the capping then of course it's a four-point capping which is much more complicated on but it can get a lot of information from the three-point capping and so this is what I wanted to look at how we now make things a bit more complicated I will introduce some more much more general formalism where write this expression essential like that I introduced operator is an exponentially this there Bart would help us With factory so what I'm doing here is that I just expand this to power a square peg the derivatives and the I also more general expressions therefore that there pluses but don't look too much into that that's what you find is that this is a very compact way of doing things because you can do it right and it's been at the same time and it's also a very good way of doing it because all these genetic generators which we have which gives you know tells him what to listen to whatever they're automatically satisfies you only have to find what you write written like you have to write these 2 generators in a separate form and then you you you just check the algebra that it satisfies the blanket added OK what there are no indication on homage so so so so for this incident in sense in index 3 formulations and objects at this and when you do it like this it looks very unique you can so you can in principle construct ordinary gravity this way it's hard to to to get if you would like if you had to guess at 4 . cupping it it's hard and it's a lot academicians to do to check that the worst but this is a way of constructing Dukes unique but the question can ask if this is where the massive capping constant here we can also allow for terms with more derivatives so what happens if you look at the corresponding terms with 4 derivatives and then Of course you you can see immediately that what you have to do with it you have to Aberdeen 1 Del ball R and Mondale in order to have the right to this so the former the expression is found where natural knowing what I do is that this was the 6th 1 subtlety and I both at an exponent here coefficient with for the bottom of Fidel songwriters expression AC means that there are 3 the bars and 1 day so this is the expression start and then this redundant minute as we know it thinks that this is the expression for the dynamic apart so to speak in these 2 generated by the look of
this was there to go efficiency and which are in principle and all but this is to form they have to be in order to satisfy all the other generators is so that the only generators we have to check it how these 3 commute with each other so we do that then then without got a big a bit of a surprise to us because we got solutions for all values of eminent so we got infinitely many terms possible here for this so this is sort of a gains to what I was saying before that it was this was a fully fixed gage the area the fault fully fix engaged in and that's you that the underbrush should give you everything summary doesn't because of my wife we had sought general namely that we would get on 2 terms here because in Brisbane Ewing gravity that should be to Contagem said 1 of them you know this is it this famous calculation of tools development if you essentially have an hour squared and you haven't on new new army you knew you don't have your mural script because of the ghost when it ships of Asia that get to terms so we were very mystified by this and this check this calculation under times in over whether we have done in mistake with couldn't find facts another virtue of the form whether this generalized to to then a higher order can determine but you know where the only thing we have to do is that we have to add more derivatives but it's the same calculation because you know we do the calculations regardless of what is and what it says here that this week to begin you all to terms all higher-order candidates In the same calculations and "quotation mark will undoubtedly have to to variable is imminent according to these rules of this is the spin of what border I should say so if you take a 137 but you just have to put it was a 137 and this is the potential rehab I but the and you will have you know a fantastic number of derivatives but it's still doubted working so that was a problem so you have to go back to the drawing board again to see what are we missing I mean I've been telling you that you you could defeated by just 2 Inge the area and something is missing here let's go back you well I have to tell you 1 more thing was you also know that the these terms are 0 on shelves so how is that implemented here well as implemented in while it this close a bit difficult to remember that about the questions emotion fought for this field is this In my way of writing so it considered it if I look at it this box acting on an expression like that let me do it already has essentially what I'm doing is that a box on something with an age and age so this is a clear plus their minus minus there the bar acting
on but and I'm going to use the questions emotion because August will use equations I get higher-order terms but I'm not worrying about those themselves I want to have to do that is through here but I want well every time the disconnect acts here I would get correspondent on this so that will not sevilla tablets expressions their went to worry about is this I don't like that buffer that we use the questions emotions to resented that is still the born wouldn't plus and then the terms with Dale page India born like that and if you put all these terms together you see that it's again so I do it in a more general with it Is it again of all the same form
as I have I just have to change the bill pluses and which derivative exodus speed set up with this backing I would say that if I have this is still the starting In the integrations motion here that I have a typical counter germ that if it's of this form I can just use this formula to rewrite this guy as well with the box up there and then I make the redefinition this and then we absorbed To Barfield a definition we can absorb all these and counter terms that the show which is of course will we should the I mean among those
that should be the correct ones and there are 0 on ships everything is the on shell
and so I but it might tower North would ever be able to get the counter during which is that is non-zero on shell because you know at the at the next level at 2 2 distributor
which goes like essentially killed the speaker well if you go back look at this
expression here he is it in order for there is a term like that to be it specific if you'd think it would be able to write like that you have to have both of the derivatives I suppose I don't have at this derivatives said that I cannot do it because I cannot heritage Q minus 1 so the only way to have a counterterrorism if it's only written in terms of 1 type of derivative I I'm sorry where where OK so I
don't know all I knew it I missed slide toward 2 unaware got to the point is that I can have a term attempted term which but that the if on the following form I term which goes essentially like 6 of those but like but I like that that we will have a city too the letters there it is only 1 term all 1 type this is such a I write down I can check it will do that then in fact you know we will find a unique is result so we
can find the true the 2 .period Covington were confined and that's unique and it's 0 is non-zero on but is go back to this problem that we had the
sense 4 so the shaking and this is what I certainly With chose gage and we eliminated this feeds through this equation so the question is then how much I would fix the while so this is the the at the start with Of course in order to still be in that gage we have to demand that this should satisfy this condition well Saturday the man you know that any variation should have been there done that this should be married 1 that means that we have to do To solve others but this means you know that there is incidence sort of remaining engaging variance in the theater even the way it's fully fixed and youngest whenever systems so the lesson is that it there must be some some kind of of invariants in the system which found blanketed in a number of and how did Warhol would do that what before I said that there is 1 thing so if we still have some kind of gage invariance here what should be able to write it in a gage invariant way so if you go back to this expression that had appeared 1 can shake you know this invariants but can we write in equal variant form when a fact what we we can because what we can do is that we can introduces so the local variant derivative which looked like this and it has a term like that man sense this the transformation as we do satisfy this condition it doesn't really matter if it's a Delta plus their 1 of Adele so so this is certain gage covariant under this remaining engaging variants on writing like that refined expression for the who is the miners which was above the hematoma as a quadratic form like that so this it's been this way that this is a whole expression for young years in this way during which is I think very nice because we're not being able to to really use it will been trying went on is still you could always unit imagine that you have perhaps I can now make it got complicated Phaedra definition and debris it would really look like if he a free theory innocence and then you would put every all the dynamics into the measuring the function into doesn't held that but is it is it is good to know that for for amuse you have this expression and that if a bit shaky the gravity case it can also be written in this form so there must be some kind of in various left there this and then what is that so I think that that was what 1 can do it so I can start on With with variant form and tried to to really chick look at all the questions look at all the the fees set you that we have eliminated and try and To find out that you possess a very complicated way but looking like a short you because you can guess that's part of the remaining gage what it would look like what is rather natural that it because of elicited this is essentially the only former can have cited as a parameter .period side which has elicited 1 and then there then this is the form you can ride and I think use this conditionality was introduced that so this CSI success explored so this is after changes this is sort of 1 what I want to say that this is a remaining engaged in various this is not the whole but it will only need to know enough in altitude to be able to do so to select the right terms but because that's a problem with this synergy which and it up it does not close so this infinitesimal transformations ever can do the which the the theory should be invariant under effect if we try know to look at the the closure order we will get back to us into some Mass but In order to just check it is enough to know this and that I should say this is a this is that we don't know exactly what this means for businesses this is not a closing symmetry at what we cannot really is said to have sort of as a but what happens said that it will really selective right that by chance for us so I would check you know all this 1 counted as we have just sat there is just 1 left and I but so this are expressions and it's only 1 combination of N M named this 1 the survives so now we're down to just 1 term member were was supposed to have 2 and so that's could be a bit of a headache but that is a simple solution to that I said it issued the 2 however you know this is just a at the 3 point level is going to the 4 four-point level you you would see to it that the three-point and those 2 expressions would give thing so am this but it means that we have at it it works
that that's the way to do it and the as an hour here I have this expression for the for the 2 of you should this form currently an SSN there is only 1 Thai border we can have some of which were called B 6 which means that is exactly of this form serious using you know that's an age women age bondage and and this point that by algebra gives you this unique solution and if reject this good it works so sits down this is our house and we were of understood was a problem and and the it's sort of like the new lesson is that it's in it's not enough just to close .period upon carried in a report didn't know about it counted as a but said there is not enough to just use group theory in order to select the set a general theory for spin to Ohio so so what on and this other remarried in terms was what this gage this remaining gage invariance what does he do it doesn't take away you non-physical degrees of freedom but what it does is that it's selecting selecting this being too from higher speeds so bad so that's what it is that it has different role here it is assuring you that that there we get just spin too because the other ones some kind of invaders from high is that the terms that we wrote good returns coming from from higher spins and then we have to somehow get rid of them and that's what we do we have 2 so that since you might say that it sense you can go back to 2 . carrier the bribery would have sold his because the ideas of was that the way we do it is it you cannot really cannot checked in immediately that you have speed too fees in those terms so this is the sort of the criteria for keeping that but it looks like you use is remain gauging their eyes and you have to some kind of local invariants In order to do it in order to select that so that's comes on top of all that the the point agenda so when people are not interested in speeds we we were really trying to get rid of the highest in so this is a way of getting rid of the highest things but this is but you know this remaining amassed looks like you Association but it does not close however not been able to do it to turn the I find a two-dimensional local algebra that works it's not that it should exist it would be nice to know fears a recession at some stage of that them might be something there is something in the Austrian in discredited theory that we don't see by the ice which is there you know and that with we get it from From that this locally invariants but but it's haven't really presented it in the best possible form but as I said I've been optimistic for sometimes have not been able to think about it but he is certain the messaging in the in the gravity cases that that that there is something doing it in this form which is very much strip down to the bare bones should only use the L real degrees of freedom it you have to do something else this was that this was just a simple exercise exercised in started just because we wanted it to see what is the corresponding expressions in any crusade super gravity some and then I'm not going Bender says much about that In any good sensible gravitated this described by field which no innocence is just an X and 8 status and if expanded to have all these supermarket within system beautiful way American or you just have to to right the whole expression in terms of his 5 this 5 sets sort of a character the constraints and what we're calling inside of CUC that if that is so symmetric take the got covers conjugate and you turn it around and you get it back to the age quality age it's Simon amp Simon so it's it's sensor rather than the strict constraints and is can the constraint for example shows you immediately that I will not be able but to take his candidature over 2 Super gravity because that then it would mean that on this side will have a Delta flight and he will have 5 but this is satisfying a carotid condition that this has adopted Carolyn it's not going away so so overseas people know from the beginning of Super guarantee that there new to counter Johnson supra gravity Dole also works for clues but he it's obvious ```  188 ms - page object