An integral transform for elliptic four-point conformal blocks in Liouville theory

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An integral transform for elliptic four-point conformal blocks in Liouville theory
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By applying an integral transformation on a version of the Knizhnik-Zamolodchikov equation, we obtain explicit representations of Liouville theory four-point conformal blocks for values of their external dimensions on a lattice where their expressions were not previously known.
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satisfies them 2nd order differential equation in G 1 G because this distance corresponds to there is a degenerate since there is no status for that value of X and O momentum for this verdict OK now so because of that the band's acid this differences this difference elected I must say yes listening to a certain logic of this morning when he was writing something very close to this was very much afraid that he was going to describe exactly what they're going to say so and so fortunately the structure of these thinks is rich enough that there's still some place for myself so yes this I have to write some relatively tedious formula clear it is known that this country can be this fall .period functions can be set aside the differential equation for the following quantity you don't have to really do really follow exactly on this I take out some relatively trivial factors one-liners seek to minus 2 . 4 2 5 3 1 1 as you wanted to hold the beautiful In view of the ones where this function g of the ones to satisfy this that special 2nd order differential equations coming from the fact that I have here in our state which is actually a century the is a question except that it's written for you think is the same idea that we use the fact that there is no state to obtain a differential equation for this and then the soul I should say the London G 2 are their parameters the integration of whatever they United States the cross ratios corresponding to the lakes in this dual diagram representations and what we are after in principle is on this GOP you wanted to do is to take that and it is you want equal to 0 to factor out the extract and obtain ultimately the function of the four-point function is a function of the tool controlmore block as a function of the now wants you to use your old actually then we've we are shifting precisely on 1 of the boats here and we know that we have a had the geometric the question as a function of the One that and then we can try to solve and powers of the 2 and it can solve recursively which we did in those days but this is not very helpful because what we really want is the 1 you could you write a complete function complete function of the 2 so ultimately well let's let it netted right down the differential equation for suggestions so it's the following former you want one-liners you 1 . 1 as you wanted to do these things you can squeeze but then there are some acquisitions in the U 1 he wanted to do you wonder square up to 2 times the you on June 10 here is easy to derivative was quite on the you wanted the Times he once cleared of all this is nearly as this is the 2nd oldest sorry the squares of various yet the easy to point magazine too the message to the US and this was he to persuade the ones g because you know I'm ABC-TV's ABC or functions of the the external debt stands at the external momenta end of the tunnel the internal momentum here please and don't die here this Delta is the delta of the tournament and this is really what's essentially what such awards or already introducing this morning dealt on his chest and Fla commander cellphone is also accused created for minus the square and then and soul so when you actually work this thing out which is not very complicated then what you find is that this this city is tool is a derivative of inspectors to win and there was just the classical in that if you wish when he goes to 0 then this is the behind questions as is seasonal and the accessory parameter of of the Holland equation is related to the internal momentum OK no soul so what can we say about whether Wednesday do goes to 0 ah yes that 1 thing that we want to have to say that the boundary conditions if that Of the ones include is of the form euros 1 presses too GE the 1 that is the 1 that's etc. When the 1 goes to world with all of these Jeezy GI GI regularly this is the 1 that was yeah so once you do is 0 then this is the iPad geometric equations which again was used by recession and Ameritech offered 1984 94 to compute structure functions of conformal operators interviewed fury so but we are we really wanted to proceed to a full complete G different from here now look at now so the idea is that suppose that we have 1 solution of we happen to live up to lay our hands on 1 particular solution of this equation for the values of the conformal charged and their internal momentum here and then what what else can we find other solutions like this stadium at the end it would have been of course to compute a four-point function interesting for string theory but that is not the case at the but the priest of been some explicit formulas where we could have some explicit formula for for a function of the external out 1st and of the internal momentum in which we could find we could check explicitly the conformal bootstrap by integrating over the and because 94 the social and educational computer showed the conformal distracted by numerical integration for P and perhaps it would be amusing to find cases where we could do it and Ameritech so this is that this is actually the case OK soul OK so so then the basic formula for the figure I want to see 1st is
the following that so I say we are really interested in the case where Jesus regular when you 1 goes to 0 that however just like this the question has many properties which are quite close quite similar to those of the hyper geometric equations special you can you can find that forget it let me have maybe right down the formula for the iPad geometric a question to which some people would probably recognize the PCs this is standard had the genetic equation satisfies the following identity so that you see here that there is a relationship between but but with thanks to with this extra power of 1 mind is the between the behavior of the other geometric Zeke was EUR and adds equals 1 here's something similar occurs and it's interesting to see that when you do this change and the coefficients of the opera geometric and functional change a little bit this was and is here this is the same actually you can't choose what 1 9 necessary what noted I'm sorry existence is well but I think this is still true but this is a little bit I should see the different right this way because it because this is a regular none of this is exactly the thing yeah this is correct OK so this you can do the same suppose suppose forget about the boundary condition Angie and then say Well suppose that solution of this and then tried to another solution G 1 whatever some of which would be the disease 1 to some power times and try to find a power here of you want such that G 1 satisfies disintegration but with different different values of the core of the coefficients in DC truck and see what this student means in terms of the external dimensions which are here well what you find is that indeed there is this possibility that there is a power here such that G 1 satisfies the same type of the question and when you do that you find that this amounts to change the In 2 -minus to P 1 2 minus Q and so this is it this is a very simple interpretation in terms of the external momentum P 1 and then the find agenda a G 1 which would be like one-liners the G again to some power and this turns out to change the 2 . 2 9 speak to and then there is another 1 with one-liners you wanted to some power again and this change is peace to minus 3 and there is no corresponding formula for before because actually the energy question only Delta for the players so that before appears on the square OK soul these are so if I am in a some kind in it's four-dimensional space for all these lament that the went into 3 before then I see that I have symmetry some Hall do it by reflection with respect to 1 of the of Gordon a type of brains and I can use this if I have a solution if I know something in 1 case for this differential equation in order to find new solutions and booking so that's 1 part of those want to say which 1 look next well next is actually coming from 1 abatements formula suppose that I have against suppose that I have achieved you wanted to which satisfies the question that the drivers this into this interval transformation that we call this when you do you will resume 1 assumed empower text X to be determined and that because this G-7 integral I on the ones cost then I say well can I find next said strategy satisfies integration of the type Jiang satisfies the stadium type of the question sir I don't think sit here either a contour interval or it's 6 contour which ends at 1 of Boeing's where on the right-hand side there is no integer power or negative integer powers so that I can throw away their integration by parts terms can so I put that do I put this the questions and then actually when you do this in the IPO geometric equation you find that for any action you get a new hybrid-electric question of who is in different questions of course but in the case of the year .period sequence of high integration I think call it the soluble including visits to please the need to pieces then you find that X has to be 1 of 2 values which is amusing and these 2 values of actually very simple and very simple expression in terms of the moment that key expertise one-liners piece creative at 2 when is the times people want to speak to the press .period 3 before and to press miners before because obviously because again I said also 184 which appears if you were in the questioned the coefficients of most of the questions so that's 1 thing so X has satisfy this and G I sort of G 1 of the ones we do then we will depend on I PGA primes PGA J. costs 1 2 3 4 which are transformed from the please just like the geometric occur in the Abidjan Michigan question when I do the same thing than than the new coefficients depend also not only on the original coefficients of the hydrogen Madrid but also also on X soapy friendship and what it amusing is that some of the vector P 1 CAP Prime-1 from to front 3 people 4 this some new tricks which is planned to be have from out 1 got kind of ministry of mine when I went out of control of Anderson went 1 100 members times
he wants to peacefully before what sort of missing a very important thing to do for this 1 1 1 1 this sort of thing on this is just 1 the 1st of a bonus of interpretation immediately but what this is the end of this matrix and is a reflection with respect to the vector to this sector the the hyper plane perpendicular to the sector and is a reflection of the friction with respect to the type of player the that perpendicular to 1 1 1 1 so I start with an initial value the initial values of the these than reflection with respect to and then translation perpendicular the power of the particular to the type printer partners to this fact foreign foreign-funded of this version of the online Milanello is different but like human is due to the minister something like the planets another well being to minor speed that clear this is not really paean to minus who you know see here and that train changing appealed to my piece into minus P into minor speed is would be that it would be a matrix of the about minus 1 . 1 1 1 for example this would be changed B 1 2 minus Q 1 reflection to the 1 of the Code and the type of player his reflection to respect to another thing that I've playing soul the soul OK so low so this is what we find now hope all this you can repeat this provided that well if you don't hit cases where you couldn't do with reasonable and sole choice of integration the boundaries or contour so whatever and you repeat this then you find that you can go on it at this with spacing B of a formal deal be of formal on and so for example if you start from the back of Cherokee was due to be freed before and then apply the following string all of transformation 1st I this interval transformation that I wrote here then reflection with respect to some of the coordinate hyper Prince and so it's successively all 4 off 3 are too so this is changing into minor speed .period which is that the authority for 2 minutes before the 3 to manage P-3 between 2 minutes speed 2 then again I think our 4 costs 3 again see the same set of transformations began on and finally Part 1 then what you find is that this for arbitrary values of their peers this is changing to P 1 presently being proved to be a threat 4 so this means that for from starting from an arbitrary from some set of keys then I can navigate this followed by a modern was duties on fire while on the other hand find value OK because of the exchanges and exchanges of fire was my especially for those in the market OK so so obviously story that means that you can go on at this With here too good of obviously have you want place and 1 of the people to pursuant to the peace through this influence through the years that from the functions and in 1 and 2 and 3 and 4 can be negative or positive Jewish because you can convert this animal OK so that's 1 thing that's 1 of the things that you can do this is not the only 1 I mentioned perhaps some something in banking and more interesting we know the problem is what are we going to apply the set of transformations to what what this what would be the starting point for this B-1 B-2 B-3 before and OK maybe I can continue here no this event with the stopping point but I think it be still stopping 1st stopping point the user what appears in the paper that we roared 4 years ago with a variety of it Association and area shall myself and Enrico and know which users manipulation of the stopping point of view which is integration press that that easy to their knowledge you know the island's equation can be change can be changing too the NYT Dick variables and included Dick functions and I particularly love elliptic functions because more than 40 years ago when I was in Princeton judge force remembers where you assure created a lot of fun playing with elliptic functions which appeared diagrams of the strike so I have a love affair with elliptic functions and I said Well let's try let's try using a little bit but more elliptic functions to see what happens to this equation when we go to variable no Seoul used to the made the following changes and this is really .period remarkable in my opinion what happens when you go to a extraction soul define new case the competing political goal think it's a minute but this year over what do you want 1 assumed the doing completed a ticket to go here Wednesday OK it is 0 2 . 0 with 2 1 minus square see signs career the keys to the Administrator of the Duty and he do the Commission's gives clear this is take modulus Kay said that to that's where the functions common of 0 com the 3 for 4 years and also to make this a strange the change of variables Of course and then the
furthermore defined G the original gives you wanted to call it GE exists struck change or function all of new and old plants of this capital city of you and told and then you find at
this capital to satisfy the relatively simple the relatively simple equation this is we see the fact that there is this the would be going to town and this extra derivative to into to come out rather nicely you find that GE satisfies the following question squared use clear so this is what remains of the day he once cleared the you want more than you wish and then there is the need to be and the 2 is a function of Of this year didn't take total pressed for politely please cleared the too the dollar summits of already Jedi Gedi told so this looks like a time-dependent shorten questions and then there is 6 in his career focus created a function of style think the president the coup constraints in mind bleeds created Q center charge he the company as the company has a interval 10 let me this in a special case and and faultlessly 1 of 4 yen screen and now I should say that this so this is the case and fell 1 because of the concept 3 Pecos minus Q 4 minutes 1 over for being chosen this particular values of the external dimensions to see was the 2nd enough for this I choose it to be unfair for minor security to implement it and this 1 over for the Magnuson fall over to and so the FN again as I said was aware are only functions of the office this is a function of the 2 only and I have here this year and is the elliptic function the ratio of the deferred elliptic functions Indian well spent yes I guess I'm sorry I'm sorry there changed so so this is amusing because here what we find is when when the 0 here when I don't include the teller derivative then this is for integer for integer and therefore for this is a type which Verdi potential a particular 1 Lamaison any question a question in this particular case of attribute showed the potential and generalization seoul amusing by itself and so what's even more of the same using his dad we can also when when there is no tell derivative and solution of this is known for arbitrary values of the of the year all the internal momentum the horror of the supplementary parameter of the of the of the initial on the question so this is all of this is known but what's amazing is that there is a solution so you see this is a very strange because this is like it is this is like a shredding every question in the Senate that potential but find that time-dependent but the time is the year of my is related to To the elliptic potential itself so this is quite strange and it turns out that this has an explicit in this case became this but these particular values of the external dimensions this hazard an explicit solution teamed with anything explicit integral representation give and told which is exponential to the plane the 2 parties times and intercollegiate you wrote few 1 told c . 4 on the New Preston 1 university just 4 and while world of 2 1 until this city relatively strange-looking and in order which is found by specific accusations exponential to pipeline at BU Over beasts the new 1 times "quotation mark issue whether in terms which depends on the Intel which I don't really want to make explicit this is just the the Dow functions the various functions again it's a tragedy but which depend only on top well this the accessory here just as you want during the for this before and
until we have an explicit representation of the solution we don't syphilitic functions and for all values of the government moved amusing remark is that when the Eagles to 0 which is the classical limit and the soul and the P B P 1 soaring the militants please for then the solution to this new kind steepest descent and this and then you find that new new 1 is 6 to be complied by the thousands speed please she declined 1 new 1 overseas . 1 and 1 equals 0 but you here you have a 1 over the square and here you have a one-over did when you do steepest descent find that this is the solution and the solution is exactly the solution of the initial I'm question which was found by Mbeki 250 years ago something OK so we have here is an extension including the stellar derivative and an explicit integral representations so so
what else so in the paper and differ with that but yeah when you shut it enough and Michael of then we have got other cases Americans we could find explicit integral reservations so in other words specifically expressed emotions watcher said yes so sorry this is in the case in 4 equal to 1 Seoul still again so if I if I take this fell 1 minus Q and fell 1 nephew too is equal to my understanding for the general minders and won over to me and the same for the 2 of her 3 of 4 so disillusioned here this equation here was indicates and 1 close to Kosovo 3 because 0 and 4 equal to 1 then we have then you see there is a need for an integral presentations with 1 variable 1 1 integration variable France should have called it you for and what we have explicit into gold was envisions Wendy's and for arbitrary we don't to spend the an arbitrary integer and then we also have you can also have 2 of them equal to 0 and then in fall and 43 per cent and and finally 21 Nicholson to Corson civilian for the plan in Elverta integers and actually you get the but the cases from the first one by using duplication formulas for offered for the for elliptic functions of these are relatively trivial extensions of the of the of the most the scale of this case those using saying also that we have that soldiers or the usual this or that some of the traditional the potentials of where I could find extension with the tall derivative no what's amazing is that we also found more such values of the year-end where these could be solved over the world with an explicit Indigo representation and fuel we see what does this mean -minus head on over to be and what we actually find is that you can also had minor spending and 1 of the like this so this is a large potential if I may say so and this is more potential when the small but we have an integral representation for say going back to and 4 and 4 and and 4 and for and for belonging to an and here similarly could have In 1 course and include 0 and and 4 and 3 quarters for don't include and here all of them also belongs to the so we have explicit integral representations for a relatively large number of values of the and which is of course these cases where n is non-zero disappear in the classical limit and were not immediately interested and then in the classical limit and we have is and then we can go to war these particular cases we can apply the but if transformation that I mentioned in the 1st part of the talk and get even more explicit representations of conformal rocks the 1 goes to 0 at completely once you ghost as you get a conformal blood from UVF fury four-point functions the case we can get even more such explicit threatening legal representations we can't and we can't yet go everywhere because we are restricted on this at this so the conjecture that we had in the paper was that actually we could get more integral presentations involving well away from these particular cases where the movie ends of all the 4 of them or equal can go in principle we hope to get everywhere for all values of any and all values on which is not the case here even lower values and we can already reached more than thanks to the integral additional mentioning and did not finish and get on with their at the finish was 1 amusing remark apparently if that this interval representation that I wrote down it can be used to values of the lands which show
for all that it for example apparently apparently haven't checked all the details but that a reflection formula with respect to that type of plane ride perpendicular to the 1 1 1 1 actions can be used to reach the values of the ends here which all of them half integral and which is where there is not supposed to be known as solution of even the casework tore his doesn't appear accept that there is a strange remarkably paper by Castel del Buddha dating back to 1882 who says that he has found a good solution indeed all of the well not exactly the any question the case where there are or engineer or a half integer and so maybe that's what he have found in those days but hasn't published details and the populist noted the attrition sold individually maybe like commodities like film on the future of the somewhere in a footnote of 1 of the world's or what His states all what I have a sense yes good but it's not clear that his proof was correct however not correct well would have to to discuss that and I think we will have to be so the Newfoundland in and around the nation in the wake of the Government of the nations of the world and that the existence of this equation in terms of yours world Youth this is the 1st of of some of the tools of the handed out the results were announced this will suddenly realize this is the same as the original 1 of the sources of the messages on the chorus now know this isn't always so far was emotional because there is a will there is a change of regions like the only from automobile to sell gold was the president of the New England has the formula for reasonable it was initially had initially I was an ominous here I'm an endowment here here and here I am a there something of the university the Canadian yes yes yes that's what it is in this case such as this is a one-pound function of the authority this is part of the of final formal on this year's political control of the on also the story of questionable use of "quotation mark but then you will hear the sounds of the worst so inspirational Ireland it is for the citizens of this sort of action of some of those things where depends on the vertex operators have a point when you can do it and when you cannot do it probably the would if it doesn't fit with his neighbor would answer he said that was assertion I don't think it was probably too what thought knew it was going to us that this is the likely images ended up this thing you wanted by the so all I find my mind has said is that you can just read 1 of the his wife positive about on negative watch on negative because you can invert the transformation and each well because each of these units each of these operators are reflections with India's four-dimensional space and whichever Nunavut this is of phony with the the fuel was on way the importance of the lessons of life was right of the political movements and you will be proud of the Nile you on the to the human want history as well as the various of this legislation start so early in the season of the this it's not about used along know with them in their own lives and the government and really 1 man and his love of nature and is negative will and up to 1 in here and 2 1 1 and into 1 month and it's just so I understand him that he is that this additional fees and president of the jury know what we we already know this is the we will


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