Conformal Fishnet Theory in Any Dimension

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Conformal Fishnet Theory in Any Dimension
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I will review the properties and recent results for conformal fishnet theory (FCFT) which was proposed by O. Gurdogan and myself as a special double scaling limit of gamma-twisted N=4 SYM theory. FCFT, in its simplest, bi-scalar version, is a UV finite strongly coupled 4-dimensional logarithmic CFT dominated by planar fishnet Feynman graphs (of the shape of regular square lattice). FCFT inherits the planar integrability of N=4 SYM which becomes manifest in this case: the fishnet graphs can be mapped on the SO(2,4) integrable spin chain (A. Zamolodchikov, 1980). The D-dimensional generalization of FCFT, with SO(2,D) conformal symmetry can be also provided. A remarkable property of FCFT is the possibility of spontaneous symmetry breaking, which is not lifted by quantum corrections. I will also discuss the exact computation of certain anomalous dimensions and 4-point correlators, and of related fishnet Feynman graphs (of "wheel" or "spiral" type), using the quantum integrability tools: asymptotic and thermodynamic Bethe ansatz and quantum spectral curve.
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[Music] okay I'm really honored and pleased to to speak on such an event on some 60th birthday conference and I think we know each other for almost 40 years I would estimate and I always found out why someone was such a visible personality in our community in our scientific community I decided one of the reasons is that not only he he knows how to take care of the people but he also knows how to involve them into various adventures say here in this picture which I took in a play at the Chatelet he gives a talk on the thou art event where he was one of the main instigators one of the main persons having organized it and he gives scientific talk and he invited all of us asking to give really scientific talks and only during the talk little by little I understood that we were just some exhibits on this exhibition so it was quite an experience yeah so this is one of adventures little adventures in my life which was interesting okay and as concerns the science we sort of always worked with Samsung in parallel worlds which were very close to each other the same keywords gauge Theory string theory its conformal theories integrability matrix models but sort of rarely discussed unfortunately but it's actually where I remember very well and yeah it just mentioned the probably this problem and someone solved it in a few weeks I guess the correlator in the sections of our integral but the life is not over maybe one day we'll have some collaborations okay so I will speak about the subject which I am working on like four four four and a half last year's this is the conformal fishnet
Theory CFT this is the abbreviation which you know in any dimension so the main question why it was interesting to me was the following that say N equals 4 super yang-mills theory is an example of four-dimensional safety with phenomenological a very interesting properties it has it is very well defined it is strongly interacting safety well-defined on all scales from infrared to uni and also having a rich modular space and this is quite a quite a standard phenomenon for super symmetric world but unfortunately supersymmetry is not observed in nature so far and the interesting question is do we have non super symmetric safeties with these properties say in four dimensions one example which I started studying in detail was a gamma deform tentacles for gamma deformation of Cyprian meal soo-ji is completely broke they destroyed but if you want to keep it conform unfortunately the unitarity is broken so you have to pay by this or that way for for conformality and also what is nice and this theory is integrable even after gamma deformation in tough limit in large and limit so we can calculate the spectrum of animals dimensions by the method which is now called quantum spectral curve I will speak about it a little bit and but integrability is very much related to a DFT duality but both stay very mysterious we don't have any proof even in this sort of emblematic example of Radiesse safety we don't have any proof of either integrability or a DFT correspondence I mean the proof which would be even physical proof not even automatical so I wanted to find some simplified example maybe some limit in this theory which where the integrability becomes manifest statement either correct or wrong and this physical it must not cook just ok it's a correct or wrong and we know the statement is correct ok but we don't understand the reasons for it which is already a deeper question yes really we don't understand deep reasons for the theory to be integrable for any coupling ok then the proposal was to mmm special double scaled double scaling limit in echo N equals 4 young males which combines strong gamma deformations with imagine a parameters and also we weak coupling well and the integral G will become manifest namely the outcome of this limit is this fish net safety which I will which is the main subject of my talk and it appeared that it Damini it dominates by regular fishnet planar graphs sort of this final graph you take fishnet regular square lattice you put propagators say between points neighboring points to the X J minus xk ^ dimension over 2 and you integrate in each vertex so it's like a fragment of big final graph which is called 5/5 graph after the mala Jacobs paper who showed actually that this statistical mechanical system this is an integrable model and the underlying integrability is s or g - comedy conform of pin chain corresponding to conformal symmetry of this graph ok but let's start now
planing how this model how the fishnet safety can be obtained and first we have to gamma twist the yang-mills the Lagrangian of yang-mills is presented here it contains three scalar fields complex scalar fields three complex fermions Green and gluons and with some very familiar interactions here you have commutator zuv module squared of ice here commutator this fermions so there are commentators everywhere and what is the gamma twist you since all these fields are matrices and by and color matrices under trace so the order of the fields is important and you simply assign a particular factor to each order of the field say if you have other a be for two fields and we then if you interchange you get a factor QA b which is exponent of anti symmetric combination of of charges of our symmetry of this field you have so6 symmetry our symmetry so you get here this cart on charges and gamma mu is a parameter are three twists parameters so we have three more couplings here and of course what the practice the practical way to implement it here is simply to substitute commutator by Q commentators by this formula so this Q factor appears in this way and then you simply just write letter Q at which at each commutator and of course this procedure brakes completely our symmetry not to u1 to the cube I mean s over six - you want to the cube but the conformal symmetry stays there and supersymmetry is completely broken of course so I understand correctly so all these definition basically that's a curtain of your symmetry group which gets yes and then the rest of the symmetry group like that does not get quantized just the curtain yes you just point a vector with this with this coordinates in our group so in our symmetric group so it destroys our symmetry up to the rotations around curtain generators and that's what you have but conformal symmetry in principle stays and if you tune well the parameters which you get still integrable don't super symmetric logarithmic safety I'll speak about param and new parameters that will be so-called double trace couplings here I will discuss it but now what is this my double scaling limit fishnet limit I sent coupling the whole cow yeah that young Mills soft coupling to zero then the parameter gamma this parameter gamma to ie Trinity so in fact instead of unitary factor I get some big or small Q and the product of these two factors is fixed I call this new coupling sky J so in this limit interesting things happen for example from the commutator you see already that if Q goes for example to infinity then 1 or Q 2 0 you drop one of the terms of each commutator that's more or less what happens you drop half of all these commutators or sometimes all commentators for example glue ena goes away the couples that the gauge fields go away because there is weak coupling but nothing to enforce the strength of John interaction so you stay with three fermions and three bosons essentially and with some of their interactions I show you now the whole action which as it comes after the after the double scaling limit so you have bosons three bozos three fermions complex and then you have this kind of interactions which now are chiral interactions nonunitary because the whole deformation is not unitary and you have some you color terms of this kind you have a plus 5/4 button with very particular very very very particular orders of fields under the trace so that they become sort of cairo and here I discuss the underlying Fineman graphs the perturbation theory and already it shows a remarkable lattice structure regular lattice structure not completely regular so it consists of three types of lines take colors of fermions or bosons green black and red here which cross each other the lines can move so you can take this line move it through the crossings etc it changes graphs but that's the only way to change anything and also you put dotted some dotted pieces of these lines which correspond to fermions and solve it which corresponds to bosons and also and then everything is fixed because if you have for example the crossing of a sum where the crossing of bosonic and fermionic lengths you have to disentangle it in a very sane this thing in a very particular way by you kava couplings so everything is now unique once you draw this skeleton graph it's a unique way that you decorated by due to chirality it's the unique way say here it's this picture so interestingly this lattice system is another integrable system that we know for sure because the whole Jung Mills N equals four young mules gamma deformities integrable and as Samsung says it's other false all wrong but we have many many proof that it's it's actually false are true and this is true anyway this is a strange lattice system with sort of which has some dynamics you can move these lines but otherwise it should be an integrable spin chain of what kind we don't know but if we stay with only one coupling say suppose we put say once i-220 stay on excited then all pheromones go away because they have such couplings then the these interactions go away you have only this interaction and you have the model which is actually the fish the fishnet model it contains two out of three scalar fields and this interaction by one bar Phi bar Phi 1 Phi 2 now what is the perturbation theory you have two types of propagators for Phi 1 and Phi 2 they are conformal of course massless it's in four dimensions so far there must must less propagators and the only type of the vertex which is which has specific chirality if you draw it by double lines as soft Tata's then you cannot you should just draw it on the plane without turning around and then the arrows which go from Phi to Phi bar Phi 1 to Phi 1 bar Phi 2 Phi bar for example from if you go from blue to red you go clockwise and the vertex which corresponds to anti-clockwise orientation is actually missing here due to this specific limit so it shows already that the model is not unitary but and once you accept the nonunitary world you are in the paradise because the only graphs which are left are precisely this kind of fishnet graphs you simply draw for example a piece of a big graph and you cannot reorient this you have only this kind of structure if you try to turn for example this line of propagators to turn back you immediately encounter this kind of vertex so this is with appropriate boundary conditions you can only draw regular planner lodges and in a big graphs it will be always regular square
brackets so this more complicated picture becomes much simpler one and all this mess of N equals four young Mills diagram and that goes away you have this nice nice graphs there are very few of these graphs in each other sometimes zero sometimes one sometimes two depends on of course on the physical quantity on the boundary of this graph the boundary is defined by physical quantity and it's remarkable that like in yang-mills the coupling size square doesn't run the beta function is zero but for example this and this fact is important because for example you can see that you have no matter in organization this is the graph which run analyzes the mass but of course you have one chiral vertex but another one just by topology should behave should be anti Kairos so you discard this graph and this property propagates of course to higher orders and but it's well known that you generally generate so-called double double trace vertices of this kind for example this graph is not planner but trace trace still survives even under large and leave it like so we have to I could but I don't know what as it spins no here once you have a graph I mean this couplings you speak about double trace couplings or they appear only in very specific graphs which can be cut into two along this vertex which double double trace vertex otherwise they don't serve them they actually rather serve say here as regular risers in some sense I have no time to speak about it but the only thing I wanted to say that a careful choice of these couplings leads to the critical point and the theory stays conformal its drawl so for the following equals for gamma D form dynamic cross form you generate this kind of terms but you can tune to to the critical point with where the theory stays conformal and integrable ultraviolet divergences it it created of course you have ultraviolet divergences but they sort of regularize them and then everything looks nice at the end okay then actually you have to add them of course all the self has depend on X I its lines critical lines not just critical couplings excited dependent couplings and what is interesting this theory even has a space of let modeling how does it happen so as usual we can give an average to each of two fields vacuum average and look at the fluctuations then the effective action for fluctuations the effective faction will contain the original action depending on the savages and then plus 1 loop correction where m is the mass matrix it's just corresponding to one low contribution with mass this mass matrix is inserted L here of course contains both single trace and double trace terms then it happens that actually there are no other Corrections here this is the exact reaction here is exact the one loop common winder potential it's called one vendor potential appears to be exact one loop exact you can see it for example trying to draw other loops of two kinds of fields but you see the circle should cross each other both in Cairo and the Carroll direction if you have a chiral allowed vertex then there will be another one auntie Carol which is forbidden for planar graph so we is assigned correct for common there are two cases kind of write upside down Coleman wine bar and in the right form I couldn't remember proof I hope I hope so yeah I should I should warn that everything is complex already so it doesn't matter you are in the complex world so this questions doesn't make sense but okay but it's a nice the game is still nice so the flat vacuum must obey of course the condition that you are the the extremum again for complex vacuum it could be minimum maximum whatever and I can get one can give examples of such rakia for example for five one field you take zero four five two you can do you take a diagonal matrix such that it's trace of square 0 the matrix is complex so you can of course easily fulfill this condition then you get this factorized mass matrix and for example for the choice when you have this kind of breaking of the symmetry you have for similar eigenvalues which repeat many times and over four times you can you impose unimodularity on these guys you impose that classical common Weinberg zero and you also impose that quantum part this is 0 which is this condition then you find a nice solution so always you can find the obvious solution so somehow this theory inherits the vacuum and one can actually show that the vacuum of original N equals 4 gamma D forms on wheels are continuously connected to this vac you kinetic term yeah but it gave this part for example yes you integrate at our Delta Phi's okay you consider you you do this decomposition you stop integrating you look at Fineman graphs and all of them disappear so after all you you found the full effective action later you can expand around this effective does effective action has a third view v square derivative turn in phi phi is the space time dependent object right no of course you you look for the back which it's like you you look for translation anyone in variant like it was team unify squared in editor also it loops can get a correction it can be divided by square times on function of Phi I agree but that what I say you carefully write the whole action is function of Delta Phi then you start integrating then you observe these graphs and you will see what happens but of course at the a at the end you have to put them translational variant not depending on if I understand your question so I think it's all quite Auto Doc's this kind of consideration what is that vacuum you saw this this system of equations no of course it's approximation this is an algebraic system which has of course it scales you can put one g4 for example 2-1 the rest yeah it's approximation it sub computer but this is a unique nom no these are unique numbers solutions of this system with g4 equals to minus one okay now what kind of physical quantities we consider here various in this by scalar field theory various local operators and their correlators so this is the most general operator containing fields their kanji hermitian conjugate sin various powers the derivatives and then you can of course make permutations on the trace so it's picture a list leaf for example without derivatives it looks like single trace the field Phi 1 under single trace and a couple of things field Phi 2 for example which we call magnets it looks like spin chain picture and fight so Phi 1 creates a vacuum and fight to create there are two magnets to excitations for correct choice of these combinations for
operator for operators diagonalizing the the dilatation operator we get particular dimensions which we want to come compute this is the spectrum of the theory spectrum of one all those dimensions and we want to compute it the simplest possible operator is just trace Phi 1 to the L vacuum in this theory the vacuum is not not protected no supersymmetry so the a no very an organization of this operator is given by the this wheel graph so that at the center is this operator and then according to the fishnet rules since we have particular orientations of propagators this is the only graph at each order of perturbation theory which we can draw so everywhere it's that it's a part from the center everywhere it's a regular square lattice as it should be then if we compute this graph we have some in epsilon expansion we have some divergence e the power of divergence e corresponds to the number of frames here the highest power than by standard rules we can recalculate it to the dimension of this operator which is a combination of these coefficients and in this way we can generate no no it's especially done especially done to avoid logs intercept epsilon expansion okay then there are interesting generalization this is sort of a vacuum it's called wheels diagram but they're also kind of multi spiral or spider web that diagrams as I call them for when you have a couple of so when you have a couple of magnets then they start turning around because it's the only possible kind of vertices so it's Multi multi spiral configuration which is also integrable okay and in principle dimensions the dimensions of this theory can be computed by the this machine integrability machinery which was has been developed in N equals 4 super yang-mills including gamma deformation but you have to go to this special limits double scaling limit and I wanted to take a little bit of time to remind the general integrability the result shortened for integrability of spectrum of animals dimensions and N equals 4 super yang-mills because in my opinion now I need two transparencies to explain this so now it's about the quantum spec spectral curve the method which we developed after long development of integrability in N equals four yang-mills in 80s safety the final method we developed is called quantum spectral curve including gamma twist and here is how it looks like because maybe for mathematicians it should be an interesting object I feel it so it is no five everything's 5 it is for or sorry for of course and equals four young meals four-dimensional sorry sorry sir it is 554 I probably copied it okay so quanto spectral curve is an an object which involves a set of finite number of Baxter so-called Baxter functions of specter parameter u these functions are labeled by specific multi index and here how they look like for jailed jln symmetry for example jail - you have Q empty set Q naught let's call which is usually just one normalization then you have to Baxter functions which I usually the solution of second order Baxter equation and then you have Q 1 2 which is the wronskian of this tool then if you go to jail 3 you have 3 neighbors of G naught of Q naught and then you have say you continue from q3 to and from Q 1 you get q1 3 whichever on which is again over on scan etc etc up to up to the opposite corner of this cube and then for jail for it will be a hyper cube with maximum 4 in the single invitation once NIJ as large a so yeah so that you can find the whole correspondence it goes here everything is prescribed by these labels I would say it's quite usual notation for faster diagram the ordered sets where you have no in any index you cannot have equal equal number equal digits yes but so has the diagrams for Jaylyn groups represent an hypercube now we also have some structure on these q functions if we draw main diagonal from Q 0 to Q a full set then the Q functions are related by palooka relations so if you have say a face here then on this face a lot along this direction the product of these two functions is the wronskian of that two rods can with these shifts of spectral parameter and in this way you can knowing for example single index functions around Q 1 Q 2 Q 3 for example or neighbouring for Q empty set you can restore all other Q functions there are determinant function formulas which store all of them this induces a grassmannian structure on on the set of Q functions and so this is more or less the description the full description of the algebraic structure except yeah we need jl8 more or less where you have 256 you'll see so we start from this jail 8 system where this red axis shows where how these political relations I directed the flow of local relations but if you want to see possibly tries it to make JL 4/4 you simply I mean it's it depends on where you take your boundary conditions for spin chains for example you take Q naught equals 1 and this Q is due to the length of the spin chain and say single index functions are polynomials that's enough to completely fix beat equations but if you want to see parameter symmetries you simply turn you simply start from this to that I mean say for 4/4 you take this vertex you fix it and here you fix it you to the lengths of the system and this is enough to formulate the supersymmetric beta equations so you just turn it beat beat equation factors box the roof I mean everything you you just write this Q full set is equal to u to the hell that's enough importing polynomial stress I mean it's for jail eight but jail 4/4 you do the same in this direction and it's more complicated to relate these to that the equations equations what equations equation occurs blue curve structure I mean this correspondence structure is universal and only the sort of the boundary conditions how to call them boundary conditions initial conditions I different perspective car is some relation that's true so you impose polynomial ax t and you impose that for example this guy is equal to u to Dale and that fixes all better roots etc for you can restore all bit equations ok but this is also the universal picture it also should work for Sigma models for integrable quantum field theories in two dimensions and that way we can apply two ideas a gs5 string ok this we describe the algebraic structure let's go to
analytics structure and it's also quite simple you have specific 16 q functions which have nice analytics track Analytics structure on the physical sheet so I go back to to this has a diagram then first thing we fix as I said G 4/4 means that you fix here Q functions but you fix both of them to 1 in this game so this one thing and then you take single index functions on one side neighbor I mean I blow up here the vicinity of this pole so you have four plus four functions mmm we called P and Q and also on the on this opposite side we have also it's they're called Hodge dual the diameter diametrically opposite functions with upper indices and we know about them nearly everything we know that the large us sympathetics on the physical sheet is defined by cart on charges of of the state you are considering namely P functions which actually describe the dynamics on s fives here the are symmetry dynamics they have this exponential factor which are which is twist so twist explicitly I introduced here and also you have a power like factor which for various B's in depends you all have different combinations of science everything this is defined by group Theory here caused by by charges and also you know the other thing we know that there is only one single cut on the physical sheet which goes from minus two G to G where G is the yang-mills coupling so the only place when young males coupling appears is in the positions of these branch points and the only place where the twist appears is in this asymptotics and then we have also Q functions which describe the ads5 dynamics they have asymptotics related already to cartons of conformal group and they have also the same two branch points but long cut going around infinity and that's it and then we have of course to know something about monogamy around these cuts and the condition is also quite remarkable it's more or less here you relate various cue functions with from here to cue functions there by complex conjugation say cue function in this part is related to q1 on the upper sheet on the upper half-plane related to q2 r which means on the lower half plane and this is these are gluing conditions and there are more or less riemann hilbert conditions which fix everything when i described the whole scheme if you fix charges you will find discrete spectrum of animals dimensions and there have been many it's already quite heavily used this scheme like the quanta spectral curve there are many results obtained very precise numerix for animal dimensions for essentially all interesting couplings for example for Konishi and similar operators Kaneesha operator is like for s equals to l equals to here strong coupling expansion then when you see dimension perturbatively i mean it's up to its known up to 11 loops but only the size of computer the computer strength is needed to go further yeah everywhere did jitter functions here everything everything is expressed frigid of get an element it numbers and only odd arguments because even z the functions are treated like pie that's true that's true but here here there yeah there are only odd functions let's do the same thing happen for XXX correlations mm-hmm yeah we can remember mm-hmm okay and also for example this beautiful picture of analytic continuation of this operator with respect to the spin and in exact numbers but it's in the same kind of not here not here actually here it's really a quantification generate you generate these numbers there is a way to generate but there is also this this dirty methods like fitting these integers to to with did with the whole basis of data function it's it's not matter it's not meant to be seen I should have done it smaller the political recursion not that I know but maybe it does like in matrix models it's
slightly more complicated the object but it might have something to do with it at least in are always says that it should have but I don't know how okay and some is already some results in the fishnet limit I mean this is exact for the N equals four but for fishnet limits for example for this operator we compute directly the graph what is nice since at each other you have one single graph you compute directly Fineman grows very complicated twine on graphs using spectral curve so some really essentially for this operator we obtain second-order baxter equation by scaling double scaling the quantum spectral curve and works.i the coupling this fishnet theory coupling Cantor's here and then there is these gluing conditions boiled down to some quantization conditions between two solutions which are called Q 2 and Q 4 by some historical reasons here and that's and also you have asymptotics defined by charges and yourself you can solve it numerically or generate the perturbation theory inside and it gives exact values of these graphs or rather it's called periods of these girls because the graphs that themselves are diverging so here are some results also a lot of data data numbers and the what was known before quanto spectral curve is the first line this can be completely confused as my hand this one was a quite a complicated calculation by panzer at all but then you see you can have 12 loops for example in no time using quantum spectral curve on computer of course here already ok numerix is quite precise and it shows for example the animals dimension it is real up to some point when it becomes complex complexity it's you see it's real part of dimension as a function of sigh of coupling this is imaginary part so imagine it stays 0 up to some bridge up to some critical coupling where it becomes complex which is normal for nonunitary theory ok so actually at the generalization to any number of spokes of these wheels it's possible but it's not work which already I think is done for 4 spokes but would be nice to to get some idea for any number of spokes and what are magnets magnets spiral graphs you remember I inserted some fight to you you have phi1 phi1 fields which create vacuum and if you insert phi 2 so this picture is already not this real picture anymore but spirals and for spirals for Magnum she also can it's all sin yes yes but the graph completely changes changes to this by multi spiral you can compute this multi spiral graphs ok now the theory can be generalized to any dimension it doesn't have anymore it's it's for damage it's super symmetric matter theory but simply directly right this section you simply laplacian you put he'll apply phi bar laplacian phi 1 but laplacian is some power delta for another field it should be D over 2 minus Delta to make it conformal with the same interaction with the same Feynman graphs and also conformal theory Delta is any actually any any power if you take Delta over 4 it will be same any real not no but better to make it between 0 & G / - yes it's on the equivalence pin chain it will be the spin on the side on each side the value of the conformal spin on each side so you have these kind of propagators the fishnet graph and and actually if you want to to solve this problem you can start from writing such graph in terms of say for periodic boundary conditions in terms of so-called graph building operator so this is a set of product of propagators and here is the picture you have propagator is going in this direction of field safeway phi1 and in this direction of field Phi - this is an operator which projects the coordinates of the dimensional space here to the quorum XS - wise sort of spins any power you can it can be even continuous for one in principle the one day it makes sense also one-dimensional model it was used actually the similar model was used in syk it's nonlinear quantum mechanics I said that it's for many physical quantities you can you can compute something in this theory no but it's not the same but there are many different quantum mechanics if two fields with two matrices it's not so simple let's true but not every two matrix models although this one I don't know maybe maybe you have to have different methods to solve it but it's integrable for sure any DS integral model okay then you can then build the for example the periodic cylindric lattice by taking the square of this operator simply integrating all intermediate intermediate vertices up sorry then you
can take next one it will be the cube of this graph building operator etc and what happens it appeared to be the conserved charge for conformal so2 comedy spin chain so in fact this operator this graph building operator is simply one of the conserved charges of hamiltoe of Heisenberg spin chain with this non compos group and the spins are in principle series representation here an interesting limit is Delta going to 0 and G equals to 2 it actually boils down to the part of model 3G and gluons famous bf Cale so this is one of conserved charges of VF killed awfully part of Lagrangian so things are interrelated from 5 minutes I will there have been many questions I will take slightly more ok ok now now it's next to the last topic we managed to construct for this pitch spin chain for this not conformal spin chain we managed to construct the be tunnels which calculates fishnet and almost dimensions in principle for any operators but let's start from the simplest ones from these wheels so again we take this multi Magnon operators we have em nog nodes and L the length of the spin chain these are the representatives multi spiral if you have magnums and see this will operator of zero buck nodes and you see this picture repeats in each sector if you you take by this propagator stick drawn by bold if you repeat in all sectors you will build the the wheel so it's another graph building operator represented by this object which I I've drawn here I've written here and if you drag analyze it so if you have this operator you simply have to take the power this operator so if you generalize that you calculate the you can calculate the corresponding configuration Fineman graph so let's try to devise this operator and if you have only n is the number of these propagators if you have only one like an equals one it's very easy to generalize it by that by just conformal properties it's a power times some spherical harmonics and here is the eigenvalue which already depends on a specific param arbitrary parameter u which will become very soon the spectral parameter so you just generalize this kind of object to diagonalize it by by wavefunctions the great in here this kind of picture now if we go further for n equals two everything becomes more complicated and it's a real challenge because already without integrability you would never solve this problem probably you generalize I mean suppose it's only one two and there is no this upper part so this is the image of of this graph building operator we managed to construct using of course some interesting papers two dimensional papers of their catch of Menasha at all we managed to generalize it to to actually find the eigen function which consists of various propagator which I don't want to explain every detail here it's just a bunch of propagators of the type module X to the power to the powers which just presented here which consists of deltas etc L is the spin here so we have here already two coordinates and two spins and the eigen value as it should be an integrable system is the product of those eigenvalues and what is interesting when you take when you for this quite complicated function I forgot to say that you integrate over these two points you integrate here so it's a complicated integral if you study the asymptotics for these points on a very big distance you recover sort of scattering theory where the coefficient the scattering phase is OG our matrix of the Milotic of precisely depending on spectral parameters and spins so already it smells like integrability and if you go to any n you can use this to Magnum block to to actually to generalize it to any n see you can generalize to any number of these are like spins rise to any number of these spins and lambda will be a product of eigenvalues and this u 1 u 2 etcetera UN are supposed to be separated variables of Clennon in principle okay so this we can really it will be many many introduced it will be sort of a triangular a triangular lattice filled by all these squares so it's more and more integrations but in the asymptotics it's supposed to be just the factorizable in terms of to the particle me matrix okay and then we
go through standard scheme by inertia some a logic of the trick where we treat the finite system in terms of two thermal dynamic details that maybe i I don't want to go into details at the end we read the TBA for TB and the corresponding Y system so the ennoble dimension gamma is given through a set of Y functions Y functions obey Y system with this specific or to oh gee oh gee comma to diagram Y system diagram and okay the details are not so important but what is nice you can also do the particle whole transformation here and this model appears to be dual to another model Sigma model which has some illogical or D comma to actually zoom illogic of D plus 2 s matrix and it's supposed to describe the ages G + 1 Sigma model but the S matrix is the same as the mala logic of but okay there is my system and behind it but the dispersion relation for one Magnum this personal relation is quite different it has this Brillouin zone structure you have sine square PI over 2 and instead of this standard Einstein Einstein this person you get bus less I mean gapless gapless spectrum and the particle hole corresponds to filling not the filling infinite Fermi sea instead of this one you've filled the outer part parts of this and ok now I'm actually I'm almost finished and we produced in this way the asymptote exponential asymptotics of last large large fishnet graphs which were observed already by some illogical in the melodic of in any dimension ok I don't then speak about for point function this is only one transparency so we can calculate interesting for point function functions and really generate the structure constant and spare the spectrum of conform of dimensions is given by this formula for exchange operators but Dickson type
correlators computed explicitly in two dimensions box addition dixon computing demand for dimensions and finally so i
saying that fusion and safety is a unique opportunity to study non super symmetric quantum confirm world at any coupling and may give a window into origins of ideas safety integrity in addition also the physics of flat-pack you of spontaneous symmetry breaking can be studied and we want to obtain the full quantum spectral curve description of all operators of the theory in principle compute various structure constants which some of them were computed by bus that'll explain separated variables and there was a very interesting set of papers which claimed that they found the ideas dual to ideas dual of fishnet CFG of gamma fancy where would they call it fish chain because they have sort of discrete string which lives on ideas which is supposed to be the ideas duo it's still in discussion whether it's real it is dual then the last thing is there are very nice there is there are very nice fish net amplitudes which have just pieces of square check checkered paper which cut out and they then the external legs appear and you have explicit Jungian symmetry you can construct from locks operator taking its product around the boundary you can have absolutely explicit and we'll define the onion symmetry which we observed some time ago that's it thank you
[Applause] baby 1/2 1 is that if you take this dish nephew and say your computer for my aptitude how many channels do you sum over is it all in order to therefore the SCT channel also have a new channel it's four dimensions so yeah so just because it seems like it's an ordered fury ah yeah it's of course Colorado so it's Colorado all your daggers yeah that's true that's true probably SMT yes otherwise yeah you have to consider finite and ask anew [Laughter] [Applause] [Music]