 Okay, thank you very much. Yeah, it's the end of long day and my talk will be really easy, at least much much simpler than previous physics talks and I guess it will be really baby level. And my interest to the subject arise first when several years ago Joachim Andlind came and gave some talk and then Jens Hoppe gave another talk and we kind of tried to understand what is going on so it's just I'll tell you. So just it will be really for everybody, no appropriate knowledge of anything about quantization in kind of a geometric sense. Suppose we have a compact C infinity symplectic manifold or dimension will be 2D, later it will be just 2, but at the moment it's 2D and then quantization should associated a Hilbert space when maybe you have some integrality condition or you can rescale form you get scale, depending on constant Hilbert space. The dimension of the space is 1 over h bar 2 power d and integral over m omega d md factorial plus 1 plus small o1 when h bar goes to zero. So it will be roughly volume of symplectic manifold. The construction is not canonical but still and quantization means that for each function on manifold you associate a self-adjoint operator f hat from h bar to itself which is self-adjoint it's respond to that this function was real and the main thing is that if you get two functions f1 and f2 and consider product of these two functions then it will be quantization of the product plus small correction i h bar for 2, 1 hits plus on bracket plus of h bar square. Yeah, so you should get such thing but now I'll do something a little bit not standard for any function from complex valued function whose absolute value is equal to 1 so it's kind of map to u1. You should associate the unitary operator and the rules are similar and here I took h bar and consider plus on bracket of logarithms. Logarithms are not uniquely defined but locally defined up to constant plus plus on brackets as well defined so it makes sense again plus o of h bar square so you get such thing and yeah so what are kind of basic rules so h bar of plus on bracket it's more or less the same as commutator of uh operators and also error to formal for dimension one can write that integral 1 over 2 by h bar to power d integral over m omega d over d factorials some function f is again very close to trace of f hat yeah so it's a basic rules which we have to but you assume that h is five dimension yeah it's written it's it's a volume yes but I want to be sure yeah I know I start with it it's it's fine it's the dimension is all the proportion to volume and yeah so that kind of what the basic example uh want to keep in mind if manifold is u1 cross u1 two-dimensional torus uh and you have coordinates f1 exponent i theta 1 phi 2 exponent of d theta 2 and plus on bracket is and form is d theta 1 is d theta 2 divided by 2 pi uh then uh phi 1 hat yeah when I put h bar equal to 1 over n if the n is an integer then n will be dimension of Hilbert space all forms became exact and phi 1 is operator shift and by an operator and phi 2 is operator 1 exponent 2 by i o an okay yeah so this is the most basic example okay so now we have uh approximation by Hilbert spaces to of complex of a simple electric manifold uh now I will go to this thing which already mentioned several times of this conference it's shield uh functional it's something uh for the following suppose we have a map from simple electric manifold to uh to uh remaining manifold uh and there is essentially only one function which you can write which is invariant on this thing it's some sort of quartic derivatives it's quartic derivatives and uh let's assume that x is just rn rn uh then we write the action map it's kind of like phi say and then the action of say I'll write the following way it will be the volume of my manifold multiplied by integral of manifold of omega okay to power du factorial multiplied by sum over i and i less than j x i x j squared the map is given by just collection the same as collection of n function on my manifold time and x is the same as psi x is sorry is the formula for the action yes psi might be the same yeah psi okay yeah maybe psi psi yeah actually I want to know that really by axis yeah by x maybe x like like this yeah but yeah so that's that's the action in where I wrote this volume m if d is equal to 1 so a dimension of m is 2 this thing is invariant under the rescaling of omega invariant under omega goes to constant times omega uh yeah that's it's convenient sir we square yeah because it appears it's the first time power it's not the first power but here it appears twice a negative degree okay yeah so it's invariant under square and what is the quantum version we just look on the formula what is the bracket was it what's the integral and now I assume is the time dimension 2 and dm is equal to 2 then it's quantum version will be the following it's you take you replace volume by this formula this integral by this formula and eventually you get the following things you get minus 2 pi squared trade n which can be written as trace of 1 times trace of some r like j and x it will be capital x x j squared will be will be quantizations of x I had yeah yeah so I get this so this and this is action which we'll study yeah now I can write a kind of equation of motion so critical points uh critical points are if you go in classical situation there are of two types either suppose I am now speaking about surfaces only either you map surface to curve arbitrary way and embed this curve to our n and you get zero think you think it's always positive and get zero things you get kind of degenerate critical points maybe you have manifold maps to curve and curve embedded to to our n in arbitrary way yeah so it's pretty stupid huge class of critical points and other things are minimal surfaces if one write equation and the symplectic form will be proportional to pullback of volume element but because action is it is scaling scale invariance so it's it does meta and actual minimal surfaces and here's a critical value yeah here for first example the critical value zero functional source in the negative and here it will be area squared it's strictly positive um value of the action on the critical points yeah and this quantum version it's also always non-negative and it also has kind of the stupid critical points in all x i simultaneously diagonalizable it's analog of this flabby curves but the rest it's analog of minimal surfaces yeah and here there is a little uh trouble uh when you can see uh we cannot have compact minimal surface in our n it either should have boundary i don't know how to formulate theory what does mean the minimal surface is boundary in its quantum analog uh so there are several ways out as you consider non-compact minimal surfaces and this was example in uh Joachim lecture of a quantum cationoid and I will also show some even simple example or I can go tori as spheres you can yeah I can go on spheres for example you put or maybe ellipsoids you put additional constraints sum over x i square over r i square is equal to identity operator yeah you can put this constraint and study like these things like this but there's a trouble that if you can see the ellipsoids I don't know where's one can really analytically understand what are minimal surfaces in in ellipsoid I don't know whether it's kind of soluble or notates some elliptic question but can you kind of write formulas I don't know but this is a simpler case uh map to tori and yeah so my talk will be essentially kind of various version of water kind of will be maps to tori so this is what you want with the fire yeah why why I mentioned this uh um yeah if you map to torus suppose you map to some torus r n to the n yeah you map your surface classically to to the torus uh in quantum case we will describe as collection of uh n n unit operators not commuting yeah uh and now we want to write something like uh kumit product of trace of uh squares of logarithms logarithms are not really defined for unit operators because the eigenvalues will be go all around as we see from this example there's no natural cut uh thought will be how to replace uh trace of log ui log uj squared so this guys will not commute but the commutators will be very small you see it's an example the commutator has norm of something of one over n over h bar and the naive idea is to replace by the following uh think the action will be again n is this normalization constant meter to pi times trace uh somewhat i less than j and instead of making this guy you write two times identity matrix minus and you do the following you can see the ui uj u inverse uj inverse and minus inverse guy minus uj ui uj inverse ui inverse this will be self-adjoint operator and it will be very close to one over n and the trace of this guy multiply by n it will be it's positive it's like cosine yeah it's it's because when these things uh almost commute to get some think and we get some equation for motion so uh heroin can make the following conjecture this critical values when h bar goes to zero approximate critical values of uh of actual uh uh minimal surfaces centauri and this something which we can understand from algebraic geometry because minimal surfaces centauri they locally have various trace representation and uh and the description is the following it's uh what is the minimal surface on the torus uh this is uh it's the same as uh remaining complex curve yeah suppose this m is c is embedded to to some torus around some lattice suppose real torus real torus yeah uh uh and uh this can be described suppose just product of standard circles one can go to arbitrary lattice but for simplicity if you get a complex curve you get n uh uh holomorphic one forms you can have billion differentials with the property with two equations it's sum over ai squared is equal to zero a section of square of quadratic a quadratic differential you get something that has three g minus three equations if g is a generous curve and then we got also conditions that integrals over real part of alpha j belongs to integer yeah and this things has should have essentially unique solution because you get something like three g minus three this is unique choice by uh reman relations if you choose homology classes you get three g minus three conditions in the equation that gets also three g minus three conditions for a curve more or less space of curves and you get so you should get kind of a nice finite set of this minimal surfaces and they should have appear somehow the limits of critical points of this guy it's differentials differentials of real part will be angles uh theta i will be d of theta i three okay yeah so that's uh uh yeah that's kind of nice but it's you see that the equation is not really exact and here it's not uh approximate cosine by square and so it wasn't really terribly happy with it and about it and i wanted something more exact what is the meaning of shield proper name it's a name yeah yeah yeah proper name yeah of course yeah it's just yeah yeah uh just let's return to original this uh thing yeah so we get some over some constant times some trace of some of x i x j squared yeah and and consider equation for critical points uh just for this uh quantum shield action you get of course equation is double uh bracket equation for any i self-adjoint operators yeah you get this think and in compact case it's it shouldn't exist it's infinite dimensional case but in uh that's and this is young mill salge bra as you know and uh uh i it's from one to n and the dimension of the target and uh the meaning gives it is that if you consider uh uh trivial bundle on r n and consider connection constant connection d plus square root of minus one sum over x i d i know x i you can see okay this uh this uh get constant connection this is usually young mills equation for the constant connection because it's for constant derivative zero it's it's a young mill section okay but uh we know that uh supersymmetry teaches us that it's kind of second order equation for for connection and this first order equation Hermitian and mills equation it follows from Hermitian young mills uh Hermitian young mills uh for the uh it's in the following station suppose my dimension n is even and my target space is is complex vector space uh now uh Hermitian young mills equation uh in the usual geometry implies usually young mills equation here's this similar story uh now we can write complex coordinates z k is equal to x k plus square root of minus one x m plus k we replace uh two m real uh self adjoint by m just perishes without any condition and and i write uh kind of Hermitian young mills algebra as the following way so that it's free algebra generated by it'll be star algebra yeah this is by the way it's also star algebra because i said i can put this condition it's the algebra generated by m variables and variables and model as a following relations and the relations are the following oh maybe i'll just write it here model relations uh first z i commute and also z i hat commute and you have just one equation sum over commutator of z i hat z i for i is equal to zero you get this algebra and then uh you treat uh uh really imagine apart it will imply the relation for young mills equation like Hermitian and mills implies usually you really want zero into z i taker z i or constant just a second yeah for a moment zero yeah uh now so these equations imply immediately sum over j z j z j cross z i plus sum over j z j cross z j z i is equal to zero for any i and this is the same as young mills equation in kind of complex bases for real imaginary part yeah so it's uh and the meaning of Hermitian and mills equation is the following you just consider uh the same the same story consider connections and kind of meaning if you consider a trivial bundle on cm and write the same connection it will be conditioned that z i commute implies that it's holomorphic uh that means that f zero two is equal to to zero and uh this condition sum over z i bar hat z i implies that uh f f one one multiplied by omega macular form to power m minus one is equal and is equal to zero and this is exactly uh young mills equation for kind of donelson or lubeck yaw theorem one can kind of make little generalizations first one this thinks it's not necessary zero it could be constant and real constant and also uh the i not necessary commute but again uh you get some complex constants you can make kind of quantum space instead of usual space but again this will be kind of central extension it will be huge algebra it's a little bit like free algebra in some direction it's certainly certainly algebra and it maps to young mills algebra and there is kind of very pretty neat case when m equal to that means that n is equal to four and let's c equals zero c i j is equal to zero uh then we get three equations in terms of real uh real imaginary part x four x three x one x four x three x one x two yeah so it's pretty nice lee algebra these four generators and for its three relations that called bankers cyber shanker equations and uh again because it's can't have finite dimension representation because there's no compact minimal surfaces but one can make a kind of unitary analog of it in the unitary version is is the following one we consider and these things already should have many solutions you get for unitary matrices and write equation for your place comma turned by or maybe it's like u one matrices u one u four before u one before inverse u one inverse is equal to u two u three to two inverse u three inverse etc so you get some nice group which is invariant and the action of a four which is even even permutation and it looks like by one of some certain three-dimensional manifold because you get four generations and three relations one less it's yeah and such things typically like for dimensional group it has should have isolated representation so it's i think it's kind of really nice group which cup will capture minimal surfaces in uh in the product uh when large limited when the product in this you can see the Gaussian integer elliptic curve squared no i try it i no it looks it's it's has one credibility no it should be closed manifold it should be closed manifold that i think it's have one one credibility and yeah so it's some kind of pretty need group yeah so what else i can say yeah still i'm not totally happy about this groups because i write equation but i don't know what is the action for example because it's always kind of approximate and uh is other really minimal surfaces i i don't know so so what here goes on uh ah why why actually i mentioned use this Hermitian against algebra it was some kind of simple equation which implies a kind of first-order equation which implies second-order equation and if you think about of terms of minimal surfaces there is also first-order equation and consider holomorphic curves implies second-order equation minimal surfaces because in Keller manifold any complex curve is a minimal surface yeah so it's a minimal surfaces so so it's kind of a dual version of this uh relation and uh now returning to original uh equation not in Toray can we have kind of suppose we get a complex curve in let's see city can see m yeah suppose i have complex curve that's algebraic curve uh it's not compact and i should compact uh quantized by some infinite dimensional space and uh in particular uh and so it should solve from infinite dimensional space when you get this equation and because it's complex it should solve even simple equation yeah so that's there is a question about such objects uh and it will be simpler than minimal surfaces in r3 like catinoids uh which we have now explicit example but pretty complicated one and last maybe two weeks two weeks ago uh we discussed with uh Jens and realized that there is very very basic and simple example which was kind of overlooked in literature it will be simplest possible minimal surface non-compact minimal surface and it will be simplest possible quantization so so the curve which i will take will be hyperbola oh by the way just before writing example i i would write to write a question what are hermitian and mills equation you should have some Hilbert space which is can should contains like functions on my curve polynomial functions or maybe sections of some line bundle a subspace a subspace yeah and it will be and coordinates in my space should x on my Hilbert space it will be not self-adjoint operators they commute and satisfy the same relation as making the backman of the notation yeah yeah the commute yeah the satisfy the same relation that can see and the the equation it means that sum of our commutants is equal to constant yeah that's that's the equation yeah yeah so how to solve this equation and example is very simple first simplest non-trivial example the curve is c star c you can c squared so here get coordinates z1 z2 and it's given by equation z1 times z2 is equal to 1 hyperbola complex so yeah from real point of view it's some two-dimensional surface in r4 and it's minimal because it's scalar geometry and now maybe put it h bar here so so we should i can write z1 is z is z2 is z inverse yeah so we should solve this equation sum of a commutator of z hat z is some self-adjoint not self-adjoint some operator z plus commutator z inverse hat z inverse is equal to h bar the whole story it's invariant under rotation under u1 and algebra functions has basis by integers so the basis of h will be some vectors ek the k is an integer it's you'll be unitary basis and operator z will be goes to from uk goes to some constant times ek plus one where ck is positive number yeah sorry you said each contains all c but now it's it's will be dense yeah because it's Hilbert space and this is countable space that's that's a point yeah so we should make some completion and when we write this equation but you mean dense okay it will be completion of yes all c is dancing yeah and operator's multiplication will go to some unbound operators with some common domain and so and when we write this equation you get something like this and this is very easy to solve we see that it's this equal to k h plus some constant i know some kind of alpha and then ck is unique positive real solution positive real solution of these things which is you take uh solve quadratic equation yeah which is yeah so you get explicit formula for the solution so it's yeah so that's the most basic example and alpha is arbitrary sorry alpha arbitrary constant but actually defined up to h bar z so it's actually a small circle which perpetuates all solutions not a unique solution and i don't i think it's related to the fact that the curve has no trivial first homologated it's kind of a put kind of line bundle maybe secretly on this situation uh i don't really understand the meaning of this one parameter and but then it turns out that these things was kind of known 20 years ago uh there was a work by washington taylor and kornalba in about 20 years ago when they uh solve they found the same thing but for parabola parabola and the question is more complicated and i cannot solve increases closed forms so they missed as the most basic example but i think it's it exists just just one paper by taylor and kornalba and actually i think it's pretty interesting question on rig geometry you have curve in c and there should be some more or less canonical Hilbert space structure satisfies the situation which i have really no idea how to prove yeah yeah in fact this equation appears in different contacts uh uh i think in the work of nikita nikrasov uh yeah so here we get a module over the algebra functions which functions on a curve uh algebra of polynomials but functions on the curve so it's models very very thin and it's torsion model and he considers the same question to find some Hilbert space completion for completely different modules you can see the functions like an in c square when you get few points it will be not a vector bundle to the torsion free shift and on torsion free shift usually one cannot solve young mills equation does make sense and here it was some kind of regularization of young mills equation for uh for uh torsion free shifts but here it's a torsion shift and it's also makes sense and if you put constant zero get finite dimension representations like off points yeah so it's kind of very subject in algebraic geometry can cover this canonical Hermitian uh uh metric yeah yeah so the subject was started in this paper and there was one continuation by karnalberg who was a student of washington taylor which is very closely related to what also kind of uh want to tell you right now but not equivalent i think what i will say is and it will be kind of slightly better really more satisfactory version now we go back to tori uh so i explained that uh tori there should be plenty of minimal surfaces and now there is this uh supersymmetry so it should be holomorphic curves in complex tori in complex abelian varieties there are plenty of such curves and this this examples minimal sources and this should decide this first order equation kind of not second order equation how to write this thing uh i will present your construction we get holomorphic curve in complex abelian varieties then for each you get a hill with space depending on maybe h bar parameter which bars kind of roughly one over n of dimension is n plus some kind of sequence of big vector space on which acts plus action of almost commuting unitary operators will be if my abelian variety cm over gamma it will be 2m 2m unitary operators which not exactly commute and this will be uh i don't know which equations are satisfied but at least it looks uh very natural so what what does the picture and uh the story so we get this curve sitting in this variety let's go to universal cover of the abelian variety so go to cm and we go to certain cover which will be kind of like infinite periodic picture like what we saw this morning in the talk of martin borderman yeah kind of multiply periodic surface and then we can try to think in the following way so we when we try to quantize c first we quantize c hat and in a way which is compatible with shift operators so what we should have here uh first we should have infinite dimensional space no no no no no no no so we should have infinite dimensional space if you correspond to this non-compact huge non-compact curve and on this space we should have should have the solution of Hermitian mills equation should have uh z i i get the same equation from 1 to m equal to 0 of z i get uh this this condition but then also we'll get shifts by a lattice and we get operators u 1 u 2 m commuting unitary operators which are shifts and so u i u j is u j u i and if we conjugate uh maybe called something like alpha beta it's because they're completely different indices alpha beta from 1 to 2 m and we then consider u i z i u i inverse you should get z i plus some constant e i alpha it's inclusion of lattice in c m you get 2 m by m matrix that's certain 2 m by m matrix complex matrix okay i don't need it this example sorry no no z it's z z is a coordinates on c m yeah z 1 z m are coordinates kind of classical coordinates on c m and this will have a half shifts yeah and it's actually a well known observation in all the story if you look on this how we can interpret it you get Hilbert space is action of commuter cooperators you get a vector bundle on a torus and this z alpha gives you a connection here and what this gives it gives a solution of usually young mills equation for the vector holomorphic bundles for the torus so h bar will be seen in this section of some torus it will be actually dual torus with certain you hermitian vector bundle e and this uh apparatus uh u i will be coordinates on this torus and z i will be uh uh connection matrix before as our coordinates now it became uh derivatives yeah so z i goes to covariant derivatives uh for community completely classical uh young hermitian young mills equation it should exist by donson will be yaw on stable bundle yeah so the question but this is a dual torus and how we construct the bundle on a dual torus and this thing is called Fourier mokai transform namely we get curve sitting in a billion variety it's a curve and we get some inclusion we get as you see so we coherent shift on a billion variety and now we make Fourier mokai transform which is very simple we get vector bundle on dual variety what you do you uh you make a line bundle primed by dual variety restrict to the curve and take global sections and you put plant constant to make a power of these things so you get by Fourier mokai varieties yeah yeah because it's dual torus you get Fourier mokai and if you are a good station you get vector bundle holomorphic vector bundle on dual variety which will be our original my original same yeah so i get a bundle and i put again plant constant in the game and the rank of the bundle will grow linearly so i can take pick a point on this my thing ah just before before i'm going on so what i should do i i kind of make holomorphic vector bundle and now i should solve here mission and this equation i don't know actually how by Fourier mokai transform to solve this can mission question maybe it's known but if this as well was questioned and i get the representation of this algebra immediately now pick a point now i pick a point in my dual torus and point it means it still be eigenvalue of my some eigenvector of source of choose eigenvalues from commuting unit operators so i fix eigenvalues for commuting unit operators and i claim it will be still some new operators commuting with them and what are what are they it's not the i the i do not commute with them but you can see the certain linear combinations of a real imagined part of the i i take in normal sense here mission and here mission part a certain linear combinations will satisfy this by things that if you apply all shifts you get here integers commutators use your alpha belongs to in two pi i integers and now i can take exponent of this linear combination you get some unit operators yeah so get on finite dimensional space which will be fiber of my bundle you get a bunch of not not commuting unitary approaches and i think it's really kind of ideal situation that should be should correspond to this quantization of a curve in a billion variety and there is still something which i don't understand and all this story which is the following for these things called bps inequalities in physics which for which have to mean like young mean action of of some any connection is bounded below by some integral of second term class and the quality it means that you have hermitian in this equation and yeah and this similar story if you get a a curve you're going to get surface then this area as opposed to get surface which has come all the class of some complex curve and then the area is great equal to in absolute value of integral of scalar form of this surface and you get a calibration of condition you get equality if you know leave the curve is complex surface is complex curve yeah so i would like to have some notion of degree here and degree is actually something pretty funny so we want to degree which sits in a h2 of my target space torus say z coefficient okay let's try to seek first and classical example suppose we get map from surface symplectic surface to torus to the two-dimensional torus and degree f and degree which is just two s1 valued function and degree is equal to one over two pi squared integral of boson bracket now now we're replaced by two unitary matrices instead of two I think there's no surface it's two unitary matrices and suppose the comitant is less than constant divided by n it's like happens in for this generation of quantum torus when we get identical map of degree one so we get such thing so and it's and capital n is size of matrices yeah yeah so it's that's a regime when we study limits and want to get a minimal whatever surfaces yeah so the degree is defined in the following way kind of lemma if you get just two almost commuting unitary matrices then trace of u1 u2 inverse u2 u1 inverse u2 inverse minus one is equal to two pi i some integer plus o1 the k is integer and k is less than constant divided by two pi c is the same constant which appear here one with respect to capital n of course the no it's of one when n goes to infinity yeah yeah yeah so it's and this will be called a degree a degree of of my quantum surface this is a kind of means there is some kind of continuity of the map and then there will be a degree and why is this it's it's extremely easy it's kind of one line proof we know that this operator has determinant one and has eigenvalues lambda one lambda n but eigenvalues by this condition very close to c over n so it means that eigenvalues are close sits in a very small disc around one and one can have canonical logarithms so it's like psi will be exponent to the numbers equal to one exponent of i square root of one one c to i and then I get the c some of c to i is equal to two pi i integer because product is equal to one it's two two pi integer and this is a differ from this is basic quadratic term so it will be of one yeah so get this notion of degree from up to the torus and similar things one can uh in in fact if you analyze the story it's not necessary to be so smooth map to this commutative very close to one it's enough uh not for this statement to define the degree enough to assume that uh this commutator this spectrum doesn't contain one number minus one it's then we can get uh canonical choice of our argument of logarithms then we define so this is degree so this so it means it's a set of pairs of unitary matrices that commutative does have anti-invented elements has many many connective points which are degrees and uh similar thing one can map m to not to torus by s2 so I don't no longer speak about minimal surfaces you also get something like x1 square plus x2 square plus x3 square yeah suppose you get uh if you get such map uh then degree is equal one over five four pi integral over x1 x2 3 plus cyclic order yeah we get this thing and the same kind of quantum analog of the story if you've got three Hermitian matrices and again I assume that if consider any commutants I get operator norm is less than constant over n uh then the rays of x1 I just take first term here they all give the same trace is below is equal to 2 uh square root of minus one over three times integer plus o1 okay is an integer and again bounded by 3 half over c so it will be degree of map to the torus yeah so it's kind of nice I think it's I think it's something about c star algebra when I play put this inequality you can generate c star algebra so that will be some class anthropological case use very freestyle yeah maybe yeah yeah it's actually it's very easy to proof one can make some projector and blah blah blah yeah yeah yeah yeah yeah so yeah so those uh so there's something to get unfinished in this story uh I could I could really have to eventually to make this inequality that this action is greater than square of area and blah blah blah but no not yet no no no yeah I'd refer it right can set maybe I don't define left hand side yeah yeah but at least I know what the minimal surface is yeah okay that's it thank you very much questions comments maybe it will not be phrased in the language of minimal surfaces that you are discussing but so the action which you have is the one placate the action for your mystery so that action is this no this one placate ah it's kind of like discord here sure yeah it's okay and then if you consider the interviewer with that action yeah that has been studied and evaluated at large and yeah okay now now that has that large and answer can be obtained by a stationary matter meaning there is a configuration you want you too yeah so that it gives you the larger value of the thing to be is that related to the solution I don't know and I'm thinking about only classical not taking integral of x one two five solutions dominate partition function stationary phase matter yeah so so you can do classical I can take to use it for evaluating the station point of the integral yeah that's the use of classical yeah yeah now this answer is long yeah no no I think I don't think I'm telling many new things here just no no I meant the answer of the configuration you want you too yeah it gives the stationary phase answer but actually it was one of the origin of the conference I just I thought it's got these these formulas we were discussing that I mean in in physics they are also known yeah this is what we can call the master field um yeah one small comment maybe some more questions otherwise this this paper by Washington Taylor Conal was actually quite funny it's written in Princeton and the classic they don't use ever 37 page paper the word minimal surface yeah although the thing they describe is quantum analog of the minimal surfaces that were found 100 years ago in Princeton by Eisenhardt famous differential geometry and um yeah and I don't not clear why they missed the simple example they they they work they have one prime example that's this parabola yeah and it's pretty complicated yeah it's so far not no exact solution and they do mention these background yeah so they noted that yeah yeah in fact carnalbo wrote a paper when he was done essentially the same things but he used a bergain projection and I think it's kind of not less canonical and says what I would describe