Classical conformal blocks and Painleve IV
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00:00
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Transcript: English(auto-generated)
00:15
It's my pleasure and honor to speak here at this remarkable occasion and I
00:24
would like to first to thank organizers Sasha and Samson and Nikita and the Institute for this opportunity. Yeah I had a privilege of working
00:45
closely with Vadim and actually collaborating with him and this occasion brings memories, a lot of memories, fond memories and sad memories obviously and
01:01
particular memories are related to the to the way Vadim was sharing his remarkable ways of, his remarkable new ways of looking at things and not only
01:23
in physics and also his particular joy of doing physics. Well we all enjoy doing physics but I think everyone who knew him would would recognize that he had
01:40
some particular flavor of joy of approaching physics that I will, I will not forget. And well the Vadim's life in physics was short in the clock
02:05
time but it is obviously very long in terms of impact and he he was at the origin of many things including in particular the conformal field
02:23
theory and that's that will be, so this is now the the important part of mathematical physics and and pure mathematics and it's still developing
02:43
and so that my talk will be devoted to this subject so it's a it's some part of this talk would be some development of the subject that concerns conformal blocks. Most of the talk will be in the way of extended introduction
03:03
whatever the original part is based on the on the joint work with Alexei Litvinov, Sergei Lukianov and Nikita and of course all mistakes and
03:21
misconceptions are entirely mine and so this is the title of the work Classical Conformal Blocks and Penlevy 6 and so before starting I probably although conformal blocks I think it's now sort of well established notion
03:44
in in well mathematical physics, mathematics, I still I think I will need to I I will say few words in the way of introduction so classical
04:00
conformal blocks appear a certain limit of the Verasoro conformal blocks when you send the Verasoro central charge to infinity together with the conformal dimensions and the Verasoro generally the Verasoro conformal blocks are are associated with Riemann surfaces with punctures and the
04:29
Riemann surface with punctures but I will limit the tension and that will make the exposition much simpler I will limit attention to and actually
04:42
okay I will limit attention to the sphere with n punctures here is the sphere and z i will always mean denote the positions of these punctures in terms of some complex coordinate on this Riemann sphere and actually I you will
05:07
see that I will be mostly talking about sphere with four punctures but but for some time we will consider so consider n punctures. So what is the conformal block? Roughly speaking that's some sort of correlation function
05:25
of chiral primary vertex operators the primary means the standard primary type of commutation relations with Verasoro generators or operator product expansions with energy momentum tensor but because they are these are
05:44
chiral fields, galomorphic fields, they are not local fields and so the notion of correlation function is ambiguous and to to give a meaning to the correlation function one needs to specify what is what is known as a dual
06:03
diagram or or equivalently append the composition of this sphere with n punctures here and I will basically consider I will consider only this kind of a hairbrush type of the dual diagrams and in this case we can think
06:30
of it as a correlation function of the or expectation value of this vacuum vacuum expectation value of the product of this vertex operator with the
06:42
instruction to put in between the operators the intermediate state decompositions in terms of the irreducible representations of the Verasoro algebra with the dimensions with certain fixed and formal
07:01
dimension Delta of P so this is the this diagram is representing is the way shortcut representing this or the or equivalently this is the the instruction to use the operator products expansions in certain order
07:26
well in in the order which is which is dictated by the dual diagram now Delta of P is Delta of P specific parameterization of the conformal dimensions in terms of in terms of this parameters P alpha at this moment this
07:45
is not very important this parameterization is motivated originally by the Fagin-Foux representations of Verasoro algebra and and was very convenient in Liouville theory although I will not I
08:04
will not actually refer in any systematic way to a Liouville theory although I will mention it all right so coming back to to this picture we see that there are n minus 3 intermediate legs here so there are n minus 3 of
08:26
these parameters so the conformal block F depends on n minus 3 parameters P and it also depends on this positions the alpha and because of SL 2
08:43
transformations the dependence is essentially up to projective transformations so essentially there are we can just use the cell to transformation to send three of these points to pre-designed position usually
09:03
0 1 or infinity 0 1 and infinity and after that we are left with n minus 3 coordinates in this independent coordinates within this on this model a space and so it depends on n minus 3 coordinates in the model a space and
09:27
also of course it depends on this external dimensions associated with the external legs in the dual diagram again the but but I will I will omit them in
09:41
this notation because they will be regarded as fixed numbers although of course the conformal block depends on so this is conformal block I will be just like I said I will be dealing mostly with the four-point conformal block which now depends on a single intermediate state parameter P a single
10:06
coordinate acts in the model a space and the notation will be F sub P of X all right so the the conformal blocks are fundamental objects and conformal
10:29
field theory they were introduced in the eighties in that time when which we fondly remember when Vadim was with us and this conformal the basics of
10:46
conformal field theory was originating so but it's it's a basic object in the sense that correlation functions of conformal field theory are built are are built in terms of conformal blocks through the
11:03
colomorphic factorization so I will produce just an example how it's it looks in the Louisville conformal field theory where let's say four-point correlation function that's simplest example where it enters non-trivially
11:20
is expressed in terms of the absolute the absolute value squared of the of normal block so you have two copies of conformal block depending of X and X
11:43
bar and it's integrated over this parameter P with some coefficients which are known as the structure constant of of Louisville theory these are explicitly known and one of the reasons I write down this formula is to
12:01
introduce this parameter B which is a certain parameterization of the central chart which I will be using every now and then in the in the course of this discussion Nikita when I need to to to stop it's not that I going to stop now
12:27
I just need to 1250 okay that sounds good all right so so this is a this is how 50 minutes okay so this is a this is the way how originally conformal
12:55
blocks appeared in conformal field theory but but I mean it turned out
13:02
that they are more the their emergencies seem to be much more general in in in in mathematical physics and lately they they they attract attention in relation well it's quite unexpected to me I mean people who are much
13:23
smarter than myself probably expected that but I to me it it it came as a miracle that the same functions appeared in relation to the four-dimensional supersymmetric conformal field theories and well in particular this conformal
13:43
blocks appeared as a well they take precisely coincide with the instant on parts of the necrosis partition function of the of the four-dimensional
14:02
n equals 2 supersymmetric gauge theory with certain well if you are talking there are sort of conformal blocks that's where that's a supersymmetric gauge theory with certain content of of product of SU2 gauge group and and
14:27
the remarkably the the parameters which are which originate from the moduli space on this Riemann surface plays the role in of the gauge
14:42
coupling constant relates to the gauge coupling constants in this in this gauge theories and the parameters P relate to the to the vacuum moduli in this gauge theory so it all to me it looks sort of sort of fear sort of remarkable I
15:07
would say but nonetheless it's a it's a it's a so and and I will it's now it's rigorously proven and I will mention that but but so perhaps this
15:27
functions play some sort of wide role in in all these correspondences but the point is that these functions have some we have some control on this
15:44
functions in particular the conformal in very conformal symmetry generally speaking the conformal symmetry completely fixes these functions similarly on the in the in the supersymmetric gauge theories of course
16:01
well I mean conformal symmetry fixes power series expansion and and by implication of course fixes this function so if I write down power series expansion of this function in terms of the moduli all the coefficients are fixed by conformal symmetry here and similarly this is a
16:22
instanton expansion in the in the n equals 2 supersymmetric gauge theory and this these coefficients are given by necrosis integrals of the moduli of the instantons of course they are fixed but that is a power series and
16:43
we would like to have a better analytic global analytic control of over this functions and because they play this important role and that's one of the motivations of this this this work which I am going to report is gaining
17:06
some better analytic control there are some things which are known or or or assumed about this I will be speaking about a four-point conformal block but the generalization to to the end point conformal blocks more or less
17:25
trifor so the conformal block is analytic on the universal cover of the of the of the model well basically yeah on the on the universal cover
17:44
wonderless space and it as a function of this P parameter it's meromorphic function and it abates the the the crossing relay what I call crossing
18:00
relation this is a relation when you interchange the interchange the position of the punctures by braiding transformations and then the conformal blocks are related through this integral transformation you transform a
18:24
sort of conformal block through the conformal block in the cross-channel with the integral with certain kernel which is which is related with the kernel here is essentially a succession symbol 6j symbol of continuous
18:47
presentations of Q deformed in universal in the loping algebra of SL 2 so this is a part of the I think most essential part of the knowledge about
19:06
these two dysfunctions now classical limit and I'm going to talk to speak about classical conformal water that emerges when you send the central charge to infinity and the interesting limit here appears when when you also
19:27
send I will but before before proceeding I will again write the central charge in this Louisville inspired or Fagan Fuchs inspired actually because that appeared before a parameter ization of the central charge
19:49
in terms of this param coupling parameter B this is not essential but but but I will use it because I like it that's the only the most important
20:01
reason so so I parameterize the central charges in terms of this B and the classical limit will be the limit of B going to 0 B is like a Planck's constant actually B squared will be analog of the Planck's constant so
20:23
interesting limit emerges when you send central charge to infinity in which case of course we we know there are sort of algebra converts reduces to Poisson bracket algebra but also you need to send the dimensions to
20:40
infinity so that the ratios of the dimensions and central charge remain fixed are kept fixed and I will refer to this Delta's the ratios as the classical dimensions is not capital there but lowercase deltas as classical
21:02
dimensions and again I will use the parametrization of this of this classical dimension Delta in terms of this parameters lambda and nu depending off of either I am speaking about the deltas associated with the external legs
21:23
of the dual diagram or internal internal lines and well I mean the the dimensions could be any so generally well I could think if they are real I could think of lambda as a real or pure imaginary but generally I
21:44
will think of this lambda Cisco generic complex numbers now when I perform this limit the conformal blocks exponentiate in this form that's
22:03
something which semi classical intuition makes us to expect but mathematical status is to me at least is not is not completely it ultimately clear so it exponentiates so that it it is exponentials of 1
22:24
over b squared times F which is this F lowercase F is called classical conformal block now the there is no doubt that this is indeed correct
22:41
statement and actually the the I think the proof proof is sort of exists it's just my my brain doesn't didn't yet incorporate all the details of it it will happen eventually it probably consists of mixture of the results
23:02
which come from different sides of this aldiagayo toshikawa correspondence which involved involves the then the form of necrosis for a form of necrosis representation of the coefficients of the of the power like
23:21
expansion which which looks like which look like a coefficients of the virial expansion of the of certain gas and then the classical limit is like a thermodynamic limit of this gas and then exponentiation is is usual thing
23:41
which is and then there is a now a sort of rigorous proof of this correspondence at the level of this four-point function so so I think the combination of this statement if we absorb it so properly would lead to but
24:01
I know I'm not particularly interested in in rigorous proofs so let me just proceed now there is an eye then there is a neat and well-known relation I think it's it's again a game we we we all played 30 years ago we
24:23
relation of this of this thing to to the to the monodromy properties of second-order differential equation so consider differential equation of this type which was a was a parameter with a variable Z regarded as a coordinate on
24:45
the Riemann sphere this is a differential equation with n regular singularities on the sphere and and so to make it precise from the very
25:03
beginning I will regard so the the the potential term sort of in this equation involves two kinds of parameters delta I and CI the Delta I will be regarded as fixed as fixed numbers and actually I will I will take them equal to the
25:25
classical dimensions which it appeared in the previous transparencies and identify them with the parameters and and CI are the celebrated accessory parameters and they so so this is basically this differential equation is
25:47
for me tries by by the position so the I and and CI well in fact there are only n minus 3 independent accessory parameters for obvious reasons which
26:04
which I I sort of explained by this line there are three if we don't want an additional singularity to hide it infinity then there are three elementary relations between them so that basically there are n minus three of
26:21
these independent accessory parameters and also the position zi because the the form of the differential equation doesn't change under SL 2 transformation again the zi's are defined model SL 2 transformations
26:49
so there are essentially again we can send three of them to pre-designed favorite positions and so there are n minus 3 parameters of this sort so
27:04
this this differential equation is is parameterized by 2 times n minus 3 complex parameters of course this differential equation defines some anodraming group which is a homomorphism of fundamental group of a
27:25
sphere with and punctures into SL 2 SL 2 is is a transformation of the basis of of two independent solutions of this equation under the analytic
27:41
continuations at long contours representing the elements of the fundamental group and and so because because the I say the that's lower case
28:03
delta are fixed we are dealing with the representation in which in which the conjugacy classes of the matrices M I which are representations of the of the basic elementary elements basic elements associated with the with the
28:26
path elementary paths around individual punches are fixed and actually related to this parameter lambda which is the parameterization of the Delta and also
28:43
there is of course ambiguity in the choice arbitrage in the choices of base overall choice of basis and so we have this object which is a space of of such a homomorphism with fixed conjugacy classes we of the
29:03
elementary of the elementary matrices and defined up to overall conjugation and that is roughly speaking the the modulus moduli space of the flat connections over the Riemann surface over the sphere with the with the end
29:25
punctures well this is well known object it can be caught in the parameterized by by various invariants like traces of this matrices with some relations between them it's well known that it's there are exactly two times
29:44
n minus three independent objects of the sort independent parameter independent invariants of this sort and that basically means that that the
30:02
differential equation of this second order differential equation of this sort once the the parameters Delta are fixed it doesn't have continuous isomandronic deformations or currently the parameters the positions of the
30:22
punctures and accessory parameters I denote this n minus 3 G's and then minus 3 C I's indicated by primes that are n minus 3 independent of them they can be regarded as local coordinates in this moduli space of
30:44
flat connections and and also it's well known that there is a natural symplectic form on this moduli space due to ITI bot and this coordinates are Darbu coordinates and this and this moduli space all right so how the the
31:10
how the classical conformal block is related to this is is again very well known I'm still in the middle of extended introduction and that is
31:25
related through through the the special cases of the of the conformal blocks when you have a degenerate representations with null vectors and
31:42
there are I will actually explicitly refer to two of them especially today in which no vectors appear on this on the level two and I will this I think is a standard notation by now these are Delta 1 2 and Delta 2 1 these are
32:02
the associated conformal dimensions expressed in terms of this parameter B they are expressed in a very simple way that's why I like this parameterization and these are actually the coordinates of the null vectors in explicit form and from this it follows that the code that the conformal
32:25
block in which you I'm going to but I went back to the quantum case where B is fine the the conformal block which involves the the insertion of the
32:44
is degenerate vertex operator with the null vector in this in a state that representation obeys the second-order differential equation of this form and
33:02
this is called no vector decoupling equation right and and the second one I don't write down for the two comma one it's the same equation with B replaced one by one or four be there is this symmetry so I will be it will
33:22
appear shortly in this discussion all right so so this is a this is equation and now I want to send B to zero again with this Delta divided by
33:40
B squared Delta also going to infinity with with lowercase Delta kept finite but the insertion is the dimension of this Delta one two doesn't go to
34:01
infinity it remains fine at minus one half tends to minus one half and as usual in in classical limit we expect by by usual semi classical expectation we expect to have a exponential factor and the pre-exponential factor which
34:23
depends now the only effect of this of this additional insertion is going to to be on the pre-exponential factor and from that and from that the differential equation in the previous section of the previous slide this one
34:48
transforms into into the differential equation which which appeared here with the accessory parameters being the derivatives of the classical conformal
35:05
law this of course is very similar to the expression for the accessory parameters which appeared in the uniformization problem and this was conjectured by Polakov long ago again this early eighties and then
35:26
proven by Tectajan and Zograf in the but the associated monodromy problem is different as I am going to explain as I am going to explain and now
35:43
in the uniformization problem we are dealing with the with the with the condition that the monodromy group is going to be function that means the monodromy group associated with this second order differential equation is
36:03
going to be we have to impose the condition that it is embedded into a real subgroup of SL2 that means I wanted to put SL2R but but it's still
36:23
it must be SL2R here either SL2R or SL2 that's the the fault of doing this thing with mouse all right anyways it's either SL2R or SL2 and in that
36:47
case the accessory parameters are become gradients in terms of Z of the of the classical action calculated on on the solution of classical Liouville
37:03
equation which depend on Z not in a galomorphic way but it depends on on Z in non-galomorphic phase so it depends on Z and Z bar all right and this is going to be compared to what we have now this accessory parameters which we
37:23
come up with are gradients of the classical conformal block and and of course these things are somehow related in particular the the the Liouville action the Liouville action the which the Liouville action which is
37:46
related to the solution of the monodromy problem associated with the informisation problem are can be expressed and can be solved in terms of the conformal block I write down schematically the solution the the
38:03
Liouville action is expressed in terms of a combination of conformal blocks plus some capital psi which is known function but you need to to evaluate it at nu which is in turn depends on Z and Z bar so it it's not simply a
38:23
sum of the the functions of Z and Z bar because nu are determined by extremization of this function in terms of of nu so in in in in fact the the Liouville action is related to to the conformal blocks but by some
38:43
sort of a gender transformation but what I want to say is that is that it looks like the conformal blocks are more basic object these are morphic objects and and how to solve how to solve the inverse problem how to
39:01
express the conformal blocks in terms of Liouville action this problem is not solved and I suspect it is impossible to solve this problem and so I am not sure I understand why people sometimes call problem of conformal
39:25
blocks the complex Liouville theory so I think it's it's quite different problem so we can we can ask now the questions this F appears as a classic in the form of classical in the exponential so it looks like it is a
39:44
classical action of some system of classical mechanics so natural question is what classical mechanical system is behind all that and and this is a this
40:02
is a question which I I think I will produce maybe not entirely ultimately satisfactory answer but but some answer to and also but let me first start with much simpler question which is easy to answer and this is
40:22
what kind of monodromy problem this equation that accessory parameters are gradients of the informal block which kind of monodromy problem is solved by this equation that that's that's pretty obvious because we
40:43
remember that the conformal classical conformal block well conformal block itself and classical conformal block is defined relative to the pen decomposition of this dual diagram and just like I said one of the ways
41:02
to interpret dual diagram to read the dual diagram is the the way the succession in which operator product expansion are used and because we are dealing with the insertion of the vertex degenerate vertex operator v12 we
41:23
need the operator product expansion of u12 and those these this expansion be the well-known fusion rules from which one immediately read out the answer and it basically says that if we fix the the accessory parameter
41:42
according to this gradient formula or the accessory parameter are gradients of the derivatives of the classical conformal block that fixes the monodromy around the succession of contours which are exactly associated
42:00
with the pen decomposition corresponding to this dual diagram that means if I take this contours and then shrink them it would correspond to the degeneration of a sphere in which it splits into succession of of three punctures spheres in which these three punctures spheres are
42:22
exactly the vertices of this of this dual diagram okay so so yeah now let me sort of put this in the in the this thing in the neat form and in general
42:45
form which is due to to the the work of Nikrasov, Rosli and Tashvili of couple of years ago and and well first of all one immediately absorbs because
43:02
because this contours can be this this path here can be chosen to be non-intersecting associated parameters new which also can be part of coordinates and in this modular space they are Poisson commuting in the in the
43:22
Atiobot and in the with respect to the Atiobot symplectic form and so one can one can define another set of double coordinates which I denote now
43:41
new this one of the parameters associated with this with this contour and the and the conjugated parameters mu which conjugated parameters mu and then the the classical conformal block becomes important part of the
44:06
associated generating function of a society conformal canonical transformation so the the new book coordinates are related to the C and Z which has also the book of the net which are which I already mentioned so
44:26
we have a generating function actually it's it's it's convenient to add a certain term this there is ambiguity of course defining this canonical conjugated variables and and one can convenient certain term well which is
44:49
convenient both from geometric construction of the canonical conjugated variables and also it has nice interpretation in terms of supersymmetric
45:01
gauge theories all right I will come back a little bit so I I have there is something interesting things to say about about the canonical
45:20
transformations between different their book coordinates associated with different dual diagrams and that's closely related again to this work of foot across the floor really but I think the time is running out and they
45:41
will better skip this part anyway let's try to move forward and try to explore the second null vector equation which is related to the
46:00
to the to the null vector which obtained from the know the degenerate representation which is obtained from that one by replacing B by Y over B now this decoupling equation takes this form the Delta now is becoming large
46:22
in the classical limit and therefore the decoupling equation converts in the classical limit converts not in the in the second-order ordinary differential equation but instead instead it becomes a Hamiltonian
46:45
kobe like equation of this form this is of course very well known and so it's a it's a it's still a partial differential equation but but in
47:01
fact if you have only four four points it's a it's simply a Hamiltonian kobe equation for one-dimensional system so if you take four point conformal block and take a limit classical limit here the equation which
47:21
you have is a Hamiltonian kobe equation of one-dimensional system with with a coordinate which I denote Y with that's where they sort of we insert the two one degenerate operator and just like I said three of the
47:42
points I sent to to certain position that's my SL to freedom and one of the moduli is still is still one of the points is still there that's the cross-ratio independent cross-ratio so I denoted T because it's going to play
48:04
the role of time so then the classical limit generates this gamut on Jacobi equation with this Hamiltonian it may looks look ugly but everyone who is familiar with pin the way equations would recognize that this is a
48:24
associated with the pin level six equation within the basics of course is one most general and and of the of the six equations which produce
48:46
transcendental solutions and have this pin level property that they have no movable singularities that means there there are no the solutions if you take a solution Y of T as regarded as a function of complex time there are no
49:07
singularities which depend on the initial data alright so so but the but simple poles but here because Y is by by by construction Y is better
49:23
considered as a as a coordinate on Riemann sphere simple poles are also not really singularities so we don't have a singularities movable singularities at all and of course there are fixed singularities at this points one zero at infinity which are some power like singularities which I
49:48
will mention soon alright so this is I think this is also a rather straightforward thing I think many people knew that but but it turns out
50:03
that this is quite useful in in analysis of the of the conformal this is quite useful in actually evaluating the conformal block and and I will basically I will not go into many details but but it's based on very
50:22
straightforward elementary observation that that by by definition of the classical action consider consider some trajectory some solution Y of T of the six equation such that it passes through at some times t1 and t2 it passes
50:42
through two points Y and 2 then by definition of classical action we have this relation between the classical limits of this conformal blocks at Y and T associated with these two points with the coefficient which is the
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classical action evaluated at this piece of the classical trajectory so what remains is to to choose properly t1 and t2 to choose properly the solution and the initial point and the final point so the solution but we take generic
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solution right and the generic solution is known to behave like a power near any of the singular point and to be definite I choose 0 as the singularity it behaves like that with some power nu the nu will be identified with the
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parameter in monodromy parameter associated with this intermediate intermediate classical dimension and and so and so and so if I take this
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solution then and then we see that that as T goes to 0 Y also goes to 0 and in this correlation function in this in this conformal block at T goes to 0 this point and this and this point merge with this point so three of
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the insertions merge and we are we have a three point conformal block which is a constant alright so if we take t1 equals 0 then the starting point here is just it's just some constant okay so now if we look at the
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trajectory Y of T then a certain complex generally complex T it hits one
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of others other points where other insertions sit and so if we take t2 the other T in this expression over yeah if I take t2 in such a way that
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that t2 equals to some equals to some X at which Y hits one of the other
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insertions here again for definiteness I I take it that Y there hits the point infinity then this point that this reduces to four point conformal
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block with some shifted dimensions associated with the infinity which can be also with the shift can be computed using using the diffusion rules standard fusion rules and so forth I skip the details but but basically the bottom
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line is with this particular choice the the solution certain solution of the six equation defined defined by by this initial conditions which eventually
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would hit would hit infinity one of the other insertion basically interpolates between the three point conformal block and and four point conformal block and basically we need to calculate the classical action on this
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solution there are some subtleties because although the the the solution itself is regular because of the lever properties regular the action has some simple logarithmic singularities one need to make a realization and so forth so that I have to put this is something which I may put in the frames
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that's when the level of four point function that's the answer how the classical conformal block is expressed in terms of them of the classical action of the evaluated on the evaluated on certain on the
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classical on certain classical solution of the universe six equation so in some sense it answers the question of what classical problem is behind this I'm not sure it's completely satisfactory I will say probably one of word about
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that but at least it could be called some sort of lazy answer to this question there is a little bit to say about the well long standing of century
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old problem about the monodromy problem of well basically the first non-trivial problem after the Riemann equation of the differential equation with regular singularity where you have four singularities and and the
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monodromy problem then is is related to the through this is related to the connection problem often the v6 the connection problem is again if you start with the solution given in terms of that kappa and nu at the singular
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point then it eventually arrives at at certain point X where it at certain time X it says hits infinity or any other of this of these points but let's say infinity and you want to connect the initial data nu and kappa
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at one end to the to the parameters X and Y naught and if you manage to do that then the accessory parameter which solves the monodromy problem as I mentioned is explicitly related to that I don't want to enter the details here
57:47
this is fairly straightforward well of course the the one can compare that to known things about the power series this goes through to known expansions
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about the pen livre and this is just just taking in grinding some power series expansion that's simple let me make some remarks here one is that
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well everyone who who looked at the pen livre and whatever all this kind of things would recognize close relation to the monodromy preserving the formations of function systems in this case SL 2 differential equation you
58:45
have now matrix differential equation where again with regular singularities where now AI are two by two matrices traceless matrices and now we have a monodromy preserving deformations which are described by this Schlossinger
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system of Schlossinger flow and the pen livre appears in the simplest non-trivial case when the number of regular singularities is four and and
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well it was as was demonstrated by by Roger Tikhin long ago this Schlossinger flow is just a classical limit of of the system which bears was Vadim's name and so perhaps the true understanding of all this thing and not
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only the classical limit but the quantum thing lays through this this system and also
01:00:00
Another kind of incarnation of... another manifestation of Pin Le Ver was discovered not so long ago by this... in the remarkable series of papers by this... this physicist from Kyiv,
01:00:23
who discovered that... that this time the... not the classical action, but famous tau function, which is more or less the same tau function which Stashe mentioned in the first part of the...
01:00:40
in the first talk to this morning, appears as the generating function of the conformal blocks, but not the classical conformal blocks, but quantum conformal blocks at the central charge equals to one.
01:01:01
And... and just lately Nikita suggested that it might be closely related again through this Fagim's system. All right, so that's kind of what I wanted.
01:01:22
Questions? Yes. I was... I wrote some supporting letter for these guys who found this nice expansion for this, so I know this paper.
01:01:40
And surprising... really surprising that there we have C equal one and to here we have C equal infinity. May it happen that if we expand in one over B square that the next term of expansion will coincide more or less with C equal one.
01:02:04
And probably with other C and other terms of this expansion. I don't know about another terms of this expansion, but... but it might be so, yes.
01:02:22
Because a little bit miraculous. It is a little bit miraculous, but... but it might be that just... just like I mentioned that Nikita suggested that through Fagim's system there could be a sort of direct relation.
01:02:46
You consider the limit C goes to infinity. Do you have some structure if C goes to minus infinity? Why does this... It doesn't matter. The limit... the limit... from what... from the point of view I
01:03:01
am discussing the limit is regular. It doesn't depend from which side you approach it. So then the C equals minus infinity as an interpretation in terms of processes. It's a small noise limit of SLE. And do you have an understanding of why you expand V in SLE or anything?
01:03:25
I'll say it again. C equals to minus infinity and the second order differential equation is a Fokker-Planck equation in the small noise limit of SLE. Yes, yes. So you mean that L is somewhere in the SLE processes? Must be, yes. Must be... it must be possible to fish it out from there.
01:03:45
Yeah, there is a gazillion of questions about that emerging from this observation. In particular, in particular I gained back related to Liouville theory.
01:04:01
Then there is a question, how come the... what is the meaning of the spin of the equation? Because it should show up in the Liouville theory and what is the meaning of it there? Well, I mean, it definitely is there and people know it is there, but what is the meaning of it?
01:04:21
That's an interesting question. I think no one even raised it. Let's thank Sasha again.
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