 Yeah, so I want to go back to why we left off yesterday and well, and again you are free to ask me questions and so please stop me whenever you want to ask the questions. So now I want to give a formula for the hemisphere function but yesterday actually I had typos which Bruno pointed out. So I want to correct. So the mistake, the type was in the formula for the one loop determinant. So the vector contribution was fine. So there is some product over positive roots and okay, and sin pi alpha sigma. So this is a vector contribution. So here sigma is meant to be minus i times l times sigma 2. Now there is a contribution from the chiral multiple obeying the Neumann boundary conditions. So by noi I mean the set of irreducible representations of chiral multiple obeying the Neumann boundary conditions. Then there is a product over the weights in the representation. Now the argument is w times sigma or sigma is given there and so here is the correction. So the argument in the current convention, in the convention of this lecture it should be q over 2 and the sign is opposite from the convention of my paper actually. Okay, and then similarly for the Dirichlet boundary condition, so by Dir I mean the set of irreducible representations with Dirichlet boundary condition. And there is a similar product and as I explained yesterday there is minus 2 pi i e to the pi i w sigma plus q over 2. So again this again correction and gamma 1 minus w sigma minus q over 2 correction. Yeah and if you have twisted mass, you are supposed to replace what I say. Okay so w sigma to w sigma plus m a okay with suitable definition of m a. Okay so this is a one loop contribution and I want to denote by script b the correct to be the data necessary to specify the b b b b b b. So okay so the no way and Dirichlet I explained v is a champaeton vector space. A row is representation of the gauge group and the flavor group carried by the champaeton space. R star is the representation of the R symmetry group and q is a matrix factorization. So this is what we call boundary data and it corresponds to b-brand. And now okay I am going to write the formula for the MTS function. So okay so gauge group is g MTS function which depends on the boundary data okay boundary data and also the also it depends hormonically on the two complex f i parameter t. I think I wrote 2 pi xi xi minus i theta okay topological. So this is given as 1 over the size of the wide group of the gauge group. So for u n this will be n factorial and there is a u joe integral d so rank g is the rank of the gauge group and sigma 2 pi i raised to the power rank of g and now I am going to as a contour I just simply write the most naive one okay. So this is just the imaginary axis raised to the power the rank of the gauge group and this is what you get from the boundary condition on the vector marketplace I described yesterday and actually depending on the representations row and R star carried by the champaeton space you may need to and also depending on the twisted masses I guess you may need to choose you may need to impose a more refined boundary condition on the vector marketplace and one way to think about is the following. So the vector marketplace from the vector marketplace you can construct the twisted color marketplace which is sort of mirror to the color marketplace and then in order to preserve V type supersymmetry you need to choose a Lagrangian sub-manifold on the space of sigma 1 and sigma 2. So more generally this imaginary axis raised to the power rk times g rank of g should be replaced by some Lagrangian, appropriate Lagrangian sub-manifold which is almost of this form but it might get tilted in some directions. So Lagrangian we direct to the nai symplectic structure on sigma 1 and sigma 2 just as a flat space and d sigma 1 with d sigma 2. So and yesterday I evaluated the classical contribution so okay t times I think for you and I wrote t times trace of sigma but in the general case let me write it just t times sigma and in general in general the complex of the parameter gets renormalized so we need to use a renormalized one. Okay probably Benini discussed this so you can you probably remember it. And then as part of classical contribution we have a super trace in Champlain space e to the minus 2 pi i sigma and then okay there is a two important okay one loop factor yeah it's that. So this is a formula. So so okay this is basically the main result for my talk so far. Is there any question? Okay then let me continue. Okay now so then you can ask okay what is the hemispherical function good for? What's the meaning? And one meaning is the following. So this is a this was a conjecture put forward enough paper and also holy and romo okay and this is for CFT so I mean in the case okay I didn't explain the Caribbean condition but for example for the if the gauge group is Arbelian so if the gauge group is a product of U1 then the sum of gauge charges has to be zero for the actual asymmetry to be preserved and in that case it is at least in some region of the FI parameter it is believed that the gauge theory flows to a non-linear sigma model whose target space is a Caribbean manifold and therefore the theory would become so the theory is believed to become conformal and in that case our conjecture says that the hemispherical function is the unnormalized. This was implicit in a paper in the papers but unnormalized version of central charge of the D-brain. Well in in string theory you can use the B-brain boundary condition or B-brain data to describe a D-brain in the in a compact on compactification of type 2 super string theory to lower dimensions like four dimensions so for example you can you can you can have a Caribbean or typically you can have a Caribbean you have a Caribbean or compact creation and you consider a holo-rick brains or sheaves object in the derivative of coherent chips and so on and then anyway so if you have a D-brain inside the Caribbean then you get a particle right you get a particle in a lower dimensional theory like in four dimensions and if you for example start with type 2 string theory you get n equals to super gravity theory and you have particle and in four dimensional n equals to super algebra you have a central charge you have a central charge in the super algebra and this is the central charge that appears in the four dimensional n equals to super algebra. Good but the central charge of D-brain actually so there is a more intrinsic definition in terms of a two-dimensional CFT n equals to super formula theory and this is given by the partition function on an infinitely long sort of okay cigar okay I think they call it cigar so this is a this is a cap this is a sort of hemisphere and there is a flat region and there is a boundary okay so and the length is meant to be infinite you and you consider here a twist in the cap region you might call it the TT star amplitude topological anti-topological amplitude and yeah yesterday actually there was a question about how to go to CFT right so so this boundary condition indeed describes about defines a boundary state and because the length here is infinite the state created by by this a hemisphere region is is projected to to the ground state so this the function function on this geometry which is which is which is a really different from the function function I described this does compute this does compute the overlap between the Ramon Ramon ground state and the B-brain boundary state so the equality here is a was a conjecture and the conjecture was based on the explicit calculations and knowing comparison with no examples and I say unnormalized because in order to really get the central charge of the D-brain we need to normalize by the sphere partition function computed computed using the same renormalization procedure so the same realization the same counter terms okay and then this conjecture was actually proved I mentioned this already yesterday this was proved by backers and plankton using super wire anomaly okay so so in the safety case the meaning of the hemisparsion function is now clear but by the way it's the hemisparsion function function and compute the unnormalized central charge and often it is good enough because typically what you want to use the central charge is to compare the stability to study the stability of D-brains and then what you need what you really need is the relative the relative what's really important the relative phase and of course also the the the ratio of the absolute value is also important because it's from that you compare you can compare which part is more is more massive which part is more is lighter and so on but so anyway this unnormalized central charge is already very useful information this one yes this is computed from safety yeah that's right but yeah but actually this quantity can also be defined in massive theory so in non-conform non-conformal theory and this is something I was trying to mention so so you can ask what's the relation between the hemisparsion function and this quantity in the non-conformal case and in that case there is a conjecture by Chekotin-Guyotin-Waffer based on on a computation in an explicit example which says that the hemisparsion function is a limit of the this quantity t t star amplitude where the anti-hormonal part of the mass mass parameter is set to zero but the hormonal part of the mass mass parameter is kept finite so so but it I believe it's still a conjecture there is no proof okay any question okay now I want to discuss very explicit probably the most important example so which is a Caribbean hypersurface in the projective space yes yes I think more general setting but yeah yeah but in that case like for example the projective space but on the other hand then it's not clear really clear what what what it means to set the mass parameter to zero so yeah okay the conjecture I think can be the conjecture can be straightforwardly formulated for massive deformation of safety yeah the example they had was yeah massive the Landau-Ginzburg model yeah so I want to discuss an example which is okay Caribbean hypersurface in projective space P n minus 1 yeah so this is important so if n is 3 the space is torus so elliptic curve if it n is 4 this is k theory surface okay polarized k theory surface if n is 5 this is a quintic Caribbean so you see that this is an important example and the gauge theory for this Caribbean is described by gauge group so the surface is described by gauge theory with gauge group u1 and n chiral multiplets which I denote by phi i and one chiral aspect which people usually denote by P and okay gauge charge is here plus one and here minus n so they sum up to the gauge charges sum up to zero so it starts by the Caribbean conditions I mentioned okay and okay I omit specifying the R charges and the super potential is P times polynomial in phi which is assumed to be homogeneous of degree n and you probably know that yeah for positive phi parameter the theory flows to nonlinear sigma model this target space the Caribbean and I want now I want to give you a matrix factorization for the basically the D6 frame in other words the structure sheet of the Caribbean maybe with some fluxes turned on so choose the boundary condition so that all chiral fields obey the Neumann boundary so in my notation this means that noise is noise is sorry everything and there is empathy and let us introduce fermionic oscillators well in the context of matrix factorization or algebraic geometry this construction is known as I don't know actually I don't know how to pronounce this but Kozoo I say Kozoo construction okay I mean Europe so somebody is going to explain how to pronounce it and so we introduce fermionic oscillators like this and eta and eta bar square to zero and choose a Creeford vacuum yeah Creeford vacuum such that okay eta annihilates the vacuum okay so then as champeton space we get two-dimensional space so spanned by the vacuum and this excited state okay and then as matrix factorization let us consider g times eta plus p times eta bar right then it's I think it's easy to see that this is p times gn so this is a super potential or times the identity of it okay good and then then you can assign appropriate gauge and R charges so that the the conditions on the matrix factorization I described yesterday satisfied now I want to compute the oh but I think but but you know to really complete the specification of matrix factorization I need to choose the gauge charges the gauge charge and the asymmetry charge of the vacuum so yeah let me choose the gauge charge of the vacuum to be um little n plus capital N over 2 where little n is an integer and let me assume that the R R charge of the vacuum is 0 then you can come then okay then the claim of first actually okay okay sorry the claim okay I don't know who found this originally so I cannot give you the original reference but the claim is that this describes the matrix factorization uh describes the the sheaf we could say the line bundle uh all this all n of n over the calabia hypersurface n so n equals 0 corresponds to the structure sheaf and uh for non-zero n this this sheaf is a structure sheaf twisted by line bundle by some line bundle which comes from the projective space I think most of you know that on the projective space there are various standard line bundles okay so then you can apply the formula for the hemisphere function and compute it you get a contribution from the boundary interaction which is also called the brain factor it is t sigma yeah in the calabia case it's a twisted f i it's a complex body parameter is not really normalized gamma of sigma to n gamma one minus n sigma um and and we can keep rewriting it we can keep rewriting it and I can give I could give explicit expressions but I want to explain that there is a sort of nice well if you rewrite it in a certain way you get geometric expression which involves the novel concept so if you keep rewriting it in some way in some way then yeah and this is for large real part of p so which is okay 2 pi psi so the in the large volume limit this can be written as the char the char character of the sheaf or the line bundle times e to the b plus i omega times some characteristic class called the gamma class of the tangent bundle now b plus i omega so in in the convention here it it is minus t over 2 pi i times e where e is the pullback of the hyper plane class of the projective space so it's a generator of the second cohomology group and and and there I can write I think I can write like this is or not yeah so now I need to explain this gamma hat this gamma character equals gamma hat so for for a general vector bundle e gamma hat of e is defined in terms of churn roots so I assume that okay you know characteristic classes and and x j are called churn roots and churn roots are such that the churn character is given by it is some of it is x j so now this characteristic class defined also this is the definition this is called the gamma class the gamma class was known or invented earlier by basically mathematicians people like Kostani, Irritani, Kacarkov, Konsevich, Panthev, these people and yeah and their considerations were actually motivated very much by the central charge that's a formula for the central charge obtained by solving the carfix equation but okay what Holly and Romo notice but that hemispathion function the localization computation hemispathion function naturally gives rise to the gamma function and therefore explains the appearance of the gamma class in the formula for the central charge so that's a nice story okay any questions so independence only comes from the exponential also so this is just the exponential I think okay and the remaining part for example above there is a minus sign as a sign and how does it come from the below difference of exponential how does it fit into the expression below oh this one yeah so this can be written as a sign right then the sign can be written in terms of gamma functions okay product of gamma function so then uh yeah everything yeah yeah it combines yeah these things combine um yeah what you have to do is that uh yeah so rewrite it and then take the evaluate the integral by residues and I think you then focus on the the leading leading pole and and and then then use using properties properties of E I think let's see so this is a pullback pullback of the okay hyper plane class in the predictive space and and let's see H satisfies some some condition this is yeah yeah so so basically this course this condition corresponds to evaluating the residue with some powers of sigma uh yes okay here here yes yes basically uh uh what I basically this what I explain so uh it's just a computation of the residue and yeah as I said okay you rewrite it in terms of sign and gamma functions uh and then okay then you so you have some residue integral and use Cauchy theorem but then you try to interpret the residue computation as an integral over of this uh in kohomology integral kohomology element you got this geometric expression uh large number of elements yes so it's possible to compare with some super gravity result uh right yeah so I said so this is uh this is a sort of an improved version of a previously known formula and the previously known formula had instead of gamma heart I think instead of gamma heart uh it had I think it's okay it was probably square root of a heart region I think and uh in the large volume limit let's see so up to the highest so up to several reading orders uh the the two expressions give the same result but I think like uh like to the reading like n minus two degrees or something they they give the same result but but if you look at higher corrections then they start to deviate but this is okay this is this explained well in my paper and also I think in the paper for holy and roman but but but basically uh but this story is due to holy and roman yeah I need to exercise okay any more questions okay now now what can we can change boundary condition of one field to deletion right so yeah so uh if you if you use deletion boundary condition of course we change the we need to change the one of the determinant so we get one of a gamma and uh but it's actually enough to in some sense it's enough to consider normal boundary conditions because as I said yesterday deletion there is a duality between boundary condition so the deletion boundary condition is a dual to no one boundary condition with some boundary interactions therefore if you include appropriate boundary interactions then uh then you can describe arbitrary boundary condition just using no man okay any other question okay very good um now okay I want to oh I want to say just a little bit about the interface just a little bit and leave everything else to the exercise so what I described is um basically path integral over the hemisphere but uh you may want to consider a situation where you have something that divides the spacetime into two regions so for example the sphere into two hemispheres so you have an interface here and you have some theory t1 and you have some theory some other theory t2 now you can apply some transformation you might call it time reversal or you might also call it parity transformation and map it to to this system t1 times t2 bar so now you have a single product theory with some boundary or boundary conditions or whatever uh so so uh in general the the statement is that the interface preserving b type supersymmetry uh can be described as a b brain in the product theory uh can you come up with an interesting uh interface in this way um I don't I don't know if you agree but I think the interface uh the identity interface it is already rather interesting and it's actually uh there's some work and in one of the exercises actually which was given yesterday uh I asked asked you to consider a matrix factorization for the product theory actually the the this is the uh product of two copies of the gauge series I just described and so I give you so I in the exercise I give you an example of matrix factorization and my the claim is that if you compute the hemispheres partial function for that interface then you get precisely the sphere partial function okay now I want to finish the two-dimensional story uh any any more questions okay now uh now I switch the I change the topic and I'm going to discuss uh four-dimensional line half bps line operators in four-dimensional n equals two supersymmetric theories and and as usual I give uh plan uh for this part of my lecture so I'm going to discuss line operator charges do I really have time probably don't yeah so uh I will probably not be able to cover everything in full detail but I'm going to discuss line operator charges uh 2d4d relation uh s4 versus s1 times r3 the relation between monopoles and instantons monopoles screening bubbling monopole partial function webs and so and uh as references I give you uh first of all this one uh this is a review paper I wrote about this subject and the actual calculations uh okay I give uh the paper from 2011 and there is also a recent paper by a group at Rutgers uh and this is also quite relevant okay so now I want to discuss uh basically the definition of line operators and also the charges uh line operators are very basic uh operators in gaseous because they can be defined for any uh gauge group or gauge any gauge cell so so so let g be the gauge group which is a compact regroup and let me denote by t script uh okay ball t uh the carton carton sub algebra and t star okay duo now uh I think you know that uh inside the duo of carton uh sit the weight lattice and there is a sub lattice which is generated by root so it's a root lattice inside the carton sub algebra sub algebra uh there is co weight lattice which is defined to be the duo of the root lattice and there is co root lattice which is defined to be the duo of the weight lattice now uh half bps this on line operator in four-dimensional n con n cos 2 theory is defined the expression trace in representation r of the path ordered exponential of the integral of i times the gauge field plus okay this this this is a convention but uh dr part of five you might you might put some faith here if you like some people do uh line element ds defined by the by the metric and the the integral is taken along for example a straight line straight or circle yeah so this restriction is necessary and okay here i'm i'm on for example r4 and this restriction is necessary to to preserve uh one half of the full super symmetry um and it's a it's actually in interest i think open question it's an interesting open question to clear out classify on which curves you can have a supersymmetric Wilson loops for n cos 2 theories for four-dimensional n cos 4 theory uh the classification of such curves uh was done by demasquish and piston using the pure spina formalism and i think it's an interesting question to extend the result to four-dimensional n cos 2 okay um now this is a okay this i take to be an irreducible representation irrep uh specified by the highest highest weight w in lambda w i think i'm going to have too many w's but okay um sometimes i use w for the highest weight that's why the Wilson Wilson operator and physically this uh uh represents um infinitely heavy half pps electrically charged particle i mean the world line world world line of such a particle so that's the Wilson loop and now uh you can try to consider the magnetic duo of this operator and so the so the magnetic duo will be a monopole and so so we should consider an infinitely heavy magnetic monopole that propagates around the contour in spacetime and that's the uh to hoof to hoof to line operator which i denote by t of b so b is the magnetic charge i'm going to explain yeah so so this is an infinitely heavy magnetic monopole and uh uh and this is defined by by singular boundary condition so so benny explained uh last week that uh in quantum theory there are order operators such as this uh Wilson line operator which is uh which is just a function function or functional of the fields in in the path integral now a total line total line operator is a prime example of disorder operator so it's defined by a singular boundary conditions and the field strength goes like b over 2 epsilon ijk xi divided by r times dxk which dxl you might also write it as minus b over 2 times the uh volume form of a two sphere using a polar coordinates yeah so yeah so so xi and xi and okay r theta phi so these are these are okay locally defined locally defined Cartesian and polar coordinates okay so so so so this describes the transverse direction to the to the line or the loop and the scalar in the vector marketplace goes like i times b over 2r now these expressions are valid for the vanishing topological theta angle if the theta angle is non-zero we need to turn on the electric field in order to uh i think in order to be consistent with the written effect right any question okay but now there there are very interesting yeah so so i'm going to explain uh in several steps very interesting i think i think is very interesting story about the charges about the charges of the line operators so there is a restriction on the magnetic charge b there is a direct quantization condition on the magnetic charge b uh to understand this well this is a standard thing so i think you you you've heard this in similar time probably uh so you consider monopole so so you only consider the transverse direction to the to the line so then there is a there is an s2 that surrounds the monopole and there is a Dirac singularity well if you use the gauge such that a goes like minus b over 2 times 1 minus cosine theta times d phi so the Dirac string is along theta equals pi okay and you can parallel transport a magnetic field around around this Dirac string and then you require that the field is or the wave function to be single valued and the condition is that the pairing the natural pairing between the magnetic charge b which is in the in the okay co-weight lattice well which which is in the cartoon so the pairing between this b and and some weight so this is not the highest weight of the rhythm but this is a this is an arbitrary weight for of a matter representation so this is taken from the weight lattice so the condition says that this the pairing this pairing is an integer yeah so it's important that w depends on the matter content so this is a weight of matter representation or or or a root because okay we always have a gauge field which transform in the adjoint representation so there is a restriction on b and let me call the lattice of such b lambda m so so let the lambda m denotes the lattice of b satisfying this this condition okay okay so this is the so this is the restriction on the charge of the total operator you can also consider dionic line operators so dionic operators are defined by first considering a total operator tv and then inserting a Wilson loop for the unbroken part of the gauge group so so once you insert the total operator this there is a choice of b and this b locally breaks the gauge group to the subgroup so so the the unbroken part is a commutant okay of the of the magnetic charge b yeah so so the the dionic operators are specified the charges are the charges are given by a pair the magnetic lattice times the weight lattice okay but then they are subjected to the wild group action so at the level of the gauge algebra i mean the the algebra of the gauge group the line operator charges are classified by lambda m times lambda w divided by the wild group so this classification is due to essentially due to Kapustin well he was really considering n equals 4 but so so this is the classification at the level of the algebra but there is a there is an interesting refinement of this classification which reflects the global structure of the gauge theory and in order to understand that we need to consider the we need to consider what happens when you have more than more than one line operator so consider two line operators okay two line operators specified by pairs of charges b 1 w 1 and b 2 w 2 and imagine that one operator moves around the other so that the surface sweeped by the moved line operator has a linking number one with the line operator that is sitting okay i'm not sure if i'm explaining it very clearly but basically this is of course the this is something originally considered by truth okay okay no no no i can't draw yeah yeah so i can't try so you for example you you can have a sort of consider a line and you consider the transverse direction and then the transverse three-dimensional space so one operator is represented by by a point right and then you can you can consider a line the other operator that for example goes like this and then you can you can rotate this part to to this and then come back i think this represents the motion of one line operator i described then the claim is that this picks up a face basically will will some line that goes around the drag string picks up a face so the condition that this is one this is one and when this is satisfied the two operators are said to be mutually local so originally told to introduce told operators to classify possible phases or get serious and in that context to talk about actually not really considering mutually mutually non-local operators okay but the the modern the modern point of view put forward by okay aharoni, zibern and tachikawa the the modern point of view is that in order to specify theory what we need to choose choose a maximal mutually local line operators so i think this paper is of fundamental importance in my personal opinion and yeah so the claim is that in order to really specify theory yeah you need to choose a maximal set of line operators that obey this constraint okay and they argue that the cell has to be maximal in order for you know for modular invariance of the the forward major theory to hold for example so if you put put the forward major theory on the forward major torus and you can describe it using some coordinates but then you want to choose different set of coordinates and you do large different morphisms large gauge large coordinate transformation and and then the the invariance of the description i mean the lack of gravitational anomaly requires that you choose maximal set of mutually local line operators that's that's that's their claim there are other characterizations of this choice for example they they show that this choice is equivalent to a choice of discrete set angle and also you can replace this in terms of the centers in terms of the center of the gauge group and so the choice is corresponds to a choice of maximal isotropic subgroup of yeah so okay maximal isotropic subgroup the center of the gauge group times center of g g star so there is a there is a pairing between the center and its duo so and the isotropic isotropic means this condition any question yes yes you have a simple information for why the wild group acts diagonally well in principle yeah while you can choose wild group to act on lambda m and lambda w independently but i think it's more natural for the wild group to act on both um and also i i said that lambda w specified the Wilson loop for the unbroken path so the choice of the Wilson loop is correlated with the choice of b so so it must be simultaneous action okay any other question good okay i have 20 minutes just okay i want to say a little bit about the 2d 4d relation so from fielders uh we had a we had an explanation of the agt correspondence in particular he explained how the agt correspondence works or the 2d 4d relation works for uh the a1 theory of class s and uh okay i would not be very quantitative because i don't have much time but uh basically yeah so i want to consider uh so a1 a1 type a1 theory of class s and this corresponds to 2 m5 range or 6 men general n equals 0.2 theories on uh puncture removal surface cgn so g is the genus and n is the number of punctures okay and you can consider uh so for example you you can have a surface like this you can so you can introduce a plant decomposition and then you get some generalized quiver gays theories with gauge groups as you do and the plant decomposition and actually uh you also need to draw a trivalent graph but basically my plant's graph gives you gives you some some gays theory uh Lagrangian description of the quantum field theory uh and for example change of plant decomposition corresponds to the general this in reality that's what Peter explained right and uh now there is a sort of a topological story for the charges of line operators and the claim is that uh homotopy class would be class of the closed curve on the ribbon surface corresponds to a charge of a half pps line operator yeah so this correspond is actually very uh quantitative explicit and also there is it is known how s s or modular transformation acts only on the parameters of the curve curves called the Dinterson parameters yeah so the the classification classification of line operator charges it has a nice correspondence with the classification of closed curves on the ribbon surface uh yeah so you you can you can read my review for explicit explanation and uh this correspondence was uh found by Jerker, Morrison, Morrison and myself but then um after the paper by Afharoni Zaba no Tachikawa that Tachikawa had a single authored paper uh where he showed that uh the refinement actually uh also applies to to here and uh in order to specify the the theory in order to specify the four-dimensional theory you need to choose you need to choose an isotropic subgroup in this case in terms of the ribbon surface uh of the first cohomology well he was considering the case with no puncture and the coefficient group is 3g minus 3 so uh so there is a correspondence between between line operator charges and line correspondence between line operator charges and closed curves and the refinement of the classification also goes through okay this this is homotopy so the line operator charges to make some algebra don't come into the register but but then you wrote the homology before you know so yeah so so this somehow this has something to do with the the center and the yeah and the algebra line operator is the structure of the the algebra line operator is more complicated than cohomology or these topological things i don't see a direct suggestion okay uh right yeah so so so so okay so this part this this part uh is by uh this paper from 2009 deluca morisan myself uh this this part is a single author paper by tachikawa i think it ended soon after the paper with aharoni ziba ziba okay i have 15 minutes okay so now now because this is a school on localization i want to say something about localization of line operators local localization correction with line operators especially tohuf's operators now line operators half bp's line operators can be placed supersmetrically supersmetrically on s4 or its deformation s4b which uh pilas discussed or s1 times s1 times r3 let's see so you already know okay what s4b is but okay somehow it's better to write entity understanding uh you can have two places where you you can have two you have two locations where you can place line operators so you can have uh one s1 and you can have another s1 which i denote s1b and s1b inverse so in the 1 2 and 3 4 planes um and and uh you can put loop or line or line operators okay and localization has been essentially localization calculation has been done so for the western loop the localization was done by okay uh peston hammer hosomichi and uh for with the total operator uh gomis peston and myself did the localization calculation for b cos 1 and for b not equal to 1 there is a guess for what the answer should be based on the agt correspondence so uh yeah so to some extent there are results and now i said uh you can use agt for the computation of the uh expect loop operator expectation values on s4b right and this is because there is a there there's something called valende operator valende loop operator in c of t for example in duville and toda theory and you can you can use i don't i'm not going to explain what it is but you can use this this gadget to compute the expectation value of line operators so this was done okay for duville this was done by julka uh gomis uh myself intentional and also uh are the guy got to go off uh tachikawa valende and also for for for toda theory uh this was the calculation was done by gomis and reflock gomis and bruno okay uh now so so s4 s4b is good s4b it it can be done um now remember that uh s4 s4b function function contains uh instant function function right and actually also is the anti-instant function right but the instant function function was originally defined by nikita uh as we heard uh this morning uh on on the omega deformed flat space so r4 epsilon 1 epsilon 2 omega omega omega background on r4 and similarly you can ask uh what is the most natural space time to where to compute the expectation value of the line operators and the and this picture this picture suggests that we should just look at the local neighborhood of each s1 here and so then the geometry locally looks like s1 times r3 but with some twist parameter so r3 is now twisted around s1 okay uh so let's see so then the if you compute the line operator or total total expectation value of s4b this contains some contribution from the from the the circles which i call equators and uh and there are some contribution which would be the analog of the instant of partial function and i i call the contribution z mono so because okay monopole partial function so i want to explain briefly how to compute the partial functions associated with uh monopoles actually a singular monopole and the useful trick the useful method is the correspondence between uh monopoles and instant ones and this correspondence was found by Cronheimer his master thesis at oxford Cronheimer his master thesis was not available for a long time and uh pester and i uh kept bugging him and he well he and he never replied but he eventually uh posted his uh paper on his web page so it's available now Cronheimer's correspondence so uh so so monopoles are solutions to bogomolny equations star 3 f equals d phi where phi is an adjoint scalar now instantons are solutions to anti self-futurity equations which looks like okay script f plus star f equals 0 and Cronheimer uh showed that instantons tap not space invariant under under u1 action corresponds to uh solutions to the bogomolny equation with direct singularities so singular monopoles actually i gave details of this correspondence uh again in one of the exercises so you can so for equations you can look at the uh exercise i just maybe i just explain what top not space is what top not space is yeah so top not space what is top not space actually multi-center top not multi-center top not so it is a scalar a hyper scalar manifold which has a metric i think gibbons hooking form v inverse times d psi plus omega square so in Cronheimer's convention psi is an angular variable with with characteristic 2 pi rather than 4 pi it's nothing physics and v is some harmonic function r free omega and v actually satisfies the bogomolny equation yeah so this is uh this is uh the space with this metric called multi-center top not space and in one of the exercises i give you yeah what the correspondence is basically and ask you to show that the bogomolny equation and the anti-safety equations uh equivalent for the given expressions okay so this is uh Cronheimer's correspondence now i don't have much time left so i'm explaining explaining these parts just in words for the instant partial functions contributions uh came from small instantons so so instant of zero size so in the edgman construction these instantons correspond to the fixed point of the torus action okay this is what we heard this morning and the same story holes in the in the monopole case namely we have direct singularity so we have a talked singularity and on top of that we can have smooth particles talked monopoles and these particles also monopoles can get close to the singular monopole and and and get attached and eventually they can at least they can partially screen the the original magnetic charge so the magnetic charge i mean b magnetic charge b can get get weaker and i call yeah so the original magnetic charge is b and uh there is a screening by smooth particle monopoles and it gets weaker and the smaller magnetic charge i call v uh little v now this phenomenon is uh actually corresponds by a cron cronheimer's correspondence to the small instantons so small instantons in top nut they reduce to they descend to uh monopole yeah so monopole monopole screening which also sometimes which is also sometimes called monopole bubbling okay that's explanation uh now uh you can then yeah so then cronheimer's correspondence can be used to give a description of the monopole modular space because uh for for instantons there is okay for instance on c2 there's adhm construction now we are interested in the modular space of instantons on top nut but one can show that in localization calculations only components uh only components with fixed points contribute and and uh circuit circuits showed that uh the component of monopole component of instanton modular space space with appropriate fixed points uh isomorphic as a complex manifold to the instanton modular space on c2 well in the case of a single deluxe singularity so we can actually use the adhm construction for instantons and apply uh this cronheimer's correspondence so we can take a invariant part of the adhm data uh under this u1 now in the it's different it's different yeah so num construction is it involves an well infinite dimensional data so it's not yeah so far it's not it's not useful so in the recent paper uh by brendan day and moore they showed that uh this u1 invariant part of the adhm instanton modular space is actually nakajima quiva variety and so you can uh use this description and as in the discussion of pierers uh where uh he used uh quiva supersymmetric quantum mechanics to compute the instanton partial function it's again possible to use supersymmetric quantum mechanics obtained by uh d-brain construction to compute uh uh monopole partial function good uh good now uh there are yeah i was planning to give some general formulas for the expectation values of line operators but but uh i just want to give one example of the result of localization calculation which again appears in one of the exercises so if you consider s u2 if you want we can give you maybe tomorrow no no thursday maybe half an hour more so that you can actually you do this a little bit better in the exercise session to have shorter exercise session somebody asked me about the the algebra of wilson and to hoofed loops as originally described by to hoofed so i want to mention uh i want to comment on that yeah so um in 1978 in one of the papers in nuclear physics b uh to discussed an algebra of wilson and to hoofed operators talked to loops and uh well the the the statement of the uh the algebra is as follows so let's consider wilson loop in some gay theory well okay let's say uh in in an s un gay theory and uh wilson loop in the fundamental representation okay and uh and this is put place on some curve on some curve and let's consider a to hoofed loop uh on some okay other curve and okay and he was to hoofed was considering a non super symmetric theory okay um i think he can have some matter of use and the definition of to hoofed operator was more okay somewhat different but essentially okay so i what i explained was a more modern definition of the operator now um now we want to consider the following situation so so c w and c t are curves one dimensional curves but uh let's first consider the situation where these curves are hope hope linked r3 three dimensional space which you can think of it which you can think of as uh constant time slice so t at t equals zero so x1 x2 x3 yeah so what is hope link hope link is something like this and i want to consider the hope link so hopefully in the constant time slice so this is t this time and this time t equals zero right and we consider wilson loop here and uh to hoofed operator here and let's see then he claimed okay then then we can consider uh this uh small displacement of the curve c w uh in the now uh positive time direction or negative direction so then uh if you do that then the wilson operator and the toto operator uh not in not on the constant time slice so then so there is an ordering and so now the statement is that as an as operators acting on the hillbill space the wilson wilson loop and the tohoofed loop uh obeys the following commutate well sort of commutation relation so this is for uh su and real theory with minimal charges good so this actually means that the the fundamental wilson loop and the fundamental tohoofed loop are not mutually local so uh so so this may look like a contradiction to what i said yesterday right so so uh on each so so a very fine uh counterfeit theory should have only mutually local uh line operators and uh so the the situation toto uh was considering does not correspond to uh to such operators so so what's what's happening what's the modern interpretation of his discussion uh so the claim so so so the modern point of view is that uh uh w and t uh obeying this commutation relation uh cannot be uh both genuine line operators so so at least one of them has to be the boundary of a topological surface operator so i want to explain that so i've been i've been saying uh su and gauge theory but it's what i really meant was uh gauge algebra the gauge algebra was uh su n and now i want to uh consider the concrete situation where the gauge groups really uh group su n so capital su n and then then well the obviously i think it's obvious obviously the wilson loop is a well-defined line operator and then the total operator t should be the boundary of a topological surface operator and how do we actually see see this let's see so the the way to see this is to draw uh to use uh yeah to draw some figures here so okay so we have a total operator okay we are going to consider yeah okay so so i said total operator is fixed at t equals zero and we want to move wilson loop to the positive time direction or in the negative time direction and this total operator uh so in some gauge this actually creates creates a sheet of dirac strings so let's see so you might call it uh dirac this this color is probably hard to see i'm sorry but so this is dirac sheet so so there is some sheet now uh actually so i can even now uh explain how to get this algebra so uh the dirac sheet create so so dirac sheet is uh okay sheet of dirac string and then when the wilson loop go around the dirac string then you get phase factor and that's the origin right so if the wilson loop is uh let's see the wilson loop is placed uh around time earlier than the total loop then uh there is no phase but if the wilson loop is placed well after time equals zero then it picks up an extra uh phase here so that's how you get this this to algebra and uh and you you also see that the dirac string it's uh so in some sense a topological object so because by changing the gauge you can move the location of the dirac string or dirac sheet so so indeed in this picture total operator is uh is the boundary of a topological surface operator so that's good um well then let's consider another situation for example uh g equals su n mod uh z n z n uh okay so in this case uh uh uh uh in this case uh what happens is that the wilson loop becomes boundary over topological okay surface operator and okay and and i'm restricting to the case where the topological theta angle is zero now okay how do you see that the wilson loop is at the boundary of a topological surface operator so so we are in the uh no supersymmetric situation so we have the wilson loop in so it's in the fundamental representation now um and the fundamental representation right so the fundamental representation is not a very fine representation of this gauge group okay uh because uh yeah because it transforms under the this action of the center uh so so so so naively you would think that uh this object is not allowed and that i think that's what usually people say what written in textbooks but actually yeah um but if you think about it this this uh expression makes sense even in the case uh yeah so the representation is not is not a very fine representation of a group because uh the whole thing is express expresses just in terms of the the connection which takes values in the rearrangement so so and you do expansion right so there is nothing wrong with this expression in some sense and indeed for example if you are on r4 uh if you are on r4 and you consider uh some some circle you can consider such a wilson loop and it's a gauge invariant nothing wrong with that but uh but when i say this uh is gauge invariant the reason it's gauge invariant is that the curve you consider so c w this is uh so so if this is contractable then it's gauge invariant so so it's so this the gauge invariant of this operator is actually uh a background dependent statement so so if you if you uh in a spacetime which has non-trivial one side one cycle and if you put a wilson loop along non-trivial along along the one non-trivial one cycle then there is a large gauge transformation which transforms this operator so so then it's not it's no longer gauge invariant so so so you see that uh so in order to define this object in for for a representation that is not well defined for the group then you have to you need to specify in something so how to fill in fill in this uh yeah so of which surface this curve is a boundary so so in this sense uh this this wilson loop is uh is a boundary over topological surface operator okay does this make sense okay uh yeah so so so so you might have thought that okay maybe uh yes why the the representations are not well defined because there are some parts of the some representations of the s u n will be sent to the representations of of the quotient uh okay so so the question was why is the fundamental representation ill-defined for fundamental representation ill-defined for uh s u n uh for s u n mod z n okay so so what is s u n mod z n uh so so so so an element of this quotient group is an equivalence class right so so g is an element of s u n but g is identified with itself multiple by by this phase right now so so so so consider consider okay something phi uh in in the fundamental representation so so phi transforms by g times phi but but this is not this is not a very fine this is not a very fine representation of s u n mod z n because the result of the action of the group element so this is not the same as the the result of the action of g so so so the action of the result of the action of the group element depends on the choice of representative in the equivalence class so this means that uh uh uh yeah the fundamental representation of s u n is not a very fine representation of s u n mod z n uh am i clear okay very good yes so are you saying so in g equals s u n mod z n some some rules should be regarded as top-level service right yeah so in the s u n theory some tools should be regarded as yeah not all not all right what is the condition which one should be so for the for the uh so it it has to do with an artist so for example for the wilson loop yeah basically the wilson loop for the wilson loop that the the number of boxes in the young diagram it gives a condition for the appropriate to be well defined and there should be a corresponding statement for the uh toad loop so so toad loop is specified by b right and it's an element of the uh co weight lattice and then uh right so so so for given theory uh such a total operator is a genuine line operator if uh the charge is uh in some sub lattice of the co weight co weight lattice yeah so and and this sub lattice is this sub lattice is uh uh this sub lattice is picked by the choice of the the global global property of the gauge group so um so yesterday the choice of the precise gauge theory yeah the choice of the precise gauge theory was uh uh explained in terms of the choice of uh mutually maxima maximally mutually local line operators and so so uh in some in that language yeah the the choice of sub sub lattice is the choice of uh of the theory let's see um uh and and there are other characterizations in terms of this discrete theta angles and so yeah um and of course the there is direct direct condensation condition so the sub lattice has to be inside the uh magnetic lattice i mentioned yesterday and but the but the sub lattice is not necessarily the same as the magnetic lattice i defined yeah uh yeah so so i want to say that uh yeah so to toad was not stupid okay um so so i mean what he did was very physical and uh what he did was much more important than the mathematical physics we are doing here and and and and and also on his argument his result uh demonstrates that uh yeah it's also important to take into account uh uh line operators that are risers it's also non-genuine line operators that are risers the boundary of uh surface operators okay good uh now i want to uh come back to yeah what i was discussing yesterday and uh yes yes yes yes so the drag string is uh uh so it's a sheet so it's a two dimensional object right and in the time so in the time direction it uh goes in the positive time directions but it also has a spatial component along the city so so so topologically this drug sheet is just uh yeah half line times ct okay and uh now yesterday i so so in words actually i covered most of what i wanted to say and uh so i think i'll be repeating myself uh to some extent yeah okay what the and actually i don't have too much time anyway um yeah maybe uh i want to say that uh so yesterday i explained the cronheim as correspondence so there's a correspondence between instanton zone top-notch invariant under some ur action and uh singular monopoles on r3 and but i i think i did not really say uh what this ur action is and okay i'm not i think i'm not giving you the complete uh description uh now but uh i think i should at least say that this u1 which i denote by u1k acts by shifting the variable psi by a constant and remember that the the the metric of the multi-center top-notch takes this uh given hooking form and psi is is an angular coordinate is periodic to pi and yeah so the u1 action the relevant u1 action acts on the angular coordinate coordinate psi and it also acts on the gauge bundle equivalently good and then um yesterday i did not have time to write down some formulas which i was planning to write so um now i i want to use this occasion to write the formulas so for the truth to live truth to live on uh s4b so deformed force here uh yeah so uh so this is a guess for b not equal to 1 but for b cos 1 this was shown by uh uh gomis uh pester and myself uh so this takes the form so there's an intake the usual integral uh for uh is for partial function and then there is uh summation over v so so this is a uh screening or a bubbling contribution so yesterday i explained the rough idea for how this comes about so smooth monopoles get attached to singular monopoles and weaken the singularity and this little v is uh the magnetic charge after such screening occurs and uh there is uh contribution from the from the region from the region uh in the neighborhood of the the loop operator and which and it locally looks like s1 times r3 so uh i write it in this way and there is contribution that that is an uh analog for monopoles of the instant of partial function and then there is a usual uh sphere uh one loop part and the instant partial function mode squared so so on the deformed force here uh this uh okay this is a form of the uh expectation value of the torque operator uh now i also want to give a formula for s1 times r3 and uh there is actually a omega deformation sort of omega deformation parameter on s1 times 3 because r3 is fibered over s1 so so when you go around s1 there is some special rotation along uh third axis now uh this can be written as as a supersymmetric trace in some hillbill space so maybe i should emphasize this also um line operator which is inserted in the time along along the time direction uh so so that is different from tohoots case to for tohoots loop operators were uh in the spatial direction but now i'm considering a line operator along the time direction which is this one uh so so so line operator such a line operator modifies the hillbill space okay so it's not an operate line operator in that situation is not an operator that acts on the hillbill space but it rather changes the hillbill space and and we take the trace there and uh uh so so the path integral we are considering can be interpreted as a supersymmetric trace so here h is the Hamiltonian r uh r is the radius of the circle uh lambda is some lambda is basically the omega deformation parameter j3 is uh rotation operator i3 is the su2r rotation operator and the capital f sub f is uh the flavor generator and m sub f is in the fugaz now this if you do localization on s1 times r3 what you find is the following structure again there is the same sum that uh that take account that take into account the screening of babbling contributions and there's 2 pi i b b there's some pairing oh i don't okay and okay and then this b this b depends on one of the uh real scalars in the vector marketplace and also uh it contains the basically the graph photon so the the dual of the gauge field the three-dimensional gauge field and then there is a one loop part okay times the monopole function function part okay so what i okay maybe i still be important but okay i can i i give the parameter dependence yeah so it looks like this so in particular okay so capital b enters here but not here and and and little a is a combination of one uh the other uh real scalar in the vector marketplace and and uh the wilson wilson line so i mean the horn me along this one no it's not over counting uh so for example if you use the the equivalent index theorem the one loop determinant uh receive contributions from from specific locations on s s for b so uh yeah so so s for b one loop determinant this is contribution from the north and the south poles and uh yeah s one times r three one loop determinant this is a contribution from the well sort of the equator or the location of the loop operator so the equity contribution does it appear on the second line it's a look ah so so you mean yeah so basically this is the same yeah so so yeah yeah thank you so the point is that uh these these quantities appear uh on s one times r three with some okay with arguments shifted yeah and then i i give one okay a couple of examples concrete examples i think yesterday right and they are also in the exercise and and yeah you can also compute the model product okay maybe i should say okay you get the model product because these line operators realize deformation quantization of the hitch in modular space and there's an interesting story when you dimensionally reduce to three dimensions okay thank you