Daj. Zvom, ki sem poznala, sem poznala, da je vse klasifuje supersymmetrično geometrije, ko vse izgleda, od tajsvom z rovnju spinor izgleda z minimi guts, in drugih, da sem poznala, da sem poznala, korespondin to a left-ended spinor, whenever the manifold is complex. Or you could also have a right-ended supercharge when n is complex, but if the complex structure is anti-serve tool. I should point out some references about this. So there is a paper by Claret Tomasiello in the Farone, FTH, and also another reference by Dumitrescu, Cyberg and myself. So one comment that we should make about this case of one supercharge on some generic complex manifold is that we have to turn on some, generically we have to turn on trivial U1r gauge field and therefore the resulting theorem might only make sense if the r-charges of the fields are quantized in some way. So for instance it definitely makes sense if all the fermions have r-charges which are odd integers and the bosons have r-charges which are even integers. But depending on the manifold it might be that other quantizations are allowed. So generically the r-charges should be quantized. And then the other comment is that there are also manifolds which we don't have to turn on some non-trivial U1r bundle. One example is a three times this one. And then the r-charges of the fields can be whatever they want. But there might still be more data that one has to give, one has to choose some spin structure or some other similar discrete data. We also discussed a little bit what happens if one requires more supercharges. So I am not going to comment very much on the case of two queues with the same corresponding to two left-handed spinors or two right-handed spinors but you can find some, well, the relevant discussion in this paper. As I said, in the non-compact case we don't have an answer for what is the classification of the corresponding manifolds and for the compact case one is left with very little namely, like the four torres with the flat metric, the s3 times this one with the round metric and the k3. So that's a very short list. And for the case of s3 times this one there can be some quotients of it. And the other case I looked at just yesterday was the case when one has one solution zeta and one solution zeta tilde. So the four correspondingly there will be one charge, one supercharge of r charge minus one and one supercharge of r charge plus one. And then as I said this case is better analyzed thinking about the killing vector zeta sigma mu zeta tilde which turns out to biolomorphic both the complex structure arising from zeta and the complex structure arising from zeta tilde. And I also brought down the form of the metric which is the most general metric compatible with this structure. Ok, so next we'd like to make some discussions about background vector multiplets. So as we discussed when your theory has some u1 symmetry some flavor symmetry then you can couple it to background gauge fields. So what I'd like to discuss is suppose we are on a manifold which allows for one supercharge q which is corresponding to a left handed spinor so then let's suppose also that our theory has some let's say u1 flavor symmetry we want to know what kind of background gauge fields we can introduce on top of this background which would preserve the same supercharge. So we have already discussed what the background gauge fields so in one of the past lectures we already discussed what is the relevant equations that we have to satisfy and so our background gauge field is comprised of the gauge field mu and an auxiliary scalar d there are fermions but those are set to zero in the background and the supersymmetry the fact that this background preserves supersymmetry stems from the variation of the gauge genus which should be zero and that gives we only have one zeta corresponding to one q we only have one equation that we have to satisfy where f mu nu is the field strength corresponding to mu so this should be zero and now using the fact that you know that there is a complex that this z gives rise to a complex structure you can rewrite this equation as telling you that the anti-olomorphic part of f and that very good I can just am unable to copy from my own notes then ok so we have so this is the first condition and the second condition is that d will be determined by contracting the field strength with the complex structure ok so if we are in a situation where there is also another super charge and we want to our background gauge field to also preserves that one then we would have more restrictive conditions so let's give an example so in particular there is some interesting family of complex manifolds which are called op surfaces so let's talk about op surfaces so these are complex manifolds that are difumorphics to s3 times this one and they can all be described as the a quotient of the two-dimensional complex plane the c2 minus the origin by secret group and now we can describe what this quotient is so there are two cases the first one which I think goes by the name of primary op surface so we will introduce complex coordinate w and z on the two complex planes and then these are identified as pw and qz p and w are complex numbers which are subject to this constraint so they are less than 1 so this is a contraction and p is less or equal than the absolute value of p is less than equal than the absolute value of q and they are both greater than 0 so this give rise to have two complex dimensional family of complex manifold all difumorphic to s3 times this one and p and q are the complex structure moduli of this family so different p and q correspond to distinct complex structures then there is so this is something that we will discuss more then there is the other case which is and I want to have a lot to say about this maybe we can have some discussion after the lecture with people that are interested so in this case the identification is the following so there is a positive integer n and q again has to be such that this norm is in between 1 and 0 lambda is any complex parameter and n is a positive integer so natural number so one comment here is that it would appear that again this would be a two parameter family of different well that the complex structure would be parameterized by two moduli one being lambda and the other one being q but that is actually false because as you can realize very easily if I rescale the coordinate w and z I can rescale away lambda so lambda is a fictional parameter it can be whatever it cannot be 0 so maybe I should take out 0 from here but otherwise you can set it to be whatever you want but q is instead complex modulo and there are several of these branches and they are like indecised by this integer ok so for the let's discuss a little bit more the primary case so case a so we can write down what p and we can give some parameterization for p and q in terms of some real positive number beta and some angle theta and similarly for q oops e to the minus beta q plus i theta q so again because of this conditions we have that beta q and beta p are positive and that they have to satisfy the following relation while p and theta q are just angles and then in terms of this in terms of this beta p and beta q we can write down some parameterization for the coordinates w and z so we can write w as e to the minus e to the minus beta p plus i theta p times x times cosine of theta over q e to the i phi and z will be similar e to the minus beta q plus i theta q x times sine of theta over q and then there is some other angle so where phi and chi are angles in between 0 and 2 pi and theta goes from 0 to pi so you can check that this is a parameterization of c2 and under this under this identifications what you have is that x is identified with x plus 1 so x is a coordinate along a circle and a while phi chi and theta parameterize s3 so indeed this this thing is well defined because so this gives a parameterization so when you impose these conditions and the fact that these are angles this parameterizes a fundamental plane under this identification and you can check that this is like indeed as 3 times s1 because it satisfies e to the 2 beta p x times the absolute value of w square plus e to the 2 beta q x times the absolute value of zeta square equals 1 we can also write down a metric on our up surface and we can actually check that we can write metrics which are of the form that I wrote yesterday for this manifolds in which you can preserve two supercharges, actually not just one of opposite R charge so indeed we can write something as follows and you can check that this metric is invariant in identifications that are written above so this is a good metric on the up surface are there any questions so from this there is a killing vector dx this just corresponds to translations along the s1 and you can also write down anulomorphic killing vector actually I am going to write down the corresponding one form which therefore will look anulomorphic so this is the anulomorphic killing vector that we called k yesterday and you can indeed check that k square is equal to 0 and that it commutes with k bar so then you can also check that by using this k and this formulas that I gave yesterday reproduce you can find out complex structures which do correspond at least one to that thing over there ok, so we discovered that we can all on any primary of surface so on any primary of surface let's call it mpq we can preserve two supercharges q and q bar with oppositor charge and now if you think about this a little bit more you can convince yourself that these parameters p and q's are actually the geometrical implementation of the two parameters p and q which appeared in the formula for the index so remember we had a formula for an index this was a trace of minus 1 to the f times e to the minus so the partition function z on mpq will be the index that we defined yesterday with fugacity is q and p and moreover if I have some other flavor symmetries then I could add background gauge fields along this one and those will give other fugacities to the index yes, so there is there is a question about ok, let's be maybe a little bit less so there can be subtleties with perfectors ok, so now I wanted to say this just because somehow is I think a good introduction to the next topic of discussion which would like to understand how various supersymmetrical observables depend on the geometries of these spaces that we can place them in so for instance there is one choice of metric which works I could choose many many other metrics which are compatible with any given complex structure on some hop surface but this argument tells me that in the end the result is just going to depend on this p and q which are the modulate that characterize the complex structure and moreover it's not just any dependence on p and p of complex parameters and they appear holomorphical in the answer so, well this tells us that maybe a good hint that the correct answer is that at least in this class of theories the partition functions will depend only on the complex structure and not on say the choice of metric and moreover that the dependence on the complex structure should be holomorphic in the complex structure modulate so, given this example this is just a wild guess but as we will see this is actually what is going to happen but in order to make a sensible state but we need some definition of this index where both q and q bar will appear on p and p bar and then we will somehow demonstrate the dependence on p bar and q bar but the formula which you wrote last time only contained p and q so there was no question could you just write again the definition of I yeah, I only contained q and p I agree yes, yes, yes ok, so if you want this partition function like which in principle could depend also on p bar and q bar like turns out to be holomorphic and and then I guess that tells us that this index does not have dependence on I mean I don't know, I guess this comment maybe if I could ask a phrase question differently so how do I know that the partition function in the manifold mpq is actually an index of something yes, that's right I would like to see a circle which is separated and then some so so you have a separated circle so you could define a theorem whichever squash fears remains but does pq act on this s3 what do you mean act they define the geometry of the s3 they define the geometry of the s3 so once you reduce along this one so you will consider some trace of the Hilbert space of states on some squash sphere where the geometry of the squash field depends somehow on p and q now I'm not sure it depends I think it will depend on the ratio between p and q bar if I remember correctly and and then where the other parameters enter is in the way you identify the fields it's going around the s1 so then is that trace on the Hilbert space which itself depends on p and the operator which acts there also depends on p that's by some miracle which is not let's say in the elliptid genus where we have this standard Hilbert space an operator which depends on q bar but the index does not depend on q bar so that's more subtle any other question on this before I go into ok, so I guess it's time to erase I don't want to erase that ok, so I'm just taking this as a definition so this is c2 minus the origin quotiented in this way so the interesting fact that I actually is that like these are all these are like this classification there is like a classification of all the four-dimensional complex manifolds with a difomorphic to s3 times s1 that is not very obvious statement I think it was due to codira ok ok, so we would like to understand how this partition functions or principle some other supersymmetric observables depends on the geometry of the space say m so first of all what could it depend on for instance it could depend on the choice of metric or because we are dealing with complex manifolds it could depend on the choice of complex structure moreover I'm not sure you remember this but at some point it could depend let's say on other background other sugra background besides the metric for instance we had a formula for v mu this conserved background one form which appeared in mm supergravity and this was determined in terms of the divergence of the complex structure plus some piece which we called u mu and we said that u mu had to be polymorphic and conserved so now the question is can the partition function depend on what we decide to use as u it's a freedom that we have to that we can make if you want some non universal coupling to the geometry and you need to know if the partition function depends on it or not and moreover we could also have background gauge fields so we could also ask how does it depend on background gauge fields or other backgrounds so in order to study these questions so I will not present here the more rigorous proof I guess which I can tell you about if you are interested but one thing you could do is somehow try to introduce appropriately twisted fields, well at least for theories with some Lagrangian descriptions which will be like forms with which will be olemorphic or antelomorphic depending on the fields you are dealing with and then rewrite the Lagrangians that come from supergravity in terms of these twisted fields and check that somehow all the terms, if I do a variation of say the metric keeping the complex structure fixed then this is always q-exact and so on instead what we'll do is something more pedestrian which is we'll just consider this manifold to be the manifold where we are to be very very large so that it's very given point it's more or less like flat space and then we will analyze this perturbation from flat space using linearized supergravity and then you could say well this doesn't prove anything and well indeed it's just an indication that the results that I'm going to present are correct but well if one weighs one's arm very robustly one can say that the fact that we have a scalar supercharge means that we can just analyze this problem near flat space and that should be enough okay but this is maybe an argument which is not very it's not very good so you can also do this other twisting procedure which however I don't have time I think to discuss okay so so the idea is that we take this manifold to be very very large and then we study how the Lagrangian of our theory responds to this small change of the geometry and we know what it does so there will be the change in the metric which will come to the energy momentum tensor then we will have the change in the connection for the asymmetry which will couple to the conserved arc current then we will have the change in this conserved vector field V mu which will couple to the connection which would give rise to the string to this f which enters into the string current and well then if we also have auxiliary if you also have background gauge fields then like we could imagine varying those and this will couple to the flavor currents and this will come with the variations of the auxiliary components of the gauge fields which couple to the appropriate scalar operators in the linear multiple for the conserved current so these are terms that arise in background gauge fields and those are terms that arise from coupling to supergravity at the linearized level so again these are not really independent because we want our geometry to be super symmetric so that implies that delta mu and delta v and delta g are all related to each other in the way that we discussed yesterday so there are explicit formulas v, a in terms of the geometry ok So what about the domain wall current? Yeah so that, no the domain wall current is not there because we are considering theories which couple to which have an R-multiplet and which couple to minimal supergravity if you were to redo this in old minimal supergravity well then in this you will also have this complex scalar which couples to something related to the domain wall current Ok So any question? Ok so that's the logic that we would like to pursue Ok so before going to the more complicated part which is understanding what happens when one changes the supergravity background let's make some let's see what one can say about the case of the background gauge fields which is somewhat somewhat easier so so we have to remember the formulas that have wrote up there so those are the conditions for a background to be supersymmetric so that means that the antelomofic components of the field is trying to 0 and d is determined in that way so f i bar j bar is equal to 0 and d, if I write it in complex coordinates will be equal to minus 2 i f w w bar plus f z z bar Ok so these are the conditions for the background field to be supersymmetric and ok locally ok so there is a there is a Poincaré theorem, lemma whatever it's called for d bar operators so locally this thing means that the corresponding gauge field a i bar is d bar of some scalar and if I am allowed to apply the flavor gauge symmetry then this would mean that a i bar is locally trivial so I could set it to 0 locally by using complexified gauge transformations now we want to very little bit this we want to consider a small variation of this background we could start with just a flat background so there is no gauge field well then the conditions are that I don't need to rewrite them they are just that the field strength of this small variation is equal to 0 so d i of delta j bar minus d j bar of delta i is equal to 0 but not all these variations will be will be important because some of them are just trivial those are the ones which correspond to actual gauge transformations so the ones which are trivial are the ones where a i bar is of the form d i bar of lambda so ok this basically means that the this deformation of this background gauge fields are classified by looking at the homology of the bar so ok so now we want to see like we want to write down these terms and check which one of these are q-exact and which will and therefore will not change the answer for some super symmetric observable and which ones will not ok so im gonna give some more details for the case of the gauge field because it's amenable to be done in a short amount of time and then I will repeat more without many details the super gravity case ok so you need to remember what are the objects which couple to the to the background gauge fields so again these are objects which live in a multiplet which contains a scalar j and there are fermionic operator j alpha and j alpha dot and then there is the conserved flavor current so this thing is such that d mu f mu ah d mu g mu f equal to 0 ok so now we can look at how the super charge acts on these operators that again just follows from standard formula so we have our super charge which is determined by the spinner z and when it acts on the scalar component it gives i zeta times j where j is the spinner and then q zeta over j that is 0 well that is expected because q square is 0 so from this equation and then you have q acting on j tilde alpha dot that gives minus i sigma tilde mu zeta alpha dot times j mu minus i d mu j and q acting on j mu that gives minus 2 zeta sigma mu nu d nu leto j ok so these are all the variations and we can now figure out like using the fact that we have this complex structure that we can build out of z we can rewrite this variation in a slightly more convenient form by multiplying this equation so we take this thing and we multiply it by something proportional to zeta dagger sigma rho divided by this value of z squared which again is well defined because z squared doesn't have zeros and ok so here just by counting you see that there should be two q exact operators and indeed once you do this what you find out by multiplying this thing by that is that so basically the q exact operators so the things that appear here just become just are the antyolomofi components of this object here so let me write it so then j i bar minus i d i bar j are q exact ok so actually you can even check that these are not just q exact so all the q closed operators that you can write out ok so now with this information we can just rewrite this delta L over here so delta L is a mu j mu plus d j and ok so we can do some algebra which is not too complicated and what we find is sorry let me put delta mu and delta d it's not really necessary so and similarly so we have two complex coordinates so we have some other coordinates let's call it z we have a similar expression and then ok so this is a little bit and in order to derive these expressions you will have let me think do you have to integrate by part something or no I don't think you need to yes no you don't have to integrate by part j z plus i d z j ok so now you look at this and they say but this guy and this guy multiply the q exact operators so this is q exact on the nose so this piece here is q exact this piece here is q exact and the only two pieces that remain that are not q exact are the ones here which corresponds to the variation of the antihelomorphic components of A and those are the components which have to satisfy this constraint there is a plus sign missing yes and any other question you integrate by part when you convert delta d into something in the variation delta d delta d is like a plus sign of the projection of delta a ok delta j is you have a relation between d and f yes that oh yes right ok yes then I have to integrate by part so there will be plus total derivative ok so now we have to understand what happens to these two terms so first of all like it would just seem strange that this thing is even supersymmetric because we just claim that all the q exact q closed operators are just these how come that this subject after I cancel these two pieces which are q exact ends up being supersymmetric but it has to be supersymmetric because it comes from a supersymmetrical agrangian so there is you know that that has to be the case but you can do a small exercise and you can check that if I do the variation of these two terms and I integrate by parts by these places then what you get is the following I guess this was expected that's plus total derivatives so q of delta l or delta of delta l is this and then you can see that indeed this thing is zero provided that the background is supersymmetric so it satisfies this relation so indeed you recover at the linearized level the same condition so now we just have to figure out as something so we figure out that the background can only depend on delta aw bar and delta is a bar so now you can again make a clever argument saying oh but my original agrangian so the original agrangian was invariant under background gauge transformations by using the conservation of the current and therefore that means that if I take a variation of my A's which is exact that this object vanishes so this is a general argument it has to be true but let's check that it is indeed true was invariant on the unit relation well we are assuming that we are complexifying so we are allowing ourselves to complexify the flavor gauge transformations indeed if you only allow for real ones then you would not be able to say that you can set such a thing to zero so how do we do this so we set delta aw bar and delta is a bar to epsilon so we set delta ai bar to be di bar of epsilon and then we integrate by parts and when we integrate by parts we eat various terms we have two derivatives acting on j and sometimes we have derivative acting on the current so we set this then we integrate by parts then in the terms that have derivatives acting on j use the conservation of the current so then after everything is done what you are left with is q epsilon times dw of jw bar minus idw bar j plus similar expression for dz plus total derivatives but now the operators that appear here are exactly the q exact ones so you are indeed proved that this trivial gauge changes of the background do not change the answer of the partition function ok, that's everything for the gauge fields what we have proved is that the answer the partition function has to be holomorphic in these parameters that classify the backgrounds because it depends only on delta aw bar and delta is a bar and not on delta and not on the holomorphic components and secondly that it only depends on homologically on these parameters so this argument is still sort of a linear order yeah, yeah, this is a how do you come up with this? ok, so to be fair for an abelian gauge field I think this argument just is fine at all orders just, I mean because you can linearize around some non-trivial background and the equations are gonna be similar for the case of supergravity that's clearly not the case so that as I was saying so either you say that the supercharge is a scalar so working around flat space should be sufficient or you can try to better by really working at nonlinear level for that however I think it's more convenient to introduce twisted fields and so on that's it, yeah I just worry that there might be new dependence coming up if you go to second order so you won't exclude that well, indeed so also in topological twisting you can just either define the theory in using the twisted variable and then it's topological in the usual sense or you can just you can see papers where they just look at the energy momentum tensor in flat space and they check that it's q-exact and they say, okay that's an option okay but yeah, I'm not claiming that this is complete proof, I think you should really work at nonlinear level any other questions? Okay, just want to make sure so actually this argument shows that you don't need to assume that the field is invariant because of the flavor gauge fields you just prove that the variation Lagrangian under the deformation of the gauge field which corresponds to the complex gauge summation is q-exact okay, yes because the measure is not invariant in complex factor gauge summation that will not be good argument this is a good argument let me think about it okay, yeah, I think that's correct indeed, so you want to set it to zero yeah, so if you say if it's there, then it's q-exact so that is still a couple of topomorphic line bundles for the flame symmetry that's a obedience in a way right, okay, so so now we can progress to the gravity case how much time do I have, it's still 12 half an hour, more or less okay, good, I think that should be sufficient okay, so to understand what happens in the case of gravity you have to work a little bit harder so people go to gravity so just can you just repeat if my flavor symmetry wasn't on a billion well, if the flavor symmetry weren't on a billion then okay, so in the holomorphic back bundle yes, it would be I mean, either you can start from zero and then go a little bit away so you could have yeah, okay, that's sure so the current preservation will be some kind of current wait a second, but I'm working locally I mean locally the variation will be q-exact I mean, it's true that it could be around variation is small, but it doesn't change the topology in the bundle sure but since the topology is on q-exact you can't assume that you can set the background h to zero and start there no, yeah, right, okay, yeah so then the current preservation should be the covariant preservation or what kind of conservation of current linearized around trigger background background h, what kind of so you are going to use the I did indeed so this was fine for the abelian case maybe it works two things could go wrong because you use the commutation and the conservation right, and the conservation I love to think about it okay, so okay, so first so in order to understand how the answer might change, you have to understand how do you vary a complex structure, so you can start from like how do you look at variations of an almost complex structure and characterize them at the linear level so the almost complex structure is defined in the following way so okay, so therefore let's just do a linear variation of this equation then what you get is that delta j mu nu times j mu rho plus j mu nu times delta j mu rho beta b0 and therefore this means that delta j with odomorphic indices has to be zero why the delta j with one odomorphic index and one anti-odomorphic index is unconstrained so this is just at the level of an almost complex structure but we have a complex structure so that means that this j has to satisfy this differential equations this which involves products of j's and derivatives of j so we have this tensor at the mu nu rho which has to be zero and now we can do a variation of the complex structure appearing here and work out what it means so setting delta j i j equal to zero because we know that it has to be at least an almost complex structure and see what constraint do we get on these components which were otherwise arbitrary so and basically what you get if you do this computation which is not too complicated is that you can define some objects theta i which are just delta j i j bar times d z bar j bar so there are one forms with values on the odomorphic vectors and then what this condition implies at the linearized level is that d bar of theta i is zero again not all the theta i that you get in this way or if you want not all the delta j's are actually interesting because some variations of the complex structure which you could just make an infinitesimal difumorphism will change the complex structures a little bit but that is not really a change of the complex structure it's just an infinitesimal change of coordinates so you want to know what are those that correspond to an infinitesimal difumorphism so that's a good exercise actually it's just a one line so an infinitesimal difumorphism is parameterized by some vector field epsilon mu so what you want to know is the derivative of the complex structure along this epsilon mu so you want to compute the derivative along epsilon of j mu nu and then that should give you the form of the delta j mu nu which are trivial so you do this exercise and what you discover is that the corresponding theta i's must be d of d bar of epsilon i so again the problem at least at the linearized level has become a homological problem you want to find the d bar closed theta i's which are not d bar of something so this is some Dorbók homology for objects which are valued in the eulomorphic vector fields so it's a so you want to find these but no doubt by the theta i's which are d bar of epsilon i so again this is only at the linearized level so while it's true that for any deformation of a complex structure you can find the theta i with such properties the vice versa might not be correct because there might be obstruction coming up at higher levels so it's not going to be interesting in such case I'm just working at the linearized level so if there are obstruction then that would just mean that the possible variations are less than what I am considering so are there any questions on this? no, okay that's good actually it's interesting so you can deform Lagrangian interestingly by the first information of the complex structure which more or less happen to be obstructed but in theory it doesn't mean yes that's the argument it doesn't, maybe there are some so if you try to differentiate the deformation maybe in counter terms your meaning is like that the Lagrangian itself should see that the deformation makes no sense the obstruction should appear there I don't know if it's a contact term because it's maybe two of these obstructions don't appear because the obstruction is H2 of the tangent bundle it's like H2 minus 1 maybe the motion 3 may appear maybe, but in any case I said the obstruction should be something that you could see locally obstruction is also logically well, then this argument so this thing which is called differential, actually solves a nonlinear equation debartatical state of square in first linearized solve equation debartatical to 0, find linearized state then you take the square and try to solve that equation so that's the reason there is an obstruction there but then I don't think it could appear in contact term well more food for thoughts but it's true that something bad should be seen at the level of the action as well so we just need one extra ingredient which is that so we also have to talk about what variations of the metric but again variations of the metric means that if you vary the complex structure then we also need to vary the metric in such a way that it stays a mission so one of the conditions that we had for the geometry to be supersymmetric was that the metric had to be a mission so that means that we have to figure out so if we change the complex structure we have to change the metric in such a way that it stays a mission and we have to understand what that means so we have to linearize the compatibility condition for the metric so we can work at first order in delta G and delta J and what we find is there is something which is more or less obvious which is that the components of the metric which have one homomorphic index and one antheromorphic index those are unconstrained and they make a change of the Hermitian metric and that will still be compatible with the complex structure it's only the components which are zero which will have to change due to the change in complex structure so this is unconstrained if you want another way to say this is that they can always do a change of Hermitian metric before doing the change in complex structure and then that changes in the complex structure in the following way so again they involve these objects which parameterize the change in complex structure so now you almost have all the ingredients that you need the other ingredient that you need is that you have to use the formulas that they gave in the last lecture to compute given a change of metric and complex structure what are the changes in delta A that arise so that I'm not gonna show but it's more or less immediate I mean not immediate but it's clear what you have to do bunch of algebra and then after you do that you have to look at similar to what we did with the vector fields you have to look at the multiplet of currents equivalent of these equations except that now here you have to consider all the objects appearing in the current multiplet so it's much more involved and then you can play the same trick you will multiply some of these by some spinors to get expressions like this let's say the sum of the components of the operators which appear in the r multiplet are actually q-exact operators so you will do this then you can rewrite delta L of this data and what you find is that delta L is a bunch of q-exact terms plus various terms multiplying delta j i i bar times some operators of i bar i and then that's it and now you can say ok, so this already tells me that my Lagrangian will only depend on delta j i i bar and not on delta j i bar i so it's holomorphic in these parameters these delta js have to satisfy this delta closeness delta bar closeness requirement and then exactly as before you can make the short argument oh well, like if I consider a variation of the complex structure that is trivial that just arises from infinitesimal differomorphism but my original Lagrangian is is differomorphic invariant therefore that this change better be q-exact and indeed that's what happens or you can like for delta j something which is exact then integrate by part using the conservation equations for the r-current multiplet and rewrite the resulting operatories that arise here as q-exact ones so this analysis is completely parallel to the case of the background gauge field but it's just more involved so the results are that z so another thing that you notice is that there is no variation of the metric here the delta g ij bar did not appear anywhere they appear here so the partition function on M does not depend on the Hermitian metric so on the choice of Hermitian metric compatible with the complex structure and it depends or can depend on the complex structure model but only olomorphically well, olomorphically maybe it's a little bit too much it's let's say locally there could be singularities appearing somewhere indeed if you check for instance the expression for the index that I gave yesterday and you vary p and q those sorts of singularities it's a meromorphic function of p and q is it actually an option that it's not actually a function could there be anomalies so I should also say that all what I said so far is disregarding the possibilities that there are any anomalies so that's actually an interesting research project for whoever is interested just like 20 years ago it's but this is certainly a point that is worth stressing ok, so this is the this is like the main point that that I wanted to make about the dependence of the partition function on this on this geometrical object actually there is well, you could have wondered what happened to the dependence on the background fields as I said there is this this U that appears in the variation of in the formula for V so what you can check is that that U actually it satisfies some so there are some use which are exact in a sense which I have not explained that I can explain if you are interested and then you can show that the Lagrangian does not depend on those variations of U which are exact so again the dependence on this U is only comological ok, so this this concludes this topic so unless you have some question ok, so in the remaining nine minutes I think, is it nine minutes I can talk maybe say something about what happens in three dimensions instead of four so I will have to just keep my short so n equal 1 in with U and R in 4D is simply related to n equal 2 in 3D again with a U and R symmetry and indeed you can obtain a lot of information about this theories by just the dimension reducing facts that you know about the 4D case so for instance this theories also have an R something which is called an R multiplet and this R multiplet comprises conserved R symmetry current then as usual when you dimension reduce you will get also a central charge and there is a corresponding central charge current which is also conserved then there is the energy momentum tensor and then there is also something which is related to the to the string current and so there is a scalar and you can form a string current coming from it and then there are the gravitini so again the structure of this multiplet you can get dimensional reduction from the previous one and also the supergravity looks very similar there is M U R some connection for the U and R symmetry then there will be some connection for the central charge and the metric and there is a scalar H and then there are the gravitini so sorry, did I say this were the gravitini I might have misspoken before but these are the gravitini which couple to the supercurrent and so instead of so instead of working with C it actually is useful to write things in terms of the dual of the field which is which is conserved so this looks like the field V in U minimum and so you can repeat the story that we did in 4D but in 3D and you can find similar results so in particular it turns out that the geometrical structure which is parallel to a complex structure in 4D is something which goes the name of a transverse solomorphic foliation so you can find details in our paper but again a lot of the things that you know about complex structure go through for this transverse solomorphic foliation, there is some notion which is equivalent to the obok homology and then you can try to characterize what are the transverse solomorphic foliations and you find out that they are characterized very similarly in terms of this other thing which is similar to the obok homology and that these moderate spaces are usually finite dimensional et cetera et cetera and then you can repeat this analysis that I described today and you can find out that the partition functions then will only depend on the choice of metric which is compatible with this structure but only on the choice of this transverse solomorphic foliation and mathematicians have classified which manifolds admit such structures and for instance the manifolds with diffomorphic to the trisphere admit one parameter family of transverse solomorphic foliation parameter is complex and indeed that is to be related to the squashing parameters that people introduced on the trisphere except that that's one thing that you can say and actually just on the trisphere as for the op in some sense the trisphere is just what you get dimensionally reducing the op surface and so exactly in parallel where there was primary op surface so this gives this moderate space of of transverse solomorphic foliation which depends on one complex parameters which I think you can say it's p over q bar or something like so and then there are also isolated different isolated transverse solomorphic structure corresponding to the dimensional reduction of the non-primary op surfaces and so okay so that classification goes to exactly exactly in the in the same way okay so for instance one can give some one application of these ideas in 3D which I think is I can probably sketch in the remaining two minutes oh okay but I can allow some time for some questions I guess so the transverse solomorphic structure isn't different on the complex structure? yes it's different than a context structure it is related to an almost context structure and then there is an integral so as for the case of the complex structure there is an almost context structure and that's something similar to the almost complex structure but then the integrability requirement that for the complex structure gets translated in some integrability requirement for this object which makes it into a transverse solomorphic affiliation which basically means that the spaces transverse to the vector which defines the almost context structure have a complex structure and as you move along somehow these complex structures are like behave nicely I mean I could write down the equation if you want but we can also discuss it later so there is some kind of part of this it's actually much easier let's say that there is a counterpart so again also for this transverse solomorphic affiliation you can introduce a coordinate along the vector for the context structure and then in the transverse space you can introduce a complex coordinate and then the transition function is going from patch to patch so it looks very much basically for what I understand we know about complex structure as an equivalent for transverse solomorphic affiliations so as an application of this background supergravity ideas can talk a little bit of how to use them so people computed the partition function for this n equal to field theories so in this parameter b which is this parameter that I interpret as the saying which is the transverse solomorphic affiliation that you are using but the squashing parameter for the real sphere should be real no so the squashing so if you interpret this as a squashing then so there are the definition is not super easy so there are different squashings which in principle are not even related to each other and the different squashings so for instance in some cases for some of the squashing this parameter is real for some others of the squashing this parameter is imaginary so they look at the first site of the related squashings of the sphere and you can even keep the metric round and change the other supergravity backgrounds and get a squashed sphere in which the parameter appears not in the metric but elsewhere and then it's not really a squashing I don't know what you want to call it but there is some dependence on this parameter so if I give you the metric for a sphere which is this parameter you need extra information so is it some super geometry which has a parameter it's equivalent to the parameters p and q which appeared in the op surface so if you reduce that discussion there are two but when you reduce you lose one one the radius of this one you clearly lose and then I think in some sense some combination of the angles can't you go to one so that log p divided by log of q remains finite that I'm not sure but what is important is that there is some parameter and for specific cases in which this partition function is computed you can now use this fact to embed this to describe this geometry in terms of coupling to supergravity to say something so you think that b is the ratio of log p with log p ok so then is it log p over log q or log p over log q bar log p over log q is b square it should be homo, right? ok that's not what I remember but maybe there is a question of definitions but I think you can find the answer in this paper about the dimension reductions of dualities by Cyberg, Willett and Radzamat so over there I think they have some precise match between this b and the p and q so let's take the round sphere which will say corresponds to b equal 1 in some set of conventions and with some slicing of this space of squash spheres and then you can look at what happens into the Lagrangians when I vary b away from the round case so then delta l, so if I do delta b then we have some delta l and this delta l will involve these background fields in the supergravity multiplet which couple to the conserved arc current and c mu which couples to the conserved central charge current but at first order in delta b this is all what there is so the other auxiliary field doesn't enter and then there are higher orders in delta b which so there will be delta b squared which multiplets some operator so again we are considering a variation near the round sphere and actually for round sphere we have to be more precise because we have a non-trivial b even on the round sphere so what we mean is that when the sphere is round we actually take a theory and conformally couple to it so that's the definition and then you can give an interpretation for what the variation of z with respect to b at quadratic order the round case should be and there should be in principle two contributions one which involves integrating over the sphere a two point function of the conserved currents so for instance there will be some vectors which in order to find the precise expression you have to study the background which integrate j mu r at x and j mu r at y and then there could be other objects involving j mu z but ok so now let me let me say so let's suppose that we consider some theory on the sphere and then we make the sphere larger and larger then the theory would flow to some flow to the infrared and in the infrared the correlation function of j mu z j mu z becomes a redundant operator and the and therefore like its correlation functions are only non-zero at coincident points so here we'll write that there could be contact terms so we are using implicitly that the partition function does not depend on the size of the manifold so there are contact terms that I will discuss very briefly later and then at order delta b squared you would also have the expectation value of this operator but in a conformist field theory that's zero so you don't get it so you just have this and now we have to discuss these contact terms so this would like take a while to explain but so this contact terms can in principle give some non-zero contribution because they are integrating them over the entire manifold but you can prove that they only change the imaginary part of this object and not the real part so if you concentrate on the real part then it's all encoded into the separated in the two-point function at separated point of the conserved arc current and this you can just write down for a general super conformal field theory and it depends on one parameter only there are some factors and there is delta mu nu d squared minus d mu d nu of 1 over x squared and in this parameter also determines completely the two-point function of the energy momentum tensor that's related by super conformal symmetry so then you can just compute this integral by using the conformal map from flat space to the sphere and then you integrate over the sphere using this v that come from like the background and you compare with what people obtained for the partition function and in this way you can determine this number which characterizes your CFT 0 minutes and then done


There we go. Okay welcome back to Senate Education. Jim, I'm wondering if you would please bring us back to the topic when we were talking with Professor Teachout and yourself around some of the issues we're dealing with around church and state. I know you have put together some options for us to consider around what we might do to protect that line if you will. And so I'm wondering if you could just give us an overview of your thinking and why this direction you think might make the most sense and then a little walkthrough of the bill. And we're not going to be making any big decisions today but it will just help focus us I think. Sure, yeah. Thank you. So for the record, Jim, everyone, let's console. I think this area because it involves a constitutional principle which has been interpreted many times by the court is frankly quite confusing to understand. So before we go through options and language, let me just help define if I could the problem you're trying to solve here. And the problem you're trying to solve here is that the Vermont Constitution has this clause called the Republican support clause which essentially says that citizens of Vermont cannot be compelled to support any religion. So the way that's been interpreted by the Vermont Supreme Court in a case back in the 1990s called Chittenden Town, the court said yes, you can provide tuition, public tuition, to religious schools but you have to be sure, you have to take some safeguards to ensure those funds are not being used for religious instruction. So you have that guidance from the Vermont Supreme Court over 20 years ago and the problem that we have is that districts have been denying use of public tuition for religious schools like Rice High School. They've been denying it because they are religious schools. So they haven't been applying a safeguard against their use. They've been saying as a blanket statement, we can't fund you because you're religious school. So the Supreme Court has held you cannot discriminate against a religious school solely based on its stats. So the recent cases that have been coming up in Vermont, we've seen, there have been some holdings that basically have held that Vermont is discriminating against these schools based on their stats because that's what the record has shown in the court record. It shows denial by districts on that basis. What has it happened in the 20 or so years since Chittenden Town was decided, there's been no state directive going to school districts as to how they can establish safeguards against improper use. The districts have been on their own to figure that out or just to deny funding based upon stats, which is unconstitutional. So that's the problem. The problem we have is we're in a situation now, we've got court cases that are in live flow here, which are coming against state because we really haven't defined what their safeguards look like. So what the AOE did in February of this year is they issued for the first time non-binding guidance to school districts after what their safeguards might look like. The AOE says as guidance that has no authority to bind school districts because it's what a district issue, an AOE issue. So it's given some guidance and then Professor T-Shelt came in and testified as to a process for certification from him, which is part of the guidance that the AOE gave as well. But I think what an option for you is to give direction from direction. The school district is saying that you shall provide funding to these religious schools if they provide the district for the certification. The funding is not being used for the purposes of religious destruction, worship, or propagation. So what I have here for you today is a bill that gives that direction to school districts and with that direction it's now saying to school districts on a uniform basis, here's how you do it. And therefore it will allow school districts to fund religious schools if they get certification using those funds for improper purposes. They pause there before it goes on. Any questions about that framing? Any questions? So are we putting here Jim in in effect creating a certification program? Yeah. Senator, do you have a question? So let me share my screen from it. I'll come back to we talked as well about the idea of conditioning funding on application of anti-discrimination laws to these schools. I haven't put that in the bill and I'll explain why. But let me go through the bill first and then we can talk about that element of it as well. Thank you. I'm going to share my screen if I can. And show that. I'm going to bring that over. Okay. Can people see this? Not yet. There's the old windows. Is that the old windows? I don't get it. Looks like it wasn't opened up. You need to open the document on your desktop screen. I did that. Yeah. Let me find it again. You take your time while you're doing that. I just I don't know if anybody has seen the perhaps the worst or the funniest technology mistake that I've seen in a while with the the attorney who had the cat filter on. Has anyone seen that? Oh yeah. I mean there is nothing that tops tops that one. I will send it along. I will send it along. I sent it to Dick Sears. Oh you did? My daughter sent it to me like at six o'clock in the morning and then I said oh my god. Well I saw in the news I think it was no less than a week ago and I don't know if it was a senator house committee in New Hampshire but there was a legislator whose cat continued to get in the screen and that caused some frustration for committee members to the point that they're trying to do something to ensure your pets are not seen in the New Hampshire legislature. Really because one of my cats will sometimes jump behind me and Senator Pearson and Finance, his cat when I was on Finance would always pop in and other cats. Yeah that's funny where people I wonder why they want to stop seeing pets in the screen maybe just. I didn't dive into it. My kids walk in front of the screen four times six times a day so I mean look at. Right. How can you be upset? Last year in this committee we saw a lot of pictures of Senator Ingle's cats. She just was showed to us around the committee room. Senator Ingram's. Ingram yeah not Ingram. Yeah yeah. Jim. Yeah I'm having trouble here I'm not sure why I'm sorry. If it's better for you we can always follow along. Is it on the committee? I believe it's been posted. Yeah if you don't mind that would be no that's fine. This time that's probably the better way of doing it. Does everybody have it with anybody can everybody find it? Which one is it on there? Let's see. Senator Campion if you give me a couple of minutes if you want me to post it I can. I don't see it there. If you have not posted it yeah that would be great. I have posted it but you want me do you want me to try to share it? Where is it on the committee page Jeannie? It's under Jim's name. Second second. It should be under Jim's name. Okay. I think it's the second one the first one was just finished and then yeah. Okay great and great everybody have it? Okay great. Okay thank you. Okay good so I will go through this draft. It is a draft that could be conveyed though right now it's under Senator Campion's name and it prohibits it proposes to prohibit school districts from paying tuition to certain schools or programs regardless of religious status or affiliation unless it receives certification from that school or program that none of the tuition for which payment is requested has been or will be used to support religious destruction or worship or the propagation of religious views. So this findings and purposes let me get to take you through those because it emphasizes again what I said at the beginning. So first you have the Vermont Constitution has the common support clause which provides that no person can be compelled to support any place of worship contrary to the dictates of conscience and then it shouldn't in town decide back in 1999 the Vermont Student Court held that school district violates the compulsive support clause when it pays public tuition to a religious school in the absence of adequate safeguards against the use of such funds for religious worship or instruction or the propagation of religious views. And it says the purpose of this act is to define adequate safeguards that a school district must employ to ensure that public tuition is not used for religious worship or instruction or the propagation of religious views. So Section 2 amends a section of statute dealing with the payment of tuition to spokes. So A is not amended but what A says is that a school district shall not pay the tuition of a student except to a public school and approved independent school and independent school meaning education quality standards, a tutorial program approved by the state board and approved education program or an independent school in another state or country approved under laws of that state or country. So you can pay tuition to that list okay and what B says is that a school district shall not pay tuition under Subsection A of this section to any of the schools or programs identified in that subsection regardless of this religious status or affiliation unless it receives certification from that school or program that none of the tuition for which payment is requested has or will be used to support religious instruction or worship or the propagation of religious views provided that public schools that receive tuition from a school district are exempt so we're not requiring public schools to certify this because they are public schools for any independent school tutorial program etc listed in A this certification will be required and then it goes on to say instruction and religion that focuses on the history and teachings of various religions shall not be deemed to support religious religious instruction or worship for the propagation of religious views provided that the instruction is not designed to and does not support religious instruction or worship for the propagation of any one religion or theology or group of religions or theologies that is designed not to prohibit schools from offering survey courses in religion right um that language might need to be tweaked a bit but that's the idea behind that language I mean that Jim that could even be you know somebody a school offers uh you know uh Fedford Academy offers a course in the New Testament they can still do that uh or or does it have to be served well I'm not sure so that's what we in testimony I think I'm not you can test me on that because I wrote to be kind of survey like more or less okay um I'm not sure how to write write that if it's if it's a course in a specific religion I guess the point where it seems like this is instruction so I'm not sure how you how you how you try that line okay um so I wanted to mention Professor T-Shop's testimony he he recommended recommended this same certification in his testimony um and he also mentioned that the certification approach is simple and practical and he said it's a great advantage advantage of certification mechanism is that it avoids the need to get into any great detail about which expenditures might be eligible for tuition reimbursement and which might not be not be and then he was trying to say that the mechanism of certification is a familiar mechanism used by both federal government with court approval so this method we're talking about is been approved by the Supreme Court in my various courts so this is trying to method of providing a safe part so um I'll pause there before we talk about discrimination when you pause there and if you've got questions on the belt language itself where you approach any questions uh Senator Chairman Zuny thank you Senator Kimme so um Senator Hooker understands we I'm thinking of locally we have a Catholic elementary school and we have a Catholic high school and you know if you decide to send your children there you pay the tuition it's not funded by you know tax hours so we already have that in place I guess maybe I missed a part but what what are we are we just trying to strengthen that foundation or what are we trying to accomplish with this bill um Senator Campion because I already thought it sort of is already in place that we're not paying for you know for kids to go to these Catholic or private religion schools so this would this probably wouldn't impact I don't think your schools at all unless you're not in a choice district so correct me if I'm wrong Jim this is this is we're saying if somebody these are schools that if a um if you're in a choice district you now be able to right now you're able to take your choice dollars to a religious institution and we are getting sort of mixed things happening out there some people are saying yes some districts are saying no this would give some clarifying language to make sure that uh schools know under what circumstances they districts know under what circumstances they could be sending and to what schools does that make sense Jim well let me let me be afraid to answer a little bit I could um so so uh so so these schools are well and there are tuition districts nearby right um and they they could send their kids to private private schools in Rutland um uh also if you're a public operating school you can send if the district allows it uh you can send your child to a different school outside the district pay for with tuition dollars public tuition dollars um so for example our school that operates uh operates schools the district operates schools maybe doesn't have a program that a child wants but maybe it's over there um in in bed school so both operating schools in more narrow circumstances and for tuition districts um um they they can send children students to independent schools under current law um they um as you mentioned um in Rutland the religious schools are parents are paying their own other own pockets for those programs right because school districts have been saying we can't provide funds to religious schools but that's not what the Rockstone Court said Rockstone Court said that you can buy tuition to religious schools if there is a safeguard that the schools are using that funding for religious destruction so imagine that that the school in Rutland um the Catholic school in Rutland has a has a system that says okay we're going to take public tuition but we're going to use that to pay our religious teacher or our gym teacher or wherever but I can usually have to pay for religious destruction we use that in these other areas are not for that purpose that is permitted permitted under the Vermont Constitution and under the shipping case but what's happening today is districts are saying no because you're in school you can't do that and that is unconstitutional so this is a way to thread that and basically give guidance school districts to not say no but to say yes if you have safeguards in place and the way to do that is your certification so with this change um Rutland um or let's say Chittenden Chittenden is a tuition town uh Chittenden could pay tuition to your Catholic school in Rutland if the Catholic school in Rutland gives a certification to Chittenden school district that funds are not being used for religious destruction worship or propagation so we change we change what's happening on the ground now okay so that that that helps a lot so I let me just pitch this here I live in Rutland town so we don't have a high school so I had school choice I could have went to Rutland or Proctor West Rutland or wherever I could have essentially I could have gone to Mount St. Joseph's in Rutland if I wanted to um and right now for Vermont Constitution Vermont law would have authorized the tuition to be paid to MSJ on my behalf if they can certify that it's not being used for Roots use yes interesting yeah okay let's say it's 1999 but I should say there's been no guidance issues since I've dated so school districts have been on their own I think this is a you know maybe maybe senator Hooker knows more but I think this is a sort of a hidden secret to most people I mean I've never heard of this before you just live around here with Catholic schools you know if if you're going to the Catholic school you know your parents are paying for it or you're going to the public school so this is this is interesting to me but wasn't that one of the cases that was brought up in Chittenden I mean I know that there were there was um discussion about whether or not the kids from Barstow could go to MSJ yeah so the case that came up with Chittenden back in 1999 um which was exactly Chittenden involved actually going to I think March St. Joseph so that was the case actually and um there have been a number of cases since and and a year ago or so the Supreme Court in the Espinoza case said that you cannot discriminate against whether this school was based upon their sass and that's what's been happening over the month so um that the kind of urgency around this if you will is that we are not well placed right now to get on the right side of the line and if giving some guidance to districts after how to how to do this in a way that it's constitutional is the purpose of this the way it's served is that helps full Senator Karensen yeah okay but Jim is every given that the Espinoza case that was a federal case yes our are basically all the states right now in the union dealing with this same kind of thing I mean well maybe not grappling with it or trying to do what we're doing but you know if you're in in Wyoming has this opened up that Espinoza cases opened up a similar uh opportunity situation where they too can say if their version of Rutland town in Wyoming can now send their kids to their version of uh the Catholic school in Rutland yeah so what's happened it's just saying that the Espinoza case that was my town my town law actually and I think I mentioned in my slideshow a while ago many states adopted the so-called many brain brain amendments in the 1800s which were amendments to their constitution called no aid provisions those provisions said uh the state shall provide no aid to uh business organizations right and um and the Supreme Court found that's discrimination based upon status as well as it said so struck down essentially the montan constitution um provision and likewise states that have that provision the provision was adopted uh as an anti-catholic thing back in the 1800s um and so many states have that provision we don't we've got our propelled use provision which is around how you use funding so we we're a much better place in terms of constitutional law in Vermont we are consistent with the US Supreme Court but in practice we're not because in practice school there's been denying uh public funding to these schools based on the status so while we're okay constitutionally right now we're not okay on the ground and that's what this is the standard cure okay center uh center hooker oh thank you Jim um with regard to the certification I know that you know we're as long as they can certify that they're not going to use it for um religious based religion based studies or payment to staff and all of that um what else would that include I mean how do they have to itemize you know each of the things that they're not going to use it for and and what do we do we have any um sense or parameters around um what are those those functions those religious functions I mean talking about the possibility of offering say a a survey of a comparative religion that would be okay but hiring someone to teach theology would not is that correct yeah it does leave um the language in the draft bill now leaves a quite open bag for sure um and that was the recommendation of Professor Teachout because he was worried about going down the other avenue trying to define all these parameters in terms of segregation of funds and and things like that so his idea was keep it simple make a certification which has been accepted by the court um but it does open the question as to you know is there some audit audit that's done after the fact to find out how the use of funds is there's some control I mean you're basically relying on the honesty of the school to certify so you might decide to you know have the out of it you have some odd artwork done or have some parameters but this recommendation was keep it simple just do this but it's up to you of course what you do but that then begs the question of you know blended funds and and separate accounts from coming from public schools yeah it is very much it sounds like the honest you know the honor policy here you know please do this and we hope that you know you'll you'll abide by what we're saying I I still worry a little you know I well we'll get into the anti-discrimination stuff center Perswick and then center lines that was going to be my question about the anti-discrimination stuff if we were going to get into that or not yeah I think that's going to be next after we finish up with other questions center lines that's also my question but yeah a comment that gee I really hope we can trust religious schools yeah seriously yeah yeah no I the moral fiber right so before maybe you go to anti-discrimination is there a way to so let's talk about the teaching of science for example do what we should we say that science has to be followed by the Vermont standards if you know creationism versus the teaching of you know evolution I mean are there things there that are concerning to you Jim well I I guess what I'm thinking about this from um I'm trying to not go into policy and your jobs I'm trying to try to line here if you won't but what I see is the course of facing bad records the only thing I saw was in the record right now the records are showing decisions by school districts that are based upon not providing funds based on status so the goal here might be short term goal do something quickly to get the record and bear a place so of course have a record that you take an action and give them guidance it's mandatory um and at least they have a record so they know that we're not discriminating on that basis whether you go further and add um anti-discrimination language or you know various other conditions um if you do that my fear frankly is that you're going to bother to go down so much that it might not make it through and we're going to have to figure out what we have today of course we're looking at so you can always take on the discrimination point at future time in terms of what schools have to comply with um but so I was suffering that conversation in my mind a little bit because I was just concerned that that that addition could really affect the likelihood of this fix being put in place in a short term way to create a better record for the courts uh for Vermont um are you particularly concerned with I mean us getting something through the senate the house and the governor's signature if it has the anti-discrimination because I feel that that I wouldn't personally I feel pretty confident that uh we would be okay there well it's my concern is more about constitutional issues so I think once you start adding those things then you raise a host of other issues that have to be dealt with so for example we talked before about applying labor laws to religious schools and the exception they have for um when people were considered to be uh ministerial function so in the U.S. Constitution as talked about between court there is an exception where those schools can discriminate against individual whose employee um and they can discriminate on the basis of age of sexual preference so that's that's attacked by the constitution so if you try to add that as a condition here you're up against constitutional issues there and I don't know as we talked before we've got the anti-discrimination laws in Vermont based on public accommodation and I'm not clear in my mind whether those laws apply to private schools or don't because it's not clear to me that they're open to the general public because they're private schools so I think if you if you add that provision maybe it's okay but I think you have to hear from the uh the right advocates around those laws to figure out how they apply we don't apply what the issues are so I think it gets a lot more complicated once she goes down that path because it raises other issues um I'm not saying that's not a good idea from a policy standpoint I'm just saying it raises no no that's helpful I wonder uh and then Senator Hooker if uh you know we've heard from Professor Teach Out which was was helpful I'm not questioning his skills or talents but I wonder if we should also hear from another constitutional expert and that's just something that maybe we can we can talk about in terms of possibly finding one or just getting another opinion Senator Hooker. I don't know if this has to do with the anti-discrimination or what but would this be open to every religious school are there some kind of parameters around well I guess it would be discrimination if we didn't open it to everyone but you know you have some people who open a school with certain religious intent and you know it may just be in somebody's house and kids are going to that school and you know how I mean they would have to be recognized correct uh more more than recognized so schools who which are eligible for public tuition whether you're sectarian or secular or wherever have to be approved by the AOE so they have to be approved by the AOE a bunch of standards for approval based upon academics and teaching and so you have to be approved first and some of the religious schools are approved by Bryce Memorial is approved even though it doesn't take public tuition because it wants the stats to be improved but in order to take public tuition you have to be approved so it can't be anybody has to be approved in defense school. Okay and that of course has to comply with all kinds of regulations to be approved because we do have another religious school in Rowland that is you know has been running for quite some time I'm not sure what their status is. Senator Perslick anything? I'm so interested in the state's interest in ensuring that these schools that are getting public funds aren't discriminating and maybe that's through that a public accommodation you know portal if you want to call it that so I guess I'm just saying I'm interested in that conversation to hear more more about that and is there anything the state can do about that? Yeah I know I too am interested. Senator Lines? Yeah but I do think that once you consider anti-discrimination for the institution then there is let me just ask this question there is no requirement that the institution accept or allow for any certification to receive the public dollars so they wouldn't have to do that I mean they wouldn't have to right? I'm asking Jim that question. Under this bill any school, any independent school, any approved independent school whether secular or sectarian would have to certify but they're not using the funds for right but they don't have to accept any funds so then if there is an anti-discrimination policy embedded in our bill then independent schools say well look you know what we we want to continue doing what we're doing period end of conversation then the question is who suffers from that then the kids don't have the opportunities you know but but it also then it also says that they're discriminating against kids you know so it's that circular argument it's just pretty horrific. Well it sounds like it would be if you if the school said if we said you needed to follow the state's anti-discrimination policy and the religious school said no we won't accept that out LGBTQ teachers we you know we'll have students you know you know we kind of have that policy then they would just have to find another school to go to basically if if that's you know their choice district might you know if if we let's just take you know you go to Proctor or Firm Burton Academy instead of of the of the religious school. So the question is do we want to look at anti-discrimination? Do you think Jim it makes sense and committee members but Jim you might know more on this does it make sense for us to talk who else does it make sense for us to have this conversation with while we're making a decision for example does it make sense for us to have the Attorney General and does it make us make sense for us you know to hear from other individuals in particular? Well I think the Human Rights Commission the other ones to administer the the accommodations of all so definitely from them. Labor the expert in the office on labor issues is Damian now so if you want to hear more about that exception that we talked about in labor law he'd be helpful. A.G. I'm not sure A.G. has been involved in some of these cases so you might want to get the perspective I'm not sure. Chris Curtis from that office might be somebody who might be helpful there. Senator Lyons were you going to? Oh no I was thinking uh Brent Hare's been really helpful to me and looking at constitutional issues and I'm wondering if she what your thoughts are with another Ledge Council on constitutional issues I mean as compared with Peter Teachout it's kind of difficult that's a question for you Jim. I'm not sure that Brent covers this area specifically. The lawyer that covers the accommodation laws is David Hall. Oh okay. Yeah. So I'm sorry did I interrupt Senator Lyons were you going to have system else? Okay so I will if it looks like perhaps the committee is interested in hearing then more information on whether or not we should include anti-discrimination language if we were you know going forward so we will go ahead and we will set I will work reach with reach out and work with Jeannie who uh and uh continue some testimony in terms of this the urgency of this Jim um this is something that uh you think in your professional opinion we need to do something with this session this year. Well I'm I can't I can't say it that way because it's not my role to tell you tell you that. I am just seeing that that the record before these courts though looking at these cases it's not very good because the record is now based upon the status and just haven't had the guidance they need so if you want to have a better record going forward put the courts to consider this is a step to take this is a option to take in that in that regard. Yeah and it sounds like also I will talk with Senator Sears a little bit also this weekend to see if maybe as we're working on this they might want to have their eyes on it either early or they'd rather wait until we vote something out and then kick it down their way for them to have a look before it makes its way to the floor which I would feel feel a bit about. Jim anything else from you this very helpful um you know we had jumped into this a little bit challenging topic and I will work with Jeannie to bring some additional perspectives in next week. Senators anything else on this or anything else for the week before we say goodbye to one another for a few days. Senator Perslick. We could use the term science-based religious instruction. You could. Do I hear evolution? Just tie it in with the literacy science-based everything will be signed. Evidence-based religion evidence-based would really be the coolest right? Wait a minute they're all based on faith. All right nothing else on this nothing else on any other topics please I'm going to jump in soon after this and try to finish up what I'm going to talk to Senator Kitchell about and we'll send it all off to you for your okays or your edits and feel free as usual please edit freely and if you want to call and talk about it that's that's great too so okay thanks everybody thank you thank you Jim thank you send your vaccine questions in please we'll do happy Valentine's Day everybody oh happy Valentine's day


The Appalachian Advantage plan is the simpler, better way to get the phone you actually want instead of paying hundreds of dollars up front.  It's called the Appalachian Advantage and is available at Appalachian Wireless. Appalachian Wireless will be hosting their 15th annual charity golf scramble Friday, June 3rd at Stonecrest Golf Course in Prestonsburg, Kentucky. We're excited to announce that the East Kentucky Dream Center is the charity that we've chosen this year to be the beneficiary of our charity funds that are raised during the golf tournament. So typically we raise somewhere between $8,000 and $12,000, depending on how generous the players are. It is not a tournament that's technically open to the public. It's more of a thank you to our largest business customers. We have a great gift bag that we give out to them and the hope is, is that in lieu of donating or having a huge entry fee that they will generously give to the charity of choice. The East Kentucky Dream Center works to give all children and adults in Eastern Kentucky equal access to food, clothing and basic life skills. East Kentucky Dream Center believes that love, proper education and networking can change lives and combat the effects of poverty. We chose them because they're more regionally based and we try to find some place that helps certainly a wider base than just one county or something like that. The region is very important to us. This is a local company all the way through locally owned, locally operated. All the customer service is locally based here so it's very important that we give back to the region that's given us so much. So it's very important that we choose people that are also having the same kind of vision that we do and we thought the East Kentucky Dream Center certainly lined up for that. If you would like to donate to the East Kentucky Dream Center, go to Appalachian Wireless on Facebook on June 3rd and click the link to donate. For Mount Top News, I'm Breanna Robinson.


This video is brought to you in part by TrueTech Tools quality tools essential support Walking cooler. It's not working. Let's go see what we got going on. Is this the cooler that's not working right? Well, you can see how bad the other one looks this one here I can hear it running, but we have no fans It's cleaner than the other one. I've been through it a hundred mile an hour. These are those bone miser TXVs that I had to deal with not too long ago. That didn't make sense. I don't know if this was originally a freezer. Maybe that's why the fans are not running. It's getting power from up on the roof trying to maneuver around all this stuff in the way. Hey it's NorCal Dave's beer of the day. Now this is a beer cooler as you can tell it's a beer cooler which means we usually run a freezer style unit in here and means it'll have heated defrost which really makes it easy to get to. So let's get in here and see if by chance it's probably not getting cold enough. That's probably what's going on. And the temperature switch is not hitting to bring the fans on. Bar of the fan delay. Let's see if we can get up here. Do we have a fan delay on this thing? All right so here's your defrost termination switch right there. So it's going to have one. Okay I did not see a defrost termination switch on the other side but there's that big bone miser honker donker there medium temp TXV which like I said is not running very cold. I remember this being a little low. Wouldn't be surprised if that's not leaking because believe it or not that little bulb right there is literally in the suction line and is can leak around that thing because it's just got a rubber seal around it that's squeezed just like a side glass on a boiler would be. All right so I did not look very closely. There is a fan time delay termination switch right there. So your power comes in on four. In is always the other side of power so if you was to go to in which is right here and go to four you should have power which we've got on four. Okay let's see if I can make this easy enough. You have power coming in on in into the switch and out of it coming back over to f1. The other side of f2 goes to four. So four has one side of power and the other power comes out of the black lead. So if we use the jump black which is in to f1 it should run which we already checked it with the meter. So I'll go to f1 which is right here and go to in which is right there. Look at that. There's your fans. So yeah our problem is the switch is not cold enough and that's why it's not wanting to run. So that also explains why the TXVs are so noisy because we're low on refrigerant just about guarantee it. So let's just go outside and do that real quick because we'll probably have to look for a leak in here. All right what we're going to do is grab the leak detector. Miles is going in there and check for leaks before we get the fan running. That way it isn't blowing the stuff all over the place. So we're going to get out the world's greatest leak detector. So we got the old TX Stratus warming up. All right so warmed up ahead of time. They've got so many things in this building that leaks it wouldn't be a real surprise. Okay let's just go ahead and get up in here for all those elbow joints are. Go over here on this other side where the TXV is checking around the suction on the TXV. We already scanned all this. Stuck it in to the fan area there which you can't do with an H10. And getting those nice tight spots. So I've got nothing not granted it's running so the suction pressure is going to be in the 40s probably depending on what refrigerant we got. Let's go outside and get this thing charged up and get it running. Okay obviously that's right there coming out of the very back one. So it's going to be this first one. I think the one I worked on the last time was one back here at the back. So we're going to go ahead and focus on this one right here. Let's get it opened up so we can see what we're doing. Yeah I remember there being a lot of oil in here and not knowing where it came from and not sure if it was like from another compressor change or something. But that's an 05 compressor so this thing's been in at least 17 years, 18 years. So it's just like a brand new spring chicken. Let's go to our front sight glass here. I think we're going to see what's obvious. They put the disconnect box down there where it's real convenient. Let's scan it over real quick if we need to make a fix. We can do it before we have to pull it back out. Worst case scenario with it being that's 220. We could always come back or whatever. See if that one controls the back when that would be logical. Yep it looks like that's the one. It's slowing down. Lucky here nailed it in a couple seconds. Right there. Never fails on a discharge valve. They always leak.  Go ahead and scan the rest of it. But I've had that leak multiple times. It never gets tightened up usually at install and they get left loose from the factory. I'll like scan on the backside of the condenser coil. Usually cleaner and it's got that shroud around it to help keep the wind out which helps out. But right there seems to be the main one. Let's grab some soapy joke. Get some solid stream on there if possible. Let's see if it's leaking on the threads. Hopefully that way we can just tighten it up and not have to cut it out. Now we should be able to tighten that up and not have to deal with it. Listen that up and then look at how bad that's leaking. Yeah somebody's toasted that. Oh wow it's going nuts up here.  She's going nuts. Look at that. It's leaking out of the bottom even. Oh yeah look at that. Look at that. Look at them cookies. Chocolate milk and cookies for everybody. Heck yes.  Or chop it out. I'd prefer not to have to chop it out. Let's grab some nylog. Goop in it. Maybe get the wire brush out here. Let's see if we can get away with not having to go berserker. You can see some like rust color. Yeah she's going nuts. She don't like that. Probably not the right cap. Yeah cleaned up that side. Somebody probably cranked down on way too tight. It has a little red raised edge that generally just needs slightly tighten like a eighth of a turn and usually it'll be fine but unfortunately somebody donkey conged it and got her done. Let's grab a little bit of my favorite nylog. It gets a little messy. I think that's silly. It's messy. I don't want to get messy. I don't want to get messy.  Put a little bit in here on the threads. That part there. She's she's she dizzles and diggles. Put that right like that. See if I did it. Not seeing anything. Now granted it's not running either with 250 plus pounds of pressure on it. I think I would feel a little better if we added a little bit of teflon tape to it maybe. Get in there. Wrap it the right direction. I hate getting rid of it because you can actually isolate the compressor if you need to. And I prefer not to. Plus time factor. I got to go to another one of their locations and fix another one. Okay not leaking. Okay we'll leave a note up here. Discharge valve leaks. I'll do 413. There we go. Left them a little note. That's sealed. I'm not picking up the detector or that. We'll do it when it's running with high pressure on it but it's no different than a screw in pressure relief valve for a liquid receiver tank. We usually use pipe dope of some sort. Usually like a lock tight brand white stuff. I forget the 700 number whatever it works. Either way let's go grab some go-go juice. This should be 404. Let's see that's PoE so it's going to be 404. And get this thing. Draw up an attempt. I'm sure I've said things about this before. That wireless scale which is really sweet just like all the Testo stuff. Do it right here on the display. Don't have to get out your phone if you don't want to. It has that option but you don't need to. Then again refrigerant charge. I'm going to go manual. I do have the auto capabilities on the solenoid which is out in the carrying case for this but I just don't use it very often.  Let's go ahead and kiss back on. I already rotated my clocks so that we aren't going back into a defrost anytime soon. And it just so happened to be perfectly at the three o'clock mark. All right luckily we're at 34 pound suction so that means we didn't pull anything in. The pressure switch would have shut it down so we don't have to miss with anything far as dryers or anything like that. Go ahead and get this thing added to it. The compressor doesn't sound the greatest in the world. It's not the greatest in the world that's for sure. It's like you know everything looks fairly okay. You'll spray that thing see if it's bubbling.  And we do have a accumulator here on the suction side of the compressor. So we're not going to be slamming it with liquid too bad. I mean obviously we got to charge a liquid and we're kind of moving along fairly quickly. But most part that nylon is going to fit in between the brass cap and the brass threads there at the top and it's going to seal when it squeezes down on it. It like I said we could chop it out and then you have to change the filter dryer, pull a vacuum, a lot of extra time. As you can tell generally they don't want to spend any more on they need to on it. They just want to keep the toss down and keep moving along. And baby she's coming. That'll fluctuate a little bit more. We're right there at 4.6 pounds. Pressure's dropping a little bit. She'll settle down. Probably could use a wash enough for sure. You can see the TXV starting to pull back a little bit. I've not dumped any in for probably about two three minutes so that frost back is slowly disappearing from the TXV adjusting. Run about eight degrees, 244 head. So as soon as that starts to slow down a little bit that'll also fill up our side glass. That'll back it up a little bit. Not seeing anything leaking on it. I know that's not the greatest way of doing it but if you know your customer you know what they would rather have and I know they would rather have cheaper if it works. Looks don't matter, operation does. There's other ones where absolutely take your time, chop it out. It just really depends on the customer. Went around sprayed all of our other little fittings here just now that we've got more pressure build up on it. All our fittings out here, everything looks good. Cilenoid valve's good. 267 on the head. She's rising a little bit. Let me take a look here in this coil. This looks a little jackereed. Actually ain't too bad. Okay well we're running that head pressure there of about 265 area. It's a little warmer today. I think we're going to get up to 80 which is really surprising for mid-April for us. I'm not picking up any speed ups or anything so I think we're good on that. Looking good. Not flooding back no more. Not bad. All right let's go back in here and make sure that it's dropping in temperature and the fans are on. Yeah just like that right there needs needs cleaned up. Fans are on. Looks a little better. Yeah working good. Not too horribly dirty. It's a little bit of gunk on it. Now we're near what the other one is. Let's check the thermostat. See where it's at. About 32-ish. 32-ish.  This is where all the kigs are and stuff are at so they're going to run at the colder. They have it all on tap and it all goes to the front. Plus they got meat and stuff in here too so yep and look at that right there's the thermometer running about 38-36. All right guys it's going to wrap this one up. I did check with them about washing out the coil. They usually do their own maintenance on their coils and their HVAC stuff. We usually just take care of the refrigeration and if they can't get the other stuff working then they call us for that so simple stuff like maintenance things they have in their own maintenance department for that. I forgot about that so they're going to do that here in the next month or so washing out coils and things. Like I said no reason why the fans didn't come on it's because the defrost termination switch that's used typically in a freezer is what they're using in this cooler because they do run at colder temperatures that's why it has defrost heaters on it because they want to run it just at or below 30. That's the reason why we run that type of evaporator with the cooler is just so we can run those lower temperatures. Everything else looked pretty good for the age of it. Like I said it's 2005 and I'm sure we'll be back for plenty of other things that we work on. Guys I appreciate you taking the time to watch the video if you would consider subscribing. Check me out on Instagram and until next time we'll catch you on the next one later.