 Thank you ok. So, as you can see from the title. So, this will discuss some aspects of super strain partner version theory ok. And since these this topic is somewhat technical what I plan to do in this first lecture is to try to give you a general overview of the program ok or what we will try to achieve in these lectures ok. And then from second lecture onwards will become a little more technical ok. So, today I will try to keep it as non-technical as possible. So, the plan for today will be that I will briefly review the conventional approach and the shortcomings of the conventional approach ok. I will assume that more or less you are familiar with the way super strain partner version theory is done, but I will just say a few words about it ok. We will see that this these shortcomings ok will naturally lead us to consider off shell amplitudes ok. And I will discuss specifically what we mean by off shell amplitudes ok. And from this will be led to 1 by effective strain field theory ok. And this in turn will help us resolve the shortcomings that the conventional approach to super strain partner version theory have ok. So, this is the general program and I will try to summarize this development today. So, the standard formulation of heterotic type 2 strain theory. In fact, also Bosonic strain theory, but that of course, you know is not fully consistent ok. So, the standard formulation of strain theory in a given background is based on some two dimensional super conformal worksheet field theory ok. So, for Bosonic strain theory it is just a conformal worksheet field theory for heterotic it is super conformal, but it has super symmetry only on the right moving sector ok. And for super strings type 2 and type 2 super strings it is super conformal with super symmetry both on the left and right moving sector of the voice sheet theory ok. So, these super conformal field theories typically contain a matter system and also a ghost system ok, but the total central charge of the matter and ghost system always adds up to 0 ok. This is in fact, one of the consistency conditions that lead to the critical dimension of strain theory. So, in this formalism we introduce on-shell states ok. So, a generic state in the matter ghost conformal field theory of course, is not physical ok. We put some additional condition to identify which operators in this combined conformal field theory describe physical states and that is described by imposing the condition for BRST invariance ok. So, the on-shell states in this system are described by BRST invariant vortex operators ok. And unless we are at some special values of momentum like 0 momentum we can choose this BRST invariant vortex operators to be dimension 0 primary operators ok. So, they are conformal invariant vortex operators in this conformal field theory ok. So, this is the standard formulation of strain theory and once we have this background namely a specific super conformal field theory which satisfies the required consistency conditions ok. We can construct the G loop n point S matrix elements ok. There is a standard procedure for describing these ok and this involves first of all computing certain CFT correlation functions on a genus G Riemann surface with n punctures ok. So, n punctures are n mark points where the vortex operators are inserted. So, n point amplitude means they are n external vortex operators and you insert them as these at these n punctures ok. So, that is what I have said we insert vortex operators at the punctures ok, but as it turns out you also need to insert a set of ghost operators following specific rules ok. And if you have not seen this earlier I will describe these rules in more detail in lectures 2 and 3 as to how the string perturbation theory is built. And in super strain theory one also needs to insert a certain set of picture changing operators which are made of ghost fields and super stress tensor of the matter fields ok. Again if you have not seen this before I will describe this picture changing operators in the third lecture ok when I discuss super strain perturbation theory in detail ok. And finally, after you have computed this correlation functions we have to integrate this over the 6 G minus 6 plus 2 n dimensional moduli space of the corresponding Riemann surface and that is what is supposed to give the on shell amplitudes ok. So, here this number 6 G minus 6 this is the dimension of the moduli space of genus G Riemann surface and 2 n refers to the locations of these n punctures ok. So, each of these punctures require 2 coordinates because you have a complex coordinate to describe the location of the puncture on the Riemann surface. So, that is this number 2 n ok. So, integrate over the locations of the punctures you integrate over the moduli space of the genus G Riemann surface and you get your amplitude. So, are there any questions on this ok. So, what I have done is that I have given you the standard procedure that one uses for computing on shell amplitudes in strain theory. However, as you will now describe this approach is insufficient for addressing many issues even within perturbation theory ok. Of course, one of the standard criticisms is the strain perturbation theory ok is only a perturbation theory it does not tell you how to compute non-parturbative amplitudes ok. But what I am going to discuss here have nothing to do with non-parturbative aspects of strain theory just even within perturbation theory ok. This approach is insufficient for addressing many issues and these issues involve master normalization and vacuum shift ok. And I will discuss each of these in some detail ok what exactly we mean by master normalization problem and a vacuum shift problem in strain theory. So, I will begin by discussing the problem of master normalization what exactly is the problem ok. And so, for this we recall the standard LSD formalism for computing S matrix elements in a quantum field theory. So, this is the LSD formula for S matrix element in quantum field theory this is the formula for scalars there is a slightly different formula for formulas and vectors and so on. So, this formula in this formula g n is the n point rings function a 1 to n are the quantum numbers characterizing the external states ok n point rings function as n external states. So, a 1 to n tells us what those external states are ok k 1 to k n are the momenta that those external states carry the z i's at the wave function normalization factors m i p's at the physical masses of the i's external state ok. And I am underlined the word physical because these are not typically the tree level masses ok or the bare masses of the that you put in the Lagrangian these are renormalized masses ok renormalized physical masses. So, more precisely we have to compute m i p ok within field theory by looking at the locations of the poles of the 2 point function in the minus k square plane ok. So, you look at the 2 point rings function and see where the poles are in the minus k square plane that is what gives you the values of the m i p the physical masses ok. And the z i's are calculated in a similar way from the same 2 point function these are the residues at the poles of this 2 point function ok. So, that is the standard LSE procedure for computing X matrix elements in a quantum field theory. So, I will compare this with what string amplitudes actually compute ok the if you take BRSt invariant operators in conformal field theory and calculate correlation functions of Riemann surface and integrate you compute something. But what to compute is not this ok, but that is it has an analog in a field theory what to compute is this. So, what is this here? So, g n is again the n point rings function these are the external quantum number these are the moment as before ok. But what appears here m i is a tree level mass of the I f external state the limit that you take is k i square goes to minus m i square and the wave function numbers and factors are missing. So, this factor this limit the fact that you take k i square goes to minus m i square this condition in fact is needed to make the external vortex operators conformal invariant ok conformal invariance which is the BRSt condition for BRSt invariance is what you derive at tree level and that is what forces you to have to satisfy the tree level on shell condition ok. So, this is clearly not the same as what you should compute to calculate the S matrix elements in string theory. So, here I have again written down these two expressions ok this is what you should compute this is what we do compute in string perturbation theory ok and these two are clearly different. Now, it turns out that the effect of the wave function homologation factor is not hard to take care of it you still have to do some work, but it is possible to take care of this ok. But the effect of mass denomination is more subtle ok it is not an easy task to simply modify the usual perturbation theory to account for mass denomination and the reason is clear that the perturbation theory from the beginning asks you to be on shell ok because you are working with BRSt invariant vortex operators and once we have put this condition at the beginning you cannot really change it ok. So, we conclude that the string amplitudes compute S matrix elements directly only if the physical mass is equal to the tree level mass, but not otherwise ok. Now, of course there are many states in string theory which do satisfy this requirement, BPS states are examples, mass less gauge particles are examples ok and of course all amplitudes are tree level are examples ok tree level by definition MIP is the same as MI ok. So, for those you are perfectly ok, but not in general yes this one this is what it computes pardon. It is it is in fact, no it is not finite even finite it is infrared divergent ok. If there is a mass denomination you know that if you try to do this right you are not killing the pole correctly right and you will get an infrared divergent answer ok and so, the usual approach produces answers which are infrared divergent if there are a mass if the external states are mass denomination ok are there any other questions. So, this is the problem of mass denomination. Now, let me describe the problem with vacuum shift ok and here I will describe you to the example ok, but this is a generic problem ok if you do not have supersymmetry or if you have less supersymmetry ok. So, the example that I will take is compactifications of esothotic to heterotic string theory on a Calabiou 3 fold ok. Typically it turns out that when you compactify esothotic to heterotic string theory on Calabiou 3 folds ok one loop correction. So, this is an n equal to 1 supersymmetric compactification in 4 dimensions, but one loop corrections generate what is called a phylopulostat ok. Now, we do not need to really know the details ok what we need to know is what its effect is ok and one of its effect effects is to generate a potential of a charge scalar which I will denote by phi or the following form ok. So, this is a scalar that is charge under u 1 gauge field ok and you generate a potential for the scalar field of this form the c and k are constants that you can calculate using string perturbation theory ok. In fact, c you can calculate at pre level k you can calculate at 1 loop ok. G acr is a string coupling and from this it is clear that at pre level the potential is just c times phi star phi whole square ok it has a nice minimum at phi equal to 0, but once you take into account the loop corrections ok and you generate a potential like this ok phi equal to 0 is no longer the desired minimum ok and if you are a quantum field theorist you will clearly know what to do ok. You will take this potential and you will say that you will have to shift your field phi to go to the correct vacuum. So, the correct vacuum is at mod phi equal to gs root k. So, you should really quantize your theory around the vacuum gs root k you calculate all your scattering matrix by expanding the fields around that vacuum. The travel in string theory is that this vacuum is not described by a wall sheet conformal field theory ok because wall sheet conformal field theory describes classical solutions in string theory ok and the classical solution is phi equal to 0 ok not phi equal to gs root k ok. So, while from a field theory's intuition we are assured that there is a perturbative vacuum nearby ok we should be able to expand our fields around this perturbative vacuum calculate our s matrix elements ok. If you follow the standard rules of string theory string perturbation theory you cannot calculate s matrix elements around this vacuum ok. So, the usual conventional perturbation theory fails are there any questions? Well you can try to do that, but there is no conformal perturbation theory of course, requires regularizations and so on ok and there is a fixed regularizations you will not get an unambiguous answer by trying to use conformal perturbation theory ok. We will see in string theory ok there is a way to get unambiguous answer but not you have to go beyond the conventional perturbation theory because in conventional perturbation theory two of these operatives will collide ok when you try to do this and you have to know how to regularize it ok. Now, it turns out that even when all these problems are absent in conventional perturbation theory we have to deal with vacuum shift ok we have to deal with infrared divergences ok when we try to integrate over the modulized space at intermediate stages ok even in the absence of master or marginal vacuum shift ok and again it is easy to illustrate this by you drawing a field theory analogy. So, let us suppose that you have a field theory in which there is a Tadpole diagram like this and suppose further that this state that is propagating around the vertical line is massless ok. So, since this state carries 0 momentum you get a 1 over 0 that is clearly divergent ok. So, this diagram will be divergent in conventional field theory. Now of course, in conventional field theory you know what to do ok I will come to that the best possible scenario ok is perhaps this Tadpole cancels ok indeed if you have a supersymmetric theory often formulas propagating in the loop and bosons propagating in the loop cancels and gives you 0 result ok but that happens after you integrate over momentum. So, clearly at some point ok we need an infrared regulator ok we have to make sure that this 1 over k square which is 1 over 0 ok we regulate it ok do all or the rest of the calculations to see that this diagrams cancel and then we can set the regulator to infinity if you want. Now in strength theory the analogs of these divergences ok, but of course, you do not have diagrams like this you have a single Riemann surface of which you are on which you are integrating ok. So, what this kind of divergence translates to or what kind of this regularization translates to ok is that you have to use some kind of regularization procedure for carrying out the integration over the moduli space of Riemann surfaces ok. Essentially what you have to do is to put an upper cut of L on certain moduli ok which moduli these are the moduli which corresponds to the Schrodinger parameter of this vertical propagator ok in string perturbation theory each propagator is represented by in the Schrodinger representation ok. So, we regulate the infrared divergence 1 over 0 divergence by putting certain cut of L then we do do the integration over the other moduli that is the analog of doing this loop momentum integrals ok and then at the end we let L go to infinity at hope for the best that hope that eventually we will get a finite answer as you take L to infinity. So, the world. So, this works often ok when the tadpoles actually cancel nevertheless this requires an infrared cut off at the intermediate stages of calculation. So, the question that we would like to address is how do we circumvent these difficulties the difficulties remaster normalization and vacuum shift which are genuine difficulties and this kind of difficulty which requires use of an infrared cut off at the intermediate stage ok these are not genuine difficulties in the sense that you can live with it nevertheless it will be nice to not have to use our infrared cut off even at the intermediate stage of calculation ok. So, these are the questions we will try to address and the way we will try to address this is to go off shell ok and I will explain what you mean by going off shell as I go on are there any questions ok. So, let me go on. So, off shell amplitudes are of course, not new in strength only these are discussed by many authors even in fact, in the early days of string perturbation theory the one that I will be using is closest to the one which was first described by Nelson and then Zoibach used it in construction of a strength field theory. So, the goal here the what one does here is that one relaxes the constraint of conformal and BRST invariance on the vortex operators ok because this is the culprit right because if you the problem is coming because you insist from the beginning that all the external states are on shell ok. So, you just relax the constraint of conformal and BRST invariance but the price you pay is that the result now will start depending on the wall sheet metric in particular it will depend on the wall sheet metric around the punctures where the vortex operators are inserted. So, you have to first find a way to characterize this dependence ok and then see if we somehow we can get rid of these dependence at the end. And the procedure that these people used is the following. So, they chose some local coordinate system which I will denote by W this is a homomorphic coordinate system around the ith puncture for each high ok. So, around each of the punctures we will introduce some local coordinate system and take the metric around the puncture W i equal to 0 to be mod d W i square ok. So, we take the ith puncture to be situated at W i equal to 0 W i some arbitrary choice local coordinate system and you declare that the metric that we will choose is mod d W i square these of course will give a definite answer for the correlation functions ok even though the vortex operators are not BRST invariant or conformal invariant ok because I have already chosen specific metrics around the punctures. But clearly if we choose a different local coordinate system suppose you choose another coordinate Y i which is some function of W i then will this will correspond to choosing a different metric which is mod d I i square that is of course not the same as mod d W i square it is related to this by this vile factor ok and so you will give different offshore amplitudes for the same external states ok. So, the offshore amplitudes you can define offshore amplitudes in strength theory but the price you pay is that these start depending on the spurious data in this case the choice of local coordinates around each puncture. Now, for super strength theory the situation is even more complicated because we need to insert certain additional operators which will call picture changing operators on the Riemann surface and the offshore amplitudes now depend not only on the choice of local coordinates at the punctures but also on the locations of this picture changing operator. So, there is no way that we can make the offshore amplitudes independent of this external spurious data ok. But the real question that we are interested in answering is are the physical quantities that will compute from these offshore amplitudes independent of the choice of local coordinates are picture changing operator locations ok. This is perhaps not such a I mean such a surprise that it may happen that way happen this way because after all when in Gates theory when you try to define offshore amplitudes the offshore amplitudes are gauge dependent ok. Never the less we know that physical quantities like renormalized masses or S matrix elements are independent of the choice of gauge ok. So, the hope is that perhaps something like that can happen here also ok but this is what we will try to explore ok. Are there any questions? So, now I will introduce some notations as to what he how to precisely think of this choice of this extra data like local coordinates or picture changing operator locations ok. So, I will denote by m g n the 6 g minus 6 plus 2 n dimensional modular space of genus g Riemann surface with n punctures. p g n will be a fiber bundle with m g n as a base and the choice of local coordinates at punctures and PCO locations as fibers ok. So, this is an infinite dimensional fiber because the choice for example, PCO locations of course, there are finite number of coordinates that you have to choose, but the choice of local coordinates essentially involves choice of functions right and there are infinite numbers always we can choose these functions. So, fiber here is really infinite dimension. Nevertheless, we can think of this as a fiber bundle ok. And in this language a choice of local coordinate system and PCO locations correspond to is choice of a section of this fiber bundle ok. So, because for every point in m g n ok, we have to choose a specific point on the fiber which tells you how we have chosen our local coordinates and how we have chosen our PCO locations ok. So, the choice will correspond to a section of this fiber bundle ok. Now, there are various subtleties involving choosing these sections and those subtleties will be discussing as we go along ok, but for now this is a picture which is useful to keep in mind. And because this is a section the dimension of this is the same as the dimension of the base which is 6 g minus 6 plus 2 n pardon. It no it no in general it is not a vector bundle right because for example, locations of the picture changing operators right that is each picture changing operator can take value on the Riemann surface itself right. The Riemann surface is not a vector space right. So, it is in general not a vector bundle ok. Are there any other questions? Yes. Well the point is of course, you can nobody can prevent you from using the same operators right at defining the correlation function. The question is whether it is useful right and we will see whether it is useful or not right. So, that is the question of whether the dependence on this extra data go away at the end right. Because now because of what external vortex operators are not bearish invariant ok. If you change the location of the picture changing operator result will start depending on the on the choice ok. So, that is clearly not a desirable not a an encouraging situation official amplitudes will depend on the locations of the PCOs. The question of the final results physical quantities depend on them ok. So, we will be using the same PCOs. We are not going to change the prescription for the picture changing operators even though the external vortex operators are not bearish invariant. So, the procedure for constructing off shell amplitude ok can be summarized in the following way. So, for a given set of external off shell states which I will collectively denote by phi ok. So, phi can stands for a set of n external states in the Hilbert space ok. One constructs a p form omega p for every p in fact on p g n which satisfies an identity of this sort ok. So, omega p is a p form on p g n ok. You can construct this for every p ok and in fact, this is what we will do more explicitly in lectures 2 and 3 ok. What is this property? So, here QBI stands for the BRSTR charge acting on the ith state ok. For every state you can apply the BRST charge. So, you apply basically the sum of BRST charges acting on all the states ok. And the special property of omega p that we will be using is this one ok. Then omega p when you replace the state phi by argument by the BRST charge acting on the argument that is equivalent to removing the BRST charge and putting an exterior derivative on this and replace omega p by omega p minus 1. This is an identity ok that omega p satisfies ok and this identity can be derived from properties of conformal field theory the explicit construction of omega p will be describing later. And once this omega p has been constructed ok a genus J n point amplitude is defined to be the integral of omega g this omega 6 g minus 6 plus 2 n over S g n ok. So, omega you remember if we recall S g n has dimension 6 g minus 6 plus 2 n. So, we can naturally integrate a 6 g minus 6 plus 2 n form over a 6 g minus 6 plus 2 n dimensional space ok. So, even though we need omega p only for this particular value of p having this omega p is for different p's are useful for various manipulations are there any questions is this clear. Now, let us see how this helps. So, let us suppose that we have two different sections pardon p g n p g n was the fiber bundle this whole space ok. So, we concentrate omega p not on the modular space, but on the whole of p g n. So, let us suppose that we have two different sections and we want to see calculate the difference in this integral over this section and over this section ok. So, omega p over this section minus omega p over this section. Now, using Stokes theorem we can express the difference as integral of d omega over the over some region u that lies between in between two sections which fills in between the two sections ok. So, construct some space u which interpolates between these two sections. So, integral of d omega over u will be given by the difference between these two integrals ok. And now we can use the previous identity to write this in terms of omega, but with argument replaced by sum over q by i and phi. And now we see that if the external sets are BRST invariant ok. Then this is 0 ok, because this state vanishes and that immediately tells us that the result for integrating omega over this is the same as the result for integrating omega over this ok. It tells us why one shell amplitudes are unchanged when you change the section. Yes. So, typically there are you may also have to worry about the boundary conditions contributions and those have to be analyzed separately and you have to check that those vanish separately yes. What is the dimension of u? One more than the dimension of the sections. So, 6g minus 6 plus 2n plus 1 right because you have two lines imagine that you have in whatever dimension right if you have two lines you fill it by a surface right. So, the surface is not unique in general because you are embedding it in infinite dimension space. So, it is just one more than the dimension of the space on each one integrating ok. So, this vanishes for all shell states, but this clearly does not vanish for off shell states ok. So, we get back our old conclusion which you knew anyway ok, but this is just a different way of saying the same thing that the off shell amplitudes depend on the choice of local coordinates at the punctures and the PCO locations because as we change the section the result changes ok. This difference is not 0. So, our goal will be to prove that all physical quantities that we compute from off shell amplitudes are independent of the choice of local coordinates even though the amplitudes themselves are not independent ok. Amplitudes themselves clearly depend on the choice of local coordinates and PCO locations, but the physical quantities should be independent of this choice. Now, it turns out that for this we have to work within a specific specific class of local coordinates is quite natural ok. Nevertheless, we cannot allow arbitrary choice of local coordinates and this specific class of local coordinates is what I will call growing compatible local coordinate system. So, in the next two slides I will describe what exactly you mean by growing compatible local coordinate system and you will see that it is quite natural. Nevertheless, it is a choice that you have to make. So, we will describe now the growing compatible sections ok, what we mean by growing compatible sections and also consequently it will lead to the notion of one pi amplitudes, one particle irreducible amplitudes ok, but I will describe it as I go on. So, let us suppose that you have a genus G 1 remand surface with m punctures and the genus G 2 remand surface with n punctures and we have chosen local coordinate system around each of these m punctures on the first surface and around in each of the n punctures on the second surface. Now, let us take one puncture from each of them. Suppose, W 1 and W 2 are the local coordinates around these punctures and the punctures are situated by convention at W 1 equal to 0 and W 2 equal to 0. So, you always choose local coordinates ok, so that the puncture is at the origin. Now, we can glue them via an identification of this sort, W 1, W 2 is e to the minus s plus i theta ok and I will choose a convention in which I will let s run from 0 to infinity and theta run from 0 to 2 pi ok. Now, the lower limit of s is purely conventional, upper limit is not conventional, upper limit is something that you have to always choose to be infinity ok, because this corresponds to degeneration. Now, what it does, what this identification does is to glue the two remand surfaces into a single remand surface. It is not hard to see ok. So, you can see that by this identification if you take a circle on the first remand surface at mod W 1 equal to e to the minus s by 2 and the circle on the second remand surface as mod W 2 equal to e to the minus s by 2 ok, those two circles are identified right, because W 1, W 2 mod is really e to the minus s. Now, you can imagine that on the first remand surface you take this small circle, you take this circle this is mod W 1 is e to the minus s by 2 you take another circle on the second remand surface mod W 2 is e to the minus s by 2 and now imagine that on this remand surface. So, this is the exterior of the remand surface you go inside by this identification in the other remand surface you will come out ok. So, on the first remand surface as you are going inside the circle on the second remand surface are coming out ok. So, you basically glue the two remand surface into a single remand ok. So, this gives a family ok two parameter family of new remand surfaces which has G 1 plus G 2 and the total number of punctures is now m plus n minus 2 because two of the punctures have been used up ok. So, this is a standard procedure of gluing remand surfaces which is called plumbing fixture. I have told that I have already chosen local coordinate systems around each of the punctures for the first remand surface and local coordinate systems around each of the punctures of the second remand surface. So, when you glue the two remand surfaces together like this the new remand surface inherits automatically a choice of local coordinates around each of the punctures ok. And gluing compatibility real size that this is the way we should choose the local coordinates on the new remand surface ok. Now, this is a condition that you impose on the choice ok you could have this declared that we will choose the local coordinate systems on the combined remand surface independently of how we have chosen it in the original remand surfaces it will not be a natural choice, but nevertheless it would be a perfectly good choice you have a right to do so, but we will not do so ok. We will demand that whenever two remand surfaces are glued together to produce a third remand surface this way. The choice of local coordinates at the punctures are also chosen to be in the same way. And in super strength theory when you introduce PCOs picture changing operators there will be similar additional condition that the locations of the picture changing operators on the combined remand surface should be the ones that we derived from the original remand surfaces by that we will describe in more detail when we come to super strength ok. Are there any questions? Is this procedure clear? Yes, consistently in what sense? Yes, so that is why I have taken this to be from 0 to infinity ok. Now, if you find that you it does not work right you have somehow produced the same remand surface twice ok. There are two choices ok one is there ok let it be and I will tell you what to do ok how to compensate for it ok. The other choices are you basically scale your local coordinates ok, so that you you never happens ok. So, basically that corresponds to taking S not from 0 to infinity, but suppose you take S from a million to infinity right. Then you are always close to degeneration right. So, you will never reproduce two different remand surfaces this way ok, but it is not a serious issue because even if there is an overlap ok you can compensate for it ok. Are there other questions? Yes, so these are of course just coordinates right. So, what you get you should count the dimension of the moduli space. So, you see that here we get let us see. So, we get two extra right we get two moduli from S and theta ok. So, what you should do is that you count now on the first remand surface that you are gluing it was a G dash G 1 with M punctures ok. So, we had 6 G 1 minus 6 plus 2 M ok that is what we get from the first remand surface and from the second one you get 6 G 2 minus 6 plus 2 N ok. So, this is in fact 6 G 1 plus G 2 minus 6 plus 2 M plus N minus 2 ok, I think it is correct yeah minus 2 L plus 2 ok. So, these are the same dimension as the dimension of the moduli space of a 6 of the of the combined remand surface ok, but typically it will not cover the full moduli space right it will cover only part of the moduli space ok and on that part of the moduli space the choice of local coordinates should be chosen in this gluing compatible way right that is the demand. So, what this does this gluing compatibility ok, either it allows us to divide the contributions to the off shell greens functions into one particle reducible and one particle irreducible amplitudes ok. Now, this concept is a concept that is inherited from string from field theory ok. In field theory one particle reducible contributions are involved those Feynman diagrams which can be made into two separate Feynman diagrams by cutting a single line ok. So, the example of a one particle reducible diagram is this one in phi cube theory suppose you have something like this this is one particle reducible right because you can just cut this line or this line it falls apart into two parts ok. This is another example of one particle reducible diagram because you can cut this internal line and it falls apart into two parts ok. This on the other hand is one particle irreducible if you have a blocks diagram like this. So, this is one p i this is one p r. So, this one there is no internal line that you can cut single internal line that makes it falls apart into two pieces ok and one particle irreducible diagrams play an important role in field theory which I will describe later. But so, this is a field theory notion ok, but once we have this growing compatible choice of local coordinates ok. We can extend this notion into string theory by saying that when you have two Riemann surfaces that are joined by plumbing fixture that will correspond to two amplitudes joined by propagator ok. You just make this correspondence ok. Two amplitudes joined by propagator is what is represented in string theory by two Riemann surfaces joined by plumbing fixture. So, in this once you have made this transition of in the language ok. We will say that Riemann surfaces which cannot be obtained by plumbing fixture of two or more Riemann surfaces those correspond to contribute to one p i amplitudes ok. The Riemann surfaces which can be obtained by joining two lower Riemann surfaces will be declared as one p r one particle irreducible is this point clear ok. So, for example, if you have two three punctured spheres imagine that we have two three punctured spheres ok and these you have chosen your local coordinate system. Now, by gluing these we can produce four punctured spheres right take one puncture from which glue them you produce four punctured spheres ok. You just identify these each such gluing ok. There will be three different contributions like s t and h u channel diagrams each of these cover part of the modulate space of four punctured spheres ok. Those part of the modulate spaces will be declared as one particle irreducible ok. The rest which is left over will be declared as one particle irreducible ok. This is the way we divide the modulate space into one particle irreducible part and one particle irreducible part is this clear or any questions. So, this is another of saying this that once you have a growing compatible choice of sections what we can do is to identify a subspace of the full section which you can plot the one p i subspace ok. So, we had earlier denoted the full section by a line continuous line like this. Now, what we are doing is that we are identifying subspaces of this sections ok which are declared which I am declaring as one p i subspaces ok. And the idea is that all Riemann surfaces correspond to the full section s g n are given by Riemann surfaces in R g n and that plumbing fixture in all possible ways ok. This is the way we construct a one p r diagram in field theory from one p i diagrams. So, once you have know the one p i vertices you just join them together by three amplitudes and you get the full amplitude yes sounds yes because this division will depend on the choice of local coordinates right. Only after you have chosen local coordinate you can declare part of this one p i and part of this one p r. So, there are certainly a lot of arbitrariness in this division right, but that is fixed by the choice of local coordinates what will be one p i exactly yeah certainly ok. So, basically in this language right this two three point functions joining to four point function ok. What you are doing is that we suppose you have the full mobilized space ok. Now what this three so ok see here as I said there are three possible diagrams st and u channel ok. So, each of them are giving suppose you have chosen the the local coordinates on the three part or three puncture sphere ok. Now, you glue you get part of the four puncture sphere part of the four puncture sphere and part of the four puncture sphere ok whatever is left out you join in some way ok. These you declare as one p i one p i ok these are one p r one p r one p r and you join the sections whatever is left by something which will declare as one p i ok. Here of course, there is its arbitrary how you join it right you could have joined it like this or you could have joined it like this ok. And now coming back to you this question that imagine that you have covered some human surfaces more than once ok imagine that the situation is not like that, but it is like this ok. So, then you basically still can feel it right you that means you are basically subtract of what extra you have ok you declare that as a one p r one p i contribution is this ok yes yes that is right ok. So, because the sense you I mean each of this of course have an orientation right. So, the fact that it runs this way right means it will come with a negative sign. So, once this division has been made we can define the one p i amplitudes by a similar expression to what we had for the full off shell amplitude ok, but now restrict the integral over r g m ok. And once we have this off shell one p i amplitudes the generating function of this amplitudes is what is normally called the one p i effective action ok. And we explicitly construct this one p i effective action in the fourth lecture ok. And one p i action just like in ordinary field theory ok even in string theory has this special property that if we now compute the three amplitudes from this one p i action we get back the full off shell string amplitude including loop corrections which are given by integrals over the whole section s g n ok even though one p i amplitudes are computed by integrating over r g n. And now we can apply the standard field theory methods to compute renormalized masses s matrix elements etcetera from this one p i action ok. In particular the renormalized masses will be obtained by looking at the quadratic term in this one p i action and finding the inverse of the kinetic operator and its poles ok or the zeros of the quadratic kinetic operator ok. Those will be the renormalized masses s matrix elements can be computed similarly. In this approach ok that if we really have this one p i action and then you start computing amplitudes from there ok. We always have to first determine the vacuum by solving the classical equations of motion that is derived from this one p i effective action. And then we do perturbation expansion around this vacuum. As a result the perturbation expansion is free from all IR divergence associated tag poles ok. We will never get the tag pole diagrams of this kind that I described earlier because in this one p i action you have to first solve the equations of motion you have to make sure that the place around which you start expanding is a solution to equations of motion ok. So, one point functions will be automatically zero ok. So, there is no need to regulate infrared divergences even at intermediate stages of calculation ok. And this in fact, this is somewhat abstract, but this can be translated to a very specific prescription for normal perturbative strength theory. And this is also perfectly suitable for dealing with vacuum shift ok. Because if we have this kind of potential that we encountered earlier right phi star phi minus phi star phi minus a square whole square ok. In the language of this one p i action what we will find is that instead of having one solution you will have to have three solutions. Well let us suppose it is phi square that is that is let us suppose you have a real scalar function ok. Otherwise you have a continuous parameter. So, in this case the one p i action will have three defined solutions for correspond to phi equal to 0 and plus minus a ok. And you have a choice of expanding other on phi equal to 0 or phi equal to plus minus a ok. But of course, you will find eventually that the phi equal to 0 is not consistent ok. Because this is only part of the action that I have written ok. Once you include the deletion this vacuum has a deletion tackle the one point function where deletion does not vanish ok. So, deletion equation of motion is not satisfied by the phi equal to 0 vacuum ok. Because if you remember the potential it is known 0 at phi equal to 0 right. So, there will be a one point function of the deletion ok. So, if we really have this one p i effective action ok. Then we can deal with this vacuum shift problem because you simply find the correct solution expand the fields around that solution and then calculate tree level amplitudes with that action ok. So, the goal of role of string perturbation theory here is to produce this one p i action ok. Where of course, you have this integral over Riemann surfaces and so on right. One p i action is obtained by this procedure which includes doing all the stringing machinery but you do not integrate over the full modular space you integrate only over part of the modular space ok. Now, we return to our question ok. Clearly the one p i action depends on the choice of h g n ok. It depends on the choice of the section because the choice of R g n is controlled by the choice of local coordinates ok. In fact, also the choice of the PCOs. However, what one finds is that the one p i action that we get for different choices of h g n they are related by field definition ok. They are different ok, but you can construct an explicit field definition ok that relates the one p i action computed using one choice of local coordinates and PCOs to another choice of local coordinates and PCOs. And this immediately implies that all physical quantities are unchanged under this ok. Because you know in standard quantum field theory ok. In fact, this is classical field theory right because with one p i action you only do tree level analysis. So, if two theories are related by field definition the renormalized masses ok and s matrix elements remain unchanged ok. In this case renormalized mass is the classical mass because one p i action classical another results of one p i action is the full result in strength theory ok. So, this immediately proves that all the physical quantities that you get from this will satisfy the desired property any questions ok. So, we will explicitly write down this field definition ok. It is a little more complex, but as we go along in the final lecture I will write down what this field definition is. There is also a bonus ok. What one finds is that this one p i action automatically has infinite dimensional gauge invariance ok. In fact, this includes general coordinate transformation local supersymmetry and of course, much more ok. Despite the presence of this infinite dimensional gauge invariance ok one by train wonder about the gauge fixing ok to calculate s matrix about to gauge fixing, but remember that you are doing only tree level analysis right. So, no Fadyapokov ghost tested have to be introduced ok. Gauge fixing is reasonably simple and you can compute s matrix elements from these quite easily ok. And again this whole procedure can be translated back to some operation on string perturbation theory ok which gives you a result that is manifestly free from all infrared divergences ok and that manifestly tells you how to go to the correct mass cell you are not limited by this idea that you have to start with the VRST invariance state from the beginning ok. So, let me then end by listing the tasks that you have to perform ok. This is in the context of Bosonic string theory for super strings you also have an additional task in involving fixing the with the training operators. So, the first task is to find the parametrization of the space P g n and its tangent space ok. You need its tangent space also because to define forms you have to define omega p right. If you remember omega p is a p form on this infinite dimensional space. So, you not only need to know how to co-ordinateize this infinite dimensional space you also have to know how to parametrize its tangent space in a simple way ok. So, this is something that we will learn in the next lecture ok. Then for n given external states which will collectively denote as phi we need to construct this p forms omega p on P g n with the desired properties ok and the desired property is this. And finally, we will also have to find an appropriate algorithm for constructing doing compatible sections and one p i subspaces r g n of P g n. So, this is what we will try to do in the next lecture.