2/5 Curved - space supersymmetry
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Transcript: English(auto-generated)
00:15
So I will summarize briefly what we were left with during the previous lecture.
00:26
So we talked about supercurrent multiplets and specifically we focused on the case of n equal 1 field theories in 4D.
00:52
So supercurrent multiplets is the multiplet which contains the supercurrent which integrates to the supercharges and the energy-momentum tensor of the theory.
01:05
But as we saw, this multiplet contains a variety of other operators. So in particular the most general multiplet which this theory possesses, which goes under the name of the S-multiplet,
01:29
contains, besides the energy-momentum tensor and the supercurrent, contains a variety of other operators, among which there is a string current which is determined in terms of a 2-form which is closed.
02:06
Then there is also a domain wall current which is given in terms of a 1-form y which is complex and also closed.
02:35
And finally there is an R-current which is not conserved, generically conserved.
02:53
Indeed a generic n equal 1 field theory in 4 dimensions does not need to have an R-symmetry.
03:01
Then there is a real scalar and so these are 16 bosonic degrees of freedom and then there are correspondingly... These are not bosonic.
03:20
So there are 16 bosonic degrees of freedom and then there are also 16 fermionic degrees of freedom. In particular, besides the supercurrent, you also have fermions psi alpha and psi bar alpha dot. As we saw, this string current and the domain wall current will appear, so the corresponding charges will appear in the superalgebra.
03:56
We've wrote down the resulting superalgebra where we have the usual momentum appearing
04:14
and then this string charge which is obtained by integrating the string current.
04:23
Similarly, in the Q-alpha Q-beta anti-commutator you find the appearance of the domain wall charge.
04:46
We also looked at what kind of improvements the supercurrent can be subjected to.
05:00
We found out that also the improvements can be treated in a supersymmetric fashion. We have started describing possible particular instances in which this supercurrent can be improved to a shorter multiplet.
05:28
For instance, we discussed two different scenarios.
05:49
The first scenario happens when this form f, instead of being just closed, is also exact.
06:02
Then that means that the string current can be improved away and one is left with a smaller multiplet which is called the Ferrara-Zumino supercurrent multiplet.
06:27
This Ferrara-Zumino supercurrent multiplet contains less than 16 bosonic and 16 fermionic degrees of freedom. Indeed, there are only 12 plus 12 degrees of freedom. What happens is that there is no string current, there is no f and also there is a relation between a and the trace of the energy momentum tensor.
07:12
In particular, we saw that there are cases, there are examples of field theories which do not have a Ferrara-Zumino supercurrent.
07:26
Namely, if you consider a Bessumino model whose target space is compact, this cannot have a Ferrara-Zumino supercurrent because the 2-form f is going to be proportional to the scalar form and it's going to be therefore closed but not exact.
07:49
In general, whenever the scalar form of some Bessumino model is closed but not exact, then you cannot improve the s-multiplet to the Ferrara-Zumino multiplet.
08:07
There is also another example of a field theory which does not allow for a Ferrara-Zumino supercurrent and that is obtained by just taking the case of U1.
08:23
Let's do another example. If you consider some U1 gauge theory with an fi term, then your Lagrangian is going to include the
08:56
usual field strength part plus complex conjugate and then there is also the fi term, this usual superfield terminology.
09:16
Then one can show that for this theory, one can write down the supercurrent multiplet and in particular the 2-form
09:32
f which appears in the string current is proportional to the fi parameter times the field strength of the gauge field.
09:53
Hence, if it were possible to improve away this f-mu, then it would mean that
10:09
it has to be exact but it's not because A by itself is not gauge invariant. This is another example of a theory which cannot have a supercurrent which is improvable to the fc kind.
10:33
Indeed, in one of the exercises you could check that you could consider Sqld, so the U1 gauge field coupled
10:45
to matter and then you can check that this theory does indeed have string configurations which carry this string charge.
11:03
The other case that we took into consideration is the case where instead of being able to improve away the string current, you are able to improve away the domain wall current. In particular, we found that there is another multiplet which exists whenever the theory has conserved our symmetry.
11:35
When this current which appears in the multiplet is conserved, then again the multiplet
11:45
shortens and includes only 12 bosonic degrees of freedom and 12 fermionic degrees of freedom. So these theories, clearly you cannot have such a multiplet if your theory does not have conserved our symmetry.
12:11
For instance, one example could be if I take some Bessumino model with some generical cubic or even
12:24
non-cubic superpotential, this will not have an R symmetry, therefore it will not have an R multiplet. Another example which is maybe more interesting is that of pure Yang-Mills, so if you take supersymmetric Yang-Mills, the R symmetry would be anomalous and then this theory would not have an R multiplet.
12:55
So in both these cases, then when a theory of an FC multiplet, then there is no string charge in the algebra and
13:07
vice versa, in the case that the theory has an R multiplet, then there would be no domain wall charge in the supersymmetric algebra. We can also think about theories which have both an FC multiplet and an R multiplet, so certainly there are such theories.
13:32
So in these theories you wouldn't have either domain wall charges or string charges and then it may be that the improvement that you need
13:44
to make the S multiplets the FC multiplet and the improvement that you need to make the S multiplet the R multiplet actually do coincide. And then it is possible to get rid of short and even further the multiplet, so that's the third possibility.
14:04
So this happens when you can get rid of both this superfield chi alpha and the Y alpha in the definition of the S multiplet. And this happens for theories which are super conformal.
14:28
So indeed, if you look at the conditions that needed to be satisfied in order to have an FC multiplet and you combine them with those that need to be satisfied in order to have an R multiplet, then you will discover that clearly the R symmetry has to be conserved.
14:51
But also, to have an FC multiplet, this scalar field A had to be proportional to the trace of the energy momentum tensor, but to have an R
15:02
multiplet, this field A has to be zero, so that means that the trace of the energy momentum tensor is equal to A and is also equal to zero. Then you don't have a string current, nor you have a domain wall current, so FD nu is equal to zero, Y mu is equal to zero.
15:31
And there are also conditions on the fermions, namely what you find is that psi alpha is equal
15:46
to zero and sigma mu S bar is equal to zero and similarly for psi bar and S.
16:02
So these are indeed the conditions that need to be satisfied for the super current to actually be us. Well, that depends on which multiplet you want.
16:22
So if you do a soft breaking, then the trace of the energy momentum tensor won't be zero and this guy in particular will not stay zero, so you will have to fall into one of the larger multiplets.
16:48
Okay, so this is for super conformal theories. Is it possible to get rid of just the domain wall charge, for example, by keeping the J mu, for example?
17:01
What is J mu? J mu is the U and R current. Yeah, so that's the R multiplet. I'm sorry, I'm keeping that not to be non-conserved, but just to eliminate the domain wall charge. There are many ingredients, so there seems to be many options of which one to eliminate. Right, so as discussed, there are two possible ways to do it consistently with supersymmetry.
17:25
One reduces to the Farad-Zumino super current and the other one reduces to the R multiplet. Okay, but those are the only possibilities? Yeah, those are the only two sharpenings. Well, and the CFT shortening, which includes both.
17:42
So if you introduce a super potential mass, then that would break the R-symmetry. Right, so for instance, let's take this theory. So this theory is the U1 gauge theory with a phi term. It has an R-symmetry.
18:05
So indeed, you can write for this particular theory, you can write down some R multiplet. Indeed, I think the R multiplet is going to be just given by the following expression.
18:28
But then suppose you couple this theory to matter and you take some generic super potential so that the R-symmetry is broken. Then this theory will not have an R multiplet anymore, but it will only have an S multiplet.
18:41
So indeed, the most generic theory will have an S multiplet. And then there are more special theories that allow for Farad-Zumino super multiplets or for R multiplets.
19:05
And then in the intersection, there are some more even special theories that are conformal. What happens if this conservation condition is broken by the memory?
19:23
That's the same. So for instance, you could consider the case of n equal 1 super Yang-Mills, pure super Yang-Mills. So classically, it has an R-symmetry, but it's broken by the anomaly. So that means that this theory does not have an R multiplet.
19:44
I think all we use is the... No, but this is... Okay, so let me comment on this in a second. Okay, so these are the three possible shortcomings.
20:07
What happens if the f-i parameter is dynamical? Suppose it's a modular automata field, for example. Is the conclusion still the same? I'm asking partly because people discussed, for example, f-i parameter in context of d-parameter creation and cut into super graphically.
20:23
And we seem to say that... So if you have a theory which generates in the infrared some u1 gauge field with f-i parameters, that means that this theory cannot have an f-symmetry.
20:43
So this cannot happen if you start... Okay, so let me just make this comment now because it already arose twice, so I should address it. So one application of these ideas is that to put constraints on the RG flows of some theory that you are interested in or in general.
21:22
And so let me talk about constraints on RG flows.
21:55
So the idea is that the super current multiplet of a theory is some kind of short multiplet.
22:22
Or alternatively you can say that the super current is embedded in a super field which needs to satisfy some constraints. So then this means that the structure of the super current multiplet is preserved along the RG flow. So if you start from the nearly UV with the theory which has an f-symmetry multiplet,
22:49
then this theory will have an f-symmetry multiplet all along the RG flow. And the same is true for a theory which has an R multiplet.
23:00
So when you get to the... So the structure is conserved along the RG flow.
23:22
So there is a comment here that... So first of all, if you start in the actual UV, then suppose you have some asymptotically free theory, then there you will have a free theory which is also conformal. And then you might think that this does not really follow these rules,
23:44
but actually what you have to think about is the theory at any high energy, very high energy but not infinite. And then the statement applies. And the other caveat is what happens in the extreme IR. So in the extreme IR, your theory could be some SCFT.
24:07
So in that case, what happens is that some of the operators in the super current might become redundant. So they just decouple. They don't have correlators at separated points.
24:25
So but in between, just up to when you get to the deep IR, your theory will have the structure of the multiplet will stay the same. So in particular, that means that you can determine what the structure of the multiplet is
24:46
by just doing some computation in the UV. So now this computation does not just have to be a classical computation. It might include quantum effects of non-perturbative effects,
25:01
but it must be a possible computation in the UV. So for instance, for the case of super Yang-Mills, like classical, it would have an R-multiplet because it has conserved R current. But if you include quantum effects, you discover that
25:21
there is no R-symmetric current. And then the theory does not have an R-multiplet. It however does have an FC-multiplet. And indeed, in the theory, if you consider SCM super Yang-Mills,
25:42
it will have N distinct vacuum and there will be super symmetric domain walls that interpolate between the various vacuum. So that means that the theory does allow for a domain wall charge, but not for a string charge because it doesn't have an R-multiplet.
26:08
So between case 1 and case 2, is there a difference between how the theory can be coupled to super graphically? So that's the first comment.
26:21
But let me just say some more things about this. This can be used to constrain the behaviour of theories quite a lot. For instance, if you start with a theory which has a Ferrara-Zumino super current, then you know that it will not develop Fi terms for U1.
26:43
So in the IR there could be some emergent U1, some emergent U1, and then you could say, oh, maybe this emergent U1 will have Fi terms. Well, that cannot happen because then the theory will lose its FC-multiplet. And the same thing can be said for possible,
27:01
suppose that the IR description of this theory is given by some Vazumino model, then you could ask about its target space. Well, what we just said is if the theory is an FC-multiplet, then the target space of this Vazumino model that you have in the IR cannot be compact because if it were compact, then the colour form would not be exact,
27:33
and that cannot happen if the theory is an FC-multiplet. So there are all sorts of statements that you can make
27:41
by just using the structure of the multiplet on the behaviour of the theory along the RG flow. And then the other comment that we want to make is that indeed, eventually we are interested in coupling some n equal 1 supersymmetric theory to supergravity,
28:08
and then the structure of the multiplet actually dictates which supergravity you have to couple to. So in particular, there is the most common supergravity for n equal 1 field theories.
28:22
This is called old minimal supergravity. So in old minimal supergravity, if you count the bosonic degrees of freedom, there are 12 of them, and indeed this couples to theories which have an FC-multiplet.
28:44
So this is appropriate for theories with an FC-multiplet. And in particular, you cannot couple to old minimal supergravity any Vazumino model which has a compact target space.
29:10
Also, you cannot couple to old minimal supergravity a U1 gauge theory with an Fi term. Then there is another version of supergravity which is called new minimal supergravity.
29:29
I guess it's new because it was newer when it was invented. So this version of supergravity, which we will describe a little bit more of them in more detail,
29:46
this couples to the R-multiplet. So you can use it whenever you have a theory which has a conserved R-symmetry. Now, both the old minimal supergravity and new minimal supergravity have 12 bosonic and 12 fermionic degrees of freedom.
30:08
So they can couple to these shorter multiplets. And then the question arises of what happens for a theory which doesn't have either an FC-multiplet nor an R-multiplet.
30:21
As for instance, some gauge theory, U1 gauge theory with Fi terms and generic superpotential. So then, in that case, in order to couple to supergravity, you have to use a new minimal supergravity, which has more degrees of freedom in it. So this is called 16-16 supergravity and this couples to the S-multiplet.
30:58
So the price you pay is that you have a longer multiplet and therefore your supergravity contains more fields.
31:07
And you can also go in the other direction and if you have an SCFT, then you can couple it to conformal supergravity, which has less fields. Now, one important point, I guess, is that if you look at old minimal supergravity and new minimal supergravity,
31:33
then it is true that they are different, but the difference only arises in auxiliary fields.
31:41
So if you integrate out the auxiliary fields, they are actually equivalent. So, on shell, they are equivalent. These two supergravities are equivalent on shell.
32:02
But because we are interested in, like, we would be interested in, like, backgrounds, so in considering the fields of supergravity as, like, fixed backgrounds, then we will work off shell and therefore the two formulism, old minimal supergravity and new minimal supergravity, are actually not equivalent.
32:28
Okay. Are there any questions? Well, there is a similar story in 4th major, N equals 2. Yeah, so in 4th...
32:50
Okay, the N equal 2 becomes redundant. Yes, so also for N equal... So you could repeat this story for, like, theories with a different amount of supersymmetries,
33:00
and then it would be similar. Well, clearly the details would change. But yeah, so you can also have supercurrent multiplets for N equal 2 field theories. The most common one is called the Sonius multiplet,
33:20
and it contains, like, the conserved SU2R symmetry in it. But then, like, if the theory becomes superconformal, then the multiplet shortens, and indeed, like, you lose the trace of the energy momentum tensor you lose, you also acquire another conserved U1, because there is also...
33:41
then the R symmetry becomes SU2R times U1R, and so on. So in the 1616 supergravity, we are not just adding auxiliary q, we are adding... Yeah, so one way you can think about the 1616 supergravity is that
34:03
you take some... you basically... you have one of... you take new minimal supergravity, and then, like, you add an extra chiral field that you use as compensator. So indeed, yes, there is more propagating stuff.
34:24
So the 1616 super is not irreducible? Is it? Irreducible. Yeah, the 1616 super is irreducible or reducible? Well, it couples to this multiplet that, as we discussed, is generically not reducible.
34:44
The super gravity looks like irreducible. Again, this might be something which has to do with, like, imposing the equation of motion. I'm not sure. And also, there is a constraint, an equal one supermultiply,
35:02
in which you can replace one of the real fields in terms of the binomial formulas? I don't know about that. Like, after work of multiply? Mm-hmm. Okay, and we get... Yeah, we can talk about that later.
35:24
I mean, are you talking about... Okay, yeah, let's talk about that later. Okay, so are there any more questions about this different supercurrent multiplets?
35:53
Sorry, I don't understand the fact. You said that the various versions of the supercurrent multiplets dictate how we can couple the theory to supergravity. Why is that?
36:02
No, just because, like, given the structure of the supermultiply, you have to choose a different supergravity to couple to. So, for instance, at the linear level, like, you would have some coupling of the...
36:25
So, if you do linearized supergravity...
36:40
Alright, the light. So then, like, suppose you start with some theory in flat space, and then you want to couple it to linearized supergravity, so that, like, at first order, you would add a coupling of the supercurrent multiplet, S mu,
37:02
to some superfield H mu, which contains the linearized metric. And then, depending on which constraints S mu satisfies, that changes the nature of the metric superfield.
37:24
So that's one way to see the difference at linearized order. Then, I mean, if you want to do the full non-linear theory, you have to work harder. I'm sorry, one quick question. Just to make sure I understand your statement that the structure is preserved along the RG flow, does that mean if there exists a shortening condition on the S multiplet at any point on the RG flow,
37:43
then it must exist all along the flow? Is that the equivalent statement? Yeah, so, as I said, with the caveats of what might happen, like in the extreme IR where some operator might decouple. Could you repeat the argument for why that is?
38:01
Just, it's the usual argument that, like, it's in a short multiplet, so it cannot, like, the nature of the multiplet will not change. But sometimes, if you have more than one short multiplet, they can combine? Yeah, so, right. But, yeah, so you want to combine them in a longer multiplet.
38:44
I don't want to. Yeah, but I think then the other multiplets should already be there. And so I think that if there is already another multiplet which it can combine, then I think you're not in this case.
39:04
Okay, so now I can jump to something different. So what we want to put to use is, let me erase something first.
39:58
So now we want to put to use this structure that we have uncovered
40:04
to explain how to address the questions that we talked about in last lectures, that is, given some supersymmetric field theory in flat space,
40:22
we would like to understand on which manifolds it can be placed preserving some supersymmetry and what kind of properties the resulting theory will have. So the reference is for this part of the lectures. So there is this paper 1105-0689 by Seiberg and myself.
40:54
And there is also a nice review by Dumitrescu 1608-02957.
41:03
Okay, so let me start by talking about something somewhat trivial,
41:26
but which I think gives an idea of what is that we want to do in supersymmetric theory. So we'll just give an example which is without supersymmetry.
41:42
So let's consider some theory in flat space. Now you can take your favourite theory, maybe just some scalar field,
42:02
then we can couple it to gravity. So once you couple it to gravity, the metric becomes dynamical and your theory will be disinvariant.
42:34
So now the metric can be whatever you wish. So in particular you can imagine your theory being on some manifold of your choosing with some particular metric
42:47
and you can decouple gravity by sending the Planck mass to infinity. So now we can fix the metric on some manifold M
43:01
and decouple the fluctuations of the metric by sending the Planck mass to infinity.
43:22
So what I have done is that I basically took my theory in flat space and I have coupled it with this manifold with some particular metric. So now I can ask what happened to the diff invariance of the theory with gravity. Well clearly fixing the metric breaks the diff invariance of the original theory
43:47
but there are some diffeomorphism that stay. In particular all the diffeomorphism for which do not change the metric that you have chosen as background will be actual symmetries of the resulting theory.
44:05
So what are these? So if you look for a diffeomorphism for which the change in the metric is zero that means that the infinitesimal parameter for the diffeomorphism which is epsilon mu
44:22
has to satisfy the killing vector equation. So these are just isometries of your Riemannian manifold.
44:41
So this is a somewhat trivial example. You don't need to go through this series of steps in order to figure out that if you have a theory and you put it on some manifold with some isometry then it will have a symmetry corresponding to that isometry. So it's a way which explains what we want to do in the more complicated supersymmetric setting.
45:10
Any questions?
45:25
So one topic which will play a very important role in the following is that of coupling some conserved currents in your theory to background fields.
45:46
So for instance in the example above once you fix the metric that the metric becomes a background field and if you look at the deformation of the theory from flat space
46:05
then the deformation of the metric couples to the energy momentum tensor which is one of the conserved currents in your theory. So this is a theme which will recur during the next few lectures.
46:21
So I'd like to make some comments about this. What happens to this background metric under normalization? Under what normalization? How do you know? So the metric is a kind of collection coupling. So I'm fixing the metric to be some background.
46:43
We turn on the group effects. The metric might get a little more abnormal. The operators of couples to the energy flow. The first term is stress on the tensor but the higher the terms.
47:00
Yes, so it is true that in general like there are lots of I mean you can always change, so the coupling to the metric is not universal. You can always add terms which are suppressed by 1 over r where r is some scale in the manifold.
47:26
So in principle this could be generated during the RG flow. Right, so the metric itself maybe not even observable but the metric The metric is The curvature, things like that The metric is not an observable. We take it to be like a background field which we fix.
47:44
So it couples to objects in your theory. So if I change the metric this will So if I take my theory coupled to background metric and other background fields then I can compute its partition function on some manifold
48:03
then taking functional derivatives of the result with respect to the background metric this will give me information about correlators of various operators in the theory, right? I mean what's the difference between metrics and let's say the gauge coupling in Yang-Mills theory? So you could say well the Yang-Mills coupling is also a kind of background field
48:23
but we know it's not a parameter. You cannot compute the function of the gauge coupling. It becomes a scale if your theory is not conformant. So how is the metric different from that?
48:43
For example it could be that as we go down in energy flow the background metric has to become flatter and flatter. So it actually becomes just a flat space. Conversely there is some concentration of curvature and it becomes similar.
49:07
Ok, well I think this is a choice of scheme what they are doing. They fix the curvature and it can't change but then other quantities, the way they normalize will be affected and you can't really think of it as a choice of scheme
49:23
or the way you calculate your loops. It's like the opposite of two pole mass scheme. It's somehow the opposite of that. When you say the piece of curvature, I presume it's a constant curvature metric?
49:42
Constant. So that's a very special class of metrics, one feature only. I mean if I think about general form A fold it has many features. Here it's one curvature and there it's another curvature. What do I fix? Usually they will do sphere, so then very simple differences.
50:02
No, that's for specific discussion. We'll start with general discussion, so that's why I asked the general question. Ok, so let me continue but I'll go back to this question. So now where was I left?
50:23
I wanted to make some comments about coupling to background fields. As an example, again, without supersymmetry we can consider some theory with some U1 symmetry.
50:58
Then this will mean that your theory will have some conserved current Jmu.
51:10
I can imagine coupling this conserved current to a background, to a gauge field and then to make this gauge field non-dynamical.
51:21
So then the structure of your Lagrangian will have the usual Lagrangian you started with.
51:45
Then you add the coupling of the background gauge field Amu to your conserved current. In general this will not be enough.
52:02
Up to here this is invariant under gauge transformation of Amu because Jmu is conserved. This is true at first order. At higher order you might have to add terms of order A squared which are usually called siegel terms.
52:37
Again, these terms are here to preserve gauge invariance.
52:40
For instance, if you do this for a theory where the math, say you have some theory of scalars with a U1 symmetry then these are the usual siegel terms in scalar QED. Another example that we talked a little bit about is that of the energy momentum tensor.
53:13
Let's take some theory with some conserved energy momentum tensor which is symmetric. Then we can couple this to fluctuations of the metric.
53:25
So if you take Jmu to be flat space plus linearized correction then again the structure of your theory will be this theory you start with plus the coupling of the linearized metric to the energy momentum tensor.
53:48
Again, in order to preserve different variance at higher order you might in general have to add more terms. Here we can also discuss what happens to this under improvement.
54:03
As we said in the last lecture, Tmu is, even when we consider just the symmetric energy momentum tensor can still be changed by improvement transformations. For instance, it can be shifted by something of this form.
54:27
This is not the most general improvement but it is some improvement where U is some other scalar operator in the theory. Then what this does to this deformation is that it will add.
54:47
It changes T here but you can reinterpret the change in T by pulling the derivatives on H. Then this is just working us with the old T but adding a coupling of the linearized Dirichlet curvature
55:05
as a function of H to this scalar operator U. This makes the general point that by improvements I can shift the coupling to the metric
55:23
by terms which scale down like 1 over the radius of the manifold to some power. The Ricci curvature of the manifold scales like 1 over r squared. Indeed, the coupling of a theory to some curved space is not universal
55:41
but there are all sorts of 1 over r terms that can be changed. For instance, they can be changed by improvements of the conserved currents.
56:11
In the S-max set you have a real scalar A. Can you consider a similar linearized coupling for A?
56:27
Is the physical meaning clear in that case? To be fair, I don't remember what A exactly couples to in 1616 supergravity. In principle, A will couple to some operator in your theory.
56:47
Sorry, A is an operator in your theory. It will couple to some field in the supergravity. Other than that, can there be any physical meaning other than there is some coupling to some field?
57:07
If you do background 1616 supergravity, that coupling will be required to preserve supersymmetry. If it weren't there, you wouldn't have supersymmetric field theory. For other fields like C-min, C-nu, C-nu, C-nu, C-nu, C-nu, C-nu,
57:25
then the physical meaning is clear because of the brain current. For A? 16 supergravity is a little bit more complicated anyway because there are more propagating degrees of freedom.
57:43
In particular, both in the FC multiplet and the R multiplet, A is determined in terms of the... It's either zero or it's determined by the trace of T.
58:06
The other example that I wanted to make before getting to the more complicated supergravity case,
58:29
is that of the... Let's now add supersymmetry.
58:46
Again, consider some supersymmetric field theory which has a U1 flavor symmetry, some U1 global symmetry. That means that this theory will have a conserved current.
59:04
Let's call it J-mu, corresponding to the symmetry. But because the theory is supersymmetric, the current will be part of a multiplet. Exactly in the same way as the energy momentum tensor was part of a multiplet,
59:22
also just global symmetry will be part of a multiplet. Maybe this is actually an example I should have given before. This multiplet is a linear multiplet and it contains other operators beside the conserved current. It can be embedded in a superfield J, which satisfies the following constraints.
59:50
It's a real superfield which satisfies this constraint. We can write J in components. There is a bottom component.
01:00:00
which I will call j again, which is a scalar, a real scalar. Then there are fermions, little j and little j bar. And finally, in the top, in the component
01:00:24
with theta and theta bar, there is the conserved current plus other stuff. So again, there are four bosonic degrees of freedom and four fermionic degrees of freedom.
01:00:40
And then these fields can be coupled to a gauge multiplet preserving supersymmetry. So again, so the gauge multiplet
01:01:01
contains a U1 gauge field, a mu. Then there is an auxiliary field D, which is a real scalar. And then there are gauge units, lambda alpha and lambda bar a dot, alpha dot.
01:01:22
And well, also these you can embed in a super field. And you can write down a supersymmetric interaction between the fields in the gauge multiplet and the fields in the linear multiplet. So your Lagrangian will change by something,
01:01:43
which will contain the bosonic term we discussed above, the coupling of the conserved current to the gauge field. But it will also contain the coupling of the auxiliary field D to the scalar at the bottom of the multiplet.
01:02:01
And finally, it will contain fermions. And possibly, there will be siegel terms.
01:02:25
So now we can think of making this gauge multiplet non-dynamical. So we can set the fermions in the gauge multiplet to 0 and give some value to the a mu and D.
01:02:46
But these in general will break supersymmetry. So when is supersymmetry not broken? So supersymmetry is not broken when the supersymmetry
01:03:14
variation of my chosen background is 0. And what that means is that delta of a mu must be 0.
01:03:27
But that's fine, because if you look at the supersymmetry transformations, the variation of a mu is always proportional to the gauge units. So because we set them to 0, this will not give any constraint.
01:03:42
The same thing for delta of the auxiliary field. It's also going to be 0 because it's proportional to derivatives of the gauge units. So we only have to check that the variation of the gauge units themselves is 0. And that will give rise to some interesting constraints.
01:04:02
So for instance, if you look at the variation of lambda, then we get that right here. So there are two terms.
01:04:27
So you get that the variation of the gauge unit is 0 provided that this holds at least for some of the supersymmetry variation parameters. So that means that if you choose a background, which
01:04:40
means that you choose some f mu nu and some d such that this combination is 0, at least for some spin or zeta, then that spin or zeta will generate supersymmetry transformation of the resulting theory. And OK, there is a similar equation for the variation of lambda bar.
01:05:03
OK, is there any questions so far? If we don't go to the original gauge, we have a little bit more obsidian field. Can this help us a little bit to have a bigger freedom for the spin or zeta that we have to pick?
01:05:21
No, I don't think going, I mean, this, you always want to write gauge invariant interactions. So whatever brings you out of the Zumino gauge, I think, will be irrelevant.
01:05:48
Yes? So when you say, and you get another equation from lambda up to that bar, is that essentially going to be the complex conjugate, or will you get any new actual constraints? Well, OK, so if you work in COSC space, then they are related by complex conjugation.
01:06:04
OK, so you don't get essentially any new constraints on d and the gauge field from imposing that the other page in the Spanish operation. But who said that the background should be real? Again, so if you work in the Orenzian spacetime with usual reality conditions, then the background
01:06:22
will be real. But the background is just some Catholic. If you want some unitary field theory. OK, but indeed, when we go to Euclidean spacetime, then there is no reason why the background should be real. OK, so then, indeed, and besides,
01:06:42
in Euclidean spacetime, when you take the complex conjugate of zeta, you don't get zeta bar. So then the background, indeed, can be complex. And indeed, we will see it's important. So there are cases where it's useful to think about complexified background. Talking about Euclidean and COSC makes much sense.
01:07:02
Even the metric could be complex. If it's just the perturbation of flat space Lagrangian, I could have complex conditions. That's most important what I'm thinking. No, what I'm saying is that the correlation functions of the operators to which my background couples, I should allow all possible couplings to be able to really differentiate those terms.
01:07:21
Right, but in a unitary field theory in Orenzian flat space, the various operators that can couple to these currents will satisfy reality conditions. Unitarianity is a bonus of a special slice. Yes, yes, I agree. Then if you want to go beyond that,
01:07:41
yes, then you can consider also complexified deformations. Good. So what is that? Then you should write another equation as well, because they are truly independent. I agree, I agree.
01:08:01
Especially, we will write both equations. Good, so now we can go to the supergravity case. Is that it? There it is. That's much better.
01:08:35
OK, so next, let's consider the case with supergravity.
01:08:42
So what we want to do is indeed to couple some field theory with some supercurrent multiplet to supergravity.
01:09:01
So we take some n equal 1 field theory, or some SUSY field theory in general, and we couple it to the appropriate supergravity. OK, so at the linearized level, this coupling
01:09:40
can be described in quite some detail. And it will depend on the structure of the supercurrent multiplet that the theory has. So there will be the original theory you started with. And then there will be couplings of the supergravity fields
01:10:04
to the operators that appear in the current multiplet. So for instance, you will have the coupling of the linearized metric to the energy momentum tensor, but then there would be various bosonic fields in the supergravity.
01:10:25
Let's call them bi. And this will couple to various objects in the supercurrent multiplet, which we will call Ji. So again, in specific cases, this
01:10:40
could be the string current, or the domain wall current, or the arc current, and so on. Then I will have also couplings of the fermions. So there will be the gravitino, which couples to the supercurrent, the similar coupling for the other gravitino.
01:11:06
And then there could also be other fermionic fields in your supergravity. So those will couple to some fermionic operators in the current multiplet besides f.
01:11:24
And finally, there can be siegel terms, which are there. And you can discover by looking at the entire supergravity Lagrangian.
01:11:43
So some comments. So apart from the siegel terms, in order to work this out, you actually need to work out the entire supergravity Lagrangian. All these terms here are completely dictated just by the currents in your multiplet.
01:12:05
And in particular, they can be described even for theories which don't have a Lagrangian. So if you have some theory, it has some supersymmetry. It will have some supercurrent multiplet. In the supercurrent multiplet, they use these operators, and they will couple in this way to the various fields in the supergravity.
01:12:26
So this theory, it's like a supergravity theory. So it is invariant under local supersymmetry transformations.
01:12:48
And these are parameterized by some spinors, zeta alpha, which depend on the position you are in your manifold. And it's friend, zeta bar alpha dot of x.
01:13:08
So now we want to proceed exactly in the same way as in the trivial example we discussed at the beginning. So we will set our background to whatever
01:13:22
background of our choice. We will also set the, in our background, we'll set the fermions to 0. So we'll set the gravitinos to 0. And then, so we choose a background.
01:13:48
So for the metric and the various bosonic fields in the supergravity background, we'll set the fermions in the supergravity to 0.
01:14:05
And then we send the Planck mass to infinity to decouple the fluctuations of the supergravity fields. So what we are left with is some theory on some manifold with some metric. And there is going to be also various couplings of these fields that appear.
01:14:24
And in general, this procedure breaks all the local supersymmetries. Except, as we discussed in the previous cases, if there are some of the local supersymmetries that
01:14:45
keep the background invariant, then these local supersymmetries will remain in the theory once with the frozen supergravity fields. So in order to figure out which supersymmetries remain,
01:15:00
we need to figure out what are the conditions for this background to preserve some of the local supersymmetries. But luckily enough, this is not a very daunting task. What we have to check is the variations of all the various fields to be 0.
01:15:20
But the variation of the bosonic fields in the supergravity are always proportional to the fermionic fields in the supergravity, which we have set to 0. So those are not going to give us any equation to solve. So we only have to check that the variation of the fermions is 0. And in particular, for the case of the minimal supergravities that we will be concerned with, we just
01:15:42
have to check that the variation of the gravitino is 0, and its friend. So now this equation has a general structure that we will see borne out in examples.
01:16:02
So it starts with the covariant derivative of the spinor parameter zeta. And that has to be equal to some matrix M, which depends on the metric and the various other background
01:16:27
supergravity fields, which acts on zeta. And then there can also be another piece. Let's call it M tilde acting on zeta bar.
01:16:52
And there is a similar equation which comes out of the variation of the gravitino with the bars. So basically, the task of finding which manifolds allow
01:17:06
for some supersymmetry just becomes the task of finding for which values of the metric and various auxiliary fields you can solve these equations, or you can find some solutions to these equations.
01:17:21
So now I would like to make some comments on this equation and the general structure of them. So one important comment is that this equation does not really depend very much on which theory you started with, because what these objects depend on
01:17:42
are just the background supergravity fields. So there are no fields. There are no matter fields inside these matrices. Because if you work off shell in the supergravity variation of the gravitino, you only have fields in the supergravity multiple appear.
01:18:00
You don't have fields in the matter multiples. So this is an advantage of working off shell. So you can solve this equation, and then the results that you have will apply to many different theories, not just one. So in some sense, finding supersymmetric backgrounds
01:18:22
is independent of the theory you start with, except that this is not a completely correct statement, because as we saw, depending on the theory you start, you have to choose a different supergravity to work in. And the supergravities are not equivalent of shell. So depending on the theory you have, if the theory is an FC multiplet, you will use old minimal supergravity,
01:18:42
and then you have one equation. And you can find a certain set of backgrounds which are supersymmetric. If the theory is an R multiplet, then you have to use new minimal supergravity, and you will find a different set of equation and a different set of solutions.
01:19:02
Any questions on this? You could decide to just work with conformal supergravity if you wanted to couple conformal field theories.
01:19:21
But I think it's actually important to use the non-conformal supergravities.
01:19:43
First of all, you might be interested in coupling non-conformal theories. Even if you're just interested in conformal theory, many times to do some computation, you might have to introduce some regulator. And then this regulator will introduce dependence on the break, usually breaks conformal invariance.
01:20:05
So then I think it's more appropriate to use the non-conformal couplings. So for instance, if you compute some partition function using some regulator or some regulator procedure, this will depend on the background fields.
01:20:25
But the possible dependence on the background fields will have to follow from some respect gauge invariance. So they will be dictated by some supergravity.
01:20:43
And the appropriate supergravity, I think, is the non-conformal one. So it's true that you can choose the scheme carefully so that you will be left only with the couplings which are conformal. But that might be difficult in any specific scheme.
01:21:05
There are often physical questions that are asked for conformal. They're just conformal theory, right? For such questions, does it essentially try to give results or answers in terms of conformal supergravity?
01:21:22
Somehow, not many people seem to do this. I don't understand why. Is there a good reason? No, but as I said, if you are just concerned with conformal field theories, then you can use conformal superability.
01:21:40
Up to this comment that when you actually do some specific computation, you might have to introduce a regulator and this may break conformality. Is there a more supersymmetric way of analyzing this? Yes, there is. Yes, so supersymmetric, you mean
01:22:00
can you work in superspace? So there is a way to encode all this in some superspace formalism. I think you can actually find some of it in this book by Kuzenko, which is called A Walk Through Superspace or something.
01:22:21
I don't remember, but it has some comments about how to get super currents. I have five minutes, so I don't know
01:22:44
what can I do in five minutes. Let's see. Well, OK, so I can give you, I guess, an example so that this does not seem too dry.
01:23:01
So let's see what happens for a theory which has an FC multiplet. So then, as we said, this theory will couple naturally to old minimal supergravity.
01:23:22
So then we can be a little bit more specific. So first of all, let me say that I'm going to write the coupling. So you have some theory with an FC multiplet. So the FC multiplet that we discussed in the lectures, we had this superfield Y alpha.
01:23:41
But we said that in many cases, Y alpha can be written as D alpha of some kind of superfield X. So this is the case I'm going to consider. So then, the toppings that we've
01:24:01
wrote in the previous blackboard are going to be, OK, so there is going to be the coupling of the linearized metric to the energy momentum tensor. But then, you have some coupling of some auxiliary vector field in the supergravity, which we call B mu, to the non-conserved R current,
01:24:23
which appears in the FC multiplet. So this is a supergravity field, which maybe I can write the supergravity fields in a different color. So that's a field in the supergravity,
01:24:41
which is an auxiliary field. And we also have couplings of some other complex scalar auxiliary field in the supergravity to the bottom component of the multiplet X.
01:25:14
And in this particular case, we can also write down what are the equations that come from setting the variations of the Gravitino to 0.
01:25:25
So here they are, not maybe the best idea.
01:25:58
And there is another one for the right-handed.
01:26:27
OK, so here they are. So as you see, the general structure that I advertised is indeed borne out by this example. These equations, they only depend on the supergravity fields.
01:26:43
OK, so now let me make some general comments, which actually do continue to be true in other examples. So if you look at some specific manifold,
01:27:00
and you solve these equations, then you can indeed check that there is a scaling. All these fields, all the auxiliary fields in the supergravity multiplet M and B, and bar in this case, scale like 1 over R.
01:27:22
So that's just given by their dimensions. And again, this same field, M and B,
01:27:44
will also appear in the transformation laws of matter, as we will see in the next lecture. So what that means is that in the UV, the theory that we've wrote approaches a SUSY theory
01:28:11
in flat space, if you want the original SUSY theory in flat space.
01:28:25
So by using this formalism, you are not going to obtain the formations of the theory that you started with, which are present even in flat space. So there are the formations of the supersymmetry of some theory that might be there even in flat space,
01:28:43
and we are not going to obtain those in this way. And the other thing is that, again, as we discussed before, if you improve the FZ multiplet, then this will change the various currents which
01:29:01
couple to these fields, and this will introduce the difference which is going to be in the couplings which scale as the radius of the manifold becomes big. So that's exactly as in the bosonic case. And the final comment that I wanted to make, but which I've basically already discussed,
01:29:20
is that it might be important to consider cases in which the auxiliary fields or whatever, the fields in the background supergravity are not real, especially when you consider the Euclidean case where the spinors are not
01:29:42
related by complex conjugation. Now, however, to be fair, I never consider the case of a complex metric, so that might be something that some of you can think about. That's it.