1/2 4d N = 1 localization
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00:00
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21:16
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Transcript: English(auto-generated)
00:15
First of all, thank you very much for the organizer for inviting me to talk in this exciting school
00:26
And So I will talk about something about the for the integral one supersymmetric localization actually actually for the integral one supersymmetry grocery is a bit difficult to Consider using a localization, but there are some example to use a localization technique
00:46
So I'll comment But first of all, I will actually For the integral one case there are many works done by a cyborg and cyborg written or and of course if mom by
01:01
science are off there are many nice exact result you Obtained using the holomorphic of the super potential. Okay, so first review I mean So in a flat space Lorentz temperature
01:36
Okay, so I introduce what's a gauge in condensation here, okay, or something some other things
01:45
First to fix an notation So the algebra for the for the n equal ones there are no central charges so there is only the
02:05
Sushi generator and the translation Okay, and the non-trivial Commutation is like, okay
02:27
okay, so this is yeah, so okay, this is a
03:37
my normalization
03:41
okay, and I Assume you understand the Text I mean result of the giving a textbook by the best backer so so
04:05
So in for the n equal one case, there's a notion of a color super field it
04:42
okay, this is a definition of the color super field as you know
05:24
Okay, so color super field contains a complex scalar and Wyschina and auxiliary field data, okay, and Okay, super potential is
05:54
So this is the invariant. Okay, but there are no Anticarous super field, the anticarous field is defined by the dagger of this one
06:03
We include this one, then This is not Sushi invariant then should be
06:32
And then since the in our own as effective action
06:54
It's a very very complicated in use in user because of the quantum correction, but
07:02
still this should be polymorphic function and This is a very restricted because the homework function is very circuit and also in this case For example, this file contains some symmetry Index or something like that then we can construct a five five dagger or something like that and contracted say index
07:23
Then it becomes a singlet of the index. Okay, then symmetry doesn't Constrain anything about dependence of the five five dagger. Okay So but for the homomorphic case only the fire appears there's no five dagger
07:41
So we cannot construct a singlet for that Then the symmetry constant electric potential very very much and then we can Have some exact result using this holomorphic because of the Symmetry and some limiting to a studio
08:02
Constrains, no matter if extraction very very so this is a idea of the holomorphic as you know probably
08:20
okay, and Okay, so okay, so then we have a exact Desire for this doing as effective action. I mean or more precisely doing like in
08:42
This low energy super potential in a low energy because can be or sometimes can be determined sometimes cannot be but sometimes we can And next I important things is a higher angle
09:03
Okay This notion. I mean this notion are not I mean special for the for the n equal 1 But of course for the r2d2 comma 2 or 3d n equal 2 n equal 1 almost any cell we have this structure I mean that this holomorphic structure so we can use this idea for any
09:21
almost any Super symmetric service and this coloring is also a very general idea that that is a lowest component of I
09:46
satisfies
10:17
Here we denote this suji denoted by a decomposing this that into the delta epsilon direction bar. I mean
10:27
This is for the This this this epsilon bar say it's Vanishes acting on this phi phi is the lowest component of the scarf. Okay
10:44
okay, because the Transformation is this one. I mean from from this I mean super theater we can Find so the transformation is this So there's no epsilon bar Okay, so so phi is I mean phi is a delta cross
11:03
Okay, phi is delta closed and also This d mu phi Derivative of the phi is given by the delta bar delta because the delta bar
11:25
acting on phi vanishes so committed of this one gives a translation, okay So there mu phi is identify the delta bar delta phi. Okay. So this is This means this
11:43
Phi there mu phi is a data by exact so he conscious a cohomology class of this data bar then This I mean there I mean some dependence on the coordinate doesn't matter, okay
12:04
So this is called coloring Okay for any call one super here but of course we can use this one to the to the any code to come to Any other thing?
12:22
Next okay Okay, so then okay, sorry To be precise Because there a mu
12:41
Translation and committed I mean so yeah, they are committed each other. I I don't I don't so Phi is a color field. I mean, what's the comment of a chiral field then there me phi is also color field, okay So we can write down this one, so
13:04
Okay So phi sorry phi. So phi X is identified as a phi zero with this Delta bar exact term
13:23
Okay, so this gives you a color ring in a general n equal one case service and then we consider the Insurgent of this one, I mean the n phi
13:42
This one So this is a Position independent because actually this can be Then and of course this coordinator or the vacuum
14:13
Is invariant and this is passivity transformation. So this panishes, okay
14:21
Okay, okay so using this expansion to this one then This correlated phi are here this phi 1 phi 2 this insertion all insurgents are chiral in a chiral ring, okay So then the committee I mean collision function was a
14:41
chiral ring is coordinate independent okay, so so this is a
15:01
We can change this X to other places everywhere. Okay, and then we take X goes to infinity and then the superscript field theory is assumed to be Assume assumed to be Cluster
15:22
Decomposition, okay, if we if we start Some operator into here and very far away and I another Universal something somewhere then there is no correlation. Okay, so it's a factorized Okay, so then this these and coordinate independent
15:43
so this coordinator becomes Okay So for the this carding or this a color of color field
16:07
If we know the This condensate for the gauge invariant operator, then we have all information okay, so
16:23
So important things is this one point function for the R4 case Here using cluster decomposition, are you okay with that?
16:51
Here yes, that's true actually, yeah actually, you know here this correlation function can have
17:00
non-zero value for the non-compact case Okay, and that case Actually, and then usually this is non vanishing for the anomalous you won't kiss Okay, like a supersymmetric QCD or use our supersymmetry. Yes, it is not conformal
17:21
Yeah of course a conformal field theory we cannot give a Expectation value so this is for the R1, 3 or something like that Coma yes, yeah because of the asymmetry. Yes. Yeah, usually yes, but Yeah, okay. So
18:14
Finally, I I'll explain a bit about for the analytical ones super QCD very very briefly
18:22
This is a for example If you don't know much about this you consult this famous Maybe by interrogate and cyber Okay
18:40
This theory is a gage group is a CMC and NF fundamental flavors fundamental means a fundamental definition of CMC And then the color super field for this theory. So this is
19:46
Okay, this I is a flavor index we did not I Didn't write down the gage in this But this is adjoint. This is a fundamental. This is a fundamental Okay, this is called the N equal mass super QCD in general, okay, since the action
20:09
of this theory is
21:06
This is action Okay here this one Contains the F squared. I mean Yeah, I'm use coupling and also if F star F I mean topological term and this tau increase that I mean theta angle and also
21:26
Gage coupling constant for the M. You see, okay, and then this have 1 over G square times F square plus theta times F F to
21:42
as a So in the supersymmetric gauge theory these tools are combined into the one super field, okay And this is mass. Sorry. So this is a this is a mass term and
22:01
We can give only the Mass term by this Q Q tilde combination not a Q Q dagger or something like that. This is a holomorphic combination Okay, so this M is also holomorphic Okay, we don't we cannot introduce Yeah, I mean Q in a user QCD There is a mass like a Q Q Q Q dagger or something like that, but we cannot introduce here
22:24
keeping super symmetry We keep this so we need a Q and also Q tilde it's a different kind of super field so gauge invariant
22:46
What is gauge invariant kind of super field of this theory the mission and?
23:14
M tilde is a given by Q Q and also there is a volume. It's like a Q
23:28
Q times epsilon R1 to R and C but I'm not explain it and the other
23:40
There is a another one. That is a gauge, you know or glue ball glue ball or gauge, you know by linear This is given by this So this contains a sorry
24:18
This one yes, this one too. Oh, sorry
24:27
Children and the barium sorry barium and the There is a beach either for
24:51
Okay, and W contains a Filimium as a lowest component
25:03
So this component contains a trace lambda alpha lambda so this is a film in by linear and You know user QCD. Oh, yeah, miss user QCD. I mean, you know, you know, wow the film in by linear often condensate
25:24
sometimes it's called a kind of condensate and sometimes so this Gauge in by linear condensate, I mean nonzero eigenvalues and to compute it for the dl6l is very very difficult But for this types of a supersymmetric case we can compute this
25:44
gauge in condensate Exactly Actually to to compute this one instanton is not sufficient instanton have a I mean To nc0 mode, but this has a two only two so instanton cannot give a nonzero value for this one
26:02
Okay, so this is a purely strong coupling. Actually have a nonzero expectation value for this s Okay, and as I explained we
26:22
Because a correlation function factorizes because the correlation function factorized
27:03
if we know the one-point function, then it's sufficient and
27:27
In this case It's not conformal If this if nf equal to nc it's going sometimes
27:42
Sorry nf equals 3 nc. It becomes conformal, but the user is not conformal. I mean there is a Anomaly or the beta function is nonzero scaling anomaly. So
28:10
And actually using a holomorphic for the coupling constant tau
28:41
this tau tau is in a super potential so it should be holomorphic and also theta is Topological term I mean theta fun. I mean in a topological term and the theta should be periodic theta should be equal to theta plus 2 pi okay, so using that one this
29:03
Beta function should be becomes one loop exactly So this is I I said this is one little bit function, but this is exact beta function was a super symmetric case I Will not explain about that but it's known and
29:25
Then I'll not write down the symmetry of this theory in there is some flavor symmetry you want our symmetry and And As I say I'm using this anomalous symmetry also. We can fix the form of the super potential
29:50
This is a famous one
30:14
Yeah Like
30:25
Yes Okay, sorry, yeah, okay Yes, yeah, no question technique is more I mean general in but Yeah, this is a history. Okay, what we said this is very famous on under
30:45
This is a in some sense very Strong I mean this method I mean holomorphic method is very very strong and It's like a magic As I said, so this is a from symmetry and so this is a
31:04
symmetric and flavor symmetry and also Anomalous you want our symmetry because this lambda we can give us Some transmission rule for this lambda for the anomalous you and something like that then We fix this one and actually this coefficient is not fixed by the symmetry, but we should do this some
31:25
Some computation like a instanton at the somewhere but or using a cyber witness theory But anyway, we can fix this Numerical factor by some some way and then so this is given by this
31:48
okay, so So then integrating out and I J. It's a meson field. Sorry. Oh, okay. Sorry. This is for the
32:34
Then from this super potential Oh, this is a f-term
32:40
condition so M
33:04
Is given by this combination from this equation for large mass?
33:33
Then inserting this one to W effective potential Then we found
34:13
So inserting this M into this potential we have this one, okay This is just a computation and then we have this one if we define this one, this one is a
34:28
Dynamical scale I mean defined by this one for the low-energy super Yang-Mills because here much is very large So the croaks much better Q and Q should are decoupled
34:41
Completely, then there were a theory becomes a Yang-Mills cell. Okay, so then the wonder Effect the whole drama is I'm sorry is given by the NF equal zero. Yes, this one this is called a Matching condition for the dynamical scale and this is it. I mean, this is the exact result. So This is the low energy
35:03
Danica scale Okay, so then we have this effective super potential for the role as effective action and First of all, how should notice that there is one of I see here that means there is a NC branch
35:21
Okay, so this theory has a NC back here because there is a NC branch. Okay So actually we can compute that with the index. I mean, it's like a localization computation I've explained about that, but we can compute with the index of this theory on the torus then we have a NC Okay, so this is consistent with with index computation first of all
35:45
Okay, so this is a
36:02
result
36:36
Then okay, we can we can compute a gauge of condensation from that results
36:50
That is given by trace
37:03
So maybe it's a bit Difficult to Get this one from that one actually to do this we need to lift gauge coupling constant tau to the color super field and
37:26
Introduce some auxiliary field like this one and Consider this as something of your field and
37:41
only the this tau Expectation value, okay So this is IR, yes then we can so this is also a service some technique to compute Some some quantities, okay to some coupling constant
38:02
I mean if things are coupling constant into the chiral super field the chiral super field should be holomorphic I mean holomorphic, I mean superposition should be holomorphic dependent on the chiral super field Okay, so then but I mean this tau is also Appear in a chiral army a holomorphic
38:20
then the G means, okay, so So this is trace lambda lambda
38:40
because As I wrote down the super potential contains a tau times ww. Okay, and the tau contains this Theta theta term f tau so f tau coupling to the lambda lambda lowest component of the ww Okay, so if we gave it this f tau then it gives this one And then I we know now we know what is log Z in a low energy effective potential
39:06
Okay, so of course first integral after the first integral So this is given so this should be derivative of the effective super potential This one Okay, so then
39:22
Here this f tau is a derivative is tau for the lambda so then
39:43
then okay, so this Lambda is created by this relation lambda on the top. Okay, so log lambda Okay, log lambda is proportional to tau, okay So in tau, it's a becomes the correct super here, okay
40:03
Then David over the tau gives a real lambda and for this case due to this one gives Sorry, this one this one gives this actually out this is To okay, sorry
40:26
Lambda I said lambda is a Lambda is a chiral super field lambda is the lowest component of the W alpha. I mean capital R. Okay. Yeah Right, right, right, right. Actually. Yes. Yeah, because of this relation tau is
40:43
Identified as a lambda. So if tau is a chiral super field and lambda is a chiral super field or something like that Lambda zero is the lowest component of the W alpha Okay, well maybe so I I start from the
41:03
Super qcd but we can consider the unethical zero case even that case we This should be correct. I mean from the super. I am just a super. I am a theory without a flavor Then the reaction should be this one With this lambda zero is given by the original I mean yam is action and
41:24
Then the action can be lifted to the tau tater then we can such a computation
41:46
Over So that is not really The the one with W alpha W alpha because in B is a deeper B in B the real super field if we express the the super field strength in terms of the real super field
42:08
Which we are really integrating That is a theater. Oh, no, no, no, actually what we consider just tau as this This chiral super field and we we think I mean we pretend to
42:25
first integrate out of this data also, but for example, we We give us some weak coupling kinetic term for this tau then this can be a free story with this This expectation value
42:42
Okay And then nothing is nothing is defined tau is just like a spectator I mean just Just a treat or something. Okay Okay, but so actually there no Yeah, of course. Yeah, we need a kinetic term for this tau it is
43:03
I mean, no, it is a care of potential right that it true but yeah Coupling of the tau to the W is a tau WW only, okay I won't it's not only about that. Yeah. Yeah. Yeah. Okay. So for in this case we can
43:33
Compute this quantities or maybe from this quantities we can deconstruct this super potential
43:40
Using an inverse direction. Okay, it's called integrating method technique or something like that Okay, so if we know the gauge in condensation in some theory is a maximum something then we can reconstruct almost everything from this one I mean like M expectation value of the method is given by this one of is a
44:01
From this super potential so from this one we can have this one then we compute every expectation value of the coloring so it gives almost all information about the exact results Okay, but you may think This is a I mean, what do you know about this theory is probably the cyborg do up, okay
44:28
so actually I It's a I mean It's a for the this window
44:41
as you know Okay, this is for the NF is this sign C and for this region and this is of course different from this one Zanes are blow energy. This is high on trivia
45:14
Found by or claimed by cyborg and moreover I think you you know about this fact, but I'm not fact about this claim, but this is a famous
46:24
famous, I I think Sorry Okay, so here this M is This theory has a free I mean some
46:42
Some flavors user denoted by Q and Q children, I mean small small it So this M is completely different from this Q and Q children. This is just a shingled. Okay in electric. Sorry, this is Mason I mean Mason is a composite of the Q and Q children. Okay, but here this is a completely different
47:03
Thank you consider this theory. This is a very similar to the original SN super QCD, but at low energy limit The theory should be same. Okay but Maybe this is a called the cyborg dual, but it's not actually a dual or maybe it's called the IR dual
47:21
But it's not the dual just no any salary is same But the asymmetry is different so This is a sorry NSE NSE children is So gay symmetry is different, but it gives the same theory. So this is very surprising
47:46
About the cyber claims that one because of the some symmetry matching and also are called the Tofut anomaly matching condition So this satisfies This series, okay, so this is nice
48:00
but actually this entry because of this because the super kohmer index then People found actually these two series have a same Super kohmer index. This is a very nice things about that. I I probably I will talk about Okay, okay then I will move to the localization so
48:50
Okay, no question about that, okay, so
49:34
I will summarize what condition for the
49:56
Localization works. Okay. So if these conditions are satisfied, I mean action is invariant and it's no anomaly and
50:27
This is closed, I mean suji invariant, I mean and also there tabu is also Invariant, okay. This is the condition and then we can say this one G is key independent
50:46
And also if If this country is satisfied Our address there is a some bound on this one for the bosonic path, then
51:03
we can take T goes to infinity limit and first integral Localize
51:27
Phi 0 is given by this one. So this is what the Francesco and every sticker said, okay And there's a canonical choice for this way for the super symmetric. Yes, sorry. This is important
51:47
Is like that Okay then
52:05
So then let us consider the n equal 1 case only on s4
52:25
Actually this case As a grid Explained we can construct this Super simple on this s4 by using super gravity and also we can use
52:42
N equal to s4 Pakistan. I mean in general we can construct a super simple s4 Like this indeed and then the by the projection to n equal 1
53:03
And the supermarket then we have a n equal 1
53:22
How do you do it actually by I mean by hand I mean by hand Okay, we see yeah consistent, of course, yeah, there is a consistent Okay, we have this n equal 1 s4 and actually we have explicit the 3d transformation and then it's closed
53:42
So it's consistent 3d transformation and we can We can have some super symmetric action for that 3d transformation Okay, we have a I mean like a native method. We have a consistent n equal 1 Transformation. I don't know actually here. I don't yeah, actually this yeah
54:07
this method it can be worked without the super gravity just Just yeah n equal to 1 s4. Then just put it on this Yeah, and of course, yeah, there is a super gravity theory. So it also works better. Yeah about the problem is
54:28
Is as we said probably this is not unitary on s4. So it's maybe not good I mean in a Lorentzian version, but actually in only s4 actually unitary is not
54:40
Important actually only so there is no notion with new tidy because there's no big rotation Okay, so maybe the question actually eviction productivity is important, but maybe it's not I mean we can forget about that just a conscious to other toy model I was just a failure theory as as we did actually we don't care about almost Don't care about the real standard model or something like that. But so maybe we can question this one and
55:05
Actually, we have a super symmetry then we can apply this procedure so this is this works always okay, so It's nicer but problem is this case
55:40
problem is this is
55:47
This is violet suction this conical choice Seems delta lambda dagger times delta lambda. So it should be positive definite, but as a people I mean
56:01
Francesco Guido Explained this dagger is not a dagger. I mean Hamish and conjugate. This is something Some some operation, okay Okay Okay, and I use our theory case. I mean, oh, yeah, it's regrosses one or one s3 or s2 case
56:23
We can Choose what is dagger here to satisfy this one because of the probability in utility in a rotation version Okay but this case
56:41
this is not I mean This is not easy to find that this delta lambda dagger to satisfy this one. Okay, I don't have such things Okay So in this case this localization works, but it doesn't localize to the subtle point
57:01
Okay, so So it's maybe Important but I'm not sure but in this case what program is a killing vector or generalized king vector, this is
57:21
given by this So So the actually and this is a killing the killing spinner or general the king spinner and this is a king vector This is is not there Okay
57:40
Okay, this includes something some spinner in this or something like that or Yeah, yeah, yeah, yes. Yes. Yeah. This is analog of the n equal to 1 is 4 By the projection projection means we kill some some Z or something like that
58:04
So I'm more precise. Maybe the busy Should be some projected one Then this is should be complex not real by any choice Okay Actually for any if we have any code to
58:21
Then there's another type of generator Then the combination of these twos use a real one but half of them, I mean projected to n equal 1 then it Doesn't use a real one. I mean, it's very complex. Okay, so it's a complex rotation something like that
58:41
so that doesn't Yeah Doesn't make sense this Positivity probably. Okay. So this is a situation for the sushi on s4 Maybe you try
59:01
To get a nice salary on this on s4 actually many people probably try this one, but Probably no one succeed. It's nice Passion, I'm nice close the forms of passion function or the recent loop or something like that. So sorry
59:23
For any n equal 1 sorry, yes, this is about super symmetry super-symmetric transformation itself Yeah, this is this doesn't depend on the super. I mean the Saudi I mean matter content or the action three-level action or something like that so this is a problem of the this
59:42
This is a king probably this killing a spinner something. Yeah Probably it's a needed to know you type D or something like that, but I'm not sure I Have a comment
01:00:02
So are you going to use any sort of super-spinning of this way or this? I am just pointing out this general problem. Are you going to use any sort of super-spinning of this way? No, no, no, no, no. This is a, sorry, this is, yeah. Yes, yeah. It's about, yeah, oh, yeah.
01:00:20
Actually, you know, for this gauge in condensation, I mean, old program is a strong coupling instant expansion by the Amarti-Konishi-Rossi-Veneziano. It's a very famous one, but it probably doesn't give a correct one. And this, that is related to this one, probably, on S4. I mean, instant expansion on S4, I mean.
01:00:44
But it doesn't work. And probably, seems it doesn't work. Probably it's related to this, this doesn't work. You know, this propagation problem. That is my comment. But, yeah, yeah, yes. Anyway, yeah.
01:01:02
I'd like to say this, yeah, actually, this works in some sense. This, like, localize something. So this is key independent. Maybe it's, can be used to get some information. Okay. And not to localize, but, yeah. But by now, I don't know how to use this one, or how to use,
01:01:24
maybe there is something like a Hicks-Nuss localization, we have a something nice one, but I don't know. At least this doesn't work. This type canonical one doesn't work, probably. So this is, yeah.
01:01:47
So actually, for the n equals one, I mean, localization technique applies for n equals two. For n equals two, it's a bit difficult. Okay, so some difficulties.
01:02:02
But only the, yeah.
01:02:21
Yes. For a three-crosses function, it works. But actually, yes. Okay. I'll explain. Actually, there is two choices for this manifold.
01:02:40
I mean, two different choices for the localization. But first I write down the transformation.
01:05:53
Okay, so what I wrote down is the transformation for the flat space, Euclidean version.
01:06:01
So it's a bit different, different from the Lorentzian version. So, first of, oh, sorry. So here, by doesn't mean together, okay? As usual.
01:06:21
Actually, in a flat space, this, sorry. Notation symmetry is SO four, it's like SU two cross SU two. And the lambda and the lambda bar have a different SU two indices. So it's a completely different spinner. So we cannot identify this lambda dagger and lambda bar.
01:06:41
So it's completely different. In a Lorentzian case, it's not, okay. Okay, then, on S three cross S one. So this is actually, already explained, but maybe,
01:07:26
so this DM is a cover and derivative, okay? So actually, for the vector merge plate, to obtain the supersymmetry on S three cross S one is very easy. Actually, just replacing this to this.
01:07:40
I mean, including a cover and derivative, and that means, this include something, a spin connection also, okay? This include a spin connection. And VM is, sorry. And here, Q is R charge.
01:08:01
So here we consider as U one R invariant theory.
01:08:38
So this T is a Euclidean time.
01:08:40
So this is a constant background field for the R symmetry, okay? Okay, then we can check. This C transformation, it satisfies, I mean, commutation relation. It closed under the gauge symmetry,
01:09:02
local Lorentz plus asymmetry, something like that, okay? So this is consistent supersymmetric transformation rule on S three cross S one. If we replace, just replace this one. You can check that by hand. And I think, I think I agree. I'm not sure, but this is same as we did explain
01:09:23
from the supergravity. Okay, to get supersymmetry theory, the two, I mean, different method. One is a supergravity. One is just for the NATO method,
01:09:41
just constructs a consistent supersymmetry. Just, I mean, includes order by order. Order by order means one over, sorry, curvature expansion, okay? In this case, curvature expansion, I mean NATO method terminated by replacing this one.
01:10:04
And this here, yeah, Q is, here I assume R charge is non-anomalous. Yeah, R charge is symmetric. I mean, like this one, it's not anomalous. I mean, measure is invariant. Lie, lie, lie, lie, lie, lie, lie.
01:10:23
That is, yeah, actually the problem, but actually sometimes, at least in a, in a cyborg there, okay? I mean, kohmer window, okay? There's something accidental, ultimate enhancement or something like that. So this can be applied, yeah, in that case.
01:10:40
And actually, for the gauge and condensation, actually we use not our S3, but largely a symmetry of S3, I mean, R3 cross S1 case. Then we can apply something like a location method to compute the gauge condensation. Okay, this, yeah.
01:11:02
Sometimes, actually, for my case, actually for, I constructed supersymmetry on S5, N equal one, supersymmetry on S5. In that case, I used, we used supergravity and also NATO method, and we checked it's same, but in this case, I, personally I do not check that,
01:11:21
but, sorry. So here. Looks like these, you are imaginary in this.
01:11:40
You are imaginary. You are dm, dxm, equal to yi. I, yeah. That's it, yeah, yes. Yeah, this is a correct one. In a Euclidean, say. Yes. Low length sign is different, but in Euclidean, this is correct. Yes. So, yeah. In a Euclidean case, many things are strange.
01:12:04
Anyway. To, you know.
01:12:27
So here, we normalize the radius of this S1 as one. And the radius of this one is as beta. And also, of course,
01:12:46
these two are completely different, independent. Independent parameter. And, actually, you should know, of course, should be hitting spinner, or generalized hitting spinner.
01:13:22
Four is time direction. Actually, we fix zit's symmetry or something like that.
01:13:42
Okay, this is simple. Actually, for the color, color matrix is a bit complicated, but for the, I mean, yeah. For the vector matrix, it's simple. Like, it's almost, I mean, the form is look like. Flat form. But, of course, this is completely different
01:14:00
from the flat form. Okay.
01:14:29
And, yeah. For the supersymmetric action, we can write down the supersymmetric Yamir's action.
01:14:44
I mean, very familiar one.
01:15:04
Of course, in a supersymmetric, this comes from a WW. W alpha, W alpha. But here, of course, we do not have a super field on S3 plus S1. So, we need to write down the component field. And, sorry.
01:15:22
For us, this is the invariant. Suji means the transformation. I write dot term.
01:15:45
And, this is a bit different version, but we define the complex file gauge coupling
01:16:01
for this case also. There's no form of the notion in this case, but still we define this one. Okay, then the localization.
01:16:23
So, in this case, we are the canonical term.
01:16:51
Okay, so, probably there's two choices, essentially. Essentially. For, essentially, two choices.
01:17:10
So, to do the localization, we need to specify what is delta. Delta is, actually there is four,
01:17:21
I mean, there are several supersymmetric generators, and we can choose one combination with supersymmetric generator. That combination means choose epsilon and epsilon bar, satisfying this one. Okay, there are several solutions, but if we choose one choice,
01:17:42
then with this canonical term, delta is fixed and localization is fixed. But, okay, then this case.
01:18:02
Then, and, okay.
01:18:33
So, this f plus, so this is a unself-real part
01:18:46
and this is a self-real part.
01:19:06
Okay, so, actually, okay.
01:19:32
So, if both of them are non-zero, it's independent, then these terms is like, oh, sorry.
01:20:01
Plus, okay. So, adding these terms, this gives these terms,
01:20:21
if the equation is the same, but if the equation is different, then this gives a linear combination of f squared and f of theta term, okay. Or theta term, but the coefficient is not there, imaginary theta term, anyway. This gives this one, okay.
01:20:40
Then, this case, subtle point, subtle point is the three bar, because theta term is nothing. I mean, because,
01:21:02
So, this gives a free theory on three bar backing.
01:21:26
So, this is almost trivial, I mean, the computation is almost trivial, just computes a free theory, okay. Forget about the introduction, just compute the spectrum on s1 cross s3. It's very, at least it's straightforward and easy, okay.
01:21:48
And this case, at least, okay, so, no local observable, maybe,
01:22:10
and there's no supersymmetric vision loop. And, since the function,
01:22:22
what we will compute usually is a function function, okay. That is called super kohmer index.
01:22:40
So, for this case, the localization is very trivial, just computes a free theory, okay. So, it's like a weak index.
01:23:03
This is for PIMLs, but if you increase chiral multiplets, of course this is same, I mean, for the, sorry, for the chiral multiplets also, it's trivial, I mean, we should give phi, I mean, I'll show you non-zero,
01:23:20
then several points include something. So, it's, yeah. So, computation is totally around the trivial, okay. If we include the chiral multiplets.
01:23:44
Super kohmer index, what is defined for the kohmer multiplets? Is there an output? Yes, yeah, yes, yes. But, yes, yes. Yes, yes. But, probably super kohmer index is not so useful
01:24:03
for the non-kohmer field theory, but anyway. Name. Name? Name? Why this is a super kohmer? The one we have here, hesitation. Ah, hesitation. Ah, actually we need a U and R symmetry, yeah. And that is, yeah.
01:24:22
Yes, that is a, yeah, user related to super kohmer symmetry in a low energy or something like that. Yeah, so then, yeah. Like a regional condensation case or something like that. I mean, it's not so useful. Or, I don't know.
01:24:41
Yeah, yeah. At least it's, yeah, it doesn't have a, yeah. Yes. Yeah, I mean, at least some, I mean, no anomalous U1 because of this one. Yes, so. Yes.
01:25:05
Yes. Okay. And then this case. This case is a very, I mean, tricky, very strange, but anyway.
01:25:20
We can consider this choice. And then, okay, that means that this delta, I mean, with this choice, epsilon is zero. Then, delta lambda equals zero. I mean, this is a lowest.
01:25:55
Lowest component of the kohmer super theory, W.
01:26:00
Okay, that means that this is close. Are you saying that epsilon bar is nowhere vanishing or that it's non-zero? Just non-zero. I mean, it's, It can vanish? Yeah, it's like, yeah. There is nowhere vanishing. Actually, this case is also, I mean, I mean, Okay, yeah, this, yeah.
01:26:20
This epsilon doesn't have any fixed point, but it's, yeah. Yes. Yeah, if there is something, then this is not true. And the super-convocation should include some very complicated instant, I mean, if there's a fashion function like something, I don't know, but yeah. But there is no such thing. So it's, yeah.
01:26:42
In some sense, good. Okay. So then,
01:27:02
Okay, so this is observable in the localization computation. And this case, this canonical localization term
01:27:21
becomes only the delta, lambda, bar, dagger, okay? So there is, no, this term. Oh, sorry.
01:27:43
Okay, so this epsilon by, no. Sorry.
01:28:03
Okay. So this includes only the F minus square. So this other point,
01:28:26
Southern point is F minus. So this is instant
01:28:43
over AST connection. Okay, so this case, localization Southern point is not just a factor, but we need to include AST connection on S1 cross S3.
01:29:38
Okay, so then,
01:29:44
in this case, for this Southern point, M is actually, of course, zero. I mean, trivial. But for that case, lambda bar equals zero. Then, plus here.
01:30:25
So this is user instant factor.
01:30:42
So this case, including a three-level Lagrangian, it depends on the gauge coupling constant. So this is for these choices. Okay, actually, okay, so, then I will talk about
01:31:00
this case more, more about that. So then consider the gauge in condensation. Actually, to get gauge in condensation, we need to consider the R3 cross S1, I mean, R3 cross S1, and they compact this S3. But this notion can be used.
01:31:20
So this is observable, and it becomes a weak coupling instanton computation. I mean, okay. So I will talk about next lecture. So, time's up. So thank you very much.