2/2 4d N = 1 localization
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45:00
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53:57
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55:40
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Transkript: Englisch(automatisch erzeugt)
00:16
So this, so I continue the lecture I gave before.
00:23
It's about this one for the integral one, SPSS1. It's called supercomma index. This is actually the overlap thing with Guido's lecture, but maybe this is last lecture in this week, and you are very tired, so maybe it's okay.
00:40
Anyway. Okay, so I will start. Okay, so I wrote down the suji algebra on S3 cross S1.
01:04
And of course, this manifold, so this manifold, S3,
01:21
have SO4, so there is SO4. SU2 cross SU2, SU4. And there's S1, so there is a U1 direction. And this SU2 contains a 3, I mean, J3, and this also J3. And I will call this L and R. And for the U1 direction,
01:42
the generator is maybe H, Hamiltonian is more appropriate, but I will denote it as D here. And I will choose the general king spinners,
02:03
satisfying this, sorry, this equation. I mean, it's related to this 3 direction, okay. We can choose that. Then, R charge of this one,
02:27
this is the same, but... Then this is a spinner, so, I mean, delta spinner,
02:42
so it have a 1.5 momentum, I mean, angular momentum here. And delta epsilon bar is R charge 1 and J3.
03:07
J3 minus 1.5. The interesting thing with stress is this, both of them are right, not left, okay. This may be strange, but actually,
03:23
we construct a SUJI on, I mean, for example, I mean, we can consider some UV limit of this theory, and then SUJI algebra becomes a flat one, okay. So then, for that case, SO4, I mean,
03:49
but here, it's different. This is for the SO S3, okay. So this is why we have this. Okay, we can check that.
04:03
And now, because dt equals del t minus... So this q is, this is R charge of the field, okay. Then, okay, so this means minus del t
04:27
becomes d minus, okay. Here, I mean, in a previous lecture, I write down the SUJI algebra, and this one, I mean, we, I mean, from the flat space to here,
04:43
we just rewrite this del t to this one, this combination. That is the only thing. And that means this is like a twist, and this is R, so R charge. So, ah, sorry, I will call this R charge capital R, okay.
05:00
So then the derivation, this u1 direction generator is not t itself, but del t becomes d minus this one. Okay, is it okay? And then, partition function, S3 cross S1
05:37
becomes this one. This is a user, just, yeah,
05:42
equivalence between the first interval and the operator homework, okay. Here, this minus f is because we, sorry, impose the periodic boundary condition for the fair mean on this S1. Then, this partition function is equivalent to this one. Here, usually, summer partition function,
06:02
it's here, is Hamiltonian, so d itself, but because of this one, this twist, this becomes, okay. So this is just rewriting of the first integral. And of course, there is some factor
06:20
because of the ambiguity of the first integral measure and the overall constant, that is called Casimir energy, but here, we think it is absorbed in the definition of this first integral measure.
06:42
Then, we can do the further twist. It's del t to, so this is possible
07:10
because first, this combination, this xi and eta is a parameter, okay,
07:22
and m is something flavor symmetry generator. For the vector multiplication, there is no such thing, but anyway, this is zero. This is from this assignment. I mean, choice of this one, okay.
07:40
From this assignment, delta is committed with this combination. And of course, delta committed with JL3 because it doesn't charge. Only the right one. And of course, delta committed with flavor charge.
08:01
Okay, so then, this is a constant, I mean, constant matrix, a constant generator, okay. So, if we think the 3D transformation and the supersymmetric reaction and replacing this dt to this twist, then this twist, I mean, this replacing
08:22
is consistent with the closure of the 3D algebra and the supersymmetric action, okay. Okay, so then, from here to here, then still, we can, we have the supersymmetric action on this one, okay.
08:45
And then, this partial function becomes changed because this one, I mean, this is s1 and there is a helical space and x minus h or something to here to here.
09:03
Then, there is some, like Wilson charge here. Okay, then after the here, I mean, and we use a trace here, then there is some insertion by this element, exponential of this element, okay, exponential dt.
09:23
So then, the partial function, I mean, partial function with this twist is different from here to this one, okay.
09:57
So, this is an insertion on the, sorry, xq,
10:18
okay, so here, small process, data, d, Hamiltonian.
10:27
Okay, is it clear? Okay, so then the, I mean, partial function is equivalent to this, this one, in operator formalism, this is just rewriting.
10:44
And then, this, this is called super conformal index. Actually, here we do not consider the super conformal field, sorry, but it's called super conformal index.
11:02
So why? Actually, if we consider the first super conformal on flat space, then there is a user conformal map,
11:21
conformal map to the s3 cross r, okay, this is a user cylinder map, okay. And then, so there is a delta epsilon, delta epsilon bar here, okay, but what is this corresponding,
11:41
corresponding in this one? Actually, commutator of these twos include this d, I mean, s1 direction, it's a Hamiltonian in this here. plus d in this r4 is a dilatation because this conformal map is non-trivial, okay.
12:00
So, these should be, if this is a q, q and q here, then it doesn't give the dilatation, that gives a moment, okay, here. So this should be s here, because of s, s is a super conformal, super conformal generator. Super conformal generator use a function,
12:21
so the generator gives a dilatation and a special conformal transformation. this map relate this s super conformal and super conformal charge and use a super charge to this epsilon and epsilon bar.
12:41
Then, we can calculate the h, okay. So then the commutator of this one, I mean, we can compute it from here,
13:02
or some, then, we can think, I mean, from this super conformal scenario r4, we can construct just a bit index from this Hamiltonian and discharge, like a, this is a q, q, like a q, q bar, okay. So then this is called a super conformal index, okay.
13:22
This is just a generalization of the bit index to the, this super conformal here. Okay. Okay, but here, from the localization point of view, or filocellul, on s3, s3 cross s1, we don't need to impose the cellul is conformal, okay.
13:44
But if it's conformal, it's related to this conformal index. So this is called a super conformal index.
14:44
Then, okay,
15:07
by the, the change of the parameter there, we can rewrite it as a,
15:43
we can rewrite this one into this form. But here, of course, there is some, I mean, there's only a parameter p, p prime, but here there's a q, x, y, but some combination becomes proportional to h, and this doesn't contribute.
16:00
So we can neglect that. So this is called super conformal index, and this was given by the grid. So this is super conformal index.
16:32
Okay. Then, of course, there is two way to evaluate this expression.
16:40
From the pass integral, or from this operator homogen. In operator homogen, we just count the number of the state. Actually, theory is free theory. I mean, because of the localization becomes free. So we just count the free gauge theory. So it's, in some sense, trivial,
17:00
but it's, of course, not so simple. But we can count that. Or, using a partial function, in that case, actually, the theory is free, but there is one, I mean, the geometry is three cross s one.
17:21
So we can, even at the free theory, there is a Wilson loop for s one direction. So, we need to integrate over the Wilson line. So it's called a, and of course it's, there's a gauge symmetry,
17:42
so we take some a t equal constant, t is, sorry, s one direction. So then the Gaussian, around this one, we can do just a Gaussian integral. Okay, it's a free theory, so it's trivial, okay.
18:00
Then the expression is what it's like, it's like what Francesco wrote down for the two sphere.
18:35
So this is for the Fermion, for the lambda, and this is linear, so this is like Dirac equation, okay.
18:41
This is, yeah, Dirac operator on s three. This is the gauge field. Okay, so this should be big for the vector approach. Okay, fine. This is a c, because we fixed a gauge, so we need to cancel it as a ghost here, c. And prime means zero mode is
19:00
emulated, because we integrate it over this a direction, like s two case, or s three case. So actually this is very similar to the n equal two, so it's only on s three. And then, c becomes,
19:23
sorry, here, a hat equal beta a. Beta is length of the s one, and this becomes,
19:49
so this is just a fundamental determinant to fix this gauge. And the i vector, this is, come from this one.
20:01
Of course there are many, on a huge cancellation, between the denominator and denominator. And, final result is,
20:44
this, yeah, this is same. And for the chiral mass vector,
21:00
there is a same procedure, and we have these things for the psi and phi in chiral mass vector, it's just a scalar and vector, our spinner, so it's a, I mean, spinner is this same, but the difference is different. So then it becomes,
21:37
so gamma is called
21:40
elliptic gamma function. So this is what we do. Write that. Okay. So this is just a,
22:01
I mean, straightforward computation, this, this one, okay. Then it becomes same as index computation, I mean counting the number. It's a, use a free, free, free solar exponential, something like that. Then it coincides with this expression. So from this,
22:22
we can compute a super common index, for the for the if we choose our charge appropriately, then so this, this, not simple, so this complex integral
22:40
is a coincide for the Seiberg-Real pairs. Actually, so this integration, number of integration is different. Actually, this is a rank, so it's a for, it's actually it's Nc, but real, so it's a Nf minus Nc. So this integral, the integration is
23:01
completely different, but there is some identity found by the mathematician, and we found actual coincidence. So it is a very strong support for the Seiberg duality. Okay, Seiberg duality. There are not so much support before this index, I think. Actually,
23:21
there's only the two matching condition or some strings already or something like that, but here, this exception gives infinite number by expanding at p and p prime, but it's coincide. So it's nice. Okay. Okay, so this is about so this is
23:42
about Seiberg common index, and then I will move to the different one. I mean computation of the Gaussian condensation. Okay, no question.
24:20
So this case,
24:22
as I said,
24:59
okay, oh,
25:00
this is four.
25:25
So this is Debye's Horovit cause and modest. So this is, okay. So first of all, for the I don't say about Chiral merge spread. Chiral merge spread
25:41
B equal if we choose this delta psi dagger as I mean satisfying this bound, then
26:01
delta B bosonic is like so the subtle point is the same. I mean trivial one
26:21
for the for the chiral merge spread. But with this choice for the vector merge spread, so the point is unsafe there connection. About actually I will move to the
26:41
S one cross R three I mean flat space. This case B becomes the M phi squared plus M squared. So there is no this term here. So actually not phi equal zero but D M phi
27:00
I mean phi is covariantly constant is required. And this is a comment about the holomorphic but
27:25
so there's no super potential or associate super super field and yeah no super field but actually we consider
27:41
the
28:18
and sorry
28:22
okay we consider this delta exact term then we call the delta psi bar is epsilon bar x bar for this choice. You can see if this acts on this one because if we choose the eta bar as this one
28:43
then it can satisfy this combination. So it gives just del w bar del phi phi bar times f bar. So it is just a super potential term okay in a component
29:01
okay so this so this is like delta phi bar i delta w bar times f bar plus phi bar okay so this is should you remind me in terms of valorization of what this
29:24
exact delta exact delta should I be prepared to connect I don't know I just maybe it's related but sorry I don't know in variation yeah okay
29:44
so this is a super potential term so this is del delta actually it's so here this eta bar epsilon breaks Lorentz symmetry or rotation symmetry but this result is not I mean rotational invariant
30:01
and if we increase the fermion also it's rotational invariant okay I mean Can I ask you to do this like here I'm not sure I don't have one oh no no
30:21
yeah I mean just consider some function of phi and consider this combination then it's a delta exact okay so this is a form of the super potential I don't need to have the super potential in the original and not whatever I want
30:40
it's holomorphic yeah yeah yeah yeah so this combination is yeah so this is yeah I mean almost trivial I'm not sure but yeah actually we can construct this one I mean so actually so the holomorphic combination I mean ant-holomorphic combination doesn't give any contribution
31:00
for this localization computation okay What about holomorphic here? A holomorphic ah this I mean this is just yeah I mean fiber I mean this combination this only includes fiber and no phi or something ah yeah
31:24
actually yeah yeah I mean it doesn't I mean ant-holomorphic variable ant-holomorphic coupling ant-holomorphic variable cannot contribute to the these types of the computation it's almost in a flat space it's almost as a user
31:41
I mean yeah Phi is in which representation of the holomorphic and you have to the gauge invariant I guess right right right right why is it in which representation ah which yeah any any it's contracted yeah this is contracted my most general gauge invariant operator
32:02
I guess right right right right right right okay so this is like a user super potential but yes yeah but yeah usually we use a super field or something but we don't yeah this is this is delta exactly are you saying that the ant-holomorphic
32:21
parameters in the data only to that true true yeah right we can change such things that is the user's comment but here that's what you mean by holomorphic right right yeah
32:41
we just wanted to understand the statement so this no factor yeah yeah the statement is just this is delta exact so this type of the this type this type of the deformation cannot change the result how is it how is it so this is yeah
33:00
user statement of the holomorphic okay then so let's move to the gauge invariant condensation
33:21
in n equal 1 ah first of all so we consider super yamils or s3 ah sorry r3 cross s1 so this is I mean the computation is almost same as
33:40
this but okay so some something we need to care okay so gauge condensation is this one here g can be anything I mean any semi simple algebra but for definiteness or I mean
34:01
ah for the simplicial notation ah so in this case nc we choose s and c and then what I said ah localization is just adding
34:21
these terms plus fermi okay okay okay so then the theory is recoupling
34:44
limit on instant or asd or instant answer the connection okay
35:00
so this is yeah actually the classical computation on this background so it's structure okay and then this is by gross this yeah configuration
35:20
on r3 cross s1 are classified by instant on charge so this is
35:45
of course and Wilson loop
36:12
on this s1 direction ah we call it a0 direction and finally it's a monopole
36:20
charge for u1r so this u1r is because of this Wilson loop this breaks a symmetry to the u1r okay
36:40
so this is so sorry ah delta b
37:01
for the I do not write down but yeah of course it's a simple one but yeah actually for the the lambda lambda bar coupled so it gives lambda bar equals zero condition or something like that so okay
37:20
so we need two two zero model okay a zero model means a zero because there is I mean it's a lambda lambda okay
37:42
but for example instanton has two and a c zero so instanton cannot give c this one but so we need a unsafe dual
38:00
connection with two zero model it's one sorry r three
38:21
cross s one okay so then it known there is
38:41
a r plus one fundamental monopoles this actually very simple I mean well known one is a TDR over PPS monopole I mean TOS to polyac monopole
39:03
or more precisely it is fractional instanton TDR means PPS monopole is defined by r three okay r three and
39:20
there is a scalar I mean for the TDR monopole TDR means phi is can be considered as this a zero just changing say I mean that is called TDR or PPS monopole in s one cross r three I mean yeah PPS monopole is a gauge field a mu mu equal
39:41
one two three and phi okay was a scalar TOS to polyac monopole has a scalar and we can this is a non-trivial some configuration and we interpret this as a zero then it's
40:00
called TDR or PPS monopole okay it's I mean satisfies the PPS condition TDR this TDR means I will explain okay sorry not
40:20
actually it's fractional instanton okay I will explain sorry just for a mu yes yeah for a mu yes yeah
40:41
it's yeah yeah instanton but it's actually it's r3 so it's non-compactor r3 cross s1 so yes it depends on s1 direction coordinate right right because as you said previously it sounds like there is no dependence okay I mean
41:00
this is for this one yeah yeah actually yeah there's another cause culture crime monopole found by the TDR means
41:22
there's for su3 case in for this is identified as s1 and there is for example 3d4 brains wrapping this one okay then instanton
41:41
is d0 instanton this has a charge 1 I mean instanton charge 1 and then take a period then it becomes like
42:02
this one so d3 d3 d3 okay because oh sorry and phi equal for example phi 1 phi 2 phi 3 this is a so in some
42:20
coordinate it's on this one okay then this d0 of TDR becomes d1 okay d1 wrapping this dual
42:41
circle but because of this d3 we can move this d1 to the far away in our 3 direction or somewhere then okay so this is called VPS monopole so this
43:00
segment is a VPS monopole okay in this picture actually this gives monopole configuration user 1 in a 3d because we forget about the time direction okay so this because monopole this also monopole but this V1 is called this lapse
43:20
this S1 this is called the culture crime monopole okay for 3d picture it's I mean this is non-compacted limit or something like that so there is no this culture crime monopole but if it compacted so there is culture crime monopole okay so then this instance have
43:40
charge 1 but by just a replacement of this phi to the a zero
44:00
is clear okay what so forget about this one yeah we can consider this V1 okay so this is a VPS configuration and there is TDR of this one in this picture
44:21
it is called here VPS monopole or fractional instance and if we consider some bound sets of these three then it becomes an instance but or or here but we can consider
44:40
this fraction I mean just part of this only then it's called VPS okay maybe I can yeah I can write down the solution explicitly because this is normal I mean this is like
45:00
use our multiple we can write down the explicit solution in a text book but I'm just replacing phi to a zero but because it should depend on the x it's zero direction because it's but actually from this picture and other VPS are
45:20
not distinct I mean it's same so it should be same actually if we take a different and gauge for the text polygon monopole then we can think we can have this monopole is same as this one so
45:40
it's I mean it's all of them are actually same class of solution so actually VPS monopole only on R3 according to R3 so on
46:01
R3 for this one yes but your picture is clearly the things are not very okay actually for the one VPS monopole by the gauge transformation it's singular gauge then
46:21
it becomes X0 independent but this gauge is not but if becomes a similar one then there's no gauge that makes
46:41
them X0 independent only the one okay so that is strange but that is here actually yeah
47:03
this actually this is X0 dependent user in a user gauge dependence okay okay
47:24
then okay so that means of course NC or something then there is S0 embedding related to the simple one
47:43
and to R and also sorry this is so then we listen this is BPS mobile so it's
48:02
clear from this picture and this segment is correspond to the simple route also so I do not write down the explicit form but
48:20
since magnetic charge I mean magnetic charge is this alpha I I mean this is label of the magnetic charge and sorry instanton charge
48:40
instanton charge is Q equal so it's proportional to this combination and the classical action S equal
49:01
minus I tau alpha I but so then the culture cry monopole this is as you expect magnetic
49:20
charge is alpha zero it's this for the this S A N C case given by this one okay so I mean it's I mean lowest route okay so as you see in the
49:40
picture this alpha zero alpha and alpha I's are O's I mean equivalent and charge Q equal one plus alpha zero phi and S is okay
50:05
so this looks different from this one but this is just I mean coordinate dependence in the P equal
50:39
one over beta
50:40
times graphic point and alpha I sorry V this one beta is length of this one okay what is this one V sorry we define this one I mean sorry I will use
51:00
maybe sorry define this one I have yeah sorry actually I I I plan to write down the explicit solution I mean I include this but sorry I didn't do that
51:20
so okay then we here we can identify the background which contribute to this occasional condensation then what we should do is just
51:40
evaluate the lambda lambda in the background or if we need including one loop correction because L3 is non-compact and the three direction so
52:01
we need to find back here of the theory first okay so there should be for this case we need yeah there is some notion of back here so okay
52:25
but the computation is a weak limit I mean semi classical okay so then
52:43
consider the effective action for massless field here it becomes
53:00
just u1r so this is free okay free means there is only a joint matter so there is no coupling okay this is because by this
53:20
Wilson line and actually means zero culture crime moment S1 sorry R3 yeah so you first specify the boundary condition at the
53:41
infinity of R3 and deliver the path integral right but it's yeah here we take a user I mean something dumping at infinity yeah like a user in something like boundary it's okay yeah actually yeah of course if we yeah this is
54:00
I mean yeah but then what do you mean by need to find I thought that fixes the boundary condition already so yeah actually there's a choice okay this yeah sorry Wilson line is modular but there's a choice of the subtle point but there is no choice of the boundary condition yeah sorry so this is a modular I mean five is
54:20
a modular I mean modular of the bucket okay oh so okay so you need to find a subtle point yeah yeah actually some of them are lifted and actually there is only N, C points remains so we need to find yeah
54:44
if it's compact actually we should integrate over the bucket I mean some over the bucket I mean bucket it's not called bucket but here it's yeah non-compact so for the field sorry we should satisfy me specify as bucket okay
55:00
if it are one then it's okay so we need to yeah integrate over everything like integrate over this A itself like a three-position function it's okay but it's this is non-compact so we need to specify I mean to fix what is A I mean bucket
55:22
okay that's okay and it's a user I mean general general field I mean not something I mean we have two or three or four non-compact direction then we need to specify the bucket but in low dimension
55:41
over compact okay so we need to integrate over yeah but A is okay A A does I mean so A does so what boundary condition goes for the A okay or maybe it's a dump I mean dump means action is finite
56:00
or something like that or A is A is constant is okay it's like like in a user sense I mean so A goes to constant right but then you have to choose the value of the constant to fix the boundary that's true that is what
56:20
we want to do here yes okay okay I'm not okay anyway okay then actually zero constant
56:40
prime prime actually because of this weak coupling limit and this yes you actually there is no coupling between the so we can forget about I mean constant constant state it's in a 3D sense okay
57:01
so we can consider it as a 3D theory so it's in junction to 3D I mean much more decoupled you know T goes to limit infinity limit okay so then there's a 3D N equal 2 you want R vector multiple
57:23
actually it's in a literature it's N equal 1 but I think in a sense it's N equal 2 you I think but anyway then phi is this one I mean it's very some right constant mode
57:40
in C and this is a real photon actually by using a Bianchi identity or something like that by I mean adding some some terms imposing a Bianchi identity we need yeah we can do I mean actually this is U1 so we can dualize
58:01
explicitly okay this is Wilson data so this yeah can be a classical modular okay so this ZZ is maybe a classical modular so this is 3 N equal
58:21
2 okay there is a super symmetry so it should be this combination and then original action original action is just Yang-Mills action plus yeah plus some topological term
58:41
becomes oh my gosh so this is just a gauge coupling constant or something I mean it's a dual gauge coupling constant because it's a D-R photon
59:02
K-Lap okay so original Yang-Mills action is a K-Lap potential so it's no potential so it's a flat it's of course of course because it is modular so this X is a classical modular
59:27
sorry T goes to infinity limit yes actually here first we consider T goes to infinity limit circle size is finite circle size is
59:40
finite better why did you repeat because T goes to infinity means I mean externally we couple okay so if we consider some some loop correction to the in the machine mode it becomes a
01:00:00
some beta times something, but it's I'll say it by this, beta times expansion minus T because of the coupling constant. So in T goes to infinity, that vanishes always. So actually there is only the mass dimension beta and the beta times expansion minus T or something, one over beta times expansion minus T or something.
01:00:21
It's very small in T goes to infinity, so it becomes decoupled, so much more decoupled, okay? Yeah, and so R can be fixed. Actually, R goes to limit, R goes to, or beta goes to zero limit is discussed by this Davis,
01:00:42
but here it's fixed, but modify the action. Okay, then to find the back here, we need the scalar potential. Yeah, it's a, yeah, we need the scalar potential.
01:01:06
Yeah, it's a straightforward computation, we can do that, but it's for the, I mean, something easy way, I mean,
01:01:22
in principle it's okay, but easy way is the scalar potential, oh sorry, this is scalar potential is related to the super potential, I mean, ferrimium bilinear term, because of the suji, okay?
01:01:40
Because of super symmetry, phi is related to the scalar bilinear, okay? So if we compute the scalar bilinear, we have found the super potential over scalar potential, so we find the back here, okay? So this is a ferrimium bilinear, so we need the two ferrimium zero mode.
01:02:06
So we can, or we should consider the fundamental monopole.
01:02:20
If we consider, if we want to consider the scalar potential, we need a different one, I mean, different configuration, but to consider the ferrimium bilinear in the effective action, okay? If we want to compute the ferrimium bilinear in the effective action, it's, I mean,
01:02:44
necessary to compute two ferrimium zero, okay? And then, the procedure is a user one, I mean, user instanton computation, or monopole computation.
01:03:03
So first integral measure of the zero mode on this fundamental monopole is like this.
01:03:39
This Sj is a cluster action with a fundamental monopole,
01:03:42
including, I mean, yes, called the current monopole here, as I wrote down here. Actually, it becomes a, not phi, but it becomes a z, I mean, including a zero photon. It's because of the boundary term, given by the, sometimes imposing a Bianchi identity to zeroize something.
01:04:01
So this is correct one, as expected, it's actually a form of a combination. And, sorry. By fundamental monopole, do you mean the embedding of the activity for the monopole by a simple one? Yes, yes, simple one, and also a culture prime monopole. This, yeah.
01:04:21
Fundamental monopole means that these BPS monopole plus culture prime monopole. So I learned one, two, R, and zero. Alpha, zero. R plus zero. Pi-imposed? No, no, no, not superimposed. If it's superimposed, it becomes a bound state,
01:04:43
it becomes an instanton, but. fundamental monopole means that the BPS monopole? Yeah, right, right, right. Then we have a two. Fermion zero. Okay, so this A is a position in the R3,
01:05:04
position of the monopole in the R3, and this is a user, I mean, two-step level of monopole measure. Omega is a U1 phase, and this is a position in R3, and this is a Fermion zero mode. This is a bit strange, but the cutoff scale,
01:05:33
mu, and gauge coupling constant, gauge coupling G at the cutoff scale mu,
01:05:46
which is defined at T equal zero, okay?
01:06:00
So because, actually, we modify the action, but we do not modify the measure. So this measure comes from the measure of the original theory, so this measure should respect this T equal zero series.
01:06:31
So what, yeah, here we consider the T goes to infinity limit, action is different, but, okay.
01:06:49
Then, Fermion zero mode is given
01:07:01
by the supersymmetric transformation of the Fermion, and explicitly given by the, okay, here, sorry,
01:07:51
here is X infinity, special infinity, okay? And this tensor just means
01:08:00
we do not write down the gauge indices, so gauge indices like this one, I mean, okay, specified by RFRJ here and here. So this is our abbreviation, so it's not, I mean, this doesn't mean input and things. Okay, here, actually, here, we consider the different,
01:08:23
I mean, different background, I mean, we define BPS monopoles and cuts and crev monopoles, and we already summed here, okay? So this is, okay, okay, so this is a free Fermion propagator here,
01:08:43
from the instanton to the Fermion, instanton to Fermion, X goes to infinity, this is a user in a monopole or instanton computation. Okay, then, superpotential,
01:09:01
which reproduce this, I mean, that means reproducing, I mean, reproducing this one, we need some terms, Fermion variant terms in the effective action, which is given by the, this one.
01:09:39
Sorry, I'm getting lost,
01:09:40
so I thought that you were doing the localization, so you should identify the subtle point and then do the compute the one-loop determinant around it. A one-loop computation, okay. Yeah, you seem to be doing something rather different, so. Yeah, actually, this is a classical, I mean, this is a classical result, I mean, just plugging in the classical configuration into the field.
01:10:00
It's the same. Actually, I do not write down the one-loop collection, one-loop determinant around the VPS monopole. That is true, but actually, it's a bit difficult, but we can evaluate it. Okay. And then, the result doesn't change this one for the superpotential. So the one-loop determinant is bound? Sorry.
01:10:20
One-loop determinant is trivial? Not trivial, actually, it's, it changes the scalar potential on this. That doesn't, I mean, doesn't change the potential, I mean, scalar potential, something like that. Well, let's see, but I just want to compute lambda lambda, so, right, so then it's just one expression. I don't even know why you're talking about the effective potential.
01:10:41
I mean, in the localization, you specify some correlate or something. Correlate, yeah, yes, yeah. So that's lambda lambda. Lambda lambda, yes. Just the other q-exact term, and the rest is all the computations, so I don't know why you begin to talk about superpotential, et cetera, in this way, I mean. Well, you want to use chromatography. Yeah, actually, what, I mean, here,
01:11:00
just, I mean, localization just means constant theory with different action, with, which depends on t. So, here, we just constant, constant, or computes a correlator, I mean, Fermi and Ballina in some strange action, depend on t.
01:11:20
Okay, so this is just a user computation, except the action is very strange, but it's, so it becomes a weak coupling limit or something like that, so, okay. Sorry, yeah, yeah, yeah, yeah, yeah, fine, but I thought that the constant t to infinity, and then you can restrict to the saddle point locus, but you should still take into account the one root correction and everything.
01:11:41
Sorry, maybe, okay, maybe, please, please, yeah, please continue, maybe I can see after. Okay, the saddle point is AST connection, it's a BPSM report, so, yeah, and actually, with these terms, actually, there should be some correction by the one root, but it only affects the K-level potential.
01:12:00
So, this is, actually, this is just a classical result, but it's okay, I mean, one root is not. So, then it sounds like you are using the polymorphic or something, in some way. Actually, yeah, yeah, if you, okay, okay, no, yeah, here, yeah, I use one way, but yeah, of course, yeah, this is just a trick, I mean, we can compute this color potential directly in this theory, I mean, with something, okay,
01:12:22
but here, yeah, it's just a trick, I mean, to add some shortcut to compute potential, okay. I see, because if you use the polymorphic, I mean, that's what essentially, what Matt called it, or, and all of this are different. No, actually, R, yeah, R. That's why we've got smoke in this computation,
01:12:42
but then analytical continuing radius, that's what it is. Yeah, yeah, yeah, but, yeah, actually, analytic computation with R is, I mean, is different, I mean, R is not analytic, so. Yes, sir, maybe, maybe, go ahead, yes. How did you get the film, the free-tempered polymorphic?
01:13:01
How did they come along? Oh, this one, actually, the film, zero model is given by the, actually, transformation with this background. What transformation? Actually, transformation of the lambda, delta lambda. It's become a zero model, easily. Okay, around the back here. How did you get the chroma here? Yeah, no, no, yeah, actually, this is just not propagate, I mean,
01:13:21
this is zero model, and it becomes this propagator type in a large X. I mean, why? It's, yeah, I mean, if you compute it, it's like that. Actually, it's, I mean, delta lambda, so transformation is given by the something,
01:13:41
by F or something, I mean, field strength, something like that. It's like something dumping, X, one over X, X squared something, okay? So, this very, very much zero model also dumping in this term, and it becomes just a free, free program means just one over X
01:14:01
times gamma, mu, or something like that, okay? So, gamma, mu is in fairly zero mode. So, this is, yeah, yeah, okay. So, this is just the asymptotic behavior of the zero model. So, asymptotic behavior becomes just one over X something.
01:14:20
Okay, so it's not, I mean, yeah, not special, I mean, not special meaning, okay? So, actually, so, so this is X goes to infinity, so maybe there's a derivative, but to determine the potential, it's not, it's okay.
01:14:40
Then, okay, so, so the bracket is determined by
01:15:09
logic, logic. Yeah, actually, yeah, we compute some correlated, okay?
01:15:22
Then, in a low energy effective action, low energy effective action, a velocity effective action contains something lambda lambda because of the, to give this one, okay? So, this is determined, so we can
01:15:40
specify what is effective action, I mean, this coefficient in the effective action, okay? So, lambda, I mean, Fermi and bilinear in effective action. Then, then, because of the supersymmetry, we don't care about superposition and everything, but there's a supersymmetry, so we need, there should be scalar potential in effective action,
01:16:02
right? Is lambda lambda identified with X? Lambda lambda identified with X, yeah, with X, yes, yes, yes, yeah, yes. It's contains, yeah, sorry, yes. Yeah, I'm saying that if you go to three dimensions, three dimensions, they go to break supersymmetry,
01:16:22
and then you go, and then this is the portfolio of potential, portfolio of mechanism of confines, where there is a cosine for the, your photon. But this is a supersymmetric version of that, and also a little bit of finite sub-data for example. Okay, okay, then the,
01:16:43
back end is determined. So you have this, you have this use of potential coupling in the effective action, and compute the two-point function of terminals from this potential coupling, and then match it with the classical computation that is on the ground, and that's the way it is used in the action.
01:17:01
Yes, yes, yes, yes, yes. But maybe, yeah. You have one final diagram? Sorry? One final. You have one final diagram? One final diagram. Two propagators? Yeah, yeah, two, yeah, one is a two, yeah, yeah, but even bilinear, so. You have like a vertex-induced form of this terminal? Yeah, that's true, yeah, yeah, one vertex, yes, yes, yeah, two.
01:17:21
So you're mimicking that, but you want to produce the classical computation? Right, yeah, from the low energy effective action, I mean, low energy limit effective action, yeah. So yeah, we can compute the scale of potential directly. Yeah, it should be, but yeah, it's, yeah. But yeah, this is a trick, but yeah, it's easy.
01:17:47
Then, oh sorry, C2 is, sorry.
01:18:04
C2 is a real coset number, sorry. And w is a fundamental weight. Fundamental means w i times alpha j equal delta i j, or something like that, okay? Alpha zero is the sum of the minus sum of the alpha, one to alpha r, okay? So then we, you can check.
01:18:21
Actually, exponential is same for this computation, I mean, this exponential is same for this value. And yeah, it's sort of, you get, sorry, flatness condition. Okay, so, two over two.
01:19:01
Okay, then plugging this back the super potential, so this is a low energy super potential. Okay, here we consider S1, so there's a beta, but as I explained in first lecture,
01:19:23
if we found this super potential, I mean, suppose in a low energy limit, then by derivative with lambda, I mean, by taking lambda to the chiral super field, we can compute lambda lambda from this one,
01:19:44
the unnecessary case. So this becomes the correct one.
01:20:08
Here, because these are defined at t equals zero, so it's okay. And actually this, yeah, this doesn't depend on the beta.
01:20:22
That is a result. So we can take the beta goes in the limit, to the R4. Then, okay, so then in a weak limit computation,
01:20:42
we found this, six. Fundamental rate, fundamental rate, yes. So, or something like that.
01:21:04
So now, consider theory at t equals zero. So what we consider, t goes to infinity limit. It's different.
01:21:23
So first, as a query, original theory, but, okay.
01:22:02
So this means the action is, action is the same, I mean, original t equals zero, but the back here, if it's, I mean, choosing the real, I mean, correct back here for the action with t.
01:22:20
Then, use a localization computation or something. There is some terms that remains.
01:22:56
This one. Actually, because of the back here, it changes. So this doesn't vanish in general.
01:23:04
But actually here, back here are discrete. There is only, I mean, NC back here. Only NC back here, because of this fraction.
01:23:23
So if this is, this change, this divergent, I mean, it's a, I mean, if this change, it's a, I mean, up, up, I mean, sudden change. Okay, if there's a modular, then this t dependence, I mean, means, can mean the dependence on the modular
01:23:42
or something like that, but it's, back is discrete, okay? So we, so this should be zero. I mean, as a user, I mean, assumption for the index localization computation. Actually, if there is something like,
01:24:05
asymptotic behavior changing, then may, I mean, this may change, like with index can change. Well, localization doesn't work anymore. But here, we assume, as a user,
01:24:20
I mean, like a user with index in a philosophical case. There's no such things happen here. Okay, so, because of the backyard are discrete, so back, I mean, back end doesn't change. That is the conclusion, okay?
01:24:40
So that means, in this t goes to infinity limit computation can gives this gauge in a condensation. So you're already using the n backyard input? No, this is a, yeah, no, no, this is a, in a t, this is a t goes to infinity limit, okay? Okay, then it becomes, yeah.
01:25:03
If, yeah, of course, if, yeah, there's some sudden something happens, I mean, it becomes one vacuum or something like, it happens, like a wall crossing, I don't know, or with index jumps. Then, yeah, of course, this computation doesn't work.
01:25:20
But- Does the vacuum, the t equal to infinity theory, or R3 times this one? Die, die, die, die. Okay. Die, yeah. And then we change, yeah, one over t to the zero to one discrete, yeah, at least that t goes to infinity limit.
01:25:42
Okay? We compute it. Okay. I thought that you need that to compute it. No, no, we computed it for the t goes to infinity limit. The action, if action is a change, then we can compute this one, because it's a weak coupling.
01:26:03
Okay? Then we found its ANC vacuum. As it changes, t goes to infinity to infinity finite, or one over t to zero, zero to something- Okay, yeah, yeah, yeah, yeah, yeah. If we change, it's, yeah, drastic change, like jump over with index or something like that.
01:26:21
It can happen. But it's, yeah, usually assumed in a user computation, I mean, user localization technique, it doesn't happen. That is assumption. Okay. But, okay. If it happens, yeah, if it's a modular,
01:26:40
I mean, yeah, the back end is not discrete, then if we change the t, then the back end can change in a modular space, or something like that. It can happen, but it's discrete for this case. So, it's okay, I think. So then, yeah, here,
01:27:01
we consider the localization in S1, S1, sorry, S1 cross R3, but we can directly use this one to the R4 case. Then maybe it's more simple, but this doesn't work.
01:27:20
Actually, there's a two-scale for this case, and the coupling is weak for this region. This region is weak, because here, the coupling is,
01:27:41
I mean, scale is here. Above this scale, I mean, much of this scale is weak, but here, this region becomes stronger. And even when we take t goes to infinity limit, then there should be some region strong.
01:28:03
So, theory never becomes strong coupling. But, okay. For this case, for this case, this region is weak,
01:28:21
because this factor. But this region is also weak, because in this region, sorry, it becomes a 3D measure, so it becomes weak, okay. So this case, weak coupling computation is okay, but for R4, there is no this scale.
01:28:45
So, theory never becomes weak. Even if we take the t goes to infinity limit, okay. So, localization doesn't work here. So we need to compactify this one.
01:29:02
And S4 is not good, as I explained. Okay, so finally, constant n equals one for the vector.
01:29:29
And the general action is like this one. Here, this is Yamir's term,
01:29:41
but we can consider the general action. Then we can include this, the other terms. Then, following this computation, okay, and yeah, some work, then we can show. This is given by the,
01:30:12
this is given by the, sorry, this one.
01:30:22
This is called the Benecian-Yankelovitz potential. So, okay. So, for this action, the non-perturbative production is given by this one. That is, yeah, with some computation, we can show that. And then, conscious of the Kähler multiplex,
01:30:43
and then we can add the Kähler potential with the matter multiplex, arbitrary. Then it becomes a weak coupling. So we can integrate the matter arbitrarily weak. So in a perturbative, we can integrate as a matter field. Okay, then we get this action,
01:31:03
but because of the, we should integrate as a matter field with background field, S or W alpha. Okay, so then it becomes this one. Then, the gauge in condensation is given by this one, if we know this one. I mean, perturbative result, okay?
01:31:22
And if the matter cell is, I mean, adjoint case, it is given, I mean, by, this F is given by the matrix, C equal one matrix model computation. Okay, that is called a digraph buffer. And this means, there is some,
01:31:42
some proof of the digraph buffer theory, by Katya Zu, digraph, Sabag-Witten and Zanon, digraph buffer, something. But they assumed, adding this term by hand, I mean, no potential, this one.
01:32:00
Actually, we can, I mean, show this is okay for the non, I mean, non-SUJI breaking case, G to G, then it's okay. But if, for example, SUN to SU, anyone times SUN to SU or something, then we cannot trust adding this one is okay. But here, this type of computation can show this is okay.
01:32:22
I mean, this is just a straightforward computation, I mean, a gauge in a condensation. So this is okay. So that means, yeah, for with any current method, or any action, we can compute this gauge in a condensation by the perturbatory integrating us, then fixing F.
01:32:45
So this is some proof, I mean, no perturbative proof of the digraph buffer conjecture. So that's it, thank you very much.