1/2 Extended operators
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Extension (kinesiology)Physical systemQuantum field theorySurfaceMusical ensembleDifferent (Kate Ryan album)Boundary value problemInterface (chemistry)Dimensional analysisOperator (mathematics)Local ringSupersymmetryTime domainQuantumLine (geometry)Körper <Algebra>Lecture/Conference
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DistanceNichtlineares GleichungssystemBounded variationMatrix (mathematics)Hydraulic jumpMultiplication signMereologyBoundary value problemGamma functionAxiom of choiceOperator (mathematics)Alpha (investment)SpacetimeRootPhysical systemSupersymmetryVector spaceGauge theoryRotationMortality rateTerm (mathematics)Local ringLimit (category theory)Queue (abstract data type)SpinorString theoryParameter (computer programming)Transformation (genetics)SphereSquare numberGroup representationMassInterface (chemistry)Connectivity (graph theory)ThetafunktionVector fieldLimit of a functionSymmetry (physics)Potenz <Mathematik>Combinatory logicLoop (music)Lecture/Conference
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Theory of everythingSupersymmetrySpherePole (complex analysis)TheoryComputabilityMaterialization (paranormal)Function (mathematics)Bounded variationIdentical particlesTerm (mathematics)DivisorMultiplication signPotenz <Mathematik>PolynomialSquare numberTransformation (genetics)ExponentiationSpacetimeMultiplicationAlgebraic structureMereologyAlpha (investment)Symmetry (physics)Group representationMatrix (mathematics)Axiom of choicePosition operatorSign (mathematics)Nichtlineares GleichungssystemContent (media)Real numberOperator (mathematics)AlgebraGauge theoryModel theoryPartition function (statistical mechanics)Boundary value problemProfil (magazine)ModulformElement (mathematics)Körper <Algebra>Numerical analysisSineGamma functionLocal ringGroup actionCategory of beingAreaQueue (abstract data type)Right angleState of matterBlock matrixCalculationMusical ensembleVector spaceScalar fieldPresentation of a groupNetwork topologyRing (mathematics)Partial derivativeMathematical analysisCondition numberLecture/Conference
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Bargaining problemBoundary value problemGamma functionWeightMatrix (mathematics)Right angleComputabilityAxiom of choiceProduct (business)Price indexFunction (mathematics)Local ringRegular graphGroup actionBeta functionAlgebraRotationFree groupKörper <Algebra>Sigma-algebraBoundary value problemCondition numberNumerical analysisString theoryAlpha (investment)SupersymmetryMultiplication signGravitationMaxima and minimaMereologyGroup representationDimensional analysisTransformation (genetics)Computer programmingNetwork topologyDeterminantSphereConstraint (mathematics)Gauge theoryWell-formed formulaSupergravitySign (mathematics)ThetafunktionLimit of a functionOcean currentRule of inferenceResultantParameter (computer programming)Curve fittingKontraktion <Mathematik>Vector spaceSymmetry (physics)Normal (geometry)SpacetimeReduction of orderGraph coloringScalar fieldDirac-MatrizenLecture/Conference
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Interior (topology)Right angleTerm (mathematics)Gamma functionLimit of a functionConjugacy classSphereConfiguration spaceFunction (mathematics)Group actionTheoryVector spaceString theorySymmetric matrixBounded variationVector potentialBoundary value problemLoop (music)Körper <Algebra>Group representationGauge theoryLocal ringCorrespondence (mathematics)MassComputer programmingFunktionalintegralOrder (biology)Potenz <Mathematik>CurveSigma-algebraSupersymmetryMultiplication signState of matterLecture/Conference
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AsymmetryBounded variationGroup representationSpacetimeTerm (mathematics)Boundary value problemComputabilityGroup actionKörper <Algebra>Vector potentialSign (mathematics)Operator (mathematics)Potenz <Mathematik>Multiplication signIdentical particlesLinear mapCategory of beingMatrix (mathematics)Gauge theoryElement (mathematics)DivisorAxiom of choiceSquare numberNichtlineares GleichungssystemLecture/Conference
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Block matrixLocal ringHolomorphic functionDivisorMatrix (mathematics)Group actionMereologySpacetimeMultiplication signConstraint (mathematics)CalculationLecture/Conference
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Local ringBoundary value problemLoop (music)Gauge theoryDeterminantSphereMultiplication signAngleSigma-algebraResultantNetwork topologyComplex (psychology)ThetafunktionGroup actionPhysicalismHiggs mechanismString theoryVector spaceConstraint (mathematics)Gamma functionFunction (mathematics)Regular graphInfinityProduct (business)WeightModel theoryLecture/Conference
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Multiplication signDivisorDirichlet-ProblemDeterminantLecture/Conference
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Boundary value problemMultiplication signDirichlet-ProblemDuality (mathematics)DeterminantOrder (biology)ComputabilityDerivation (linguistics)SphereLecture/Conference
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Gauge theorySupersymmetryRegular graphSymmetry (physics)ExpressionEnergy levelSymplectic manifoldLecture/Conference
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Vector spaceExpressionAlpha (investment)Product (business)Function (mathematics)Position operatorResultantSymmetry (physics)Gauge theoryBoundary value problemRegular graphTerm (mathematics)Root systemAxiom of choiceCombinatory logicArithmetic meanPhysicalismTheory of relativityDuality (mathematics)Euclidean vectorValidity (statistics)Stochastic kernel estimationLocal ringDeterminantLoop (music)Lecture/Conference
01:28:43
Diagram
Transcript: English(auto-generated)
00:16
Okay, thank you. So I'm Takuya Okura from the University of Tokyo and the organizers asked me, actually
00:23
not just boundary, but also extended operators in actually supersymmetric localization with extended, extended operators in
00:42
supersymmetric quantum field theories. So I think okay by extending what is meant is that the operators are not necessarily local so they can some non-trivial dimensions and the examples include
01:02
line operators such as Wilson-Hust line operators and there are also surface operators and then
01:25
here come boundaries interface interfaces also known as domain walls and
01:47
we can also have coupled system of QFTs of
02:01
different dimensionalities. I think I can include them as examples of extended operators in quantum field theories.
02:22
And we can do various things and we can indeed do supersymmetric localizations with these extended operators and also they play significant roles in the 2D, 4D or AGT correspondence or related 3D, 3D correspondence and so on and
02:44
these objects are also important in other contexts such as conformer bootstrap, holography and so on and maybe most significantly especially the line operators or loop operators were first introduced as order parameters of
03:04
gauge theories so the behaviors the behaviors of their expectation values or correlation functions as functions of some parameters can be used to classify possible phases of quantum field theories and
03:25
so today I will focus on the boundaries and interface, well the mostly boundaries in two dimensions so this is a topic of today and
03:42
well at least my current plan is to talk about line operators based on truth operators in four dimensions tomorrow, that's my plan.
04:03
Okay, so today's topic, this is about in three dimensions, so today's topic is closely related to what Benigni discussed last week and okay, the topic of tomorrow tomorrow the topic will be about four dimensions, so it will be closely related to what
04:26
pillars discussed last week. Okay, and I do believe that the examples I will discuss are important examples, but I also want to
04:42
emphasize and illustrate the general principles that that can be learned from the examples I concretely discussed so these are so well I believe my examples will be interesting and important, but I also want to emphasize the general aspects.
05:03
Okay so I usually I write down the plan of my presentation so for today's lecture here is the plan so I first well, I'm already giving the
05:22
giving an overview of my lectures and then I will discuss the so-called b-type boundary conditions and then I'll explain that something called matrix factorization will be necessary and then I will actually
05:42
perform super symmetric localization and explain how to compute the so-called hemispherical partition function. Okay, and then I will tell you that the hemispherical partition function has a meaning as central charge over D-brane when
06:02
the two-dimensional theory is used as a word series theory of type two string theory and also as an application of the hemispherical partition function I plan to discuss the interface. Okay, and of course the questions are welcome so please feel free to
06:23
feel free to stop me and ask questions anytime. Okay, so I want to introduce
06:42
b-type boundary conditions and in order to do that I need to explain the setup. So this is basically the same as what Beni discussed so we are going to consider two dimensional n equals two comma two
07:02
theory on the two sphere or well or hemisphere.
07:21
All right, actually I want to give you references so as a general reference, elementary
07:44
pedagogical reference, I give you section 39 of a textbook mirror called mirror symmetry written by
08:07
holy other people. Well you can google you can google mirror symmetry holy
08:21
and then you can find the PDF from the Kray mathematical institute. That's true. Yeah, but this is the book. Yes, okay, maybe you can put the Kray and then you'll find PDF or you can you can get it from INSPI.
08:42
So this is a textbook and the actual content of my lecture today will be based on on my paper on the hemispherical partition function. And actually on the hemispherical partition function there are three papers.
09:06
One is by Sritishthana Teresima. Teresima was here last week. Another Onda and myself and
09:20
yet another one by Hori and Roma. These are all from 13 August 2013. So basically these papers were
09:41
posted on the archive simultaneously. Of course, naturally I would be mostly using the conventions or my own paper. Yes, and our convention is very similar to
10:01
Dorud Benini and sorry Dorud Reflog Gomis and Riva were using. Yeah, okay. So so the setup is a two-dimensional n equals 2 comma 2 and
10:27
because we are on a curved line fold we want to consider some background, super gravity background. And okay
10:47
I'm I'm interested in a particular version of two-dimensional super gravity called U1b and this is a dimensional reduction of four dimensional n equals 1 super gravity, the new minimal super gravity, which is future discussed
11:03
last week. Okay, so in particular vector R symmetry. Yeah, so this super gravity can be coupled to a theory with vector R symmetry. And I want to consider
11:24
the following metric.
11:46
So f So theta and phi are the usual polar coordinate on the truth here and f is it behaves like like this for theta
12:05
close to zero and okay because we are interested in essentially the hems here. Let's say I want to restrict to Well, actually, okay
12:22
Yeah, okay, so this is the behavior for theta near zero and okay. On the truth here theta takes value between zero and pi and phi takes values between zero and two pi as usual Okay, and
12:41
sometimes I will use field by so and I will be using the hat to denote the frame index. So this is the index for the field by so e1 hat is f times d theta and e2 hat is
13:04
L sine theta d phi Okay well if f is simply L then this is this corresponds to the round metric
13:23
of radius L and We may also take L squared cosine theta cosine square theta plus L tilde squared
13:42
sine square theta where both L and L tilde are some parameters cosine parameters and this is also a good example of f. I need to experiment a bit and
14:25
Okay, we have we also have your asymmetry gauge field So V R. So this is a one form one minus
14:40
L over f of theta d phi like this And if you remember I think from Benin's talk this super gravity this version of super gravity also contains something I think He denoted by H. So this is a
15:07
gravity photon field strength and this is okay, this is a proportional to one over f
15:22
Okay Am I clear? Is everything clear? No question Okay Do you know what data the partition function would depend on? I mean for example in the 4D case grid was explained that it depends only on the complex structure etc. So do you know if the similar analysis is done specifically in 2D and what the conclusion is?
15:45
Well, I know the result of localization calculation and Let's see Yeah, it depends on the context but for example for the Hentia Poisson function in the conformal case there is an analysis using the super wire anomaly by Bacchus and Plankner and
16:05
They showed that it depends holomolecularly on the Kähler Yeah, so and here I introduced deformation so L and L tilde
16:25
So ST partial function There is no analog of B for ST partial function. Yeah, so here I introduced deformation But the partial function actually is independent of that deformation So it's actually enough to
16:42
compute a partial function on the round sphere I'm asking you if you take a more general metric, are you sure that this is the only answer you get or you take a crazy some complex structure etc. Is there a possibility of getting another answer? For the two spheres, I don't think there are many
17:07
Okay, sorry say that again You say when theta goes to zero, f approaches n
17:22
Yes So then it becomes zero This one? Yeah, so right so Yeah, indeed f goes. Yeah, this part becomes zero at theta equals zero and that necessary for the for this gauge field to be smooth
17:40
because this is a angular variable Yeah, I think I'm going too slowly so Okay Now I should now write down the generalized
18:02
Cheating Spinner, can you read? So So this is a condition for super symmetry to be preserved. Epsilon is a
18:28
two component spinner and I'm going to define the gamma matrix So there are two kinds of
18:43
Spinners epsilon and epsilon bar minus one over two f gamma mu gamma three epsilon bar epsilon and epsilon bar have opposite R charges and
19:01
Okay, my convention for gamma matrices is Theta heart corresponds to okay one gamma one heart This is okay the first sigma powered matrix gamma five heart is
19:23
minus I I and Gamma three just denotes the chirality matrix in three dimensions Yes Okay
19:40
and on the on the round sphere we have SU trash SU two slash one symmetry, but The deformation and also when we put a boundary around the equator theta equals pi over two The symmetry SU two slash one he gets broken to SU one slash one
20:05
so so that's the symmetry We will be interested Interesting So I'd be considering very explicit
21:01
Spinners supersymmetric parameters, so I'm just writing down the solution to the killing killing spinner equations general killing spin equations so I'll be considering explanation minus i over two theta gamma two heart
21:22
So this is a matrix and it acts on exponential of I over two times Phi Zero and epsilon bar equals I over two
21:40
theta Gamma two heart zero It is minus I over two phi. Yeah, so so these are the general killing spin equations, which I wrote down and They all
22:07
And if you compute the super symmetry transformations and then you find that the super symmetry squares to bosonic symmetries and And the bosonic symmetries involve some killing killing vector field
22:23
So it gives a rotation of the sphere and the rotation is such that The the boundary at theta equals phi over two is preserved. Okay. Yeah, so
22:45
so I will be putting the boundary at Theta equals phi over two and in this limit the spinners the supersymmetric parameters becomes
23:02
both basically one one up to Arithmetic transformation. This is because gamma two
23:24
Heart is a two by two matrix and We see and if we set
23:45
X 1 to be L tilde times theta minus I over 2 and X 2 is a L times phi then locally near the boundary The system looks like this
24:03
so there is a boundary at X X 1 equals 0 or theta is in pi over 2 and The the remaining part is yeah, so the the region on the left is the bulk of the hemisphere
24:21
okay, and if you okay, I haven't told you the rule, but if you compute the Supercharge, yeah, so so in terms of the spinners they look like this and
24:48
In terms of supercharges looks like
25:01
well, so in the in the combination of the combination of the textbook which it takes back it looks like this and these are and this such a
25:39
supersymmetry is known as
25:42
b-type supersymmetry So, yes
26:02
Well, this is this is simplicity and It may be possible it may be possible to okay, I think it's an interesting question whether one can actually put a boundary or some somewhere else and Actually some interfaces can be put
26:22
At places other than the equator like I know that we're going to put the Wilson loop can be placed away from the equator so so I think it's an interesting question whether one can put a boundary somewhere else and and for example do supersymmetric localization
26:42
But I don't yeah, I don't know for sure with a One can do localization For a boundary or a general marginal location but this is a choice and And this choice is nice because the supersymmetry looks like a b-type and I think
27:10
Benny told you that near the north and the south poles of the Two sphere the supersymmetry looks like a-type. So so the theory looks like a model and
27:22
But now If you consider a boundary along the equator then the supersymmetry there is a b-type so if you consider so for me the the aim of this talk is to compute the hemispherical function and
27:41
So this hemispherical function then gives an interesting
28:30
The hemispherical function gives a Interesting coupling between a model observables at the North Pole and
28:43
B-type observables at the equator, so let me write it equation so just by analysis of supersymmetry, it looks like the hemispherical function
29:02
is Is of this form so so So something that depends on the boundary or boundary condition it Brain it's a d-brain preserving B-type supersymmetry. So it's called b-brain and
29:23
On the North Pole we have a model so that means that if we do not insert any operator then it corresponds to the identity operator the algebra of operators of
29:41
Topologically twisted a model is called the AC ring So the hemispherical function appears to be computing a coupling coupling like this Coupling a pairing like this and I think I will come back to to this later
30:00
So so my my goal is to compute this this partial function Okay, so I so far I have only
30:22
Explained the background and I want to Tell you about the matter content the field content. So we have We have a vector multiplete for gauge group G and this includes the gauge of a mu
30:54
Two real scalars sigma 1 and sigma 2 and there are gauge any
31:00
Lambda alpha and lambda alpha bar. So alpha takes one and values one and two and There is a real auxiliary field D and then we also consider the chiral
31:21
Earth breath and it's well in representation in the presents R of G and the fuse are Phi
31:41
Psi alpha and F. Okay. So these multiplets have already appeared in the lectures Last week, I think many times Okay
32:01
Yes I mean
32:27
So you're asking about the computation from here to here, yeah, okay
32:45
Yes, and You're asking how to get rid of this Yes, this is by our symmetry Gate transformation. Yeah. Yeah, so the
33:17
You'll be interested in
33:22
vector and the chiral Okay, so now I want to say a little bit about a super symmetry
34:10
Transformations, I'm not going to write all of the transformation
34:26
So For example for the victim are spread we have Delta of a mu Going to like minus i over 2 It's number gamma mu
34:43
lambda plus lambda bar Epsilon and my convention is something like this. So For example, if I write epsilon bar gamma mu lambda lambda then this means
35:00
that so it's from bar alpha with upper index now gamma matrix has Lower left upper right lower right an index for the spinner and The for the spinners the indices are placed and lowered by charge congregation
35:27
Matrix, of course, these are details, but I Gave some exercises to check boundary con the super symmetry of boundary condition. So so
35:43
You are going to use Some transformations like this Okay, and then there are other transformations For the color of much breath we have
36:01
So delta phi so the transformation of the complex scalar is epsilon times psi where the contraction rule is basically, okay gamma removed And Okay gamma psi is
36:22
I times gamma mu epsilon d mu phi Plus I epsilon sigma phi plus gamma 3 epsilon
36:42
sigma 2 phi plus actually so There is some conventional dependence on the of the sign of the archers I'm choosing I think Choosing okay some convention
37:01
This is which is actually the opposite from the paper I think and then okay, there is some formula for F like this Okay
37:24
Yes, so this is exactly the transformation on the on the sphere, that's right So yeah, so the hemisphere is just a half of the sphere. So we are using the same Backgrounds and same transformations
37:55
Yeah, and I know I'm doing I'm considering the boundary in two dimensions, but the story
38:01
Can be repeated in other dimensions like three and four dimensions and the people have indeed studied Such spaces People have considered hemchairs in four dimensions definitely and
38:20
In three dimensions, I think people have considered It's one times two dimensional hemisphere. So say again Part of it, that's right. Yeah, so
38:48
Yeah, so beta beta by beta I mean really a super sub algebra of full n equals 2 comma 2 and in particular B-type sub algebra has two
39:01
Superchances and it's also possible to consider a type and a type and B type are rated by mirror symmetry and And from four dimensions These setups arise by dimensional reduction of okay either new minimal gravity new minimal super gravity or
39:24
minimal super gravity Okay Now, okay. Now I want to write down the super symmetry
39:41
Boundary conditions of a b-type before the vector marketplace. I will consider the following
40:08
At theta equals pi pi over two, so Since we are considering the field here, okay
40:26
So we require that He'll certify the following so here a a a one really means
40:40
Okay, I can also write a a theta and by boundary boundary condition We require that the field strength is zero at the boundary. So so
41:34
You see that Yeah, a one is zero, but I don't I don't do anything on a two or a five
41:45
Well except that okay, essentially this is a normal boundary condition So so this preserves this actually preserve gauge symmetry along the along the boundary So so this is a boundary condition I want to consider but it's also
42:01
possible to consider a Boundary condition that breaks symmetry, but I'm not going to do that today Okay So this is for the vector multiple. Now, I'm going to describe
42:45
boundary conditions In the chiral marketplace The main the main boundary condition on the chiral marketplace is a No, no, no, no boundary condition on the chiral marketplace
43:09
Which says that d1 phi equals 0 epsilon bar gamma 3 psi equals d1
43:22
epsilon bar psi equals 0 But it's also possible to consider an alternative boundary condition which Says that the phi is constant. Okay, I choose the constant to be zero. So phi is zero
43:48
epsilon bar psi equals d1 epsilon bar gamma 3 psi equals 0
44:02
d1 hat e to the minus i phi F i d1 phi equals 0. Okay This the appearance of e to the minus i phi may be annoying but it can be removed by again
44:23
Automatically gauge transformation. So so To some extent this is a matter of convention. Well, so if you go to a different gauge this disappears and then Boundary condition looks more natural. Yeah. So in one of the exercises for this lecture is to check
44:44
that this one, okay, this boundary condition actually in flat space the flat space version of these boundary conditions is It preserved B type supersymmetry, so that's one of the exercises
45:10
I see. Okay, so The way it's written these are not elliptic I think because I think imposing
45:21
These are differential boundary conditions on the field means is I think it's too strong if you try to solve the If you try to formulate a variation of principle Yeah, but for supersymmetry organization. This is this is enough and that's
45:44
What I'm doing Yes But you can Yes, I think in this case you can actually just remove the well at least for the norm I think you can just remove this
46:05
Yeah, the thing that If you have this equation, then you can do a super Smith transformation and I think you find you find this
46:27
Right. So yeah, so if you if you want to do perturbation theory then You need to be more careful about the boundary condition, okay
46:47
Now, okay, so these are the boundary conditions, okay any more questions Okay, so so these are Boundary conditions and I'm still in the process of defining the theory
47:01
So defining the theory on the manifold with boundary. Yes, I don't know what I don't
47:26
Yes, right. I get Oh AC ring Yeah, so Yeah, so the question is
47:43
the following so so I wrote that the hemispherical function equals the overlap between the b-brane state and Identity operator in the AC ring and But she pointed out that I only have a gate theory and I don't have a super conformal symmetry so
48:04
How does it make sense? So that's the question right now Yeah, so Did I did I well what I What I meant was So that was a rough equality and actually I probably should have a put on a question mark whether that
48:26
Equation is true or not Yeah, I was going to mention it today later, but I can I can tell you now So we are considering a gate theory and in some cases so we saw with the right
48:44
matter content and charge assignment that this gate theory can flow to super conformal field theory and then the The thing well historically that once you get the formula for hemispherical function
49:05
From no examples it was apparent that hemispherical function Compute something called the central charge, which can also be written as the overlap between the b-type B-brane boundary state and the identity element
49:21
Yeah in the AC ring or the Raman Raman ground state so so so when we wrote These papers it was a conjecture that the hemispherical function computes such an overlap and more recently I've already mentioned there's a There is a work by Bakas and Planck which use uses
49:44
The super viral anomaly to show that this conjecture is actually true. So so it's not trivial It's not trivial that Calculation in gate theory actually computes the overlap between the
50:00
boundary state and The identity operator, but it can be shown independently. Yes. Yes to be a state
50:22
right, uh-huh okay, so the question is why is a hemispherical function is Is a Number rather than a state. Yeah. So I think what is meant is that yeah, so path integral on a
50:42
Manifold with boundary can indeed Represent it can you represent a wave function, right? So yeah, so but but So in general in quantum field theory, so it's good because I need to emphasize the general principle In quantum field theory, we need to distinguish the two kinds of boundaries
51:03
so one Yeah, I forgot the general name, but okay one type of boundary condition supports us State actually normalizable state in the silver space. Okay, so some some type of boundary manifold represents
51:20
Space time represents a state on the other hand. There are boundaries that Really boundaries, so these are not An intermediate state But rather it corresponds to the end of the world okay, so some some some some boundaries in quantum field theory end of the world and
51:44
So and here we are Yeah, yeah, okay, okay Now I'm not completely sure if there is a clear distinction, but but that's a general story and here I'm considering the
52:01
boundary that is the The end of the world and in that case Yes, an example is a d-brane. Okay in string theory Well, I mean, I mean the boundary in the sense of the world sheet So if you have a d-brane boundary condition then you can
52:28
Construct a State which is a boundary state. So it's a state but it's not normalizable Okay, so if you have a genuine boundary in quantum field theory, then you can formally construct a state but it's not normalizable
52:40
So if you compute a normal square it diverges Yeah, so in that sense I do not regard the hemispherical function as Wave function, it's more like a partial function. Okay, any other question?
53:03
Okay Now, yeah, so, you know, so I want to define a theory on Space time with boundary and in order to do that, of course, I need to specify
53:20
the Lagrangian Lagrangians and actions Okay I'm going to write action Actually, yeah, so so and there are two types of actions so one is a physical action and the other one is a localization action and
53:45
The theory is specified by the physical action and it takes the form So the vector martinite part the chiral martinite part the super potential term
54:01
Theta the topological term and the Fi parameter Okay I think yeah, so basically these are I Think I am I'm going to omit giving these explicit expression because they are essentially given in Benigni's talk
54:25
I think but There is one point Yes Okay, yes, okay good, yeah, so the the only important part is
54:42
The super potential term which looks like a Fi Ti W W is a super potential. So it depends on Phi looks like looks like this
55:19
I Think the remaining terms. I think you mostly know and the important thing is that
55:27
If you compute that through the variation of the physical action Then Horvit vanishes except the super potential term
55:48
So you can Okay Yes, also essentially so the remaining in terms are Can can be written as through the exact term plus boundary term and the boundary terms vanish by the boundary condition
56:07
Yeah, that is true for vector and chiral Okay, the theta term actually need to be super symmetrized and if you do that It's super symmetric and the Fi term is also well usually
56:24
On the truth here also We use a super symmetric version Well, I'm at a super symmetric. Yeah Super symmetric action on the truth here. Yeah. So anyway, what's important is that the super potential
56:40
the variation of the super potential term Is It's not automatically 0 and this right this looks like
57:03
epsilon gamma mu psi I divide W plus Conjugate so it's a Yeah, so it becomes a boundary term. Okay, and this means
57:26
What does This means a foreign this implies
58:02
So the hemispherical function well, it's supposed to be You are given by the path integral over field configurations Subject to p-type boundary conditions and the theory is defined by
58:25
The physical action and we might put a localization action also, but the problem is that This is This does not represent a super symmetric system because of the because
58:41
The variation does not vanish. So we need to do something about it So this is a well-known program and this term is called Warner Warner term and This one the term needs to be cancelled
59:03
And in order to cancel it we include The following term trace in some vector space V Passover of exponential of some Some version of okay script this script a script a phi
59:27
So these are some version of Wilson loop But this Wilson loop. So this is called the boundary interaction
59:47
and this Looks like the usual super symmetric Wilson loop a phi plus i sigma 2 in some representation For this
01:00:00
for this vector space V. Yeah, and this vector space V called the champeiton. So this is familiar from string theory. Champeiton vector space. And it carries some representation of the gauge group, flavor group, and the earth material group.
01:00:24
Let's see. And then there's a contribution corresponding to earth charge. If there is flavor's material, there is also twisted mass. So this is, okay, I think I forgot to mention. Twisted mass is a scalar component
01:00:43
corresponding for the vector multiplet, background vector multiplet for the flavor's material. Well, this was explained in Ben-Yin's lectures. And now there is, there are some terms that depend on what I call Q
01:01:04
and its conjugate Q bar.
01:01:38
And here, so Q is some operator that depends on phi,
01:01:45
and it's an operator which maps, which it's a linear operator that maps V to V, champeiton space to itself. Okay. So what I want to do is to, yeah.
01:02:03
So the role of Q is that if you compute the supersymmetry variation of the integral, of the path integral, then the variation of the boundary, but the variation of the boundary interaction should cancel the Voronoi terms.
01:02:21
And that's the role of the Q and boundary interaction. Let's see what's next.
01:03:04
It's actually an interesting computation that you can, it's a interesting computation, also it's relatively non-trivial. And if you do it, it's actually rather long computation. And for details, you can look at that,
01:03:24
you can look at a paper by Herbst, Hori, and Page from, I think, 2008. So anyway, the upshot is that the supersymmetry variation
01:03:46
of the super potential term, e to the minus super potential terms, times trace of, tracing V of P of exponential of I D phi A phi.
01:04:04
So this is zero if Q phi squares to the super potential times the identity operator of V.
01:04:20
Is that statement clear? Okay.
01:04:42
Yeah, so this is the most important property of this operator Q, where it's also called the Tachyon profile, or is also called the matrix factorization.
01:05:01
But for this physically to make sense, Q has to obey some constraints, and okay, let me write it this way. So suppose that, okay, so G is an element of the, okay,
01:05:22
gauge group and the flavor asymmetry group, which I denote by GF. And suppose that rho is a representation of a gauge group and the flavor group on the champetron space V. Then we need that it satisfies the following.
01:05:56
So this is necessary for this to be symmetry.
01:06:01
And we also need that for asymmetry, yeah, so there is a representation of, there is a representation of asymmetry
01:06:23
on the champetron space V, and then Q has to satisfy this.
01:06:41
By capital R, I mean the action of asymmetry on the chiral multiple field. The sign convention here is a little strange, a little strange, but this is a standard choice.
01:07:04
So anyway, so these equations need to be satisfied. And in the exercise, in the exercise, I give at least one example,
01:07:20
and okay, I didn't ask you to do this, but you can also try to check. With some appropriate choice of representations, the example I gave in the exercise actually satisfied this. And also I want to explain
01:07:41
why this is called matrix factorization. So if the champetron space has a structure, so if it's Z2 graded, and actually it has to be Z2 graded, and yeah, okay, maybe this is part of the constraint.
01:08:01
So Q has to be odd, so this is one requirement. And then we can, okay, using this decomposition, we can write Q as block diagonal matrix,
01:08:21
is non-zero entries, non-zero over-diagonal entries, A and B, and then Q squared equals B, can be written as AB equals BA equals W times identity. So you see that, yeah, so A and B,
01:08:45
indeed give a factorization of this holomorphic function W, which is usually polynomial. Okay, so I have 18 minutes, I see.
01:09:12
Okay, so I explained the matrix factorization.
01:09:26
Now I want to perform a localization calculation.
01:10:00
And basically I'm going to recycle what Benigni did on the two-sphere. So, okay, first of all, I'm going to use the same localization action. So I'm going to omit a lot of things.
01:10:21
I can omit a lot of things because Benigni did most of the work. Yeah, so we use the same localization action. And, but actually we are going to specialize. And here I'm going to consider
01:10:40
a Courant branch localization. Higgs branch localization should also be interesting, but it has not been done yet. So now for the vector merit bet, remember that we require that F1, so the few strings is zero on the boundary.
01:11:01
So this gives a strong constraint on the localization locus. In particular, it means that, okay, so the few strings is zero on the localization locus, sigma one is zero, and then what remains is a zero mode of sigma two.
01:11:21
So the constant mode of the sigma two. So we get the path integral reduces to finite mental integral over sigma two. And for the Courant merit bet, we get if phi equals zero, I think F is also zero. And of course the Fermions are zero.
01:11:42
Then we can evaluate physical action on the localization locus. And we find that e to the minus physical action is given by e to the, okay,
01:12:03
t times trace of sigma. Okay, so I'm writing, so I'm writing the result as if the gauge group is, well, I'm writing the result for the gauge group un, for gauge group un.
01:12:21
And t is the complex wide Fi parameter, which, okay, I think I could write two pi, so Benigni was using xi, so I think in his combination,
01:12:41
t would look like two pi xi minus i theta, theta, or theta, the topological theta angle. Okay, and the boundary interaction,
01:13:07
we also need to evaluate the boundary interaction on the localization locus. And if you do that,
01:13:21
you find that this is given by e to the minus two pi i times sigma plus m over, okay, here by sigma, I really mean minus iL times sigma two.
01:13:45
And here m is, yeah, okay, minus L times sigma two times the flavor path.
01:14:05
So this is the classical path, okay.
01:14:21
I'm going to discuss the one loop, one loop determinant.
01:14:44
Yeah, so the zero principle is the following. So without the boundary, people have already computed the one loop determinants. And the localization locus is also a specialization of the sphere case.
01:15:03
And now, when computing the one loop determinant, we only need two contributions. We only need to keep contributions from the modes that obey the boundary condition. So that's what we're going to do, okay. And, okay, for the karaov multiplet,
01:15:24
with no non-bound boundary condition, so this, so, okay, formally this can be written
01:15:42
as a product of weights in the model representation, and the infinite product, which is divergent.
01:16:06
And, okay, as Benigni did, we might do zeta function regularization and get like this.
01:16:31
And if you have a flavor symmetry and a twisted method, we also need to include them. But the way it enters is the same as sigma.
01:16:44
Okay, Dirichlet boundary condition. This may be a bit interesting.
01:17:04
Again, we have a product of weights, and there is an infinite product.
01:17:25
Now, if you do zeta function regularization, you get something, okay, one over gamma function, one over gamma function.
01:17:47
But I claim that this is wrong, this is wrong, okay. So I claim that zeta, so this is hard to read, right?
01:18:05
So zeta function regularization is, okay, I say not always correct. More precisely, okay, zeta function regularization
01:18:23
does not always automatically give the correct result. And now I'm going to give you the correct result and explain how to justify or derive it
01:18:44
from first principle.
01:19:06
We were correct, one determinant for the Dirichlet boundary condition includes extra factors. Minus two pi i, so very specific factors,
01:19:23
pi i times w times sigma, I think I need to include minus i over q over two divided by one minus w sigma minus i,
01:19:40
q over two. So then why do we need to include those factors, right? So that's not a crucial question, right? So there are two explanations. So one explanation which we originally had in a paper was that there is a duality
01:20:06
between Dirichlet boundary condition and Neumann boundary condition
01:20:20
plus some boundary condition, boundary interactions. So yeah, so there is something called Attiabott-Shapiro construction of diesel brains. And there is unknown duality between the boundary conditions.
01:20:41
And in order for the one-loop determinant to be consistent with such duality, we need to include these factors. By now, there is, okay, I have not published it, but there is a first principle derivation of this,
01:21:03
which is Parevilla's regularization, yeah. Yeah, so the computation for the hemispheres
01:21:21
has not been done, but Parevilla's itself, I wrote a paper one year ago. So you can take a look if you're interested. So if you apply Parevilla's, then the ratio between the Dirichlet and Neumann
01:21:45
one-loop determinant turns out to be one minus e to the e to the two pi i times W sigma minus i.
01:22:06
I times Q over two. And Parevilla's, Parevilla's, this preserves gauge and flavor symmetry.
01:22:21
So if you demand that flavor symmetry, okay, gauge symmetry is preserved, gauge symmetry and R symmetry is preserved, then Parevilla's regularization gives gauge invariant, gauge invariant of the manifested supersymmetric regularization, and this ratio is unambiguous.
01:22:42
It's not ambiguous, okay. So this is fungal, and it's consistent with these expressions. Yes? So if you, what is the Parevilla's? You already have the Parevilla's at the Lagrangian level before organization, or you have Parevilla's afterwards. But if you add the Parevilla's from the outside,
01:23:01
then supersymmetry is not consistent with supersymmetry. At which stage do you introduce the Parevilla's radiator? Well, it's a general story, I think. So we should start from the beginning by including the Parevilla's goals. So then is it consistent with the supercharge you're using? Yes.
01:23:20
Is it? Okay. Yeah, there is a supersymmetric version of Parevilla's. That's why. Oh, okay, that's what you're saying. That's what I'm saying. Yes, yes. So we have what's known as Zeta function? Yeah, so in principle, there is no reason
01:23:43
that Zeta function regularization should work. What I'm saying is not again published, but in general, you can relate Parevilla's regularization, sheet kernel regularization,
01:24:00
and Zeta function regularization by using the, well, the sheet kernel method. And one can check how these regularizations compare with each other. But usually what people have done in localization
01:24:20
is Zeta function regularization is not very systematic. And so I would say that in general, there is no a priori reason that Zeta function regularization should work. So it's valid issue to be checked by other methods like duality. And the relation between the Zeta function
01:24:44
and the, yeah, and Parevilla's is clear for bosons, but for fermions and spinors, this situation is more, yeah, so more care is necessary,
01:25:03
and especially with in the presence of boundary, the situation is more subtle. How it works, which one is correct and which one is wrong? Yeah, so I mean, Parevilla's regularization
01:25:21
preserves the symmetries, gauge symmetries. So if you are interested in results that are comparable with gauge symmetries, this is a correct result. But actually, okay, the Neumann Dirichlet boundary function themselves, actually they,
01:25:41
the boundary function themselves break some other symmetries, like charge conjugation symmetry. And if you want to charge conjugation symmetry, if you give more priority to charge conjugation symmetry, then you should combine some regularization method
01:26:02
with the proper counter terms. So in general, so two different regularizations are related by a choice of counter terms. And regularization and counter terms, their combination is the invariant thing. The physical meaning thing is a combination of regularization, UV regularization, and counter terms.
01:26:24
And the correct choice should be made according to which symmetry you want to preserve. And if you want to preserve gauge symmetry, this is a, so this is a correct result.
01:26:40
Yeah, okay. I think I want to give an expression for the vector method and then stop for today.
01:27:37
So for the vector method, there is a product of positive roots.
01:27:46
And again, there is an infinite product, this alpha. Alpha is a positive root, like this.
01:28:02
And okay, for this, this is a function of regularization, which is compatible, okay, a consistent result. And it is, it's like this.
01:28:31
Yeah, so I finished giving one loop determinant and yeah, I want to stop for today.