2/2 Extended operators
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00:00
Gamma functionSigma-algebraMassGroup actionProduct (business)Function (mathematics)ThetafunktionParameter (computer programming)Complex (psychology)Boundary value problemPower (physics)RankingPartition (number theory)Mortality rateMultiplication signWeightPresentation of a groupGroup representationObservational studyCondition numberSpacetimeAlgebraic structureNumerical digitSet theorySummierbarkeitVector spaceMathematical optimizationMaterialization (paranormal)Well-formed formulaImaginary numberSineGauge theoryDeterminantManifoldNormal (geometry)AngleDirection (geometry)ModulformMultiplicationClassical physicsSampling (statistics)MereologyNetwork topologyOrder (biology)Many-sorted logicMusical ensembleMathematicsAlpha (investment)DivisorRoot systemSphereDirichlet-ProblemMinkowski-GeometrieSymplectic manifoldFactory (trading post)Lecture/Conference
09:22
SummierbarkeitLimit (category theory)Regulator geneState of matterArithmetic meanPosition operatorString theoryBoundary value problemD-braneFunction (mathematics)Gauge theoryProduct (business)Fraction (mathematics)Conformal mapLengthOrder (biology)ComputabilityGroup actionSpherePartition (number theory)TheoryAbsolute valueDimensional analysisNumerical analysisMassCentralizer and normalizerParameter (computer programming)MereologySupergravityManifoldSpacetimeModel theoryRootRight angleSweep line algorithmStability theoryTotal S.A.ResultantGeometryCompactification (mathematics)MultiplicationTheory of relativityMany-sorted logicGoodness of fitSpherical capCondition numberDifferent (Kate Ryan album)AlgebraEqualiser (mathematics)Two-dimensional spaceTerm (mathematics)Partial derivativeRenormalizationCompact spaceObject (grammar)AsymmetrySigma-algebraConformal field theoryCalculationPairwise comparisonDivisorCoherent sheafLecture/Conference
18:44
Mortality rateMatrix (mathematics)Gauge theorySpacetimeDivisorOscillationSurfaceLatent heatHyperflächeNoise (electronics)Condition numberTwo-dimensional spaceBoundary value problemShift operatorFluxVacuumSummierbarkeitProof theoryPosition operatorMassParameter (computer programming)TheoryPolynomialMultiplication signDegree (graph theory)MereologyVector potentialProjektiver RaumModel theoryGoodness of fitAlgebraic structureIdentical particlesSigma-algebraPolarization (waves)Direction (geometry)String theoryTransverse wavePhase transitionGeometryProjective planeHyperplaneAlgebraElliptic curveSeries (mathematics)Operator (mathematics)Musical ensemblePoint (geometry)Dimensional analysisThree-dimensional spaceMass flow rateLine (geometry)Right angleOrder (biology)Positional notationSquare numberTime zoneSet theoryLecture/Conference
26:51
Tangent bundleCharakteristische KlasseGamma functionLine bundleComplex numberSocial classFiber bundleComputabilityVektorraumbündelSphereProjektiver RaumWell-formed formulaRight angleConvolutionPartition (number theory)SpacetimeHyperplaneLogarithmStandard deviationMultiplication signVacuumCorrespondence (mathematics)Sigma-algebraGauge theoryCohomologyProduct (business)Boundary value problemDivisorFraction (mathematics)Function (mathematics)SummierbarkeitTerm (mathematics)RootCharacteristic polynomialHyperflächeParameter (computer programming)Matrix (mathematics)Complex (psychology)Nichtlineares GleichungssystemAlgebraic structureSheaf (mathematics)ChainVolume (thermodynamics)Invariant (mathematics)2 (number)TheoryMereologyQuantum field theoryGoodness of fitSurfaceOperator (mathematics)Dressing (medical)Helmholtz decompositionShift operatorGroup actionCoordinate systemSet theoryClique-widthOrder (biology)CurveLoop (music)Axiom of choiceEigenvalues and eigenvectorsLocal ringLine (geometry)Cubic graphMany-sorted logicSubgroupCausalityVotingIndependence (probability theory)Condition numberExpressionAngleThetafunktionProcess (computing)Different (Kate Ryan album)Graph (mathematics)Descriptive statisticsMathematicsGravitationSymplectic manifoldMaxima and minimaNumerical analysisMathematicianLecture/Conference
34:58
ResultantFunction (mathematics)Boundary value problemGroup actionProduct (business)T-symmetryMultiplication signTheoryDegree (graph theory)Correspondence (mathematics)Condition numberTerm (mathematics)SurfaceExpected valuePartial derivativeSpacetimeResidual (numerical analysis)Interior (topology)Normal (geometry)Power (physics)Operator (mathematics)Local ringInterface (chemistry)MathematicsDivisorSign (mathematics)Order (biology)TheoremParameter (computer programming)Classical physicsRight angleMass flow rateMereologyRiemann surfaceCategory of beingGamma functionHyperplaneExpressionComputabilityClosed setWell-formed formulaINTEGRALGraph coloringMany-sorted logicSphereSineHomologieExtension (kinesiology)CubeFlock (web browser)Parity (mathematics)Transformation (genetics)Physical systemAlgebraStatistical hypothesis testingLine (geometry)Sigma-algebraIdentical particlesCalculationDuality (mathematics)CurveSupersymmetryMusical ensembleTotal S.A.Quadratic equationSocial classGeometrySupergravity2 (number)Square numberRootDirichlet-ProblemFunktionalintegralDifferent (Kate Ryan album)CohomologyKörper <Algebra>Lecture/Conference
43:05
Gauge theoryExpected valueCalculationTheoryCorrespondence (mathematics)CurveConservation lawAmsterdam Ordnance DatumNichtlineares GleichungssystemPoint (geometry)InstantonTorusGroup actionSpacetimeLattice (group)WeightPartition (number theory)Line (geometry)Formal power seriesMusical ensembleMultiplicationSimilarity (geometry)Multiplication signGeometryProduct (business)Neighbourhood (graph theory)Boundary value problemAlgebraFunction (mathematics)Metric systemMereologySinc functionDoubling the cubePrice indexManifoldWeltlinieGroup representationTheory of relativityRootSeries (mathematics)Right angleDirection (geometry)Latent heatGeometric quantizationDeterminantOperator (mathematics)Open setAnalogyDimensional analysisPhysicalismCausalityGoodness of fitNetwork topologyPotenz <Mathematik>Körper <Algebra>CircleTotal S.A.SupersymmetryHypothesisEquivalence relationElement (mathematics)Interface (chemistry)ExpressionResultantGlattheit <Mathematik>Materialization (paranormal)Parameter (computer programming)Matrix (mathematics)Boom barrierDivisorModal logicPresentation of a groupCombinatory logicModulformPole (complex analysis)Four-dimensional spaceCovering spaceLoop (music)Phase transitionOrder (biology)Root systemCompact spaceRepresentation theoryHamiltonian (quantum mechanics)QuantumDegree (graph theory)Lecture/Conference
51:12
Function (mathematics)Körper <Algebra>Glattheit <Mathematik>Operator (mathematics)FunktionalintegralModel theoryLocal ringMultiplication signString theoryVector spaceCorrespondence (mathematics)Transverse waveBoundary value problemInstantonSpacetimeSphereTheoremConnectivity (graph theory)ModulformScalar fieldCalculationVolume (thermodynamics)Descriptive statisticsPoint (geometry)MultiplicationGroup actionDirection (geometry)ThetafunktionExpressionPolar coordinate systemTotal S.A.Mathematical singularityCoordinate systemLine (geometry)EntropyLoop (music)Prime idealLecture/Conference
52:46
Condition numberGroup actionLine (geometry)WeightRootString theoryGroup representationOrder (biology)Wave functionElectric fieldKörper <Algebra>AngleMultiplication signOperator (mathematics)Direction (geometry)Lattice (group)Identical particlesGauge theoryGeometric quantizationModule (mathematics)IntegerSpacetimeWaveSupersymmetryInstantonRight angleAlgebraVariety (linguistics)Function (mathematics)IsomorphieklasseContent (media)Mathematical singularityPoint (geometry)CalculationMereologyGoodness of fitResultantCorrespondence (mathematics)Cone penetration testSymmetric matrixComplex manifoldFilm editingInvariant (mathematics)Local ringThetafunktionClique-widthExpected valueQuantum mechanicsTrigonometric functionsLecture/Conference
56:10
CurveCommutatorGroup actionAlgebraObject (grammar)Multiplication signString theoryReal numberPhase transitionOrder (biology)Gauge theoryCanonical commutation relationFeldtheorieLogical constantFundamental theorem of algebraOperator (mathematics)Reduction of orderSurfaceMereologyGroup representationExpressionLink (knot theory)Direction (geometry)Line (geometry)Energy levelLoop (music)Model theorySubgroupAlgebraic structureDimensional analysisMaxima and minimaTotal S.A.Hilbert spaceRight angleTheoryLattice (group)Network topologyBoundary value problemMany-sorted logicNegative numberSupersymmetryFigurate numberDivisorAxiom of choiceNumerical analysisPoint (geometry)Local ringInvariant (mathematics)Coordinate systemWeightThree-dimensional spaceAngleModule (mathematics)Set theoryDrag (physics)Different (Kate Ryan album)Term (mathematics)Transverse waveConstraint (mathematics)Lecture/Conference
01:05:45
Gauge theoryTerm (mathematics)Different (Kate Ryan album)Multiplication signQuotient groupCircleAxiom of choiceTheoryLattice (group)Computer programmingRight angleWeightMaxima and minimaOperator (mathematics)Position operatorResultantGroup representationDirection (geometry)Mathematical physicsCondition numberDiagramFocus (optics)Two-dimensional spaceCuboidString theoryBoundary value problemMereologyConnectivity (graph theory)ExpressionExtension (kinesiology)Thermal expansionElement (mathematics)Connected spaceFundamental theorem of algebraGroup actionObject (grammar)Equivalence relationDrop (liquid)Loop (music)Numerical analysisSocial classOrder (biology)SpacetimeCurveCondensationCharacteristic polynomialSurfaceAngleCategory of beingLocal ringCubic graphParameter (computer programming)CohomologyTheory of relativityTransformation (genetics)HomotopieLine (geometry)Goodness of fitModule (mathematics)Graph (mathematics)Special unitary groupCorrespondence (mathematics)HomologieDescriptive statisticsMathematicsSymplectic manifoldFeldtheorieRiemann surfaceMany-sorted logicHelmholtz decompositionAlgebraic structureSubgroupAlgebraArithmetic meanLecture/Conference
01:15:20
InstantonComputabilityCalculationKnotCorrespondence (mathematics)Scalar fieldHyperplaneParameter (computer programming)Nichtlineares GleichungssystemNeighbourhood (graph theory)GeometryLine (geometry)Expected valueLocal ringGroup actionMinkowski-GeometrieInvariant (mathematics)Loop (music)Square numberSpacetimePhysicalismOperator (mathematics)Total S.A.Multiplication signPartition (number theory)Mathematical singularityAnalogyResultantHypothesisNatural numberLimit of a functionPlane (geometry)Fiber (mathematics)Musical ensembleGauge theoryRotationHamiltonian (quantum mechanics)TheoryDirection (geometry)KinematicsWell-formed formulaAlgebraic structureEquivalence relationModulformSweep line algorithmFiber bundleSphereMereologyFunktionalintegralFocus (optics)Metric systemRadiusCartesian coordinate systemThree-dimensional spaceStatistical hypothesis testingMultiplicationGradientVector spaceSpecial unitary groupMathematicsCircleDerivation (linguistics)StatisticsFunction (mathematics)Descriptive statisticsVariable (mathematics)SummierbarkeitAdditionMany-sorted logicLecture/Conference
01:24:55
Descriptive statisticsResultantLoop (music)Correspondence (mathematics)Operator (mathematics)ExpressionCurveMultiplication signInstantonLine (geometry)SupersymmetryInfinityNichtlineares GleichungssystemInvariant (mathematics)MereologyLocal ringGoodness of fitAlgebraSpacetimeConnectivity (graph theory)Nuclear spaceKörper <Algebra>Quantum mechanicsPartition (number theory)Complex manifoldPoint (geometry)TorusExpected valueAdditionIsomorphieklasseSpecial unitary groupLogical constantKnotHarmonic functionGlattheit <Mathematik>MultiplicationComputabilityFundamental theorem of algebraMathematical singularityLink (knot theory)Lecture/Conference
01:34:30
Multiplication signDirection (geometry)Position operatorCurveNegative numberLine (geometry)Mathematical physicsVector spaceGauge theoryScalar fieldVektorwertiges MaßFunktionalintegralExpected valueCombinatory logicParameter (computer programming)Loop (music)Group representationFundamental theorem of algebraGroup actionSet theorySocial classQuotient groupTheoremFigurate numberOperator (mathematics)Price indexBoundary value problemSurfaceNetwork topologyEquivalence relationSpecial unitary groupCanonical commutation relationThetafunktionAngleAxiom of choiceLocal ringResultantAlgebraSpacetimeTerm (mathematics)Characteristic polynomialElement (mathematics)TheoryMereologyMaxima and minimaString theoryCategory of beingMetric systemDescriptive statisticsPoint (geometry)ModulformWeightOrder (biology)Phase transitionObject (grammar)Correspondence (mathematics)RotationExtension (kinesiology)RootFiber (mathematics)InstantonMany-sorted logicMathematical singularityInvariant (mathematics)Cartesian coordinate systemDimensional analysisExpressionConnected spaceCondition numberWell-formed formulaTwo-dimensional spaceCuboidGraph coloringDrag (physics)Connectivity (graph theory)QuotientSummierbarkeitAlgebraic structureSphereProduct (business)CircleFunction (mathematics)RadiusRight angleNumerical analysisGoodness of fitSquare numberGeometric quantizationNeighbourhood (graph theory)Lattice (group)AnalogySupersymmetryLatent heatDeterminantPartition (number theory)Nichtlineares GleichungssystemLecture/Conference
Transcript: English(auto-generated)
00:15
Yeah, so I want to go back to where we left off yesterday.
00:20
And again, you are free to ask me questions. And so please stop me whenever you want to ask a question. So now I want to give a formula for the hemicapartition function.
00:41
But yesterday, actually, I had a typo, which Bruno pointed out. So I want to correct. So the mistake, the typo was in the formula for the one-loop determinant. So the vector contribution was fine.
01:01
So there is some product of positive roots. And OK, and sine pi alpha sigma. So this is a vector contribution.
01:20
And so here, sigma is meant to be minus i times l times sigma 2. Now, there is a contribution from the kara-martipates obeying the Neumann boundary conditions.
01:40
So by Neumann, I mean the set of irreducible representations of kara-martipates that obey the Neumann boundary conditions. Then there is a product over the weight in the representation. Now, the argument is w times sigma,
02:05
where sigma is given there. And so here is the correction. So the argument in the current convention, in the convention of this lecture, it should be q over 2.
02:22
And the sine is opposite from the convention of my paper, actually. OK. And then similarly, for the Dirichlet boundary condition. So by Dir, I mean the set of irreducible representations
02:43
with Dirichlet boundary condition. And there is a similar product. And as I explained yesterday, there is minus 2 pi i, e to the pi i w sigma plus q over 2.
03:06
So again, this again, correction. And gamma 1 minus w sigma minus q over 2, correction.
03:29
Yeah, and if you have twist in mass, you're supposed to replace, OK, what I'm saying?
03:44
OK, so w sigma to w sigma plus ma, OK, with a suitable definition of ma.
04:00
OK, so this is a one-loop contribution. And I want to denote by script B, correct B, the data necessary to specify the B-brane, B-brane.
04:26
So OK, so the Noe and Dir I explained, v is a champeyton, a champeyton vector space. Rho is a representation of the gauge group
04:40
and the Freiburg group, carried by the champeyton space. R star is a representation of the R-symmetric group. And q is a metric factorization. So this is what we call boundary data.
05:04
And it corresponds to B-brane. And now, OK, I'm going to write the formula
05:23
for the hemispherical partition function. So OK, so gauge group is g, hemispherical partition function, which depends on the boundary data.
05:41
OK, boundary data and also it depends homomorphically on the complex-wide Fi parameter, t. I think I wrote 2 pi xi xi minus i theta, topological.
06:04
So this is given as 1 over the size of the y group, of the gauge group. So for un, this would be n factorial. And there is a usual integral, d.
06:24
So rank g is the rank of the gauge group. And sigma 2 pi i raised to the power rank of g. And now I'm going to, as a contour,
06:41
I just simply write the most naive one. So this is just the imaginary axis raised to the power, the rank of the gauge group. And this is what you get from the boundary condition on the vector marketplace I described yesterday.
07:03
And actually, depending on the representations, rho and r star, carried by the shunt pattern space, you may need to, and also depending on the twisted masses, I guess, you may need to impose a more refined boundary
07:21
condition on the vector marketplace. And one way to think about it is the following. So the vector marketplace, from the vector marketplace, you can construct the twisted chiral marketplace, which is sort of mirrored to the chiral marketplace. And then in order to preserve B-type super-symmetry, you need to choose a Lagrangian submanifold on the space of sigma 1 and sigma 2.
07:43
So more generally, this imaginary axis raised to the power rk times g, rank of g, should be replaced by some Lagrangian submanifold, which is almost of this form, but it might
08:00
get tilted in some directions. So Lagrangian with respect to the naive symplectic structure on sigma 1 and sigma 2, just as a flat space.
08:25
So I'm d sigma 1 with d sigma 2. So and yesterday, I evaluated the classical contribution. So t times, I think for UN, I wrote t times trace of sigma,
08:48
but in the general case, let me write it, just t times sigma. And in general, the complex phi-de-phi parameter gets denormalized.
09:01
So we need to use a denormalized one. Probably Benigni discussed this. So you can probably remember it. And then as part of classical contribution, we have the slipper trace in champion space, e
09:22
to the minus 2 pi i sigma. And then, OK, there is this important one factor. Yeah, it's that. So this is a formula.
09:44
So OK, this is basically the main result for my talk so far. Is there any question? OK, then let me continue. OK, so then you can ask, OK, what
10:24
is the hemispherical function good for? What's the meaning? And the one meaning is the following. So this was a conjecture, good for inner paper,
10:46
and also Hoare and Lomo. And this is for CFT. So in the case, OK, I didn't explain the Karabayo condition.
11:07
But for example, if the gauge group is abelian, so if the gauge group is a product of u1, then the sum of gauge charges has to be zero for the actual asymmetry to be preserved.
11:23
And in that case, it is at least in some region of the F5 parameter, it is believed that the gauge theory flows to a nonlinear sigma model whose target space is a Karabayo manifold. And therefore, the theory is believed to become conformal. And in that case, our conductor says
11:42
that the hemispherical function is the unnormalized. This was implicit in the paper, but unnormalized version of center of charge of a D-brane, of the D-brane.
12:09
I should say the D-brane. Well, in string theory, you can use B-brane boundary condition
12:23
or B-brane data to describe a D-brane in a compactification of type 2 superstring theory to lower dimensions,
12:41
like four dimensions. So for example, typically, you have a Karabayo compact creation and you consider homorrhic branes or sheaves or object in the derivative coherent sheaves and so on.
13:00
And then anyway, so if you have a D-brane inside the Karabayo, then you get a particle. You get a particle in a lower dimensional theory, like in four dimensions. And if you, for example, start with type 2 string theory, you get N equals two supergravity theory.
13:22
And you have particle and in four dimension, N equals two super algebra, you have a center of charge. You have a center of charge in the super algebra. And this is the center of charge that appears in the four dimension N equals two super algebra.
13:41
Good, but the center of charge of D-brane, actually, so there is a more intrinsic definition in terms of a two dimensional CFT, N equals two comma two super comma three theory. And this is given by the partition function
14:03
on an infinitely long sort of, okay, cigar, okay, I think they call it cigar. So this is a cap, this is a sort of hemisphere and there is a flat region and there is a boundary. Okay, so and the length is meant to be infinite.
14:21
You consider here a twist in the capped region. You might call it the TT star amplitude, topological, anti-topological amplitude. And yes, actually, there was a question
14:43
about how to go to CFT, right? So this boundary condition indeed describes, defines a boundary state. And because the length here is infinite, the state created by this hemisphere region
15:02
is projected to the ground state. So this, yeah, the partial function on this geometry, which is a priori different
15:20
from the hemisphere partial function I described, this does compute, this does compute the overlap between the Raman ground state and the D-brane boundary state. So the equality here was a conjecture and the conjecture was based on the explicit calculations
15:43
and no comparison with no examples. And I say unnormalized because in order to really get the central charge of the D-brane, we need to normalize by the sphere partial function,
16:01
computed using the same renormalization procedure. So the same regularization, the same counter terms. Okay, and then this conjecture was actually proved. I mentioned this already yesterday.
16:20
This was proved by Baccus and Plankton using super wire anomaly. Okay, so in the CFT case, the meaning of the hemisphere partial function is now clear.
16:45
Oh, by the way, it's, so hemisphere partial function and compute the unnormalized central charge and often it is good enough because typically what you want to use the central charge is to compare the stability,
17:02
to study the stability of D-branes. And then what you need, what you really need is relative, yeah, the relative, what really important is the relative phase. And of course also the ratio of the absolute value
17:22
is also important because from that you compare, you can compare which particle is more massive, which particle is more, is lighter and so on. But so anyway, this unnormalized central charge is already very useful information.
17:49
This one? Yes, this is computed from CFT. Yeah, that's right. But yeah, but actually this quantity can also be defined
18:02
in massive theory. So in non-conformal theory and this is something I was trying to mention. So you can ask what's the relation between the hemisphere partial function and this quantity in the non-conformal case.
18:22
And in that case, there is a conjecture by Chakoty, Guyot and Wafer based on a computation in an explicit example, which says that the hemisphere partial function is a limit of this quantity, t t star amplitude, where the anti-hollmolecular part
18:40
of the mass parameter is set to zero. But the holomolecular part of the mass parameter is kept finite. So, but I believe it's still a conjecture. There is no proof. Okay, any question?
19:02
Now I want to discuss very explicit, probably the most important example, which is a carabiau hypersurface in this case. The projective space.
19:26
Yes. When you say non-conformal case. Yes. You mean a master formation of a safety or a general? I think more general setting,
19:43
but yeah, but in that case, like for example, the projective space, but on the other hand, then it's not clear what it means to set the mass parameter to zero. So yeah, okay.
20:00
The conjecture I think can be straightforwardly formulated for massive deformation of safety. Yeah, the example they had was, yeah,
20:20
mass safety, Landau-Ginzburg model. Yeah, so I want to discuss an example, which is a carabiau hypersurface in projective space Pn-1.
20:52
Yeah, so this is important. So if n is three, the space is torus. So elliptic curve.
21:02
If n is four, this is K3 surface, polarized K3 surface. If n is five, this is a quintic carabiau. So you see that this is an important example.
21:20
And the gauge theory for this carabiau is described by gauge group. Yeah, so the surface is described by gauge theory with gauge group U1 and n chiral multiplets,
21:47
which I denote by phi i and one chiral multiplet, which people usually denote by P. And okay, gauge charge is here plus one
22:06
and here minus n. So they sum up to, the gauge charge is sum up to zero. So it's such by the carabiau conditions I mentioned. Okay, and okay, I omit specifying the R charges.
22:24
And the super potential is P times polynomial in phi, which is assumed to be homogeneous of degree n.
22:49
And you probably know that, yeah, for positive Fi parameter, the theory flows to non-linear sigma model,
23:01
this target space is a carabiau. And now I want to give you a matrix factorization
23:21
for the, basically the D6 brain. In other words, the structure shift of the carabiau maybe with some fluxes turned on. So choose the boundary condition
23:41
so that all carafews obey the Neumann boundary condition. So in my notation, this means that noise is of, noise is, sorry, everything and there is empty. And let us introduce fermionic oscillators.
24:09
Well, in the context of matrix factorization or algebraic geometry, this construction is known as, I don't know, actually I don't know how to know, how to pronounce this, but COSU, I say COSU,
24:26
COSU construction. I'm in Europe, so somebody is going to explain how to pronounce it.
24:40
And so we introduce fermionic oscillators like this. And eta and eta bar squared to zero. And choose a Kirchhoff's vacuum. Yeah, Kirchhoff's vacuum such that, eta annihilates the vacuum.
25:05
Okay. So then at Champetian space, we get a two-dimensional space. So spanned by the vacuum and this excited state. Okay. And then as matrix factorization,
25:29
let us consider G times eta plus P times eta bar.
25:44
Right, then it's, I think it's easy to see that this is P times GN. So this is a super potential or times the identity. Okay, good. And then you can assign appropriate gauge and R charges
26:05
so that the conditions on the matrix factorization I described yesterday are satisfied.
26:26
Now I want to compute the, oh. But you know to really complete the specification of matrix factorization, I need to choose the gauge charges.
26:45
The gauge charge and the asymmetry charge of the vacuum. So yeah. Let me choose the gauge charge of the vacuum to be little n plus capital N over two,
27:07
where little n is an integer. And let me assume that the R charge of the vacuum is zero.
27:21
Then you can, okay. Then the claim of first, actually, okay, okay, sorry. The claim, I don't know who found this originally. So I cannot give you the original reference, but the claim is that this describes
27:42
the matrix factorization describes the sheaf, we could say the lime bundle, all this O, all M of N over the carabia hypersurface N.
28:04
So N equals zero corresponds to the structure sheaf. And for non-zero N, this sheaf is a structure sheaf twisted by a lime bundle,
28:21
by some lime bundle which comes from the protective space. So I think most of you know that on the protective space there are various standard lime bundles. Okay, so then you can apply the formula for the hemisphere fractional function and compute it.
29:06
You get the contribution from the boundary interaction, which is also called the brain factor.
29:23
It is T sigma. In the carabia case, complex value of parameter is not to be normalized. Gamma of sigma to N, gamma one minus N sigma. Okay, and we can keep rewriting it.
29:48
We can keep rewriting it and let me see. I can give, I could give explicit expressions, but I want to explain that there is a sort of nice,
30:09
well, if you rewrite it in a certain way, you get a geometric expression, which involves a novel concept.
30:22
So if you keep rewriting it in some way, in some way, then yeah, and this is for large real part of T, so which is two pi psi. So in the large volume limit.
30:41
This can be written as the chain character of the C for the line bundle times E to the B plus I omega times some characteristic class called the gamma class of the tangent bundle.
31:06
Now B plus I omega. So in the convolution here, it is minus T over two pi I times E where E is the pullback of the hyperplane class
31:21
of the projective space. So it's a generator of the second cohomology group. Well, and there are, I guess I can write,
31:45
I think I can write this or not.
32:19
Yeah, so now I need to explain
32:28
this gamma hat, this characteristic is gamma hat. So for general vector bundle E, gamma hat of E is defined in terms of churn roots.
32:43
So I assume that's okay. You know characteristic classes. And XJ are called churn roots.
33:00
And churn roots are such that the churn, character is given by sum of E to the XJ. So this characteristic class defined, so this is the definition.
33:20
This is called the gamma class. The gamma class was known or invented earlier by basically mathematicians, people like Kostanyi, Yuritanyi, Kaczalko,
33:44
Kontsevich, Panteff, these people. And yeah, and their consideration was actually motivated very much by the central charge. There's a formula for the central charge obtained by solving the Tcl-frequency equation.
34:06
But okay, what Hori and Romo noticed was that Hamilton partition function, the logarithmic computation of Hamilton partition function naturally gives the right to the gamma function and therefore explains the appearance of the gamma class
34:23
in the formula for the central charge. So that's nice, nice story. Okay, any questions? Did we compute the two formulas, pitch-fast and pitch-fast? Gamma hat corresponds to the product of gamma function
34:43
and sharpness corresponds to pitch-fast. So N-dependence only comes from the exponential, so this is just the exponential, I think. And the remaining part.
35:02
For example, above there is a minus sign, there's a sign, and how does it come from below? Difference of exponential, how does it fit into the expression below? Oh, this one? Yes. Yeah, so this can be written as a sign, right? Yes. Then the sign can be written in terms of gamma functions. Oh, okay, product of gamma.
35:21
Product of gamma functions. So then, yeah, everything fits here. So the sign is the common that goes to above. Yeah, that's right. So that factor comes from gamma hat zero?
35:44
Yeah, it combines, yeah, these things combine. Yeah, basically what you have to do is to, yeah, so divide it and then take, evaluate the integral by residues, and I think you then focus on the D-ring port,
36:05
and then use properties of E, I think, let's see. So this is a feedback of the hyperplane class
36:24
in the predictive space, and let's see, H satisfies some condition. This is, yeah, yeah, yeah.
36:45
So basically, this condition corresponds to evaluating the residue with some powers of sigma. Yes, okay. Probably more basic than that, so I understand how you went from the first to the second.
37:05
Here to here? Yeah, basically, what I explained, so it's just a computation of the residue,
37:34
and yeah, as I said, you rewrite it in terms of sine and gamma functions,
37:41
and then, okay, then you have some residue, residue integral, and you use Cauchy theorem, but then you try to interpret the residue computation as an integral of this, in cohomology, integral of cohomology element, yeah.
38:10
You got this geometric expression, large quadratic limit? Yes. So is it possible to compare with some super gravity result? All right, yeah, so I should have said,
38:25
so this is a sort of an improved version of a previously known formula, and the previously known formula had instead of gamma hat, I think, instead of gamma hat,
38:43
it had, I think it's, okay, it was probably square root of a-hat, g, I think, and in the large body limit, let's see, so up to the highest, so up to several reading orders,
39:04
the two expression give the same result, but I think like to the reading, like N minus two degrees or something, they give the same result,
39:20
but if you look at higher corrections, then they start to deviate, but this is, okay, this is explained, well, in my paper and also, I think, in the paper of Hori and Ramon, but basically, but this story is due to Hori and Ramon. Yeah, I need to emphasize, yeah, okay.
39:47
Any more questions? Okay, now, now, what can, we can change boundary condition of one field each one?
40:03
To Dirichlet, right, so, yeah, so if you use Dirichlet boundary condition, of course, we need to change the one-loop determinant, so we get one over gamma,
40:21
and but it's actually enough to, in some sense, it's enough to consider normal boundary conditions, because as I said yesterday, Dirichlet, there is a duality between boundary condition, so the Dirichlet boundary condition is a dual to normal boundary condition with some boundary interactions. Therefore, if you include appropriate
40:43
boundary interaction, then you can describe arbitrary boundary condition just using normal. Okay, any other question? Okay, very good. Now, okay, I want to,
41:03
I want to say just a little bit about the interface, just a little bit, and leave everything else to the exercise.
41:22
So what I described is basically path integral over the hemisphere, but you may want to consider a situation where you have something
41:42
that divides the space time into two regions. So for example, the sphere into two hemispheres. So you have an interface here, and you have some theory, T1, and you have some theory, some other theory, T2. Now, you can apply some transformation.
42:04
You might call it time reversal, or you might also call it a parity transformation, and map it to this system, T1 times T2 bar.
42:22
So now you have a single product theory with some boundary or boundary conditions or whatever. So in general, the statement is that the interface preserving B-type supersymmetry can be described as a B-brain in the product theory.
42:51
Can you come up with an interesting interface in this way? I don't know if you agree,
43:02
but I think the identity interface is already rather interesting, and it's actually, there are some work. And in one of the exercises, actually, which was given yesterday, I ask you to consider matrix factorization
43:21
for the product theory. Actually, this is the product of two copies of the gauge theory I just described. And so I give you, so in the exercise, I give you an example of matrix factorization. And the claim is that
43:41
if you compute the Hamiltonian partition function for that interface, then you get precisely the Fourier partition function. Okay, now I want to finish the two-dimensional story. Any more questions?
44:02
Okay, now I changed the topic, and I'm going to discuss half BPS learning operators in four-dimensional N equals two supersymmetric theories. And as usual, I give a plan for this part of my lecture.
44:24
So I'm going to discuss line operator charges. Do I really have time? Probably don't. Yeah, so I will probably not be able to cover everything in full detail, but I'm going to discuss line operator charges,
44:41
2D-4D relation, S4 versus S1 times R3, the relation between monopoles and e-stantons, monopole screening, bubbling, monopole partition function waves, and so on. And as references, I give you, first of all, this one.
45:02
This is a review paper I wrote about this subject. And the actual calculations, okay, I gave a paper from 2011. And there is also a recent paper by a group at Rutgers,
45:21
and this is also quite relevant. Okay, so now I want to discuss
45:43
basically the definition of line operators and also the charges. Line operators are very basic operators in gauge theories because they can be defined for any gauge group.
46:05
Okay, any gauge theory. So, let G be the gauge group, which is a compact degree group, and let me denote by T, the script, okay, bold T,
46:22
the Cartan, Cartan sub-algebra, and T star, okay, duo. Now, I think you know that inside the duo of Cartan sit the weight lattice,
46:43
and there is a sub-lattice, which is generated by root, so it's a root lattice. Now, inside the Cartan sub-algebra,
47:00
there is co-weight lattice, which is defined to be the duo of the root lattice, and there is co-root lattice, which is defined to be the duo of the weight lattice.
47:23
Now, half-VPS Poisson line operator in four-dimensional n-cos-2 theory
47:41
is defined in the expression trace in representation of R of the path order exponential of the integral of I times the gauge field plus,
48:03
okay, this is a convention, but DR path of phi, you might put some phase here, if you like, some people do, non-element DS, defined by the metric.
48:20
And the integral is taken along, for example, a straight line, straight or a circle. Yeah, so this restriction is necessary, and okay, here I'm on, for example, R4,
48:43
and this restriction is necessary to preserve one half of the full supersymmetry. Okay, and it's actually, I think, an open question, it's an interesting open question to clarify, on which curves you can have
49:02
three-class metric wheels on loops for n-cos-2 theories. For four-dimensional n-cos-4 theory, the classification of such curves was done by Dimaski and Piston using the Peir-Spina formalism, and I think it's an interesting question to extend the result to four-dimensional n-cos-2.
49:25
Okay, now this is a, okay, this I take to be an irreducible representation, irrep specified by the highest weight, W, in lambda W.
49:43
I think I'm going to have too many Ws, but okay. So sometimes I use W for the highest weight, that's this one, the Wilson loop, Wilson operator. And physically this represents infinitely heavy half-PPS electrically-charged particle,
50:02
I mean, the world line, world line of such a particle. So that's the Wilson loop. And now you can try to consider the magnetic dual of this operator.
50:26
And so the magnetic dual would be a monopole. And so we should consider an infinitely heavy magnetic monopole that propagates around the contour in space-time, and that's the 2-5th line operator,
50:46
which I denote by T of B. So B is the magnetic charge, I'm going to explain. Yeah, so this is an infinitely heavy magnetic monopole.
51:03
And this is defined by singular boundary condition. So Benin explained last week that in quantum theory, there are other operators such as this Wilson operator, which is just a function of the field in the path integral.
51:26
Now a total line operator is a prime example of disorder operator, so it's defined by a singular boundary conditions.
51:40
And the field strength goes like B over two epsilon ijk xi divided by R times dxk which dxL, you might also write it as minus B over two times the volume form of two sphere
52:06
using the coordinates. Yeah, so xi and r theta phi, so these are locally defined,
52:22
locally defined Cartesian and polar coordinates. So this describes the transverse direction to the line or the loop. And the scalar in the vector monopole goes like i times B over two R.
52:44
Now these expressions are valid for the vanishing topological theta angle. If theta angle is nonzero, we need to turn on the electric field
53:01
in order to, I think in order to be consistent with the Witten effect. Right, any question? Okay, but now there are very interesting,
53:27
yeah, so I'm going to explain in several steps. Very interesting, I think it's very interesting story about the charges, about the charges of the line operators. So there is a restriction on the magnetic charge B.
53:49
There is a Dirac quantization condition on the magnetic charge B. To understand this, well, this is a standard thing,
54:03
so I think you've heard this several times already. So you consider monopole, so we only consider the transverse direction to the line. So then there is an S2 that surrounds the monopole
54:20
and there is a Dirac singularity, well, if you use the gauge such that A goes like minus B over two times one minus cosine theta times d phi. So the Dirac string is along theta equals pi. And you can parallel transport a magnetic field
54:44
around this Dirac string and then you require that field is, or the wave function to be a single valued. And the condition is that the natural pairing between the magnetic charge B,
55:04
which is in the, okay, covalent lattice, well, which is in the carton, so the pairing between this B and some weight,
55:21
so this is not the highest weight of the realism, this is an arbitrary weight of a matter representation. So this is taken from the weight lattice. So the condition says that the pairing,
55:44
this pairing is an integer. Yeah, so it's important that W depends on the matter content, so this is a weight of matter representation or a root because, okay,
56:02
we always have gauge fields, which transform in the identity representation. So there is a restriction on B and let me call the lattice of such B, lambda M.
56:27
So lambda M denotes the lattice of B satisfying this condition, okay? Okay, so this is a restriction on the charge
56:49
of the total operator. You can also consider a dionic line operators. So dionic operators are defined
57:00
by first considering a total operator, TB, and then inserting a Wilson loop for the unbroken part of the gauge group. So once you insert the total operators, there is a choice of B
57:21
and this B locally breaks the gauge group to a subgroup. So the unbroken part is a commutant, okay, of the magnetic charge B. So the dionic operators specify,
57:48
their charges are given by a pair,
58:01
the magnetic lattice times the weight lattice, okay, but then they are subjected to the wide group action.
58:25
So at the level of the gauge algebra, I mean the d-algebra of the gauge group, the line operator charges are classified by lambda M times lambda W
58:40
divided by the wide group. So this classification is due to, essentially due to capacity. So this is a classification at the level of d-algebra. But there's an interesting refinement
59:02
of this classification, which reflects the global structure of the gauge theory. And in order to understand that, we need to consider what happens when you have more than one line operator.
59:20
So consider two line operators, okay? Two line operators specified by pairs of charges, B one, W one, and B two, W two. And imagine that one operator moves around the other
59:44
so that the surface sweeped by the moved line operator has a linking number one with the line operator that is sitting. Okay, I'm not sure if I'm explaining it very clearly, but.
01:00:01
Basically, this is something originally considered by Toulouse, okay? No, no, no, I can draw, yeah, yeah. So, I can try. So, for example, you can have,
01:00:21
so consider a line, and you consider the transverse direction, and then the transverse three-dimensional space. So one operator is represented by a point, right? And then you can consider a line, and then the other operator,
01:00:42
that, for example, goes like this. And then you can rotate this part to this, and then come back. So, I think this represents the motion of one line operator I described.
01:01:06
Then, the claim is that this picks up a face, basically Wilson line that goes around the drag string picks up a face.
01:01:35
So, the conclusion that this is one, this is one, and when this is satisfied,
01:01:41
the two operators are said to be mutually local.
01:02:13
So, originally, Toulouse introduced two operators to classify possible faces of gate series, and in that context, Toulouse was actually
01:02:21
not really considering mutually non-local operators, okay? But the modern point of view, put forward by Aharoni, Ziberg, and Tachikawa.
01:02:46
The modern point of view is that in order to specify a theory, we need to choose,
01:03:01
choose maximum set mutually local line operators.
01:03:32
So, I think this paper is of fundamental importance in my personal opinion. So, the claim is that in order to really specify a theory,
01:03:43
yeah, you need to choose a maximum set of line operators that obey this constraint. And they argue that the set has to be maximum in order for modular invariance
01:04:04
of the four dimensional theory to hold. For example, so if you put the four dimensional theory on the four dimensional torus, you can describe it using some coordinate, but then you want to choose different set of coordinate, and you do large different morphism, large coordinate transformation.
01:04:21
And then the invariance of the description, I mean, the lack of gravitational anomaly requires that you choose maximum set of mutually local line operators. That's their claim. There are other characterizations of this choice.
01:04:42
For example, they show that this choice is equivalent to a choice of discrete set angle. And also you can replace this in terms of the centers,
01:05:03
in terms of the center of the gauge group. And so the choice corresponds to a choice of maximal isotropic subgroup of, yeah, so, okay, maximal isotropic subgroup,
01:05:31
the center of the gauge group times center of g, g star. So there is a pairing between the center and its duo.
01:05:48
So, and the isotropy means this condition.
01:06:01
Okay, any question? Yes. In your few statements in classifications. Of the charges, do you have a similar definition of why the wild group xi is? Why, sorry? Why the wild group xi? I might expect that these boxes are much better.
01:06:34
Well, in principle, you can choose wild group to act on lambda m and lambda w independently, but I think it's more natural
01:06:42
for the wild group to act on both. And also, I said that lambda w specifies
01:07:04
a Wilson loop for the unbroken part. So the choice of the Wilson loop is correlated with the choice of b. So it must be simultaneous action. Okay, any other question?
01:07:23
Good. Okay, I have 20 minutes left. Okay, I want to say a little bit about the 2D, 4D relation.
01:07:45
So from Peters, we had an explanation of the AGT correspondence. In particular, he explained how the AGT correspondence works, or the 2D, 4D relation works
01:08:02
for the A1 theory of class S. And, okay, I will not be very constricted because I don't have much time, but basically, yeah, so I want to consider, so A1, A1 type, A1 theory.
01:08:23
Of class S. And this corresponds to two M5 brains, or six dimensional n equals zero comma two theories on punctured Riemann surface, C, G, N. So G is a genus,
01:08:43
and N is the number of punctures. Okay, and you can consider, so for example, you can have a surface like this.
01:09:05
So you can introduce a punct decomposition, and then you get some generalized quiver gauge theory,
01:09:21
gauge groups, SU2. And the punctured decomposition, and actually, you also need to draw a trivalent graph. But basically, my puncture graph gives you some gauge theory, Lagrangian description of the content field theory.
01:09:42
And for example, change of punctured decomposition corresponds to the general, this duality, that's what Peter explained. And now, there is a sort of a topological story
01:10:05
for the charges of line operators, and the claim is that homotopy class, homotopy class
01:10:21
of the closed curve on the Riemann surface corresponds to a charge of a half BPS line operator.
01:10:49
Yeah, so this corresponding is actually very quantitative, explicit, and also there is, it is known how S or modular transformation acts on the parameters of the curves,
01:11:05
called Dean-Thirson parameters. Yeah, so the classification of line operator charges, it has a nice correspondence with the classification of closed curves on the Riemann surface.
01:11:23
Yeah, so you can read my review for explicit explanation. And this correspondence was found by Jelker, Morrison, Morrison, and myself.
01:11:45
But then, after the paper by Afharoni, Zaben, and Tachikawa, Tachikawa had a single author paper where he showed that the refinement actually also applies to here.
01:12:01
And in order to specify the theory, in order to specify the four dimensional theory, you need to choose, you need to choose a isotropic subgroup.
01:12:28
In this case, in terms of the Riemann surface, of the first cohomology, he was considering the case with no puncture,
01:12:41
and the coefficient group is three G minus three. So there is a correspondence between line operator charges and line, correspondence between line operator charges and closed curves, and the refinement of the classification also goes through.
01:13:08
Here? This is homotopy. Homotopy. So the line operator charges, they make some algebra, homotopy charges.
01:13:21
Okay. But then you wrote the homology before. Yeah, so this, somehow this has something to do with the center. And the algebra of line operator is,
01:13:40
the structure of the algebra of line operator is more complicated than cohomology and all these topological things. I don't see a direct reaction. Okay, so this part is by,
01:14:11
take this paper from 2009, and drew from myself. This part is a single authored paper by Tachikawa.
01:14:24
I think we'll have to do this soon after the paper with Aharoni-Zayevova. Okay, I have 15 minutes.
01:14:50
Okay, so now, because this is a school on localization, I want to say something about localization of line operators,
01:15:00
localization of characterization with line operators, especially Tohoku operators. Now, line operators, half-B-piece line operators can be placed supersymmetrically on S4, or its deformation, S4B, which Peter's discussed,
01:15:27
or S1 times, S1 times R3, let's see. So you already know, okay, what S4B is,
01:15:43
but okay, somehow it's better to write entity understanding.
01:16:03
You can have two places where you, you can have two, you have two locations where you can place line operators. So you can have one S1, and you can have another S1, which I denote S1B and S1B inverse,
01:16:24
so in the one, two, and three four planes. And you can put group or line operators, okay? And localization has been, essentially localization calculation has been done.
01:16:42
So for the vision loop, the localization was done by, okay, Pest, Hama, Hosomichi, and with the Tohoku operator, Gomis, Pest, and myself, these localization calculation for B equals one. And for B not equal to one, there is a guess for what the answer should be
01:17:03
based on the AGT correspondence. So yeah, so to some extent, there are results. And now I said you can use AGT for the computation of the operator expectation values on S4B, right?
01:17:20
And this is because there is something called the Barinde operator, Barinde loop operator in CFT. For example, in D'Youville and Toda theory, and you can use, I'm not going to explain what it is,
01:17:43
but you can use this gadget to compute the expectation value of line operators. So this was done, okay, for D'Youville, this was done by Dzurka, Gomis, myself, and Teschner,
01:18:00
and also Ardaigai, Yotogukov, Tachikawa, Barinde. And also for Toda theory, this was, the calculation was done by Gomis and Refloc. Gomis and Bruno, okay?
01:18:24
Now, so S4B is good. And S4B, it can be done. Now, remember that S4B partition function contains an instant of partition function, right?
01:18:43
And actually also is the anti-instant function, right? But the instant partition function was originally defined by Nikita, as we heard this morning, on the omega deformed flat space.
01:19:02
So R4, epsilon one, epsilon two, omega background on R4. And similarly, you can ask, what is the most natural space time to where to compute the expectation value of the line operators?
01:19:20
And this picture, this picture suggests that we should just look at the local neighborhood of each S1 here. And so then the geometry locally looks like S1 times R3,
01:19:40
but with some twist parameter. So R3 is now twisted around S1, okay? So let me see. So then if you compute the line operator
01:20:00
or total expectation value on S4B, this contains some contribution from the circles, which I call equators. And there are some contribution, which would be the analog of the instant of partition function.
01:20:21
And I call the contribution z mono. So because, okay, monopole partition function, okay?
01:20:47
So I want to explain briefly how to compute the partition functions associated with monopoles, actually a singular monopole.
01:21:06
And the useful trick, the useful method is the correspondence between monopoles and instantons. And this correspondence was found by Cronheimer
01:21:22
in his master thesis at Oxford. Cronheimer. His master thesis was not available for a long time. And Peston and I kept bugging him.
01:21:41
And he, well, and he never replied. But he eventually posted his paper on his webpage. So it's available now. Cronheimer's correspondence.
01:22:00
So monopoles are solutions to Bogomolny equations. Star three F equals D phi, where phi is an adjoint scalar. Now instantons are solutions
01:22:24
to anti-Sellufi-Thierty equations, which looks like a script, F plus star F equals zero.
01:22:44
And Cronheimer showed that instantons top knot space invariant
01:23:08
under U1 action corresponds to solutions
01:23:21
to the Bogomolny equation with Dirac singularities. So singular monopoles.
01:23:43
Actually, I gave details of this correspondence, again, in one of the exercises. So you can, so for equations, you can look at the exercise. I just, maybe I just explain what top knot space is.
01:24:07
What top knot space is. Yeah, so top knot space, what is top knot space? Actually, multi-center top knot. Multi-center top knot.
01:24:26
So it is a scalar, a hyper scalar manifold, which has a metric, I can give on the hooking form, V inverse times d psi plus omega squared.
01:24:43
So in Cronheimer's convention, psi is an angular variable, which is characteristic two pi, rather than four pi, in physics. And V is some
01:25:01
harmonic function of R field. Omega and V actually satisfies the Bogomolny equation.
01:25:22
Yeah, so this is, this is the space with this metric called multi-center top knot space. And in one of the exercises, I give you, yeah, what the correspondence is, basically, and ask you to show that the Bogomolny equation
01:25:42
and the anti-safederi equations are equivalent for the given expressions. Okay. So this is Cronheimer's correspondence.
01:26:02
Now I don't have much time left, so I'm explaining, explaining these parts just in words. For the instanton function, contributions came from small instantons. So instantons of zero size.
01:26:21
So in the addition construction, these instantons correspond to the fixed point of the torus action. Okay, this is what we heard this morning. And the same story holds in the monopole case. Namely, we have Dirac singularity,
01:26:42
so we have a torus singularity, and on top of that, we can have smooth polycov talk to monopoles. And these polycov talk to monopoles can get close to the singular monopole and get attached. And eventually, they can partially screen
01:27:04
the original magnetic charge. So the magnetic charge, I mean, B, magnetic charge B can get weaker. And I call, yeah, so the original magnetic charge is B, and there is a screening by smooth polycov monopoles,
01:27:24
and it gets weaker. And the smaller magnetic charge, I call V, little v. Now this phenomenon actually corresponds via Krone-Heimers correspondence to the small instantons.
01:27:41
So small instantons in top-notch, they reduce to, they descend to monopole, yeah, so monopole screening, which is also sometimes called monopole bubbling.
01:28:02
Okay, that's explanation. you can then, yeah, so then Krone-Heimers correspondence can be used to give a description of the monopole modularized space, because for instantons, there is,
01:28:21
okay, for instantons on C2, there's ADHM construction. Now, we are interested in the modularized space of instantons on top-notch, but one can show that in localization calculations, only components with fixed points contribute,
01:28:41
and Sergei Cherkis showed that the component of instanton modularized space with appropriate fixed points isomorphic as a complex manifold to the instanton modularized space on C2,
01:29:02
well, in the case of a single Dirac singularity. So we can actually use the ADHM construction for instantons and apply this Krone-Heimers correspondence. So we can take an invariant part of the ADHM data under this U1. Now, in the-
01:29:23
Is this different from the non-construction? It's different, it's different, yeah. So non-construction, you cannot use for this purpose? So non-construction is waiting for, well, infinite dimensional data, so it's not,
01:29:40
yeah, so far, it's not useful. So in a recent paper by Brennan, Day, and Moore, they showed that this U1 invariant part of the ADHM instanton modularized space is actually a Nakajima-Kruva variety, and so you can use this description.
01:30:01
And as in the discussion of Peters, where he used Kruva supersymmetric quantum mechanics to compute the instanton partition function, it's again possible to use supersymmetric quantum mechanics obtained by de Bruijn construction to compute multiple partition functions.
01:30:24
Good, good. Now, there are, yeah, I was planning to give some general formulas for the expectation values of line operators, but I just want to give one example
01:30:48
of the result of localization calculation, which again appears in one of the exercises. So if you consider, if you want, we can give you maybe tomorrow,
01:31:03
no, on Thursday, maybe half an hour more, so that you can actually do this a little bit better in the exercise session, to have shorter exercise session. Somebody asked me about the algebra of Wilson and Tohut loops, as originally described by Tohut.
01:31:22
So I want to mention, I want to comment on that. Yeah, so in 1978, in one of the papers in nuclear physics, we discussed an algebra of Wilson and Tohut operators.
01:31:45
Tohut loops, and well,
01:32:02
the statement of the algebra is as follows. So let's consider Wilson loop in some gate theory. Well, okay, let's say in SUN gate theory, and Wilson loop in the fundamental representation, okay?
01:32:20
And this is put place on some curve, on some curve, and let's consider a Tohut loop on some other curve. And okay, Tohut was considering a non supersymmetric theory, okay?
01:32:41
I think it can have some matter fields. And the definition of Tohut operator was more, okay, somewhat different, but essentially, okay, so what I explained was a modern definition of the operator. Now, we want to consider the following situation.
01:33:03
So CW and CT are curves, one dimensional curves, but let's first consider the situation where these curves are Hopf linked, R3,
01:33:22
three dimensional space, which you can think of it, and constant time slice, so T equals zero. So X1, X2, X3, yeah, so what is Hopf link?
01:33:44
Hopf link is something like this, good? And I want to consider the Hopf link,
01:34:04
so Hopf link in the constant time slice, so this is T, this is time, and this time T equals zero, right?
01:34:22
And you consider Wilson loop here and Tohut operator here. And let's see, then he claimed, okay, then we can consider
01:34:42
small displacement of the curve CW in the positive time direction or negative time direction. So then, if you do that, then Wilson operator and the Tohut operator, not on the constant time slice,
01:35:01
so there is an ordering. And so now, the statement is that as operators acting on the Hilbert space, the Wilson loop and the Tohut loop obey the following sort of commutation relation. So this is for SU and Gal theory with minimal charges.
01:35:27
Good, so this actually means that the fundamental Wilson loop and the fundamental Tohut loop are not mutually local. So this may look like a contradiction
01:35:41
to what I said yesterday, right? So on each, so it's a very fine counterfeit theory should have only mutually local line operators. And so the situation Tohut was considering does not correspond to such operators.
01:36:01
So what's happening? What's the modern interpretation of his discussion? So the claim, so the modern point of view is that W and T obeying this commutation relation
01:36:21
cannot be both genuine line operators. So at least one of them has to be the boundary of a topological surface operator. So I want to explain that.
01:36:49
So I've been saying SU and Gal theory, but what I really meant was gauge algebra. The gauge algebra was SU n.
01:37:01
And now I want to consider the concrete situation where the gauge group really group SU n, so capital SU n. And then, well, obviously, well, I think it's obvious, obviously the Wilson loop is a very fine line operator. And then the top operator T should be the boundary
01:37:25
of a topological surface operator. And how do we actually see this?
01:37:40
So the way to see this is to draw some figures here. So we have a top operator. Okay, we are going to consider, yeah, okay. So I said, top operator is fixed at T equals zero,
01:38:01
and we want to move this only to positive time direction or in the negative time direction. And this top operator in some gauge, this actually creates a sheet of drag strings.
01:38:25
So let me see. So you might call it Dirac, so this color is probably hard to see, I'm sorry, but this is Dirac sheet, so there is some sheet.
01:38:46
Now, actually, so I can even now explain how to get this algebra. So the Dirac sheet create, so Dirac sheet is a, okay, sheet of Dirac string,
01:39:00
and then when the Wilson loop go around the Dirac string, then you get phase factor, and that's the origin, right? So if the Wilson loop is, let's see, so Wilson loop is placed around time earlier than the two loop, then there is no phase, but if the Wilson loop is placed,
01:39:21
well, after time equals zero, then it picks up an extra phase here. So that's how you get this truth to algebra. And you also see that the Dirac string, it's in some sense a topological object,
01:39:41
so because by changing the gauge, you can move the location of the Dirac string or Dirac sheet. So indeed, in this picture, top operator is the boundary of a topological surface operator. So that's good. Well, then let's consider another situation.
01:40:00
For example, G equals SU n mod Zn, Zn, okay? So in this case, in this case, what happens is that the Wilson loop
01:40:23
becomes boundary. Of a topological surface operator. And okay, and I'm restricting to the case where the topological theta angle is zero.
01:40:41
Now, okay, how do you see that the Wilson loop is the boundary of a topological surface operator? So we are in the non supersymmetric situation, so we have the Wilson loop in the fundamental representation. Now, and the fundamental representation, right?
01:41:02
So the fundamental representation is not a very fine representation of this gauge group, okay, because it transforms under this action of the center. So naively, you would think that this object
01:41:21
is not allowed and I think that's what usually people say, what's written in textbooks. But actually, yeah, but if you think about it, this expression makes sense even in the case, yeah, so the representation is not
01:41:42
a very fine representation of the group, because the whole thing is expressed just in terms of the connection which takes values in the algebra, and you do expansion, right? So there is nothing wrong with this expression in some sense. And indeed, for example, if you are on R4,
01:42:01
if you are on R4 and you consider some circle, you can consider such a Wilson loop and it's gauge invariant, nothing wrong with that. But when I say this is gauge invariant, the reason it's gauge invariant is that
01:42:21
the curve you consider, so Cw, this is, so if this is contractable, then it's gauge invariant. So the gauge invariant of this operator is actually a background dependent statement. So if you, in a space time
01:42:41
which has non-trivial one cycle, and if you put the Wilson loop along the non-trivial one cycle, then there is a large gauge transformation which transforms this operator. So then it's no longer gauge invariant.
01:43:00
So you see that, so in order to define this object in for representation, that is not very defined for the group, then you have to, you need to specify, in some sense, how to fill in this, of which surface this curve is a boundary.
01:43:25
So in this sense, this Wilson loop is a boundary of a topological soft operator. Okay, does this make sense? Okay, yeah, so you might have thought that,
01:43:41
okay, maybe, yes? Why the representations are not well defined because there are some parts of the, some representations of the SUN will be sent to the representations of the quotient. Uh-huh, okay.
01:44:01
So the question was, why is the fundamental representation is defined for, yeah, fundamental representation is defined for SUN mod ZN.
01:44:23
Okay, so what is SUN mod ZN? So an element of this quotient group is an equivalence class, right? So G is an element of SUN, but G is identified with
01:44:45
itself multiplied by this phase, right? Now, so consider, okay, something, phi, in the fundamental representation,
01:45:03
so phi transforms by G times phi, but this is not a very defined, this is not a very defined representation of SUN mod ZN because the result of the action
01:45:21
of the group element, so this is not the same as the result of the action of G. So the action of, the result of the action of the group element
01:45:41
depends on the choice of representative in the equivalence class. So this means that, yeah, the fundamental representation of SUN is not a very defined representation of SUN mod ZN. Am I clear? Okay, very good.
01:46:00
Yes? So are you saying, so in G presents SUN mod ZN, some, some Balsamos should be regarded as topological surface objects. For example, like in hadron Balsamos in K, that I have a dot here. Right. That would just be a general line operator. Yeah. So in the SUN theory, some Balsamos should be regarded
01:46:21
as topological surface objects. Yeah. Not all. Not all, right. Well, what is the condition, which one should be regarded as surface objects? Yeah, so for the,
01:46:45
so it has to do with NRT. So for example, for the Wilson loop, yeah, basically, the Wilson loop, for the Wilson loop, the number of boxes in the Young diagram, it gives a condition for the loop operator to be very defined,
01:47:02
and there should be a corresponding statement for the total loop. So total loop is specified by B, right, and it's an element of the core weight lattice,
01:47:22
and then, right, so for a given theory, such a total operator is a genuine line operator if the charge is in some sub-lattice of the core weight lattice.
01:47:43
Yeah, and this sub-lattice is, this sub-lattice is, this sub-lattice is picked by the choice of the global property of the gauge group.
01:48:05
So yesterday, the choice of the precise gauge theory, yeah, the choice of the precise gauge theory was explained in terms of the choice of mutually, of maximally mutually local line operators,
01:48:24
and so in that language, yeah, the choice of sub-lattice is a choice of the theory. Let's see, and there are other characteristics
01:48:48
in terms of discrete set angles, and so, yeah, yeah.
01:49:03
And of course, there is direct quantization condition, so the sub-lattice has to be inside the magnetic lattice I mentioned yesterday. But the sub-lattice is not necessarily the same as magnetic lattice I defined.
01:49:23
Yeah, yeah, so I want to say that, yeah, so truth was not stupid, okay? So what he did was very physical, and what he did was much more important
01:49:41
than the mathematical physics we are doing here. And also, his argument, his result demonstrates that it's also important to take into account line operators that arise as, it's also non-genuine line operators that arise as the boundary of surface operators.
01:50:03
Okay, good. Now I want to come back to what I was discussing yesterday, and, yes, yes, yes, yes.
01:50:25
So the Dirac string is, so it's a sheet, so it's a two-dimensional object, right? And in the time, so in the time direction, then it goes in the positive time directions, but it also has a special component along the CT.
01:50:47
So topologically, this Dirac sheet is just, yeah, half line times CT, okay?
01:51:06
And now yesterday, I, so in words, actually, I covered most of what I wanted to say, and so I think I'll be repeating myself to some extent.
01:51:27
Yeah, okay, what the, and actually, I don't have too much time, anyway.
01:51:41
Yeah, maybe I want to say that, so yesterday, I explained Krone-Heimers correspondence. So there's a correspondence between instanton zone, top net, invariant under some U1 action, and singular monopoles on R3. But I think I did not really say what this U1 action is
01:52:00
and I'm not, again, I'm not giving you the complete description now, but I think I should actually say that this U1, which I denote by U1k, acts by shifting the variable psi by a constant.
01:52:22
And remember that the metric of the multi-center top net takes this Givens-Hulking form
01:52:40
and psi is an angular coordinate, it's periodistic to pi.
01:53:05
And yeah, so the U1 action, the relevant U1 action acts on the angular coordinate psi and it also acts on the gauge bundle, equivalently.
01:53:25
Good, and then yesterday, I did not have time to write down some formulas, which I was planning to write. So now I want to use this occasion
01:53:41
to write the formulas. So for the truth to do it,
01:54:02
truth to do it on S4B, so deformed four sphere. Yeah, so this is a guess for B not equal to one, but for B equals one, this was shown by Gomis, Pester and myself.
01:54:24
So this takes the form, so the usual integral is for Poisson function and then there is summation over V.
01:54:46
So this is a screening or a bubbling contribution.
01:55:00
So yesterday I explained the rough idea for how this comes about. So smooth monopoles get attached to singular monopoles and weaken the singularity. And this little v is the magnetic charge after such screening occurs. And there is a contribution from the region,
01:55:30
from the region in the neighborhood of the loop operator and it locally looks like S1 times R3, so I write it this way and there is contribution.
01:55:48
That is an analog for monopoles of the instant of Poisson function. And then there is a usual sphere one loop path
01:56:04
and the instant Poisson function mod squared. So on the deformed full sphere, this is a form of the expectation value of the total operator.
01:56:22
Now I also want to give a formula for S1 times R3 and there is actually a omega deformation parameter from S1 times three because R3 is fiber over S1.
01:56:42
So when you go around S1, there is some special rotation along the third axis. Now this can be written as a super symmetric trace in some Hilbert space. So maybe I should have emphasized this also.
01:57:03
Line operator which is inserted along the time direction. So that is different from Tohut's case. For Tohut loop operators work in the special direction but now I'm considering a line operator along the time direction which is this one.
01:57:23
So line operator, certain line operator modifies Hilbert space. So it's not an operate, line operator in that situation is not an operate data from Hilbert space but it rather changes the Hilbert space. And we take the trace there. And so the path integral we're considering
01:57:51
can be interpreted as a super symmetric trace.
01:58:12
So here H is the Hamiltonian, R is the radius of the circle. Lambda is basically the omega deformation parameter.
01:58:22
J3 is a rotation operator, I3 is the SU2R rotation operator and the capital F, sub F is the flavor generator. And M sub F is in the fugacity.
01:58:40
Now this, if you do localization on S1 times R3, what you find is the following structure. Again, there is a same sum that take into account a screening of bubbling contributions.
01:59:01
And this is two pi I VB, there's some pairing. Oh, I don't, okay. And okay, and then this B, this B depends on one of the, a real scalar in the vector measurement
01:59:21
and also it contains basically the glide photon. So the dual of the gauge field, the three dimensional gauge field. And then there is one root part, okay,
01:59:41
times the monopole partition function part. Okay, maybe, no, it's a little bit important, but okay. I can, I give the parameter.
02:00:00
dependence Yeah, so it looks like this So in particular, okay, so capital B enters here, but not here
02:00:24
And on and the little a is a combination of one the other real scalar in the vector multiplet and and The Wilson
02:00:41
Wilson line, so I mean the whole economy along this one. No, it's not over counting
02:01:00
So for example, if you use the the equivalent index theorem the one-loop determinant receive contributions from From specific locations on S4B, so Yeah, so S4B one-loop determinant, this is contribution from the north and the south poles and
02:01:23
And Yeah S1 times R3 one-loop determinant, this is contribution from the equator or the location of the loop operator Yeah, so basically this is the same yeah, so so yeah, yeah, thank you so the point is that
02:01:53
These these quantities appear On S1 times R3 with some Okay with some with arguments shifted. Yeah, and then I gave one
02:02:05
Okay, a couple of examples of concrete examples, I think yesterday right and it's they are also in the exercise and and Yeah, you can also compute the Moiré product Maybe I should say okay, you get the Moiré product because these line operators
02:02:21
Realize deformation quantization of the heat in modularized space and These are the interesting story when you dimensionally reduce to three dimensions. Okay