3/4 Motivic periods and the cosmic Galois group
This is a modal window.
The media could not be loaded, either because the server or network failed or because the format is not supported.
Formal Metadata
| Title |
| |
| Title of Series | ||
| Part Number | 03 | |
| Number of Parts | 4 | |
| Author | ||
| License | CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | |
| Identifiers | 10.5446/16353 (DOI) | |
| Publisher | ||
| Release Date | ||
| Language |
Content Metadata
| Subject Area | ||
| Genre | ||
| Abstract |
|
00:00
Local GroupFrequencySpecial unitary groupDegree (graph theory)Simplex algorithmFunctional (mathematics)Numerical analysisConnected spaceMassProduct (business)Right angleLogistic distributionNetwork topologyLine (geometry)Rule of inferenceAntiderivativeMultiplication signMomentumGraph (mathematics)PolynomialINTEGRALVotingSpacetimeCycle (graph theory)Associative propertySummierbarkeitAreaSquare numberNatural numberDifferential formFilm editingGoodness of fitFamilySurgeryFrequencyDimensional analysisGroup actionCohomologyFood energyPositional notationAlpha (investment)Time domainSigma-algebraString theoryLoop (music)Observational studyGamma functionDivisorCondition numberVertex (graph theory)Entire functionDifferent (Kate Ryan album)Projektiver RaumLecture/Conference
09:23
Uniformer RaumModulformGraph theoryTime domainFamilyGraph (mathematics)Plane (geometry)INTEGRALDimensional analysisTotal S.A.Parameter (computer programming)Line (geometry)Multiplication signDifferential formAreaQuotientKontraktion <Mathematik>HomologiegruppePoint (geometry)Arithmetic progressionVertex (graph theory)Wellenwiderstand <Strömungsmechanik>Simplex algorithmSheaf (mathematics)SubsetSurfaceSubgraphSigma-algebraSpacetimeSet theoryMathematical singularityBoundary value problemCategory of beingFood energyAlgebraic structureMereologyArithmetic meanHexagonDifferent (Kate Ryan album)Thermal expansionConnected spaceLimit of a functionAlpha (investment)PolynomialHyperplaneNormal (geometry)Lattice (order)Gamma functionFigurate numberGraph (mathematics)Square numberRootFiber (mathematics)Coordinate systemOrder (biology)MomentumCanonical ensembleLecture/Conference
18:46
SubsetForcing (mathematics)Kontraktion <Mathematik>Numerical analysisGamma functionSubgraphState of matterIdentical particlesDoubling the cubeSurfaceForestMassQuotientPositional notationFormal power seriesMultiplication signConnectivity (graph theory)DivisorClosed setAlpha (investment)Right angleVertex (graph theory)Graph (mathematics)Nichtlineares GleichungssystemModulformParameter (computer programming)Plane (geometry)Nominal numberResultantDivision (mathematics)Connected spaceMomentumRecursionCorrespondence (mathematics)Many-sorted logicPolynomialPoint (geometry)Loop (music)Well-formed formulaSummierbarkeitTerm (mathematics)GeometryGraph (mathematics)Flow separationHyperflächeKinematicsLecture/Conference
28:08
3 (number)AxiomWell-formed formulaAxiom of choicePolynomialDegree (graph theory)Series (mathematics)Alpha (investment)Social classOrder (biology)Variable (mathematics)SubgraphRule of inferenceGraph (mathematics)QuotientDivisorCategory of beingDifferent (Kate Ryan album)RenormalizationIdentical particlesMomentumExistenceNichtlineares GleichungssystemMultiplication signProof theoryProduct (business)Gamma functionLimit of a functionMereologyGeometryAlgebraVertex (graph theory)Total S.A.TheoryPoint (geometry)Game theoryCondition numberNumerical analysisGroup actionMany-sorted logicRoundness (object)Moment (mathematics)FactorizationMathematical optimizationLimit (category theory)Standard deviationEvent horizonDivergenceConnected spaceNormal (geometry)Graph (mathematics)Equivalence relationResultantLecture/Conference
37:31
Curve fittingDegree (graph theory)Multiplication signMomentumVertex (graph theory)KinematicsLink (knot theory)Greatest elementNormal (geometry)FamilyExt functorSet theoryWell-formed formulaSeries (mathematics)DivisorMoment (mathematics)Square numberCausalityConnected spaceFrequencyQuotientGraph (mathematics)Dimensional analysisProcess (computing)Euklidischer RaumSummierbarkeitCondition numberMassExtension (kinesiology)Variable (mathematics)Complex (psychology)Conservation lawGamma functionConnectivity (graph theory)Covering spaceRight angleSubgraphVector spaceVotingSubsetTotal S.A.Graph (mathematics)Decision theoryHand fan1 (number)Many-sorted logicMereologySpacetimeFunctional (mathematics)Neighbourhood (graph theory)Graph coloringQuaternion groupCarry (arithmetic)Positional notationFactorizationTriangleMultilaterationGeometryComplex numberAffine spacePolynomialResultantAsymmetryLecture/Conference
46:54
Maxima and minimaPoint (geometry)Combinatory logicPropositional formulaWell-formed formulaDegree (graph theory)Physical lawDependent and independent variablesObservational studyMathematical singularityProjektiver RaumDivisorMomentumMultiplication signSocial classRenormalizationDifferent (Kate Ryan album)SubgraphDiscounts and allowancesLine (geometry)AsymmetryFocus (optics)MassQuotientGamma functionArithmetic meanGame theoryCondition numberKinematicsContrast (vision)Kontraktion <Mathematik>Monster groupFactorizationClosed setProjective planeParameter (computer programming)Alpha (investment)Series (mathematics)TheoryGeometryAlgebraPositional notationAbsolute valuePolynomialVertex (graph theory)Symmetry (physics)Evelyn PinchingSubsetSummierbarkeitINTEGRALLecture/Conference
56:16
Graph (mathematics)SpacetimeSubsetFamilyDivisorMoment (mathematics)Greatest elementGroup actionDifferent (Kate Ryan album)Complete metric spaceGamma functionDegree (graph theory)FactorizationMomentum1 (number)CircleWell-formed formulaSurfaceMassMany-sorted logicRepresentation theoryDimensional analysisEvent horizonNear-ringCondition numberNumerical analysisSubgraphResultantValuation (algebra)Term (mathematics)Goodness of fitOrder (biology)Variable (mathematics)Food energyMultiplication signCollatz conjecturePoint (geometry)KinematicsAlpha (investment)PolynomialSet theoryAnalytic continuationLoop (music)Equaliser (mathematics)Series (mathematics)Linear subspaceDivisor (algebraic geometry)Fiber (mathematics)Lecture/Conference
01:04:52
MassAlpha (investment)Condition numberPoint (geometry)Different (Kate Ryan album)MomentumDivergenceMultiplication signStandard deviationExistenceLoop (music)SubgraphSet theoryGraph (mathematics)Gamma functionRepresentation theoryCategory of beingCompactification (mathematics)Order (biology)Maxima and minimaGeometryAbsolute valueHyperflächeKinematicsLinear subspaceCoordinate systemGraph (mathematics)PolynomialValuation (algebra)Well-formed formulaQuotientSpacetimeDimensional analysisLatent heatFilm editingNumerical analysisArithmetic meanFrequencyState of matterFree groupAreaInferenceVariable (mathematics)1 (number)Element (mathematics)Proper mapLecture/Conference
01:13:27
Category of beingDimensional analysisSpacetimeGrothendieck topologyGamma functionSubsetElement (mathematics)Inverse elementSet theoryOrder (biology)Kontraktion <Mathematik>Numerical analysisFrequencyExtension (kinesiology)Open setSpecial unitary groupEvent horizonStaff (military)Physical lawQuotientCondition number1 (number)Point (geometry)Line (geometry)Projektiver RaumTransformation (genetics)Finite setMereologyIntegerStandard deviationPositional notationAlpha (investment)Absolute valueDivisor (algebraic geometry)Cartier-DivisorGoodness of fitAlgebraic closureGraph (mathematics)Power setLecture/Conference
01:22:03
Graph (mathematics)Line (geometry)HyperplaneRecursionSpacetimeTransformation (genetics)CircleSurfaceCondition numberSubgraphCategory of beingKontraktion <Mathematik>Projektiver RaumMultiplication signSet theoryAlgebraic structureWell-formed formulaDirection (geometry)Product (business)Point (geometry)Divisor (algebraic geometry)QuotientSubsetExtension (kinesiology)Cartier-DivisorTriangleGraph coloringKodimensionDivisorRight angleOperator (mathematics)Graph (mathematics)Gamma functionPlane (geometry)Food energyGradientModel theoryTotal S.A.Block (periodic table)Division (mathematics)Lecture/Conference
01:30:38
GeometrySubgraphDivisorAlgebraSet theoryProof theoryArithmetic meanGraph (mathematics)Inclusion mapRight angleWell-formed formulaDifferent (Kate Ryan album)SpacetimeNormal (geometry)Order (biology)1 (number)PolynomialMathematical singularityNumerical analysisMassMany-sorted logicSphereFamilyMaxima and minimaDivergenceConnectivity (graph theory)Representation theoryPhysicistGamma functionAlgebraic structureMultiplication signCategory of beingProduct (business)Nichtlineares GleichungssystemVariable (mathematics)Degree (graph theory)Point (geometry)Observational studyTensorCartier-DivisorHyperflächeQuotientRenormalizationCompactification (mathematics)Open setLogarithmMomentumGraph (mathematics)Divisor (algebraic geometry)Limit (category theory)Transformation (genetics)Coordinate systemLecture/Conference
01:39:13
Order (biology)GeometryPoint (geometry)SubgraphModule (mathematics)Algebraic varietyRenormalizationRing (mathematics)Graph (mathematics)Free groupAlgebraProduct (business)PolytopAlgebraic structureGamma functionBijectionRecursionDimensional analysisDifferential (mechanical device)Limit (category theory)Motif (narrative)Network topologySigma-algebraMassRootDiagramInverse elementVarianceINTEGRALTensorLinearizationCombinatory logicTetraederGraph (mathematics)SpacetimeLine (geometry)Dot productConnected spaceAssociative propertyHyperplaneMultiplicationAbsolute valueNumerical analysisGradientDifferent (Kate Ryan album)VotingRight angleExplosionLikelihood functionFigurate numberSimplex algorithmFilm editingTriangleConnectivity (graph theory)10 (number)Food energyMereologyMomentumMultiplication signMany-sorted logicVector spaceLecture/Conference
01:47:49
Boundary value problemDifferential (mechanical device)Graph (mathematics)Near-ringProof theoryInterior (topology)FunktionalgleichungHomologiegruppeVektorraumbündelNichtlineares GleichungssystemKontraktion <Mathematik>Algebraic structureArithmetic progressionNormal (geometry)Functional (mathematics)KnotNormal subgroupSimplicial complexKozyklusProduct (business)DivisorConnectivity (graph theory)Different (Kate Ryan album)ResultantMany-sorted logicPoint (geometry)Category of beingGeometryMaxima and minimaExtension (kinesiology)AlgebraMultiplication signKinematicsSpacetimeMassBounded variationNumerical analysisDiagramWell-formed formulaFrequencyFamilyMomentumLimit (category theory)Observational studyCorrespondence (mathematics)Pairwise comparisonRenormalization groupMathematical singularityConcentricInfinityLocal ringFinitismusPhysical systemParameter (computer programming)Confidence intervalConjugacy classSequenceWater vaporBetti numberDegree (graph theory)Sheaf (mathematics)Matching (graph theory)Group actionRule of inferenceEnumerated typeSummierbarkeitRecursionQuotientLogarithmCohomologyTransverse waveINTEGRALRegular graphVector spaceSpectrum (functional analysis)Sigma-algebraHomologiePresentation of a groupGraph (mathematics)KodimensionPolytopRenormalizationOpen setLecture/Conference
Transcript: English(auto-generated)
00:31
So first of all, I'll remind you about graph polynomials from the first lecture.
00:50
So the data of a Feynman graph was a connected graph, G. And it had some external momenta,
01:10
which were called qi. And each qi is in typically real space, the dimension d. But we assume the momenta are non-zero.
01:24
And we had masses, a certain number of non-zero masses, sitting on internal edges of the graph. So the edges of the graph will always
01:43
be labeled in this talk. And this dimension of spacetime is an even positive integer. So over here, I'll draw an example that will be considered throughout today's talks.
02:06
So we'll study this in detail. So G will be the following graph.
02:28
So it has two momenta, q1, q2 coming in. Edge is labeled 1, 2, 3, 4. And the thickened edge means that it carries a mass. So we have m1, non-zero mass.
02:44
But the masses m2, m3, and m4 are zero. We just ignore them. This external momentum is considered to be zero. So I don't write it. And we had momentum conservation q1 plus q2 equals zero.
03:04
So two such a graph. In the first lecture, I explained how to associate graph polynomials. There were three of them. There was the Kirchhoff polynomial,
03:23
which was the sum of all spanning trees. And I called them spanning one trees. And then the product of the edge is not in that tree, alpha e.
03:42
Then that's also called the first semantic. This is the second semantic polynomial, is the sum over spanning two trees. And you take the product of the same thing again.
04:03
But this time you multiply by the total momentum coming into one of the trees, t1 or t2. It makes no difference. Total momentum entering t1.
04:23
And then the thing we're most interested in is polynomial I called psi, which depends on the masses and the external momentum. And it is defined to be phi gq plus the sum of the squares
04:44
of the masses times psi g. So in our favorite example, we have psi g equals and phi gq.
05:16
So here we have, let's write q squared equals q1 squared
05:21
equals q2 squared equals q squared alpha 1 alpha 2 alpha 3 plus alpha 1 alpha 2 alpha 4 plus alpha 1 alpha 3 alpha 4. OK, so we were interested in the Feynman integral, i gq
05:45
m, as a function of masses and momenta. And it had some gamma factors that I'm just going to ignore. So it's an integral over a certain domain sigma, 1 over the first semantic to the d over 2.
06:06
And then this ratio, and here we have ng minus hdd over 2 omega g.
06:21
So let ng be the number of edges of the graph. And hg is the number of loops, the first Betti number. Oh yes, and so as I explained in the first section,
06:42
so psi g is homogeneous of degree equal to the loop number. And the degree of phi and hence psi in the alpha variables, the string of parameters, is hg plus 1. That's very important. OK, so what was sigma? Sigma was the domain in projective space of dimension
07:06
ng minus 1. And it was the coordinate simplex. So sigma, it's the region with projective coordinates
07:21
in the Anglo-Saxon notation where alpha i are real and positive. And omega g is the sum minus 1 to the i alpha i d alpha 1 omit d alpha i d alpha ng.
07:45
So the goal then is to, as I explained in the last lecture, is we want to understand this as a period of some cohomology. And that will take the entire talk today
08:01
and a bit of the beginning of the talk next week. So we want to write igqm as a period, or family of periods, of cohomology. So we want to think of it as a pairing between a duram class, a differential form, and some cycle of integration.
08:26
And so what is this letter, your sine? Psi, yeah. So a capital psi is like this. And I hate writing that because it doesn't look like anything. So on the board, I prefer to write a line through it. Otherwise, it just looks like three random lines.
08:48
I know, my Greek teachers as well would be appalled. One question about your graph. Is it allowed to have one edge connecting one vertices to itself? Yes. That's called a tadpole. Yes. And do you assume your graph to be primitive or not
09:03
for the moment? Primitive, in what sense primitive? No, general. No condition whatsoever. No condition whatsoever. And in fact, I won't consider primitive until much later. What's the motivic thing you're going to work suddenly for d even?
09:22
What happens? Yeah, so it works for d even because that's the case we're most interested in because, as we all know, we live in 11 dimensions, which is, yeah, OK. I will spare you the lame jokes. So obviously, when d is odd, you get a square root here. You have to do something. That's no problem.
09:40
I think one can work with that. But I haven't thought about it. I won't. No, for what you do, you maybe need d equals 3. Is that a dimension or a figure derivation? Yeah. I believe that people who sometimes do three dimensions work in four and do a perturbative expansion from four
10:01
dimensions down to three. But I'm sure that everything works for odd. It's a different kettle of fish. Anyway, so the issue here is the same issue as I explained on the example of zeta 2 last time, is that we want to interpret the domain of integration
10:21
as a homology class. So we look at its boundary, which is contained in the place where all the alpha i's vanish. So these are called the coordinate hyperplanes. And the problem is, which is exactly the issue with the zeta 2 integral
10:42
last time, is that the domain of integration itself, not necessarily its boundary, we can just describe that, meets the singularities of the integrand
11:01
of the differential form. And this poses problems. So compare the zeta 2 example from last week. So now, to explain this issue, let me do an example.
11:27
So let's look at this graph. And for reasons that will become clear later on, I'm going to label the edges 2, 3, and 4. And so the graph polynomial here
11:40
is alpha 2, alpha 3, plus alpha 2, alpha 4, plus alpha 3, alpha 4. So let's call x subscript psi g to be the vanishing locus of this graph polynomial in P2.
12:09
This is the zero locus where the graph polynomial vanishes. And we call Li the i-th coordinate hyperplane. It's where alpha i vanishes.
12:21
And so we have the following picture. The graph hyper surface looks something like this. Well, because it will appear as a quotient of this graph when I contract the edge 1. And so later on, this example will serve a purpose later on.
12:42
That's why. The coordinate hyperplanes look like this. And so let's call that L1. Sorry, I've made L2, L3, and L4.
13:04
This is x psi g. And the domain of integration is this simplex here. So in this situation, I'm assuming, say, that ng equals hd over 2.
13:20
So I'm not considering the polynomial psi. But that's just a toy example. And exactly as in the last lecture, there are bad points here which are not normal crossing. In fact, there are three bad points. So the bad points where the domain of integration
13:45
meets the graph hyper surface are the intersection sigma psi psi g. And they are exactly the three vertices
14:00
of the simplex. So as I explained last time, the solution is to blow up these points to make the situation normal crossing. And the picture we get after blow up is this.
14:26
And then the three bad points become copies of P1 that, sorry about that, I should have drawn it like this.
14:46
And the white lines no longer meet graph hyper surface in the new picture. We get a new domain of integration which is no longer a triangle, but a hexagon. So I'm going to call this sigma tilde g,
15:02
which I'll define in general. And these are exceptional divisors, epsilon 2, 4, epsilon 3, 4, epsilon 2, 3. So what do I want to do today?
15:20
We need to understand the general geometric situation. So first of all, we want to determine the bad loci in general. Then we want to blow them up.
15:44
And then the third part will be to study the structure of the resulting space. OK, so I did in the first lecture.
16:01
I'll say it again, the alphas are called Schrodinger parameters. Oh yeah, so the alphas are called Schrodinger parameters. And you want to do this in a canonical way in order to get a well-defined motive? Exactly, I want to do it in a canonical way uniformly for all Feynman graphs.
16:22
So I'm very slightly, well, we'll get to this point. But of course, these are families of hypersurfaces, depending on mass as momenta. I'm just going to work fiber by fiber, because I don't want to say a lot about families. But it poses no problem whatsoever. And one can be very precise about how it works. But I'll just pretend that q and m are fixed and generic.
16:45
And one can be very precise about what generic means. OK, so first of all, a lot of properties of graph polynomials. And the first one is called contraction deletion.
17:03
So this is going to be half an hour of just graph theory, or maybe more. So if e is an internal edge of our graph, then define g minus e, which is the deletion of edge e.
17:29
So you delete the edge e, but you retain its endpoints. So for example, if we had a graph like this,
17:45
and this is the edge e, then deleting the edge e gives you a new graph, which now has two connected components. So if this is g, this is g minus e. OK, so for most of the rest of the lecture,
18:08
so I may be a little bit sloppy about this, I'm going to consider subgraphs defined by sets of edges. So an edge subgraph.
18:27
So that means you take a subset of edges and you include all the endpoints as vertices. So sometimes I may forget to write gammas contained in the edges. But we think of it as a subgraph of g. So in this case, we define the quotient or the contraction.
18:44
What is the edge with the vertices at the end of the edges? You always take the vertices at the end of all the edges. And no more. To remove the edges. Exactly, you take the minimal subgraph that contains those edges, that's a graph.
19:01
So what is the contraction? What you do is you delete all the edges in gamma. And then you identify all the vertices lying in each connected component.
19:29
So if gamma has a bunch of connected components, each connected component gets contracted down to a single vertex, gets squashed down to a point. No masses for now, they're going to come later.
19:45
But actually, thank you, that's my next comment. So the quotient, yeah, you're right, thank you. The quotient, first of all, the momenta, quotient g gamma will inherit the external kinematics.
20:12
And in general, it will inherit the massive edges. If they were massive before, they will remain massive. It actually depends on the situation.
20:21
I'll say more about that later. So here's an example. Let's take g and contract the subgraph 3, 4. This is just this graph.
20:41
And g, contract the edge 1, is the other example, which is why I considered it. What you do if one of the mass that you contract,
21:00
one of the edges you contract has a mass? You ignore it, it pays no role. Well, that corresponding edge does not appear in the quotient. So for now, I'm just talking about the quotient. So that mass does not appear, it is not on the edge of the quotient, so it's gone. But later on, I'll explain exactly what
21:20
happens to the mass as a momenta, and it's quite subtle. But that will come. It's because the internal momenta goes to infinity, so the mass is a division. I hadn't thought of that, but that's probably a good explanation. Yeah. OK. So unfortunately, there's another notion of contraction
21:41
for sort of stupid reasons to do with this contraction deletion identity. And with hindsight, I realized that I should have defined things slightly differently, but it's OK.
22:07
So define another notion of contraction is g double slash gamma. This is not a standard notation, but I think it's helpful. So it's g slash gamma if gamma is a forest,
22:24
so it has no loops. But it's the empty graph, or zero graph, whatever you prefer, if the subgraph you contract contained a loop, at least one loop. And then the contraction deletion identity
22:47
is the following. We take any internal edge in the graph, and then the first graph polynomial satisfies you delete the edge e and multiply
23:00
by the corresponding Schrodinger parameter. And the constant term is the contraction in this sort of brutal sense. And the second semantic polynomial satisfies a similar identity. So here it's clear, when we delete an edge, you have the same external vertices.
23:21
So they keep their external legs. So there's no ambiguity about what this means. What can happen, of course, though, is that when you delete an edge, the graph is no longer connected. And in this case, this graph polynomial will vanish.
23:41
And that's what you want. The equations are correct precisely because of that fact. OK. So the only thing that we're going to need from this is the following identity. So the first graph polynomial restricted to the locus alpha
24:01
equals 0 is just psi double slash e. And from the definition of psi, the restriction to alpha equals 0 is psi g double slash e qm.
24:26
So this tells us something about the geometry. It tells us what happens to this graph hypersurface when we intersect with a coordinate hyperplane. There's a beginning of a recursive structure because you get the corresponding hypersurfaces
24:41
of quotient graphs where you contract an edge. So that will be important.
25:01
So now I'm going to state some what I call factorization theorems. These are absolutely crucial for everything that I'm going to say. The entire crux of the story is hidden in two or three identities that I'm going to write down now. And everything relies on that. So g be connected again, connected Feynman graph.
25:25
And gamma, a subgraph defined by a subset of edges with several connected components, gamma up
25:45
to gamma n. So the number of loops in gamma is just the sum of the number of loops of each connected component. And the first result, so this is not new
26:06
because it's only in a paper of mine with Dirk Kreimer, and I'm sure these identities are, in principle, been known for a long time, but not necessarily been written in this form.
26:22
So we can write the first semantic polynomial ultraviolet. Don't ask me what it means.
26:56
Actually, this formula is in a paper by Bloch, Inor, and Kreimer. The following one is in a joint paper with Kreimer.
27:08
And I'll explain in a minute why these, in principle, have been known for a very long time.
27:24
R phi uv gamma g. So here we have two approximate factorizations of the graph polynomials. And I need to explain what that means, what R is.
27:42
Do you want to put the first label uv on the first R? I've deliberately not done that for a subtle reason. I'm glad you spotted that.
28:11
Where R dot, in either case, gamma g is of degree strictly larger than h gamma.
28:24
So h gamma is the degree of this part. This has degree h gamma in both cases. But the R is the remaining term. It has higher order in the edge variables alpha e for e
28:40
in the subgraph gamma. So let's do an example. So in this example, we have, let's just look at the first graph polynomial, R.
29:12
So here we can write the first graph polynomial. You can check that it can be written alpha 3 plus alpha 4 times alpha 1 plus alpha 2 plus alpha 3 alpha 4,
29:26
where the subgraph is 3, 4 and the quotient graph 1, 2. The external momenta play no role for the first graph polynomial.
29:40
And it's also equal to, so it factorizes in two different ways, alpha 4 plus alpha 1 alpha 3 plus alpha 2 alpha 3. And here the subgraph is, and the quotient is a tadpole.
30:06
So here we have, the subgraph here has variables alpha 3 and alpha 4. And the leading order in these variables, this is of degree 1, but the remainder is quadratic, so degree 2. So if we think of alpha 3 and alpha 4 going to 0,
30:22
this goes to 0 at order of epsilon squared, and this goes to 0 at order epsilon. So in the limit, as these subgraph variables go to 0, the remainder terms will disappear, and we'll just get the product on the left. That's very important.
30:42
So a couple of remarks. So in the very special case, so in the very special case where gamma is subdivergent, then these factorizations
31:05
are sufficient to derive all the main, well, most of the main results, certainly, of renormalization theory. So you can prove BPHZ theorem, and you can derive the Canon-Somansic equation just
31:24
from the existence of these factorizations in the very special kit for this very special family of subgraphs. So this was done in a paper with myself and Dirk a couple of years ago. And that's why I say physics knows about these factorizations and has done for a long time,
31:43
because it's essentially equivalent to being able to renormalize. But the funny thing is that renormalization doesn't use the full strength of these equations. It only uses the case where the subgraph is graph is subdivergent. And the general case of these identities is not used,
32:06
but it will be absolutely crucial in the construction of the cosmic Galois group. A third remark is that you can ask, since this factorization property is very closely
32:24
related to renormalization, you can ask what are the possible polynomials that satisfy these factorization identities? And if you think about it a little bit, you can check that they uniquely determine the graph
32:41
polynomials. So the factorization formulae, plus very little input, in fact, uniquely determines. So there are remarks on this in my paper with Kreimer.
33:00
But one thing we never got around to was to explore this idea and see how rigid the Feynman rules are if you demand that they be renormalizable in this algebraic geometry sense. It can be anything. It's just any polynomial that will be homogeneous.
33:21
But in those variables corresponding to the subgraph, it has higher degrees. And would you call input? Input, well, so for psi. So it depends what you can choose what axioms you want. If you give yourself a class of graphs, say these graphs, and you want this formula to hold,
33:40
and you want this formula to hold, and you specify what psi of an edge is and psi of a tadpole, then those two axioms will uniquely determine the first semantic polynomial. So there is no choice. Physics gives us the unique possible choice for which these equations hold. And so this sort of photo experiment that we never
34:00
got around to is to try to, well, there's an obvious game to play and say how unique are the Feynman rules if you impose a certain number of desired conditions. OK, so here comes an important point that will seem trivial at first,
34:22
but will be important next week. The quotient graph polynomial is never 0, but it can happen that the second semantic polynomial
34:47
of the quotient graph is 0. And so it was the case here because here
35:00
I was a little bit sloppy. This quotient graph here really has two internal momenta, q1 and q2. But because q1 plus q2 is 0, this is equivalent, as I explained in the first lecture, to a graph with no incoming momenta. And so this has vanishing second semantic polynomial.
35:21
So this is an example where the quotient has vanishing second semantic polynomial. Oh, I'm ahead. Wow. OK, so we want to understand this phenomenon now, which is related to infrared subdivergences.
35:44
So definition, a subgraph gamma, which is connected.
36:04
So I take a connected subgraph, is momentum spanning. So this is a phrase I made up. It's not standard terminology, as far as I'm aware. Momentum spanning, if for every vertex, v in the big graph,
36:30
which carries some momenta, such that the total incoming momenta, so this is the total momenta entering,
36:42
total momentum entering that vertex. It's momentum spanning. If every vertex, which carries some momentum, then that vertex is in the graph, is in the subgraph. So it's a subgraph that touches every external leg that
37:03
carries some momentum. And a general graph, gamma, an arbitrary subgraph not necessarily connected, is momentum spanning, if some connected component, if it has at least
37:29
one connected component, and hence exactly one connected component of gamma, is momentum spanning.
37:45
So a momentum spanning subgraph is a subgraph that has exactly one component that touches all the external legs, the non-zero external legs. And all other components do not touch any of the external legs.
38:01
Example, OK, example. So here in this case, a momentum spanning graph would be a subgraph, because it meets the only two momentum carrying vertices, here and here.
38:25
Let me draw them in a different color. Here and here. Another momentum spanning graph would be 2, 3,
38:47
because it has a connected component that meets everything. I don't know what else. What is not momentum spanning? So this subgraph here, 3, 4, is not momentum spanning,
39:06
because it doesn't see this. But you could also take a triangle, yes? Sorry? You could also take a triangle. Triangle, yeah, sure. Yeah, I'm not going to write the full list of all momentum spanning. It'll take time. And I need to refine the condition later.
39:21
So that'll be another condition to do with masses. And then I will give the complete list of those subgraphs a little bit later on. Where do you have to see some of the one with momenta? The vertices are the one with non-trivial momenta. So in this case, this vertex does not carry momenta,
39:44
because the incoming momentum is 0 by momentum conservation. So the little gamma is momentum spanning precisely on the phi of the quotient 0. That's my next comment. Thank you very much. Yeah, so yeah, with a caveat that we
40:04
have to assume generic momenta. Because you could have all, if all q squared equal to 0,
40:23
for example, you can easily construct examples for which that's just not true. But we don't consider that. Yeah, so with sort of generic kinematics, I'll be vague, but actually it's very easy
40:40
to write down the condition. It's actually that the partial sums should be non-zero. That's the only condition. For generic kinematics, gamma is momentum spanning
41:05
if and only if the quotient has 0 vanishing second semantic polynomial. OK, so now this leads me to the infrared factorization
41:20
formula, and I'm led to believe that this is new, in fact. I've certainly never seen it anywhere in the literature. I don't understand the formula that you wrote under general kinematics. What do you mean here? Yeah, put the square, the square momenta. The norm, this is a sum of vectors in d-dimensional space, and this is the Euclidean norm.
41:43
For all i, for any subset of external legs, you take the sum of the momenta and you take the Euclidean norm. But if you take a subset of all the vertices, then the sum of qi is 0?
42:00
Absolutely right. So it's all i strict. Thank you. I was not planning to talk about this, but yeah, you're right, absolutely right. But here, what's with the square, you're in the Euclidean space, remaining Minkowski. No, no, I'm not. OK, so I'm using algebraic geometry. So the masses and momenta are variables
42:24
in a complex space, an affine space. Everything makes sense. The geometry makes sense over complex numbers. But the square is the Euclidean norm. When you say generic kinematics, you assume that each edge has a non-zero momenta or not?
42:43
Each edge. No, I said at the beginning, some vertices have zero momentum. And we specify which vertices have zero momentum. Like here, it's the bottom one. And the others, which are non-zero. Not for all i inside of e, g, then? Oh, sorry, sorry, sorry, yeah. See, this is what happens when I don't use my,
43:01
when I improvise. For all i contained e, I think it was called e xed. Well, I think I defined a set of vertices which carried kinematics in the first lecture. So that's the notation I used with the external half edges
43:20
that carried. Yeah, I didn't want to get into these details. It's very simple, but it's just fiddly to get there. So to answer Thiebaud's question, I've forgotten all the questions. What were you talking about, the square and the Euclidean? Yeah, so what will happen is we'll have a family over some space.
43:41
And the Euclidean region will be the region where the q's are real or something like this. And in a neighborhood of that space, everything will behave beautifully. And we can have well-defined analytic functions. And then we can extend over a complex space, and then we'll have some terrible discriminant that we don't understand. q's are real, the square is not useful.
44:04
Yeah, so this is a trivial condition. Yes, if they're real, this condition is trivially implied by the ones I had in the first lecture. OK, so then infrared factorization theorem. So let gamma and edge subgraph, which
44:25
is momentum spanning, with components gamma 0, gamma 1, up to gamma n, where gamma 0 is the connected component,
44:40
which is momentum spanning. And I should add a remark that when you have a momentum spanning subgraph, of course it inherits the external kinematics, external legs.
45:02
External kinematics, because it's precisely connected to every single external leg that carries momentum, by definition. So then we have the following factorization formula. Phi gq equals phi gamma 0 q.
45:24
That makes sense, because gamma 0 inherited the external legs. Then we multiply by the first semantics of the other graphs. And then we take the first semantic of the quotient graph and a remainder term, which I now call the infrared
45:41
remainder term, where the same story as before, the remainder term has higher degree than the leading piece. So it has degree bigger than h gamma plus 1,
46:02
because this time, this part here has degree h gamma plus 1. So r has strictly higher degree in the edge variables of the subgraph gamma.
46:20
So as I briefly mentioned, I don't think this is in the literature anywhere. And to my surprise, I think it's a new result that's maybe implicit in some of the very old literature, though not stated explicitly. So is there a similar result for psi sub g?
46:41
No. So there's an asymmetry in this formula, because psi always is sort of the factorization of psi only involves psi. The factorization of phi involves psi on one side and phi on the other side. And it by contrast with the contraction deletion,
47:00
which involves phi on both sides. And as an exercise, you can try to prove the contraction deletion from the factorization and vice versa. So that's quite a nice exercise. And when you do it using this, you realize very quickly that to get the symmetry in the two phis here tells you that there must exist a symmetric formula with the phi on the other side.
47:24
And so they have to both exist. And it's this asymmetry that the phi involves the psi's and the phi's that means that they're two different formula for phi, but only one for psi.
47:40
Yes, I was a bit surprised when I saw this. OK, so back to this example. Let me erase this.
48:01
Now I can put infrared. So I'll just take a connected subgraph. So let's take the subgraph 2, 3, whose quotient is 1, 4.
48:34
And then we get q squared alpha 2 plus alpha 3
48:40
times alpha 1, alpha 4. And the remainder term is alpha 1, alpha 2, alpha 3. And let's do another one. We can take the subgraph given by this massive line. Of course, masses play no role in this current discussion.
49:05
And here we get q squared alpha 1, alpha 2, alpha 3 plus alpha 2. And here for degree reasons, the remainder happens to be 0.
49:23
OK, so there we have two different factorizations. So of course, this begs the question whether what one can do with regularizing infrared singularities. And I have no idea. But I am tempted to think that the study we did with Dirk for ultraviolet, some of it
49:43
at least can carry through verbatim by using this formula in the infrared. You said the masses play no role in the infrared. Why do you mean phi singularities is crucial in the infrared? For now, in five minutes, I'll answer that question. Because we haven't considered phi doesn't occur on its own in the Feynman integral. It occurs in this combination.
50:01
Well, let me write it. It occurs in this combination, psi g qm equals phi q plus sum e and eg m e squared alpha e psi g.
50:20
So of course, as you rightly say, we now need to consider masses. And we want to understand factorization formally for psi. So that's what I'm going to do now. So another definition. We take a subgraph defined by a subset of edges.
50:40
And it is called, again, I've invented this phrase. So it should be taken with a pinch of salt. Mass momentum spanning, and I will abbreviate this by mm.
51:02
It's mass momentum spanning if, first of all, one, the edges of gamma contains all the massive edges, and two, if gamma
51:25
is momentum spanning. So basically, the vertices contain all the momenta, and the edges contain all the non-zero masses. Sort of obvious meaning.
51:40
And again, if we have generic kinematics, again, it's a very weak condition on the external momenta, then gamma is mm if and only if the psi polynomial of the quotient
52:05
vanishes. Here, do you assume gamma to be connected or not? No, absolutely not. So for example, the masses could be on one component? Absolutely. Absolutely. That's absolutely right, and that's a very good remark.
52:20
It's an excellent remark. Yep. And it's for that reason that I'm going to state a slightly simplified version of the next proposition to make sure I don't have very cluttered notation. But it works fine. I'll just write it here, and then I'll take a break.
52:41
So the proposition. So let's take gamma contain eg, but I'm going to suppose connected just for simplicity. It really makes no difference.
53:04
Then we have two factorization formally for psi, which is the thing we're most interested in. It's psi gamma psi g mod gamma qm plus a remainder term.
53:22
This is the ultraviolet factorization. And psi gqm. Oh, sorry, well, this is for any graph, but let's just put gamma 0 mass momentum spanning.
53:40
And the infrared factorization occurs when gamma is momentum spanning, ir gamma g. So when gamma is mass momentum spanning.
54:01
Of course, so where the degree of r psi uv gamma g in the alpha e for e in the subgraph is strictly greater than h gamma. And the degree as before for the ultraviolet
54:26
in the same thing is bigger than h gamma plus 1. So this proposition follows from the previous propositions. You just have to take the definition up here.
54:41
Definition of psi, and you plug in the previous factorization formally, and you get this coming out. And you notice that in, to answer a question, that in this second formula, so the first formula uses the two ultraviolet factorizations for psi and for phi. The second formula uses the infrared factorization
55:02
for phi, but the ultraviolet factorization for psi. So the factorization of psi is both infrared and ultraviolet in some sense, because it appears in both. I'm a bit confused. The way the Schrodinger parameters appear, which is that 1 over p squared plus x squared is integral of x minus 1 minus alpha.
55:22
In general, the uv is alpha going to 0, and the infrared is alpha going to 3. Here we're in projective space, so it's meaningless. So from an algebraic geometry point of view, this is why I said I don't understand what uv or ir means, because we have a picture like this,
55:43
and we have some bad points. And that's why I'm not, so the renormalization only cares about ultraviolet, but from the mathematical point of view, the infrared is in there as well, and the theory works fine without any problem covering all cases. So I'll explain that after.
56:00
Projective line, and we could be down or up. Exactly. So mathematically, we don't see the difference. So these letters uv and ir are artificial. It's just for physics intuition. So we'll stop here, and then I'll continue in five minutes. OK, so let's continue. I think this is number five.
56:21
Now we want to determine the bad low psi, which are the bits we're going to need to blow up. So let's take i any subset strictly contained in the set of internal edges, and sort of here we're
56:44
working with generic momenta, which is again a precise but weak condition that I don't particularly care to write down. So let Li contained in projective ng minus 1 space
57:06
be the locus where alpha e vanishes for all e and i. So it's a coordinate linear subspace.
57:23
Then we have two hypersurfaces, what's often called the graph hypersurface, x psi g equals v psi g, then in projective ng minus 1 space.
57:46
And we have a family of psi hypersurfaces, which I want to call x psi g.
58:02
And this is v, the vanishing locus of the polynomial qm. Of course, it's a family, but we're going to think of it fiber by fiber. So it's a fiber over some point in some appropriate space
58:29
of kinematics that I don't care to write down. It's not very difficult. And so the remark is that subsets i contained,
58:47
in fact, any subset contained in the set of edges of g, as we remarked earlier, this is equivalent to looking at certain families of subgraphs, edge subgraphs, gamma and g defined by a set of edges.
59:05
So now let vi, for example of the first semantic polynomial, be the order of vanishing of psi g along the divisor Li.
59:31
And so we know exactly what these are from these factorization formulae. So factorization formulae, sorry?
59:45
Li is not a divisor. No? No, did I say divisor? Oh, sorry. Yeah, thank you. Another comment from non-native English speakers. Factorization is written on s, but I think the z is an endangered letter. So I take it upon myself to.
01:00:00
for z's in words, otherwise there's no use for letter z in English if you never use it. So let's compute the valuation along a subgraph of psi g, where we apply the factorization formula for the first polynomial.
01:00:21
And it's psi gamma psi g mod gamma plus the remainder term. But by definition of the remainder term, the remainder term vanishes to higher order than psi. So this is, recall that this is of degree hg.
01:00:44
And this thing, it was that little remark I made earlier, is non-zero. So that means that the valuation is just the valuation psi gamma psi g mod gamma. And what's the order of vanishing of psi gamma? Well, when all the variables go to 0, it's just the degree.
01:01:02
So this is just h gamma. And likewise, the valuation of the psi polynomial is h gamma if gamma is not mass momentum spanning.
01:01:30
But it goes up by 1 and exactly 1 when gamma is mass momentum spanning. So these are sort of an ultraviolet subdivergence
01:01:42
if h gamma is positive, well, in some dimension. And these are sort of the infrared ones, and they have one degree. And i and gamma can be changed. i and gamma are interchanged. I think, yeah, so i. A subset i and a subset gamma is, I think of a subset as a subgraph.
01:02:03
OK. So now I need a definition. So right, I'm a little bit embarrassed because I needed a word to define the bad loci. And I thought about it.
01:02:22
And the adjectives I could think of were worse than the one that I'm going to propose. So I'm going to try this out, and we can see if you think it's ridiculous. We can abandon and think of a different word. But the motivation is there's a word in English called a moat, which is a speck of dust or a particle.
01:02:49
This leads to notions of irreducibility. And there is no word in English. There's no adjective relating to moat in English, but the letters M-O-T-I-C stands for members
01:03:04
of the inner circle by a coincidence. So this lends to notions of interconnectedness. And of course, the motive of the graph
01:03:22
will be related to its moatic subgraphs. So I will define a moatic. Again, so this word does not exist, but let's try it out. A moatic subgraph gamma contained in G,
01:03:40
it's defined by a subset of edges, is a subgraph such that for all strict edge subgraphs, oh sorry, this should be gamma, a moatic subgraph big
01:04:01
gamma is a subgraph such that for all strict subgraphs defined by a strict subset of edges, which is mass momentum spanning, then the number
01:04:21
of loops of the subgraph is strictly less than the number of loops of the moatic graph. So in this sense, it is irreducible. So it can't be made smaller without changing something. And then it's empty.
01:04:43
It's empty, yeah, it's an equality. So if there is no, for all subgraphs which is mass momentum, there is no such graph. Oh yeah, then, sorry, there's no such graph which is mass.
01:05:02
You want to say that instead of which. Any idea about that? What difference does that make? It's the same thing. Because it's including the definition of gamma anyway. OK. So if there's no graph that is mass momentum spanning,
01:05:21
let me think. It should be true, yeah, because. Oh, yeah, yeah. But yeah, that leads me to an important point. So remark, if there are two cases, so if it's not quite an intrinsic notion, sorry, yeah, sorry, mass momentum spanning in gamma.
01:05:42
Sorry, I apologize, I forgot the key point. Mass momentum spanning as a subgraph of gamma, it's a relative notion, then this equality holds. So it's a slightly subtle notion. But so if gamma is mass momentum spanning in G,
01:06:01
apologize, this is slightly technical, then we think of it as a Feynman subgraph. So it inherits all the masses and momenta from the big graph. And then this condition is non-trivial. But if gamma is not mass momentum spanning,
01:06:25
we think of it, we give it no masses or momenta. So it is considered with no kinematics. And in which case, every subgraph is mass momentum spanning.
01:06:48
So for physicists, if we forget the condition about mass and momenta, so if gamma had no masses and no momenta, then this condition is equivalent to being one particle irreducible.
01:07:02
So I can erase this now. So in this case, so if gamma, gamma not, gamma has no masses or momenta, then this
01:07:23
is gamma emotic if and only if it's one particle irreducible. So that means that it doesn't have an edge that you can cut. So here's an edge, a graph that's not one particle irreducible.
01:07:40
There's an edge that you can cut, and the corresponding subgraph has the same number of loops. It has two loops. So this is a slightly more subtle notion related to the existence of masses. So this is a physics terminology. It's called one particle irreducible.
01:08:03
So this is when you consider ultraviolet divergences. This is a very standard notion in physics because it just comes up all the time. But because we're considering infrared stuff as well, the definition has this extra condition of mass momentum
01:08:21
spanning. And the point is that gamma emotic subgraph in G implies that, from the previous formulae here for the valuations, implies that the valuation along gamma of the graph polynomial psi
01:08:47
is strictly bigger than 0 because H gamma is strictly bigger than H lower gamma. And that implies that the coordinate subspace L gamma
01:09:08
is contained in the graph hypersurface. And so we get a bad locus.
01:09:21
So that's something we need to blow up. Absolutely. So the emotic subgraphs are precisely the minimal set of subgraphs that we need to blow up in order to get a good compactification. And because I was worried I wouldn't have enough time,
01:09:43
I actually prepared some pictures on some handout. So if you'd like to pass them around. It saves you having to. So now some properties of emotic graphs.
01:10:13
Oh yeah, sorry. Of course, what's on that sheet is on this board here. So ignore this bottom thing.
01:10:20
That's not yet. These are the sets of emotic subgraphs in the graph G. So let's give them names. This one is mass momentum spanning because it contains the massive edge. There's only one of them. And it meets all the external momenta. This one is not mm.
01:10:43
This is mm. This is mm. And this is mm. So when the variable alpha 1 goes to 0, you can see in this polynomial alpha 1 divides every monomial. So it vanishes along alpha 1 equals 0.
01:11:01
And that's an infrared subdivergence. That's an infrared subdivergence. This one, when alpha 3 and alpha 4 go to 0, again, you can check that the whole thing vanishes. This is an ultraviolet subdivergence. And to go back to the question earlier,
01:11:21
these graphs here are both infrared and ultraviolet. I don't know what to call them. We don't really see the difference in this geometric picture. So they're sort of mixed. They're both infrared and ultraviolet, whatever that means.
01:11:43
And you can check that they all have this minimality condition. So if you cut an edge here in this graph, if you cut an edge, the edge is massive. So you can't cut that edge.
01:12:03
And it's motic. In these graphs, if you cut this edge, then the number of loops goes down. And you can check the definition on there. So finally, I'm confused. Motic means infrared safe or infrared problematic? Motic means either UV or infrared problematic. It means problematic at all in any dimension of spacetime,
01:12:23
sufficiently large dimension of spacetime. And it's minimal with that. It's minimally problematic. So it's not just the problematic subgraphs. It's the minimally problematic subgraphs for some arbitrarily high dimension.
01:12:41
Wherever you have to do the geometry, the capacity of geometry. Yeah, and that's why I avoid the use of the word divergent, because divergent refers to a specific, well, for me, it refers to a specific dimension of spacetime. So a theorem, you take two edge subgraphs,
01:13:10
property Q for quotients. If gamma 1 is contained in gamma 2, and gamma 2,
01:13:24
motic in G, then gamma 2, a quotient of a motic graph is motic.
01:13:41
Extensions, if gamma 1 contained in gamma 2, and both gamma 1 and the quotient gamma 2 mod gamma 1 are motic, then this implies that the big graph,
01:14:00
gamma 2, is motic. Unions, if gamma 1 and gamma 2 are motic, then gamma 1 union gamma 2 is also motic.
01:14:23
And then contraction, if E is an edge in G, and the edge contraction is motic,
01:14:43
then it comes from a motic graph. Then either, then at least one of gamma or gamma union E is motic. By the way, I have a suggestion, so why don't you
01:15:03
call them problematic? Very good. Problematic, very good. Yeah, problematic. I like that very much, problematic. So that deserves a citation.
01:15:36
I wish I had thought of that, but now I can.
01:15:41
Now I will. OK, so now blowups. So linear blowups.
01:16:00
I've forgotten what number this is, six maybe. Linear blowups and projective space. OK, so S, a finite set.
01:16:23
And PS is a projective space of dimension mod S minus 1 with coordinates. That's exactly what we've been doing all along. Alpha S for S and S.
01:16:42
So now let's choose any subset B be a subset of the power set of S. Sorry, an element of the power set. So the first notation is 2 to the S. So this is a set of subsets of S with a property
01:17:02
that it's closed under unions, which is if I and J are in B, implies I union J is in B. OK, so now we define the iterated blowup.
01:17:30
So call it P B. It's a blowup of projective space PS. And what we do is we blow up, following
01:17:40
an absolutely standard procedure, along all the Li where I is in our set. So B stands for bad loci, or loci bad, or blowup, are the things you want to blow up. And you do it in the following way.
01:18:03
So first of all, we blow up the points. So the linear subspaces, which are in the set to be blown up,
01:18:21
such that the dimension of that space is 0. So we blow up first all the points. And then after that, we then blow up in the new space, in the new blown up space.
01:18:40
The new space consists of one element, a whole element, one element. Yeah, yeah, yeah. So the number of elements is the number of elements of S minus 1. Yeah, so this is S minus I equals 1.
01:19:04
And then we blow up the strict transforms. Sorry? It takes all such I. Oh, then we skip this step. We move straight onto the dimension 1. So we blow up. So this could be an empty set.
01:19:21
Then blow up all strict transforms of the Li, I and B, such that the dimension of Li equals 1. So I don't know, think of these as the lines, the P1s. And you keep going. So keep blowing up in order of the dimension 2 and so on.
01:19:47
What makes the strict transforms? Oh, so when you blow up, like here, so when you blow up, you can look at the complement in projective space
01:20:02
of all the bad points. It's an open subset. In that open subset, you have some divisor like this, this x psi g. There's a map that way. Look at its inverse image on that open space. It's just this part here. I think it's a risky closure. So it's exactly this part here.
01:20:23
The full inverse image also involves these red exceptional divisors. So the part up here that maps down to here is all the red stuff. The strict transform is the red stuff that does not involve the exceptional divisors. So it's the bit that's strict. I don't know.
01:20:42
So the total transform is everything, including the exceptional divisors. The blow stops here, this divisor after the blow. So you direct the blow to the small projective project, right? Absolutely. And the first point is that it's well-defined.
01:21:01
The key point is because of this condition star, at the k-th stage, the strict transforms of Li1 Li2
01:21:23
for dim Li1. So there are two things we want to blow up. And because we've already blown up their intersection by the condition star, they are disjoint.
01:21:42
So they're far apart from each other, and it does not depend on the order. So as long as you blow up in the strict order of dimension, it doesn't matter in each step which order you choose. So that's a very well-known fact.
01:22:03
So PB has a stratification. I have to accelerate a little bit. So it's codimension one strata. So it's the red stuff in that picture.
01:22:20
Codimension one strata are following.
01:22:41
They are the di, which are the, sorry, it's not the red picture, sorry. My colors are mixed up. I take that back. Now the stratification of the di, which is the total inverse image of the original coordinate
01:23:07
hyperplanes, Li. So that's what the sides of this triangle we started off with. And then the di, we also have new sides, new facets,
01:23:20
which are the total inverse image of the Li corresponding to everything that was blown up. So not, yeah, these are divisors.
01:23:41
No, sorry, this is great. I big in the one. And there's a product structure. So this divisor, di, is isomorphic to P, a space of the same type, which
01:24:07
is a blow up of Pi cross Ps minus i, where Bi, B upper i,
01:24:23
is the set of subsets in B, which were contained in i. And B lower i is the set J minus i, where J is in B, and G contains i.
01:24:44
So it has this nice recursive structure that every divisor is a product of spaces of the same type. I'll rub this out.
01:25:04
Yeah, I was going to talk about operands, but I'm out of time. So I'll just talk about hopfars instead. And then the di that often gets neglected in this business is isomorphic to a point cross P, B subscript i.
01:25:22
And that's a blow up of Ps minus i. So it has this nice recursive structure. And of course, the union of these divisors, of all the d's, is strict normal crossing.
01:25:44
OK, so let's apply this for Feynman graph definition. So this is why I mentioned this example last time about these moduli spaces, because they have this very nice operatic structure.
01:26:00
But I actually forgot to mention this product structure, which can be thought of as particles going in different directions. That was an omission last time. In any case, it doesn't matter, because if we take G Feynman graph in general, we get the same structure,
01:26:23
because we can set P G to be the projective space blown up in all the motic subgraphs.
01:26:46
And there's this condition of being closed on the unions, and that was a property that was satisfied by motic subgraphs.
01:27:04
So actually, what I should have done here was to draw these circles in white. So, so far, we've been discussing the whites, but we've defined this blow up.
01:27:21
It's got nothing to do with the red line yet. It's just the white stuff. So there's this recursive structure in the white divisors, which are just related to projective space, and have nothing to do with graph hyper surfaces for now.
01:27:47
OK, so now, let y psi G and y psi G QM be contained in this
01:28:10
compactified space for a graph to be the strict transforms.
01:28:20
So in this picture, it's the red line now, now that I've changed the colors, of, respectively, the graph hyper surface, psi G, and the graph hyper surface. So it's what remains upstairs. And the theorem, which is absolutely key to everything,
01:28:44
is the following. So first of all, this space PG is well-defined, and that was the property that motic subgraphs were closed under unions. It has this stratification by divisors.
01:29:06
So what are they? They are of two types. They are the subgraphs, which are motic, with at least two edges.
01:29:21
And they are just the edge divisors. That's just the edges of this triangle, in this case. And so again, we had this recursive structure in the general setting. So d gamma, the facet corresponding to gamma,
01:29:44
is itself a product of the same structure for the subgraph and quotient graph. And that uses the quotients and the extension property of motic subgraphs. And the edge divisor is a point cross PG double slash E.
01:30:06
And that uses the contraction property of motic subgraphs. And now we want to use the factorization formulae. So why the graph hypersurface, when
01:30:20
it intersects with a coordinate hyperplane, or this version upstairs, then by the contraction deletion formulae, that's exactly the same situation but for the contracted graph. So that was contraction deletion. Then the factorization properties
01:30:43
said that when you intersect this graph hypersurface with one of these facets, you get two irreducible components, which have this nice product structure.
01:31:10
And likewise for the other hypersurface, and when we intersect with one of the blown-up facets,
01:31:25
we get, sorry, one of the blown-up facets, we get the other graph hypersurface corresponding to the first semantic.
01:31:50
So this is if gamma is not mm, so it's a pure ultraviolet subdivergence, or potentially ultraviolet subdivergence.
01:32:04
And in the case when it is mm, we get this. And so if gamma is, so this is an infrared thing.
01:32:25
OK, so as Pierre said, essentially this defines a sort of operand structure on this geometry. I'm in the generic situation. I'm always in the generic situation.
01:32:40
So this is a family, and we blow up the fibers, and it will be nicely behaved on some big open set. How does this hypersurface work? No, no, so that's the point. They're absolutely highly singular. They're extraordinarily singular. And it's good that they should be singular, because otherwise,
01:33:02
I should say, these are y psi g and y psi g are very singular, and we don't understand the singularities in general. And in fact, they need to be singular, because if they were smooth, the Hodge numbers would be all wrong,
01:33:22
and we wouldn't get multiple zeta values. So because we see multiple zeta values, we should expect these to be very singular. And you can also, in the setting, you can ask what is the discriminant.
01:33:41
And again, it's something fiendishly complicated, and it's related to the study of Landau singularities in physics. And it's a tricky business. What's the statement about divisors? So there's divisors of two types. There are six divisors.
01:34:02
There are the ones corresponding to each edge. So this is edge two. And so I should draw the quotient graph here. So I contract the edge two, and I get the quotient here. Here I contract the edge three, and I get this graph.
01:34:21
And here I contract the edge four. And then there are other divisors which come from blowing up. And they corresponded to subgraphs. So here we get the subgraph was three, four. And I will write it three, four tensor two.
01:34:42
And it will be a product of the corresponding spaces of the corresponding graph. So this one is two, three tensor four. That's how we label. So the D is the set of all these white divisors. There are six of them here, and they fall into two sets. Those that corresponded just to edges,
01:35:02
so contracting an edge in the graph, and those which we actually blew up. And they correspond to the modic subgraphs. OK, so very quickly, the idea of the proof of this is rather simple.
01:35:20
Let's just look, for example, at the local coordinates for a single blowup, Li, where for convenience, i is just the set one up to i in P n minus one.
01:35:44
Then the coordinates for a single blowup on some affine chart are given by just changing variables. So as I explained last time, we can blow up and see what happens locally by passing to new variables.
01:36:09
And what we do is we write psi g, let's say, for example, equals psi i psi g mod i plus r.
01:36:22
Take the factorization formula, and you write it in the new coordinates. So write these in the new coordinates. So that's how you compute the strict transform. And then you take the limit.
01:36:41
Well, you don't need to take the limit. You just get rid of the exceptional divisor. Something will factor off. No, actually, in the limit, you will get the r will go to 0 because it's of higher degree in the first variables. And it will become psi i psi g mod i.
01:37:04
And so the zero locus, so that the intersection of the strict transform with this corresponding facet will have this equation. And the zeros of this equation is a union of two pieces,
01:37:21
and it's exactly of this type. So that's the essential reason why this works. It follows from the factorization formulae. And for physicists acquainted with the old literature, they should recognize these types of ideas in a different language were used by Hepp.
01:37:42
So I think, if I understand correctly, he blew up everything. But there's subsequent work by Speer, where he defined a sort of minimal set of coordinates to understand the singularities of these polynomials. And I suspect that it's very similar to the story
01:38:01
I'm telling here. So remark, in the case where g has no masses or momenta
01:38:23
and primitive, and log divergent, then this is the same space. Pg is the same space as defined by Bloch and Onkreimer.
01:38:43
So they did this first in this particular case. There's another case that I did with Onkreimer for something called single-scale graphs in order to understand renormalization
01:39:01
in the context of algebraic geometry. And actually, it's slightly different from this one. It's a different compactification in algebraic interpretation of renormalization. But I presume one can renormalize directly
01:39:27
in this geometry by jazzing it up a bit. OK, so here's the example of this situation.
01:39:42
This is the graph. This is the list of emotic subgraphs. And so it tells us we should blow up first the points, which are the subvarieties defined by these. So we blow up the points. First of all, the points L123, L124, and L134 in any order.
01:40:12
And then secondly, we should blow up the line L34.
01:40:20
And this subgraph plays no role. So here's a picture of the corner hyperplanes in P3. The three bad points are red dots here. We blow them up. They're disjoint, so it doesn't matter which order we do it.
01:40:41
You can think of a blow-up as simply taking a piece of cheese and cutting off the corners. So after blowing up, the piece of cheese will look like this. That's the first step where we've blown up these three red corners. And then there remains to blow up this line, which is the red line here.
01:41:00
And then the second step, we blow up that. So we cut this edge off the cheese. And we get a figure, a polytope of this shape. And you can check that. So it's a picture of this space PG. And it has facets in one-to-one correspondence
01:41:22
with either edges of the graph or motif subgraphs. No, no, no. No. It is not. It has triangles.
01:41:53
OK, so what happened to the coordinate simplex? We had sigma in P and G minus 1 R
01:42:05
was the coordinate simplex. And so it's just the locus where all the projective coordinates are real and positive.
01:42:21
So it's the tetrahedron on the left in that figure. And we define the Feynman polytope, for want of a better name, Feynman polytope,
01:42:42
which is sigma tilde G, which is the inverse image of sigma. And it's contained in the real points of this blown up space. And so pi G is the map from PG to P and G minus 1.
01:43:05
So sigma tilde is this polytope. And the polytope is a nice way to visualize all the geometry that's going on in the Feynman integral and hence the motif.
01:43:23
So the facets of sigma tilde G are in one-to-one correspondence with either the edges E of the original graph. When you write pi G inverse of sigma, do you mean you take the inside of sigma,
01:43:40
you take the inverse image, and then you take the root of it? Yes, that's what I mean. Yeah, do you want me to write? Apology diagram from Grotius. Yeah, I'll put analytic, analytic topology. Yeah, sorry, I've been sloppy, yeah, sigma. That's the interior, and that's, yeah.
01:44:05
So the facets are the edges of the graph and the motif subgraphs. And of course, it has this nice structure.
01:44:22
So sigma G intersected with D gamma is isomorphic to products of Feynman polytopes. We have this recursive structure.
01:44:40
And so from this, we get a Hopf algebra. So I think the correct concept here is that of an operad, but I'm slightly out of time, so it's quicker to define the Hopf algebra.
01:45:01
So let F be the ring generated, the vectors, the free Z module generated by disjoint unions of Feynman graphs.
01:45:27
So it's graded. So normally, when you do Hopf algebra of graphs, you grade with respect to the loop number. That's possible, but I prefer to grade
01:45:40
by the number of edges for reasons that will become clear. And this is an algebra where the multiplication is disjoint union of graphs. You just put them side by side. I should perhaps say that all my graphs are always labeled.
01:46:04
The edges are labeled. And on this, we have a? Absolutely. So we have a co-product following Kahn and Kaimer,
01:46:22
which, to a connected graph, associates the linear combination gamma tensor G mod gamma, or gamma-motic subgraph in G. So this is much more general than the Kahn-Kramer-Hopf algebra because it contains the infrared. So my understanding is that the Kahn-Kramer-Hopf algebra
01:46:43
sees the ultraviolet information. And maybe I'm ignorant of the literature, but it seems to me that the infrared case hasn't been considered. This contains all the infrared. Gamma is motic. Sorry? Gamma is motic.
01:47:00
Yeah. Which is a variance on the common kaimer. Yeah, so in the special case. So a special case with no masses and no momenta, just ultraviolet, then we get this
01:47:22
is called the core Hopf algebra, defined by Bloch and Kaimer, and is, in some sense, the limit of the Kahn-Kramer-Hopf algebra is in all, for all dimensions.
01:47:41
And then on this, we have a differential. So the fact that this is co-associative follows from the properties of quotients and extensions of the motic graphs are stable on the quotients and extensions.
01:48:02
We also have a differential for every edge. We have a differential f to f, which sends g to g contracted e. And then we can define d to be the sum following
01:48:22
the maximum minus 1 to the i dEi. And then the result of the corollary is that f is a differential graded Hopf algebra.
01:48:41
I wouldn't actually need this, but it's just enumerated. Enumerates and ordering, yeah. Yeah, yeah. Otherwise, you have irritating. You have to worry whether this is plus or minus. Because when I apply, the enumeration doesn't work here.
01:49:01
No, so I just take a disjoint union of graphs which have ordered edges. And then the Leibniz rule tells me how to differentiate a product. So I just start with the lowest edge in each component. There are different ways to get around this. That's one way to do it. So what that means is that one delta is co-associative,
01:49:24
which is the key fact for the renormalization group equation. It's not co-commutative. No, no, no, it's not. So this is commutative, but absolutely not co-commutative.
01:49:44
Yeah, so it's a function on some groups. So as Dirk explained to us, the Hopf algebra of graphs gives you the theoretical explanation for the first formula and all the recursive structure renormalization and the renormalization group equation.
01:50:04
But bizarrely, this holds in the infrared case as well. And I don't know what that means. The differential structure is this equation. And of course, these differentials,
01:50:21
edge contractions commute. So the proof, actually, what's nice about this geometric picture is the proof of this is completely obvious if you have in your mind this polytope. Because if we think of this as a simplicial complex,
01:50:42
there's a map. There's the boundary map. So the interior, the interior stratum, is indexed by the graph G. And we want to take the boundary of this polytope. So the boundary of the polytope is the sum of all the facets,
01:51:00
the facets being, so the facets of two type, either they're the edges, and that gives us the differential, or they are of the blown up facets that correspond to multi-subgraphs. I'm slightly annoyed with this labeling. So a finite graph for you has labeled edges?
01:51:23
Yes. So for example, if you have various labeling of the same graph, you take it many times, you mean? Yes. Yeah. It's heavy-handed, but yeah. And so the remark is that the boundary operator,
01:51:41
D squared, equals 0. So if you take the boundary of the boundary, you get 0. So this statement, the fact that you've got a differential graded Hopf algebra, is just this fact. And in the handout, I explained how to see that. We can look, for example, at the two skeleton of this polytope.
01:52:01
We look at this edge here, and this edge can be viewed either in the boundary of this facet or in the boundary of that facet, and it gives the same edge. And that's exactly the co-associativity. So once you have this geometric picture, this statement is completely trivial. So proof, look at the co-dimension two skeleton
01:52:30
of sigma tilde G. And so next time, we want to study the graph motive, so Mg,
01:52:46
which will be a relative cohomology, Pg minus Yg, relative to D minus D intersection Yg.
01:53:04
So here, Yg is the union of these. In the generic case, we'll need both hypersurfaces. And so the first thing to do next time is to define a Betty
01:53:21
homology class, which will be the sigma tilde, the Feynman polytope, and from that, define empathetic period. And so then we can then speak about the motivic period associated to Feynman amplitude. And the main theorem, which says that most of the conjugates of the same type
01:53:43
will follow from some spectral sequence argument applied to the geometry of this. Will you see your period inside of the big category that you called H? So I only had time to define the category H, which is when you think of a period as a number
01:54:01
in the classical sense. So here, what you need to do is you need a good notion of a period function, a family of periods. And you need a Tanakian category of not mixed-hot structures, but variations of mixed-hot structures. So the Betty will be the category
01:54:22
of local systems of finite dimensional q vector spaces, equipped with an increasing filtration. The Duran will be integrable algebraic vector bundles with regular singularities at infinity, equipped with an increasing filtration and a decreasing Hodge filtration, satisfying Griffith's transversality. The comparison will be given by the Riemann-Hilbert correspondence.
01:54:41
And you can set up, this is a Tanakian category, and you can set up a notion of motivic periods in families in exactly the same way. One knows that this Tanakian category goes to it by a fully faceful function. So it goes to what? Oh, no, so if you specialize at a point, no, you can't take limits.
01:55:01
So I don't understand what happens. If a momentum goes to 0, the cohomology can jump. So I don't know how to do that. That's something that someone needs to look at. That could be very complicated. But over the generic point, yes, superficially, the formula for the motivic correction
01:55:21
will be the same, though you're in a different ophthalmology person. So you could have something like the motivic logarithm that you view as a function. And then you look at the motivic logarithm at a point. And at this particular rational point, it could have some special functional equations that aren't true in general.
01:55:41
So specialization is a tricky business. But with generic, the idea is that we'll have a space of, we look at all Feynman diagrams with a certain number of incoming momenta and a certain fixed set of masses. That's a space of kinematics. And on some huge open set in this,
01:56:00
we have a Tanachian category of variations of mixed-hot structure. And every graph with that kinematics defines a motivic period on this Tanachian category. And so we have a cosmic Galar group acting on all those Feynman diagrams. And it works fine. But I won't talk about that because it's not enough time.
Recommendations
Series of 4 media