Extended HFTs in dimension 2
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Correlation and dependenceManifoldHomotopieAlgebraKörper <Algebra>Category of beingSymmetric matrixCircleMorphismusPoint (geometry)Object (grammar)Social classExtension (kinesiology)Element (mathematics)Group actionBoundary value problemFlow separationTheoryTerm (mathematics)Invariant (mathematics)Frobenius methodProduct (business)Equivalence relationEnergy levelTheoremSurfaceTwo-dimensional spaceSpacetimeResultantMonoidModel theoryGoodness of fitFeldtheorieLatent heatTowerConjugacy classAutomorphismAlgebra over a fieldVertex (graph theory)Module (mathematics)Square numberMass flow rateCW-KomplexDivisorHydraulic motorComplex (psychology)FunktorIdentical particlesTopologischer KörperMultiplicationChaos (cosmogony)1 (number)Table (information)DiagramCoordinate systemDimensional analysisCausalityShift operatorLecture/Conference
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Category of beingMorphismusAlgebraObject (grammar)Many-sorted logicLattice (group)MonoidObservational study3 (number)Module (mathematics)Equivalence relationRight angleFrobenius methodIdentical particlesConnectivity (graph theory)SkalarproduktraumMultiplicationAnglePrincipal component analysisSymmetric matrixTransformation (genetics)Devolution (biology)Lecture/Conference
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Fiber bundleConnectivity (graph theory)AnglePrincipal component analysisCoordinate systemCircleLecture/Conference
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Point (geometry)LengthProof theoryConnectivity (graph theory)Correspondence (mathematics)Principal idealPairwise comparisonAlgebraic structureMereologyCategory of beingInfinityGradientAlgebraObject (grammar)MonoidVertex (graph theory)HomotopieCommutatorUnendliche GruppeArithmetic meanPrincipal component analysisEquivalence relationFunktorSpacetimeManifoldGroup actionHypothesisFiber bundleExtension (kinesiology)Condition numberModule (mathematics)ResultantGamma functionState of matterSymmetric matrixWater vaporComputabilityLine (geometry)Körper <Algebra>Model theoryGroupoid1 (number)Glattheit <Mathematik>StatuteStructural loadLecture/Conference
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DiagramTheory of relativityCycle (graph theory)Projective planeGroup actionCovering spaceFeldtheorieSimilarity (geometry)TheoremTheoryEquivalence relationObject (grammar)Sigma-algebraTwo-dimensional spaceCategory of beingSpacetimePresentation of a groupManifoldHypothesisFunctional (mathematics)Proof theoryMathematicsSurgeryOrientation (vector space)MorphismusSurfaceKörper <Algebra>Social classGenerating set of a groupModel theoryDifferent (Kate Ryan album)Curve fittingDimensional analysisFunktorHelmholtz decompositionHomotopieLinear algebraMonoidMorse-TheorieSymmetric matrixRing (mathematics)Planar graphComplete metric spaceDiffeomorphismSet theoryEqualiser (mathematics)Fiber bundleResultantLecture/Conference
30:06
Musical ensemble
Transcript: English(auto-generated)
00:16
First of all, I would like to thank the organizers for giving me a chance to talk in this conference.
00:22
I'm going to talk about two-dimensional extent homotopy field theories. More precisely, I will tell what they are and let me know about their classification. Let me start with two-dimensional topological field theories. If you remember Claudia's talk, she mentioned two ways of studying topological field theories.
00:43
One is the functorial field theories and the second one is factorization algebras. This is completely just a functorial study of topological field theories. Let me remind you what is a two-dimensional topological field theory. It's just a symmetric monoidal functor from the coborism category to any other symmetric monoidal category.
01:05
Let's take this to be a category of vector spaces and linear transformations. What is an object of this category? It's just oriented circles. And morphisms are just oriented. They form some class of oriented cobortisms between circles.
01:20
An example of a cobortism or a morphism in this category is something like this. The first thing to realize is that this category is not hard to understand. Basically, objects are just a disjoint union of circles.
01:43
And the cobortisms are just like this. And just writing, it is not very hard to write generators and relations for the cobortisms of this category. And then, under disjoint union, it is really well understood and they are classified by commutative frobenius algebras.
02:04
Like 2-D TFTs, Abram-Kolosvaitin-96 by commutative frobenius algebras.
02:24
And we want to understand homotopy field theories, which are introduced by Triab in 1999, which are basically just applied to axioms of topological field theory to manifolds and cobortisms which are equipped with maps to some fixed target space.
02:44
Therefore, to define HFT, homotopy field theory, we need to fix a target space. We call it X.
03:01
And for this talk, it will always be a KG1 space, which is the homotopy type of KG1 space. Basically, we want to understand those two-dimensional TFTs with principal G bundles for discrete group G. And how do we do this? We just take homotopy class of maps to this KG1 space, which is a pointed space.
03:29
And now, we define this new category, X cop 2, whose objects are pointed oriented circles, with homotopy class of pointed maps to this space.
03:46
And morphisms are cobordisms equipped with homotopy class of maps to this pointed homotopy class of maps to this target space. And we can think of just taking the universal cover of this by pulling back along this homotopy class,
04:03
we have a principal G bundle over this thing. And try to classify such TFTs by crossed Frobenius algebras. 2D HFTs with target X, which is a KG1 space, by crossed Frobenius algebras.
04:33
What is crossed Frobenius G algebras? What is crossed? Crossed replaces the commutativity. What do I mean?
04:40
We have a G algebra. When we take two elements, AB is not necessarily BA, unless G is commutative. But if G is not commutative, we have an automorphism of algebra, which replaces commutativity.
05:05
This is the cross structure. So it's a G-graded... Yes, it's a G-graded algebra with inner product, non-degenerative product. And there's an automorphism of where rho is here. Rho is an automorphism of A.
05:24
And it's like conjugation type. We also have extended field theories, which are semitic monoidal two-functors,
05:46
two-dimensional extended field theories, semitic monoidal two-functors from extended boardism category to some semitic monoidal bi-category. Let's take the bi-category of algebras, bimodules, and bimodule maps.
06:06
Christiaan Rupis classified them in terms of separable symmetric Frobenius algebras. And he introduced this bi-category. Let me tell what it is. Objects of morphisms into morphisms.
06:27
What are the objects of this board? They're just oriented points, like this. And objects of this bi-category are just K-algebras. And one morphism are just boardisms between points, like this.
06:55
And here, one morphism are bimodules.
07:02
And here, two morphisms are a certain type of surfaces with faces. What do I mean by certain type? I'll make it more precise.
07:22
What I mean by certain type is this vertical boundaries are always trivial, or product type. There is no vertical boundary like this. They're always product of the zero manifolds and boundaries at the corners. And here, two morphisms are bimodule maps.
07:50
And Christiaan Rupis classified symmetric monoidal 2-functors in terms of specific K-algebras, which are separable symmetric Frobenius algebras.
08:01
I'll write it here. To the extended TFTs, separable symmetric Frobenius algebras.
08:32
What does this line segment mean? It's a natural question. This term is also called 0-1-2 theories.
08:42
The data that this gives us is assigned some algebraic data to points, are morphisms and boardrooms, and boardrooms between boardisms. And what this line segment means is this. We can think of HFTs as generalization of this, because for each homotopic field theory, by just taking trivial homotopic classes,
09:03
we just get a topological field theory. And similarly, in a similar way, if you start with an extended field theory, and we restrict ourselves to just forget the points, but circles and boardrooms between circles, we again get non-extended TFT.
09:25
Once we have this diagram, it's an interesting question, I believe it's to understand what are two-dimensional extended HFTs. The goal of this talk is to define these things, these gadgets,
09:45
and classify what they are. Any questions so far? Can you vary the target category at all, or will you stick to that? No, to define HFT, we just fixed point to target. I mean, but you could change it to something else.
10:02
Yes. There is a Christian properties, has a classification for any symmetric model by category. We have a similar... It is not as clear as this case, but it is given by some data in this category, symmetric model by category, and we have a similar result generalizing this thing.
10:24
And definition of two-dimensional extended homotopic field theory is fairly straightforward. We do the same thing as we did here. What we do, we fix this KG1 space, point it, and consider point to homotopic classes.
10:48
So what do we have? Since these are all point to homotopic classes, we can just associate, we can canonically assign some group elements to these things.
11:03
But we want specifically for two morphisms to be these vertical faces, boundaries to be identity. E is identity. Yes. We always want X to be KG1.
11:23
HFT's TRIV defines for arbitrary CW complex, but for general arbitrary CW complex, there is not a classification theorem. Once we take KG as a KG1 space, we have a good classification theorem.
11:42
And that is where this homotopy name comes from. When we take space X as any arbitrary CW complex, we just consider maps of these things, not homotopic class of maps. But once we take KG1, there is no higher K invariance. Maps can be replaced by homotopic class of pointed maps.
12:06
This is the definition of, we define this symmetric model by category of this type. Could you replace X instead of by just a KG1, by a KG1 in a cosmological tower to level 2 or something?
12:21
There are works, not in this way, but there are works for such things like KG2 in non-extended case. But in this case, we just stick to the KG1. Yes, this is the definition of two-dimensional extended homotopic field theory.
12:41
And now, what is the classification? The theorem is this. There is an equivalence of bi-categories.
13:07
Let me tell what these bi-categories are.
13:39
Objects of this bi-category are just extended HFTs.
13:46
One morphism are just symmetric monoidal transformations between them. And two morphisms are symmetric monoidal modifications.
14:02
And these bi-category objects are what we call quasi-bi-angular G-algebras. And one morphism are compatible G-graded monoidal contexts.
14:34
And two morphisms are equivalences of G-graded monoidal contexts.
14:53
Let me tell you what is this classifying object, quasi-bi-angular G-algebra. Its name is a bit strange, but it is just some sort of Frobenius G-algebra.
15:05
We call this quasi-bi-angular algebra because it generalizes bi-angular algebras introduced by Triad in the study of lattice HFTs. Their definition is just not that complicated.
15:20
Quasi-bi-angular G-algebra is Frobenius G-algebra. That means a G-algebra with a non-degenerate inner product,
15:42
such that the principal component, the identity component, AE, is separable. And each component, AG, is both left and right rank-1 AE modules.
16:11
In short, in here we have quasi-bi-angular algebras, G-algebras.
16:22
What they are, they are basically just a copy of this for each component. And here, again, we have to go here and go here. Here we just restrict to those maps with trivial maps. It gives here, which just restricts to this principal component.
16:42
And here we just restrict to just circles and keyboards in between them. A G-grade mortar context, this is, G-grade mortar context is defined by Boyzan in 1994.
17:05
It's just a, it's a generalization of usual mortar context in the way that modules become G-graded. What does it mean? Basically, G-graded mortar context between G-algebras is a quadruple.
17:59
It's a unit and co-unit of the A-junction.
18:02
They're the usual things by module map.
18:29
And these both are invertible. And this M and N are G-graded modules. Like what do I mean by this?
18:44
Like B-G, X1 M-H, M-G-H, and similarly for N. Does this mean that you have a 2-1 category there? A 2-0 category? Yes. Yes, it is. It is 2-0.
19:06
Yeah, these are all invertible. And the coolness of grade mortar context is just a straightforward thing. If you have another grade mortar context, it's just a map of bimodule maps, which commutes with these things.
19:20
Vertical bimodule maps. Any other questions? The main corollary of, the main corollary is the structured cobord is my part. This is in a very special case.
19:42
The corollary of the bank. When K, let me take this algebra K. When K is algebraically closed field of characteristic 0,
20:08
the SO2-structured cobord is my part. This holds for 2-D SO2-structured K-valued extended TFDs.
20:41
It holds true. The proof of this is basically just a comparison with Orit Dovidovic's homotopy geo-soto fixed point computation,
21:16
where she has just some simple G-graded algebras over this field,
21:24
and each component of the thing is just rank one principal component. Separable algebra is just the same as semisimple, but under this condition Cartesian zero fields.
21:40
Maybe I can remind people what the structured cobord is my part is. Briefly, for a group gamma, gamma-structured cobord is my part. This is due to Lurie. It is stated as follows.
22:02
Semitic monoidal functors to any semitic monoidal n-category. I need to explain what these are.
22:22
This is an equivalence of semitic monoidal infinite n-categories. For any semitic monoidal infinite n-category C, this board n is fully extended boardism category with structure gamma,
22:42
which means we have principal G bundles over all manifolds, principal gamma bundles over manifolds, and these are all TFTs. The infinite n-category of semitic monoidal infinite n-functors. Here we have this full subcategory of infinite n subcategory of C,
23:01
consisting of fully dualizable objects. Once we take the subcategory, we take down the length infinite group point, and we take the homotopy fixed points. In this specific case, we have the correspondence.
23:29
Does Davidovich compute homotopy fixed points and classify such extended HFTs with the structures? Our proof does not use any cobord as my part. We get the same result in this case.
23:42
Not exactly. I can tell a couple of words about the proof of this trick.
24:03
Basically, proof, in a sentence, we just generalize some of the techniques which were introduced in this thesis to these manifolds equipped with maps to this KG1 space. How do they go? This is this way.
24:20
He has a planar decomposition theorem, and we make it into G-planar decomposition theorem. What it is, it is basically replacing those objects with diagrams,
24:44
with some combinatorial data. What I did is just encode this homotopy class of maps, or just these labelings into these diagrams consistently, and to get the G-planar decomposition theorem. What are these diagrams in dimension one?
25:04
It is basically just Morse theory. It's just a diagram consisting of critical values of Morse functions, in this case projection, and some nice open cover. Here we have some G-label,
25:25
and we encode this, adding this G-data in here, with some additional data on this diagram. In dimension two, he stratifies these jet bundles for a surface sigma,
25:42
and he gets the similar data in some diagram in R2. Since we take equal diffeomorphism class of surfaces, this is like two-dimensional Morse theory. But since we have diffeomorphism classes,
26:00
he also stratifies these jet spaces, like two-dimensional Z-theory, how these diagrams are related. Understand? We do similar things in his diagrams. We add some additional data, which just encodes these characteristic maps of the manifolds,
26:26
and we get the G-planar decomposition theorem, and his theorem, and why these are useful, then once we have this G-planar decomposition theorem, we basically replace this category with equivalent category,
26:47
category P-lanar diagrams. This category just consists of diagrams, and this is freely generated. And Schumacher has a coherence theorem for symmetric monoidal two-functors,
27:03
out of freely generated bicatagories. Like any, it says, that means, what I mean by freely generated, there is a list of generators for objects, one-morphisms, two-morphisms, and relations between two-morphisms. And the coherence theorem says,
27:21
symmetric monoidal two-functors out of such freely generated bicatagories precisely determines up the equivalence to where generators go subject to relations. And that gives the classification, exactly what he did. And the additional data that we encode in this diagram does not change anything in these algebraic methods.
27:47
Yeah, that's it. Can you modify your proof to also get like a framed version,
28:02
or an unoriented version or something else? Framed version is done by Piotr, and unoriented, we have the same thing. I have an unoriented case, just generalizing the Schumacher piece unoriented case. Similarly, we have kind of G-stellalogies of this.
28:20
Again, up to G-graded, more to cohesives. More to context. Like a general statement for general? Yeah, we have a result for any target, just as in this, which just consists of some data, which is the image of the generators and relations in here.
28:44
But for any group mapping set of O2 or something? Could you do it on a spin case? A spin case, hopefully next project. There is some work, not complete work by someone else.
29:01
The related two-dimensional spin field theories with homotopy field theories. Yeah, I'll try to do this. There is no work yet for spin case, but yeah, possibly. Do you have a favorite example? Favorite example of G-algebra?
29:22
It's just matrix algebras works for this one. Matrix algebras works, or group rings also work. Or normalized two cosycles works. Other examples. Or just taking anything with this and copies of G copies of this.
29:43
It's not a boring example. The speaker again?