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added at cobordism hypothesis a pointer to
where the case for $(\infty,1)$-categories is spelled out and proven in detail.
At cobordism hypothesis the section titled For noncompact cobordisms used to contain nothing but a link to Calabi-Yau object. I have now added a few lines of text at least, trying to convey the rough idea.
added also a section For cobordisms with singuarities (boundaries/branes and defects/domain walls) with just a few lines on the idea for the moment, just enough to highlight theorem 4.3.11.
(Which is easily missed; much of the best magic happens on the last 10 pages of 111…)
What would be the established term for these “diagrams indicating types of singularities” on which the cobordism-with-singularities hypothesis/theorem says that the cobordism-with-singularities $(\infty,n)$-category is freely generated from as a symmetric monoidal $(\infty,n)$-category with all duals?
So I mean for instance the simple diagram
$0 \longrightarrow \ast$indicating a domain wall separating the left phase (“0”) from the right (“\ast”). The archetypical path in a fundamental category crossing a stratum.
May these be called “catastrophe diagrams”? Is there any half-way established term available?
So paths which go up and down through the strata, like we discussed once? I wonder if there’s a term in a paper like Diagrammatics, Singularities, and their Algebraic Interpretations.
Taking a look at the workshop you’re attending, I see Catherine Meusburger is speaking on ’Diagrams for Gray categories with duals’, which is based on Gray categories with duals and their diagrams. Todd gets a mention:
The definition of a diagrammatic calculus for Gray categories follows the pattern for categories and 2-categories. The diagrams are a three-dimensional generalisation of the two-dimensional diagrams defined above, and were previously studied informally by Trimble [31].
added to cobordism hypothesis in the section on the framed version a brief paragraph Implications – The canonical O(n)-action on fully dualizable objects
(this statement used to be referred to further below in the entry, but wasn’t actually stated)
Added corresponding cross-pointers to dual object ($n = 1$) and to orthogonal spectrum ($n = \infty$). Somebody needs to create an entry for Serre automorphism ($n =2$).
How does that result fit with the $(\mathbb{Z}_2)^n$ automorphism $\infty$-group of $(\infty, n)$-Cat? What are the fully dualizable objects of the latter?
On the other hand, what would happen if we opted instead for a profunctor or span-like approach? Objects there are generally fully dualizable?
I see just out there is Rune Haugseng’s Iterated spans and “classical” topological field theories. Urs and Joost get cited.
In §6 we then prove that $Span_n(C)$ is symmetric monoidal and that all its objects are fully dualizable
Oh, but
Conjecture 1.3 (Lurie). The $O(k)$-action on the underlying ∞-groupoid of $Span_k(C)$ is trivial, for all ∞-categories $C$ with finite limits.
Anyway, I’m more interested with my first question.
Christopher Schommer-Pries has some useful notes – Dualizability in Low-Dimensional Higher Category Theory.
I guess it’ll only be the terminal object which is dualizable.
Thanks for highlighting Haugseng’s article! Would have missed that otherwise.
Have added pointers to his results to (infinity,n)-category of correspondences and elsewhere.
added a tad more on the definition of cobordisms with (X,xi)-structure and then added in particular the proof idea of how the cobordism hypothesis for $(X,\xi)$-structure follows from the framed case
added now also the proof of 2.4.26 from the $(X,\xi)$-version, i.e. the reduction to the special case that $X = B G$. This is of course just a straightforward corollary, but I have added a line discussing how the result is really the correct concept of homotopy invariants in the sense defined/discussed at infinity-action.
added now also the proof of 2.4.26 from the $(X,\xi)$-version, i.e. the reduction to the special case that $X = B G$. This is of course just a straightforward corollary, but I have added a line discussing how the result is really the correct concept of homotopy invariants in the sense defined/discussed at infinity-action.
What’s the status of the generalized tangle hypothesis?
This appears as theorem 4.4.4 in Lurie’s writeup.
Started a section on the case of (un-)oriented field theories.
After recalling some of the statements from Lurie’s article, I am after making explicit the following corollary. While being a simple corollary, this way of stating it explicitly is immensely useful for the study of unoriented local prequantum field theories. I wonder if this has been made explicit “in print” elsewhere before:
Let $Phases^\otimes \in Ab_\infty(\mathbf{H})$ be an abelian ∞-group object, regarded as a (∞,n)-category with duals internal to $\mathbf{H}$.
At least if $\mathbf{H} =$ ∞Grpd, then local unoriented-topological field theories of the form
$Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}$are equivalent to a choice
of $X \in \mathbf{H}$ equipped with an $O(n)$-∞-action
a homomorphism of $O(n)$-∞-actions $L \colon X \to Phases$ (where $Phases^\otimes$ is equipped with the canonical $\infty$-action induced from the framed cobordism hypothesis), hence to morphisms
started disucssion of some simple but interesting examples at local prequantum field theory – Higher CS theory – Levels.
But I am being interrupted now…
I followed up that proposition Exchanging fields for structure with a remark amplifying its relevance/meaning.
Given a “structure”, i.e. an $(X,\zeta)$-structure in the terminology of Lurie’s writeup, and hence given $Bord_n^{(X,\zeta)}$, what is actually a direct way (i.e. not via the full cobordism hypothesis) to define the “$(X,\zeta)$-diffemorphism group” of an $n$-dimensional manifold $\Sigma$, i.e.
$\Pi(Diff_{(X,\zeta)}(\Sigma)) \coloneqq \Omega^n_{\Sigma} Bord_n^{(X,\zeta)}$?
(Notice: no geometric realization on the right.)
I think I know what it is, but I am a little vague on how to formally derive this from the “definition” of $Bord_n^{(X,\zeta)}$.
I think the right answer is to form the homotopy pullback along the canonical map
$Diff(\Sigma) \longrightarrow \mathbf{Aut}_{/BO(n)}(\Sigma)$of automorphisms in the “slice of the slice” over the classifying map $X \to BO(n)$ of $\zeta$.
I have spelled this out now as def. 3.2.9 on page 34 at Local prequantum field theory (schreiber). After that definition there are spelled out proofs that with this defintion we do get the expected higher extensions of $B Diff(\Sigma)$.
So this looks right. But if anyone cares to give me a sanity check, that would be appreciated.
I have forwarded that question to MO
Shouldn’t the statement of the cobordism theorem either assume that $\mathcal{C}$ has duals, or alternatively have the map $\mathrm{pt}^*$ land in the core of $\mathcal{C}^\mathrm{fd}$ rather than in the core of $\mathcal{C}$ itself?
(I would have fixed this myself if I was absolutely sure about it.)
Woops. Yes, you are absolutely right. I have added the qualifier “with duals” at the beginning of the Statement.
But the entry would deserve some further polishing. If you feel energetic about this at the moment, you should edit it.
Added the reference
This claims to have the proof modulo conjecture 1.2, which is to appear shortly. Has it appeared?
Is there as yet a published complete proof of the cobordism hypothesis. The page says
This is almost complete, except for one step that is not discussed in detail. But a new (unpublished) result by Søren Galatius bridges that step in particular and drastically simplifies the whole proof in general.
Do we know if the status has changed?
Yes, it’s time this was sorted. Lurie having proved it but then not supplying all details I’m sure killed off other people’s incentive to work on it, since the credit had already been claimed.
Last I checked, the proof of that conjecture by Ayala-Francis has not appeared. That was a few months back and I should check again. But clearly there is no evident public announcement of the proof.
On the culture of announcing conjectures in homotopy theory as theorems, see also Clark Barwicks’s The future of homotopy theory (pdf)
The politically efficient way to proceed is shown by number theory: Huge excitement built up by conjectures, even if their actual content doesn’t mean much to most researchers.
added pointer to today’s
By the way, the following might be interesting to compare to:
The FRS-theorem on rational 2d CFT says essentially that a rational 2d CFT is equivalently
local geometric data encoded by any of chiral algebra, vertex operator algebras or whatever one uses;
global topological data encoded, holographically, by Reshetikhin-Turaev-style 3d TQFT but with coefficients not in the category of vector space but in the modular tensor category of representations of the chiral/vertex local data.
This is somewhat reminiscent of Theorem 1.0.4 on p. 3 of Grady & Pavlov 2022, where the right hand side decomposes the geometric FQFT into local geometric data ($\mathcal{S}$) with coefficients in purely global topological data ($\mathcal{C}^\times_d$).
Are there any direct implications for physics that come to mind?
I see you were answering this in #29 as I was asking in #30.
The potential application of all extended functorial QFT to physics is in it being a non-perturbative definition of QFT. For example, one way you might go about claiming the mass gap problem is to produce a 4d extended functorial QFT which locally and perturbatively reduces to QCD or similar, and then to show that this exhibits the “mass gap”.
Accordingly, the definition of EF-QFT shares with that of AQFT the issue that it provides a definition for non-perturbative QFT without however, in itself, getting us any closer to actually constructing any interesting examples.
That being so, classification results like a geometric cobordism hypothesis-theorem serve to at least break down the open problem of constructing examples into small sub-problems, which may be more tractable.
one way you might go about claiming the mass gap problem…
Is there a route from Hypothesis H to an extended TQFT? In Mathematical Foundations of Quantum Field and Perturbative String Theory, you have the sections:
I. Cobordism representations; II. Systems of algebras of observables; III. Quantization from classical field theories,
so I guess I’m wondering about a route from III to I. KK-reductions don’t do this, I take it.
Yes. I have indicated this before, let me say it again:
Hypothesis H says that quantum states/observables are the co/homology of (the loop space of) the (twisted+equivariant+differential-)Cohomotopy cocycle space of some background spacetime super-orbifold (thought of as an asymptotic boundary of a black brane spacetime).
In simple examples corresponding to asymptotic boundaries of codimension 3-branes (MK6s/M5s) intersecting MO9-planes and for trivial twisted+equivariant+differential structure, Segal’s theorem identifies (around p. 15 of arXiv:1912.10425, recalled on p. 18 of our arXiv:2105.02871) this Cohomotopy cocycle space with the configuration space of ordered points in the 3-space transversal to these branes, and hence in this situation the states/observables are the (co)homology of configuration spaces of points in Euclidean space.
It is “well known” (to those who know it well), that the (co)homology of such configuration spaces of points is an alternative way of encoding quantum field theory (see at correlator as differential form on configuration space of points). Here the cohomology in degree one less than the dimension of the ambient space encodes the usual propagators, while the cohomology in lower degrees encodes higher order observables as expected in extended QFT.
In less simple examples this picture will pick up bells and whistles. For example, if the branes sit at an orbifold singularity, then the Rourke-Sanderson theorem – refining Segal’s theorem to the equivariant case – says that we get the evident equivariant version of the configuration space of points, instead, and the quantum states/observables will be its equivariant cohomology. This is much richer in detail, but the general picture remains the same.
This is something I hope we might look at eventually: There will be a resulting equivariant version of the algebra of horizontal chord diagrams, etc.
Incidentally, just last night we received message that our proposal for an NYUAD Research Institute Center “for Topological and Quantum Systems” has been approved, which, in one of its four sub-clusters, is to be concerned with this and related questions. There will be openings for 6-7 postdoc positions advertized soon, to start by Sept. 2022. I’ll post links as soon as the new Center’s webpage has gone live.
Congratulations on the funding, Urs! That’s great news.
Thanks for the explanation, and well done! If you are ever looking for a philosopher to be briefly in residence,…
and then to show that this exhibits the “mass gap”.
Assuming such a fully extended functorial field theory has been constructed, can we say what it means for such a field theory to have a mass gap, specifically in the language of functorial field theory?
That being so, classification results like a geometric cobordism hypothesis-theorem serve to at least break down the open problem of constructing examples into small sub-problems, which may be more tractable.
We are already working on computing the right side of the GCH in a rather general setting. Remark 3.0.8 gives a glimpse, but really we will be treating almost completely arbitrary targets C, not just B^d(A). This will recover nonabelian differential cohomology with coefficients in the tangent Lie-∞ algebroid of C.
Assuming such a fully extended functorial field theory has been constructed, can we say what it means for such a field theory to have a mass gap, specifically in the language of functorial field theory?
While there are bound to be subtle technical details, the broad idea is simple:
The “masses” exhibited by any $d+1$-dimensional FQFT on a spacetime manifold $\Sigma^d$ without boundary are the elements of the operator spectrum of its Hamiltonian over $\Sigma^d$, which is the derivative with respect to $t$ of the FQFT’s value on the cobordisms of the form $\Sigma^d \xrightarrow{\; \Sigma_d \times [0,t] \;} \Sigma^d$.
Here the geometric structure must be such that one can make sense of $[0,\epsilon]$. The default would be (pseudo-)Riemannian structure, with $[0,\epsilon] \subset \mathbb{R}^1$ regarded as the evident (pseudo-)Rimannian manifold with metric $d s^2 = \pm d t \otimes d t$ and volume $\pm \epsilon$.
A “mass gap” means that this Hamiltonian operator does not have continuous spectrum around 0, but a “spectral gap” between 0 and the next smallest eigenvalue. (Googling for a reference, I see that the first lines of Wikipedia’s Spectral gap (physics) has the right keyword combination. I can try to dig out more authoritative/original references, if desired.)
Thanks, this clarifies it a lot!
What is the value of d for physically realistic cases? You say that Σ is a spacetime, which would seem to imply d=4, but then to evaluate on Σ⨯[0,t] we would need a 5d-FQFT, since dim(Σ⨯[0,t])=d+1=5.
Or perhaps Σ is just a space, and then d=3 and dim(Σ⨯[0,t])=4? The line segment [0,t], is it supposed to be interpreted like time?
Oh, I see, I misspoke, right: $\Sigma^d$ is meant to be space, and then $\Sigma^d \times [0,\epsilon]$ is a slab of spacetime, with $[0,\epsilon] \subset \mathbb{R}^1$ an interval inside “the time axis”. So for the Clay Millennium problem, taken at face value, $d = 3$.
I see, thanks for clarifying this.
Do we know what the classical (or rather prequantum) Yang–Mills theory is supposed to assign to manifolds of positive codimension?
For the classical Chern–Simons theory this is clear: the action on manifolds of codimension 0 is given by the holonomy of the Chern–Simons 2-gerbe, so we now know how to construct a fully extended functorial field theory: the geometric cobordism hypothesis computes the space of FFTs with geometric structure B_∇(G) and target B^3(U(1)) as the derived hom of simplicial presheaves Hom(B_∇(G), B^3_∇(U(1))) (now with ∇ on the right side also), and from the work of Freed–Hopkins we can compute this as the space of invariant polynomials on the Lie algebra of G.
Is there any similar description for the classical Yang–Mills theory? Something in terms of differential cohomology, perhaps?
We have written about this in “The stack of Yang-Mills fields on Lorentzian manifolds” (arXiv:1704.01378).
If you are serious about attacking Yang-Mills via extended FQFT, it might make sense to try to get into contact with Alexander Schenkel, who is running a program on the analogous attack via homotopy AQFT. It should be fruitful to think about the two in parallel.
Re #42: Thanks for the reference!
Did I gather it correctly from your paper that B^4 R is (or would be) a natural target for the 4d prequantum Yang–Mills theory (as a functorial field theory), just like B^3 U(1) is a natural target for the 3d prequantum Chern–Simons theory?
I would say that the $\mathbf{B}^3 U(1)$ in 3d Chern-Simons theory is the coefficients which classify the higher pre-quantum line bundle, the one whose transgression to a spatial slice gives the actual (traditional) pre-quantum line bundle on the space of on-shell fields over that spatial slice.
In this sense any $(d+1)$-dimensional field theory would replace this coefficient by $\mathbf{B}^{d+1} U(1)$. Taking it to be $\mathbf{B}^{d+1}\mathbb{R}$ is to pre-suppose that the pre-quantum line bundles will be topologically trivial. That is unlikely to be the case in full generality but may be reasonable to assume for starters and for non-topological field theories under assumptions that suppress topological effects in favor of the local geometric structure.
In the article “The stack of Yang-Mills fields” we discuss just what would be the base of the pre-quantum line bundle, namely the stack of fields and on-shell fields.
After that I had then started a project with Igor Khavkine on constructing higher prequantum line bundles in generality from Lagrangian data in the guise of “Lepage gerbes” on jet bundles. It looks like we never put our stack of notes online, but there is talk notes (pdf) at Prequantum covariant field theory. This project was interrupted when the gods called me to leave Prague for Abu Dhabi. But I know that Dave Carchedi has been building on this, at least in parts, though I’d have to contact him to know if there is anything in writing.
Re #44:
I see, so would it be correct to say that although we may expect the classical Yang–Mills theory to involve gerbes and similar stuff, this has not yet been figured out in details?
Your project on Lepage gerbes looks incredibly interesting.
Here is my line of thought on constructing quantized FFTs:
Produce the prequantum data (some form of nonabelian differential cohomology) from the classical data.
Convert the prequantum data in (1) to a map like in the right side of the geometric cobordism hypothesis.
Use the geometric cobordism hypothesis to convert (2) to a fully extended functorial field theory.
Integrate the prequantum data in (1) using pushforwards in nonabelian differential cohomology, producing another map like on the right side of the GCH.
Use the GCH to convert (4) to a fully extended functorial field theory, which is the quantization of (3).
Status so far:
(3) and (5) are supplied by the geometric cobordism hypothesis.
(2) is current work in progress (the third paper in the series), and should be out soon.
(4) is planned (the fourth paper in the series), in principle we know what to do.
(1) was missing so far, but it seems like your work with Khavkine provides a complete solution.
Yes, I think the topic is very much open. Mostly because essentially nobody seems to be attacking along that path.
That’s maybe not too surprising, since people tend to work on small steps for which there are existing hints of success, instead of embarking on one long journey through deserts and over mountains, from which one has not yet seen anyone come back.
And this is certainly somewhat puzzling about the heads-on approaches to non-perturbative Yang-Mills via AQFT or FQFT: that there are next to no hints for that or how it will work, when it works. There is no partial result and few heuristic arguments for what it is that will eventually make the zoo of hadron masses come out by these approaches. If and when it eventually works, it is going to be a dramatic success of pure abstract thinking.
That’s why I came to feel that the alternative approach via holographic QCD inside M-theory is more promising: even though it superficially sounds more crazy, there is a wealth of hints for how it makes things work (all those strings are, after all, the original and still the best idea for how confinement works, namely via tensionful color flux tubes) and, more importantly, hints that it works (from the close match of the zoo of predictions/measurements of hadron masses here).
$\,$
By the way, it should be only the moduli stack of fields which is a stack of non-abelian differential cocycles. But the higher pre-quantum gerbe on that stack should be abelian, and the quantization should be by push-forward in some abelian cohomology twisted by that pre-quantum gerbe.
This is, in any case, what happens in familiar examples, notably this is how the quantization of 3d Chern-Simons theory works (geometric quantization by push-forward) where the push-forward is in differental K-theory but computes the quantum states for non-abelian CS theory.
It is this picture of quantization via push-forward in abelian (“linear”) cohomology over non-abelian stacks which I was after in Quantization via Linear homotopy types.
We should add something about Daniel and Dimitri’s paper to the section For cobordisms with geometric structure.
Out of interest, is there a form of generalized tangle hypothesis of which this is the stabilization?
I recall we spoke about such things back at the n-Café here, following a conversation on $n$-categories of tangles as kinds of fundamental $n$-category with duals of stratified spaces. But the question was how to deal with geometric structure not just on the normal bundle.
I admit not to have read the new article beyond the introduction, but without going into details, can one say in a few words what the key new insight is that makes the new proof happen?
Given that a fair bit of high-powered effort by several people had previously been invested into a proof of just the topological case while still leaving gaps, it’s a striking claim that not only this but also a grand generalization now drops out on a few pages. What is the new insight which makes this work and that previous authors had missed?
Re #48:
The locality paper is inseparable from the geometric cobordism hypothesis paper. So it’s not a “few pages”, but 40+41=81 pages (using version 2 of the GCH paper that will be uploaded soon).
For comparison, the sketch of a proof in Lurie’s paper is in Section 3.1 and 3.4 (and some fragments from 3.2, 3.3), which occupy 5+9=14 pages, plus a couple more for the relevant parts of 3.2, 3.3.
Some insights:
The locality property is invoked in the very first step of the proof to reduce to the geometric framed case. (In Lurie’s paper, Remark 2.4.20 instead deduces the locality principle from the cobordism hypothesis. However, very roughly this step corresponds in purpose to Section 3.2 there, which reduces to the case of unoriented manifolds instead of framed, although the actual details are completely different.)
The geometric framed case produces a d-truncated bordism category (before adding thin homotopies) because there are no nontrivial structure-preserving diffeomorphisms of d-dimensional bordisms embedded (or immersed) into R^d. This is used in the proof many times to simplify the arguments. (As far as I can see, there are no analogues in Lurie’s paper.)
The site FEmb_d (which encodes the geometric structures) plays a crucial role. In particular, encoding the homotopical action of O(d) using the site of d-manifolds and open embeddings is crucial for simplifying our proofs. (As far as I can see, there are no analogues in Lurie’s paper.)
Invariance under thin homotopies is encoded using a further localization of simplicial presheaves on FEmb_d. This is new and important for our proofs. In the topological case, this recovers precisely the (∞,d)-category of bordisms of Hopkins–Lurie, as opposed to just the d-category of bordisms. (As far as I can see, there are no analogues in Lurie’s paper.)
The machinery of the locality paper is used to establish the filtrations and pushout squares for the geometric framed case. (Very roughly, corresponds in purpose to Section 3.3 in Lurie’s paper (only the small part that is actually used in 3.4) and Claims 3.4.12 and 3.4.17, which are stated without proof (and without a sketch of a proof). The actual details are completely different, though.)
As an additional remark, all of these insights also apply to the topological cobordism hypothesis, even if we are not interested in the geometric case.
And even with these insights, the proof takes more than 80 pages.
Thanks. So is it right that you are saying the geometric framed case is actually more amenable to direct proof than the topological case, but then implies it?
I have a vague memory that the remaining gap in the existing proof had to do with showing (or showing convincingly) that some space of Morse functions/handle decompositions is contractible or something like this. Is this an issue you solve or circumvent?
So is it right that you are saying the geometric framed case is actually more amenable to direct proof than the topological case, but then implies it?
I would say that the geometric case inspired ways of thinking that we may not have encountered otherwise. For example, the necessity of incorporating the site Cart in the picture also led to consider the site FEmb_d as a natural extension. If we stayed in the purely topological case, then we could have tried to use spaces with an action of O(d) and may not have noticed the site of d-manifolds and open embeddings, which is more convenient to use in practice.
I have a vague memory that the remaining gap in the existing proof had to do with showing (or showing convincingly) that some space of Morse functions/handle decompositions is contractible or something like this. Is this an issue you solve or circumvent?
This was resolved by Eliashberg and Mishachev in 2011, who proved that the space of framed generalized Morse functions is contractible: https://arxiv.org/abs/1108.1000. This replaces Section 3.5 in Lurie’s paper.
We also develop a tool with similar functionality, in our locality paper this is Section 6.6 (the 1-truncatedness of our bordism categories corresponds to the contractibility of the space of framed generalized Morse functions used by Lurie to cut bordisms).
However, it is not quite accurate to say that this is “the remaining gap”, since some of the more important parts of Lurie’s argument in the other sections (3.1–3.4), such as Claim 3.4.12 and Claim 3.4.17, have their proofs omitted altogether, not even a sketch is present.
the necessity of incorporating the site Cart in the picture also led to consider the site FEmb_d as a natural extension. If we stayed in the purely topological case, then we could have tried to use spaces with an action of O(d) and may not have noticed the site of d-manifolds and open embeddings, which is more convenient to use in practice.
Thanks, that’s really interesting. I’ll try to find time to look at your locality article in more detail (when I got that darn proof typed up that’s absorbing my time and energy… ;-).
re #34:
Just to say that our new Center for Quantum and Topological Systems is now live: here.
Part of granted activity is related to the intersection of
a) quantum programming languages
b) linear & modal homotopy type theory
c) twisted generalized cohomology theory
as indicated in the refined trilogy diagram that I am showing here. [edit: now I see the typo, will fix…]
We’ll be hiring a fair number of postdocs fairly soon. I’ll post the job openings as they become public.
I guess the citation (I think?) to “Topological and Quantum Systems” is to a document about the Center?
Just to say that the postdoc job advertisements for our new “Center for Topological and Quantum Systems” (as per comment #53) is now out, see:
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