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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 12th 2012

needed to be able to point to duality in physics, so I created an entry. For the moment just a glorified redirect.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeOct 12th 2012

I have added a link to AGT correspondence, the entry which I have extended in the meantime.

• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeOct 12th 2012
hope you will point out that some dualities in physics involve more than two
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 13th 2012

Thanks.

which I have extended in the meantime.

Which entry have you expanded? I am not sure if I see it.

hope you will point out that some dualities in physics involve more than two

I already did, I thought. Maybe I should expand on it furhter.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 14th 2012

which I have extended in the meantime.

Which entry have you expanded? I am not sure if I see it.

Ah, you haven’t expanded AGT correspondence, you have expanded AGT conjecture!

I am merging the two entries now…

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeOct 15th 2012
• (edited Oct 15th 2012)

That is strange. I thought there was only AGT conjecture page which I created on June 25 and reported. I have just added there the redirect to it in the moment when I referred to it in duality page. Now I see you created different entry on July 18 (search AGT). AGT conjecture is much more quoted phrase than AGT correspondence which I have seen just few times, but probably it will change in future, regarding that AGT conjecture is already proven in many important cases.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 25th 2012
• (edited Oct 25th 2012)

added to the list of examples at duality in physics also the entries open/closed string duality and (subordinate to that) KLT relations

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeSep 7th 2014

I see this conference I’m attending soon is mostly attended by people coming from the physics side. Since I’m coming from the mathematics side, it would be good to say a few words about what kind of mathematical duality (if any) is involved in certain cases.

So how’s this?

• holographic duality: not really a duality. For a boundary theory, need there even be a dual bulk theory, and if so need it be unique?
• T-duality: exchange of data inducing equivalent QFTs. But we wouldn’t call 2x3 the dual of 3x2 just because they both result in 6. Also, correspondence inducing integral transform isomorphism.
• Mirror symmetry: case of T-duality. Exchange of data induces equivalent QFTs but swaps A and B models. But then I wouldn’t call reflection in $x = y$ such a big deal just because it swaps coordinates.
• S-duality: vestige of higher SL(2, Z) symmetry. But E-M duality related to Hodge duality from pairing.

Hmm, they’re not like good old dual adjoint equivalences in maths, are they? Then again I guess one’s happy enough to call projective duality a duality which is the consequence of an outer automorphism shuffling of a group which reorders the dimensions of the constitutent elements.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeSep 8th 2014

I see Urs has already given an opinion:

I think this is really all there is to it: in physics they discovered interesting equivalences and called them dualities.

E.g., mirror symmetry as an equivalence.

If that’s the case, and since speakers are interested in what dual theories tell us about the way the world is, perhaps I could play a more useful role in expounding the HoTT approach to equivalence.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeSep 8th 2014

Yes, to a large extent the term “duality” in physics is used somewhat thoughtlessly.

There is electric-magnetic duality which, when it really does work, is a $\mathbb{Z}_2$-action on the parameter space of (super-)Yang-Mills theory and hence is close to justifiably be called a duality. Notice that this is thought to really be part of an $SL(2,\mathbb{Z})$-action, though, called S-duality.

T-duality with its similarity to Fourier-Mukai duality (see at topological T-duality) comes pretty close to being a duality in the sense of mathematics. However, there are extra choices involved and indeed any given string background may have none, one or more than one “T-duals”.

As you notice, since mirror symmetry is by and large a version of T-duality, same comments would apply there, though here the situation is a bit more rigid with the action on the Hodge diamond really being a $\mathbb{Z}_2$-action.

The proliferation of dualities in physics came with the “second string revolution” in the 90s, when type II A type II B, type I, Het E, Het O, and Sugra 11 were all realized to be equivalent to each other, at least in certain limits. Since then one speaks of “duality in string theory”, even though this web of equivalences is much more like a bunch of different coordinate charts on a big moduli space than anything that one would call “duality” in mathematics.

So, yes, I think if one grants different meaning of the same words across discipline borders, then then real point of “duality in physics” is that (local Lagrangian gauge) QFT is a pretty rich beast, and that its “moduli space” contains some obvious and some rather subtle equivalences. The more subtle such an equivalence is, the more likely is it going to be called a “duality”.

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeSep 8th 2014
• (edited Sep 8th 2014)

Varghese Mathai and collaborators have released a couple of papers on ’spherical T-duality’, really a variant of the usual with the following replacements: $S^1 = U(1) \mapsto S^3=SU(2)$, 3-form on total space of bundle $\mapsto$ 7-form (or rather, cocycles). The subtitle of the second paper is telling: An infinity of spherical T-duals for non-principal SU(2)-bundles. Really there’s some sort of groupoid of duals, with some as-yet unidentified automorphism group.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 8th 2014
• (edited Sep 8th 2014)

Maybe one should keep in mind other instances of misleadingly similar but actually different use of terminology across the math/physics border. For instance when physicists speak of “covariance” they mean something that may with some effort be traced back to what would be called “covariance” in mathematics, but which really has a meaning of its own.

Other instances of such an almost similarities is “model”, “theory” and probably many more. As always, one should pay attention to the difference between what somebody says, and what he or she actually means.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeSep 8th 2014

All that said, it should maybe be amplified that of course the genuine duality in the sense of mathematics (dualizing objects) also plays a huge role in physics, even if it is not what the physicist thinks about first when saying “duality”.

For instance Dirac’s famous bra-ket notation formalism which has become the hallmark of much of quantum physics is of course a formalism for dealing with dualizing objects. And via the theory of cobordism categories we know that key concepts such as traces producing partition functions etc. are based on the existence of dual objects. Dualizing objects play a pivotal role in quantization, often implicitly so, but very explicitly for instance in what I wrote at “Quantization via Linear homotopy types”.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeSep 8th 2014

Thanks. Very useful for the talk.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeNov 16th 2014
• (edited Nov 16th 2014)

Meanwhile David and I have been chatting in person more on the issue of what it means mathematically when one speaks of duality in physics. What I had suggested to David in person I have now briefly recorded in the entry in a new section Formalization.

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeNov 16th 2014
• (edited Nov 17th 2014)

Great you’re adding this, and fun to have chatted about it in Paris. It seems I have a further month to add things to the final draft of the paper I’m writing on duality, so it would be very good to continue. There’s something I haven’t quite grasped yet about groupoid objects and effective epimorphisms, etc. But post-Paris I have some catching up to do, so haven’t time just now to see what it is I’m missing.

We should also remember some of the other questions we discussed:

• Could one notice $\hbar$ in classical mechanics? Would this need a compact phase space?
• How would one describe the rise of quantum mechanics had one had the presentation of classical physics done à la Urs.
• Riemannian geometry has 2 descriptions, as Cartan geometry for Euclidean space, and in terms of reduction of structure group $GL(n)$ to $O(n)$. What’s special about the Euclidean-Riemannian relation that allows this, where perhaps Cartan geometries for other Klein geometries don’t?
• Related to this thread, could one give a HoTT account of T-duality for QFT rather like that for covariance in GRT?

There may have been others.

• CommentRowNumber17.
• CommentAuthorDavidRoberts
• CommentTimeNov 17th 2014

This formalisation makes the two-ness of duality clear but also leaves space for “dualities” that are not binary, which is nice.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeNov 17th 2014

re #16: thanks for compiling that list. Yes, I will try to get back to all of these items once I am not under insane time pressure anymore. (That’ll be by next week).

re #17: yes, while talking to David Corfield it occurrred to be that this may just the be good way to say it. It’s excellent how talking with David C. brings out all these points. You would think that it would have bee the task of mathematical physicists to ask what the formal definition of such dualities is. But they didn’t, only David C. did. We were talking about this over lunch and it so happened that next to us is an expert on mathematical mirror symmetry. But he hadn’t thought about it either.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeNov 17th 2014

Ah, one more quick remark: regarding the third item in the list in #16: that 2-group for T-duality which I mentioned is that discussed at T-duality 2-group.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeNov 17th 2014

one more,but then I really need to run, regarding item #1: while not a full answer, at least one thing to mention are the classical anomalies an example of which (see there) appears already in plain Galailean mechanics.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeNov 18th 2014

Re #20, you mean mass? We had a discussion about that back at the cafe:

DC: I see Santiago García in Hidden invariance of the free classical particle writes that mass “has a cohomological significance, it parametrizes the extensions of the Galileo group.” Is this an interesting point of view?

JB: Yes! The locus classicus for this viewpoint may be Guillemin and Sternberg’s book Symplectic Techniques in Physics.

Briefly: in classical mechanics, the Galilei group acts on the symplectic manifold of states of a free particle. But in quantum mechanics, we only have a projective representation of this group on the Hilbert space of states of the free particle. The cocycle is the particle’s mass.

Switching to a much more lowbrow way of talking: you can’t see the mass of a free classical particle by just watching its trajectory, since it goes along a straight line at constant velocity no matter what it’s mass is. But you can see the mass of a free quantum particle, because its wavefunction smears out faster if it’s lighter! So there’s some difference between classical and quantum mechanics. Ultimately this arises from the fact that the latter involves an extra constant, Planck’s constant.

Working out how the last two paragraphs are related is a fun exercise in taking some ideas from cohomology and seeing what they amount to in ‘real life’.

Hmm, what John’s saying about the difference between classical and mechanics, the Planck constant, is the very issue that started off my query in that it seemed to have already appeared according to the account of classical mechanics you gave last week in Paris.

US: In slight disguise, one can see this cocycle also controls already the classical free non-relativistic particle, in the sense that its action functional is of the form of a 1d WZW model with that cocycle being the “WZW term” that however comes down to be the ordinary free action.

• CommentRowNumber22.
• CommentAuthorDavid_Corfield
• CommentTimeNov 22nd 2014

Re #15, when you have time to come back to this, at Formalization, the next step up from dualities in the diagram with three arrows down to it is what? A triple of Lagrangians and equivalences between their QFTs? What does such a tower give us?

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeNov 24th 2014
• (edited Nov 24th 2014)

Yes, this continues in the way that Cech covers do (whence the name): above “$Dualities$” there is the space of triples of Lagrangian data that all have the same quantization, equipped with dualities between any two of them, and equipped with an equivalence of dualities between the composite of two of these and the third:

$\array{ && \vdots \\ && LagrangianData \underset{LagrangianQFT}{\times} LagrangianData \underset{LagrangianQFT}{\times} LagrangianDat \\ && \downarrow \downarrow \downarrow \\ Dualities &\simeq& LagrangianData \underset{LagrangianQFT}{\times} LagrangianData \\ && \downarrow \downarrow \\ && LagrangianData }$

Such towers are to be thought of as the incarnation of equivalence relations in homotopy logic. A plain equivalence relation is just the first stage of such a tower

$\array{ Dualities \\ \downarrow \downarrow \\ Lagrangians }$

The conditions on an equivalence relation (reflexivity, transitivity, symmetry) may be read as those on a groupoid object (identity, composition, inverses). So now in homotopy logic this is boosted to an $\infty$-groupoid object by relaxing all three to hold only up to higher coherent homotopies.

The bottom-most arrow

$\array{ Lagrangians \\ \downarrow \\ LagrangianQFTs }$

is the quotient projection of the equivalence relation. In 1-logic this would be its “cokernel”, here in homotopy logic it is the homotopy colimit over the full simplicial diagram.

So the perspective of the full diagram somehow gives the usual way of speaking in QFT also a reverse:

a) that two Lagrangians are dual if there is an equivalence between the QFTs which they induce under quantization,

we may turn this around and say that therefore

b) quantization is the result of forming the homotopy quotient of the space of Lagrangian data by these duality relations.

It is one of the clauses of the Giraud theorem in $\infty$-topos theory that these two perspectives are equivalent.

• CommentRowNumber24.
• CommentAuthorDavid_Corfield
• CommentTimeNov 24th 2014

Excellent. I’ll just add that to the page.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeNov 24th 2014

Thanks for doing that! I should have, but am under time pressure.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeNov 24th 2014

I have edited a tad more at Formalization and Relation to mathematical duality in order to streamline more.

• CommentRowNumber27.
• CommentAuthorDavid_Corfield
• CommentTimeNov 25th 2014

The only mention of three things being related in this area that I can recall is Frenkel in Gauge Theory and Langlands Duality on p. 18 relating A-Branes on something to B-branes on something else to D-modules on yet something else. He says there that the dualities along each side are Homological Mirror Symmetry, categorical Langlands correspondence, and an unnamed one worked on by Witten and Kapustin, and Nadler and Zaslow.

Beyond simple $Z_2$ actions there is also the SL(2, Z) symmetry of S-duality. Then there’s T-duality where something may have #10

none, one or more than one “T-duals”,

as also appears to be the case with the spherical T-duals appearing in the papers David R. mentions above in #11.

I wonder how these cases could be formalised.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeNov 25th 2014
• (edited Nov 25th 2014)

That there are triples appearing here is mostly an artifact of the simplicial language. What counts really is that there are 2-morphisms involved, namely 2-equivalences of the form

$\array{ && L_b \\ & \nearrow &\Downarrow& \searrow \\ L_a && \longrightarrow && L_c }$

We may consider just the special case that $L_b \to L_c$ is the identity, then this reduces to the more vivid globular 2-cell

$\array{ & \nearrow \searrow \\ L_a & \Downarrow & L_c \\ & \searrow \nearrow }$

So these are “dualities of dualities” in that they are equivalences between equivalences between quantizations of Lagrangians.

Similarly the $n$th stage in the Cech nerve of the map $quantization : LagrangianData \to LagrangianQFT$ is only superficially about (n+1)-tuples of Lagrangian all dual to each other, while its real content is that it exhibits $n$-fold higher order dualities.

Notice that we already captured the aspect that “duality” in physics is not something that goes back and forth between just two objects when realizing that it’s really an $\infty$-groupoid object. This takes care of any numbers of objects being equivalent to each other. But we discovered something new, too, that is not in the literature, namely that there are also higher order dualities.

• CommentRowNumber29.
• CommentAuthorDavid_Corfield
• CommentTimeNov 25th 2014

So I added at g+ that the triangle on p. 18 of Frenkel should involve a 2-morphism. Is that the kind of thing we’re looking for at the next higher level?

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeNov 25th 2014

Yes, absoluteley, one would be inclined to ask if that triangle on p. 18 of arXiv:0906.2747 may be filled by a 2-morphism! If so, this would be a “duality of dualities” in the sense we are discussing here.

What I meant to highlight is that this question appears more generally, independently of whether there is a triangle of equivalences – though of course that is an excellent example to look at.

• CommentRowNumber31.
• CommentAuthorDavid_Corfield
• CommentTimeNov 25th 2014

I get that now that triangles aren’t important. It’s just that I know of no other candidate 2-morphisms. Do you?

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeNov 25th 2014

The construction that we are looking at will consider all 2-equivalences that are present. We have heard over lunch the other day that not much is known about which 1-morphisms actually implement the mirror symmetry between two Calabi-Yau $A_\infty$-categories. Suppose somebody constructs any one, then the immediate candidates for interesting 2-morphisms to consider are the auto-equivalences of that 1-morphism. Next, sombody else constructs another one, then either it is equivalent to the previous one or not. If it is, there is another 2-equivalence, and so forth.

Does Frenkel or anyone make any comment on whether that triangle is suppose to commute, up to equivalence?

• CommentRowNumber33.
• CommentAuthorDavid_Corfield
• CommentTimeNov 25th 2014

Let’s see.

Thus, according to Kapustin and Witten, the (categorical) geometric Langlands correspondence (3.1) may be obtained in two steps. The first step is the Homological Mirror Symmetry (5.2) of the Hitchin moduli spaces for two dual groups, and the second step is the above link between the A-branes and D-modules.

So that’s getting at the B-brane/D-module correspondence via the other two sides, i.e., using A-branes to mediate.

But also,

There is actually more structure in the triangle (5.3). On each of these three categories we have an action of certain functors, and all equivalences between them are supposed to commute with these functors.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeNov 25th 2014

The second comment is, I think, about everything respecting the action of Hecke operators or similar, on top of being a suitable equivalence.

I would tend to go hunting for explicit constructions of (just) homological mirror symmetry as equivalences of $A_\infty$-categories. I’ll see if I find something…

• CommentRowNumber35.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014

So now in homotopy logic this is boosted to an groupoid object in an (∞,1)-category

Did you mean ’an $\infty$-groupoid object’?

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

Yes!

• CommentRowNumber37.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014

OK. I was going to change it, but I see it points to groupoid object in an (infinity,1)-category. There it says

By the logic of vertical categorification, an internal ∞-group or internal ∞-groupoid may be defined as a group(oid) object internal to an (∞,1)-category C with (∞,1)-pullbacks.

So one does drop the $\infty$ when speaking of what kind of object is internal?

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

Yes, the default meaning of “groupoid” internal to an $\infty$-topos is that which externally is an $\infty$-groupoid. If one actually means a 1-groupoid one should speak of a truncated groupoid or the like.

This is in the spirit of homotopy logic: if we say “groupoid” in the internal language then it comes out as an $\infty$-groupoid. Well, at least up to the point where one says “associativity” internally, beyond that the coherence for associativity is something one might forget to say. It is presently an open problem how to bring higher coherences systematically into an internal homotopy logic such as HoTT.

• CommentRowNumber39.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014

Along with finding examples of things in the higher levels of the diagram, such as dualities of dualities, there’s the philosopher’s question of which level is the ’real’ one. Of course, they’re just working with the Lagrangian Data and QFT levels. The perplexity, if you remember, comes from one QFT arising from two or more Lagrangians which appear to say very different things about the world.

What should we say now? Presumably that Lagrangian Data level is not privileged in any way. Is it that in HoTT spirit, we’ll end up saying something very simple, but which has a complicated external interpretation, such as when ’There is a path’ becomes ’There is an object of paths with superposed actions, up to higher-gauge equivalence,…’

Back to T-duality, is there any way we can see T-folds and T-dual 2-group relating to the tower?

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

It seems to me that this is much unexplored, and I am not aware of anyone besides you and me who has really raised this issue.

My whole point with “local pre-quantum field theory” etc. is that the Lagrangian data may well matter, in that just quantizED field theories are not an accurate description of reality, but that the information of what they are the quantization of is part of what it needs to have a model in physics. There are lots of indications that this is so, but when it comes to “saying things about the world” of course we are in deep water. It’s my long term strategy to develop some formal foundations of physics to the point where we may talk about this with more confidence, but maybe that still needs more work.

I’d just generally be careful with making naive identifications between the symols that we bring to paper and “the world”. When considering dualites of the kind we are considering here, then we are talking about concepts of fundamental physics at length scales many orders of magnitude smaller than what present day accelerators may see. It seems wrong to worry much about whether on that scale there “really is” that Calabi-Yau manifold of internal degrees of freedom or rather its mirror dual, because there is no reason to expect that the concept “spacetime manifold” by itself is of much meaning at that scale. I would turn this around and think that the duality between descriptions of reality by such manifolds is exactly what characterizes the actual reality at such tiny scales. Much like the “wave-particle duality” of plain quantum physics is not something where we are to choose between one picture or the other to get a picture of the world, but rather is a reflection of the fact that only some fusion of both pictures is the real picture.

The concept of T-fold gets to this point a bit more. Worrying for a T-fold whether it locally “really” is that torus bundle or rather its T-dual seems to be the same kind of wrong worry that our ancestors had when looking at points in a manifold acted on by diffeomorphisms and finding a “hole paradox”.

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeNov 26th 2014
• (edited Nov 26th 2014)

Back to T-duality, is there any way we can see T-folds and T-dual 2-group relating to the tower?

The trouble is that T-duality is about “geometric” strings described by genuine 2d CFT, and for these much less is known than for their topological counterparts. For the case of homological mirror symmetry between topological string models we have a very clear mathematical picture of what that map $quantization \colon LagrangianData \to LagrangianQFTs$ is, as a map of moduli spaces (namely the one from CY manifolds to their CY $A_\infty$-categories), but for genuine CFTs nothing of this mathematical precision is in reach. Most of what is understood about T-duality mathematically is accordingly topological T-duality which is really a bit of educated guesswork for extracting the underlying math from fully fledged T-duality.

• CommentRowNumber42.
• CommentAuthorDavid_Corfield
• CommentTimeNov 26th 2014

…a map of moduli spaces (namely the one from CY manifolds to their CY $A_\infty$-categories)

I thought it was the case that there were two maps from CY manifolds to $A_{\infty}$-categories, such as here:

for two Calabi–Yau manifolds X and Y which are mirror, the bounded derived category of coherent sheaves on one variety should be equivalent to the derived Fukaya category of the other one.

Oh but at mirror symmetry you factor these maps via $X \mapsto SCFT(X)$, then different maps go out from $SCFT(X)$ to its A and B models. Hmm, how does that fit in with the tower diagram?

• CommentRowNumber43.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

Right, the “Lagrangian data” here is that of a certain type of sigma-model with target space a given Calabi-Yau manifold. The types of sigma models here determine the actual Lagrangian built from the CY. For the present purpose one should say that a point in “$LagrangianData$” is a pair consisting of a CY manifold and an element in $\{A,B\}$ indicating which of two possible Lagrangians we build from this.

• CommentRowNumber44.
• CommentAuthorUrs
• CommentTimeNov 26th 2014

I have added the missing $A$,$B$ at mirror symmetry. This entry deserves some more detail, eventually.

• CommentRowNumber45.
• CommentAuthorDavid_Corfield
• CommentTimeNov 28th 2014

Where in #23 you say

Such towers are to be thought of as the incarnation of equivalence relations in homotopy logic,

is there a formulation of this construction in the internal logic?

• CommentRowNumber46.
• CommentAuthorDavidRoberts
• CommentTimeNov 28th 2014

@David wouldn’t it be a groupoid object in an infty,1 category?

• CommentRowNumber47.
• CommentAuthorUrs
• CommentTimeNov 28th 2014
• (edited Nov 28th 2014)

Yes… except possibly for the fact mentioned elsehwere recently, that higher coherences are presently hard so say all the way to infinity in the internal logic.

But in low stages the idea of a groupoid object is just that embodied in univalence: there on the type $Type$ one considers the equivalence relation $\underset{X,Y \colon Type}{\sum} Equiv_{Type}(X,Y)$ and the univalence axiom in this case comes down to saying that this is the equivalence already encoded by the morphisms (“paths”) inside $Type$, hence that $Type$ is already itself the (homotopy) quotient under this relation.

Only that actually the full concept of homotopy quotient is presently not expressed in HoTT, because taken at face value it is the colimit over a simplicial diagram, and it is problematic to encode these.

On the other hand, if we are happy with using the statement of the Giraud theorem that holds externally in $\infty$-toposes, then we know that the simplicial description of groupoid objects is equivalently encoded in the 1-epimorphism which is their homotopy quotient map, and “1-epimorphism” one may easily say in HoTT. From this perspective saying (internally) that

$quantization \colon LagrangianData \longrightarrow LagrangianQFT$

is a 1-epimorphism already is equivalent to saying that there is an equivalence relation up to coherent homotopy on $LagrangianData$ given by dualities.

This may seem like a cheap trick, but I actually think this is a useful perspective. Something similar happens with the classification of principal $\infty$-bundles and with internal (co-)limits: externally the theory in both cases a priori needs homotopy colimits over non-finite diagrams, but since these turn out to be equivalent to something internal and “elementary”, we may just take that elementary characterization as the defining one and happily proceed internally.

And indeed, sticking to the internal logic tends to make this more true than it might seem from the external perspective. For instance here the fact that any 1-epimorphism $X \to Q$ may be regarded as an equivalence relation becomes manifest internally once one regards it as exhibiting $X$ as a $Q$-dependent type. Then one may ask what $X(q)$ is in the context of a term $q \colon Q$ and one finds that $X(q)$ is the homotopy fiber of the morphism over $q$ hence is the elements of $X$ together with the $\infty$-equivalence relations between them as encoded by the original 1-epimorphism.

And hence we find as a theorem in the internal logic:

Duality is the equivalence of Lagrangians in the context of a joint quantization.

:-)

• CommentRowNumber48.
• CommentAuthorDavid_Corfield
• CommentTimeNov 28th 2014
• (edited Nov 28th 2014)

This may seem like a cheap trick, but I actually think this is a useful perspective.

That’s the nub of it. What’s the perspective that will continue to hold that this is a cheap trick, whatever you go on to say, because of some principles which, say, would require the expression of that colimit?

And conversely, can we understand your perspective to be more than just ’useful’, but getting things ’right’?

• CommentRowNumber49.
• CommentAuthorUrs
• CommentTimeNov 28th 2014

The perspective that an equivalence relation is a 1-epimorphism is close to the traditional naive idea: the map says which elements sit in the same equivalence class. It seems to me that this is plausible to every kid, while the definition as an $\infty$-groupoid object in comparison is rather wild. What we learn in the lack of internal simplicial diagrams is that many things we do with them are equivalent to elementary operations that are much easier.

By the way, it is generally thought that it ought to be possible to express simplicial types and simplicial colimits in HoTT, just that due to some technicalities it hasn’t been done yet. It’s one of the most annoying gaps in present HoTT. But maybe we learn from it to stick, where they exist, to elementary concepts equivalent to concepts that would need simlicial constructions.

• CommentRowNumber50.
• CommentAuthorUrs
• CommentTimeNov 28th 2014

So on reflection, that would be my suggestion for how to present the situation in the most evident way in internal logic:

start with the 1-epi $quantization \colon LagrangianData \to LagrangianQFT$;

then “dicover” $n$-fold dualities for any finite $n$ by forming the $(n+1)$-fold homotopy fiber product of that with itself. Which is just the $(n+1)$-fold product of $LagrangianData$ with itself, regarded in the context of $LagrangianQFTs$.

One won’t get the full simplicial diagram of higher dualities this way, but in fact that’s not what most physicists or philosophers would expect to see or want to see anyway. It’s maybe not really close to “elementary” thinking. Nevertheless, one discovers all the higher dualities to any degree and one is guaranteed, by the original map being a 1-epi, that everything is “right” from a mathematical point of view, too.

• CommentRowNumber51.
• CommentAuthorMike Shulman
• CommentTimeDec 1st 2014

maybe we learn from it to stick, where they exist, to elementary concepts equivalent to concepts that would need simplicial constructions.

An interesting point, that perhaps one of the things HoTT (and our current inability to deal with ∞-coherences therein) teaches us is to avoid higher homotopy coherences whenever possible.

Of course, now I can hear my advisor saying “we knew that decades ago!”…

• CommentRowNumber52.
• CommentAuthorUrs
• CommentTimeDec 20th 2014

added pointer to the new preprint Polchinski14 (which I suppose is to go along with David’s contribution into the proceedings of the meeting that David is writing his contibution for, too)

• CommentRowNumber53.
• CommentAuthorDavid_Corfield
• CommentTimeDec 20th 2014

Yes, that’s right. I included a brief mention of those ideas on effective epimorphisms from Lagrangian data. We’ll have to spot a duality between dualities one day.

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeDec 20th 2014

We’ll have to spot a duality between dualities one day.

We know they are there, it’s maybe more a matter of telling an interesting story about them. Namely given any one explicit equivalence between the quantizations of two different Lagangians, then all elements in the same connected component of its hom-space will be dual dualities.

I expect that if one would write an MO/PO message that briefly explains what we are after, people would recognize some examples. But myself, I don’t feel I have time for writing this at the moment.

• CommentRowNumber55.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 10th 2015

There’s a good talk by Vafa On Mathematical Aspects of String Theory in which he discusses dualities as ’highly nontrivial isomorphisms’.

• CommentRowNumber56.
• CommentAuthorUrs
• CommentTimeFeb 10th 2015

You mean around 12:00? I have added the link to the entry.

• CommentRowNumber57.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 10th 2015

At 3:30 he writes up what I said. But the whole talk is an illustration of these isomorphisms.

• CommentRowNumber58.
• CommentAuthorUrs
• CommentTimeFeb 10th 2015

Ah, thanks. I have added that “3:30” to the entry, too. Too bad that he leaves it that vague.

• CommentRowNumber59.
• CommentAuthorDavid_Corfield
• CommentTimeMar 23rd 2015
• (edited Mar 23rd 2015)

I added a section Relation to Morita equivalence to jot down some references.

• CommentRowNumber60.
• CommentAuthorDavid_Corfield
• CommentTimeAug 29th 2015

There’s a nice paper on dual descriptions in physics recently out by Mina Aganagic, String Theory and Math: Why This Marriage May Last, so I started a page for her.

Presumably she’s alluding to Dyson’s 1972 jest:

I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.

• CommentRowNumber61.
• CommentAuthorDavid_Corfield
• CommentTimeDec 12th 2016

We were speaking about “Duality of/between dualities” above, and in the entry duality in physics. I see Duff speaks of “duality of dualities” on p. 26 of arXiv:hep-th/9805177,

It occurred to the present author that string/string duality has another unexpected payoff. If we compactify the six-dimensional spacetime on two circles down to four dimensions, the fundamental string and the solitonic string will each acquire a T-duality. But here is the miracle: the T-duality of the solitonic string is just the S-duality of the fundamental string, and vice-versa!

But I guess that’s not in the same sense as here.

It did not take long to realize that 6-dimensional string/string duality (and hence 4-dimensional electric/magnetic duality) follows from 11-dimensional membrane/fivebrane duality.

Does something of that duality of dualities exist up at the level of 11-dimensional membrane/fivebrane duality?

This from the abstract of arXiv:0906.3013 sounds like Duff’s sense too:

we find an interesting duality of dualities: a pair of theories related via mirror symmetry can be mapped, via geometric duality, into a pair of gauge theories related by Seiberg duality. This network of dualities can be understood as the simple result that all of these theories are different realizations of one and the same system in M-theory.

and

Toric duality is Seiberg duality and toric duality is mirror symmetry: a duality of dualities

• CommentRowNumber62.
• CommentAuthorUrs
• CommentTimeDec 5th 2017
• (edited Dec 5th 2017)

First time that I see somebody pick up the perspective from the entry duality in physics:

• Yuji Tachikawa, What is Quantum Field Theory? (pdf)

from slide 39 on.

• CommentRowNumber63.
• CommentAuthorDavid_Corfield
• CommentTimeDec 5th 2017

Oh, I was hoping from that that they’d really picked up our perspective. So nothing on dualities between dualities, and most attention on it not being an epimorphism to all QFTs in

$quantization \;\colon\; LagrangianData \longrightarrow LagrangianQFTs \hookrightarrow QFTs \,.$
• CommentRowNumber64.
• CommentAuthorUrs
• CommentTimeDec 5th 2017

Oh, I was hoping

One step at a time. Now that they re-discovered that not every field theory is Lagrangian (understood by Helmholtz in 1887) they are asking for a definition of QFT without a Lagrangian, understood by Haag in 1959.

At this pace, it will take a bit until they get to simplicial sets and dualities between dualities ;-)

• CommentRowNumber65.
• CommentAuthorDavid_Corfield
• CommentTimeDec 5th 2017

You had the idea (#54) that dualities between dualities might already be out there. The trouble is that the literature often means by that a pair of dualities, so two dual theories which are dual to another pair of dual theories. That seems to be happening in Hidden Finite Symmetries in String Theory and Duality of Dualities.

Might “symmetries of symmetries” throw something up? But you use that expression to speak of gauge-of-gauge transformations in Lie n-algebras of BPS charges. I guess in a similar vein, there is CP and other Symmetries of Symmetries:

This work is devoted to the study of outer automorphisms of symmetries (“symmetries of symmetries”) in relativistic quantum field theories (QFTs). Prominent examples of physically relevant outer automorphisms are the discrete transformations of charge conjugation (C), space–reflection(P), and time–reversal (T).

• CommentRowNumber66.
• CommentAuthorDavid_Corfield
• CommentTimeDec 5th 2017

Shouldn’t our duality story be extended to include non-Lagrangian theories? e.g., as it stands we aren’t representing dualities between Lagrangian and non-Lagrangian theories, as in in AdS-CFT.

Is it that we ought to have a larger domain than ‘Lagrangian data’ on which to construct the homotopy quotient?

• CommentRowNumber67.
• CommentAuthorUrs
• CommentTimeMar 26th 2019

(we had it at duality, but not here)

• CommentRowNumber68.
• CommentAuthorDavid_Corfield
• CommentTimeJan 2nd 2020

Added a quotation from lectures by Vafa

Since we have seen that the full string theories are all interrelated by a sequence of dualities, one would expect that their compactifications are also related by dualities. As it turns out, these relations are so abundant that we can make the following observation:

“Conjecture”: Whenever the dimension, number of preserved supercharges, and chiralities of two different compactifications of string theory match, there are choices of compactification geometries such that they are dual descriptions of the same physical theory.

Surprisingly, we are aware of no known counter examples. In this sense, dualities in lower dimensional theories are not hard to find, but rather are hard to prevent! One rationale for the existence of dualities is as Sergio Cecotti puts it, “the scarcity of rich structures”. In particular the very existence of quantum systems of gravity is hard to arrange and if we succeed to get more than one theory with a given symmetry, there is a good chance we have landed on the same theory.

• CommentRowNumber69.
• CommentAuthorUrs
• CommentTimeJan 2nd 2020

Thanks!

I have touched the hyperlinking, to be like so:

Whenever the dimension, number of preserved supercharges, and chiralities of two different compactifications of string theory match,

• CommentRowNumber70.
• CommentAuthorUrs
• CommentTimeJan 2nd 2020

By the way, with the two diagrams displayed at Rozansky-Wiiten theory we have now a concrete example of the fiber-product formalization of dualities that is indicated in the entry. Will add something on that…

• CommentRowNumber71.
• CommentAuthorUrs
• CommentTimeJan 2nd 2020

By the way, with the two diagrams displayed at Rozansky-Witten theory we have now a concrete example of the fiber-product formalization of dualities that is indicated in the entry. Will add something on that…

• CommentRowNumber72.
• CommentAuthorDavid_Corfield
• CommentTimeJan 2nd 2020

So Vafa’s conjecture

Whenever the dimension, number of preserved supercharges, and chiralities of two different compactifications of string theory match, there are choices of compactification geometries such that they are dual descriptions of the same physical theory,

could be said to specify parameters of some type of theories.

Does your example fit somehow with Vafa’s conjecture if

Rozansky-Witten theory may be identified with topologically twisted KK-compactification of the D=6 N=(2,0) SCFT on the M5-brane,

and the D=6 N=(2,0) SCFT emerges from M-theory via a process of compactification and the AdS-CFT correspondence?

• CommentRowNumber73.
• CommentAuthorUrs
• CommentTimeJan 2nd 2020

Okay, so I have added a remark (here) on how a concrete class of examples of that quantization map from Lagrangian data to quantum observables is given by the assignment of a) Lie algebra weight systems for Chern-Simons theory and b) Rozansky-Witten weight systems for Rozansky-Witten theory.

• CommentRowNumber74.
• CommentAuthorUrs
• CommentTimeJan 2nd 2020

The interpretation of that statement by Vafa is complicated, as the full KK-compactification is a hugely intricate object, and any of the usual approximations requires making choices. On top of that, the RW-theory involves twists.

I am generally sceptical that the approach of replacing detailed analysis by sweeping vague conjectures (as in the swampland conjectures) is going to help much. Better to make a hypothesis of what the theory actual is and then crank out the consequences…

• CommentRowNumber75.
• CommentAuthorDavid_Corfield
• CommentTimeJan 3rd 2020

I guess at least we might expect a more diverse range of equivalent/dual descriptions the deeper the compactification. Is it even thinkable that there be other points in LangrangianData equivalent to your version of M-theory?

What of Vafa’s ’missing corner’ idea in Lecture 3 (p. 35)?:

we want to know fundamentally, what is quantum gravity? It should describe the quantum fluctuations of the metric. From a brief analysis of the standard Einstein-Hilbert action, we see that fluctuations of the metric at the Planck scale should become very violent, leading to potential changes in the topology of the spacetime [101, 102]. This leads naturally to the idea that quantum gravity should be equivalent to summing over all spacetime topologies and geometries…

In general we have no idea about what description will lead to the correct sum over geometries and topologies. We only do know that there should be some mechanism that washes out the Planck scale fluctuations to produce a smooth space at lower energies. It seems that this description must come from some new fundamental principle, rather than from some duality such as mirror symmetry or AdS/CFT. This lack of knowledge of describing the gravity side quantum mechanically is “the missing corner” in our understanding of string theory.

• CommentRowNumber76.
• CommentAuthorUrs
• CommentTimeJan 3rd 2020
• (edited Jan 3rd 2020)

It’s interesting to see quantum geometry appear here:

With Hypothesis H the space of observables on intersecting $D6\perp D8$-brane intersections contains $\underset{n}{\prod}\big( \mathcal{W}^n\langle \hbar^n \rangle\big)$. We need to invoke some coordination to identify this abstract space with physical content.

One way to coordinate is to compose with the maps $\mathcal{L}_{\mathfrak{su}(2)} \to \underset{n}{\prod}\big( \mathcal{W}^n\langle \hbar^n \rangle\big) \overset{\simeq}{\to} \mathcal{V}$. This identifies the observables with Wilson loop observables of $\mathfrak{su}(2)$-Chern-Simons theory on the 3-sphere containing a Wilson loop labeled $N$.

If the Wilson loop knot is hyperbolic we find (with the volume conjecture) that in the limit $N \to \infty$ our observables see the volumes of hyperbolic 3-manifolds. So some classical geometry has emerged. And by extension, it means that for finite $N$ we see quantum-corrected volumes of quantum hyperbolic 3-manifolds.

• CommentRowNumber77.
• CommentAuthorDavid_Corfield
• CommentTimeJan 3rd 2020

Interesting!

I see we don’t have any quantum geometry entries, but then one often sees this term used by proponents of loop quantum gravity.

• CommentRowNumber78.
• CommentAuthorUrs
• CommentTimeJan 3rd 2020
• CommentRowNumber79.
• CommentAuthorUrs
• CommentTimeDec 20th 2020
• (edited Dec 20th 2020)

In doing so, I noticed that the list of references here (which is far from satisfactory) ended with these four items:

It seems unclear what any one of these has specifically to do with the topic of the entry.(?) If there are specific passages in these articles that should be brought to the attention of a reader of our $n$Lab article, then let’s add them back in with concrete comment on what we suggest the reader might want to look at.

• CommentRowNumber80.
• CommentAuthorDavid_Corfield
• CommentTimeDec 20th 2020

Well, we have Section 5. ’Duality after compactification’ which includes a quotation from the final one of the articles you’ve removed, Brennan et al.

• CommentRowNumber81.
• CommentAuthorUrs
• CommentTimeDec 20th 2020

I see, thanks for saying. So I have copied Brennan et al to duality in string theory and made the reference pointer go there.

I forget what happened here, whether we first had a single entry which was then, incompletely, split into duality in physics and duality in string theory.

But I find that random-seeming references reduce the value of the list they populate. If we do remember what the “blob complex” is meant to have to say about duality in physics, let’s add it back in, too, with due commentary.

• CommentRowNumber82.
• CommentAuthorUrs
• CommentTimeDec 20th 2020

Oh, I see, all these references are called in the one paragraph starting out as “Morita theoretic ideas are also involved in…”.

That’s really vague, I find, but at least it’s the kind of comment that puts the point of these references in perspective. So I have re-organized the references-section a little now, in order to better indicate what is meant for what.

• CommentRowNumber83.
• CommentAuthorUrs
• CommentTimeDec 20th 2020
• (edited Dec 20th 2020)

have turned to pointer to Schwarz’s scientific autobiography into a proper item in the list of references:

• CommentRowNumber84.
• CommentAuthorDavid_Corfield
• CommentTimeDec 20th 2020

Re #81, 5. Duality after compactification is just an extract from Brennan et al., so now the reference has moved, the link to it doesn’t work. Presumably the section itself should be moved.

• CommentRowNumber85.
• CommentAuthorUrs
• CommentTimeDec 20th 2020