I must have forgotten to refresh.

]]>The link does work (on my machine at least!?), I had made it point across pages. Also had added a minimum of words for context.

]]>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.

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

- Albert Schwarz,
*My Life In Science*, 2004 (pdf)

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.

]]>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.

]]>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.

]]>I have added pointer to

- Luis Alvarez-Gaumé, Frederic Zamora,
*Duality in Quantum Field Theory (and String Theory)*, AIP Conference Proceedings 423, 46 (1998) (arXiv:hep-th/9709180)

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

David Ben-Zvi, Adrien Brochier, David Jordan,

*Integrating quantum groups over surfaces: quantum character varieties and topological field theory*, (arXiv:1501.04652)Scott Morrison, Kevin Walker,

*The blob complex*, (arXiv:1009.5025)Claudia Scheimbauer,

*Factorization Homology as a Fully Extended Topological Field Theory*, (pdf)T. Daniel Brennan, Federico Carta, Cumrun Vafa,

*The String Landscape, the Swampland, and the Missing Corner*(arXiv:1711.00864)

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.

]]>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.

]]>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.

]]>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.

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…

]]>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.

]]>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?

]]>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…

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…

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,

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.

added pointer to

- David Corfield,
*Duality as a category-theoretic concept*, Studies in History and Philosophy of Modern Physics Volume 59, August 2017, Pages 55-61 (doi:10.1016/j.shpsb.2015.07.004)

(we had it at *duality*, but not here)

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?

]]>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).

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 ;-)

]]>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

$quantization \;\colon\; LagrangianData \longrightarrow LagrangianQFTs \hookrightarrow QFTs \,.$ ]]>