highercategorytheory

Why Higher Category Theory in Physics?

[Total: 5    Average: 5/5]

 

This here is my personal story. For an alternative introduction see my talk: Higher Structures in Mathematics and Physics.

Initially I discovered higher category theory and higher homotopy theory for myself in my PhD work, in the course of analyzing the supersymmetric quantum mechanics of the superstring on loop space. Driven, as I am, by the conviction that fundamental physics requires following the lead of fundamental mathematics, this made me learn and apply higher category theory and homotopy theory to prequantum field theory and string theory. Eventually it became clear that higher category/higher homotopy theory is strictly necessary to understand what modern physics in general and string theory/M-theory in particular actually is all about.

This curious story goes as follows.


In my Diplom (~MSc) thesis

  • Supersymmetric homogeneous quantum cosmology, Diplom thesis 2003 (pdf)

I had studied supersymmetric quantum cosmology viewed as supersymmetric quantum mechanics on the configuration space of gravity on a time slice, i.e. on Wheeler’s “superspace” — hence mechanics on super-superspace, for both meanings of “super” used in physics.

My subsequent PhD work began with the task of applying the same techniques to the worldvolume of the superstring, describing its dynamics as supersymmetric quantum mechanics on the smooth free loop space of spacetime.

It seems to be a little appreciated fact that this is where supersymmetric quantum mechanics originates from in the first place, due to

  • Edward Witten,  Supersymmetry and Morse theory, J. Differential Geom. Volume 17, Number 4 (1982), 661-692

The reason is clearly that Witten’s Fields-medal winning article didn’t dwell on its origin from string theory. This origin he revealed later, somewhat hidden in

  • Edward Witten, from p. 92 (32 of 39) on in Global anomalies in string theory, in W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, pp. 61–99. World Scientific, 1985 (pdf)

The idea is exactly that of cosmological models, but now with the superstring worldsheet theory regarded as 2-dimensional quantum super-gravity coupled to worldsheet “matter” fields (the string’s “embedding” fields): One considers a foliation of spacetime/worldsheet by spatial slices/loops, then makes a Fourier mode expansion of the fields along the spatial slice, discards higher modes for the desired quasi-homogeneous approximation, and then propagates the degrees of freedom that remain via their quantum mechanics on their effective configuration space.

Anyway, what I considered in my PhD thesis work was the algebraic deformation theory of this setup, and this is where I found higher category/higher homotopy theory show up.

Namely, one punchline of Witten’s “Supersymmetry and Morse theory” was that susy quantum mechanics has interesting deformations by scalar function on configuration space (those “Morse functions”). I wondered what these deformations gave when applied not to a finite dimensional manifold, but to loop space, hence to the configuration space of the superstring. It turns out, unsurprisingly but neatly, that they provide an alternative way to discover the various possible string background fields and their relations:

But interestingly, at this level there are more deformations possible on loop space than correspond to the usual background fields. In particular there is also a deformation induced by the function on loop space which is the supersymmetrized Wilson loop observable of a non-abelian 1-form gauge field on loop space, coupled to a non–abelian 2-form ##B##, i.e. a “higher gauge field“. Back then I was surprised to find that consistency required that these are related by the condition that

$$ F_A = B $$

a condition that these days you may find under the name of “fake flatness of higher gauge connections”.

This algebraic deformation result I had made public as:

  • Urs Schreiber, Nonabelian 2-forms and loop space connections from SCFT deformations (arXiv:hep-th/0407122)

based on the results in

  • Urs Schreiber, Super-Pohlmeyer invariants and boundary states for non-abelian gauge fields, JHEP0410:035,2004 (arxiv:hep-th/0408161)

Back then, the referee of the “Nonabelian 2-forms” article wrote something like this (paraphrasing from memory):

It looks okay in itself, but I checked with my colleagues, and none of them knows what to make of this higher nonabelian gauge field in string theory. Therefore this cannot be recommended for publication.

Think of this style of reasoning what you will, but in any case my next task was to figure out what in fact this all means. And it meant that higher category theory enters the picture.

Namely by a lucky coincidence, just when I worked out these results in the deformation theory of susy quantum mechanics on loop space, John Baez was popularizing the idea that there ought to be a consistent and interesting theory of nonabelian higher gauge fields, obtained by “categorifying” ordinary gauge theory in a suitable way:

In this context just that “fake flatness condition” above had arisen, from a 2-categorical constraint:

  • Florian Girelli, Hendryk Pfeiffer, equation (3.25) in Higher gauge theory — differential versus integral formulation J.Math.Phys.45:3949-3971,2004 (arXiv:hep-th/0309173)

Here the statement is that if one “categoifies” gauge groups to categorical groups, also called “2-groups“, then a consistent concept of “higher gauge fields with Wilson surfaces” categorifying the familiar concept of “gauge fields with Wilson lines” requires a fake flatness condition to hold.

This was all announced in

  • John Baez, Urs Schreiber, Higher Gauge Theory, in Categories in Algebra, Geometry and Mathematical Physics,  A. Davydov et al. (ds.), Contemp. Math. 431, AMS, Providence, Rhode Island, 2007, pp. 7-30 (arXiv:math/0511710)

but for some reason the publication of the fully rigorous discussion took until

  • Urs Schreiber, Konrad Waldorf, theorem 2.21 (see p. 4) of Smooth Functors vs. Differential Forms, Homology, Homotopy Appl., 13(1), 143-203, 2011 (arXiv:0802.0663)

(which gives the local statement) and

  • Urs Schreiber, Konrad Waldorf, Connections on non-abelian Gerbes and their Holonomy Theory Appl. Categ., Vol. 28, 2013, No. 17, pp 476-540 (arXiv:0808.1923)

(which gives the extension to the complete global statement).

So that’s how I got into higher category theory: I studied the superstring, considered an algebraic deformation that had not been considered before, and found that the mathematical explanation of a funny constraint appearing thereby is provided by 2-category theory  — or really by 2-groupoid theory, which is homotopy 2-type theory.

This became my Phd thesis:

Incidentally that thesis ended (in its section 13.8) with a brief outlook on higher gauge fields for gauge 3-groups with Lie 3-algebras suitable for the analogous discussion of membranes. This was before Bagger-Lambert 06 caused the “membrane mini-revolution” with Filippov-style “3-Lie algebra” . In contrast, I was and am looking at Stasheff-style “Lie n-algebras” . See Christian Saemann’s discussion of how the “Filippov 3-Lie algebras” on the membrane are equivalently “Stasheff Lie-2 algebras” with metric.

The theory touched on in that outlook at the end of my PhD thesis — the theory of higher nonabelian gauge fields of arbitrary deree, via Lie ##n##-algebras for arbitrary ##n## — eventually materialized in these articles:

Digging deeper in this direction turned out to be necessary to finally fully understand the initial open question:

Where is that nonabelian 2-form field in string theory?


The answer to this is beautiful, but its full appreciation does require some genuine higher homotopy theory to fully appreciate, such as an understanding of PostnikovWhitehead towers: Nonabelian 2-forms are a subtle beast.

Roughly, the punchline is that higher non-abelian gauge fields provide the unification of higher abelian gauge fields with certain twists and actions of ordinary non-abelian gauge fields on them. Hence — and this keeps being a source of confusion to people — for every higher non-abelian gauge field there is a kind of choice of coordinate decomposition that makes it appear as an abelian higher gauge field over a background of ordinary non-abelian gauge fields: But the full gadget is more than this local data.

This phenomenon has been called the Whitehead principle of nonabelian cohomology in

  • Bertrand Toën, Stacks and non-abelian cohomology” (pdf)

Here are the pertinent examples:

 

Example 1 — Orientifold B-field

In string theory on an orientifold, then there is a subtle twist of what is locally an abelian 2-form B-field by an action of the orientation cover ##\mathbb{Z}/2##-bundle of the orbifold spacetime. The joint structure unifying the B-field and its orientifold twist is a “Jandl gerbe

  • Urs Schreiber, Christoph Schweigert, Konrad Waldorf, Unoriented WZW models and Holonomy of Bundle Gerbes, Communications in Mathematical Physics August 2007, Volume 274, Issue
    1, pp 31-64 (arXiv:hep-th/0512283)

Writing this out in detail is a lot of data, as you may see form the formulas in the above article. But using nonabelian higher gauge theory it all unifies to the following elegant statement: there is a nonabelian smooth 2-group called ##\mathbf{B}U(1)/(\mathbb{Z}/2)##. A string orientifold background is precisely nothing but a higher gauge field for this 2-group as gauge group.

This example is instructive for getting de-confused about the issue of (non-)abelianness in higher gauge theory: both groups ##U(1)## and ##\mathbb{Z}/2## are ordinary abelian groups, clearly. Nevertheless, the 2-group ##\mathbf{B}U(1)/(\mathbb{Z}/2)## is not abelian as a 2-group. Accordingly, in terms of local data a higher gauge field for this 2-group is given all by ordinary abelian 2-forms, locally, but nevertheless the global structure is not that of an abelian U(1)-bundle gerbe. Instead, the higher non-abelian nature of the higher gauge field induces the peculiar “orientifold twist” structure on what look like abelian gauge fields.

This plays a role on orbifolds even without the peculiar orientifolding business. This was first understood by

 

Example 2 — Heterotic B-field

A richer example of the same kind is the all important Green-Schwarz anomaly cancellation in heterotic string theory (the source of the “first superstring revolution”). Here the locally abelian B-field with (locally abelian) field strength 3-form H is subject to the famous constraint

$$d H = \langle R \wedge R \rangle – \langle F_A  \wedge  F_A \rangle  $$

where ##R## is the Riemann curvature 2-form and ##F_A## is the gauge field strenght 2-form, and where ##\langle -,-\rangle## indicate suitably normalized bilinear invariant pairings.

Again, even though it locally looks like we are dealing with abelian higher form fields, one readily sees that globally something more subtle is going on, as there is an interplay between the non-abelian 1-form data and the 2-form data.

Indeed, passing to the global picture and unifying all the data into a single structure reveals that the Green-Schwarz anomaly-free background field content of heterotic string theory is a single higher gauge field for a gauge 2-group that is famous as the (twisted) String 2-group. This we demonstrated in

 

Example 3 — M-theory C-Field

Yet another example of this “Whitehead principle of non-abelian cohomology” appears when passing from string theory to M-theory: the C-field in 11-dimensional supergravity also looks locally like just an abelian 3-form field, but again it is subject to some twists and turns by nonabelian 1-form field data. Namely in this case there is the “Witten flux quantization condition” which very roughly says that the 4-form field strength ##G_4## is related satisfies

$$ 2 G_4 = \langle R \wedge R\rangle  + \langle F_A \wedge F_A\rangle $$

with an innocent-looking but very subtle factor of 2, and for F_A some auxiliary ##E_8##-gauge connection (which is unphysical in the bulk of the 11d-spacetime).

Making global sense of this requires some serious machinery, which was mostly done in the remarkable article

  • M.J. Hopkins, I.M. Singer, Quadratic functions in geometry, topology,and M-theory, J. Diff. Geom. 70 (2005) 329-452 (arXiv:math/0211216)

We used this to show that in fact the full field content here is a single higher gauge field for a certain non-abelian gauge 3-group:

This has implications: if one makes this higher gauge field explicit, then one realizes that the non-abelian piece of the Chern-Simons term in 11-dimensional supergravity necessarily contains a contribution by a higher gauge field for the twisted String 2-group (a non-abelian 2-group, remember). After compactifying on a 4-sphere, this yields turns the naive 7d Chern-Simons theory into a non-abelian higher gauge field theory, as we showed in this article:

Here it is good to recall that this is not proposed or introduced by hand, but that this follows, by applying, if you wish, the “Whitehead principle of nonabelian cohomology” in reverse, to lay bare the higher non-abelian gauge field theory structure hidden in the flux quantization constraint on the supergravity C-field.

Notice the implications: to the extent that AdS7/CFT6 applies, this means that the infamous 6d SCFT in the nonabelian case must be dual to a 7d theory with a topological sector given by a non-abelian higher gauge field Chern-Simons theory. There is confusion and debate as to whether the worldvolume theory of the 6d theory itself contains a nonabelian higher gauge field, before or after quantization. But the statement here is different: its *holographic dual* is provably a higher nonabelian gauge field. What that implies for the 6d theory itself remains to be investigated. But it seems before this happens on a larger scale, some more basics on the nature of higher gauge fields need to percolate further through the community.

Meanwhile, with the M-theory C-field, we have arrived at 3-group higher gauge theory. But it does not stop here, in general one is faced with ##n##-groups for n going to infinity:

 

Example 4 — RR-Fields

In String theory the passage to unbounded higher categorical/homotopy theoretic degree occurs at the very least with the RR-fields. Again, there is a local abelian picture where the RR-fields are higher gauge fields with coefficients in the “abelian infinity-group” (called a “spectrum“) known as KU. But in general there is again a twist: The B-field with its field strength H locally interacts with the RR fields strength ##C## (in every degree for type IIA string theory) by the famous relation

$$ d C = H \wedge C$$ .

Globalizing this, one finds that the unified structure is the “non-abelian” homotopy quotient ##KU/BU(1)## (technically this now is a “parameterized spectrum“), in higher generalization of the simple case of the orientifold 2-group  ##BU(1)/\mathbb{Z}/2## that we saw above. A derivation of this fact “from first principles” at the rational level is in

For a discussion showing the fully fledged use and need of stable higher homotopy theory in the accurate description of type II string backgrounds see also

 

These example show that higher category theory/higher homotopy theory is not only a useful tool, but is indispensable for an accurate understanding of what string theory actually is. Constructions as the above are simply unthinkable without higher structure tools


Finally, it is fascinating to see that the higher category/higher homotopy theory is not just descriptive, but there are indications that it is in fact constitutive for string theory.

We may consider the “atom of superspace”, namely the superpoint, and then put it under the magnifying class of higher categegory/homotopy theory, namely the Whitehead tower construction.  Remarkably this reveals that “inside” the superpoint all the spacetime and brane content of string/M-theory homotopically appears all by itself. I had talked about this before here in the series Emergence from the Superpoint.

A more detailed exposition of how this works is in this talk that I gave recently:

based on our articles

In conclusion, it is clear that to understand what string/M-theory really is, it is necessary to speak higher category/higher homotopy theory. That’s why I am interested in it.


But string theory is not the only the place in physics where higher category/higher homotopy theory appears, it is only the most prominent place, roughly due to the fact that higher dimensionality is explicitly forced upon us by the very move from 0-dimensional point particles to 1-dimesional strings.

But in fact higher category/higher homotopy theory is right at the heart of variational local field theory itself. There are surprisingly many types of ordinary familiar physical systems whose full and accurate understanding (as opposed to some perturbative approximation or other) necessitates higher category/higher homotopy theory. I recently delivered a gentle exposition as to how that is, here:

For yet more basic exposition of this important point, you might also see this popular discussion forum explanation:

as well as the previous installment in this very series: Higher Prequantum Geometry I: The Need for Prequantum Geometry

Hence more broadly speaking the answer to “Why are you interested in higher category theory?” is simple: “Because this is what is at the foundations of physics..”

Finally, to be completely honest, the issue ranges deeper still. At times I am interested in metaphysical questions, such as “Why Lorentzian spacetime?”, “Why local Lagrangian densities?” in the first place. It may sound outrageous, but I claim that higher category/higher homotopy theory yields explanations here, too. How this comes about I have tried to lay out in

To my mind, the considerations discussed in this note are the deepest reason to be interested in higher category/higher homotopy theory in physics. But it’s a little esoteric. That’s how it goes.

 

 

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague. Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.
19 replies
  1. mitchell porter
    mitchell porter says:

    Just a few days ago, for the first time, a paper by Patricia Ritter gave me an explanation that I could understand, for the relevance of n-categories to objects like branes – the categorical identities express the equivalence of different ways of doing certain integrals over a volume, e.g. where in effect you might first integrate in the x direction, then along the xy plane, then throughout the xyz volume; but you might have done all that for a different order of x,y,z… the result needs to be the same for all orderings, and that leads to the categorical formulation of higher gauge theory.

    I want to emphasize, that's not exactly what she says, that's me trying to dumb it down to the simplest way of saying it. But am I even approximately correct in this interpretation?

  2. Urs Schreiber
    Urs Schreiber says:

    Just a few days ago, for the first time, a paper by Patricia Ritter gave me an explanation that I could understand, for the relevance of n-categories to objects like branes – the categorical identities express the equivalence of different ways of doing certain integrals over a volume, e.g. where in effect you might first integrate in the x direction, then along the xy plane, then throughout the xyz volume; but you might have done all that for a different order of x,y,z… the result needs to be the same for all orderings, and that leads to the categorical formulation of higher gauge theory.

    I want to emphasize, that's not exactly what she says, that's me trying to dumb it down to the simplest way of saying it. But am I even approximately correct in this interpretation?

    Yes, that's one good way of thinking about it. This is the motvation from "higher parallel transport".

    Like so: the structure of a group (an ordinary group) is exactly what one needs in order to label the edges in a lattice gauge theory: the group product and associativity give that edge labels may be composed, and inverses gives that going back and forth along the same edge picks up no curvature. Of course this is not restricted to the lattice. In general, group-valued gauge fields are exactly the right data to have consistent Wilson line observables

    Now a 2-group (categorical group) is, similarly, exactly the data needed to consistencly label edges AND plaquettes in a consistent way (with possibly different labels for each). For instance associativity now includes a 2-dimensional codition which says that with four plaquettes arranged in a square, then first composiing horizontally and then vertically is the same (in fact: is gauge equivalent to) first composing vertically and then horizontally.

    Again this is not restricted to the lattice. Generally, 2-group valued gauge fields are exactly what one needs for consistent Wilson surfaces.

  3. Crass_Oscillator
    Crass_Oscillator says:

    This is not even unconvincing, to coin a phrase. If you cannot connect this mathematics to a physical problem other than gravity, it's almost certainly useless, since you are unable to make tangible statements about experimental reality.

    There's lots of physics out there in need of new mathematical models. Non-equilibrium quantum/classical physics of few or many bodies, soft matter, fluids, strongly correlated materials etc. Why, a great example would be in 2D materials; we already know that some exhibit Dirac excitations, and speculation has been ongoing for years on condensed phase simulations of gravity. Hawking radiation in dumb holes has already been verified or at least strongly suggested in Bose gases if I am not mistaken.

    Without that, there is absolutely no reason for any physicist to take this seriously.

  4. MathematicalPhysicist
    MathematicalPhysicist says:

    It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?

    BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.

    Will it work?

    Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.

    BTW, @Urs Schreiber do you work at the maths or physics department? :-D

    Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".

  5. Crass_Oscillator
    Crass_Oscillator says:

    It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?

    BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.

    Will it work?

    Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.

    BTW, @Urs Schreiber do you work at the maths or physics department? :-D

    Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".

    Well, Feynman is also reputed to have said that "Math is to physics as masturbation is to sex." ;)

    However that's not really the point of my gripe. Urs is trying to sell higher category theory to physicists judging by the title alone, but he doesn't really seem to grasp how to make a compelling case to physicists. I could have been polite, but instead I decided to give him a hard prod. I'm very mathematically conservative and am no fan of math driven physics research, but I do recognize the value of more sophisticated mathematics. He needs to make a much better case.

    Mathematicians often get lost in how, for instance, the machinery they've constructed shortens previously elaborate proofs, or paves the way to proving that certain mathematical structures have exciting, exotic properties, but I use these mathematical structures to construct models. I don't care about proofs, save quick and dirty ones, and much of physics still doesn't get far past super-charged 19th century calculus, with topology and advanced algebra sprinkled in here and there.

    Perhaps starting with Lawvere's book would be the place to begin. And it had better simplify calculations, not make them more complicated. Or, allow me to calculate something that previously was difficult. If Lawvere's book suggests algorithms for solving continuum mechanics problems that nobody had thought of which are competitive with simpler 20th or 19th century algorithms or allows you to build models that obtain experimental observables that are hard to describe otherwise, it's interesting to a physicist. Otherwise, it's a waste of time.

  6. Urs Schreiber
    Urs Schreiber says:

    I understand that not everybody is inclined to follow and follow through the arguments and pointers that I gave, many of them related to string theory. Were it not for the fact that this particular article originates in a personal reply to a question in an interview that Greg was (trying to) do with me, as briefly explained at the beginning, I should have given a more broadly targeted exposition which would have pre-empted some of the misunderstandings that are surfacing above. I have to apologize for this neglect. I believe though that I had included pointers to more general expositions which I have produced elsewhere, a good point to start may be my Oberwolfach talk Higher Structures in Mathematics and Physics A conspiring phenomenon which I don't feel responsible for is that not everyone cares about the fundamental issues at stake in the first place, and ignorance of a problem may cause underestimation of its solution.

    But even string theory with its explicit higher gauge fields aside, there is no room left for the standpoint that higher homotopy structure may be ignored in the formulation of accurate physical theory, certainly not in fundamental physics, but increasingly also in physics relevant for desktop experiments.

    Regarding the former I now use the occasion of this addendum to highlight what in a more pedagogical and less personal account would have been center stage right in the introduction, namely the developments propelled by A. Schenkel and M. Benini in the last years, regarding the foundations of quantum field theory Curiously, it had been a well kept secret for more than half a century that the mathematical formulation of Lorentzian QFT in terms of the Haag-Kastler axioms (AQFT) is incompatible with local gauge theory. At the QFT meeting in Trento 2014 I had pointed out (here) that this may be seen irrespective of details of formulation from basic principles of gauge fields, which is what in mathematics is the principle of "stacks" (higher sheaves). By a curious coincident, at the same meeting Alexander Schenkel presented (here) a detailed analysis of the AQFT construction of free QED (without matter) showing explicitly how it fails the locality axioms. As I had explained (here, see also this BA thesis for a still simple but more technical introduction ) the solution to this problem is higher homotopy/category theory, namely the local net of quantum observables has to be promoted to its homotopy version, sometimes called a co-stack or similar. Since then Beninin, Schenkel at al have be been showing this is increasing detail, I recommend to try to look at least at the introductions of these articles:

    Marco Benini, Alexander Schenkel, Richard J. Szabo
    "Homotopy colimits and global observables in Abelian gauge theory"
    Lett. Math. Phys. 105, 1193-1222 (2015)
    https://arxiv.org/abs/1503.08839

    Marco Benini, Alexander Schenkel
    "Quantum field theories on categories fibered in groupoids"
    https://arxiv.org/abs/1610.06071

    Next, regarding the second point of higher mathematical structures required in solid state physics, I'd just draw your attention to a little mini revolution in the field that has been going on the last years, and which is reflected in the last round of Physics Nobel Prizes: the understanding of topological phases in solid state physics. The influential publication here is

    Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen,
    "Symmetry protected topological orders and the group cohomology of their symmetry group",
    Phys. Rev. B 87, 155114 (2013) arXiv:1106.4772; A short version in Science 338, 1604-1606 (2012)
    http://dao.mit.edu/~wen/pub/dDSPTsht.pdf

    which spurred much activity in the use of higher mathematical structures for the description of topological phenomena in solid states, such as notably certain configurations of Graphene. It turns out that the stable homotopy theory of twisted generalized cohomology theory is required to understand the special topological behaviour of these gapped physical systems

    Daniel S. Freed, Gregory W. Moore,
    "Twisted equivariant matter",
    Annales Henri Poincaré December 2013, Volume 14, Issue 8, pp 1927–2023
    arxiv/1208.5055

    You see these solid state physicists indulge in higher category theory for their solid needs, such as

    Liang Kong, Xiao-Gang Wen
    "Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions"
    arXiv: 1405.5858

    To dispel the idea that this is maths too far ahead of its physics development, it may be instructive to see the evidence that instead the maths is lagging behind, see Edward Witten's question to the maths community Group cohomology and condensed matter

    So much for tonight. If you have further questions or remarks, I'll offer to react, but please take a moment to make sure that before you go the easy route and conveniently declare as irrelevant what is unfamiliar and potentially scary to take the chance to read up and learn first of all about open problems in physics that you may have been unaware of, and second about their mathematical answers.

    Beware the instructive events in history where this attitude of ignorance backfired. There was a time when people rejected complex numbers as overly fancy mathematics. Interestingly, it was largely the observation of complex numbers in physics experiment, namely in the guise of quantum mechanical phases, which revealed this attitude as born out of ignorance and laziness. What complex numbers were for the physics of the beginning 20th century, so higher homotopy/category theory is for the physics of the beginning 21st century. Don't be left behind.

  7. RockyMarciano
    RockyMarciano says:
    Urs Schreiber

    Regarding the former I now use the occasion of this addendum to highlight what in a more pedagogical and less personal account would have been center stage right in the introduction, namely the developments propelled by A. Schenkel and M. Benini in the last years, regarding the foundations of quantum field theory. Curiously, it had been a well kept secret for more than half a century that the mathematical formulation of Lorentzian QFT in terms of the Haag-Kastler axioms (AQFT) is incompatible with local gauge theory. At the QFT meeting in Trento 2014 I had pointed out (here) that this may be seen irrespective of details of formulation from basic principles of gauge fields, which is what in mathematics is the principle of "stacks" (higher sheaves). By a curious coincident, at the same meeting Alexander Schenkel presented (here) a detailed analysis of the AQFT construction of free QED (without matter) showing explicitly how it fails the locality axioms. As I had explained (here, see also this BA thesis for a still simple but more technical introduction ) the solution to this problem is higher homotopy/category theory, namely the local net of quantum observables has to be promoted to its homotopy version, sometimes called a co-stack or similar. Since then Beninin, Schenkel at al. have be been demonstrating this in increasing detail, I recommend to try to look at least at the introductions of these articles:

    Diagnosing that there is a problem(I'd say a serious one), like the mentioned about gauge fields and locality in the way they are usually displayed, and finding a good path towards its solution are two processes that not always come together.
    How would you convince a physicist that at least gets to glimpse the issue (the well kept secret) that the solution lies in higher homotopy/category. I mean I guess going to higher homotopy allows you to obtain finer distinctions and gives you more flexibility to accommodate locality in a way that the more rigid lower categories can't, but how is this actually done and connected to the actual physics? what is the physical correlate of the 2-bundle?, how do the different physical interactions fit in all this?
    On the other hand it seems this is a movement in the direction of greater complexity, while traditionally generalizing theories have worked in physics in the direction of simplifying the local apparent disparately unrelated observations(well admittedly this trend is not so clear in the case of QFT). It would seem that going to higher categories adds in complexity, so it would be great if examples were given about how it could be applied to specific physical problems and the unification of observations or maybe give more details about the above mentioned applications to solid state physics.

  8. Urs Schreiber
    Urs Schreiber says:
    RockyMarciano

    How would you convince a physicist that at least gets to glimpse the issue (the well kept secret) that the solution lies in higher homotopy/category.

    By explaining it, as I did. Do have a look at the slides . They are expository. They are aimed at a QFT audience. They were invited at a QFT meeting. Do have a look. It's not black magic.

    RockyMarciano

    I mean I guess going to higher homotopy allows you to obtain finer distinctions and gives you more flexibility to accommodate locality in a way that the more rigid lower categories can't, but how is this actually done and connected to the actual physics?

    The point is that homotopy in mathematics is exactly the formalization of gauge transformation in physics. If you want to be serious about describing a system with gauge symmetries, the relevant mathematics is, by necessity, homotopy theory. See the slides.

    RockyMarciano

    what is the physical correlate of the 2-bundle?,

    It's the correct field bundle such that the field content does contain the global instanton sectors, and not just the patchwise gauge field information. See towards the end of the slides where this is explained.

    RockyMarciano

    how do the different physical interactions fit in all this?

    They are encoded by the Lagrangian density, as usual. The difference is this: textbooks say that the Lagrangian density is a horizontal differential form on the jet bundle of a field bundle. But this is just not true in general. For gauge theories there is no field bundle that captures the full global field content. Instead the correct field bundle is a 2-bunde. So is it's jet bundle. And the Lagrangian density which encodes the gauge QFT (interactions and all) is a differential form on that jet 2-bundle.

    RockyMarciano

    On the other hand it seems this is a movement in the direction of greater complexity,

    This is an illusion, coming from confusing unfamiliarity with complexity. Similarly, originally people said that complex numbers are overly complex, whence the name. Later they realized that, on the contrary, many a thing in real analysis becomes simpler when passing to the complex domain.

    Homotopy theory is conceptually most simple. But rich in phenomena. It is just mathematics with the gauge principle natively built in.

  9. Debaa
    Debaa says:
    MathematicalPhysicist

    It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?

    BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.

    Will it work?

    Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.

    BTW, @Urs Schreiber do you work at the maths or physics department? :-D

    Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".

    [emoji23] [emoji23]

  10. RockyMarciano
    RockyMarciano says:
    Urs Schreiber

    By explaining it, as I did. Do have a look at the slides . They are expository. They are aimed at a QFT audience. They were invited at a QFT meeting. Do have a look. It's not black magic.

    The point is that homotopy in mathematics is exactly the formalization of gauge transformation in physics. If you want to be serious about describing a system with gauge symmetries, the relevant mathematics is, by necessity, homotopy theory. See the slides.

    It's the correct field bundle such that the field content does contain the global instanton sectors, and not just the patchwise gauge field information. See towards the end of the slides where this is explained.

    They are encoded by the Lagrangian density, as usual. The difference is this: textbooks say that the Lagrangian density is a horizontal differential form on the jet bundle of a field bundle. But this is just not true in general. For gauge theories there is no field bundle that captures the full global field content. Instead the correct field bundle is a 2-bunde. So is it's jet bundle. And the Lagrangian density which encodes the gauge QFT (interactions and all) is a differential form on that jet 2-bundle.

    This is an illusion, coming from confusing unfamiliarity with complexity. Similarly, originally people said that complex numbers are overly complex, whence the name. Later they realized that, on the contrary, many a thing in real analysis becomes simpler when passing to the complex domain.

    Homotopy theory is conceptually most simple. But rich in phenomena. It is just mathematics with the gauge principle natively built in.

    Actually I had read the slides. The problem with slides separated from their oral presentation is that they are not nearly as explanatory as the actual talk. I understand you might be rationing your efforts to spread higher category among physicists though, it is a long distance race. Hope it gets somewhere, I think it is a step in the right direction.
    Thanks for bothering to answer, had more questions but I guess I'll just see the slides again.

  11. RockyMarciano
    RockyMarciano says:
    Urs Schreiber

    It's the correct field bundle such that the field content does contain the global instanton sectors, and not just the patchwise gauge field information.

    Some would argue that it is the demand of locality that prevents from having a field that contains global instanton sectors.

    They are encoded by the Lagrangian density, as usual. The difference is this: textbooks say that the Lagrangian density is a horizontal differential form on the jet bundle of a field bundle. But this is just not true in general. For gauge theories there is no field bundle that captures the full global field content. Instead the correct field bundle is a 2-bunde. So is it's jet bundle. And the Lagrangian density which encodes the gauge QFT (interactions and all) is a differential form on that jet 2-bundle.

    SCNR. Extraordinary claims require extraordinary proofs. The above are very strong statements, the physical validity of wich can only be demonstrated by proving how the 2-bundle contains the global instanton sector and captures the global field content. First you have to prove that the usual textbook procedure can't do it(in other words that the Yang-Mills existence and mass gap problem cannot be solved within the current framework), this by itself is a highly prized enterprise, and then you have to prove that the 2-bundles of higher category do indeed give rigorous path integrals(that need to capture that global content) and solve the Yang-Mills problem.

    Without all this backing the above statements, certain scepticism from physicists is granted, not towards the milder claim that this COULD be a good research program to eventually be able to make them , but about doing it as of now.

  12. Urs Schreiber
    Urs Schreiber says:
    RockyMarciano

    Some would argue that it is the demand of locality that prevents from having a field that contains global instanton sectors.

    Sorry to say this once more, but this is just the point explained in the slides.

    If you tell me to which point you follow the argument, and where you first feel you're thrown, I'll help you out with further comments at that point.

    RockyMarciano

    Extraordinary claims require extraordinary proofs. The above are very strong statements, the physical validity of wich can only be demonstrated by proving how the 2-bundle contains the global instanton sector and captures the global field content.

    That part is easy. For more details than in those slides, you could see the review

    A higher stacky perspective on Chern-Simons theory
    Domenico Fiorenza, Hisham Sati, Urs Schreiber
    https://arxiv.org/abs/1301.2580

    RockyMarciano

    First you have to prove that the usual textbook procedure can't do it(in other words that the Yang-Mills existence and mass gap problem cannot be solved within the current framework), this by itself is a highly prized enterprise, and then you have to prove that the 2-bundles of higher category do indeed give rigorous path integrals(that need to capture that global content) and solve the Yang-Mills problem.

    This paragraph is mixing up a few things. What is your background? Do you know the concept of a fiber bundle? Maybe at the level of Fiber bundles in physics?

    RockyMarciano

    Without all this backing the above statements, certain scepticism from physicists is granted,

    I don't think it's fair towards the physics community to continuously refer to them as a crowd unable to understand some basics. None of the authors or references that I listed here in this thread are in pure mathematics, all of them are physicists who understand this business.

  13. RockyMarciano
    RockyMarciano says:
    Urs Schreiber

    If you tell me to which point you follow the argument, and where you first feel you're thrown, I'll help you out with further comments at that point.

    Thanks for the offer. I've found a video with the presentation of the slides, I haven't finished it yet.

    This paragraph is mixing up a few things.

    My background includes fiber bundles. I might be mixing up things but if the claim is that the usual QFT textbooks have a wrong mathematical framework(Yang-Mills gauge fields and gauge bundles) to handle locality, that should have some consequence in the form of rigorous proof, I would honestly like to know how is this reasoning confused?

    (P.S.:Maybe it's just my perception but your tone in part of your replies to my comments comes across as defensive/dismissive, not sure why. If I ask all these questions is because the approach interests me, anyway I'll keep leaving out the condescending parts).

  14. RockyMarciano
    RockyMarciano says:

    So after watching the talk I have a very basic question. When going from the usual gauge fields with their gauge equivalence relations and with the gauge principle defined informally as the fact that these gauge equivalence relations that allow the concept of gauge transformations are supposed to leave the physics unaffected,
    to the gauge fields as groupoids in which the choice of equivalence(g,h,…) is "remembered" and matters physically(in the talk it is stressed that instantons are physically important in baryogenesis and the observed manifestations of the QCD vacuum), the added step about the choice of gauge actually being physically important and observable, even if it looks as a good step locality-wise as explained in the talk, appears to me go counter the very gauge principle it is trying to extend.

    I mean if the gauge principle is about redundancy and gauge equivalence as "physical undistinguishability" of the gauge fields under gauge transformations, I'm not sure how introducing formally(the move from pre-stack to stack and from bundle to 2-bundle) this "memory of the choice of equivalence" that actually is inserting physical distinguishability, can keep the original notion o gauge equivalence. Is this modification perhaps changing the meaning of gauge symmetry, so it doesn't require for the gauge fields under gauge transformations to be physicall indistinguishable anymore? But this seemed to be the essence of the gauge principle.

    This is a very basic question that might come from some deep confusion of mine but I think is worth clarifying.

  15. Urs Schreiber
    Urs Schreiber says:
    RockyMarciano

    I mean if the gauge principle is about redundancy

    That's the thing, it is not. Only globally, when all the dust has settled, at the very end of a computation, we are intersted in passing to gauge equivalene classes. But if you insist on doing this throughout, then either locality goes out of the window or else all topological effects such as instantons go away.

    Consider the simple case of an SU(2)-instanton; its instanton number is all in the gauge transformation, it's the winding number of the gauge transformation. If you forget which specific gauge transformations you use, then you destroy all nontrivial instanton sectors.

  16. Urs Schreiber
    Urs Schreiber says:
    RockyMarciano

    To be more specific,

    That would be good

    RockyMarciano

    there is a subtle mathematical nontriviality in applying the cocycle condition to fix the above that seems to be overlooked in the slide presentation.

    That's not specific at all, it is impossible to tell what you mean. Try to ask a concrete precise question regarding the first point in the presentation where you are not following.

  17. RockyMarciano
    RockyMarciano says:

    Thanks for your replies.

    Urs Schreiber

    That's the thing, it is not. Only globally, when all the dust has settled, at the very end of a computation, we are intersted in passing to gauge equivalene classes. But if you insist on doing this throughout, then either locality goes out of the window or else all topological effects such as instantons go away.

    I can see this. By redundancy I meant that global passage to gauge equivalence at the end that you are also assuming when the dust settles. Now the intermediate steps where you are very reasonably demanding locality have to be compatible with the end global result. I was stuck trying to justify the compatibility of these two realms(local or intermediate and global or final) in a fundamental level. But I guess in doing this you are simply relying(as everyone) on the usual concept of actual infinity and the axiom of choice, right? And for this the whole apparatus of higher categories and homotopies ad infinitum fits really well so it makes sense to promote it.

    Urs Schreiber

    Try to ask a concrete precise question regarding the first point in the presentation where you are not following.

    Hope I managed above.

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