Why Higher Category Theory in Physics?

 

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 Ph.D. 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.

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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 Ph.D. work began with the task of applying the same techniques to the world volume 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 originate 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 with 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 Ph.D. thesis work was the algebraic deformation theory of this setup, and this is where I found higher category/higher homotopy theory shows 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:

• Urs Schreiber, On deformations of 2d SCFTs, JHEP 0406:058,2004 (arXiv:hep-th/0401175)

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:

• John Baez, Higher Yang-Mills Theory (arXiv:hep-th/0206130)

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 Ph.D. thesis:

• Urs Schreiber, From Loop Space Mechanics to Nonabelian Strings, PhD thesis 2005 (arxiv:hep-th/0509163)

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 equivalent to “Stasheff Lie-2 algebras” with metric.

The theory touched on in that outlook at the end of my Ph.D. thesis — the theory of higher nonabelian gauge fields of arbitrary degree, via Lie $n$-algebras for arbitrary $n$ — eventually materialized in these articles:

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_\infty$ algebra connections and applications to String- and Chern-Simons $n$-transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 (arxiv:0801.3480)
• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Čech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1, 2012 (arXiv:1011.4735)

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?

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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 Postnikov-Whitehead 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 from 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 looks like abelian gauge fields.

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

• Eric Sharpe, Discrete torsion and gerbes (1999)

 

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 strength 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

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted differential String and Fivebrane structures, Commun. Math. Phys. 315 (2012), 169-213 (arxiv:0910.4001)

 

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:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The  moduli 3-stack of the $C$-field, Communications in Mathematical Physics, Volume 333, Issue 1, 2015 (arXiv:1202.2455 )

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 turn the naive 7d Chern-Simons theory into a non-abelian higher gauge field theory, as we showed in this article:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory, Advances in Theoretical and Mathematical Physics, Volume 18, Number 2, 2014 (arXiv:1201.5277)

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 world volume 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 a 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 national level is in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Rational sphere valued supercocycles in M-theory and type IIA string theory, Journal of Geometry and Physics 2017 (arXiv:1606.03206)

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

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.)
  Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795)

 

These examples 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

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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 category/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:

• Urs Schreiber, Super Lie $n$-algebra of Super $p$-branes, talks at the Fields, Strings, and Geometry Seminar, Surrey Dec. 5 – 9, 2016

based on our articles

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie $n$-algebra extensions, higher WZW models and super $p$-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018  (arXiv:1308.5264)
• Domenico Diorenza, Hisham Sati, Urs Schreiber, T-Duality from super Lie $n$-algebra cocycles for super $p$-branes (arXiv:1611.06536)

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.

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But string theory is not the only 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-dimensional 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:

• Urs Schreiber, Higher Structures in Mathematics and Physics, an introductory talk, held at a) Meeting of Maths@CAS Brno, 2016 Nov 9-11 b) Oberwolfach Workshop 1651a, 2016 Dec. 18-23

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

• Intuitively, why are bundles so important in physics?

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 layout in

• Urs Schreiber, Modern Physics formalized in Modal Homotopy Type Theory, to appear in Stefania Centrone, Deborah Kant, and Deniz Sarikaya (eds.) What are Suitable Criteria for a Foundation of Mathematics?, Synthese Library, Springer

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.