Stringification as categorisation

In quantum field theory one is typically taught to use perturbation theory when the equations of motion for the fields are nonlinear and weakly interacting. For example, in \phi^4  theory one can use a formal series as described by Rosly and Selivanov [1]. Perturbative theory is about mastering series expansions. The basic idea, upon constructing some correlation function in the full nonlinear model, is to expand in powers of \alpha  , namely the interaction strength. In the language of perturbative physics, Feynman diagrams give a representation of each term in the expansion such that we use them to illustrate linear operators. This ultimately enables us to obtain a good approximation to the exact solution. Needless to say, there is a real power and usefulness about perturbative methods and the sum of Feynman diagrams.

When computing amplitudes with Feynman diagrams, the amplitudes depend on various topological properties (i.e., vertices, loops, and so on). Although not always made explicit in the perturbative view, from the Fenynman diagrams of 0-dimensional points with 1-dimensional graphs (to use the language of p-branes, which we’ll get to in a moment), we have topologies that describe linear operators: i.e., what Feynman diagrams start to make explicit is the deeper role of topology in physics [2]. This was summarised wonderfully in a lovely article by Atiyah, Dijkgraaf, and Hitchin [3]. Mathematically, and from the perspective of geometry, the main idea is that a linear operator behaves very much like an n-dimensional manifold going between manifolds of one dimension less, which we may define as a cobordism (i.e., think of a stringy ‘trousers’ diagram) [2,4].

Now, consider the story of p-branes, in particular the perspective as we pass from standard quantum field theory to string theory. The language of p-branes as first described by Duff et al [5] may be reviewed in any introductory string theory textbook. We can, from first-principles, motivate string theory thusly: in a special, if not unique way, we may generalise the point-like 0-dimensional particle to the 1-dimensional string, which is made explicit when we generalise the action for a relativistic particle to the Nambu-Goto action for the relativistic string. In the language of p-branes, which are p-dimensional objects moving through a D(D \geq p)  dimensional space-time, a 0-brane is a (0-dimensional) point particle that that traces out a (0+1)-dimensional worldline. The generalisation of the point particle action S_0 = -m \int ds to a p-brane action in a D(\geq p) -dimensional space-time background is given by S_p = -T_p \int d\mu_p . Here T_p is the p-brane tension with units mass/vol, and d\mu_p is the (p + 1)-dimensional volume element. For the special case where p=1 such that we have 1-brane, we obtain the string action which sweeps out a (1+1)-dimensional surface that is the string worldsheet propagating through space-time. We can also go on to speak of higher-dimensional objects, such as those that govern M-theory. For instance, a 2-brane is a membrane. Historically, these were considered as 2-dimensional particles. There are also 3-branes, 4-branes, and so on.

This generalising process, if we can describe it that way, is what I like to think of as stringification. For the case where p=1 , Feynman diagrams of ordinary quantum field theory with 2-dimensional cobordisms represent world-sheets traced out by strings. The generalising picture, or stringification, show these 2-dimensional cobordisms equipped with extra structure give a powerful mathematical language (describing the relation between physics and topology, as string diagrams enable us to sum over the various topologies and provide a valuable mathematical tool for thinking about composition). But of course this picture can still be extended. Not only does the important analogy between operators and cobordisms come directly into focus, it is also, in some sense, where stringification meets categorification. That is, from the maths side, we arrive at the logic of higher-dimensional algebra and the arrows of monoidal and higher categories. In each, physical processes are describe by morphisms or functors (functors are like morphisms between categories). This generalising picture toward higher geometry, higher algebra, and, indeed, higher structures is called ‘categorifying’ or ‘homotopifying’ (my notes on which I have started to upload to this blog). In this post, I want to think a bit about this idea of stringification as categorification.

***

There is a view of M-theory, and I suppose of fundamental physics as whole, that I find fascinating and compelling: stringification as the categorisation of physics. The notion of stringification is not formal, but captures if nothing else an intuition about a certain generalising process or abstract story, or at least that is how I presently see it. It is a term I have picked up that used to float around in different contexts a couple of decades ago. As described through the language of p-branes, the story begins with the generalisation or stringification of point particle theory (and all that it implies) toward the existence of the string and eventually other extended objects in fundamental physics. Meanwhile, the notion of categorification is certainly formal, signalling, at its origin, the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories. This process, when iterated, gives definition to the notion of n-category theory, where we also replace functions with functors, and equations between functions by natural isomorphisms between functors [6]. As Schreiber pointed out in 2004, there is a sort of harmony between these two processes – stringification and categorification – which has certainly started to clarify over the last decade or more.

As one example, the observation that Schreiber describes in the linked post refers to boundaries of membranes attached to stacks of 5-branes, which conceptually appear as a higher-dimensional generalisation of how boundaries of strings appear.

To understand this think, firstly, of the simple example of the existence of D-branes (Dirichlet membranes) and how the endpoints of open strings can end on these extended objects. In fact, an introductory string textbook will guide one to see why the equations of motion of string theory require that the endpoints of an open string satisfy one of two types of boundary conditions (Dirichlet or Neumann) ending on a brane. If the endpoint is confined to the condition that it may move within some p-dimensional hyperplane, one then obtains a first description of Dp-branes. (I think this was one of the first things I calculated when learning strings!). For the sake of saving space I won’t go into the arrangement of D-branes or other related topics. The main point that I am driving at, the technicalities of which we could review in another post, is how these branes are dynamic and as such they may influence the dynamics of a string (i.e., how an open string might move and vibrate). Thus, the arrangement of branes (e.g., we can have parallel branes or ‘stacks’) will also impact or control the types of particles in our theory. It is truly a beautiful picture.

In p-brane language, if you take the Nambu–Goto action and for the quantum theory study the spectrum of particles, you will see that it exhibits what we may describe as the photon, which of course is the fundamental quantum of the electromagnetic field. Now, what is nice about this is that, the resemblance of the photon is actually a p-dimensional version of the electromagnetic field, so it is in fact a p-dimensional analogue of Maxwell’s equations.

What Schreiber is highlighting in his post is not just that in string theory, the points of the string ending on a Dp-brane give rise to ordinary gauge theory. (One could even take the view that string theory predicts electromagnitism such that string theory predicts the existence of D-branes. It is by their nature that these extended objects all carry an electromagnetic field on their volume, i.e., what we call the brane volume). The point made is that, given there is reason to extend the picture further – the picture of stringification so to speak – to higher-dimensional generalisations, we can then replace strings with membranes, and so on. From the maths side, it was realised that from the perspective of categories, something analogous is happening: replacing points with arrows (i.e., morphisms) one finds the gauge string may be described by the structure of nonabelian gerbes (a gerbe is just a generalised analogue of a fibre bundle), and so on.

When I first learned strings, the picture of stringification was in my mind but I didn’t yet have a word for it. I also didn’t possess category theoretic language at the time; it was really only a vague sense of a picture, perhaps emphasised in the way I learned string theory. So when I discovered and read last year about the idea of stringification as categorisation [7] in Schreiber’s thesis, I was excited.

A nice illustration comes from the first pages of this work. Take some ordinary point-particle, which traces out a worldline over time t . The thrust of the idea is that, given some charge, there is a connection in some bundle (yet unspecified) such that, locally, a group element g \in G is associated to the path. Diagrammatically this may be represented as,

Now consider some time t^{\prime} , where t^{\prime} > t . The particle has travelled a bit further,

We can of course compose these paths. The composition is associative and the operation is multiplication. In fact, what we’re doing is multiplying the group elements. We can also define an inverse g^{-1} . The punchline is that, from the theory of fibre bundles with connection, we can consider how this local picture may fit globally. If g is an element in a non-abelian group, the particle we are generalising is non-abelian. Generalise from a point-particle to a string, and the diagrammatic representation of the world-sheet takes the form

Ultimately, we can continue to play this game and develop the theory of non-abelian strings (and on to higher-dimensional branes), which, it turns out, corresponds with a 2-category theory [7,8]. Sparing details, in n-category theory a 2-category is a special type of category wherein, besides morphisms between objects, it possesses morphisms between morphisms. What is interesting about this example is how we can go on to show the idea of SUSY quantum mechanics on loop space relates to ideas in higher gauge theory, particularly in the sense of categorifying standard gauge theory. For example, John Baez’s paper on higher Yang-Mills [9]. But even before all of that, from the view of perturbative string theory being the categorification of supersymmetric quantum mechanics, we can play the same game such that the generalisation of the membranes of M-theory are a categorification of the supersymmetric string, and so on. The intriguing and, perhaps, grand idea, is that this process of stringification as categorification can be utilised to describe the whole of physics, or, so, it is suspected.

***

I’ve been thinking about this picture quite a bit recently, perhaps spurred by all of my ongoing studies in M-theory. The view to be encircled, as the notion of categorisation enters the stringy picture, also marks for me the beginning of the story about higher structures in fundamental physics (in terms of the view of category theory and higher category theory). In a sense, as much as I currently understand it (as I am very much in the process of studying and forming my thoughts on the matter) we are encircling not much more than an abstract story; but it is one in which many tantalising hints exist about a potentially foundational view.

The history of this higher structure view is rich with examples [10, 11], and, for many reasons, it leads us directly to a study of the plausible existence of M-theory. From the use of braided monoidal categories in the context of string diagrams through to knot theory (See Witten’s many famous lectures); the notion of quantum groups; Segal’s famous work on the axioms of conformal field theory (described in terms of monoidal functors and the category 2Cob_{\mathbb{C}} whose morphisms are string world-sheets such that we can compose the morphisms, and so on); and of course the work of Atiyah in topological quantum field theory (TQFT) followed by Dijkgraaf’s thesis on 2d TQFTs in terms of Frobenius algebras – the list is far to big to summarise in a single paragraph. All of this indicates, in some general sense, a very abstract story from basic quantum mechanics through to string theory and, I would say, as a natural consequence M-theory.

It is a fascinating perspective. There is so much to be said about this developing view, including why higher geometry and algebra seem to hold the important clues of M-theory as a fundamental theory of physics. What is also interesting, as I am beginning to understand, is that in the higher structure picture, a striking consequence from a geometric persective is that the geometry of fundamental physics (higher geometry and supergeometry) may not be described by spaces with sets of points. And, in fact, we start to see this for each value of p  . Instead of a traditional notion space associated with the definition of topological spaces or differentiable manifolds, the geometric observation is that what we’re dealing with is functorial geometry of the sort described by Grothendieck, or synthetic differential geometry of the sort described by Lawvere, or a variation of them both.

Anyway, this is just a short note of me thinking aloud.

References

[1] Rosly, A.A., and Selivanov, K.G., On amplitudes in self-dual sector of Yang-Mills theory. [arXiv:9611101 [hep-th]].

[2] Baez, J., and Stay, M., Physics, Topology, Logic and Computation: A Rosetta Stone. [arXiv:0903.0340 [quant-ph]].

[3] Atiyah, M., Dijkgraaf, R., and Hitchin, N., Geometry and physics. Phil. Trans. R. Soc., (2010), A.368, 913–926. [http://doi.org/10.1098/rsta.2009.0227].

[4] Baez, J., and Lauda, A., A Prehistory of n-Categorical Physics. [https://math.ucr.edu/home/baez/history.pdf.].

[5] M. J. Duff, T. Inami, C. N. Pope, E. Sezgin [de], and K. S. Stelle, Semiclassical quantization of the supermembrane. Nucl. Phys. B297 (1988), 515.

[6] Baez, J., and Dolan, J., Categorification. (1998). [arXiv: 9802029 [math.QA]].

[7] Schreiber, U., From Loop Space Mechanics to Nonabelian Strings [thesis]. (2005). [hep-th/0509163].

[8] Baez, J. et al., Categorified Symplectic Geometry and the Classical String. (2008). [math-ph/0808.0246v1].

[9] Baez, J., \textit{Higher Yang–Mills theory}. (2002). [hep-th/0206130].

[10] Baez, J., and Lauda, A., A Prehistory of n-Categorical Physics. [https://math.ucr.edu/home/baez/history.pdf.]

[11] Jurco, B. et al., \textit{Higher structures in M-theory}. (2019). [arXiv:1903.02807v2].