Doubled diffeomorphisms and the generalised Ricci curvature

On October 7, 2021October 8, 2021 By Robert C. SmithIn Physics

I was asked a question the other week about the idea of doubled diffeomorphisms, such as those found in double field theory. A nice way to approach the concept is to start with dualised linearised gravity [1]. That is to say, we start with a theory considering only the field $h_{ij}(x^{\mu}, x^a, \tilde{x}_a)$ . This field transforms under normal linearised diffeomorphism as

$\delta h_{ij} = \partial_i \epsilon_j + \partial_j \epsilon_i \ \ (1)$

and, under the dual diffeomorphism as

$\tilde{\delta} h_{ij} = \tilde{\partial}_i \tilde{\epsilon}_j + \tilde{\partial}_j \tilde{\epsilon}_i. \ \ (2)$

Now, take the basic Einstein-Hilbert action

$S_{EH} = \frac{1}{2k^2} \int \ \sqrt{-g} \ R, \ \ (3)$

and expand to quadratic order in the fluctuation field $h_{ij}(x) = g_{ij} - \eta_{ij}$ . Just think of standard linearised gravity with the following familiar quadratic action

$S^2_{EH} = \frac{1}{2k^2} \int \ dx \ [\frac{1}{4} h^{ij}\partial^2 h_{ij} - \frac{1}{4} h \partial^2 h + \frac{1}{2} (\partial^i h_{ij})^2 + \frac{1}{2} h \partial_i \partial_j h^{ij}]. \ \ (4)$

This is the Feirz-Pauli action and it is of course invariant under (1). But we want a dualised theory. The naive thing to do, for the field $h(x, \tilde{x})$ , is to add a second collection of tilde dependant terms. In comparison with (4), we also update the integration measure to give

$S^2_{EH} = \frac{1}{2k^2} \int \ dx d\tilde{x} \ [\frac{1}{4} h^{ij}\partial^2 h_{ij} - \frac{1}{4} h \partial^2 h \\ + \frac{1}{2} (\partial^i h_{ij})^2 + \frac{1}{2} h \partial_i \partial_j h^{ij} + \\ \frac{1}{4} h^{ij} \tilde{\partial}^2 h_{ij} - \frac{1}{4} h \tilde{\partial}^2 h \\ + \frac{1}{2} (\tilde{\partial}^i h_{ij})^2 + \frac{1}{2} h \tilde{\partial}_i \tilde{\partial}_j h^{ij}]. \ \ (5)$

If you decompose $x, \tilde{x}$ such that $h_{ij} (x)$ no longer depends on $\tilde{x}$ , then this action simply reduces to linearised Einstein gravity on the coordinate space $x^a.$ Similarly, for the dual theory.

When the doubled action (5) is varied under $\tilde{\delta}$ , the second line is invariant under (2). However, the first line gives

$\tilde{\delta} S = \int [dx d\tilde{x}] [h^{ij} \partial^2 \tilde{\partial}_i \tilde{\epsilon}_j + \partial_i h^{ij} (\partial^k \tilde{\partial}_{k})\tilde{\epsilon}_j \\ - h \partial^2 \tilde{\partial} \tilde{\epsilon} + h(\partial_i \tilde{\partial}^i)\partial_j \tilde{\epsilon}^j \\ + \partial_i h^{ij} \partial^k \tilde{\partial}_j \tilde{\epsilon}_k + (\partial_j \partial_j h^{ij})\tilde{\partial} \tilde{\epsilon}. \ \ (6)$

As one can see, the terms on each line would cancel if the tilde derivatives were replaced by ordinary derivatives. Rearranging and grouping like terms, and then relabelling some indices we find

$\tilde{\delta} S = \int [dx d\tilde{x}] \ [(\tilde{\partial}_j h^{ij})\partial^k (\partial_i \tilde{\epsilon}_k - \partial_k \tilde{\epsilon}_i) \\ + (\partial_i \partial_j h^{ij} - \partial^2h) \tilde{\partial} \tilde{\epsilon} \\ + (\partial^i h_{ij} - \partial_j h)(\partial \tilde{\partial})\tilde{\epsilon}^j. \ \ (7)$

For this to be invariant under the transformation $\tilde{\delta}$ we have to cancel each of the terms. In order to cancel the variation, new fields with new gauge transformations are required. For the first term, a hint comes from the structure of derivatives, namely the fact we have a mixture of tilde and non-tilde derivatives. The Kalb-Ramond b-field mixes derivatives in this way, and, indeed, for the first term to cancel we may add $b_{ij}$ . We denote this inclusion to the action as $S_b$

$S_b = \int [dx d\tilde{x}] \ (\tilde{\partial}_j h^{ij})\partial^k b_{ik}, \\ with \ \ \tilde{\delta}b_{ij} = - (\partial_i \tilde{\epsilon}_j - \partial_j \tilde{\epsilon}_i). \ \ (8)$

The second term can similarly be killed upon introduction of the dilaton $\phi$ . It takes the form

$S_{\phi} = [dx d\tilde{x}] (-2) (\partial_i \partial_j h^{ij} - \partial^2 h) \phi, \ \ \text{with} \ \ \tilde{\delta}\phi = \frac{1}{2}\tilde{\partial} \tilde{\epsilon}. \ \ (9)$

This is quite nice, if you think about it. It is not the full story, because in the complete picture of double field theory we need to add more terms and their are several subtlties. In the naive case of dualised linearised gravity, we find in any case that linearised dual diffeomorphisms for the field $h_{ij}$ requires, naturally and perhaps serendipitously, a Kalb-Ramond gauge field and a dilaton – i.e., the closed string fields for the NS-NS sector.

We are now only left with one term, which is the one with curious structure on the third line in (7). To kill this term, we can observe that the gauge parameter $\tilde{\epsilon}$ satisfies the constraint $\partial \cdot \tilde{\partial} = 0$ derived from the level matching condition. This constraint says that fields and gauge parameters must be annihilated by $\partial \tilde{\partial}$ , and it is fairly easy to find in an analysis of the spectrum in closed string field theory.

So that is one way to attack the remaining term. But what is also interesting, I think, is that it is possible to accomplish the same goal by adding more fields to the theory. This is a non-trivial endeavour, to be sure, as the added fields would need to be invariant under $\delta$ and $\tilde{\delta}$ transformations. Ideally, one would likely want to be able to generalise the added fields to the formal case of the duality invariant theory. But it presents an interesting question.

***

From the perspective of string field theory, double field theory wants to describe a manifestly T-duality invariant theory (we talked about this in a number of past posts). The strategy is to look at the full closed string field theory comprising an infinite number of fields, and instead select to focus on a finite subset of those fields, namely the massless NS-NS sector. So DFT is, at present, very much a truncation of the string spectrum.

As a slight update to notation to match convention, for the massless fields of the NS-NS sector let’s now write the metric $g_{ij}$ , with the b-field $b_{ij}$ and dilaton $\phi$ the same as before. The effective action of this sector is famously

$\displaystyle S_{NS} = \int dX \sqrt{-g} e^{-2\phi} [R + 4(\partial \phi)^2 - \frac{1}{12}H^2] + \text{higher derivative terms}. \ \ (10)$

As one can review in any string textbook, this action is invariant under local gauge transformations: diffeomorphisms and a two-form gauge transformation. The NS-NS field content transforms as

$\displaystyle \delta g_{ij} = L_{\lambda} g_{ij} = \lambda^{k} \partial_k g_{ij} + g_{kj}\partial_i \lambda^k + g_{ik}\partial_i \lambda^k,$

$\displaystyle \delta b_{ij} = L_{\lambda} b_{ij} = \lambda^k \partial_k b_{ij} + b_{kj}\partial_i \lambda^k + b_{ik}\partial_i\lambda^k,$

$\displaystyle \delta \phi = L_{\lambda} \phi = \lambda^k \partial_k \phi. \ \ (11)$

We define the Lie derivative $L_{\lambda}$ along the vector field $\lambda^i$ on an arbitrary vector field $V^i$ such that the Lie bracket takes the form

$\displaystyle L_{\lambda} V^i = [\lambda, V]^i = \lambda^j \partial_j V^i - V^j \partial_j \lambda^i. \ \ (12)$

For the Kalb-Ramond two-form $b_{ij}$ , the gauge transformation is generated by a one-form field $\tilde{\lambda}_i$

$\displaystyle \delta b_{ij} = \partial_i \tilde{\lambda}_j - \partial_j \tilde{\lambda}_i. \ \ (13)$

One way to motivate a discussion on doubled or generalised diffeomorphisms in DFT is to understand that what one wants to do is essentially generalise the action (10). This means that at any time we should be able to recover it. The generalised theory should therefore possess all the same symmetries (with added requirement of manifest invariance under T-duality), including diffeomorphism invariance.

In the generalised metric formulation [2] the DFT action reads

$\displaystyle S_{DFT} = \int d^{2D} X e^{-2d} \mathcal{R}, \ \ (14)$

where

$\displaystyle \mathcal{R} \equiv 4\mathcal{H}^{MN}\partial_M \partial_N d - \partial_M \partial_N \mathcal{H}^{MN} \\ - 4\mathcal{H}^{MN}\partial_{M}d\partial_N d + 4\partial_M \mathcal{H}^{MN} \partial_N d \\ + \frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL} - \frac{1}{2} \mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}. \ \ (15)$

This action is constructed [2] precisely in such a way that it captures the same dynamics as (10). Here $\mathcal{H}$ is the generalised metric, which combines the metric and b-field into an $O(D,D)$ valued symmetric tensor such that

$\displaystyle \mathcal{H}^{MN}\eta_{ML}\mathcal{H}^{LK} = \eta^{NK}, \ \ (16)$

where $\eta$ is the $O(D,D)$ metric. We spoke quite a bit about the generalised metric and the role of $O(D,D)$ in a past post (see this link also for further definitions, recalling for instance the T-duality transformation group is $O(D,D; \mathbb{R})$ , which is discretised to $O(D,D; \mathbb{Z})$ . If $O(D,D)$ is broken to the discrete $O(D,D;\mathbb{Z})$ , then one can interepret the transformation as acting on the background torus on which DFT has been defined). Also note that in (15) $d$ is the generalised dilaton. In the background independent formulation of DFT [5], $e^{-2d}$ is shown to be a generalised density such that the dilaton $\phi$ with the determinant of the undoubled metric $g = \det g_{ij}$ on the whole space is combined into an $O(D,D)$ singlet $d$ establishing the identity $\sqrt{-g}e^{-2\phi} = e^{-2d}$ . We’ll talk a bit more about this later.

There are a number of important characteristics built into the definition of the generalised Ricci (15). Firstly, it is contructed to be an $O(D,D)$ scalar. One can show that the action (14) possesses manifest global $O(D,D)$ symmetry

$\displaystyle \mathcal{H}^{MN} \rightarrow \mathcal{H}^{LK}M_{L}^{M}M_{K}^{N} \ \ \text{and} \ \ X^{M} \rightarrow X^{N}M_{N}^{M}, \ (17)$

where $M_{L}^{K}$ is a constant tensor which leaves $\eta^{MN}$ invariant such that

$\displaystyle \eta^{LK} M_{L}^{M} M_{K}^{N} = \eta^{MN}. \ \ (18)$

Importantly, given $O(D,D)$ extends to a global symmetry, we may define this under the notion of generalised diffeomorphisms. Unlike with the supergravity action (10), which is invariant under the gauge transformations (11) and (12), in DFT the metric and b-field are combined into a single object $\mathcal{H}$ . So the obvious task, then, is to find a way to combine the diffeomorphisms and two-form gauge transformation in the form of some generalised gauge transformation. This is really the thrust of the entire story.

To see how this works, as a brief review, we define some doubled space $\mathbb{R}^{2D}.$ To give a description of this doubled space, all we need to start is some notion of a differential manifold with the condition that we have a linear transformation of the coordinates $X^{\prime} = hX$ , where $h \in O(D,D)$ (similar to the transformation we defined in the post linked above). We will include the generalised dilaton $d$ and we also include the generalised metric $\mathcal{H}$ , although we can keep this generic in definition should we like. For $\mathcal{H}$ we require only that it satisfies the $O(D,D)$ constraint $\mathcal{H}^{-1} = \eta \mathcal{H} \eta$ , where, from past discussion, one will recall $\eta$ is the $0(D,D)$ metric. It transforms $\mathcal{H}^{\prime}(X^{\prime}) = h^{t}\mathcal{H}(X)h$ . We now have everything we need.

Definition 1. A doubled space $\mathbb{R}^{2D}(\mathcal{H},d)$ is a space equipped with the following:

1) A positive symmetric $2D \times 2D-\text{matrix}$ field $\mathcal{H}$ , which is the generalized metric. This metric must satisfy the above conditions and transform covariantly under $O(D,D).$

2) A generalised dilaton scalar $d$ , which is a $2D$ scalar density such that $d = \phi - \frac{1}{2} \ln \det h$ (we’ll show this in a moment).

a) The generalised dilaton is related to the standard dilaton as already described above.

With this definition, we can then advance to define the notion of an $O(D,D)$ module, generalised vectors and vector fields, and so on. To keep our discussion short, the point is that in defining an $O(D,D)$ vector we may combine from before the vector $\lambda^i$ and one-form $\tilde{\lambda}_i$ as generalised gauge parameters

$\displaystyle \xi^M = (\tilde{\lambda}_i, \lambda^i). \ \ (19)$

One can see how this is done in [2,3]. In short, the combination of the gauge transformations into the general gauge transformation with parameter $\xi^M$ is defined under the action of a generalised Lie derivative. The result is simply given here as

$\displaystyle \mathcal{L}_{\xi}A_M \equiv \xi^P \partial_P A_M + (\partial_M \xi^P - \partial^P \xi_M)A_p,$

$\displaystyle \mathcal{L}_{\xi}B^M \equiv \xi^P \partial_P B^M + (\partial^M \xi_P - \partial_P \xi^M)B^p. \ \ (20)$

From this definition, where, it should be said, $A$ and $B$ are generalised vectors, we can eventually write the generalised Lie derivative of $\mathcal{H}$ and $d$ .

$\displaystyle \mathcal{L}_{\xi} \mathcal{H}_{MN} = \xi^P \partial_P \mathcal{H}_{MN} + (\partial_M \xi^P - \partial^P \xi_M)\mathcal{H}_{PN} + (\partial_N \xi^P - \partial^P \xi_N)\mathcal{H}_{MP},$

$\displaystyle \mathcal{L}_{\xi}(e^{-2d}) = \partial_M(\xi^M e^{-2d}). \ \ (21)$

What we see is that, indeed, the generalised dilaton, which we may think of as an $O(D,D)$ singlet, transforms as a density. This means we may think of it as a generalised density. It can also be shown that the Lie derivative of the $O(D,D)$ metric $\eta$ vanishes and therefore the metric is preserved.

What we want, for the purposes of this post, is the generalised Lie derivative of the generalised scalar curvature (15). What we find is that, indeed, it transforms as a scalar provided that the definition of (15) includes the full combination of terms.

$\displaystyle \mathcal{L}_{\xi} \mathcal{R} = \xi^M \partial_M \mathcal{R}.$ (22)

Or, looking at the action (14) as a whole, the subtlety is that the generalised dilaton forms part of the integration measure. The action does not possess manifest generalised diffeomorphism invariance in the typical sense that we might think about it, but it is constructed precisely in such a way that

$\displaystyle \mathcal{L}_{\xi}(e^{-2 d})\mathcal{R} = \partial_I (\xi^{I} e^{-2d}\mathcal{R}) \ \ (26)$

vanishes in the action integral (due to being a total derivative). So we find (14) does indeed remain invariant.

As a brief aside, from the transformations of the generalised metric and the dilaton, we can define an algebra [4]

$\displaystyle [\mathcal{L}_{\xi_1}, \mathcal{L}_{\xi_2}] = \mathcal{L}_{\xi_1} \mathcal{L}_{\xi_2} - \mathcal{L}_{\xi_2} \mathcal{L}_{\xi_1} = \mathcal{L}_{[\mathcal{L}_{\xi_1} \mathcal{L}_{\xi_2}]_C}, \ \ (23)$

where we find first glimpse at the presence of the Courant bracket. Provided the strong $O(D,D)$ constraint of DFT is imposed

$\displaystyle \partial_N A_I \partial^{N} A^J = 0 \ \forall \ i,j, \ \ (24)$

then the Courant bracket governs this algebra such that

$\displaystyle [\mathcal{L}_{\xi_1}, \mathcal{L}_{\xi_2}]^{M}_{C} = \xi_{1}^{N}\partial_{N}\xi_{2}^{M} - \frac{1}{2}\xi_{1N}\partial^{M}\xi_{2}^{N} - (\xi_1 \leftrightarrow \xi_2). \\ (25)$

An important caveat or subtlety about this algebra is that it does not satisfy the Jacobi identity. This means that the generalised diffeomorphisms do not form a Lie algebroid. But nothing fatal comes from this fact for the reason that, whilst we may like to satisfy the Jacobi identity, the gauge transformation leaves all the fields invariant that fulfil the strong $O(D,D)$ constraint.

In closing, recall that DFT starts with the low-energy effective theory as a motivation. It is good, then, that a solution of (24) is to set $\tilde{\partial} = 0$ giving (10). The Ricci scalar is the only diffeomorphism invariant object in Riemannian geometry that can be constructed only from the metric with no more than two derivatives. In DFT, we have an action constructed only from the generalised metric and doubled dilaton with their derivatives.

References

[1] Hull, C.M., and Zweibach, B., Double field theory. (2009). [arXiv:0904.4664 [hep-th]].

[2] Hohm, O., Hull C.M., and Zwiebach, B., Generalized metric formulationof double field theory. JHEP, 08:008, 2010. [arXiv:1006.4823 [hep-th]].

[3] Zwiebach, B., Double field theory, T-duality, and Courant brackets. [arXiv:1109.1782 [hep-th]].

[4] Hull, C.M., and Zwiebach, B., The gauge algebra of double field theory and courant brackets. Journal of High Energy Physics, 2009(09):090–090, Sep 2009. [arXiv:0908.1792 [hep-th]].

[5] Hohm, H., Hull, C.M., and Zwiebach, B., Background independent actionfor double field theory. Journal of High Energy Physics, 2010(7), Jul 2010. [arXiv:1003.5027 [hep-th]].

Stringification as categorisation

On September 27, 2021September 27, 2021 By Robert C. SmithIn Physics

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].

Start of new semester, thinking about double field theory cosmology

On September 13, 2021September 13, 2021 By Robert C. SmithIn Physics

I haven’t added much to my blog in the past weeks. With university kicking off again, and with Tony and I having our first work sessions of the semester, it has been quite busy. I’ve also been adjusting to being back at university after summer holiday, and with being back on campus for the first time since lock down due to the pandemic. So I’ve been finding my feet again with new daily structure and routine.

I’ve also been working on a number of projects, some short-term and some long-term, which have kept me quite occupied. It is the battle of constantly balancing enticing questions and ideas that define the day. It’s what makes life exciting and keeps me coming back to physics, I suppose.

In the last week or so we’ve been talking more about double field theory cosmology, mainly from the perspective of how matter couples. As a developing area of research there are many interesting questions one can ask. It’s quite interesting stuff, to be honest, and I’m looking forward to potentially pursuing a few side projects in this area. As it relates, I’m interested in higher ${\alpha^{\prime}}$ corrections, non-perturbative solutions, and ${\alpha^{\prime}}$ deformed geometric structures.

To share a bit more, one thing that is quite neat about DFT cosmology is how, under a cosmological ansatz [1,2], the equations coupled to matter take the form

$\displaystyle 4d^{\prime \prime} - 4(d^{\prime})^2 - (D-1)\tilde{H}^2 + 4\ddot{d} - 4 \dot{d}^2 - (D - 1)H^2 = 0$

$\displaystyle (D - 1)\tilde{H}^2 - 2 d^{\prime \prime} - (D - 1)H^2 + 2\ddot{d} = \frac{1}{2}e^{2d} E$

$\displaystyle \tilde{H}^{\prime} - 2\tilde{H}d^{\prime} + \dot{H} - 2h\dot{d} = \frac{1}{2} e^{2d}P. \\ (1)$

Here ${E}$ and ${P}$ denote energy density and pressure, respectively. These equations are duality invariant provided ${E \leftrightarrow -E}$ and ${P \leftrightarrow -P}$ . The approaches that make use of these equations are typically restricted to dilaton gravity. That is to say, the B-field is switched off. From what I presently understand the reason for this is because it is generally unknown how proceed with the full massless string sector explicit in the theory.

For a homogeneous and isotropic cosmology the metric takes the form

$\displaystyle dS^2 = -dt^2 + \mathcal{H}_{MN} dx^M dx^N$

$\displaystyle = -dt^2 + a^2(t) dx^2 + a^{-2}(t) d\tilde{x}, \ \ (2)$

where ${t}$ is physical time, ${a(t)}$ is the cosmological scale factor, ${x}$ denote are co-moving spatial coordinates. In general, the basic fields reduce to the cosmological scale factor ${a(t, \tilde{t})}$ and the dilaton ${\phi(t, \tilde{t})}$ .

Most pertinently, as we are dealing with a manifestly T-duality invariant theory, what one finds is that T-duality results in scale factor duality. In some ways, this is expected. With the B-field off, the background fields transform

$\displaystyle a(t, \tilde{t}) \rightarrow \frac{1}{a(\tilde{t},t)},$

$\displaystyle \phi(t, \tilde{t}) \rightarrow \phi(\tilde{t}, t). \ \ (3)$

The T-duality invariant combination of the scale factor and the dilaton is

$\displaystyle \phi \equiv \phi - d\ln a, \ \ (4)$

where ${d = D-1}$ is the number of spatial dimensions with D space-time dimensions.

It will be interesting to read more about the work that has so far been done in this area. One thing that is very clear, the approaches to DFT cosmology that I have so far looked at ultimately go back to Tseytlin and Vafa [3], and, also, of course, to efforts in string gas cosmology.

The main thing about these types of approaches behind (1) is that, rather than using T-duality variables, they leverage T-duality frames. The assumption, again, is the use of the section condition (conventional in DFT), which states the fields only depend on a D-dimensional subset of the space-time variables. We’ve talked about this in the past on this blog. There are different, often arbitrary choices, of this condition – what we call frames – and these different frames are related by T-duality.

The most basic example is the supergravity frame with standard coordinates transformed to the winding frame with dual coordinates. And so, what one can do, is calculate supergravity and winding frame solutions of the cosmological equations (1), with these solutions being T-dual to each other [4].

In review of ongoing efforts, it will be interesting to see what ideas might arise in the coming weeks.

References

[1] H. Wu and H. Yang, Double Field Theory Inspired Cosmology. JCAP 1407, 024 (2014) doi:10.1088/1475- 7516/2014/07/024 [arXiv:1307.0159 [hep-th]].

[2] R. Brandenberger, R. Costa, G. Franzmann and A. Welt- man, T-dual cosmological solutions in double field theory. [arXiv:1809.03482 [hep-th]].

[3] A. A. Tseytlin and C. Vafa, Elements of string cosmol- ogy. Nucl. Phys. B 372, 443 (1992) doi:10.1016/0550- 3213(92)90327-8 [hep-th/9109048].

[4] H. Bernardo, R. Brandenberger, G. Franzmann, T-Dual Cosmological Solutions of Double Field Theory II. [ arXiv:1901.01209v1 [hep-th]].

The language of morphisms and the notion of a diagram

On August 24, 2021August 24, 2021 By Robert C. SmithIn Mathematics

In category theory, different types of notation are common. Ubiquitous and important in the formalism is diagram notation. I like to think of it as follows: the diagram finds natural expression in category theory because, as emphasised in the first entry of my notes, in approaching the idea of a category $\mathcal{C}$ we may take the view that what we are defining is the language of morphisms. Indeed, it is the primacy of morphisms, and not the objects of a category, that is emphasised.

Recall that morphisms – what we have also described as structure-preserving maps – are represented graphically as arrows. From this, the next important idea is composition – that is, a view of all of the ways in which we may compose arrows. As discussed, composition is built into the definition of a category, and one observation that may be proffered is that, as a language of morphisms, when working toward category theory we may extend our view that it is furthermore a minimalist language of composition. (Eventually, once we discuss categorical products and functors (a kind of morphism of categories), we will extend this view of category theory as fundamentally the language of duality). In a sense, it is as though we are building the ideas in stages. Putting these two things together, the primacy of morphisms and the rule of composition, we also arrive at the notion of how we may express the equality of arrows using diagrams.

Categorical diagrams are powerful for many reasons. One reason has to do with how a new proof technique is devised: i.e., what is called the diagram chase [1]. For example, commutative diagrams are utilised as a new technique of proof in homology theory. In this note, as a matter of introduction, we will think about diagrams and morphisms in a basic way. Saunders Mac Lane [2] motivates it concisely, ‘Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows’. Furthermore, in these notes, while ‘[m]any properties of mathematical constructions may be represented by universal properties of diagrams’, eventually the motivation is to study physical objects like branes.

The notion of a diagram

What follows is a very brief introduction to the notion of a diagram. In particular, we are talking here about commutative diagrams. We will be somewhat informal in definition. The reason is as follows: there are two ways to approach the concept of a diagram and to give it definition. One way is through the notion of a functor, which is the way I prefer, but it is less intuitive and the topic of functors has been saved for more advanced discussion. Another way comes from a more direct and perhaps intuitive representation, namely how we may approach the commutative diagram as a directed graph of morphisms in a category.

Let’s start with something basic and familiar, an example of structured sets. A lot of objects in mathematics may be thought of as structured sets, and we can begin with a basic illustration of a structured graph with arrows from one vertex to another. This can be thought of as extra structure, or, indeed, a set with structure. A simple graphical representation is given below.

Here we have a vertex set of some graph mapped to vertex set of another graph. The arrows represent a function that preserves the graph structure. That is, there is a graph (homo)morphism such that, if vertex $A$ is linked to $B$ then $f(A)$ is linked to $f(B)$ . That is to say, if $A$ and $B$ are linked in one graph, then the image of $A$ is linked to the image of $B$ . (In fact, if we were working in category theory it would be said that this forms another category).

Now, let us advance this idea. Consider a set $A$ and a set $B$ . Define an arrow $f: A \rightarrow B$ representing a function from $A$ to $B$ , following the rule that $a \rightarrow f(A)$ assigns to each element $a \in A$ an element $f(a) \in B$ . A diagram of sets and functions may be written as below.

Here we have a commutative triangle, which we may define as a category $\mathcal{C}$ (suppressing identity arrows, etc. in the diagram) in which the hypotenuse $h$ equals the composite $g \circ f$ . In other words, this diagram is commutative when $h = g \circ f$ , where $g \circ f : A \rightarrow C$ is the composite function $a \rightarrow g(f a)$ . Generally, a diagram commutes if any two paths of composable arrows in the directed graph with common source and target have the same composite. The vertices in the graph are labelled by objects in the category, and the edges are labelled by morphisms (arrows) in the category.

Definition 1. A category diagram is commutative if for every pair of objects, all directed paths from one to the other give rise to the same morphism, i.e., the composite arrow along the first path is equal to the composite arrow along the second path.

As a matter of convention, unless explicitly stated otherwise, in these notes (and in many texts) a diagram is commutative.

The language of morphisms

It is beneficial to think a bit more about different kinds of arrows and how they interact with other arrows in the relevant category. (This is a very brief review, and more detail can be found in [1, 2, 3, 4]).

In set-theoretic language, when we speak of sets we can of course speak of elements of those sets. In category-theoretic language, however, when speaking of the category Set it is no longer possible to speak of the elements of sets as each set is simply an object in Set. More pointedly, objects in categories have nothing ‘inside’ of them in the way we may think of elements of a set. In this sense, it is nice to think of the idea of a category as a representation of structure. One may then be inclined to ask: where is the information? How does one know how many elements there are in these sets? Is there a loss of information? It turns out that we don’t really lose information. In the example of Set, we can recover information about the sets just from knowing the object that represents the sets and arrows between the objects that represent the functions. The main tool at our disposal, again, is the morphism. What categorical language and logic gives us is not so much a disadvantage as a valuable shift in perspective.

As Emily Riehl writes, ‘A category provides a context in which to answer the question, “When is one thing the same as another thing?”. Almost universally in mathematics, one regards two objects of the same category to be “the same” when they are isomorphic’.

Isomorphism

Take a simple set theoretical example of an isomorphism.

Here we have a one-to-one map. The property of having an inverse $g$ is what makes this map an isomorphism. Notice, if we compose $f$ and $g$ we obtain an identity $id_A$ . Similarly, if we compose $g$ and $f$ we obtain an identity $id_B$ . Now, compare this with a category-theoretic view of an isomorphism.

Here we have an arrow $f: A \rightarrow B$ , which is an isomorphism if there exists $g: B \rightarrow A$ such that $g \circ f = id_A$ and $f \circ g = id_B$ .

Definition 2. In a category $\mathcal{C}$ , two objects $A,B \in \mathcal{C}$ are isomorphic (to each other) if there are $f \in \mathcal{C}(A,B)$ , $g \in \mathcal{C}(B,A)$ such that $g f = id_A$ and $f g = id_B$ . In this case, we say that $f$ and $g$ are isomorphisms, and write $A \equiv B$ .

If $f$ is an isomorphism, the morphism $g$ such that $g f = f g = id$ is uniquely determined. We write $g = f^-1$ , and say that $g$ is the inverse of $f.$

Different categories possess different notions of isomorphism. Standard examples include: in the category Set, isomorphisms are bijective functions; in Grp they are bijective group (homo)morphisms; in the category of metric spaces Met (with non-expansive maps), they are isometries; in Vect they are invertible linear transformations; and in the category of topological spaces Top, they are (homeo)morphisms (i.e. continuous bijections with continuous inverses). In the language of category theory, it is generally not the case that one might distinguish between isomorphic objects. Instead, objects are determined ‘up to isomorphism’, which is a pertinent point that will be highlighted again when we start discussing categorical homotopy theory among other things.

Monomorphisms

Definition 3. A morphism $f \in \mathcal{C}(x,y)$ is a monomorphism if it satisfies the property: for every $w \in \mathcal{C}$ , $g_1, g_2 \in \mathcal{C}(w,x)$ , $f_{g_1} = f_{g_2} \implies g_1 = g_2$ . This property is called left cancellation.

If $f$ is a monomorphism, this is conventionally denoted $\hookrightarrow$ or $\mapsto$ . In adjectival form, an monomorphism is mono.

Example. Consider the category Top of locally path-connected, pointed topological spaces (a pointed space is a space together with a choice of basepoint). Let $X$ be such a space and $\pi : \tilde{X} \rightarrow X$ a covering map (which may or may not be injective). This covering map $\pi$ is a monomorphism. Having restricted to the case where we can lift maps to $X$ back up to the covering space $\tilde{X}$ , any map $f$ which can be factored through $\tilde{X}$ as some map $\pi \circ \tilde{f}$ can be lifted. Since the basepoint is fixed, it can be lifted uniquely back to $f$ , picking out arrows $g,h$ from some space $Y$ to $\tilde{X}$ . Since these maps are both lifts of the maps $\pi \circ g = \pi \circ h$ , uniqueness of lifts gives $g = h$ .

Epimorphism

Definition 4. A morphism $f \in \mathcal{C}(x,y)$ is an epimorphism if it satisfies the property: for every $z \in \mathcal{C}$ , $h_1, h_2 \in \mathcal{C}(y,z)$ , $h_1 f, h_2 f \implies h_1 = h_2$ . This property is called right cancellation.

When $f$ is an epimorphism, it is convention to represent it as $\twoheadrightarrow$ . In adjectival form, an epimorphism is epic.

Example. Consider the category TopHaus of Hausdorff topological spaces. Let $\mathcal{C}$ be the category of TopHaus, and $i : Q \rightarrow R$ be the standard inclusion. If $X$ is another Hausdorff topological space, $g_1, g_2 : R \rightarrow X$ are continuous functions with $g_1 i = g_2 i$ . Then it can be argued $g_1 = g_2$ using the fact that in Hausdorff spaces, convergent sequences have at most one limit, and continuous functions preserve limits: for any $x \in R$ , $x = \lim_{n \rightarrow \infty} q_n$ where $q_n \in \mathbb{Q}$ . It follows,

$g_1(x) = g_1 (\lim_{n \rightarrow \infty} x_n)$

$= \lim_{n \rightarrow \infty} g_1(x_n)$

$= \lim_{n \rightarrow \infty} g_2(x_n)$

$= g_2(\lim_{n \rightarrow \infty} x_n)$

$= g_2(x).$

The injection from $Q$ to $R$ is not a surjection, even though it is an epimorphism.

Sections and retractions

Definition 5. Let $\mathcal C$ be a category. A section is a morphism $f \in \mathcal{C}(x,y)$ such that there is some morphism $g \in \mathcal{C}(y,x)$ with $gf = id_x$ . Sometimes it is said that $f$ is a section of $g$ , or a right inverse to $g$ . Given $g$ , if such an $f$ exists we say that $g$ admits a section. All sections are monomorphisms.

Definition 6. A retraction is a morphism $g \in \mathcal{C}(x,y)$ such that there is some morphism $f \in \mathcal{C}(y,x)$ with $gf = id_x$ . Sometimes it is said $f$ is a retraction of, or left inverse to, $g$ . Given $g$ , if such an $f$ exists we say that $g$ admits a retraction. All retractions are epimorphisms.

If a morphism is both a section and a retraction, then it is an isomorphism.

Comments

Monomorphisms and epimorphisms should be regarded as categorical analogues of the notions of injective and surjective functions (see Section 1.2 in [1]).

Though we have yet to study duality in a category-theoretic context, it is interesting that notions of monomorphism and epimorphism are dual, which means their abstract categorical properties are also dual (see Lemma 1.2.11. in [1]).

It is fairly straightforward, I think, to see that identity arrows are always monic. Dually, they are thus always epic as well. A number of theorems also follows, which, for intuitive explanation, can be reviewed in Section 5.1. in [3].

References

[1] E. Riehl, Category theory in context. Dover Publications, 2016. [online].

[2] S. Mac Lane, Category theory for the working mathematician. Springer, 1978. [online].

[3] P. Smith, Category theory: A gentle introduction [online].

[4] D. Epelbaum and A. Trisal, Introduction to category theory. [Lecture notes].

Introduction to category theory

On August 12, 2021August 19, 2021 By Robert C. SmithIn Mathematics

This is the first entry in my notes on category theory, higher category theory, and, finally, higher structures. The main focus of my notes, especially as the discussion advances, is application in string / M-theory, concluding with an introduction to the study of higher structures in M-theory. We start with basic category theory roughly following the book ‘Category Theory in Context’ by Emily Riehl (online version here), as well as the perspective of a selection of other texts and lectures cited throughout. For the engaged reader, I recommend reviewing the respective pages on nLab for further references.

Introduction

There is a line by Wilfrid Sellars: ‘The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term’. The things we must come to know ‘in the broadest possible sense’ – at its most abstract, a type of conceptual modelling – must in some way be classified such that we may distinguish the type of thing, the relation between thing of similar and dissimilar class, and its particular properties or attributes. For example, think of basic biological nomenclature going back to Aristotle. Another example would be the standard model of particle physics. (For the time being, we will put aside philosophical issues going back to Hegel, Russell, and others, as well as broader debates having to do with process vs. substance metaphysics, and so on).

From a mathematics and physics point of view, if we take Sellars’ statement seriously, then, at the highest level in the conceptual hierarchy what we begin to contemplate is a way to think about what Peter Smith describes in his notes on category theory as, ‘structured families of structures’. That is to say, we naturally come upon the need for some systematic framework for the study of abstract structures, how we may define a family of such structures, and their interrelation. We take as a starting point in these notes motivation from both foundational mathematics and fundamental physics.

A simple example of a structure is a topological space. Simpler still, take an example from group theory. Any group may be described as a structure, which comprises a number of objects equipped with a binary operation defined on them that obeys well-defined axioms. Now, what of a family of groups? We can of course also define a family of groups with structure-preserving homomorphisms between them (for a review of groups and sets leading up to the basic ideas of category theory, see Chapter 2 in the above notes by Smith). This gives an example of a structured family. This reference to groups is apt, because as we will see later in these notes: classically, a group is a monoid in which every element has an inverse (necessarily unique). A monoid, as we will review in a future entry, is one of the most basic algebraic examples of a category.

More generally, when looking at a family of structures along with the structure-preserving maps between them, our goal will be to reach an even higher level of abstraction that takes the form of a further structure: i.e., a structure-of-structures. We can then continue this game and ask, what is the interrelation of this structure-of-structures? From this question we will look to climb to another level and speak of operations that map one functor to another in a way that preserves their functorial properties.

When I think of the idea of a category, this increasing picture of generality and of climbing levels of abstraction is often what I like to picture. To use the words of Emily Riehl [1], ‘the purview of category theory is mathematical analogy’. While some give it the description, however affectionately, of ‘abstract nonsense’, I prefer to think of category theory – and, more broadly, the category theoretic perspective – as very much akin to the geologist constructing a topological map containing only vital information. This notion of climbing levels of abstraction, is, in many ways, simplifying abstraction. What use would it be to perform analysis within the framework of these increasing levels of simplifying abstraction? In foundational mathematics, the motivation is quite clear. In fundamental physics, on the other hand, it may at first seem less obvious. But as we will discuss in these notes, particularly in the context of quantum field theory and string / M-theory, there is quite a lot of motivation to think systematically about structured families of mathematical structures.

What is a category?

One way to approach the idea of a category is to emphasise the primacy of morphisms. In the paradigm view, in contrast to set theory, category theory focuses not on elements but on the relations between objects (i.e., the (homo)morphisms between objects). In this sense, we may approach category theory as a language of composition.

Let us build toward this emphasis on composition in a simple way. Consider some collection of objects $A, B, C, D$ with a structure preserving morphism $f$ from $A$ to $B$ , another structure preserving morphism $g$ from $B$ to $C$ , and, finally, a structure preserving morphism $h$ from $C$ to $D$ . (In a handwavy way, this is how we motivated the idea of a category in a previous post). In diagrammatic notation we have,

$\displaystyle A \ \xrightarrow[]{f} \ B \ \xrightarrow[]{g} \ C \ \xrightarrow[]{h} \ D$ .

It is fairly intuitive that we should be able to define a composition of these maps. All we need, as an axiom, is associativity. For example, we may compose $f$ and $g$ such that we obtain a map from $A$ to $C$ . We may write such a composition as $g \circ f$ . Similarly for all the other ways we may compose the maps $f, g$ , and $h$ . This means that we ought to be able to then also compose a map for the entire journey from $A$ to $D$ . Diagrammatically, this means we obtain:

One sees that we can apply the structure preserving map $f$ followed by the composite g-followed-by-h. Alternatively, we may just as well apply the composite f-followed-by-g and then afterwards apply the map $h$ . This very basic picture of a collection of objects $A,B,C,D$ , the maps between them, and how we may invoke the principle of composition for these maps already goes some way toward how we shall formally define a category. One will notice below that we need a bit more than associativity as an axiom, and along with the objects of a category we will speak of morphisms simply as arrows. From now on, if $A \in \text{Ob}(\mathcal{C})$ we write $A \in \mathcal{C}$ .

Definition 1. A category $\mathcal{C}$ consists of a class of objects, and, for every pair of objects $A,B \in \mathcal{C}$ , a class of morphisms, $\text{hom}(A,B)$ , satisfying the properties:

Each morphism has specified domain and codomain objects. If $f$ is a morphism with domain $A$ and codomain $B$ we write $f: A \rightarrow B$ .

For each $A \in \mathcal{C}$ , there is an identity morphism $id_A \in \text{hom}(A,A)$ such that for every $B \in \mathcal{C}$ we have left-right unit laws:

$\displaystyle f \circ id_A = f \text{for all} f \in \text{hom}(A,B)$
$\displaystyle id_A \circ f = f \text{for all} f \in \text{hom}(B,A)$

For any pair of morphisms $f,g$ with codomain of $f$ equal to codomain of $g$ , there exists a composite morphism $g \circ f$ . The domain of the composite morphism is equal to the domain of $f$ and the codomain is equal to the codomain of $g$ .

Two axioms must be satisfied:

For any $f: A \rightarrow B$ , the composites $1_B f$ and $f1_A$ are equal to $f$ .

Composition is associative and unital. For all $A,B,C,D \in \mathcal{C}$ , $f \in \text{hom}(A,B)$ , $g \in \text{hom}(B,C)$ , and $h \in \text{hom}(C, D)$ , we have $f \circ (h \circ g) = (g \circ f) \circ h$ .

Further remarks may be reviewed in [1, 2, 3]. We emphasise that for any mathematical object there exists a category with objects of that kind and morphisms – i.e., structure-preserving maps denoted as arrows – between them. The objects and arrows of a category are called the data. The objects of a category can be formal entities like functions or relations. In many examples of a category, the arrows represent functions, but not all cases of an arrow represents a morphism. These subtitles will be saved for future discussion.

An important notational point is that one should keep close attention on morphisms. Often categories with the same class of objects – e.g., a category of topological spaces compared with another category of topological spaces – may be distinguished by their different classes of morphisms. It is helpful to denote the category as $\text{hom}_{\mathcal{C}}(A,B)$ or $\mathcal{C}(A,B)$ to denote morphisms from $A$ to $B$ in the category $\mathcal{C}$ .

Importantly, to avoid confusion, we speak of ‘classes’ or ‘collections’ of objects and morphisms rather than ‘sets’. One motivation is to avoid confusion when speaking of $\text{Set}$ , which is the the category of all sets with morphisms (as functions) between sets. If a set of objects were required, instead of a class, then we would require a set of all sets. As it will be made clear when we reach the discussion on how to consider categories of categories, we may speak of sets of sets but, as Russell’s Paradox implies, there is no set whose elements are ‘all sets’. So we cannot speak of a set of all sets or a category of all sets. Likewise, it is conventional when we consider categories of categories to avoid the notion of a category of all categories (see Remark 1.1.5. in [1]). Instead, we speak of a limit in the form of a universe of sets and, in more advanced discussion, we will come to consider categories as universes.

Related to this concern about set-theoretical issues, it is important to note that we work with an extension of the standard Zermelo–Fraenkel axioms of set theory, allowing ‘small’ and ‘large’ sets to be discussed. In category theoretic language, we invoke similar terminology:

Definition 2. A category $\mathcal{C}$ is finite iff it has overall only a finite number of arrows.

A category $\mathcal{C}$ is small iff it has overall only a ‘set’s worth’ of arrows – i.e. the class of objects is a set such that the arrows of $\mathcal{C}$ can be put into one-one correspondence with the members of the set.

A category $\mathcal{C}$ is locally small iff for every pair of $\mathcal{C}$ – objects $A,B$ there is only a ‘set’s worth’ of arrows from $A$ to $B$ , i.e. those arrows can be put into one-one correspondence with the members of some set.

Examples of categories

What follows are a few examples illustrating the variety of mathematical objects that assemble into a category:

Set, the category of sets where morphisms are given by ordinary functions, with specified domain and codomain. There is a subtlety here in that the view of Set as the category of all sets becomes paradoxical, so, typically, we limit to a universe of sets (more on this in a separate entry).

Example. In this category the objects are sets, morphisms are functions between sets, and the associativity of the composition law is the associativity of composition of functions.

We may define the category Set (The category of sets): $\mathcal{O}$ (Set) is the class of all sets, and, for any two sets $A,B \in \mathcal{O}$ (Set) define $\text{hom}(A,B) = f: A \rightarrow B$ as the set of functions from $A$ to $B$ . The composition law is given by the usual composition of functions. Since composition of functions is associative, and there is always an identity function, Set is a category. This ends the example.

Other categories of note:

Grp, the category of groups where morphisms are given by group homomorphisms.

Vect_k, the category of vector spaces over some fixed field $k$ , where morphisms are given by linear transformations.

Ring, the category with rings as objects and ring homomorphisms as morphisms

Top, the category of topological spaces where morphisms are given by continuous maps

Met, is the category with metric spaces as objects and continuous maps as morphisms.

Meas, is the category with measurable spaces as objects and measurable maps as morphisms.

Graph, the category of graphs as objects and graph morphisms (functions carrying vertices to vertices and edges to edges, preserving incidence relations) as morphisms. In the variant DirGraph, objects are directed graphs, whose edges are now depicted as arrows, and morphisms are directed graph morphisms, which must preserve sources and targets.

Man, the category of smooth (i.e., infinitely differentiable) manifolds as objects and smooth maps as morphisms.

All of the above examples are concrete categories, whose objects have underlying sets and whose morphisms are functions between these underlying sets (what we have called ‘structure-preserving’ morphisms). We will speak more about concrete categories, including formal definition, in a later note. For the sake of introduction, it is also worth noting that there are also \textit{abstract categories}. One example is as follows:

BG, the category defined by the group $G$ (or what we will describe as a monoid in the next entry) with a single object. The elements of $G$ are morphisms, with each group element representing a distinct endomorphism of the single object. Here composition is given by multiplication. There is an identity element $e \in G$ that acts as the identity morphism.

Closing comments

In the next post, we will review some other category definitions, review diagrammatic notation, and discuss in more detail the important role and subtlety of morphisms. In a closely followed entry, we will then finally turn our attention to monoids, groupoids, pre-ordered collections, and other related concepts, as well as start discussing examples in string theory.

References

These notes primarily follow a selection of lectures and texts:

[1] E. Riehl, Category theory in context. Dover Publications, 2016. [online]

[2] S. Mac Lane, Category theory for the working mathematician. Springer, 1978. [online].

[3] P. Smith, Category theory: A gentle introduction [online].

[4] J. Baez, Category theory course [lecture notes].