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].

Conference: Higher structures in quantum field theory and string theory

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

This week I am attending a conference on higher structures in quantum field theory and string theory. It’s an event that I have been excited about since the new year. So far there have been some very nice talks, with interesting ideas and calculations presented.

There is the expression about going down a rabbit hole. In the world of mathematical concepts and fundamental physics, it is easy to get excited about an especially stimulating talk and follow down several rabbit holes. I’m trying to stay especially focused on presentations that are more directly related with my current research, but sometimes the excitment and sense of interest in the discussion topic becomes too strong! This afternoon I am looking forward to Bob Knighton speak on an exact AdS/CFT correspondence and Fiona Seibold talk about integrable deformations of superstrings. The rest of the week should also be a lot of fun.

Meanwhile, in the background I’ve been working on my PhD research (even though I don’t formally start until 1 August) and some double sigma model stuff. I’m hoping to also have my next post on categorical products, duality, and universality finished, which, as it is currently drafted, also talks a bit more about M-theory motivations but I may save this part for a detailed entry of its own.

(n-1)-thoughts, n=4: A return to the North Sea, new string papers, and Strings 2021

On June 30, 2021July 1, 2021 By Robert C. SmithIn Philosophy and General Reading

Our return to the North Sea

Beth and I frequently talk about how we miss the North Sea. We lived on the coast and I think it is our nature that we both prefer its unique countryside. But now that we’re living in East Midlands, landlocked and busy at university, we haven’t been back for a couple of years. So for our summer holiday we’ve travelled to North Norfolk, a place that for many reasons became an adopted home for both of us, to smell the sea again and enjoy the beautiful sights.

It may perhaps sound a bit mawkish, but in many ways North Norfolk provided an opening for discovery, not just in a literal sense but also in a philosophical sense as a space to reflect on the world of ideas. There is a line by John Berger that speaks of a place from which the world can be discovered – that is, a foundation from which one can venture forward but always return if needed. I think this is what the North Norfolk countryside came to represent for us: at the time a much needed place of peace and calm, but also a place of thought and reflection, where time runs slow and where we could gather ourselves, find our footing in life, and sometimes spend entire afternoons contemplating life. If for Plato and Socrates our true home is the eternal world of ideas, the North Sea, with the calm and quiet nature of the hills, valleys, woodlands, and beaches unfolding alongside it, is the continuum from which philosophy and mathematical realism can be pursued. Its the total landscape, the geography, and the open horizon that is grounding.

The little cottage where we used to live, just off the main road which the Romans would have similarly ventured when travelling to the coastal towns, was our first proper home. It was maybe the first place where we both found lasting comfort, coming from difficult situations and experiences. It was modest accommodation – a flintstone cottage, with an old fireplace, pinewood shelves, and steep narrow stairs sharply turning from the kitchen at the back up to the bedroom facing the main road. It was a tiny dwelling, perhaps a bit too cramped at times. But with our library of books, regular philosophical discussion, and no shortage of slow days of reflection, maths, and hobby, it was the perfect place at the right time. On the most difficult days, we could just listen to the birds in the back garden, find peace in our thoughts, or write to our contentment.

The main town, Holt, the origin derived from the Anglo-Saxon word for ‘wood’, is a place defined very much by its surrounding line of trees and small forest areas. It is situated on a hill, and, although a couple miles from the coast, a fresh sweeping wind from the North Sea can often be felt. Out here, the air is fresh. Space is wide and open.

One of my favourite places was, and remains to be, the woodland and heath not far from where we lived. It’s not marked on the map, so it is truly a hidden place to be discovered. Beth and I would take regular walks and spend many afternoons sitting among the heather – purple flowers of Calluna vulgaris – and yellow flowers of Ulex gallii. One year, I remember wild ponies were introduced to help maintain the land, and often we would see them hanging out by the brook or underneath a tree on a hot summer day. Upon our return this week, it was a joy to visit this place again.

I’m not very good at creative writing, although I’ve tried to learn and experiment with it. One thing that we used to do is, during our walks in the heath or through the many wonderful nature reserves, we would treat them like field expeditions. And, like a keen biologist or natural scientist, we would take our time for the finer inspection of all species of plant and animal. I would practice writing in my journal in an observational and documentative way. Being back here, I was inspired to dig up one such piece of writing. At the time I was experimenting with phenomenological-style notetaking, trying to intricately describe my experience of the countryside. I think I also took inspiration from the structure of a number of writings by various authors that I was reading back then.

To the left of me dense thicket carries into the distance, a rolling plain of healthy evergreen and intricate pathways of rotting needle, brown and dank, align in close order row upon row upon row. To the right, at the nearest edge of the forest, golden rays of sunlight spray across an open valley. Its radiating warmth and all-consuming light illuminate the young grasses breaking into the soil, filling the land with a rich, unfolding spectacle of colour. Looking beyond these trees, it is readily noticeable that there is an abundance of wildflower and bracken, a diverse quality of dazzling tones and subjects, which harmonize in a single unified phenomenal pallet like one massive, entanglement of earth. Ramshackle and unexpected, diverse and revealing, from endless rills and rivulets, from ditches to dells, from hedgerows to underbrush, with each new experience pressing deeper into this landscape and, ever from the background of this vast horizon of rolling hills before me, entire swells of breathing life continuing to reveal themselves. From the rabbits and wood pigeons who rather be hidden; the swaying fields are a thoroughfare for creatures of various kinds, from field mice to deer and the odd passing fox; song thrush, jays, long-tailed tits, and spotted woodpeckers - what grows and lives in this place is truly possessed of a beauty all its own.

As for the very belly of the forest, off the heath, there is a rich vision of evergreen, each swirling pine looks entirely similar to the other. But upon closer inspection we see that each particular pine tree is distinguishable and, indeed, of unique character. The pine itself is of course home to many things. Birds, insects, a peering squirrel; all find comfort in these dense woodlands. But row upon row, with its dwarf shoots that spiral from off the axils of scaly bracts, such a dense growth of pine, whose intricate branches are like a massive conic arrangement of narrow needles bundled together by both bark and sap, is a marvel in itself. Occasionally stepping on fallen seed or the coned fruit, my senses are overwhelmed by the spatially sweet and particular fragrance that lingers throughout the air.

The countryside here is in many ways a place of Tolkein description. It has been nice sitting again by the cliffs and walking through the overgrown footpaths. As I tried to capture, it is Shire-like in its beauty. With its salt marshes; winding roads lined by hedges, wild flowers, sedges, and rushes; and rolling hills demarcated by broad leaved woodland, towering at times with veteran oaks, birch, and, my favourite, Scotch pines – there is so much to be admired in this part of the country. Down by the sea, fishermen sit with lines cast, birds circling overhead. Again, it is perhaps more than a bit mawkish, but it is for me one of the places in our beautiful country that speaks a bit to old Romanticism, with every brook and winding turn outlined by hedgerows evoking a scene from a classic Keats poem.

It is my nature to be reclusive. No doubt, there are many other reasons why I find home by the sea and in the countryside. But as I write from the cottage where we’re staying, I remember why North Norfolk represents more than a place of stillness and beauty. With a cup of tea and some maths by the window, in the quiet thoughtfulness, the appearance and seeming order of the world of phenomena, mental idealisations or not, rushes forth some profound reality.

New string papers

As I was preparing to leave for holiday, three papers appeared of significant interest. I haven’t had a chance to work through them all yet, between being strict with my holiday time and with String 2021 ongoing, but I felt motivated over a cup of tea to take note:

Heterotic duels of M-theory

A nice paper by Bobby Samir Acharya, Alex Kinsella, and David R. Morrison on the non-perturbative heterotic duels of M-theory was released. This is of particular interest to me as it relates to the wider study of the non-perturbative aspects of M/heterotic duality.

This duality was discovered in the mid 90’s in which one can take M-theory compactified on a $K3$ and find it relates to the $E8 \times E8$ heterotic theory compactified on a three-torus. When you look at the 4D picture, we may instead compactify M-theory on a $G2$ manifold (equipped with a K3 fibration), which is a seven-dimensional Riemannian manifold that is special because it comes with the holonomy group in the exceptional simple Lie group $G_2$ . For the $E8 \times E8$ , it gets compactified on a Calabi-Yau threefold equipped with a three-torus. I haven’t had a chance to read through and consider the paper in any great detail, but it is noticeable that it starts with a similar set-up, taking low-energy M-theory with $G2$ orbifolds as the choice of compactification, with choice of equipped K3-fiberation to enable comparison with the dual heterotic string spectrum. A key observation, I take it, is that for the heterotic background there is a subtlty with the gauge bundle on $T^3$ such that, when it comes to the non-perturbative physics, there are point-like instantons on orbifold points of the geometry. This is where things get both interesting and complicated, and I’m not sure in what way these instanton effects in the spectrum relate to M-branes. I am keen to read the second half of the study.

Higgs mass in string theory

Another paper that appeared looks at calculating the Higgs mass. It’s by Steven Abel and Keith R. Dienes. This paper is quite the joy, and I’m sure anyone with interest in string theory will enjoy it over a cup of tea. Abel and Dienes harnesses the powers of the world-sheet theory to perform some proper stringy calculations, developing a framework that presents a relationship between the Higgs mass and the cosmological constant. What is neat about the computation is that this connection is generic for all closed string theories and provides a bit of a platform for future studies on gauge hierarchy problems.

Double sigma models and geometric quantisation

With a rush of papers leading up to my holiday, this one immediately caught my attention and got me excited. Luigi Alfonsi and David Berman study geometric quantisation in double field theory and double sigma models. From what I have seen, it is grand.

I was actively thinking about quantisation of double sigma models, as this is one area in which I have been working. In fact, I recall a few discussions a year or more ago about a project looking into the quantisation of the doubled string. In parts, from working in the area, what we see in this paper is kind of what one would expect in that, to start, the zero-mode sector for the closed string is intrinsically non-commutative. This alone is an interesting fact with some deep implications. Commonly, in the set-up where the target-space is treated as a phase-space, one will also equip a symplectic form $\omega$ , and one will can construct a theory with an action following Tseytin (we talked about this in a past post). What is found with the inclusion of $\omega$ is an interesting connection with Born geometry (maybe I’ll write about this in a future post) and, furthermore, one will often find discussion on symplectic structures as it relates to Poisson geometry which has some deep relation with T-duality.

In short, in the quantisation procedure there is a choice of polarisation, and the authors want to make a choice of polarisation in conjunction with the strategy for geometric quantisation. What happens, in any case, is that T-duality will give polarisations. And then what one wants to study is the noncommutative algebra associated to the doubled phase space. What the paper shows is that there are, in essence, two types of quantisations going on, because there is one coming from the usual phase space and then another from the duality frame (i.e., what in the formalism is understood in terms of the Lagrangian submanifold).

A deeper idea here has to do with the doubled phase space and para-Hermitean geometry, which I think I’ve mentioned a wee bit in the past. On that note, it is also interesting to think about the findings in this paper as it relates to the idea of metastring theory and quantisation.

As an aside, I’ve been working on a draft essay about a series of papers by Luigi. I wanted to write a bit about double sigma models and double field theory before finishing this essay, with a mind toward giving the reader some reference. They are fantastic papers on the global double space of double field theory, among other things. I also have Luigi’s PhD thesis on hand, which I think is great. There is a lot to discussed here in the context of the doubled geometry of double sigma models and higher structures.

Strings 2021

The annual string conference, Strings 2021, is ongoing (21 June – 2 July). It’s always an event that I look forward to, as it brings together the entire string theory community. Among a large list of great and usual names, my eye immediate caught an anomalous speaker amongst the expected and anticipated: namely, Roger Penrose. I will be most eager to hear what he has to say during his presentation on Friday 2, July. The topic is on gravitational singularities. There are of course a number of talks that I am looking forward to – too many to list! For now, here is the schedule with list of speakers, including links to notes and recordings. If I find the time and motivation, I’ll write a summary of my favourite talks next week.