O(D,D) and Double Field Theory

1. Introduction

In continuation of a past entry, this week I was intending to write more about double sigma models. I wanted to offer several further remarks on the intrinsic aspects of the doubled world-sheet formalism, and also give the reader a sense of direction when it comes to interesting questions about the geometry of the doubled string.

However, I realised that I have yet to share on this blog many of my notes on Double Field Theory (DFT). We’ve talked a bit about the Courant Bracket and the strong constraint and, in a recent post, we covered a review of Tseytlin’s formulation of the duality symmetric string for interacting chiral bosons that relates to the formulation of DFT. But, as a whole, it would be useful to discuss more about the latter before we continue with the study of double sigma models. There is a wonderfully deep connection between two, with a lot of the notation and concepts employed in the former utilised in the latter, and eventually a lot of concepts become quite interrelated.

We’ll start with some basics about DFT, focusing particularly on the T-duality group {O(d,d)} and the generalised metric formulation. In a later entry, we’ll deepen the discussion with gauge transformations of the generalised metric; generalised Lie derivatives; Courant brackets, generalised Lie brackets, and Dorfman brackets; among other things. The endgame for my notes primarily focuses on the generalised Ricci and the question of DFT’s geometric constitution, which we will also discuss another time.

For the engaged reader interested in working through the seminal papers of Zwiebach, Hull, and Hohm, see [1,2,3,4].

2. What is {O(d,d)} ?

As we’ve discussed in other places, DFT was formulated with the purpose of incorporating target space duality (T-duality) in way that is manifest on the level of the action. One will recall that, in our review of the duality symmetric string, the same motivation was present from the outset. I won’t discuss T-duality in much depth here, instead see past posts or review Chapter 8 in Polchinski [5]. The main thing to remember, or take note of, is how T-duality is encoded in the transformations R \leftrightarrow \frac{l_s}{R} , p \leftrightarrow w  , which describe an equivalence between radius and inverse radius, with the exchange of momentum modes {p} and the intrinsically stringy winding modes {w} in closed string theory, or in the case of the open string an exchange of Dirichlet and Neumann boundary conditions. More technically, we have an automorphism of conformal field theory. In the case of compactifying on S^1  for example, as momentum and winding are exchanged, the coordinates {x} on {S^1} are exchanged with the dual {S^1} coordinates \tilde{x}  .

When T-duality is explicit we have for the mass operator,

\displaystyle  M^2 = (N + \tilde{N} - 2) + p^2 \frac{l_s^2}{R^2} + \tilde{w}^2 \frac{R^2}{l_s^2}, \ (1)

where the dual radius is {\frac{R^2}{l_s} \leftrightarrow \frac{\tilde{R}^2}{l_s} = \frac{l_s}{R^2}} with {p \leftrightarrow \tilde{w}} . Here {l_s} is the string scale. One may recognise the first terms as the number operators of left and right moving oscillator excitations. The last two terms are proportional to the quantised momentum and winding. Compactified on a circle, the spectrum is invariant under {\mathbb{Z}_2} , but for a d-dimensional torus the duality group is the indefinite orthogonal group {O(d,d; \mathbb{Z})} , with {d} the number of compact dimensions.

And, actually, since we’re here one can motivate the idea another way [6]. A generic aspect of string compactifications is that there exist subspaces of the moduli space which feature enhanced gauge symmetry. The story goes back to Kaluza-Klein. Take an {S^1} compactification and set {R = \sqrt{2}} , one finds four additional massless gauge bosons that correspond to {pw = \pm 1} , {N + \tilde{N} = 1} . One can combine these states with the two {U(1)} gauge fields to enlarge the {U(1)^2} gauge symmetry in the form

\displaystyle  U(1) \times U(1) \rightarrow SU(2) \times SU(2). \ (2)

If we want to generalise from the example of an {S^1} compactification to higher-dimensional toroidal compactifications, we can do so such that the massless states at a generic point in the moduli space include Kaluza-Klein gauge bosons of the group {G = U(1)^{2n}} and the toroidal moduli {g_{ij}, b_{ij}} , parameterising a moduli space of inequivalent vacua. This moduli space is {n^2} -dimensional coset space

\displaystyle  \mathcal{M}^{n} = \frac{O(n,n)}{O(n) \times O(n)} / \Gamma_T, \ (3)

where {\Gamma_T = O(n,n; \mathbb{Z})} . In other words, it is the T-duality group relating equivalent string vacua. (In my proceeding notes I sometimes use O(d,d) and O(n,n) interchangably).

But the example I really want to get to comes from the classical bosonic string sigma model and its Hamiltonian formulation [7]. It is fairly straightforward to work through. Along with the equations of motion, constraints in the conformal gauge are found to be of the form

\displaystyle  G_{ab} (\partial_{\tau} X^{a} \partial_{\tau} X^b + \partial_{\sigma} X^a \partial_{\sigma} X^b) = 0

and

\displaystyle  G_{ab}\partial_{\tau}X^a \partial_{\sigma} X^b = 0, \ (4)

which determine the dynamics of the theory. Then in the Hamiltonian description, one can calculate the Hamiltonian density from the standard Lagrangian density. After some calculation, which includes obtaining the canonical momentum and winding, the Hamiltonian density is found to take the form

\displaystyle H(X; G,B) = -\frac{1}{4 \pi \alpha^{\prime}} \begin{pmatrix}\partial_{\sigma} X \\ 2 \pi \alpha^{\prime} P \end{pmatrix}^T \mathcal{H}(G, B) \begin{pmatrix} \partial_{\sigma} X \\ 2\pi \alpha^{\prime} P \end{pmatrix}

\displaystyle = -\frac{1}{4\pi \alpha^{\prime}} \begin{pmatrix}  \partial_{\tau} X \\ -2\pi \alpha^{\prime} W \end{pmatrix}^T \mathcal{H}(G, B) \begin{pmatrix}  \partial_{\tau}X \\ -2\pi \alpha^{\prime} P \end{pmatrix} \ (5).

This {\mathcal{H}(G,B)} is what we will eventually come to define as the generalised metric. Keeping to the Hamiltonian formulation of the standard string, the appearance of {O(d,d)} follows. We first may define generalised vectors given some generalised geometry {TM \oplus T \star M} , in which the tangent bundle {TM} of a manifold {M} is doubled in the sum of the tangent and co-tangent bundle. The vectors read:

\displaystyle  A_{P}(X) = \partial_{\sigma} X^a \frac{\partial}{\partial x^a} + 2\pi \alpha^{\prime}P_a dx^a

and

\displaystyle  A_W(X) = \partial_{\tau} X^a \frac{\partial}{\partial x^a} - 2\pi \alpha^{\prime}W_a dx^a. \ (6)

Now, in this set-up, {O(d, d)} naturally appears in the classical theory ; because we take the generalised vector (6) with the constraint (4) and, in short, find that the energy-momentum tensor can be written as

\displaystyle  A^T_{P} \mathcal{H} A_P = 0 \ \ \text{and} \ \ A^T_P L A_P = 0. \ (7)

The two constraints in (7) tell quite a bit: we have the Hamiltonian density set to zero with the second constraint being quite key. It will become all the more clear as we advance in our discussion that this {L} defines the group {O(d,d)} . Moreover, a {d \times d} matrix {Z} is an element of {O(d,d)} if and only if

\displaystyle  Z^T L Z = L \ (8),

where

\displaystyle  L = \begin{pmatrix} 0 & \mathbb{I} \\ \mathbb{I} & 0 \\ \end{pmatrix}. \ (9)

The moral of the story here is that the generalised vectors solving the constraint in (7) are related by an {O(d,d)} transformation. This transformation is, in fact, T-duality. But to formalise this last example, let us do so finally in the study of DFT and its construction.

3. Target Space Duality, Double Field Theory, and {O(D,D,\mathbb{Z})}

From a field theory perspective, there is a lot to unearth about the presence of {O(d,d)} , especially given the motivating idea to make T-duality manifest. What we want to do is write everything in terms of T-duality representations. So all objects in our theory should have well-defined transformations.

We can then ask the interesting question about the field content. What one will find is that for the NS-NS sector of closed strings – i.e., gravitational fields {g_{IJ}} with Riemann curvature {R(g)} , the Kalb-Ramond field {b_{IJ}} with the conventional definition for the field strength {H=db} , and a dilaton scalar field {\phi} – these form a multiplet of T-duality. From a geometric viewpoint, this suggests some sort of unifying geometric description, which, as discussed elsewhere on this blog, may be formalised under the concept of generalised geometry (i.e., geometry generalised beyond the Riemannian formalism).

Earlier, in arriving at (1), we talked about compactification on {S^1} . Generalising to a d-dimensional compactification, we of course have {O(d,d)} and for the double internal space we may write the coordinates {X^i = (x^i, \tilde{x}_i)} , where {i = 1,...,d} . But what we really want to do is to double the entire space such that {D = d + n} , with {I = 1,..., 2D} , and then see what happens. Consider the standard formulation of DFT known as the generalised metric formulation (for a review of the fundamentals see [8]). The effort begins with the NS-NS supergravity action

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

In the case of toroidal compactification defined by {D} -dimensional non-compact coordinates and {d} -dimensional compact directions, the target space manifold can be defined as a product between {d} -dimensional Minkowski space-time and an {n} -torus, such that {\mathbb{R}^{d-1,1} \times T^{n}} where, as mentioned a moment ago, {D = n + d} . We have for the full undoubled coordinates {X^{I} = (X^{a}, X^{\mu})} with {X^{a} = X^{a} + 2\pi} being the internal coordinates on the torus. The background fields are {d \times d} matrices taken conventionally to be constant with the properties:

\displaystyle G_{IJ} = \begin{pmatrix} \hat{G}_{ab} & 0 \\ 0 & \eta_{\mu \nu} \\ \end{pmatrix}, \ \ B_{IJ} = \begin{pmatrix} \hat{B}_{ab} & 0 \\ 0 & 0 \\ \end{pmatrix}, \ \ \text{and} \ \ G^{IJ}G_{JK} = \delta^{I}_K. \ (11)

We define {\hat{G}_{ab}} as a flat metric on the torus and {\eta_{\mu \nu}} is simply the Minkowski metric on the {d} -dimensional spacetime. As usual, the inverse metric is defined with upper indices. In (11) we also have the antisymmetric Kalb-Ramond field. Finally, for purposes of simplicity, we have dropped the dilaton. Of course one must include the dilaton at some point so as to obtain the correct form of the NS-NS supergravity action, but for now it may be dropped because the motivation here is primarily to study the way in which {G_{IJ}} and {B_{IJ}} come together in a single generalised geometric entity, which we begin to construct with the internal metric denoted as

\displaystyle E_{IJ} = G_{IJ} + B_{IJ} = \begin{pmatrix} \hat{E}_{ab} & 0 \\ 0 & \eta_{\mu \nu} \\ \end{pmatrix} \ (12)

for the closed string background fields, with {\hat{E}_{ab} = \hat{G}_{ab} + \hat{B}_{ab}} as first formulated by Narain et al [9]. It is important to note that the canonical momentum of the theory is {2\pi P_{I} =  G_{IJ}\dot{X}^{J} + B_{IJ} X^{\prime J}} , where, in the standard way, {\dot{X}} denotes a {\tau} derivative and {X^{\prime}} denotes a {\sigma} derivative. Famously, the Hamiltonian of the theory may then also be constructed from the expansion of the string modes for coordinate {X^{I}} , the canonical momentum, and from the Hamiltonian density to take the following form

\displaystyle  H = \frac{1}{2} Z^{T} \mathcal{H}(E) Z + (N + \bar{N} - 2). \ (13)

Or, to write it in terms of the mass operator,

\displaystyle  M^{2} = Z^{T} \mathcal{H}(E) Z + (N + \bar{N} - 2). \ (14)

The structure of the first terms in (14) should look familiar. In summary, in an {n} -dimensional toroidal compactification, the momentum {p^{I}} and winding modes {w_{I}} become {n} -dimensional objects. So the momentum and the winding are combined in a single object known as the generalised momentum Z = \begin{pmatrix} w_{I} \\ p^{I} \\ \end{pmatrix}  . This generalised momentum Z is defined as a 2D  -dimensional column vector, and we will return to a discussion of its transformation symmetry in a moment. Meanwhile, in (13) and (14) N  and \bar{N}  are the usual number operators counting the excitations familiar in the standard bosonic string theory. One typically derives these when obtaining the Virasoro operators. We also see the first appearance of the generalised metric \mathcal{H}(E) , which is a 2D \times 2D  symmetric matrix constructed from G_{IJ}  and B_{IJ}  with E = E_{IJ} = G_{IJ} + B_{IJ}  . We will discuss the generalised metric in just a few moments.

As is fundamental to closed string theory there is the Virasoro constraint {L_{0} - \bar{L}_{0} = 0} , where {L_{0}} and {\bar{L}_{0}} are the Virasoro operators. This fundamental constraint remains true in the case of DFT. Except in DFT this condition on the spectrum gives {N - \bar{N} = p_{I}w^{I}} or, equivalently,

\displaystyle  N - \bar{N} = \frac{1}{2} Z^{T} L Z, \ (15)

where

\displaystyle L = \begin{pmatrix} 0 & \mathbb{I} \\ \mathbb{I} & 0 \\ \end{pmatrix}. \ (16)

This is, indeed, the same {L} we defined before. Given some state and some oscillators, the fundamental constraint (15) must be satisfied, with the energy of such states computed using (13). For the time being, we treat {L} somewhat vaguely and simply consider it as a constant matrix. We denote {\mathbb{I}} as a {D \times D} identity matrix.

Continuing with basic definitions, the generalised metric that appears in (13) and (14) is similar to what one finds using the Buscher rules [10] for T-duality transformations with the standard sigma model [11,12]. That is to say, {\mathcal{H}} takes a form in which there is clear mixing of the background fields. It is defined as follows,

\displaystyle \mathcal{H}(E) = \begin{pmatrix} G - BG^{-1}B & BG^{-1} \\ -G^{-1}B & G^{-1} \\ \end{pmatrix}. \ (17)

One inuitive motivation for the appearance of the generalised metric is simply based on the fact that, if we decompose the supergravity fields into the metric {G_{ij}} and the Kalb-Ramond field {B_{ij}} , in DFT these then must assume the form of an {O(d,d)} tensor. The generalised metric, constructed from the standard spacetime metric and the antisymmetric two-form serves this purpose. On the other hand, the appearance of the generalised metric can be approached from a more general perspective that offers a deeper view on toroidal compactifications. In (13) what we have is in fact an expression that serves to illustrate the underlying moduli space structure of toroidal compactifications [9,13], which, as we have discussed, for a general manifold {\mathcal{M}} may be similarly written as (3).

The overall dimension of the moduli space is {n^2} which follows from the parameters of the background matrix {E_{ij}} , with {n(n+1)/2} for {G_{ij}} plus {n(n-1)/2} for {B_{ij}} . The zero mode momenta of the theory define the Narain lattice {\Gamma_{n,n} \subset \mathbb{R}^{2n}} , and it can be proven that {\Gamma_{n,n}} is even and also self-dual. These properties ensure that, in the study of 1-loop partition functions, the theory is modular invariant with the description enabling a complete classification of all possible toroidal compactifications (for free world-sheet theories). The feature of self-duality contributes {O(n, \mathbb{R}) \times O(n, \mathbb{R})} . The Hamiltonian (13) remains invariant from separate {O(n, \mathbb{R})} rotations of the left and right-moving modes that then gives the quotient terms. As for the generalised metric, we may in fact define it as the {O(n,n) / O(n) \times O(n)} coset form of the {n^2} moduli fields.

4. {O(n,n,\mathbb{Z})}

In a lightning review of certain particulars of DFT, we may deepen our discussion of the T-duality group by returning first to the generalised momentum {Z} as it appears in (14). If we shuffle the quantum numbers {w,p} , which means we exchange {w} for {p} and vice versa, the transformation symmetry of {Z} is well known to be

\displaystyle  Z \rightarrow Z = h^{T}Z^{\prime}. \ (18)

For now, {h} $ is considered generally as a {2D \times 2D} invertible transformation matrix with integer entries, which mixes {p^{I}} and {w_{I}} after operating on the generalized momentum. It follows that {h^{-1}} should also have invertible entries, this will be shown to be true later on. Importantly, if we have a symmetry for the theory, this means a transformation in which we may take a set of states and, upon reshuffling the labels, we should obtain the same physics. Famously, it is indeed found that the level-matching condition and the Hamiltonian are preserved. If we take {Z \rightarrow Z^{\prime}} as a one-to-one correspondence, the level-matching condition (15) with the above symmetry transformation (18) gives

N - \bar{N} = \frac{1}{2} Z^{T}LZ = \frac{1}{2} Z^{T \prime}L Z^{\prime}

\displaystyle  = \frac{1}{2} Z^{T \prime} h L h^{T} Z^{\prime}. \ (19)

For this result to be true, it is necessary as a logical consequence that the transformation matrix {h} must preserve the constant matrix {L} . This means it is required that

\displaystyle  h L h^{T} = L, \ (20)

which also implies

\displaystyle  h^{T} L h = L. \ (21)

These last two statements can be proven, producing several equations that give conditions on the elements of {h} . The full derivation will not be provided due to limited space (complete review of all items can again be found in [1,2,3,4,8]); however, to illustrate the logic, let {a, b, c, d} be {D \times D} matrices, such that {h} may be represented in terms of these matrices

\displaystyle h = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}. \ (22)

The condition in which {h} preserves {L} demands that the elements {a, b, c, d} satisfy in the case of (20)

\displaystyle  a^{T}c + c^{T}a = 0, \ b^{T}d + d^{T}b = 0,

and

\displaystyle  a^{T}d + c^{T}b = 1. \ (23)

Likewise, similar conditions are found for the case (21), for which altogether it is proven that {h^{-1}} has invertible entries. What this ultimately means is that although we previously considered {h} vaguely as some transformation matrix, it is in fact an element of {O(D,D, \mathbb{R})} and {L} is an {O(D,D, \mathbb{R})} invariant metric. Formally, an element {h \in O(D,D, \mathbb{R})} is a {2D \times 2D} matrix that preserves, by its nature, the {O(D,D, \mathbb{R})} invariant metric {L} (16) such that

\displaystyle  O(D,D,\mathbb{R}) = \bigg \{h \in GL(2D, \mathbb{R}) \ : \ h^{T}Lh = L \bigg \}. \ (24)

Finally, if the aim of DFT at this point is to completely fulfil the demand for the invariance of the massless string spectrum, it is required from (13) for the energy that, if the first term is invariant under {O(D,D)} then we must have the following transformation property in the case {Z^{T} \mathcal{H}(E) Z \rightarrow Z^{\prime T} \mathcal{H}(E^{\prime}) Z^{\prime}} :

\displaystyle  Z^{\prime T}\mathcal{H}(E^{\prime}) Z^{\prime} = Z^{T}\mathcal{H}(E)Z

\displaystyle  = Z^{\prime T} h \mathcal{H}(E)h^{T} Z^{\prime}. \ (25)

By definition, given the principle requirement of (25) it is therefore also required that the generalised metric transforms as

\displaystyle  \mathcal{H}(E^{\prime}) =  h\mathcal{H}(E)h^{T}. \ (26)

The primary claim here is that for the transformation of {E} we find

\displaystyle (E^{\prime}) = h(E) =  \begin{pmatrix}  a & b \\ c & d \\ \end{pmatrix}(E) \equiv (aE + b)(cE + d)^{-1}. \ (27)

One should note that this is not matrix multiplication, and {h(E)} is not a linear map. What we find in (27) is actually a well known transformation in string theory that appears often in different contexts, typically taking on the appearance of a modular transformation. Given the notational convention that {\mathcal{H}} is acting on the background {E} , what we end up with is the following

\displaystyle (E^{\prime T}) = \begin{pmatrix} a & -b \\ -c & d \\ \end{pmatrix}(E^{T}) \equiv (aE^{T} - b)(d - cE^{T})^{-1}, \ (28)

where in the full derivation of this definition it is shown (E^{\prime T}) = \begin{pmatrix} a & -b \\ -c & d \\ \end{pmatrix} E^T.

Proof: To work out the full proposition with a proof of (26), we may also demonstrate the rather deep relation between (26) and (28). The basic idea is as follows: imagine creating {E} from the identity background {E^{\prime} = \mathbb{I}} , where conventionally {E = G + B} and {G = AA^{T}}. Recall, also, the definition for the generalised metric metric (17). Then for {E = h_{E}(\mathbb{I})} , what is {h_{E} \in O(D,D, \mathbb{R})} ? To answer this, suppose we know some {A} such that

\displaystyle h_{E} =  \begin{pmatrix} A & B(A^{T})^{-1} \\ 0 & (A^{T})^{-1} \\ \end{pmatrix}. \ (29)

It then follows

\displaystyle  h_{E}(I) = (A \cdot \mathbb{I} + B(A^{T})^{-1})(0 \cdot \mathbb{I} + (A^{T})^{-1})^{-1}

\displaystyle  = (A + B(A^{T})^{-1}) A^{T} = AA^{T} + B = E = G + B. \ (30)

This means that the {O(D,D)} transformation creates a {G + B} background from the identity. Additionally, the transformation {h_E} is ambiguous because it is always possible to substitute {h_E} with {h_E \cdot g} , where we define {g(\mathbb{I}) = \mathbb{I}} for {g \in O(D,D, \mathbb{R})} . In fact, it is known that {g} defines a {O(D) \times O(D)} subgroup of {O(D,D)} {g^{T}g = gg^{T} = I} .

In conclusion, one can show that {\mathcal{H}} transforms appropriately, given that up to this point {h_{E}} was constructed in such a way that the metric {G} is split into the product {A} and {A^{T}} , with the outcome that only {A} is entered into {h_{E}} . To find {G} we simply now consider the product {h_{E}h_{E}^{T}} ,

\displaystyle h_{E}h_{E}^{T} = \begin{pmatrix} A & B(A^{T})^{-1} \\  0 & (A^{T})^{-1} \\ \end{pmatrix} \begin{pmatrix}  A^{T} & 0 \\ -A^{-1}B & A^{-1} \\ \end{pmatrix}

\displaystyle  = \begin{pmatrix} G - BG^{-1}B & BG^{-1} \\ -G^{-1}B & G^{-1} \\ \end{pmatrix} = \mathcal{H}(E). \ (31)

If we now suppose naturally {E^{\prime}} is a transformation of {E} by {h} , such that {E^{\prime} = h(E) = hh_{E}(\mathbb{I})} , we also have {E^{\prime} = h_{E^{\prime}}(\mathbb{I})} . Notice that this implies {h_{E^{\prime}} = hh_{Eg}} up to some ambiguous and so far undefined {O(D,D,\mathbb{R})} subgroup defined by {g} . Putting everything together, we obtain the rather beautiful result

\displaystyle \mathcal{H}(E^{\prime}) = h_{E^{\prime}}h^{T}_{E^{\prime}} = hh_{Eg}(hh_{Eg})^{T} = hh_{E}h^{T}_{E}h^{T} = h\mathcal{H}(E)h^{T}. \ (31)

\Box

Thus ends the proof of (26). A number of other useful results can be obtained and proven in the formalism, including the fact that the number operators are invariant which gives complete proof of the invariance of the full spectrum under {O(D,D,\mathbb{R})} .

In conclusion, and to summarise, in DFT there is an explicit restriction on the winding modes {w_{I}} and the momenta {p^{I}} to take only discrete values and hence their reference up to this point as quantum numbers. The reason has to do with the boundary conditions of {n} -dimensional toroidal space, so that in the quantum theory the symmetry group is restricted to {O(n,n,\mathbb{Z})} subgroup to {O(D,D,\mathbb{R})} . The group {O(n,n,\mathbb{Z})} is as a matter of fact the T-duality symmetry group in string theory. It is conventional to represent the transformation matrix {h \in O(n,n,\mathbb{Z})} in terms of {O(D,D,\mathbb{R})} such that

\displaystyle h = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}

with,

\displaystyle a = \begin{pmatrix} \tilde{a} & 0 \\ 0 & 1 \\ \end{pmatrix},

\displaystyle b = \begin{pmatrix} \tilde{b} & 0 \\ 0 & 0 \\ \end{pmatrix},

\displaystyle c = \begin{pmatrix} \tilde{c} & 0 \\ 0 & 0 \\ \end{pmatrix}

and

\displaystyle d = \begin{pmatrix} \tilde{d} & 0 \\ 0 & 1 \\ \end{pmatrix}. \ (32)

Each of {\tilde{a}, \tilde{b}, \tilde{c}, \tilde{d}} are {n \times n} matrices. They can be arranged in terms of {\tilde{h} \in O(n,n,\mathbb{Z})} as

\displaystyle \tilde{h} = \begin{pmatrix} \tilde{a} & \tilde{b} \\ \tilde{c} & \tilde{d} \\ \end{pmatrix}. \ (33)

Invariance under the {O(D,D,\mathbb{Z})} group of transformations is generated by the following transformations. To simplify matters, let us define generally the action of an {O(D,D)} element as

\displaystyle \mathcal{O} = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} = \mathcal{O}^{T}L\mathcal{O}. \ (34)

Residual diffeomorphisms: If {A \in GL(D, \mathbb{Z})} , then one can change the basis for the compactification lattice {\Gamma} by {A \Gamma A^{T}} . The action on the generalised metric is

\displaystyle \mathcal{O}_{A} = \begin{pmatrix} A^{T} & 0 \\ 0 & A^{-1} \\ \end{pmatrix}, \ \ A \in GL(D, \mathbb{Z}), \ \ \det A = \pm 1. \ (35)

B-field shifts: If we define {\Theta} to be an antisymmetric matrix with integer entries, one can use {\Theta} to shift the B-field producing no change in the path integral. For compact d-dimensions, this amounts to {B_{IJ} \rightarrow B_{IJ} + \Omega_{IJ}} . It follows that the {O(D,D)} transformation acts on the generalised metric,

\displaystyle \mathcal{O}_{\Omega} = \begin{pmatrix} 1 & \Omega \\ 0 & 1 \\ \end{pmatrix}, \ \  \Omega_{IJ} = - \Omega_{JI} \in \mathbb{Z}. \ (36)

Factorised dualities: We define a factorised duality as a {\mathbb{Z}_2} duality corresponding to the {R \rightarrow \frac{1}{R}} transformation for a single circular direction (i.e., radial inversion). It acts on the generalised metric as follows

\displaystyle \mathcal{O}_{T} = \begin{pmatrix} 1-e_{i} & e_{i} \\ e_i & 1-e_{i} \\ \end{pmatrix}, \ (37)

where {e} is a {D \times D} matrix with 1 in the {(i, i)} -th entry, and zeroes elsewhere {(e_{i})_{jk} =  \delta_{ij}\delta_{ik}} . Altogether, these three essential transformations define the T-duality group {O(D,D,\mathbb{Z})} , as first established in [14,15]. To calculate a T-dual geometry one simply performs the action (26) or (28) using an {O(D,D,\mathbb{R})} transformation and, in general, one may view the formalism with the complete T-duality group as a canonical transformation on the phase space of a given system.

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[11] Mark Bugden. A tour of t-duality: Geometric and topological aspects of t-dualities, 2019.

[12] T.H. Buscher. Path Integral Derivation of Quantum Duality in Nonlinear SigmaModels.Phys. Lett. B, 201:466–472, 1988.

[13] Daniel C. Thompson. T-duality invariant approaches to string theory, 2010.[14] Alfred D. Shapere and Frank Wilczek. Selfdual Models with Theta Terms.Nucl.Phys. B, 320:669–695, 1989.[15] Amit Giveon, Eliezer Rabinovici, and Gabriele Veneziano. Duality in StringBackground Space.Nucl. Phys. B, 322:167–184, 1989.

[14] Alfred D. Shapere and Frank Wilczek. Selfdual Models with Theta Terms. Nucl. Phys. B, 320:669–695, 1989.

[15] Amit Giveon, Eliezer Rabinovici, and Gabriele Veneziano. Duality in String Background Space. Nucl. Phys. B, 322:167–184, 1989.

*Cover image: ‘Homology cycles on a torus’. Wikipedia, Creative Commons. *Edit for spelling, grammar, and syntax.

Institute of Physics scholarship award and full interview

I’m proud and honoured to share that I’ve been awarded a PhD research scholarship by the Institute of Physics. An announcement by my university can also be found here.

As I’ve written about elsewhere on this blog, my PhD research focuses on M-theory and the question of string theory’s non-perturbative completion. To be a recipient of the Bell Burnell Graduate Scholarship Fund on the basis of my planned research in mathematical physics, which included having to deliver a presentation on my topic and an interview with the physics panel, is quite satisfying. Admittedly, I was a bit nervous knowing that I can be maths heavy and that this might not be recieved too well in-front of a well-distinguished panel comprised mostly of experimental physicists! But I am delighted that the wonderful physics content of my research was acknowledged.

In conjunction with the announcement of my scholarship award, I was invited to participate in a more personal interview designed for the non-physics reader. Within I answer a variety of questions, including about my current research. I also share a bit more about myself, my upbringing, and other personal stories and reflections.

Below is the complete version of the interview I gave for the Institute of Physics. There is also a shortened, edited version that can be found here.

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1. Tell us about your work – and what drives you (We want to know about your area of physics and why you’re passionate about it. What does it mean to you? Why is it important? Imagine the reader is not a physicist).

Firstly, thank you for inviting me to answer questions.

The situation today in fundamental and high-energy physics is incredibly interesting. A lot has happened since the 1950s or so, with many great successes. Just think: essentially all observable phenomena are well described, on the one hand, by quantum field theory and the Standard Model of particle physics, and also by Einstein’s theory of general relativity on the other. We have established tremendously accurate descriptions of the very small – quantum theory – and the tremendously massive – cosmology and astrophysics. Modern physics has made some remarkable achievements, both in advancing human knowledge, and in supporting how we may apply the laws of nature to develop important technologies. Having said that, it is almost certain that these fundamental theoretical frameworks are incomplete. For example, general relativity and quantum field theory break down when we start to study situations at the centre of a black hole or close to the big bang. Many readers will likely also have heard of concepts like dark energy and other things, which also currently remain unknown.

A big question in fundamental physics, perhaps the deepest and most important, has to do with what we call quantum gravity. This represents the unification of general relativity with quantum field theory. I work in mathematical physics, and, in particular, my research is focused in string / M-theory. Today, this is the most promising and indeed leading theory of quantum gravity.

One of the great successes of string theory is how, in a single consistent mathematical framework, we have a theory that combines gravity with the quantum laws of nature. This means that at very large scales, we find gravity as Einstein described it in his general theory of relativity. But on very small scales, in which space-time is discretised, we have a theory that captures the idea of quantised units of gravitational energy. We think of these quantised units as particles that we call gravitons. In practical language, string theory describes how the curvature of space-time emerges from the existence of gravitons. Thus, we have a quantum theory of gravity.

Despite the many successes of string theory, we still face some open problems and challenges in formulating the complete theory. It is not possible, at this point, to speak of such challenges without a degree of technicality as this is a highly technical subject. What I will say is that, in keeping to practical language, one of the biggest and most important questions we face concerns what may be described as the non-perturbative completion of string theory. This is what my PhD research is focused on understanding.

To explain this, allow me to share a bit of history. As late as 1995, we had five perturbative string theories – type I, type IIA, type IIB, and the two flavours of heterotic string theory (SO(32) and E8 × E8) – and these were seen to be distinct. Much of modern physics is built using tools and approaches that deal with what we may describe here as local, approximate, perturbative descriptions of reality. And these perturbative theories of fundamental physics – the five string theories – are remarkably successful and beautiful. Just from the humble idea of the extended of object of the string, which is a generalisation of point particle theory (which one may have some familiarity with going back to undergraduate or A-level physics), we can generate some brilliant results like Einstein’s gravity. But just think of the situation in the mid-1990s: in quantum gravity, we had five theories without a way of knowing how to select the correct one. This is quite a messy situation! But one of the amazing qualities of string theory is that it comes with a wealth of symmetries.  And it was following a very important proposal by Edward Witten that the five perturbative string theories were found to be deeply related by a number of non-trivial dualities, or, for the sake of practicality, what we may describe here as symmetry relations. So rather than being distinct, the five string theories were found to represent different limits of an overarching theory.

This is quite an evocative idea, namely that there is some deeper underlying structure to quantum gravity, from which things like space-time may even emerge! This overarching theory is known as M-theory, and the non-perturbative completion of string theory to M-theory is specifically what my PhD research seeks to investigate.  M-theory is truly remarkable for several reasons. Although the five perturbative string theories exist in ten space-time dimensions, M-theory exists in 11 space-time dimensions. So it is a higher dimensional theory. Given that one can think of it as the parent theory to string theory – i.e., as a single mathematical structure that unifies the zoo of perturbative string theories, with its low-energy effective action being what we call supergravity – it represents a unique theory of quantum gravity. More than that, M-theory is, in every sense, the leading candidate for a Theory of Everything (ToE). It is also the mathematical theory that makes sense of the dynamical physical objects we call branes (objects that, again, emerge in higher dimensions as a generalisation of point particle theory), which propagate through space-time according to the rules of quantum mechanics.

However, although there presently exist many hints and plausibility arguments in support of the proposed existence of M-theory, a systematic formulation of the non-perturbative theory remains an open problem. There are many reasons why a fundamental and rigorous formulation of M-theory is important. Not only do we expect to find new physics and new mathematics – in fact, it has been described as a great unexplored ocean in this regard – it will also help provide a final say on things like fundamental string cosmology.

As you may be able tell, it is an area of research I find to be incredibly exciting, not least because it is potentially so fundamental. In recent years a number of especially exciting developments have begun to crystalise in how we may attack the question of a rigorous formulation of M-theory, including the study of what we call higher structures. The presence of higher structures – or what we may summarise as higher homotopy theory – in fundamental physics is in and of itself a super interesting fact. And, as I mentioned at the outset, a lot of what we talk about and study also has to do with combining spacetime geometry and quantum field theory defined as generalised geometry. As I have alluded, some of the implications are grand, including the extension of spacetime itself, with a further consequence being the possibility that geometry and gravity – indeed, space and time – are emergent concepts.

As I work in mathematical physics and find great interest in both foundational maths and fundamental physics, I enjoy this area of research because there is a wonderful interplay between the two. My PhD research is positioned at this interface.

2. What drew you to this area of physics? (Tell us a little about your physics journey and how you ended up focusing on this area. When did you first become excited about physics? What was it that excited you? What led you to where you are now?).

By most accounts within the formal parameters and constraints of mainstream education, my physics journey up to this point has been described as highly unconventional. This is certainly largely owed to the fact that I have Asperger’s, and as a person with autistic spectrum disorder (ASD) I experience a lot of difficulties and unique challenges. Formal education environments are certainly something that I did not cope with well in the past, and something I continue to struggle participating in today.

Due to the challenges that come with my Asperger’s and also the difficult conditions I experienced growing up, I was often in and out school for many years. To be completely honest, there were a lot of times when I tried to go to school or participate at a university, and it was just never sustainable for me. The classrooms or lecture halls were too overwhelming; the curriculum was too slow or uninteresting; time pressures were too difficult to manage; the lectures or lessons were not fundamental enough or too restrictive for my interest. It was often, I suppose, the case that I would rather be given the textbook and left on my own to work through and derive everything. I am much more comfortable with that independence. I have also grown used to being in my own space, with all of my books, with my maths and physics, working and studying on the things I find meaningful. So, I think there are a lot of reasons, from my perspective, as to why formal education has always been in some ways inharmonious or discordant.

In truth, without the right support, I probably wouldn’t have successfully joined the University of Nottingham and still be working at the university today. It was a massive personal step for me, one that we worked up to over a couple of years, and once I arrived at university it was incredibly challenging on so many levels. It required a lot of support, patience, and understanding. But I was also incredibly fortunate to have landed at such a fantastic school, with great support staff for people like me with ASD. The same can’t be said in all cases, and there are a lot of brilliant people out there with ASD that don’t receive the right support, or who don’t have the opportunity or foundation to pursue a formal university education for social, economic, or cultural reasons. I think I am decent at maths and physics, capable enough to teach myself string theory for example, and I was super close to not being a university student because, in a multitude of ways, I generally don’t ‘fit’ in the way that is expected.

Needless to say, and to return to the question, so much of my life so far has been outside of formal education. As a result, I have self-studied almost everything I know. When I was younger, I taught myself calculus and eventually expanded my self-learning to higher mathematics. The same with much of physics, from classical mechanics through to quantum field theory – I had already taught myself a lot of this prior to entering university as a first-year undergraduate. This is why my physics background may be described as unconventional. But to be honest, from my perspective, teaching myself maths and physics without relying on a teacher or sitting a class, seems like quite a normal and reasonable thing to do. I work at my own pace and ask my own questions. I can explore and enjoy maths and physics in my own way, giving myself as much time as I deem necessary to explore a topic fundamentally. You know, I think in a lot of ways maths and physics have become some of the only things in the world that truly make sense to me.

When I joined the University of Nottingham, I had been developing a lot of interest in general relativity, quantum field theory, and quantum gravity. I read about a number of different theories of quantum gravity, many of which I found to suffer mathematical inconsistency among other things. This is how I found my way to string theory, also with the encouragement and support of my professor Tony Padilla, who is now also my supervisor.

During the first weeks of my undergraduate, I took my string studies very seriously, and around this time my interest in non-perturbative theory began to crystalise. In that first year, the School of Physics obtained permission from the university to accelerate me to a Master of Research degree. My thesis involved the study of double sigma models in string theory. I am now looking forward to studying for my PhD within the Particle Theory Group at the University of Nottingham.

3. What does winning the scholarship mean to you – and what difference will it make? (How do you feel about winning? How do you feel about taking on a PhD? Would you be able to take on a PhD without it? In what ways will it help/make a difference?)

I’m both proud and honoured to have been awarded the scholarship for the duration of my PhD. I’m also incredibly proud to have affiliation with the Institute of Physics and to take on an ambassadorial role, something I take very seriously.

Coming from where I do, outside of formal education, I used to sometimes sneak into university lecture halls and, well, there were times when I would think to myself that perhaps I could do a PhD in physics or be a good researcher in a formal academic environment. I don’t always think about it so explicitly these days, but it truly means everything to me to be doing my PhD in mathematical physics. One could speak to the depth of the notion of existential meaning here, in terms of one’s projects and interests in life. But it is more than that for me. I would be working on my maths and physics no matter what, because it is what I know and it is what interests me; but now I have the opportunity to do so within a formal environment without financial concern, social judgement, or other pressures and worries.

To formally pursue a PhD at such a wonderful university and as part of a very cool research group, to get to continue working with Tony Padilla and to talk strings every day, and really to be able to study my maths and physics in an encouraging environment, is kind of life-changing. I am very grateful and I look forward to the future, where, hopefully, after a good PhD I can continue to contribute quality work and carve out a formal research career, maybe even teach strings one day. The scholarship has helped provide a good foundation in pursuing these ends.

4. What challenges have you faced to get to this point? (Any barriers/challenges that you have had to overcome that you feel comfortable talking about. Has anyone discouraged you? Have your personal circumstances made it harder? Have societal barriers/conditions had an impact? How have you overcome these challenges?).

In addition to my lifelong struggle with my Asperger’s, which, clinically, has been diagnosed as severe, I also had a very difficult childhood and experienced a lot of bad stuff growing up. I grew up in an environment that was incredibly dysfunctional, hostile, and in many moments scary. There was a lot of abuse and neglect, periods in and out poverty, with no heat in the winter – just not very nice things for quite a long time. By the time I was 14 or 15, without the right support, I could barely function, let alone cope. And in these circumstances, the pursuit of one’s interests and intellectual passions were rarely permitted. Instead, there were many times in life that were largely about survival and trying to escape. These were times that were generally quite debilitating. For years I also struggled with my mental health. I still do, although there is always an aspect of that owed to my ASD.

I can talk about it all now because I’ve had a lot of time and support in working through the traumatic events and the terrible stuff I witnessed and experienced. Growing up, I moved to different families, which offered great reprieve, and there have been so many extraordinary people that brought me into their homes, sometimes for years, and supported me as I slowly found my feet in life. These are individuals and families who intervened to fill the gap and take on abandoned parental responsibilities. They did so much for me, helping enable a positive foundation to grow and develop in life, to self-actualise, and to be able to pursue my interests. So, despite having to face a lot of challenges in life, some of which are quite extreme, I am also very thankful today. My life could have turned out differently on many occasions. And partly why I share that here is because, well, maybe someone will read this and take something from it. There are a lot of people that have ASD or that grow up in bad conditions and are never given the proper support they need as human beings – a positive and healthy foundation to life from which one can then begin to move. It’s Maslow’s hierarchy of needs, which, while problematic in places, serves as a reminder that the meaning of society is to help foster the conditions in which all citizens can realise their full potential. As a society, I think from a fundamental humanistic perspective we ought to never stop demanding socio-economic, cultural conditions that give as broad a scope of people as broad of a horizon of opportunity as possible.

It is also the case for me that in growing up in incredibly difficult conditions, I have come to recognise that I owe a lot to my Asperger’s and my sense of personality. I think it enabled me in many ways to survive the incomprehensible by maintaining a presence of mind. I eventually learned to cope with and understand life through my studies. Not only did my books come to provide a welcomed and flourishing space, they offered explanation and detailed insight into all that I had observed: psychologically, sociologically, economically, and so on.  One way that it is described is that, for some people, they learn to manage their present experience by thinking of their past experiences as reference points; for me, this is how I use books. So, aside from my beloved physics and maths, I have studied everything with great interest: from the whole of psychology and human behaviour to our best current theories on social structures and relations, history, anthropology, economics, philosophy, and in many ways across the social and natural sciences. I have never been dissuaded by the challenges I have faced. As a young adult, outside of formal education, I spent my days alone at public and university libraries, or sitting at the back of university lecture halls that I had snuck into.

I’ll share a story that is quite personal to me. As a child, when times were especially difficult, I remember sneaking away to the far and unvisited corner of a local park. Lying there, on the opposite side of the hill that faced away from everything and everyone, I would stare at the clouds and contemplate existence. Particles, birds, planets, and stars. Why do clouds exist and why are they shaped the way they are? Why do they move as though they are moving through fluid, floating without support? This was a site of one of my first philosophical and scientific reflections. And, really, despite my many difficulties in formal education environments, some of which are ongoing, science and academics has played an incredibly important role in my life. It has become a natural extension of myself as a person with Asperger’s who is driven to understand in accordance with my life spent with my books. I learn about human relations and behaviour through their empirical study just as I learn about quantum fields through my physics. I think with that inquisitiveness, one of my earliest memories of being excited about physics was when I was no more than 6 or 7 years old. I saw a photograph of a professor standing in front of a chalk board, and written on the board was sigma notation. Looking back, it was likely generality relativity that was being taught, but the mystery of the language, the power of physics that we may describe the nature of reality, it always stuck with me.

As I said before my ASD also brings many of its own unique struggles and daily challenges. I require a lot of support. I can compute scattering amplitudes but struggle to manage a calendar or money. I sit here writing because I am fortunate to have received support with my Asperger’s, to have a stable home environment, and to have a loving and caring partner, Beth. There is a lot of well-defined research which, last I checked, showed that about 80% of people with ASD struggle to hold down a full-time job or be independent, and it was estimated that suicide rates are 10 times more than average. Not all autistic people can work, and, for sure, I know that struggle to maintain my own independence. There were times when I was ashamed or pressured because I couldn’t maintain a job or understand how to pay rent, because I couldn’t maintain independence, understand how to manage my bills, and organise my life. Prior to intervention, I was kind of just left to work it out. Now, of course, that is my experience – everyone will have their own. But the point is there are so many simplistic narratives about autistic people and even just about poverty in general. In education, I was once deemed a troublemaker! Another lost soul and statistic.

I think we need to do more as a society to understand the complexity of individual situations, and we absolutely need to do more to combat ongoing prejudices and to support people with ASD.

5. What would you say to those who have also faced barriers to following their dreams to pursue physics at university and beyond? (Any advice/encouragement would be great).

I don’t want to be naïve and just say “go for it”. The reality is that different people have different challenges with different barriers. If a person loves physics and it is their main passion in life, but at the same time facing homelessness or a precarious existence, it is not just a matter of saying “go for it” and “you can do anything”. Poverty and class can be barriers. Racism, too, can be a significant barrier. Disability, mental health, physical health – people face all sorts of different challenges.

What I am trying to say is that if someone dreams to study physics, that is amazing because physics can offer a person so much in life. They should do so regardless of age, gender, race, disability, class, and so on. Absolutely. But saying that is not enough. People also need support, and there is absolutely nothing wrong with that. It is up to our institutions – university, government, etc – and it is up to us as a physics community to identify where support may be lacking. If you are a person wanting to pursue your passion for physics but struggling with personal circumstances or barriers to doing so, don’t be ashamed to seek support. Universities have advice and support services who can often help you to find ways forward.

6. Why do you think diversity in physics is so important?

Unfortunately, I am not familiar with the data and the mathematics, so I can only present my personal thoughts which I don’t think have much or any value. What I can perhaps share, as a physicist who also has interests in biology, especially mathematical biology, and who enjoys thinking about life, is a broader or perhaps more fundamental perspective about the concept of diversity that I find inspiring.

Let me put it this way: when we speak of diversity, what are we actually (in a fundamental sense) speaking of? I find, for me at least, that discussion about diversity can sometimes result in confusion. The word is used in many contexts with many meanings. In political and sociological language, the structure of the language often implies an antonym of homogeneous groups, or, sometimes, as an antonym of a specific individual as a group: a white male with certain physical attributes, class distinction, and heterosexual orientation. From the dictionary definition of diversity, on the other hand, we find that it means difference or variance. I think science has an important perspective to offer in this precise sense.

From a genetic-centric view, and certainly also in other parts of biology, we have a concept called normal human variety. To that, I am not just speaking here of race and racial diversity. I am talking about differences in people of all kinds, including what today is called neurodiversity, which is meant to describe people like me with autism. My point is that, prior to the development of genome sequencing, people would use phenotypic characteristics (skin colour, bone structure, head size, etc.) to assess things like racial differences and even to attempt to define the concept of race, smartness, and so on. Indeed, still today cognitive bias and other prejudices are based on phenotypic characteristics of human beings and, at times, quite archaic ways of thinking. We see it every day.

I won’t get into here the debates between realism, anti-realism, and constructivists, although the limited attempts to argue the former in this context are like Swiss cheese while the latter can also be too one-dimensional. In any case, what I am driving at is how, in the past (unfortunately these attitudes still seem to manifest in the minority) what was thought of as different species or races among humans and other biological organisms was determined by phenotypic characteristics and as a result, phylogenetic trees and different groupings of humans and other organisms were often incorrectly constructed. But with technological advancements and our ability to sequence genomes quicker, more efficiently and cheaply, we have been able to compile larger genome databases with some powerful algorithms that can compare genomes more accurately. Thus, in biology, phylogenetic trees, the relationships between species, and divergences within species, can be more accurately assessed and drawn. What we find is that a lot of things that may have been thought to have been be related are not and vice versa. When comparing genomes of people from different parts of the world, we have found that although there are many minor variations between the genomes of humans as a whole, there is not enough difference to define different races. That is to say, there is no evidence for taxonomic delineation according to any definition of species or sub-species within humans, such that phylogenies inferred from mitochondrial DNA do not show any clear distinctions.

This is what I find inspiring and what I think about when thinking of diversity. The story of human beings, of our evolution, and of the universality we all share on this rock in some isolated region of the universe – it is quite beautiful. The Homo sapiens lineage has relatively recent origin when compared to other evolutionary timescales, like the planet Earth we call home, and our cosmic insignificance couldn’t be more pronounced. The universality to this reality is one that I think supports a critical humanistic vision, a perspective that, from an objective standpoint, also celebrates the incredible genetic diversity among local populations. It gives us fundamental perspective about the arbitrary nature of geographic boarders, racist attitudes, tribalism, and the many needless wars and suffering that have been waged and inflicted throughout human history on the basis of such arbitrary identifications.

In other words, while there is this incredible universality to human beings, and the similarities among people is something to be celebrated, there is so much to also celebrate about our differences – what we can call the particular as it emerges from the general. So, for example, people with ASD and the different perspectives we may offer as individuals, which, currently, is described under the heading of neurodiversity. Or, for example, the different perspective we may all offer, given our geographies and our own psychological histories. Or the diversity in our skin colours and other phenotypic differences that have come about rapidly in our evolutionary history. One of the great things about humanity is owed to the fact that as human beings we come in different shapes and sizes, we have different facial characteristics, varying eye colour, different finger prints, and different skin colours. For me, it gives perspective on how irrational our social history has also been – the needless suffering that people have faced and continue to face as a result of grim prejudice. Recently, for example, the daily prejudice black people continue face has been a renewed subject of discussion in the media. I think also of people with autism or other disabilities.

When I think of modern science, like in my area of fundamental physics, I think of the conferences I’ve attended, and the wonderful diversity of perspectives that combine. Modern science can be a fantastic representative of a more rational world, where people from many different geographic regions and backgrounds work together to solve difficult problems and to contribute to the scientific body of knowledge.

7. The IOP is committed to encouraging participation in physics among people after the age of 16 – especially those from under-represented backgrounds. How do you think we can better support others from under-represented groups who are considering studying physics? Is there anything you want to do as an ambassador?

From the little I have shared about myself, it is obvious that I think education is important. Life-long learning, for me, is a process whose end is defined only by one’s mortality. It doesn’t matter if you’re 16, 25, 55, or 80 years of age, one can always decide to take up an interest in physics or whatever else. In fact, I would encourage anyone interested in studying science to take up physics, even if they don’t plan to pursue it as a career; because regardless of what area you find interest in, physics can offer an important perspective in life. But to anyone from an under-represented background considering studying physics, or who has a passion for studying physics, I would say keep trying, and ask for support. If you are facing challenges and barriers in life, think about who might be best placed to support you through these and contact them to ask for help. This might be a teacher at school, your local council or social services department, your GP or another healthcare professional, the Citizen’s Advice Bureau, charities and support groups set up for people like you, family or friends, or the university you would like to attend. There is support out there, and it can take time to find the right support and to work through challenges, so being patient with yourself and others can help. There will be better days and harder days, so take one day at a time. Try to learn from setbacks and if things don’t quite go to plan, try to rest, regroup, and get back up the next day and try again. This is after all what a scientist often has to do!

As someone from an underrepresented background, it can sometimes be hard to feel confident that you could go to university, or be a scientist, if you don’t see or hear about people who are similar to you doing the same thing successfully. Sometimes it can feel isolating, wondering whether others have experienced the challenges that you have in pursuing their interests and goals in science, and if so, how they might have overcome these, or even whether it is possible to overcome such challenges at all. Sometimes it can be difficult to find easy answers to questions about whether a particular environment (e.g. a university or workplace) will be welcoming and accessible to someone like you. In this respect, I think organisations like IOP can help by working with universities to make them more welcoming, accessible and supportive environments for people from underrepresented backgrounds. IOP can also work to increase the visibility of people from underrepresented backgrounds who are studying and working in Physics. This could include such individuals sharing both their successes (to show that success is possible!) and the challenges they have faced, including how they have worked through these challenges. These real-life examples can be much more helpful to people who may be facing challenges and barriers of their own in pursuing physics at university or as a career, than a rose-tinted success story that leaves out the challenges and bumps along the way.

As an ambassador this is something that I would like to contribute to, and I hope that by sharing my story, perhaps it might encourage others who have faced similar challenges to keep trying and working towards their goals.

8. What would you say to someone thinking about applying to the fund? (Would you encourage them to apply? If so, why? What advice would you offer?)

I would encourage anyone to apply. Unfortunately, I am not one to give advice about applications and things, because I tend to struggle a lot with these procedures. What I can say is make sure your application meets all of the criteria, and, if your application is not successful, don’t be discouraged. Take it as a learning experience – ask for advice about any areas in which you can improve your application for next time, and try again either with another funder and/or with the Bell Burnell fund at the next application round.

9. What message do you have for Professor Dame Jocelyn Bell Burnell – and other supporters who have made this funding possible?

I suppose I would just like to thank Prof. Dame Burnell and the others involved in the scholarship. It means a huge amount to me to be able to pursue my PhD, and I hope that my research in the next few years helps to repay the support and belief in me, and that, moving forward, I can be a good ambassador and help contribute a meaningful voice in the British scientific community.

(n-1)-thoughts, n=6: Asperger’s and writing, Lie 2-algebroids, linguistics, and summer reading

Asperger’s, studying, and writing

As a person with autistic spectrum disorder (ASD), I’ve learned that writing plays an important and meaningful role in my life. I write a lot. By ‘a lot’ I mean to define it as a daily activity. Sometimes I will spend my entire morning and afternoon writing. Other times I will be up through night because my urge to write about something has kept me from sleeping. Most often I write about maths and physics, keeping track of my thoughts and ideas, planning essays, or writing about my work. But I also make it a principle of life to read widely. Indeed, I enjoy reading – studying – as much as I enjoy writing, and this often motivates me to write about many other topics. The two go hand-in-hand.

One reason writing has become important for me has to do with how, as a person with Asperger’s, social communication (by which I mean verbal, but of course also entails other forms like sign) is a source of struggle. I don’t often write about my Asperger’s, mainly because I find it a difficult process. It is hard to organise my thoughts about it, and I am never sure what is appropriate to share. In formal language, my Asperger’s is described clinically as high-functioning but severe. A big part of my life is about learning new strategies to cope. Some of the strategies may even be familiar to others without ASD, like learning to talk in front of others in ways that minimise anxiety and stress, or without completely freaking out (what we call in my language ‘red card’ moments). Or, to give another example, we work on finding strategies for the times I am at the office, so my brain doesn’t go into hyperdrive and so I can focus on discussion and also things like writing on the whiteboard. Another thing about my Asperger’s is that it can be hard adjusting to new people and it can be very stressful acclimatising to new environments. I’ve been working with Tony, now my PhD supervisor, for two years or more and I have only recently started to acclimatise and find our engagement a bit easier to manage. Indeed, in the same time I’ve been at the University of Nottingham, it remains an ongoing process adjusting to this new environment and to being on campus. Like with my close friend, Arnold, who, even after seeing him everyday for years, it was often still a challenge for me to engage with him socially and to visit his house. There is a lot to my experience, not just the social aspect of experience, that can be difficult and demanding as well as overwhelming. I also struggle a lot with anxiety and other things, in addition to extreme sensory sensitivity. So I require a lot of time and space for stillness in my own environment, with my own structure and routine – usually in my own space with my books and other comforts – because sensory overload can easily overwhelm.

In my one attempt to write about living with ASD I expressed how it can be difficult to understand cultural meanings as another example. This is a way of describing orientation to many of the ‘codes’ or behavioural routines that normalise in society. For example, I remember when I was a teenager being pressured a lot to establish the same routine economic patterns as others, or blamed because I didn’t have a job or couldn’t maintain one. I find it difficult to compute things like why daily life is the way it is for most individuals or why people behave as they do. What motivates daily behaviour and routine? How do people make decisions or direct the future course of their lives? Science, textbooks, and studying fervently became, at least in part, a survival-based mechanism. There is no instruction manual about humans; or about why history has taken the path it has in the course of human and societal development; or why many arbitrary social customs have come to be the way they are; or why my father acted and behaved the way he did; among many other things that come to be a feature of life. Studying became my way to cope and to understand, and writing became an extension of that. For instance, I studied every aspect of psychology to help better understand my experiences growing up or why, at least in part, people act violently or use violent language. I’ve read and written across most of philosophy; the same for economics, certainly enough to understand the fundamental debates; and also a lot of sociology. At one point I read a lot of political history, with history one of my favourite subjects. While all of this has a purpose in aiding my attempt to try and understand the world I am a part of, it also supports my passion for studying, my focused interests, and provides the stimulation I need.

On top of it all, living with Asperger’s can be quite exhausting. Indeed, one thing that is common for people diagnosed with autism is the experience of a certain type of fatigue, or what, in my house, we call ‘crashes’. These are a daily experience, where I need to put on my headphones and sit in my own (still and comfortable) space for however long it takes to calm my brain. For these reasons, day to day life is often spent in controlled environments, because it helps ease the red card moments, reducing stress and anxiety, and thus also helps combat the amount of crashes.

I think it all adds up in some sort of complicated sum as to why I find writing an important outlet. But even writing has its own difficulties. I remember my teacher, when I was 6 or 7 years old, say that my brain runs faster than my pen. I think this is true. I think of the sluggish pen effect as the difficulty in converting the internal representations of whatever concept or idea into concise written form at the pace I wish to feed ink to paper. So even though I write everyday and have been practising for many years, the usual result of my writing is typically permeated with errors. The process can be disabling and discouraging, to be honest, with many moments of frustration and failure; but, I’ve also learned that when I battle through and produce something I am happy with, the moment of victory is worth so much.

For many personal reasons, I’ve been regularly encouraged to write more and share more on my blog, and this is something I’ve been working toward. I think that, over a couple of years, I’ve grown more and more comfortable sharing essays and technical notes, although perhaps that is especially true in recent months; but I am also practising writing in other ways, like more personally and less formally. Technical writing is much easier than informal discussion, although a definition of the latter still seems somewhat unclear.

So as one step, this is a new blog post format that I may start experimenting with over the coming weeks, in addition to my usual research entries, essays, and technical notes. Although I prefer to keep my blog focused on my maths and physics research, which of course is mainly string related, allowing from time to time the inclusion of the odd bit of academic diversion, I think this (weekly or fortnightly) format of (n-1)-thoughts may be a fun space that allows me to practice writing in different ways, to share disconnected thoughts or random interests, outside of the formal essay or technical structure.

Generalised geometry, higher structures, and some John Baez papers

Another gem by, Urs! In a recent post on higher structures and M-theory, I made a comment recommending that people read Urs Schreiber’s many notes over the years. In my own research, I’ve found them to be invaluable. The most recent example relates, in some ways, to what I also mentioned in that post about how we may motivate the study of higher structures in fundamental physics: namely, how the Kalb-Ramond 2-form can be seen as an example of a higher structure as it is generalised from the gauge potential 1-form. I won’t go into details here, but the other day I was thinking about such generalisations, and I was thinking about Hamiltonian mechanics in the process. As I’ve mentioned before, if I were to teach string theory one day I would take this approach, emphasising at the outset the important generalisation from point particle theory to the extended object of the string.

Thinking of higher structures, I knew there were many connections here, and I was wanting to fill out my notes, for instance from how in generalised geometry the algebraic structure on TX \otimes T*X is a Courant Lie 2-algebroid. Those who study DFT will likely be quite familiar with Courant algebroids, and, certainly from a higher structure perspective this line of study is interesting. I also knew there was an original paper, which I had seen in passing, talking about this and the relation to symplectic manifolds, but I couldn’t find it. Then, bam! As Schreiber notes in a forum reply, ‘Courant Lie 2-algebroids (standard or non-standard) play a role in various guises in 2-dimensional QFT, thanks to the fact that they are in a precise sense the next higher analogue of symplectic manifolds and thus the direct generalization of Hamiltonian mechanics from point particles to strings’.

The part ‘from point particles to strings’ was hyperlinked to an important paper, the very paper I was looking for! The paper is Categorified Symplectic Geometry and the Classical String by John C. Baez, Alexander E. Hoffnung, Christopher L. Rogers. I look forward to working through this.

I also want to highlight several other papers from around the same time by Baez, including one co-authored with Schreiber, that I think are also foundational to the programme:

Categorification co-authored with James Dolan;

Higher-Dimensional Algebra VI: Lie 2-Algebras co-authored with Alissa S. Crans;

Lectures on n-Categories and Cohomology co-authored with Michael Shulman;

and, finally, Higher Gauge Theory co-authored with Urs Schreiber.

Thinking about my summer reading list

My summer holiday is in June this year, as I have a conference in mid-July and then I am scheduled to return back to university 1 August. I think Beth and I are going to spend a week in a North Norfolk, one of our favourite places, which has also sort of become a home for both of us. In anticipation of my break, I’ve started putting together my summer reading list, as I do every year. To be honest, there are so many good books right now, it is difficult to choose.

Although my list isn’t complete, one book that I’m already looking forward to is Jennifer Ackerman’s ‘The Genius of Birds‘. I had this book on my Christmas break reading list but, unfortunately, I didn’t have enough time to get to it.

I recently purchased ‘Explaining Humans: What Science Can Teach Us About Life, Love and Relationships’ by Camilla Pang, and I think I will add this to my list. Camilla has a PhD in biochemistry and, as she also has ASD, my interest in this book is more so about her personal journey coming to grips with the complex world social around her through the lens of science. It sounds, on quick glance, that we’ve come to cope with the world in similar ways and share an interest in understanding human behaviour and development. Having said that, I think there is a bit of a risk that people might read this book and conflate it with some sort of autistic worldview, which is completely incorrect, or, equally incorrect, as a scientific view of human behaviour. Contrary to some reviews, I wouldn’t read Pang’s book looking for a strictly scientific view (else one will be disappointed). I could be wrong, but I think ‘Explaining Humans’ may have potentially been mispromoted, hence some of the confused feedback. I approach this book as I would when reading someone’s memoirs, like ‘Diary of a Young Naturalist‘ by Dara McAnulty, ‘Lab girl‘ by Hope Jahren, or ‘Letters to a Young Scientist‘ by Edward O. Wilson. With topics including the challenges of relationships, learning from mistakes, and navigating the human social world by finding tools in things like game theory and machine learning, my interest is in the fact that this is another author with autism and, for myself, I similarly use textbooks and my studies to understand and manage my experience the world. Even on a purely phenomenological level, it will be interesting.

Another book that I may add is of a completely different tone: namely, Saul David’s ‘Crucible of Hell’. I’ve been enjoying reading about WWII again, and, as noted in this post on Dan Carlin’s podcast series on the events in the Asiatic-Pacific theatre, the battle of Okinawa (and others) I haven’t read much about. A few more books I have been thinking about: Douglas R. Hofstadter’s ‘Gödel, Escher, Bach: an Eternal Golden Braid‘, ‘The Deeper Genome‘ by John Parrington, ‘King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry‘ by Siobhan Roberts, ‘Decoding Schopenhauer’s Metaphysics’ by Bernardo Kastrup, ‘Quantum Computing Since Democritus‘ by Scott Aaronson, Jared Diamond’s ‘Guns, Germs, and Steel: The Fates of Human Society‘, and Daniel Kahneman’s latest ‘Noise: A Flaw in Human Judgement‘. Tough decisions.

Linguistics

I’ve been short on time this week finishing some calculations and working on a paper, prior to receiving my second Covid jab. But the other afternoon I thoroughly enjoyed this article. It’s on the Galilean challenge and its reformulation, wherein discussion unfolds on why there is an emerging distinction between the internalised system of knowledge and the processes that access it.

As alluded a moment ago, a general theory of development has interested me for a long time. For my book published by Springer Nature, a lot of the study and references were originally motivated by this interest. When I last did an extensive read on the topic, there was a lot of progress in developmental models – biology, bio- and neuro-linguistics, child psychology, and so on. The summer when I was writing my book, I had already compiled all of my research and I was running short on time in terms of the writing process (I wrote the book in the span of two weeks). Around the time of my research, if I recall there was discussion in biolinguistics regarding the hypothesis of ‘[t]he fibre tract [as one reason] for the difference in language ability in adults compared to pre-linguistic infants’. I remember noting that interesting ideas were developing, and this is a nice article on that front. What is particularly fascinating, I would say, is how language design appears to maximise computation efficiency, but ‘disregards communicative efficiency‘ [italics mine].

This certainly runs directly counter to common belief, as mentioned in the article, namely the established view that communication is a basic function of language. For a long time, as I understand it, there was belief that there was an experiential component to early language formation; but what current research suggests is that, an experiential component is not fundamental at all. Of course an experiential component plays a role, in some capacity, when it comes to externalisation processes, such as in development of variances in regional accent here in England as an example. I mean, the subject is mediated (to whatever degree) by his/her sociohistorical-cultural circumstances, but, unless I am misunderstanding (I need to read through the research more deeply) language itself is not some purely social construct.

Regarding reference to the evolutionary record, I wonder how the developing view in the article relates to ongoing research concerning, for example, certain species of birds, their migratory paths, and the question of inherited or genetic knowledge. It’s an absolutely fascinating area of study, something I’ve been reading about with my interests in mathematical biology, and of course there is very apt analogy here also with broader developments in microbiology.

One last thing of note from reading the article, as I have written quite a bit about the enlightenment philosophes and the start of modern science, it is notable how they sought to ask the question of language. Descartes’ fundamental enquiry into language – the Cartesian question – remains interesting to this day, and I was delighted to see it referenced at the outset. I recommend reading Descartes’ meditations plus other contributions to the enlightenment philosophes – Kant, Spinoza, Hume, to name a few. There is so much here that remains relevant to our modern history and to the development of the contemporary social world. For a few years I’ve been writing a series of essays on Hegel’s science of logic and his epistemology, which is notably relevant today in my area of work in fundamental maths/physics.

Mental health awareness week

Finally, it’s mental health awareness week in the UK. Often these sort of campaigns can be incredibly superficial, failing to look at root causes or ask fundamental questions about well-being and support, but they don’t have to be. Mental health awareness is something that I’ve always taken seriously, not least because I have experienced many challenges with my own mental health throughout my life. The last time I did research and wrote on the subject, suicide statistics in many leading Western countries were significant. I know, too, that for people with autism, like myself, mental health can present a significant challenge in addition to the other challenges one may face. People with autism are much more likely to die by suicide than the general population, as many cause-specific analyses of mortality for people with autistic spectrum disorder (ASD) indicate. Sometimes these facts are overlooked when we talk about mental health as a society, and often I find it important to highlight. But mental health doesn’t discriminate, it affects all people from all backgrounds, and weeks like this one are a good time to help foster discussion, combat stigma, and to think about mental health in all of its facets.

*Image: ‘Streams of Paint‘ by markchadwickart (CC BY-NC-ND 2.0).

Mathematical physics and M-theory: The study of higher structures

In recent posts we’ve begun to discuss some ideas at the foundation of the duality symmetric approach to M-theory. As we started to review in the last entry, one of the first goals is to formulate and study a general field theory in which T-duality is a manifest symmetry. It was discussed how this was the first-principle goal of double field theory, and it was similarly featured as a motivation in our introductory review of double sigma models. There is a lot to be discussed about the duality symmetric approach moving forward, including the effective theory for this doubled string prior to ultimately looking at lifting to M-theory, where, instead of double field theory we will be working with what is known as exceptional field theory. What also remains an important question has to do with obtaining a global formulation of such duality symmetric actions. What is clear is that higher geometry and algebra are important to achieving such a formulation, and there is much ground to cover on this topic.

Meanwhile, in the present entry I would like to share what I have been studying and learning about as it relates to the other side of my PhD research: the higher structure approach to M-theory. If the duality symmetric approach is a sort of bottom-up way to attack the M-theory proposal, particularly insofar that we are building from the field theory point of view, the higher structure approach can be looked at here as a sort of top-down way to access the question of string theory’s non-perturbative completion. Although this language is a bit schematic, as there is a lot of overlap between the two approaches and their machinery, it does lend some intuition to the different perspectives being undertaken.

***

In William Thurston’s 1994 essay, ‘On proof and progress in mathematics‘ [1], it was argued that progress in mathematics is driven not only by proof of new theorems. Progress is also made by aiding in human beings ways to think about and understand mathematics. Emily Riehl made this a point of emphasis at the beginning of her notes on categorical homotopy theory [2], including on the usefulness of qualitative insights, and I think a similar emphasis may be made here in the context of our focus in mathematical physics and particularly M-theory. A further point of philosophical emphasis in this essay is Eugene Wigner’s article on the unreasonable effectiveness of mathematics in physics and, finally, the more recent presentation by Robbert Dijkgraaf on the unreasonable effectiveness of string theory in mathematics. In my view, M-theory represents one of a few research topics at the frontier of mathematical physics. What parametrises the boundaries of this frontier is the interface between foundational maths and fundamental physics. Indeed, I take this as Dijkgraaf’s point in his presentation at String Math 2020, namely both the need for this engagement and how, historically, progress is often made when the two sides (mathematics and physics) interact. For myself, I almost joined the maths school prior to deciding my future was in mathematical physics, and I find great interest in working at this interface, where, furthermore, when thinking of M-theory Thurston’s notion of progress appears particularly apt.

The motivation may be stated thusly [3]: there presently exists many interconnected hints in support of the proposed existence of M-theory. But a systematic formulation of the full theory – i.e., string theory’s non-perturbative completion – remains an important open problem. A key issue here ultimately concerns the lack of clarity about the underlying principles of M-theory (there are many references on this point, but as one example see [4]). I look at the current situation as a puzzle or as a patchwork quilt. There are pieces of the total picture that we can identify and start to fill in. There are others that remain unknown, leaving empty spaces in our picture of M-theory. And then, finally, how all of the pieces relate or connect is another question that we need to answer but cannot currently access.

To advance the problem, there is ample reason to suggest and to argue that what is needed is new mathematical machinery. As a new researcher, this need was something that I started thinking about a year or more ago. Let me put it this way: our world is best described by quantum field theory. If M-theory is the correct description of fundamental physics, we should end up with a quantum field theoretic description. But it seems unlikely that M-theory will be captured or defined by some Lagrangian, or some S-matrix, or other traditional approaches [3]. Indeed, the tools we need are more than just fibre bundles, standard topology, or differential geometry. Although much of modern physics is built using tools and approaches that deal with local, approximate, perturbative descriptions of reality, in investigating the M-theory problem we need to find ways of dealing with the global and non-perturbative structure of physical fields, and thus we are dealing with the difficulty of employing non-perturbative methods. Entering into my PhD, this is the challenge that I see. I also see this challenge, from the perspective of fundamental physics, as being similar to the situations that have historically arisen many times. A large part of the history of fundamental physics is described by the search for new mathematical language required to aid the modelling of physical phenomena. Hisham Sati and Urs Schreiber [5] presented the argument well, describing the situation explicitly, when discussing the motivation for pursuing a rigorous mathematical foundation for quantum field theory and perturbative string theory. As an example, they cited the identification of semi-Riemannian differential geometry as the underlying structure of gravity. Or, think of the use of representation theory in particle physics. In truth, there are many examples and, to Dijkgraaf’s point, we should embrace this history.

I think this is why, as I prepare to start my formal PhD years, the 2018 Durham Symposium seems momentous, particularly as I begin to generate my thoughts on M-theory and what sort of research I might find meaningful. Although it was slightly before my time, as I was only a first-year undergraduate when the Durham symposium had taken place, I was already developing an interest in non-perturbative theory and I remember learning of the symposium with enthusiasm. It gave me confidence and, I suppose, assurance that my thoughts are moving in the right direction. I’ve also taken confidence from many other important conferences, such as the 2015 conference organised around the theme of new spaces in mathematics and physics. But, for me, the Durham symposium has become a tremendous reference, because the culmination of this search for new mathematical language is apparent, organised under the study of higher structures, and I find this programme of research immensely stimulating.

Similar to the situation in QFT where, over the past decade or more, progress has been made to understand its fundamental nature – for instance, efforts to define QFT on arbitrary corbordism – higher structures provides a concise language of gauge physics and duality that has seemed, in recent years, to open pathways to rigorously attack the M-theory question. Indeed, efforts toward an axiomatic formulation of QFT (for instance, see recent developments in the area of algebraic and topological QFT [6]) and those toward string theory’s full non-perturbative completion to M-theory have a lot in common. Furthermore, an important motivation for the study of higher structures (and higher differential geometry, higher gauge theory and symmetry algebras, and so on) comes directly from decisive hints about the inner workings of M-theory. Hence, the title of the Durham Symposium and its guiding document, ‘Higher structures in M-theory‘.

To give some immediate examples and sketch a few more introductory thoughts, the higher algebraic structures we know to govern closed string field theory is something I started to investigate as related to my recent MRes thesis. But the most basic example of a higher structure in string theory arguably goes back to the first quantisation of the bosonic string. Indeed, as I described in a past note (I think from my first-year undergrad), if I were to teach strings one day my opening lecture would be on generalising point particle theory and emphasising the motivation on why we want to do this. From this approach, I think one can show in a wonderfully pedagogical way that, when generalising from 0-dimensional point particle theory to the 1-dimensional string (and so on), higher dimensionality is a natural consequence and is essentially forced upon us. (As an aside, I remember reading a comment by Schreiber about this very same point of introduction. I recommend reading Schreiber’s many notes over the years. For instance, here is a forethoughtful contribution from 2004 that begins to motivate some of the concepts we will discuss below. A helpful online resource is also ncatlab that covers many of the topics we will be discussing on this blog, along with appropriate references). And, it turns out, this is one way we might also motivate in fundamental physics the study of higher structures; because, in this picture, the Kalb-Ramond 2-form can be seen as an example of a higher structure as it is generalised from the gauge potential 1-form [3]. Of course, since the mid-1990s, a growing body of evidence urged the string theory community to study extended objects of dimension > 1  , and around the same time attempts were already developing to use category theory (more on categories in a moment) to study string diagrams [7], as one can certainly see that string diagrams possess a powerful logic when it comes to composition.

***

So what do we mean by higher structures? From my current vantage, I would describe a higher structure as a categorified mathematical structure, which I also take to mean higher homotopy theory. But we can perhaps begin to build toward the idea by reviewing briefly two main ingredients: category theory and homotopy theory. As a matter of correspondence between mathematics and physics, category theory is the mathematical language of duality and homotopy theory is the mathematical language of the gauge principle.

We may think of category theory as being positioned at the foundations of modern mathematics [8], but, in many ways, it is quite elementary. Similar to the use of a venn diagram when teaching basic set theory, we can build the idea of a category in a fairly intuitive way.

A category {\mathcal{C}} consists of the following data [9]:

* A collection of mathematical objects. If {X} is an object of {\mathcal{C}} , then we write {X \in \mathcal{C}} .

* Every pair of objects {X, Y \in \mathcal{C}} , we may define a set of morphisms {X \rightarrow Y} denoted as {\text{Hom}\mathcal{C}(X,Y)} .

* For every {X \in \mathcal{C}} , there needs to exist an identity morphism {Id_{X} \in \mathcal{C}(X,X)} .

* For every triple {X,Y,Z \in \mathcal{C}} , we may define a composition map {\circ : \mathcal{C}(X,Y) \times \mathcal{C}(Y, C) \rightarrow C(X, Z)} .

* Composition is associative and unital.

If category theory is the mathematics of mathematics, I would currently emphasise in a physics context [10] the approach to category theory as the language that describes composition. Think of the trivial example of moving in some space (let’s not get too stuck on definitions at this point). We can compose the journey from points A to B to C to D in the following way,

\displaystyle  A \rightarrow B \rightarrow C \rightarrow D \ (1).

We can also compose the same journey in terms of pairs of vertices or what we are presently calling points such that

\displaystyle  A \rightarrow C, B \rightarrow D \ (2)  ,

and then we may write the entire journey as {A \rightarrow D} giving the same description in (1).

The idea of a category can be constructed using similar logic. Given a collection of objects {A,B,C,D} , paths {A \rightarrow B \rightarrow C \rightarrow D} denoted by the arrows may be defined as the relation amongst the objects in terms of structure preserving maps {f,g,h} called morphisms.

M-theory

So at its most basic, a category is a collection of objects and arrows between those objects. It is, in some sense, a relational set, which must follow the conditions stated above.

Example. The category of sets, denoted by Set. The category of R-modules, denoted by RMod. A morphism {f : X \rightarrow Y} is said to be an isomorphism if there exists {g : Y \rightarrow X} such that {g \circ f = Id_{X}} and {f \circ g = Id_{Y}} . In the category Set, isomorophisms are bijections.

The concept of functors is of deep importance in this language. In short, a functor is a morphism between categories. If {\mathcal{C}} and {\mathcal{D}} are categories, we may define a functor {F : \mathcal{C} \rightarrow \mathcal{D}} such that it assigns an object {FX \in \mathcal{D}} for any {X \in \mathcal{C}} , and a morphism {Ff : FX \rightarrow FY} for any {f : X \rightarrow Y} , where associativity and unitality are preserved. So, for instance, if {f : X \rightarrow Y} , {g : Y \rightarrow X} , associativity is preserved such that

\displaystyle  Fg \circ Ff = F(g \circ f) \ (3).

We may also define the notion of a natural transformation as a morphism between functors. If {F,G :  \mathcal{C} \rightarrow \mathcal{D}} define two functors, then a natural transformation {F \implies D} assigns any {X \in \mathcal{C}} a morphism {FX \rightarrow GX} .

There is a lot to be said about functors, categorical products, and also the important role duality plays in category theory. In the next entries, we will formally define these ideas as well as many others. For now, I am simply trying to provide some sense of an early introduction into some of the machinery used when we speak of higher structures, such as by giving an intuitive example of a category, with a mind toward formal definition in a following post. The same can be said for all ideas presented here, as, in the present entry, we are simply encircling concepts and sketching a bit of land, similar as a geoscientist would do when first preparing to sketch a topological map.

What one will find, on further inspection, is that category theory is deeply interesting for a number of reasons. At its deepest, there is something to be said about it as a foundational framework. One of the most inspiring realisations about category theory comes from something that seems incredibly basic: the idea in set theory of taking the product of two sets. Indeed, one may have seen this notion of a product as fundamental. But what we observe is that this most basic concept of taking a product of two sets is not fundamental in the way we may have been used to thinking, because one of the amazing things about the story of category theory is how the idea of products is more deeply defined in terms of a categorical product. The reward for this realisation, aside from shear inspiration, is technically immense.

Indeed, a category can contain essentially any mathematical object, like sets, topological spaces, modules, and so on. In many constructions, one will seek to study very generally the products of these objects – so, for example, the product of topological spaces – and the concept of a product in category theoretic language can capture all such instances and constructions. In later discussions we will see how this language allows us to look at mathematics at a large scale, which is to say that, in the abstract, we can take any collection of mathematical objects and study the relations between them. So if the goal is a completely general view, using category theory we are able to strip back a lot of inessential detail so as to drill fundamentally into things.

Additionally, there is a deep relationship between category theory and homotopy theory, which, in this post, I would like to highlight on the way to offering a gentle introduction to the concept of a higher structure. Down the road we will discuss quite a bit about higher-dimensional algebra, such as n-categories and operads, which are algebraic structures with geometric content, as we drive toward a survey of the connection between higher categorical structures and homotopy theory. In physics, there is also connection here with things like topological quantum field theory. Needless to say, there is much to cover, but when thinking of homotopy theory at its most basic, it is appropriate to go all the way back to algebraic topology.

The philosophical motivation is this: there are many cases in which we are interested in solving a geometrical problem of global nature, and, in algebraic topology, the method is generally to rework the problem into a homotopy theoretic one, and thus to reduce the original geometric problem to an algebraic problem. Let me emphasise the key point: it is a fundamental achievement of algebraic topology to enable us to reduce global topological problems into homotopy theory problems. One may motivate the study of homotopy theory thusly: if we want to think about general topological spaces – for example, arbitrary spaces that are not Hausdorff or even locally contractible – what this amounts to is that we relax our interest in the notion of equivalence under homeomorphism (i.e., topological equivalence) and instead work up to homotopy equivalence.

Definition 1 Given maps {f_0,f_1: X \rightarrow Y} , we may write {f_0 \simeq f_1} , which means {f_0} is homotopic to {f_1} , if there exists a continuous map {F : X \times I \rightarrow Y} , called a homotopy, such that {F(x,0) = f_0(x)} and { F(x,1) = f_1(x)} . We may also write {F: f_0 \implies f_1} to denote the homotopy.

As suggested a moment ago, a homotopy relation {\simeq} is an equivalence relation. This is true if {F_{01} : f_0 \implies f_1} and {F_{12} : f_1 \implies f_2} for the family of maps {f_i : X \rightarrow Y} , then

F_{02} (t,x) =  \begin{cases} F_{01}(2t,x)  :  0 \leq t \leq 1/2 \\ F_{01}(2t-1,x)  :  1/2 \leq t \leq 1 \\          \end{cases} \ (4)

gives a homotopy {F_{02} : f_0 \implies f_2} .

As an aside, what is both lovely and interesting is how, from a physics perspective, we may think of homotopy theory and ask how it might relate to the path integral; because, on first look, it would seem intuitive to ask this question. There is a long and detailed way to show it to be true, but, for simplicity, the argument goes something as follows. Think, for starters, of what we’re saying in the definition of homotopy. Given some {X} , which for now we’ll define as a set but later understand as a homotopy type, let us define two elements {x,y \in X} such that we may issue the following simple proposition {x = y} . The essential point, here, is that there may be more than one way that {x} is equal to {y} , or, in other words, there may be more than one reason or more than one path. Hence, we can construct a homotopy {\gamma} such that x \xrightarrow[]{\gamma} y  is a homotopy from {x} to {y} and then an identity map {Id_{X}(x,y)} for the set of homotopies from {x} to {y} in {X} . One can then proceed to follow the same reasoning and construct a higher homotopy by defining a homotopy of homotopy and so on.

The analogy I am drawing is that, in the path integral formalism, given some simply-connected topological space, recall that we can continuously deform the path {x(t)} to {x(y)} . In this deformfation, {\phi[x(t)]} approaches {\phi[y(t)]} continuously such that, taking the limit, we have

\displaystyle  \phi[y(t)]=\lim\phi[x(t)]=e^{iS[y(t)]}, \ \text{as }x(t)\rightarrow y(t) \text{continuously}. \ (5)

The principle of the superposition of quantum states, or, the sum of many paths, in a simply-connected space can be constructed as a single path integral; because, when all of the dust settles, the paths in this space can be shown to contribute to the total amplitude with the same phase (this is something we can lay out rigorously in another post). The result is that we end up with the Feynman path integral.

In homotopy theory, on the other hand, the analogous is true in that paths in the same homotopy class contribute to the total amplitude with the same phase. So, if one defines the appropriate propagator and constrains appropriately to the homotopy class, an equivalent expression for the path integral may be found. And really, one can probably already start to suspect this in the basic example of homotopy theory of topological spaces. Typically, given a topological space {X} and two continuous functions from this space to another topological space {Y} such that

\displaystyle f,g : X \rightarrow Y \ (6)  ,

it is straightforward to define, with two points in the mapping space, {f,g \in \text{Maps}(X,Y)} a homotopy {\eta}

\displaystyle  f \xrightarrow[]{\eta} g \ (7).

This is just a collection of continuous paths between the points.

But I digress. The focus here is to build up to the idea of higher structures.

The reason that a brief introduction to homotopy theory aids this purpose is because, if we think of a higher structure as a categorified mathematical structure, what we are referring to is a phenomenon in which natural algebraic identities hold up to homotopy. In other words, we’re speaking of mathematical structure in homotopy theory and thus of higher algebra, higher geometry, and so forth. Higher algebra consists of algebraic structure within higher category theory [11, 12]. As we discussed earlier, categories have a set of morphisms between objects, and, so, in the example of the category of sets, elements of a set may or may not be equal. Higher categories, much like higher algebra, are a generalisation of these sort of constructions we see in ordinary category theory. In the higher case we now have homotopy types of morphisms, which are called mapping spaces. And so, unless we are working with discrete objects, we must deal with homotopy as an equivalence relation should two so-called elements of a homotopy type, typically represented by vertices, be connected in a suitable way.

When we speak of higher structures as mathematical structures in homotopy theory, this is more specifically a mathematical structure in {(\infty, 1)-\text{category theory}} . This is a special category such that, from within the collection of all {(n, r)-\text{categories}} , which is defined to be an {\infty-\text{category}} satisfying a number of conditions, we find an {(\infty, 1)-\text{category theory}} to be a weak {\infty-\text{category}} in which all n-morphisms for {n \geq 2} are equivalences. I also think of a higher structure almost as a generalisation of a Bourbaki mathematical structure. But perhaps this comment should be reserved for another time.

In summary, if as motivation it is the case that we often want to study homotopy theory of homotopy theories, for instance what is called a Quillen model category, what we find is a hierarchy of interesting structures, which is described in terms of the homotopy theoretic approach to higher categories. And it is from this perspective that homotopy theories are just {(\infty, 1)-\text{category theory}} , where {\infty} denotes structure with higher morphisms (of all levels) and the 1 refers to how all the 1-morphisms and higher morphisms are weakly invertible. Hence, too, in higher category theory we may begin to speak of {(\infty, n)-\text{categories}} , which may be described as:

1. An n-category up to homotopy (satisfying the coherence laws, more on this in a later post);

2. An {(r, n)-\text{categories}} for {r = \infty} ;

3. A weak {\infty-\text{category}} or {\omega-\text{category}} where all k-morphisms are equivalences satisfying the condition {k > n} .

There are different ways to define {(\infty, n)-\text{categories}} , and their use can be found in such places as modern topological field theory. If category theory is a powerful language to study the relation between objects, n-categories enables us to then go on and study the relations between relations, and so on. As an example, consider the category of all small categories. For two categories {\mathcal{C}} , {\mathcal{D}} , whose morphisms are functors, the set or collection of all morphisms hom-set {\text{Fun}(C, D)} are then functors from {\mathcal{C}} to {\mathcal{D}} . This forms a functor category in which all morphisms are natural transformations, given that the natural transformations are morphisms between morphisms (functors). Hence, in this way, we scratch the surface of the idea of higher categories, because, taking from what was mentioned above, these are categories equipped with higher {n} -morphisms between {(n-1)}-morphisms for all {n \in \mathbb{N}} .

Moreover, if in ordinary category theory there are objects and morphisms between those objects, from the higher category view these are seen as 1-morphisms. Then, we may define a 2-category, which is just a generalisation that includes 2-morphisms between the 1-morphisms. And we can therefore continue this game giving definition to {n} -category theory. We will eventually get into more detail about the idea of {n} -categories, including things like weak {n} -categories where associativity and identity conditions are no longer given by equalities (i.e., they are no longer strict), instead satisfied up to an isomorphism of the next level. But for now, in thinking of the basic example of a composition of paths and this notion of generalising to 2-morphisms between the 1-morphism, the emphasis here is on the idea that the two conditions of associativity and identity must hold up to reparameterisation (the topic of reprematerisation being a whole other issue) – hence, up to homotopy – and what this amounts to is a 2-isomorphism for a 2-category. If none of this is clear, hopefully more focused future notes will help spell it all out with greater lucidity.

***

In using the language of higher structures in M-theory, there have been many promising developments. For instance, it can be seen how core structures of string/M-theory emerge as higher structures in super homotopy theory [4, 13], leading to a view of M-theory beginning from the superpoint in super Minkowski spacetime going up to 11-dimensions. An interesting part of this work was the use of Elmendorf’s theorem on equivariant homotopy theory. It has led to exciting new developments in our picture of brane physics, with an updated brane bouquet.

Of course, the higher structures programme is far-reaching. From double and exceptional field theory and the global formulation of such actions to the study of homotopy algebras in string field theory, M-branes, sigma models on gerbes, and even modern views on anomalies in which field theories are treated as functors – this merely scratches the surface. Some nice lecture notes on higher structures in M-theory, focusing for example on M5-brane systems and higher gauge theory were recently offered by Christian Saemann [14]. Hopefully we will be able to cover many of these ideas (and others) moving forward. Additionally, I am currently enjoying reading many older works, such as Duiliu-Emanuel Diaconescu’s paper on enhanced D-brane categories in string field theory [15], and I’ve been working through Eric Sharpe’s 1999 paper [16], which was the first to explicitly draw the correspondence between derived categories and Dp-branes in his study of Grothendieck groups of coherent sheaves. These and others will be fun papers to write about in time.

To conclude, we’ve begun to introduce, even if only schematically, some important ideas at their most basic when it comes to studying higher structures in M-theory. In the next entries, we can deepen our discussion with more detailed notes and definitions, perhaps beginning with a formal discussion on category theory and then homotopy theory, and then a more rigorous treatment of the idea of a higher structure.

References

[1] William P. Thurston. On Proof and Progress in Mathematics, pages 37–55. Springer New York, New York, NY, 2006.

[2] Emily Riehl. Categorical Homotopy Theory. New Mathematical Monographs. Cambridge University Press, 2014.

[3] Branislav Jurco, Christian Saemann, Urs Schreiber, and Martin Wolf. Higher structures in m-theory, 2019.

[4] Domenico Fiorenza, Hisham Sati, and Urs Schreiber. The rational higher structure of m-theory. Fortschritte der Physik, 67(8-9):1910017, May 2019.

[5] Hisham Sati and Urs Schreiber. Survey of mathematical foundations of qft and perturbative string theory, 2012.

[6] J. Baez and J. Dolan. Higher dimensional algebra and topological quantum field theory. Journal of Mathematical Physics, 36:6073–6105, 1995.

[7] Daniel Marsden. Category theory using string diagrams, 2014.

[8] Birgit Richter. From Categories to Homotopy Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2020.

[9] Carlos T. Simpson. Homotopy theory of higher categories, 2010.

[10] Bob Coecke and Eric Oliver Paquette. Categories for the practising physicist, 2009.

[11] T. Leinster. Topology and higher-dimensional category theory: the rough idea. arXiv: Category Theory, 2001.

[12] J. Baez. An introduction to n-categories. In Category Theory and Computer Science, 1997.

[13] John Huerta, Hisham Sati, and Urs Schreiber. Real ade-equivariant (co)homotopy and super m-branes. Communications in Mathematical Physics, 371(2):425–524, May 2019.

[14] Christian Saemann. Lectures on higher structures in m-theory, 2016.

[15] Duiliu-Emanuel Diaconescu. Enhanced d-brane categories from string field theory. arXiv: High Energy Physics – Theory, 2001.

[16] E. Sharpe. D-branes, derived categories, and grothendieck groups. Nuclear Physics, 561:433–450, 1999.

*Image: ‘Homotopy theory harnessing higher structures’, Newton Institute.

*Edited for spelling, grammar, and syntax.

History of Japan from the Heian period through the Second World War

For readers who like to study history, whether rigorously or simply for the enjoyment of historical discussion, last week I finished listening to a series of podcasts – much more like a series of extensive lectures, with each entry spanning 4 to 5 hours in length – on the history of Japan and its involvement in the Second World War. The talks by Dan Carlin (see bottom) have been published over the course of the past two years, and serve almost as a narration of Japanese history (from Carlin’s view) beginning roughly from the Heian period through to the events of the Asiatic-Pacific theatre.

Carlin, it must be said, is not a historian (it is fair to say that he is an amateur historian). And while it is generally the case that many historians applaud his podcast and popular engagement with history, it is important to approach his presentation as a form of popular history, firstly. That is to say, I take it as history as a way of seeking and exploring lessons, and thus, too, as a way of speculative theorising connections of factual historical events. This is not to say that Carlin is not brilliant at presenting history. He is in every sense one of the best popular history presenters, who, as I see it, has a first-principle motivation to give context to historiography by highlighting the human experience of an event. History is bloody, absolutely; and the ‘human factor’ that Carlin ensures is not lost in history is deeply important, not least philosophically. It must also be said that he provides plenty of support for his views and never fails to provide his full list of references, which usually includes both primary and secondary sources; but, again, the insisted nuance is that rigorous historical study and popular history are two very different things. The point of discretion here is just to say that one must approach each talk critically, for example discerning when Carlin is presenting his own theory or views and when he is directly citing a primary or secondary reference.

In more ways than one, listening to Carlin’s historical presentations – especially his emphasis on the human aspect of history – reminds me of an allegory on history by Albert Camus. This is something I should maybe return to and write about sometime.

***

Admittedly, the history of Japan is something I know of in discrete, disconnected pieces. It’s just not something that has been a focus in my history studies. Like a puzzle picture, some parts I have filled in but mostly in passing or in unconcentrated ways. For example, I have some understanding of its pre-historic period, mainly from books covering our best known research on early human migration that happened to include the Japanese archipelago. Over time I have also picked up some bits on ancient Japan and things like Heian culture, famously the era in which the samurai emerged. I’ve also read bits on Japan’s involvement in World World II but, again, my focus has largely been on the European theatre. Both of my grandfathers, one on my Scottish side and the other on my English side, were involved in the war. I grew up with the Second World War being a regular topic of discussion, with the Battle of Britain and other notable events often a focal point. As a kid, I also studied planes and I really liked the old British war planes, like the Spitfire, and used to build models of them as a hobby. All of this is to say that I’ve never focusedly studied the Asiatic-Pacific theatre in the same way I’ve done the European.

This is perhaps one reason why I found myself thoroughly enjoying Carlin’s series. One can approach his telling of the history with the aim in mind being a study of Japan’s involvement in both world wars. For this reason the focus is narrowed on pertinent historical and cultural developments preceding the great wars, before finally covering the events in the Pacific theatre. There is far too much to comment on, as the range in subject matter is vast. One thing that I found interesting is Carlin’s emphasis on colonialism as it relates to Japan’s motivation, military emergence, and ultimately resource-focused campaign in Asia. But before this, there were so many pertinent socio-cultural and historical developments in Japan’s history, as Carlin tells it, which contribute to what is described as a certain cultural and behavioural fanaticism. This fanaticism is expressed, in one way, through the eyes of the Japanese soldier of the time and finally culminated in an extreme barbarity that very much defined the Asiatic-Pacific theatre.

Carlin starts by first examining the phenomenon of Hiroo Onoda, the last Japanese soldier to come out of hiding from a Philippine jungle and surrender in 1974. What drove Onoda to behave in a way that, in one frame, may be described as going beyond the valour of duty, or in another frame may be described as fanatical and delusional, is a driving question in Carlin’s thesis. It is what shapes his telling of the history, because it leads Carlin, in the prologue, to introduce the observation – very much as we observe across all societies, I would argue – that human beings are malleable for better or for worse, and the ways in which we may be shaped or perhaps even deformed in extreme ways are based on our sociohistorical-cultural circumstances. So what were these circumstances? How did they develop? And what are the deep historical roots, not least related to Japan’s foray into imperialism?

Again, there is much to say, given the range of Japan’s history covered. I encourage the reader to listen to the series, because, while at times Carlin seems to make some drastic theoretical connections, the way he tells the story is absolutely gripping and, no doubt, within his recounting of many first-hand accounts, there are kernels of truth disclosed that are overwhelmingly moving in the sense that the Asiatic-Pacific theatre, in its sometimes unrelenting barbarity, was a deeply human tragedy.

***

I will leave the reader with this comment, as it is particularly on my mind. The prologue to Carlin’s series, described above, and much of how he traces key developments in Japan’s history – it reminded me very much of Edgerton’s study on social pathology, in which it is argued that a society may be more or less pathological, with the degrees of variance characteristic of the particular sociohistorical-cultural moment. This was also the thesis of my book, Society and Social Pathology, published a few years ago. Within it I argued, if we are to understand social pathology in a critical way, conceptualizing the complex interconnection between the individual subject and his/her social conditions is the first place to start. In studying the relation between one’s sociohistorical-cultural conditions and the impact those conditions have on the individual subject, my thesis argued toward a more comprehensive, systems view of society, its development, its pathology, and its discontents. As a matter of perspective, if nothing else, a number of questions that Carlin asks – for instance, what leads to the development of the sort of behaviour displayed by Onoda – reminded me of similar questions when coming to study the importance of obtaining a well-defined and rigorous concept of social pathology. Below is an excerpt from when I was thinking about such matters:

One incredibly important argument that we will discuss […] concerns how […] all societies, just like individuals, can be pathological to greater or lesser degree (Edgerton, 2010). This is an important feature of my present thesis. In a survey of literature on the history of human society, it would appear fairly safe to conclude that social pathology as defined in this book is a reoccurring characteristic across cultures and epochs.  Overcoming the pathological development of human society is, to borrow the words of Kenan Malik (2014), “a historical challenge”. That is why although capitalism […] may take a central focus in the present study, due to the fact that capitalism as a particular social formation is what defines our present social world, this particular period of human social development is also part of a significantly broader history. For this reason if the intention is to look at the facts, the realities, the many social phenomena, which defines a large part of modern life, in attempt to understand why needless social suffering persists and why irrationality prevails, to accomplish this task we must also come to grips with […] a philosophy of history [that] intersects with and combines numerous disciplines, from anthropology and archaeology to psychology. And it will help us contextualize a framework for understanding both the ongoing process of pathological development throughout history, as well as the ongoing process pertaining to our present conditions.

Episodes:

*Image: Pacific Theater Areas, Wikipedia.