Start of new semester, thinking about double field theory cosmology

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

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

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

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

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

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

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

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

For a homogeneous and isotropic cosmology the metric takes the form

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

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

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

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

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

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

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

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

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

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

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

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

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

References

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

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

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

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

Conference: Higher structures in quantum field theory and string theory

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

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

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

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.

References

[1] Chris Hull and Barton Zwiebach. Double field theory.Journal of High EnergyPhysics, 2009(09):099–099, Sep 2009.

[2] Chris Hull and Barton Zwiebach. The gauge algebra of double field theory andcourant brackets.Journal of High Energy Physics, 2009(09):090–090, Sep 2009.

[3] Olaf Hohm, Chris Hull, and Barton Zwiebach. Generalized metric formulationof double field theory.JHEP, 08:008, 2010.

[4] Olaf Hohm, Chris Hull, and Barton Zwiebach. Background independent actionfor double field theory.Journal of High Energy Physics, 2010(7), Jul 2010.

[5] Joseph Polchinski.String theory. Vol. 1: An introduction to the bosonic string.Cambridge Monographs on Mathematical Physics. Cambridge University Press,12 2007.

[6] Stefan F ̈orste and Jan Louis. Duality in string theory.Nuclear Physics B -Proceedings Supplements, 61(1-2):3–22, Feb 1998.

[7] Felix Rennecke. O(d,d)-duality in string theory.Journal of High Energy Physics,2014(10), Oct 2014.

[8] Barton Zwiebach. Double Field Theory, T-Duality, and Courant Brackets.Lect.Notes Phys., 851:265–291, 2012.

[9] K.S. Narain, M.H. Sarmadi, and Edward Witten. A Note on Toroidal Compact-ification of Heterotic String Theory.Nucl. Phys. B, 279:369–379, 1987.

[10] T.H. Buscher. A Symmetry of the String Background Field Equations.Phys.Lett. B, 194:59–62, 1987.

[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.

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.

The duality symmetric string: A return to Tseytlin

The case of the duality symmetric string is a curious one (in a recent post we began discussing this string in the context of building toward a study of duality symmetric M-theory). In this essay, which may serve as the first of a few on the topic, I want to offer an introduction to some of the characteristic features of the duality symmetric string – what I will also refer to as the doubled string – as well as discuss some of its historical connections. One thing that we will focus on at the outset is the deep connection between this extended formulation of string theory, string field theory (SFT), and the more recent development of double field theory (DFT). Such a connection is prominent not least in how we treat the string fields in constructions in which T-duality is a manifest symmetry. For the purposes of this essay, these constructions may be defined in terms of what are called double sigma models.

To help lay this out, let’s quickly review some history. In the early 1990s, a series of papers appeared by Tseytlin [1,2], Siegel [3, 4], and Duff [5]. In these papers, the important topic of string dualities was explored, particularly the fundamental role target-space duality (T-duality) plays in string theory. T-duality is of course an old subject in string theory, and we have already spoken several times in the past about its key features. Recall, for instance, that the existence of this fundamental symmetry is a direct consequence of the existence of the string as a generalisation of point particle theory. Given how for the closed string in the presence of {d} compact dimensions T-duality interchanges the momentum modes {k} of a string with its winding modes {w} around a compact cycle, one of the deep implications is that in many cases two different geometries for the extra dimensions are found to be physically equivalent.

From the space-time perspective, T-duality is a solution generating symmetry of the low energy equations of motion. However, from a world-sheet point of view, T-duality is a non-perturbative symmetry. The fact that it is an exact symmetry for closed strings suggests, firstly, that one should be able to extend the standard formulation of string theory based famously on the Polyakov action (for review, see the first chapter of Polchinski). The idea is that we may do this at the level of the world-sheet sigma-model Lagrangian density, by which I mean the motivation is to construct a manifestly T-duality invariant formulation of closed string theory on the level of the action, remembering from past discussions that we may capture T-duality transformations under the group O(D,D,\mathbb{Z}) . When we extend the theory in this way, we find that we are obliged to introduce the compact coordinates {X} and the dual ones {\tilde{X}} in the sigma model, which means we double the string coordinates in the target-space. This gives the name double string theory.

Let’s explain what this all means in clearer terms, as many of these ideas can be sketched cleanly in the context of SFT. In 1992/93, around the same time as the first duality symmetric string papers, field theory emerged as a complete gauge-invariant formulation of string dynamics [6, 7]. This led to the development of a precise spacetime action whose gauge symmetry arguably takes the most elegant possible form [8]. What was observed, furthermore, is how the momentum and winding modes may be treated symmetrically and on equal footing. For instance, let us explicitly denote the compact coordinates {X^{a}} and the non-compact coordinates {X^{\mu}} , with {X^{I} = (X^{a}, X^{\mu})} . Conventionally, we define the indices such that {I = 1,...,D} , {\mu = 1,...,d} , and {a = 1,...,n} . If the string field gives component fields that depend on momentum {p^{a}} and winding {w^{a}} , then in position space we may assign the coordinates {X^{a}} conjugate to the momentum and, as alluded above, new periodic dual coordinates {\tilde{X}_{a}} conjugate to the winding modes.

The key point is as follows: if one attempts to write the complete field theory of closed strings in coordinate space, then as stated the full theory depends naturally on dual coordinates {X^{a}} and {\tilde{X}_{a}} . This is also to say that naturally the full phase space of the theory accompanies both the momentum and the winding modes. Or, to phrase it in a slightly different manner, for toroidal compactification there is a zero mode {X^{a}} and {\tilde{X}_{a}} , and, as the expansion of a string field provides component fields that depend on both momentum and winding, we come to the statement that the arguments of all fields in such a theory are doubled. For the doubled fields {\phi(X^{a}, \tilde{X}_{a}, X^{\mu})} we may write the following seemingly simple action

\displaystyle S = \int dX^{a}  d\tilde{X}_{a}  dX^{\mu} \mathcal{L}(X^{a}, \tilde{X}_{a}, X^{\mu}) \ (1)  .

The Lagrangian in (1) may seem straightforward, but in fact it proves incredibly complicated. One issue has to do with how the physical content of the theory becomes buried underneath unphysical and computationally inaccessible data, with the full closed string field theory comprising an infinite number of fields. This is where DFT may be motivated from first-principles; because, in response, DFT answers this problem by issuing the following simplification strategy: what if we instead choose some finite subset of string fields? An obvious choice for such a subsector of the full theory is the massless sector. In the study of DFT, we may then ask, if for the standard bosonic string the low-energy effective action is famously

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

what does this action become in the case of doubled coordinates on tori? Is T-duality manifest? What about for non-trivial geometries? Historically, DFT emerged with the aim to answer such questions. In fact, following Nigel Hitchin’s introduction of generalised geometry [9, 10], itself inspired by the existence of T-duality, serious efforts materialised to incorporate this mathematical insight into the study of the target-space geometry in which strings live [11, 12, 13, 14], beginning especially with the study of phase space and invariance of respective Hamiltonians. This culminated in 2009, when Hull and Zwiebach formulated such a T-duality invariant theory explicitly [11], formalising DFT almost two decades after the original duality symmetric string papers. What one finds is a theory constructed on the product manifold {\mathbb{R}^{d-1,1} \times T^{n}} with coordinate space fields {\phi(X^{\mu}, X^a, \tilde{X}_{a})} . The torus is doubled, containing the spacetime torus and the torus parameterised by the winding modes, such that {(X^a, \tilde{X}_{a})} are periodic on {T^{2n}} . The spectrum for the massless fields is then described in terms of the supergravity limit of string theory.

By taking this approach, DFT has presented fresh insight on T-duality in string theory, leading to the development of deeper connections between frontier theoretical physics and mathematics through the appearance and use of Courant brackets, and by gaining new insight on the deepening role generalised geometry seems to play in string theory.

Much like field theory, the doubled world-sheet theory has also been reinvigorated in the last decade or more. This follows from breakthrough work by Hull [16, 17], who established the doubled formalism to define strings in a class of non-geometric backgrounds known as T-folds. These are non-geometric manifolds where locally geometric regions are patched together such that the transition functions are T-duality transformations.

***

Currently, there are primarily two doubled string actions that we may consider when constructing double sigma models: Tseytlin’s first-principle construction of the duality symmetric string [1, 2, 15] and Hull’s doubled formalism [16, 17]. Both actions satisfy the requirement of T-duality appearing as a manifest symmetry, with the former possessing general non-covariance and the latter possessing general covariance.

Hull’s doubled formalism is interesting for several reasons. In this formulation we have manifest 2-dimensional Lorentz invariance from the outset, and a notable advantage is that there is a priori doubling of the string coordinates in the target space. In other words, both the Tseytlin approach and the Hull approach are formulated such that both the string coordinates and their duals are treated on equal footing. But in Hull’s formulation, {O(D,D)} invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space takes the form {R^{1, d-1} \otimes T^{2n}} , in which we have a non-compact spacetime and a doubled torus. From the torus identifications we have manifest {GL(2n; \mathbb{Z})} symmetry. Then, after imposing what is defined as the self-duality constraint of the theory, which contains the O(D,D) metric, invariance of the theory reduces directly to O(D,D; \mathbb{Z}) . In other words, while the doubled formalism starts with a covariant action that involves doubled coordinates, the invariance of this theory under O(D,D) is generated by imposing this self-duality constraint, which, similar to DFT, effectively halves the degrees of freedom and ensures that the remaining fields are physical.

Think of it this way: in Hull’s doubled formalism the essential motivation is to double the torus by then adding {2n} coordinates such that the fibre is {T^{2n}} ; however, typically the fields depend only on the base coordinates. Finally, the strategy is generally to proceed with a patch-wise splitting {T^{2n} \rightarrow T^{n} \oplus \tilde{T}^n} so that we have demarcated a strictly physical subspace {T^n} and its dual {\tilde{T}^{n}} . For a geometric background local patches are glued together with transition functions which include group {GL(n, \mathbb{Z})} valued large diffeomorphisms of the fibre. For the non-geometric case, this is approached by gluing local patches with transition functions that take values in {GL(n, \mathbb{Z})} as well as in the complete T-duality group, such that {O(D,D,\mathbb{Z})} is a subgroup of {GL(2n, \mathbb{Z})} large diffeomorphisms of the doubled torus.

On the other hand, Tseytlin’s first-principle formulation of the duality symmetric string and world-sheet theory for interacting chiral scalars, which presents a direct stringy extension (or stringification) of the Floreanini-Jackiw Lagrangians [31] for chiral fields, does not possess {O(D,D)} by principle of construction. Instead, we find that it emerges rather organically as an intrinsic characteristic of the doubled string, with the caveat being that we lose manifest Lorentz covariance on the string world-sheet. What one finds is that we must instead impose local Lorentz invariance on-shell. The equivalence of the Tseytlin and Hull actions on a classical and quantum level has been shown in [32, 33, 34]. Like DFT, both of these approaches are constructed around the generalised metric \mathcal{H}_{IJ} which we’ll touch on later.

It is no surprise that earlier formulations of the duality symmetric string were a primary reference in the development of DFT. In [1, 2], Tseytlin argues that the existence of the intrinsically stringy winding modes, which appear in the spectrum of the closed string compactified on a torus (created by vertex operators involving both {X} and {\tilde{X}} ), can result in 2d field theories with interactions indeed involving {X} and {\tilde{X}} . Similar models have been explored in statistical mechanics, with the key point in closed string theory being how for fully-fledged local quantum field theories we are required to treat {X} and {\tilde{X}} as independent 2d fields (dual to each other on-shell). An advantage of such an extended formulation of string theory is that we may obtain more vacua than the standard formulation. Furthermore, as one may have guessed, the notion of the duality symmetric string is based on the fact that duality symmetry becomes an off-shell symmetry of the world-sheet action. Thus, T-duality for example may be made manifest in the scattering amplitudes and on the level of the effective action.

To study the construction of the duality symmetric string, we note that directly from 2-dimensional scalar field theory constructed to be symmetric in {\phi} and {\tilde{\phi}} , Tseytlin derives the Lagrangian density

\displaystyle \mathcal{L}_{sym} = \mathcal{L}_{+}(\phi_{+}) + \mathcal{L}_{-}(\phi_{-}) \ (3)

with

\displaystyle \mathcal{L}_{\pm}(\phi_{\pm}) = \pm \frac{1}{2}\dot{\phi}_{\pm}\phi^{\prime}_{\pm} - \frac{1}{2} \phi^{\prime 2}_{\pm}. \ (4)  .

Here {\mathcal{L}_{+}} and {\mathcal{L}_{-}} are the Floreanini-Jackiw Lagrangian densities for chiral and anti-chiral fields, with {\dot{\phi} = \partial /\partial_{\tau}} and {\phi^{\prime} = \partial / \partial_{\sigma}} . The total Lagrangian {\mathcal{L}_{sym}} is itself constructed so that it is manifestly invariant under the exchange of {\phi = \frac{1}{\sqrt{2}} (\phi_{+} + \phi_{-})} with its Hodge dual {\tilde{\phi} = \frac{1}{\sqrt{2}} (\phi_{+} - \phi_{-})} . Directly from the equations of motion one can derive chirality conditions for this theory (for a complete review see also [32, 33, 34]).

For our present purposes it is important to note that the goal for Tseytlin is to realise from 2-dimensional scalar field theory the corresponding formulation of string theory, which indeed proves general enough to incorporate the world-sheet dynamics of the winding sector. Writing the Lagrangian (3) for {D} scalar fields {X^{I}} and with a general background, in the Tseytlin approach we famously obtain the action

\displaystyle S [e^{a}_{n}, X^{I}] = - \frac{1}{2} \int_{\sum} d^{2}\xi  e [ \mathcal{C}^{ab}_{IJ}(\xi)  \nabla_{a} X^{I} \nabla_{b} X^{J}]. \ (5)  .

Here {I, J = 1,...,D} . We define the coordinates on {\sum} such that {\xi^{0} \equiv \tau} and {\xi^{1} \equiv \sigma} . The two-dimensional scalar fields {X^{I}} depend on {\xi} and they are vectors in {N} -dimensional target space {\mathcal{M}} . The number {N} of embedding coordinates is kept general, because the purpose of this action is to be as generic as possible while minimising assumptions for its construction. We also note that {C_{IJ}} need not necessarily be symmetric and, from the outset, we can treat it completely generically. We also have the zweibein {e^{a}_{n}} , where {e = \det e^{a}_{n}} . This term appears in the definition of the covariant derivative of the scalar field {X^{I} : \nabla_{a} X^{I} \equiv e^{a}_{n}\partial_{a} X^{I}} , where {a} is a flat index and {n} is a curved index.

In its first principle construction, which occupies the earliest sections of [2], one can recover from this generic action (5) the standard manifestly Lorentz invariant sigma model action for strings propagating in a curved background. Furthermore, if we exclude the dilaton for simplicity we may define {\mathcal{C}^{ab}_{IJ} = T(\eta^{ab}G_{IJ} - \epsilon^{ab}B_{IJ})} , where we reintroduce explicit notation for the string tension {T} , {G} is the metric tensor on the target space, and {B} is the Kalb-Ramond field.

Keeping to a generic analysis with a general {C} , after a number of steps one finds that (5) may be rewritten in the following way,

\displaystyle S = -\frac{1}{2} \int d^{2}\xi  e[ \mathbb{C}_{IJ}(\xi) \nabla_{0} X^{I} \nabla_{1} X^{J} + M_{IJ} \nabla_{1} X^{I} \nabla_{1} X_{J}]. \ (6)  .

Here it is conventional to define {\mathbb{C}_{IJ} = C_{IJ}^{01} + C_{JI}^{10}} and {M_{IJ} = M_{JI} = C^{11}_{IJ}} . The action is manifestly diffeomorphism {\xi^{n} \rightarrow \xi^{\prime n}(\xi)} and Weyl {e^{a}_{n} \rightarrow \lambda(\xi)e^{a}_{n}} invariant, but it is not manifestly invariant under local Lorentz transformations. Moreover, notice that (6) must be invariant for the finite transformation of the zweibein, because the physical theory should be independent of {e^{a}_{n}} . This means that if under such a transformation we have {e^{a}_{n} \rightarrow e^{\prime a}_{n} = \Lambda^{a}_{b}(\xi)e^{b}_{n}} , where one may recognise {\Lambda^{a}_{b}} is a Lorentz {SO(1,1)} matrix dependent on {\xi} , we also have an induced infinitesimal transformation of the form {\delta e^{a}_{n} = \omega^{a}_{b}(\xi)e^{b}_{n}} with {\omega_{ab} = - \omega_{ba}} . Now, substituting {\omega^{a}_{b}(\xi) = n(\xi)\epsilon^{a}_{b}} , we obtain

\displaystyle \delta e^{a}_{n} = n (\xi)\epsilon^{a}_{b}(\xi)e^{b}_{n}. \ (7)  ,

however, as stated, the action is not manifestly invariant under such transformations. The requirement of on-shell local Lorentz invariance is fundamental to the entire discussion at this point. As Tseytlin comments in a footnote [2], alternatively we may prefer Siegel’s manifestly Lorentz covariant formulation, but with that we obtain extra fields and gauge symmetries; whereas in extending the Floreanini-Jackiw formulation it is fairly simple to introduce interactions and, ultimately, we find that the condition in the Siegel approach that requires decoupling of the Lagrange multiplier corresponds to what we will review as the Lorentz invariance condition in the Floreanini-Jackiw approach.

For the action (6), a way to attack the requirement of on-shell Lorentz invariance is by seeing in [2] that it demands we satisfy the condition

\displaystyle \epsilon^{ab} t_{ab} = 0, \text{where} \ t_{a}^{b} \equiv \frac{2}{\epsilon} \frac{\delta S}{\delta e^{a}_{n}}e^{b}_{n}. \ (8)  .

The general idea is that the tree-level string vacua should be assumed to correspond to {S[X, \tilde{X}, e]} , which define the Weyl and Lorentz invariant quantum field theory. In performing the background field expansion, we may take the expansion to be near the classical solution of the {(X, \tilde{X})} equations of motion with the trace of the expectation value of the energy-momentum tensor as well as the {\epsilon^{ab}} trace vanishing on-shell. In Tseytlin’s formulation, {\hat{t}} denotes precisely this epsilon trace such that {\hat{t} = \epsilon^{a}_{b} t_{a}^{b}} . The vanishing of {\hat{t}} shows local Lorentz invariance. So let us now vary (6) under local Lorentz transformation, which is proportional to the equations of motion

\displaystyle t^{b}_{a} = - \delta_{a}^{b} [\mathbb{C}_{IJ}(\xi) \nabla_{0} X^{I} \nabla_{1} X^{J} \ + \ M_{IJ} \nabla_{1} X^{I} \nabla_{1} X^{J}] \ + \ \delta_{0}^{b}[C_{IJ}\nabla_{a} X^{I} \nabla_{1} X^{J}] \ + \ \delta_{1}^{b} [C_{IJ}\nabla_{0} X^{I} \nabla_{a} X^{J}] \ + \ 2\delta_{1}^{b}M_{IJ}\nabla_{a}X^{I}\nabla_{1}X^{J}. \ (9)

This equation for {t^{b}_{a}} is equivalent to equation 4.3 in [2]. In order for the variation of the action to vanish under such a transformation, we derive the condition

\displaystyle \epsilon^{ab}t_{ab} = 0. \ (10)

In other words, the condition that must be satisfied to recover local Lorentz invariance depends on the solution of the equations of motion for the zweibein. In fact, one will recognise that what is observed is completely analogous to the standard string theory formulation based on the Polyakov action, where one will recall that the equations of motion for the world-sheet metric determines the vanishing of the energy-momentum tensor [35].

This constraint must be imposed on a classical and quantum level.

The key point is that now we can choose the flat gauge {e_{n}^{a} = \delta_{n}^{a}} , thanks to the invariances under diffeomorphisms, Weyl transformations, and finally local Lorentz invariance imposed on-shell. This is crucial for the formulation of the dual symmetric string in that, using the flat gauge for the zweibein, we are effectively performing the analogous procedure as when fixing the conformal gauge in standard string theory. Keeping {C} and {M} constant, we can compute the equations of motion for the field {X^{I}} to give

\displaystyle  \nabla_{1} [e (C_{IJ} \nabla_{0} X^{J} + M_{IJ}\nabla_{1} X^{J}] = 0. \\\ (11)

In the flat gauge this result becomes

\displaystyle \partial_{1} [C_{IJ} \partial_{0} \xi^{J} + M_{IJ} \partial_{1} \xi^{J}] = 0. \ (12)  .

From (12) a now famous identity appears, where, in the flat gauge and along the equations of motion for {\xi^{I}} , the following constraint on {C} and {M} is obtained [2]:

\displaystyle  C = MC^{-1}M. \ (13)

One may recognise the tensor structure of (13) in terms of the action of an {O(D,D,\mathbb{Z})} element. The important thing to highlight is that throughout the lengthy calculation to get to this point, {C} and {M} are held constant. (When {C} and {M} are not treated as constant, a number of interesting questions arise which extend beyond the scope of the present discussion). What is also important is that, after rotating {\xi^{I}} , the matrix {C} can always be put into diagonal form such that

\displaystyle  C = \ \textbf{diag} \ (1,...,1,-1,...,-1). \ (14)

It remains to be said that {C = C^{-1}} , which means that the constraint (13) defines the indefinite orthogonal group {O(p,q)} of {N \times N} matrices {M} with {N = p + q} in {\mathbb{R}^{p,q}} . The inner product may now be written as

\displaystyle  C = MCM, \ (15)

in which the matrix {C} eventually takes on the explicit definition of an {O(D,D,\mathbb{R})} invariant metric in the 2D target space {M} . Although, admittedly, this cursory review has omitted many important and interesting details, the pertinent point in terms of this essay is as follows. The action (6) turns out to describe rather precisely a mixture of {D} chiral {\xi^{\mu}_{-}} and {D} anti-chiral {\xi^{\mu}_{+}} scalars. In demanding local Lorentz invariance and the vanishing of the Lorentz anomaly, this requires that {p = q = D} with {2D = N} . In working through the complete logic of the calculation, we observe quite explicitly that inasmuch the requirement of local Lorentz invariance is imposed through the condition (10), this leads one naturally to an interpretation of the matrix {C} as a 2D target space metric with coordinates

\displaystyle  \xi^{I} = (\xi^{\mu}_{-}, \xi^{\mu}_{+}), \ ds^{2} = dX^{I} C_{IJ} d X^{J}, \ I = 1,...,2D, \ \text{and} \ \mu = 1,...,D. \ (16)

If we make a change of coordinates in the target space, particularly by defining a set of new chiral coordinates, the matrix {C} takes on the off-diagonal form of the {O(D,D)} constant metric {L} typically considered in DFT (for review, see [36]) and elsewhere. The chiral coordinates we define are

\displaystyle  X^{I} = \frac{1}{\sqrt{2}} (X_{+}^{\mu} + X_{-}^{\mu}), \tilde{X}_{I} = \frac{1}{\sqrt{2}} (X_{+}^{\nu} - X_{-}^{\nu}). \ (17)

In this frame, the matrix {C} is then shown to be

\displaystyle C_{IJ} = - \Omega_{IJ} = -\begin{pmatrix}  0 & \mathbb{I} \\ \mathbb{I} & 0 \end{pmatrix}. \ (18)

It follows that the condition (13) transforms into the constraint

\displaystyle  M^{-1} = \Omega^{-1}M\Omega^{-1} \ (19)

on the symmetric matrix {M} , which can be parametrised by a symmetric matrix {G} and an antisymmetric matrix {B} . Therefore, remarkably, the symmetric matrix {M} takes the precise form of the generalised metric in which {M} is found to be positive definite.

To conclude, in the chiral coordinates we arrive at a famous form of the Tseytlin action,

\displaystyle  S = \frac{1}{2} \int d^{2}\xi \ e[ \Omega_{IJ} \nabla_{0} X^{I} \nabla_{1} X^{J} - M_{IJ}\nabla_{1} X^{I} \nabla_{1} X^{J}]. \ (20)

This action is manifestly {O(D,D)} invariant. When {O(D,D)} transformations are applied to (20), we obtain exactly what we would anticipate for the standard string in the sense of T-duality invariance under {X \rightarrow\tilde{X}} and for the generalized metric {M \rightarrow M^{-1}} .

For completeness, from the action (6) in arriving at (20), it should be clear that what we are working with is a sigma model for the dual symmetric string. The generalised version of the celebrated action (20) is indeed often written as

\displaystyle  S_{General} = \frac{1}{2} \int d^{2}\xi \ [- (C_{IJ} + \eta_{IJ}) \partial_{0} X^{I} \partial_{1} X^{J} + \mathcal{H}_{IJ} \partial_{1} X^{I} \partial_{1}X^{J})]. \ (21)

This final action can be argued to be a very natural generalisation for the standard string on a curved background. It not only contains the generalised metric {\mathcal{H}_{IJ}} , but also another symmetric metric {\eta_{IJ}} with {(D,D)} signature and an antisymmetric 2-tensor {C_{IJ}} . The coordinates are defined {X^{I} = \{ X^{I}, \tilde{X}_{I} \}} with the background fields in general depending on {X^{I}} .

***

In the last decade especially, Tseytlin’s formulation has been refocused in various studies concerning the nature of the doubled string and its geometry. One notable example to which we will return in a moment, pre-dates the first primary collection of DFT papers and, in many ways, can be interpreted to give a prediction to DFT. I am refering to the 2008 paper David S. Berman, Neil B. Copland, and Daniel C. Thompson [18], where they investigated the background field equations for the duality symmetric string using an action equivalent to that of Tseytlin’s but constructed in the context of Hull’s doubled formalism. In recent years, a series of publications on doubled sigma models have appeared in connection [19, 20, 21, 22], where in [20] the double sigma model is for example directly related to DFT.

Another example refers directly to both Tseytlin and DFT from a different perspective. In the years after 2009 when Hull and Zwiebach published their important paper formalising DFT, it was recognised that while a deep connection exists between DFT and generalised geometry, with the former locally equivalent to the latter, it does not completely come into contact with its formal mathematical structures. In fact, an open research question remains motivated by the unmistakeable resemblance DFT has with generalised geometry and the formal gap that remains between them. Recent work in mathematics and physics has displayed some promise, suggesting that the use of para-Hermitian and para-Kähler manifolds may be the solution [23, 24, 25]. Related to these efforts is a recent reformulation of string theory under the heading metastring theory [24, 26, 27, 28, 29], which begins, similar to the studies on double sigma models, with a generalised version of the first-principle Tseytlin action for the duality symmetric string. The metastring is therefore a chiral T-duality invariant theory that, in many ways, wants to generalise from DFT and make direct connection with things like Born geometry [26], relying on the consistency of Tseytlin’s formulation.

If a direct consequence of making T-duality manifest is that the winding modes are treated on equal footing with momentum, then for DFT all of these properties are incorporated into one field theory. The result, as mentioned, is a doubled coordinate space. In metastring theory, on the other hand, the target space of the world-sheet formulation is a phase space, much like in Tseytlin’s original construction. The coordinates of this phase space are indeed doubled, but unlike in DFT they are also conjugate such that in this case the dual coordinates are related directly to energy-momentum coordinates. In other words, {\tilde{X}} is now identified with {p} . This means that, instead of a physical spacetime formulation, the goal of metastring theory is to construct a sigma model as a phase space formulation of the string and its dynamics.

The implications of metastring theory, as they have so far been conjectured, are intriguing. For example, there have been claims toward obtaining a family of models with a 3+1-dimensional de Sitter spacetime, argued to be realised in the standard tree-level low-energy limit of string theory in the case of a non-trivial anisotropic axion-dilaton background [29]. A key statement here is that, while string theory has purely stringy degrees of freedom (from first principles consider simply the difference between the left and right-moving string modes), these are not captured by standard effective field theory approaches and their spacetime descriptions. Such approaches are usually employed when investigating de Sitter space. In the phase-space formulation of the metastring, these purely stringy degrees of freedom (generally chiral and non-commutating) are argued to be captured explictly. When it comes to the hope of obtaining an effective de Sitter background, one of the major claims in this non-commutative phase-space formalulation is how, in the doubled and generalised geometric description, the effective spacetime action translates directly into the see-saw formula for the cosmological constant. Furthermore, in this cosmic-string-like solution related to the concept of an emergent de Sitter space, it is argued that the metastring leads naturally to an expression of dark energy, represented by a positive cosmological constant to lowest order. Finally, it is argued that the intrinsic stringy non-commutativity provides a vital ingredient for an effective field theory that reproduces to lowest order the sequestering mechanism [29, 30] and thus a radiatively stable vacuum energy.

***

Building from the Tseytlin action (21), this world-sheet theory of chiral bosons not only takes the heterotic string to its maximal logical completion (a point to be discussed another time), the total doubled space that it sees naturally accomodates stringy non-geometries. With the development of DFT and Hull’s doubled formalism in mind, one interesting question that we can ask concerns whether the best features of all of these approaches can be put together under a more general formulation. There is already a lot in Tseytlin’s original first-principle construction, and so one idea is to generalise from this theory. This was one motivation for my MRes thesis. Another question concerns the presence of generalised geometry and finally how, given a completely generalised treatment of the duality symmetric string, how may we extend the ideas toward the study of duality symmetric M-theory, where exceptional field theory seeks to promote the U-duality group to a manifest symmetry of the spacetime action [37, 38].

These comments take us back to the work of Berman et al. [18], who started to point toward the same question of generalisation in their approach that combines Tseytlin’s action with Hull’s doubled formalism. It is a very interesting entry into the ideas described, and it is this paper where my own MRes thesis more or less entered the picture.

Moreover, the approach in my MRes was basically to follow the prescription first adopted by Berman et al; however, the action they used to study the doubled beta-functionals for the interacting chiral boson model was constructed in the case where the background fields depend trivially on the doubled coordinates but non-trivially on the non-compact spacetime coordinates. This means that in their approach the target-space was constructed in terms of a torus fibration {T^{n}} over a base {N} . One may think of this as a description of string theory in which the target space is locally a {T^n} bundle, while {N} is some generic base manifold that may be thought of simply as a base space.

While such constructions are important and deserve attention moving forward – we will certainly discuss cases in the future of more complicated bundles, for example – for my MRes the idea was to first strip everything back and generalise the result with minimal assumptions. The first step, for example, was to not demand anything about the dependence of the background fields. What we arrived at was an action of the form

\displaystyle  S_{Maximally \ doubled} = \frac{1}{2} \int d^{2}\sigma [-\mathcal{H}_{AB}(X^{A}) \partial_{1} X^{A} \partial_{1} X^{B} + L_{AB}(X^{A}) \partial_{1}X^{A} \partial_{0} X^{B}], \ (22)

where {\mathcal{H}} is the generalised metric and we also have a generic 2-tensor {L} (that we continued to treat generically). In doing away with a base-fibre split (we also dropped a topological term, which isn’t so important here), what we have is the sort of action considered originally by Tseytlin. In fact, (22) is the most general doubled action we can write without manifest Lorentz invariance, because it allows us to calculate the background fields in a way in which the fields maintain arbitrary dependence on the full doubled geometry. That is to say, in taking the democratic approach in which everything becomes doubled, we’re ultimately seeking an effective spacetime theory that corresponds to completely generic non-geometric geometries. At the same time, the structure of the action is precisely the sort proposed to lead directly to DFT [20], and it also remains equivalent to the Polyakov action in the standard formulation of string theory.

Due to the fact that there are papers pending on these calculations and associated topics, I will leave more details for future entries and for when they more formally appear on arxiv.

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