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.

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.

References

[1] Arkady A. Tseytlin. Duality Symmetric Formulation of String World Sheet Dy-namics.Phys. Lett. B, 242:163–174, 1990.

[2] Arkady A. Tseytlin. Duality symmetric closed string theory and interactingchiral scalars.Nucl. Phys. B, 350:395–440, 1991.

[3] Warren Siegel. Superspace duality in low-energy superstrings.Phys. Rev. D,48:2826–2837, 1993.

[4] Warren Siegel. Two vierbein formalism for string inspired axionic gravity.Phys.Rev. D, 47:5453–5459, 1993.

[5] M.J. Duff. Duality Rotations in String Theory.Nucl. Phys. B, 335:610, 1990.

[6] Taichiro Kugo and Barton Zwiebach. Target space duality as a symmetry ofstring field theory.Prog. Theor. Phys., 87:801–860, 1992.

[7] Barton Zwiebach. Closed string field theory: An introduction, 1993.

[8] Theodore Erler. Four lectures on closed string field theory.Physics Reports,851:1–36, Apr 2020.

[9] Nigel Hitchin. Lectures on generalized geometry, 2010.

[10] Nigel Hitchin. B-fields, gerbes and generalized geometry, 2005.

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

[12] 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.

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

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

[15] Arkady A. Tseytlin. Duality symmetric string theory and the cosmological con-stant problem.Phys. Rev. Lett., 66:545–548, 1991.12.

[16] Christopher M Hull. A Geometry for non-geometric string backgrounds.JHEP,10:065, 2005.

[17] Christopher M Hull. Doubled Geometry and T-Folds.JHEP, 07:080, 2007.

[18] David S. Berman, Neil B. Copland, and Daniel C. Thompson. Background fieldequations for the duality symmetric string.Nuclear Physics B, 791(1-2):175–191,Mar 2008.

[19] David S. Berman and Daniel C. Thompson. Duality symmetric strings, dilatonsand o(d,d) effective actions.Physics Letters B, 662(3):279–284, Apr 2008.

[20] Neil B. Copland. A Double Sigma Model for Double Field Theory.JHEP, 04:044,2012.

[21] Spyros D. Avramis, Jean-Pierre Derendinger, and Nikolaos Prezas. Conformalchiral boson models on twisted doubled tori and non-geometric string vacua.Nuclear Physics B, 827(1-2):281–310, Mar 2010.

[22] David S. Berman and Daniel C. Thompson. Duality symmetric string and m-theory, 2013.

[23] Laurent Freidel, Felix J. Rudolph, and David Svoboda. Generalised kinematicsfor double field theory.Journal of High Energy Physics, 2017(11), Nov 2017.

[24] Laurent Freidel, Robert G. Leigh, and Djordje Minic. Quantum gravity, dynam-ical phase-space and string theory.International Journal of Modern Physics D,23(12):1442006, Oct 2014.

[25] David Svoboda. Algebroid structures on para-hermitian manifolds.Journal ofMathematical Physics, 59(12):122302, Dec 2018.

[26] Laurent Freidel, Robert G. Leigh, and Djordje Minic. Metastring theory andmodular space-time, 2015.

[27] Laurent Freidel, Robert G. Leigh, and Djordje Minic. Modular Spacetime andMetastring Theory.J. Phys. Conf. Ser., 804(1):012032, 2017.

[28] Laurent Freidel, Robert G. Leigh, and Djordje Minic. Noncommutativity ofclosed string zero modes.Physical Review D, 96(6), Sep 2017.

[29] Per Berglund, Tristan H ̈ubsch, and Djordje Mini ́c. On stringy de sitter space-times.Journal of High Energy Physics, 2019(12), Dec 2019.

[30] Per Berglund, Tristan H ̈ubsch, and Djordje Mini ́c. Dark energy and string theory.Physics Letters B, 798:134950, Nov 2019.13

[31] Roberto Floreanini and Roman Jackiw. Selfdual Fields as Charge Density Soli-tons.Phys. Rev. Lett., 59:1873, 1987.

[32] Franco Pezzella. Two double string theory actions: Non-covariance vs. covari-ance, 2015.

[33] Gianguido Dall’Agata and Nikolaos Prezas.Worldsheet theories for non-geometric string backgrounds.Journal of High Energy Physics,2008(08):088–088, Aug 2008.

[34] L. De Angelis, G. Gionti S.J, R. Marotta, and F. Pezzella. Comparing doublestring theory actions.Journal of High Energy Physics, 2014(4), Apr 2014.

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

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

[37] David S. Berman and Malcolm J. Perry. Generalized Geometry and M theory.JHEP, 06:074, 2011.

[38] Olaf Hohm and Henning Samtleben. Exceptional Form of D=11 Supergravity.Phys. Rev. Lett., 111:231601, 2013.

M-theory, the duality symmetric string, and fundamental mathematical structure

In quantum gravity, there presently exists a tight web of hints as well as numerous plausibility arguments in support of the proposed existence of M-theory; however, a systematic formulation of the non-perturbative theory remains an open problem. Without a fundamental formulation of M-theory, all we have is a hypothetical theory of which splinters of clues intimate 11-dimensional supergravity and the five string theories are each a limiting case of some deeper structure.

Mike Duff once described the situation like a patchwork quilt. We have corners – for instance, matrix theory – and we have some bits of stitching here and there, great successes in themselves, but the total object of the quilt is not understood.

In my opinion, this is one of the most deeply interesting and challenging problems one can currently undertake. In pursuing M-theory a great ocean lays undiscovered, in the words of Duff, the depths of which we may not yet be able to fully imagine but of which we anticipate to lead to new mathematics.

We still have no fundamental formulation of “M-theory” – the hypothetical theory of which 11-dimensional supergravity and the five string theories are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. […]. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside – temporarily. But, ultimately, Physical Mathematics must return to this grand issue.’ – Greg Moore, from his talk at Strings 2014

For our present purposes, to better explain the opening paragraphs and the two programmes of research in non-perturbative theory in which I am currently most interested, we should go back a couple of decades in time. The story begins as late as 1995, when it was believed that the five superstring theories – type I, type IIA, type IIB, and the two flavours of heterotic string theory (SO(32) and E8 ${\times}$ E8) were distinct. At this time, the situation in quantum gravity appeared messy. There were five theories and no obvious mechanism to select the correct one. When it was eventually observed that these theories are deeply related by non-trivial dualities, it was proposed by Edward Witten that rather than being distinct they actually represent different limits of an overarching theory. This overarching theory, M-theory, was indeed found to possess an extraordinary unifying power, giving conception to the notion of a web of dualities based firstly on Witten’s observation that the type IIA string and the E8${\times}$E8 heterotic string are related to eleven-dimensional supergravity [1].

More specifically, it was seen that the 10-dimensional type IIA theory in the strong coupling regime behaves as an 11-dimensional theory whose low-energy limit is captured by 11-dimensional supergravity. This mysterious 11-dimensional theory was then seen to give further clue at its parental status when it was observed how supergravity compactified on unit interval ${\mathbb{I} = [0,1]}$, for example, leads to the low-energy limit of E8${\times}$E8 heterotic theory.

So far, these two examples provide only a few pieces of the web. A common way to approach a picture of M-theory today is to start with target-space duality (T-duality) and strong-weak duality (S-duality), which are two examples of string symmetries. T-duality, first observed by Balachandran Sathiapalan [2], is a fundamental consequence of the existence of the string, and we may describe it as a fundamental symmetry. Indeed, it famously constitutes an exact symmetry of the bosonic string, encoded by the transformations: ${R \leftrightarrow \frac{\alpha^{\prime}}{R}, k \leftrightarrow w}$, which describes an equivalence between radius and inverse radius, with the exchange of momentum modes ${k}$ 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. In that closed strings can wrap around non-contractible cycles in space-time, the winding states present in string theory have no analogue in point particle theory, and it is the existence of both momentum and winding states that allows T-duality.

For example, the type IIA and type IIB string theories are found to be equivalent on a quantum level when compactified on a 1-dimensional torus ${\mathbb{T}^{1}}$. T-duality also relates the two heterotic theories. In that it is closely related to mirror symmetry in algebraic geometry, which in string theory is related to the important study of Calabi-Yau manifolds, T-duality in many cases enables us to observe how different geometries for the compact dimensions are physically equivalent, and when ${d}$ -dimensions are compactified on a ${n}$ -torus, we may generalise T-duality transformations under the group ${O(n,n,\mathbb{Z})}$.

S-duality, on the other hand, may be thought of in terms of a familiar description from classical physics, notably invariance of Maxwell’s equations under the exchange of electric and magnetic fields: ${E \rightarrow B, B \rightarrow - \frac{1}{c^2}E}$. As suggested by its name, S-duality transformation displays physical equivalence between strong and weak couplings of a theory. The existence of S-duality in string theory was first proposed by Ashoke Sen [3], where he showed that the type IIB string in 10-dimensions with ${g}$ coupling was equivalent to the same theory with coupling constant ${\frac{1}{g}}$. This is quite beautiful because, from the perspective of non-perturbative theory, S-duality ${SL(2, \mathbb{Z})}$ of the type-IIB can be observed as a consequence of M-theory diffeomorphism invariance [4]. It can also be observed how S-duality relates the heterotic ${SO(32)}$ string with the type I theory.

Together T-duality and S-duality unify all ten-dimensional superstring theories. And it is the web of dualities that unifies all five string theories, which provides one of the clues that M-theory is a unique theory of quantum gravity.

As summarised in a previous post, although we do not know the degrees of freedom of M-theory, we can begin to trace a picture. Starting with a point in parameter space, noting that there are different ways we can transform in this space, we may begin for the sake of example with type IIA string theory. We may then consider another point, for instance 11-dimensional supergravity – the classical limit of M-theory. What we find is how we can move between these two theories depending on the string coupling limit. If we go to the weak coupling limit ${g_{s} \rightarrow 0}$ (or when the dilaton has a large negative expectation value), then we go to a perturbative type IIA string theory. On the other hand, when we go to the strong coupling limit ${g_{s} \rightarrow \infty}$, we have strongly coupled type IIA string theory and, in this case, we should transform to a description of supergravity. This coincides with taking the large $N$ limit of the type IIA superstring, where $N$ is the number of D0-branes. The idea is that we can similarly carry on through each corner of the theory.

From the perspective that M-theory can be obtained from strings at strong coupling, one interesting fact is that this unique theory of quantum gravity in 11-dimensions does not in itself contain strings; instead, the fundamental objects are membranes and the theory describes the dynamics of M2-branes and M5-branes (i.e., 2-dimensional and 5-dimensional branes). When we compactify M-theory on a circle ${S^1}$ it is equivalent to type IIA string theory. What we see more technically is that a fundamental string is associated to an M2-brane wrapped around the circle. The other objects of type IIA string theory like D2- and D4-branes appear similarly from the fundamental objects of the non-perturbative theory [5,6,7]. If instead we take M-theory and compactify it on a torus ${T^2}$, we find the type IIB string compactified on a circle ${S^1}$. The idea, again, is that we may continue to play this game, from the view of the underlying theory, with the limiting cases for this unique theory of quantum gravity in 11-dimensions giving the zoo of perturbative string theories.

When T-duality and S-duality transformations are combined they then define the unified duality (U-duality). At present, I’m not entirely sure how to think about the U-dualities of M-theory as it is something I am actively working through. What I can say is that there are a few ways to look at and approach them. For instance, we can approach U-duality as the hidden continuous symmetry group of supergravity [8]. It is well-known that when compactifying 11-dimensional supergravity on tori of various dimensions, we observe a wealth of symmetries. This was first observed by Julia in 1980 [9]. But it also seems widely agreed that the hidden symmetry groups often denoted under ${G}$ and their compact subgroups ${H}$ for an ${n}$ -torus are suspected to play a discrete role within the U-dualities of M-theory in its complete form. In other words, there is suspicion going back to 1989 that some appropriately discrete version of these symmetries survive, and that they define the fundamental U-dualities of M-theory [10]. This discussion deserves a separate post in order to fully lay out the hidden symmetry groups and provide greater detail in explanation; what might be said, for now, is that the content of the dualities, as well as the way in which the duality groups describe or perhaps even fundamentally define the theory, are questions still requiring unambiguous answers.

2. Approaching M-theory: Top-down and bottom-up

There are a few more or less textbook approaches to M-theory and the important study of non-perturbative duality relations, which one can easily review. For instance, one may use low-energy effective actions, which, as we have touched on, are supergravity theories (that describe massless field interactions in the string spectrum). Within a restricted regime this approach can offer great insight into the physics at strong coupling. One can also study non-perturbative duality relations by exploiting known properties of things like Bogomolny-Prasad-Sommerfield (BPS) ${p}$ -branes, utilising a technique known as the saturation of a BPS bound. In general, the idea in both cases is to extrapolate from the weakly coupled theory to the strongly coupled theory (again, why we can trust such extrapolations is touched on briefly here).

However, going back to the discussion at the outset, one issue I currently share with other researchers in the field concerns a lack of mathematical rigour in the study of M-theory and its objects. While the existence of branes was posited during the Second Superstring Revolution’, and while there are many hints toward this non-perturbative proposition, a lot about brane physics has not been proven or rigorously derived. Moreover, there is a lot about the dynamics of branes that we still do not understand, and, impliededly, the non-perturbative effects in string theory require greater knowledge and clarity. The thing about M-theory and its properties in 11-dimensions, as presently being studied, is that it governs or is suspected to impact many aspects of the lower dimensional string theories. What the completion of M-theory should mean is greater systematic understanding of non-perturbative D/M-brane physics without ambiguity, including brane dynamics, as well as many curious properties and processes in quantum gravity, like what happens in the mathematical process when 10-dimensional space-time of string theory transforms into the 11-dimensional space-time of M-theory. It should also offer insight into the structure of things like perturbative string vacua, not to mention provide a final say on fundamental string cosmology as a whole. This refers to another concern.

For me, I would say as I so far understand, there are a number of interesting approaches to the non-perturbative theory that seem to be contributing overall to the right direction. The two approaches that I find most interesting and that I am currently focusing on for my PhD are relatively new and, while quite different from each other, I think they both have tremendous potential.

The first is a systematic top-down research programme that aims to capture a complete mathematical formulation of M-theory. This is the approach of Sati, Schreiber, and others, which I will write about quite a bit on this blog. It entails some of the best and most stimulating work in M-theory that I’ve seen to date, offering some wonderfully deep and potentially fundamental insight into the non-perturbative theory, if such a theory can in fact be rigorously proven. Here we have some fantastic developments in the form of Hypothesis H, such as the observation that the M-theory C-field is charge-quantized in Cohomotopy theory, or, as I have it in my mind, in cohomology theory M-brane charge quantisation is in cohomotopy. Recent updates to the brane bouquet are also magnificent and evoke wonderful emergent images. I can’t wait to write about these sorts of things in the coming months.

The other approach that I am interested in can be pictured as almost diagrammatically opposite to Schreiber et al. In some sense, it takes a bottom-up approach to M-theory by way of the duality symmetric string. This is what I began to study and work on for my recent MRes thesis. There is so much to be said about the doubled string and its many amazing qualities, which I may break up into several posts. For now, it is worth sharing that one of a host of reasons to study the duality symmetric string is to then look at analogous extensions of ideas and techniques in the study of duality symmetric M-theory.

The theory of the duality symmetric string is importantly a chiral theory, in which T-duality is made manifest on the level of the action, and so it is one that takes a world-sheet perspective such that we want to employ a sigma model description of the maximally doubled string. This world-sheet theory of chiral bosons that sees the total doubled space – especially when treated in a very generic way – naturally accommodates stringy non-geometries. This means that from a study of the maximally doubled string, in addition to seeking very general formulations of chiral boson models for generic doubled geometries, we can also look to construct models that realise completely the full web of string dualities.

I think there is quite a bit of potential insight to be gained when building from the duality symmetric string toward duality invariant M-theory. This relates, in no small part, to non-perturbative investigations leading to new global solutions combining spacetime geometry and quantum field theory defined as generalised geometry, if we take the view of understanding such geometry in terms of a study of conventional geometry with a metric and B-field on some D-dimensional manifold ${M}$ on which ${O(D, D)}$ finds natural action. In M-theory, generalised geometry may be extended to exceptional generalised geometry, and one implication is the extension of spacetime itself, with a further consequence being the possibility that geometry and gravity are emergent concepts. Indeed, there is the lingering idea, one that was first formulated in the late 1990s, that a complete theory of quantum gravity should give access to whatever extent to pre-geometrical features of space-time – a non-commutative geometry at very short distances. Working backwards, this is almost like a disolution of space-time in the emergent picture. And, in the quilt analogy, we should see patches defined as large groups of hidden symmetries, which contain extensions of stringy dualities – what we have described as U-duality – and even potentially a new self-dual string theory. By an analogous extension of ideas, from what we learn about the duality symmetric string, perhaps we can drill a bit more into the true meaning of hidden symmetry groups in the full M-theory. What does it mean when such symmetries are made manifest? I think these sorts of approaches, questions, and conceptual possibilities are exciting.

References

[1] Edward Witten. String theory dynamics in various dimensions.Nuclear PhysicsB, 443(1):85 – 126, 1995.

[2] Balachandran Sathiapalan. Duality in statistical mechanics and string theory.Phys. Rev. Lett., 58:1597–1599, Apr 1987.

[3] Ashoke Sen. Strong – weak coupling duality in four-dimensional string theory.Int. J. Mod. Phys. A, 9:3707–3750, 1994.

[4] John H. Schwarz. The power of m theory.Physics Letters B, 367(1-4):97–103,Jan 1996.

[5] N.A. Obers and B. Pioline. U-duality and m-theory.Physics Reports, 318(4-5):113–225, Sep 1999.

[6] John H. Schwarz. Introduction to m theory and ads/cft duality.Lecture Notesin Physics, page 1–21, 1999.

[7] M. P. Garcia del Moral. Dualities as symmetries of the supermembrane theory,2012.

[8] David S. Berman and Daniel C. Thompson. Duality symmetric string and m-theory, 2013.

[9] B. Julia. GROUP DISINTEGRATIONS.Conf. Proc. C, 8006162:331–350, 1980.

[10] Bernard de Wit and Hermann C Nicolai. d = 11 supergravity with local SU(8)invariance.Nucl. Phys. B, 274(CERN-TH-4347-86):363–400. 62 p, Jan 1986.

Literature: Duality Symmetric String and the Doubled Formalism

When it comes to a T-duality invariant formulation of string theory, there are two primary actions that are useful to study as a point of entry. The first is Tseytlin’s non-covariant action. It is found in his formulation of the duality symmetric string, which presents a stringy extension of the Floreanini-Jackiw Lagrangians for chiral fields. In fact, for the sigma model action in this formulation, one can directly reproduce the Floreanini-Jackiw Lagrangians for antichiral and chiral scalar fields. The caveat is that, although we have explicit $O(D,D)$ invariance, which is important because ultimately we want T-duality to be a manifest symmetry, we lose manifest Lorentz covariance on the string worldsheet. What one finds is that we must impose local Lorentz invariance on-shell, and from this there are some interesting things to observe about the constraints imposed at the operator level.

The main papers to study are Tseytlin’s 1990/91 works listed below. Unfortunately there is no pre-print available, so these now classic string papers remain buried behind a paywall:
1) Tseytlin, ‘Duality Symmetric Formulation of String World Sheet Dynamics
2) Tseytlin, ‘Duality Symmetric Closed String Theory and Interacting Chiral Scalars

For Hull’s doubled formalism, on the other hand, we have manifest 2-dimensional invariance. In both cases the worldsheet action is formulated such that both the string coordinates and their duals are on equal footing, hence one thinks of the coordinates being doubled. However, one advantage in Hull’s formulation is that there is a priori doubling of the string coordinates in the target space. Here, $O(D,D)$ invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space may be written as $R^{1, d-1} \otimes T^{2D}$, in which we have a non-compact spacetime and a doubled torus. From the torus identifications we have manifest $GL(2D; Z)$ symmetry. Then after imposing what we define as the self-duality constraint of the theory, which contains an $O(D,D)$ metric, invariance of the theory reduces directly to $O(D,D; \mathbb{Z})$.

1. Hull, ‘Doubled Geometry and T-Folds
2. Hull and Reid-Edwards, ‘Non-geometric backgrounds, doubled geometry and generalised T-duality

What is neat about the two formulations is that, turning off interactions, they are found to be equivalent on a classical and quantum level. It is quite fun to work through them both and prove their equivalence, as it comes down to the constraints we must impose in both formulations.

I think the doubled formalism (following Hull) for sigma models is most interesting on a general level. I’m still not comfortable with different subtleties in the construction, for example the doubled torus fibration background or choice of polarisation from T-duality. The latter is especially curious. But, in the course of the last two weeks, things are finally beginning to clarify and I look forward to writing more about it in time.

Related to the above, I thought I’d share three other supplementary papers that I’ve found to be generally helpful:

1) Berman, Blair, Malek, and Perry, ‘O(D,D) Geometry of String Theory
2) Berman and Thompson, ‘Duality Symmetric String and M-theory
3) Thompson, ‘T-duality Invariant Approaches to String Theory

There are of course many other papers, including stuff I’ve been studying on general double sigma models and relatedly the Pasti, Sorokin and Tonin method. But those listed above should be a good start for anyone with an itch of curiosity.

Double Field Theory: The Courant Bracket

1. Introduction

In this post we are going to briefly and somewhat schematically discuss the appearance of the Courant bracket in Double Field Theory (DFT), following [1]. The point here is mainly to set the stage, so we jump straight into motivating the Courant bracket. In the next post, we will then study the B-transformations from the maths side and the C-bracket, following and expanding from [1] and others, with an emphasis in the end on how all of this relates to T-duality and strings.

What follows is primarily based on a larger collection of study notes, which I will upload in time.

2. Motivating the Courant Bracket

To understand the appearance of the Courant bracket in DFT, one way to start is by considering some general theory with a metric ${g_{ij}(X)}$ and a Kalb-Ramond field (i.e., an antisymmetric tensor field) ${b_{ij}(X)}$, where ${X \in M}$. The symmetries and diffeomorphisms of ${g_{ij}(X)}$ are generated by vector fields ${V^{i}(X)}$, where ${V \in T(M)}$ with ${T(M)}$ being the tangent bundle. As for ${b_{ij}(X)}$, the transformations are generated by one-forms ${\xi_{i}(X)}$, where ${\xi^{i}(X) \subset T^{\star}(M)}$ with ${T^{\star}(M)}$ being the cotangent bundle. We may combine ${V^{i}(X)}$ and ${\xi^{i}(X)}$ as a sum of bundles, such that (dropping indices) ${V + \xi \in T(M) \oplus T^{\star}(M)}$.

With these definitions, the opening question now is to ask, ‘what are the gauge transformations?’ To make sense of this, consider the following gauge parameters,

$\displaystyle \delta_{V + \xi} g = \mathcal{L}_{V} g$

$\displaystyle \delta_{V + \xi} b = \mathcal{L}_{V} b + d\xi \ \ \ (1)$

Here ${\mathcal{L}}$ is the Lie derivative. Furthermore, note given that ${V}$ generates diffeomorphisms, in (1) we get the Lie derivative in the direction of ${V}$. Also notice that ${\xi}$ does not enter the gauge transformation of ${g}$; however, for the gauge transformation of ${b}$, we do have a one-form ${\xi}$ and so we can take the exterior derivative. We should also note the following important properties of ${\mathcal{L}}$. For instance, when acting on forms the Lie derivative is,

$\displaystyle \mathcal{L}_{V} = i_{V}d + di_{V} \ \ \ (2)$

Where ${iV}$ is a contraction with ${V}$. We’re just following the principle of a contraction with a vector times the exterior derivative. It is also worth pointing out that ${\mathcal{L}}$ and the exterior derivative commutate such that,

$\displaystyle \mathcal{L}_{V}d = d\mathcal{L}_{V} \ \ \ (3)$

There are also some other useful identities that we are going to need. For instance, for the Lie algebra,

$\displaystyle [\mathcal{L}_{V_1}, \mathcal{L}_{V_2}] = \mathcal{L}_{[V_1,V_2]} \ \ (4)$

Where ${[V_1,V_2]}$ is just another vector such that ${[V_1,V_2]^{k} = V_{1}^{p}\partial_{p}V_{2}^{k} - (1,2)}$.

And, finally, we have,

$\displaystyle [\mathcal{L}_{X}, i_{Y}] = i_{[X,Y]} \ \ (5)$

Now follows the fun part. Given the transformation laws provided in (1), we want to determine the gauge algebra. To do this, we must compute in reverse order the gauge transformations on the metric ${g}$ and the ${b}$-field. For the metric we evaluate the bracket,

$\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \delta_{V_2 + \xi_2} \mathcal{L}_{V_1}g - (1,2)$

$\displaystyle = \mathcal{L}_{V_1}\mathcal{L}_{V_2}g - (1,2)$

Using the identity (4) we find,

$\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \mathcal{L}_{[V_1, V_2]} g \ \ \ (6)$

For the ${b}$-field we have,

$\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]b = \delta_{V_2 + \xi_2} (\mathcal{L}_{V_1}b + d\xi_{1}) - (1,2)$

$\displaystyle = \mathcal{L}_{V_1}(\mathcal{L}_{V_2}b + d\xi_2) - (1,2)$

$\displaystyle = \mathcal{L}_{[V_1, V_2]} + d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1} \ \ \ (7)$

It turns out that this bracket satisfies the Jacobi identity, although it is not without its problems because, as we will see, there is a naive assumption present in the above calculations. In the meantime, putting this aside until later, the idea now is to compare the above with (1) and see what ‘pops out’. Notice that we find,

$\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}] = \delta_{[V_1,V_2]} + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} \ \ \ (8)$

In which we have discovered a bracket defined on ${T(M) \oplus T^{\star}(M)}$,

$\displaystyle [V_{1} + \xi_{1}, V_{2} + \xi_{2}] = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2} \xi_{1} \ \ \ (9)$

On the right-hand side of the equality we see a vector field in the first term and a one-form given by the final two terms. This Lie bracket is reasonable and, on inspection, we seem to have a definite gauge algebra. Here comes the problem allude a moment ago: there is a deep ambiguity in (9) in that we cannot, however much we try, determine unique parameters in our theory. Notice,

$\displaystyle \delta_{V + \xi}b = \mathcal{L}_{V}b + d\xi$

$\displaystyle = \mathcal{L}_{V + (\xi + d \sigma)} \ \ \ (10)$

The point being that the ambiguity of the one-form ${\xi}$ is so up to some exact ${d\sigma}$. To put it another way, if we change ${\xi}$ by ${d\sigma}$, we’re not actually changing anything at all. We would just get ${\mathcal{L}_{V}b + d(\xi + d\sigma)}$ where, when the exterior derivative hits ${d\sigma}$ we simply get nothing. So, given that ${\xi}$ is ambiguous up to some exact ${d\sigma}$, in a sense what we have is a symmetry for a symmetry. In other words, the present construction is not sufficient.

What we can do to correct the situation is analyse the mistake in (7). Let us, for instance, look at the right-hand side of the summation sign in this equation,

$\displaystyle d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1}) = d(di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)$

The logic follows that the first term ${di_{V_1}\xi_{2}}$ is killed by ${d}$. It doesn’t make any contribution and its coefficient is just ${1}$. The trick then is to see, without loss of generality, that we may change the implicit coefficient ${1}$ in front of ${di_{V_1}\xi_{2}}$. It turns out, the coefficient that we can use is ${1 - \frac{\beta}{2}}$,

$\displaystyle = d((1-\frac{\beta}{2} di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)$

$\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \frac{1}{2}\beta di_{V_1}\xi_{2}) - (1,2)$

$\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d [i_{V_1}\xi_{2} - i_{V_2}\xi_{1}]) \ \ \ (11)$

What we end up achieving is the construction of a much more general bracket,

$\displaystyle [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d(iV_{1}\xi_{2} - iV_{2}\xi_{2}) \ \ \ (12)$

What is so lovely about this result might at first seem counterintuitive. It turns out, as one can verify, for ${\beta \neq 0}$, we do not satisfy the Jacobi identity. So at first (12) may not seem lovely at all! But it makes perfect sense to consider cases of non-vanishing ${\beta}$. In mathematics, the case for ${\beta = 1}$ was introduced by Theodore James Courant in his 1990 doctoral dissertation [5], where he studied the bridge between Poisson geometry and pre-symplectic geometry. The idea here is to forget about the Jacobi identity – consider its loss an artefact of field theory with anti-symmetric tensors and gravity – and impose ${\beta = 1}$. When we do this what we obtain is indeed the famous Courant bracket. That is, given ${\beta = 1}$, the case of maximal symmetry is described by,

$\displaystyle [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta=1} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2} d(iV_{1}\xi_{2} - iV_{2}\xi_{2} \ \ \ (13)$

Although the Jacobi identity does not hold, one can show that for ${Z_{i} = V_{i} + \xi_{i}, i = 1,2,3}$, the Jacobiator assumes the form,

$\displaystyle [Z_1, [Z_2,Z_3]] + \text{cyclic} = dN(Z_1, Z_2, Z_3)$

Which is an exact one-form. This gives us a first hint that the unsatisfied Jacobi identity does not provide inconsistencies, because exact one-forms do not generate gauge transformations.

But why ${\beta = 1}$? Courant argued that the correct value of ${\beta}$ is in fact ${1}$ because, as he discovered, there is an automorphism of the bracket. This means that if do an operation on the elements, it respects the bracket. This automorphism is, moreover, an extra symmetry known in mathematics as a B-transformation. What follows from this is, I think, actually quite special. Given the Courant bracket is a generalisation of the Lie bracket, particularly in terms of an operation on the tangent bundle ${T(M)}$ to an operation on the direct sum of ${T(M)}$ and the p-forms of the vector bundle, what we will discuss is how the B-transformation in mathematics relates in a deep way to what in physics, especially string theory, we call T-duality (target- space duality). This is actually one of the finer points where mathematics and physics intersect so wonderfully in DFT.

In the next post we’ll carry on with a discussion of the B-transformation and then also the C-bracket, finally showing how everything relates.

References

[1] Zwiebach, B. (2010). ‘Double Field Theory, T-Duality, and Courant Brackets’ [lecture notes]. Available from [arXiv:1109.1782v1 [hep-th]].

[2] Hohm, O., Hull, C., and Zwiebach, B. (2010). ‘Generalized metric formulation of double field theory’. Available from [arXiv:1006.4823v2 [hep-th]].

[3] Hull, C. and Zwiebach, B. (2009). ‘Double Field Theory’. Available from [arXiv:0904.4664v2 [hep-th]].

[4] Hull, C. and Zwiebach, B. (2009). ‘The Gauge Algebra of Double Field Theory and Courant Brackets’. Available from [arXiv:0908.1792v1 [hep-th]].

[5] Courant, T. (1990). ‘Dirac manifolds’. Trans. Amer. Math. Soc. 319: 631–661. Available from [https://www.ams.org/journals/tran/1990-319-02/S0002-9947-1990-0998124-1/].