# O(D,D) and Double Field Theory

1. Introduction

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

and

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

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

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

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

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

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

and

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

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

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

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

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

where

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where

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

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

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

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

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

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

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

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

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

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

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

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

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

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

which also implies

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

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

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

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

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

and

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

It then follows

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

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

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

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

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

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

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

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

$\Box$

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

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

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

with,

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

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

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

and

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

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

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

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

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

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

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

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

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

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

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

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

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[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.

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