\section{Glances at double field theory}
It is now time to pause the exposition of supermathematical language and to review a selection of elements of double field theory. Closed string theory with toroidal target space geometries exhibits two different sets of ``momenta''. On the one hand, there are the standard momenta canonically conjugate to the center of mass coordinates of the string. But in addition there is the possibility of a string winding around compact directions, resulting in winding degrees of freedom. Taking the latter onto a similar footing as the standard momentum, the resulting coordinates conjugate to winding give a second set of coordinates referred to as \emph{winding coordinates}. The physical fields depend on both sets of coordinates.

\begin{rem}
In DFT, it is the coordinates that are doubled and `double fields' refer to fields which depend on both sets of coordinates.
Thus we also refer to double functions, double vector fields, double forms, etc.
\end{rem}


\subsection{Winding, double fields and the strong constraint}
In order to motivate these concepts, we choose the simplest example of closed string theory with toroidal target space equipped with a metric $G$ and Kalb-Ramond $B$-field (i.e. a two-form on $T^d$), both being constant. 


The corresponding sigma model is given in terms of maps $X : \Sigma \rightarrow T^d$ from a two-dimensional domain $\Sigma = \R\times S^1$ to a $d$-dimensional torus $M = T^d$. Taking coordinates $(\tau,\sigma)$ on the world sheet $\Sigma$, the action reads:
\eq{
\label{sigmamodel}
S = \frac{1}{4\pi} \int_0 ^{2\pi}\,d\sigma \int_0^\infty\, d\tau \Bigl(h^{\alpha\beta}\partial_\alpha X^i \partial_\beta X^j G_{ij} + \epsilon^{\alpha\beta}\partial_\alpha X^i\partial_\beta X^j B_{ij}\Bigr)\;.
}
We use the two-dimensional diagonal metric $h = \textrm{diag}(-1,1)$ and the antisymmetric symbol $\epsilon$ with the convention $\epsilon^{01} = -1$. The latter is used to indicate that this term is a two-form on the world sheet.


 Furthermore $\partial_\alpha = (\partial_\tau, \partial_\sigma)$ and we assume the periodicity $X^i = X^i + 2\pi$ in terms of  the torus coordinates. 


The momentum $P_i$ canonically conjugate to the target space coordinate $X^i$ is given by 
\eq{
P_i = \frac{1}{2\pi} \Bigl(G_{ij}\partial_\tau X^j + B_{ij}\partial_\sigma X^j\Bigr)\;,
} 
and the Hamiltonian density corresponding to the Lagrangian in \eqref{sigmamodel} is determined to be 
\eq{
h = \frac{1}{4\pi}\Bigl(\partial_\sigma X, 2\pi P\Bigr){\cal H}(G,B) 
\begin{pmatrix}
\partial_\sigma X \\
2\pi P 
\end{pmatrix}
\;,
}
where we introduced the \emph{generalized metric} $\cal H$, a symmetric bi-linear form on the direct sum $TM \oplus T^*M$, depending on the metric $G$ and $B$-field in the following way:
\eq{
\label{genmetric}
{\cal H}(G,B)  = \begin{pmatrix}
G - BG^{-1}B & BG^{-1} \\
-G^{-1}B & G^{-1} 
\end{pmatrix}\;. 
}

Classical solutions to the equations of motion of the sigma model \eqref{sigmamodel} with periodic boundary conditions are given by the sum of a left- and a right-moving part, 
\eq{
\label{classic}
X_R^i =& x_{0R}^i + \alpha_0^i(\tau - \sigma) + i\underset{n\neq 0}\sum \frac{1}{n} \alpha_n^i e^{-in(\tau - \sigma)} \;,\\
X_L^i =& x_{0L}^i + \bar \alpha_0^i(\tau + \sigma) + i\underset{n\neq 0}\sum \frac{1}{n} \bar \alpha_n^i e^{-in(\tau +\sigma)}\;,
}
with constants $x_{0R},x_{0L}$, oscillator coefficients $\alpha_n^i,\bar \alpha_n^i, \, n\neq 0$ and zero modes $\alpha_0^i, \bar \alpha_0^i$ given in double momentum space by:
\eq{
\alpha_0^i =& \frac{1}{\sqrt{2}}G^{ij}\Bigl(p_j - (G_{jk} + B_{jk})w^k\Bigr)\;,\\
\bar\alpha_0^i =&\frac{1}{\sqrt{2}} G^{ij}\Bigl(p_j + (G_{jk} - B_{jk})w^k\Bigr)\;,
}
where the $w^i$ are defined by $w^i = \frac{1}{2\pi}\int_0 ^{2\pi}\partial_\sigma X^i$. The $p_j$ are interpreted as the canonical momenta similarly to the point particle case, but in addition there is the momentum corresponding to winding, given by $w^k$. The motivation to consider a doubling of coordinates is to take the two sets of momenta as canonically conjugate to two different sets of coordinates, in the sense that for canonical quantization we get the following operators:
\eq{
\label{operators}
p_i \simeq \frac{1}{i}\frac{\partial}{\partial x^i}\;, \quad \textrm{and} \quad w^i \simeq \frac{1}{i}\frac{\partial}{\partial \tilde x_i}\;.
}


Note, that this introduction of a double set of coordinates is a consequence of the appearance of winding degrees of freedom in the classical solution \eqref{classic} of the closed string sigma model. Taking this idea further, classical double fields are real or complex functions (or more generally sections in the appropriate tensor bundles), depending on both sets of coordinates. Note the coordinates are doubled, not the functions, vector fields, forms, etc.

Furthermore, physical fields in DFT are restricted by the \emph{strong constraint}\footnote{We are not going to elaborate on its origin, but refer the reader to the literature, e.g. \cite{Hohm:2010jy}.}. E.g. for scalar fields $\phi(x^i,\tilde x_i)$ and $\psi(x^i,\tilde x_i)$ this constraint is given by 
\eq{
\partial_i\phi (x^i,\tilde x_i)\tilde \partial^i \psi(x^i,\tilde x_i) + \tilde\partial^i \phi(x^i,\tilde x_i) \partial_i \psi(x^i, \tilde x_i) = 0\;,
}
where the $\tilde \partial^i$ are the partial derivatives with respect to the $\tilde x_i$-coordinates, as given in \eqref{operators}.
We will interpret this condition in terms of the Drinfel'd double in later sections. But for the next subsection, we take these ingredients more or less axiomatically and review results of double field theory.

\subsection{Spacetime action and gauge algebra}
The evolution of DFT in the last decade has many faces and different routes. We are not in a position to describe all of its physical motivations coming from areas as different as gauged supergravity or string field theory. Even citing all of the literature is a complicated task and so we refer the reader to the expositions which provided us with a trail through the world of double field theory (\cite{Hull:2009zb,Zwiebach:2011rg,Hull:2009mi}). 


One of the main results achieved so far is to formulate a spacetime action for fields depending on the doubled set of coordinates, having a global $O(d,d)$ symmetry (motivated by the action of \emph{T-duality} at the spacetime level) and reproducing the standard low energy effective string action when reduced to half of the coordinates:
\eq{
S= \int d^d x \sqrt{-\det G}\; e^{-2\Phi}\Bigl(R + 4\partial^i\phi \partial_i\phi - \frac{1}{12}H_{ijk}H^{ijk} \Bigr)\;.
}

We denote the field strength of the Kalb-Ramond field by $H = dB$, by $\Phi(x)$ the string dilaton 

and by $R$ the Ricci scalar determined by the metric $G_{ij}$. 

The double field action having these properties depends on the doubled dilaton $\mathfrak{D}(x,\tilde x)$ and the generalized metric ${\cal H}_{MN}(G(x,\tilde x),B(x,\tilde x))$\footnote{In the following, we denote pairs of indices corresponding to the doubled  coordinates by capital letters. Mathematically they correspond to  the structure of the direct sum of the tangent- and cotangent bundle of the physical configuration space, but coefficient functions depending on both sets of coordinates. As an example we have  $V^N = \Bigl(V^i(x,\tilde x),V_i(x,\tilde x)\Bigr)$ and $V_N = \Bigl(V_i(x,\tilde x),V^i(x,\tilde x)\Bigr)$.}


defined in \eqref{genmetric}, but now also allowed to depend on the doubled set of coordinates. It is given by:
\eq{
\label{dftaction}
S_{{\cal H}} = \int \, dx d\tilde x e^{-2\mathfrak{D}} \Bigl( &\frac{1}{8} {\cal H}^{MN}\partial_M {\cal H}^{KL} \partial_N {\cal H}_{KL} -\frac{1}{2}{\cal H}^{MN}\partial_N{\cal H}^{KL}\partial_L {\cal H}_{MK}  \\
&2\partial_M \mathfrak{D} \partial_N{\cal H}^{MN} + 4{\cal H}^{MN}\partial_M \mathfrak{D} \partial_N \mathfrak{D} \Bigr)\;.
}
\begin{rem}
The notation we use here makes global $O(d,d)$-symmetry manifest. On a $d$-dimensional vector space $V$, $O(d,d)$-transformations are defined by the requirement 
\eq{
A \eta A^t = \eta\;, \quad A \in O(d,d)\;,
}
where $\eta$ is the bilinear form on on $V \oplus V^*$ represented in matrix form by
\eq{
\eta_{MN} = \begin{pmatrix}
0 & \textrm{id} \\
\textrm{id} & 0 
\end{pmatrix}\;.
} 
Indices $M,N$ are raised and lowered by contraction with $\eta_{MN}$ or its inverse $\eta^{MN}$.
\end{rem}

Instead of discussing problems of this action such as  the integration measure or covariance and attempts to provide solutions, we are going to state one additional symmetry \cite{Hull:2009zb} of the action which we are going to interpret mathematically in the following section. Let $\Sigma$ be the pair $(X(x^i,\tilde x_j),\omega(x^i,\tilde x_j))$\footnote{I.e. in components $(\Sigma)^M = (X^i,\eta_j)$.}. We will refer to objects of this kind as ``double vectors'' in the following. 

In addition, locally define the \emph{generalized derivative} ${\cal L}_{\Sigma}$ to act on double scalars and double vectors in the following way:
\eq{
{\cal L}_\Sigma \phi =& \Sigma^M \partial_M \phi \;,\\
({\cal L}_\Sigma V)_M =& \Sigma^K\partial_K V_M + (\partial_M\Sigma^K -\partial^K\Sigma_M)V_K \;,\\
({\cal L}_\Sigma W)^M =& \Sigma^K\partial_K W^M -(\partial_K\Sigma^M -\partial^N\Sigma_K)W^K\;,
}

\noindent and extended as a derivation to double fields with more indices, similarly to the standard Lie derivative on vector fields.


 It turns out that the double field action \eqref{dftaction} is invariant under transformations given by the action of the generalized Lie derivative, meaning for the fields ${\cal H}_{MN}$ and $d$:
\eq{
\delta_\Sigma {\cal H}_{MN} &= ({\cal L}_\Sigma {\cal H})_{MN}\;,\\
\delta_\Sigma e^{-2d} &= \partial_M\Bigl(\Sigma^M e^{-2d}\Bigr)\;,
}

where the latter is the generalization of the transformation of a scalar density to double fields. To get information about the nature of the generalized Lie derivative and the bracket structure underlying the space of double vectors, it is important to consider the commutator of two generalized Lie derivatives. It is given for $\Sigma_1(x^i,\tilde x_j) = (X(x^i,\tilde x_j),\eta(x^i,\tilde x_j))$ and $\Sigma_2(x^i,\tilde x_j) = (Y(x^i,\tilde x_j),\omega(x^i,\tilde x_j))$ by 
\eq{
\Bigl[{\cal L}_{\Sigma_1},{\cal L}_{\Sigma_2}\Bigr] = \; -{\cal L}_{[\Sigma_1,\Sigma_2]_C}\;,
}
where the bracket determining the algebra of ``double sections'' is given by the \emph{C-bracket}. Note that the geometric meaning of ``double sections'' is not obvious at this point in our exposition of double field theory and we only use it as a name for pairs given above. We will provide an interpretation in the next section, which leads to the right bracket structure.


Because the C-bracket will be the starting point of relating the geo\-me\-try of symplectic supermanifolds to the algebraic relations for doubled sections, we state the component form for this bracket in full detail. In $O(d,d)$-notation, it is given by:
\eq{
\Bigl([\Sigma_1,\Sigma_2]_C\Bigr)^M = \Sigma_1^K\partial_K\Sigma_2^M - \Sigma_2^K\partial_K \Sigma_1 ^M -\frac{1}{2}\Bigl(\Sigma_1^K\partial^M\Sigma_{2K} - \Sigma_2 ^K\partial^M\Sigma_{1K}\Bigr)\;,
}
or, if we sort by the various components of different type, we get:
\eq{
\label{Cbracket}
\Bigl([X,Y]_C\Bigr)_i =&\; 0\;, \quad 
\Bigl([X,Y]_C \Bigr)^i =\; X^k\partial_k Y^i - Y^k\partial_k X^i \;, \\
\Bigl([X,\omega]_C\Bigr)_i =&\; \;\;X^k\partial_k\omega_i -\frac{1}{2}(X^k\partial_i\omega_k - \omega_k\partial_i X^k) \;,\\
\Bigl([X,\omega]_C\Bigr)^i =&\;-\omega_k \tilde\partial^k X^i - \frac{1}{2}(X^k\tilde\partial^i\omega_k - \omega_k\tilde\partial^i X^k) \;,\\
\Bigl([\eta, Y]_C \Bigr)_i =&\; -Y^k\partial_k \eta_i + \frac{1}{2}\Bigl(Y^k\tilde\partial^i\eta_k - \eta_k\tilde \partial^i Y^k \Bigr)\;, \\
\Bigl([\eta,Y]_C \Bigr)^i =&\; \;\;\eta_k\tilde \partial^k Y^i + \frac{1}{2}\Bigl(Y^k\tilde\partial^i\eta_k - \eta_k\tilde\partial^i Y^k\Bigr)\;, \\
\Bigl([\eta,\omega]_C\Bigr)_i =& \; \;\;\eta_k\tilde \partial^k \omega_i - \omega_k\tilde \partial^k \eta_i\;, \quad 
\Bigl([\eta,\omega]_C\Bigr)^i = 0\;.
}

Mathematically, it is not clear at this point how to interpret doubled sections, having vector- and form-like parts but depending on a doubled set of coordinates. 


The algebra of gauge transformations parametrized by these objects is governed by the C-bracket and the transformations are a symmetry of the double field spacetime action. In the following we are going to re-interpret double fields as functions on an even symplectic supermanifold and derive the expression for the C-bracket by using its Poisson algebra.
{"config": "arxiv", "file": "1406.3601/dft-fin.tex"}