In the theory of open string one imposes boundary conditions on string ends. in general there are two types of these conditions: Dirichlet and Newmann. Newmann boundary conditions effectively allow the string end to move arbitrarily. Dirichlet conditions restrict movement of the string end to some p-dimensional surface in the bulk space, that is called Dp-brane. From the point of view of external space a D-brane wrapped around internal torus looks as a point particle.
Interestingly, that in the spectrum of supergravity compactified to 3 dimensions one meets exotic states whose higher dimensional origin is not obvious. In string theory these correspond to the so-called exotic branes. A fascinating feature of these branes is that they typically generate non-geometric backgrounds, i.e. such configurations of the space-time metric and p-form fields that cannot be globally described in geometric terms [1].
Exotic branes appear as duality orbits of ordinary branes. In other words, starting from an ordinary brane one obtains (in particular) exotic branes by T- and S-duality. Lets demonstrate this idea on a toy example that captures the most important features of T-duality orbit of an NS5 brane wrapped around 5 internal coordinates [1].
Start with a background given by a flat 3-torus and a non-zero H-flux
\begin{equation}
\begin{aligned}
ds^2&=dx^2+dy^2+dz^2,\\
B&=Nz dx\wedge dy.
\end{aligned}
\end{equation}
The field strength of the 2-form $B$ is then constant $H=dB=N dx\wedge dy\wedge dz$. The coordinates $(x,y,z)$ are periodic $x\sim x+2\pi$. Going around the $z$-cycle of the torus $z\to z+2\pi$ results in a simple gauge transformation of the B-field. Hence, the metric and the gauge field are globally well defined in this example.
A little bit more subtle background appears if we T-dualize the $x$ coordinate of the 3-torus. By making use of the Buscher rules [2,3]
\begin{equation}
\begin{aligned}
&G'_{x x}=\frac{1}{G_{xx}}, \quad G'_{x \mu}= -\frac{B_{x\mu}}{G_{x x}}, & B'_{x \mu}= -\frac{G_{x\mu}}{G_{x x}},\\
&G'_{\mu\nu}=G_{\mu\nu}-\frac{G_{x\mu}G_{x\nu}-B_{x\mu}B_{x\nu}}{G_{xx}},\\
&B'_{\mu\nu}=B_{\mu\nu}-\frac{G_{x\mu}B_{x\nu}-B_{x\mu}G_{x\nu}}{G_{xx}},
\end{aligned}
\end{equation}
we arrive to the following metric and B-field
\begin{equation}
\begin{split}
ds'^2&=(dx^2-Nz dy)^2+dy^2+dz^2,\\
B&=0.
\end{split}
\end{equation}
It is easy to see, that the background we have obtained describes a twisted torus fibration. Indeed, the 3-torus $\mathbb{T}^3$ locally can be represented as $\mathbb{T}^2\times \mathbb{S}^1$, where the 2-torus is parametrized by the coordinates $(x,y)$ and the $z$-coordinate runs along the circle. To have the metric globally well defined one has to twist the 2-torus when going around the circle $z\to z+2\pi$
\begin{equation}
\begin{aligned}
& x\to x+2\pi N y,\\
&y\to y.
\end{aligned}
\end{equation}
This background has non-zero f-flux defined as $de^a=f^{a}{}_{bc}e^b\wedge e^c$ for $e^a$ being a vielbein 1-form. In the example above the only non-zero component of the f-flux is $f^x{}_{yz}=2\pi N$.
To go further along the duality orbit generated by the NS5 brane we T-dualize the remained $y$-coordinate. One may use the same Buscher rules as above with the change $x\to y$. We then end up with the following background
\begin{equation}
\begin{aligned}
ds''^2&=\frac{1}{1+N^2z^2}(dx^2+dy^2+dz^2),\\
B&=\frac{Nz}{1+N^2z^2}dx\wedge dy.
\end{aligned}
\end{equation}
Here the shift $z\to z+2\pi$ produces a complete mess and mixes the metric and the B-field. This is an example of a (globally) non-geometric background. It is important that locally this configuration is described by distinct metric and B-field, but it is not a manifold globally. One may check that going around the circle results in new metric and B-field, $\tilde{G}$ and $\tilde{B}$, that are related to the old ones by a T-duality transformation.
In the case of an ordinary manifold with non-trivial monodromy properties metric and gauge fields do not mix when going around a non-trivial cycle. This means that the monodromy is a diffeomorphism and/or a gauge transformation and transforms the metric and the gauge fields separately. In the sample above the monodromy is an element of the T-duality group, that is not a direct sum of diffeos and gauge transformations. This non-geometric background is characterized by the Q-flux, that is a sign of non-commutativity of string coordinated [4]
\begin{equation}
[X^a,X^b]=Q^{ab}{}_{c}X^c.
\end{equation}
Hence, we now recovered the following chain that represents T-duality orbit of the NS5-brane [5]
\begin{equation}
H_{xyz} \longrightarrow f^x{}_{yz}\longrightarrow Q^{xy}{}_{z}.
\end{equation}
Each action of a T-duality transformation effectively raises the corresponding index. The last step remained is to T-dualize the $z$-coordinate. Although, this procedure is not well defined it is usually said that it produces a locally non-geometric background with non-zero R-flux. In contrast to the previous example this background is not geometric even locally and the R-flux reflects non-associativity of string coordinates
\begin{equation}
[X^a,X^b,X^c]\sim R^{abc}.
\end{equation}
Then the full T-duality chain will be
\begin{equation}
H_{xyz} \longrightarrow f^x{}_{yz}\longrightarrow Q^{xy}{}_{z}\longrightarrow R^{xyz}.
\end{equation}
In the context on previous posts it is important to mention that non-geometric backgrounds find a natural description in terms of the extended geometry. In this approach T-(U-)duality transformations are good geometric transformations of the extended space, that is obviously not a manifold. The Scherk-Schwarz reduction of the extended space allows to end up with non-geometric backgrounds in a natural way.
Compactifications in the presence of (non-)geometric fluxes have found their application recently to the problems of cosmology, phenomenological models, problem of string vacuum. For example, certain models with non-zero fluxes allow to end up with de Sitter space in string compactifications, that is the relevant background for modern cosmology.
On the other hand backgrounds with non-zero fluxes allow to end up with more stabilized moduli and reduce the number of arbitrary parameter in phenomenological models.
An interesting effect related tightly to exotic branes is the so-called supertube effect. Due to this effect interacting (ordinary) branes may polarize into exotic branes. This means that one is interested not only in compactifications to 3 dimensions. For example, in such objects as black holes, that are typically constructed from D-branes, exotic branes should be taken into account as well.
Interestingly, that in the spectrum of supergravity compactified to 3 dimensions one meets exotic states whose higher dimensional origin is not obvious. In string theory these correspond to the so-called exotic branes. A fascinating feature of these branes is that they typically generate non-geometric backgrounds, i.e. such configurations of the space-time metric and p-form fields that cannot be globally described in geometric terms [1].
Exotic branes appear as duality orbits of ordinary branes. In other words, starting from an ordinary brane one obtains (in particular) exotic branes by T- and S-duality. Lets demonstrate this idea on a toy example that captures the most important features of T-duality orbit of an NS5 brane wrapped around 5 internal coordinates [1].
Start with a background given by a flat 3-torus and a non-zero H-flux
\begin{equation}
\begin{aligned}
ds^2&=dx^2+dy^2+dz^2,\\
B&=Nz dx\wedge dy.
\end{aligned}
\end{equation}
The field strength of the 2-form $B$ is then constant $H=dB=N dx\wedge dy\wedge dz$. The coordinates $(x,y,z)$ are periodic $x\sim x+2\pi$. Going around the $z$-cycle of the torus $z\to z+2\pi$ results in a simple gauge transformation of the B-field. Hence, the metric and the gauge field are globally well defined in this example.
A little bit more subtle background appears if we T-dualize the $x$ coordinate of the 3-torus. By making use of the Buscher rules [2,3]
\begin{equation}
\begin{aligned}
&G'_{x x}=\frac{1}{G_{xx}}, \quad G'_{x \mu}= -\frac{B_{x\mu}}{G_{x x}}, & B'_{x \mu}= -\frac{G_{x\mu}}{G_{x x}},\\
&G'_{\mu\nu}=G_{\mu\nu}-\frac{G_{x\mu}G_{x\nu}-B_{x\mu}B_{x\nu}}{G_{xx}},\\
&B'_{\mu\nu}=B_{\mu\nu}-\frac{G_{x\mu}B_{x\nu}-B_{x\mu}G_{x\nu}}{G_{xx}},
\end{aligned}
\end{equation}
we arrive to the following metric and B-field
\begin{equation}
\begin{split}
ds'^2&=(dx^2-Nz dy)^2+dy^2+dz^2,\\
B&=0.
\end{split}
\end{equation}
It is easy to see, that the background we have obtained describes a twisted torus fibration. Indeed, the 3-torus $\mathbb{T}^3$ locally can be represented as $\mathbb{T}^2\times \mathbb{S}^1$, where the 2-torus is parametrized by the coordinates $(x,y)$ and the $z$-coordinate runs along the circle. To have the metric globally well defined one has to twist the 2-torus when going around the circle $z\to z+2\pi$
\begin{equation}
\begin{aligned}
& x\to x+2\pi N y,\\
&y\to y.
\end{aligned}
\end{equation}
This background has non-zero f-flux defined as $de^a=f^{a}{}_{bc}e^b\wedge e^c$ for $e^a$ being a vielbein 1-form. In the example above the only non-zero component of the f-flux is $f^x{}_{yz}=2\pi N$.
To go further along the duality orbit generated by the NS5 brane we T-dualize the remained $y$-coordinate. One may use the same Buscher rules as above with the change $x\to y$. We then end up with the following background
\begin{equation}
\begin{aligned}
ds''^2&=\frac{1}{1+N^2z^2}(dx^2+dy^2+dz^2),\\
B&=\frac{Nz}{1+N^2z^2}dx\wedge dy.
\end{aligned}
\end{equation}
Here the shift $z\to z+2\pi$ produces a complete mess and mixes the metric and the B-field. This is an example of a (globally) non-geometric background. It is important that locally this configuration is described by distinct metric and B-field, but it is not a manifold globally. One may check that going around the circle results in new metric and B-field, $\tilde{G}$ and $\tilde{B}$, that are related to the old ones by a T-duality transformation.
In the case of an ordinary manifold with non-trivial monodromy properties metric and gauge fields do not mix when going around a non-trivial cycle. This means that the monodromy is a diffeomorphism and/or a gauge transformation and transforms the metric and the gauge fields separately. In the sample above the monodromy is an element of the T-duality group, that is not a direct sum of diffeos and gauge transformations. This non-geometric background is characterized by the Q-flux, that is a sign of non-commutativity of string coordinated [4]
\begin{equation}
[X^a,X^b]=Q^{ab}{}_{c}X^c.
\end{equation}
Hence, we now recovered the following chain that represents T-duality orbit of the NS5-brane [5]
\begin{equation}
H_{xyz} \longrightarrow f^x{}_{yz}\longrightarrow Q^{xy}{}_{z}.
\end{equation}
Each action of a T-duality transformation effectively raises the corresponding index. The last step remained is to T-dualize the $z$-coordinate. Although, this procedure is not well defined it is usually said that it produces a locally non-geometric background with non-zero R-flux. In contrast to the previous example this background is not geometric even locally and the R-flux reflects non-associativity of string coordinates
\begin{equation}
[X^a,X^b,X^c]\sim R^{abc}.
\end{equation}
Then the full T-duality chain will be
\begin{equation}
H_{xyz} \longrightarrow f^x{}_{yz}\longrightarrow Q^{xy}{}_{z}\longrightarrow R^{xyz}.
\end{equation}
In the context on previous posts it is important to mention that non-geometric backgrounds find a natural description in terms of the extended geometry. In this approach T-(U-)duality transformations are good geometric transformations of the extended space, that is obviously not a manifold. The Scherk-Schwarz reduction of the extended space allows to end up with non-geometric backgrounds in a natural way.
Compactifications in the presence of (non-)geometric fluxes have found their application recently to the problems of cosmology, phenomenological models, problem of string vacuum. For example, certain models with non-zero fluxes allow to end up with de Sitter space in string compactifications, that is the relevant background for modern cosmology.
On the other hand backgrounds with non-zero fluxes allow to end up with more stabilized moduli and reduce the number of arbitrary parameter in phenomenological models.
An interesting effect related tightly to exotic branes is the so-called supertube effect. Due to this effect interacting (ordinary) branes may polarize into exotic branes. This means that one is interested not only in compactifications to 3 dimensions. For example, in such objects as black holes, that are typically constructed from D-branes, exotic branes should be taken into account as well.
