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#### M-theory

Let us first recall how M-theory emerges, from the superstring point of view. The tension of a $Dp$ brane is
\begin{equation}
\tau_p = g_s^{-1} (2 \pi)^{-p} \alpha ' ^{-(p+1)/2} \, .
\end{equation}
Here $g_s$ is the string coupling constant, and $\alpha ' = l_s^2$ is the square of the string length. If we focus on the $D0$ brane, which is a particle, and for which the tension is just its mass, we find the mass $m_{D0} = \frac{1}{g_s \sqrt{\alpha '}}$. There also exist bound states of $n$ $D0$ branes, with mass $M = n m_{D0}$. Now when the string coupling becomes large, $g_s \rightarrow \infty$, those states become light particles, that can be interpreted as a Kaluza-Klein tower, with mass $M=n/R$ with a radius
\begin{equation}
R = g_s \sqrt{\alpha '} = g_s l_s \, .
\end{equation}
This means that while at weak coupling $g_s \ll 1$ the strings length is much bigger that $R$ and the string theory is effectively ten-dimensional, but when the coupling becomes large, we can no longer ignore the additional compactified eleventh dimension. At low-energy, this theory is identified with eleven-dimensional supergravity, and at any energy scale, we have M-theory.

#### Branes in M-theory and string theory

The brane content of M-theory can be inferred from the low-energy supergravity. This theory has a three-form, which should be sourced by some three-dimensional object (counting time), that is the $M2$ brane. By eleven-dimensional Hodge duality[1. Recall that a $Dp$ brane in $d$ dimensions couples to a $p+1$ gauge field, which has a $p+2$-form field strength, whose dual is a $d-p-2$ field strength that derives from a $d-p-3$ gauge field that couples to a $D(d-p-4)$ brane. Applying to $d=11$ and $p=2$, we find that the dual of the $M2$ is the $M5$. ], we should also have an $M5$ brane. Somehow less well-known is the fact that there is a third kind of supersymmetric object, called the M-wave, or sometimes the $M0$ brane. Depending on whether these two types of branes wrap or not the compact cycle, they give various ten-dimensional objects :

• The M-wave can not wrap the circle (it is space-like). It gives the Kaluza-Klein states that are identified with bound states of $D0$ branes.
• If the $M2$ wraps the circle, it gives the fundamental type $IIA$ string, sometimes called $F1$.
• If the $M2$ does not wrap the circle, it gives the $D2$.
• If the $M5$ wraps the circle, it gives the $D4$.
• If the $M5$ does not wrap the circle, it gives the $NS5$.
• The $D6$ brane, magnetic dual in ten dimensions of the $D0$, corresponds to the Kaluza-Klein monopole, or Taub-NUT space.

All this is very nice, except that we didn't explain the $D8$ brane ! This is a more complicated story, maybe the subject of an following up note.

#### The fate of the M5 brane

In his seminal paper about solutions of four-dimensional theories via M-theory, Witten uses a construction with $D4$ and $NS5$ branes as follows.
\begin{equation}
\begin{array}{c|cccccccccc}
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
NS5 & - & - & - & - & - & - & \mathrm{Re} \, s & \cdot & \cdot & \cdot \\
D4 & - & - & - & - & \mathrm{Re} \, v & \mathrm{Im} \, v & - & \cdot & \cdot & \cdot
\end{array}
\end{equation}
There are $n+1$ $NS5$ branes, and $k_\alpha$ $D4$ branes between the five-branes $\alpha -1$ and $\alpha$, for $\alpha=1,\dots,n$. The four-dimensional interpretation is the following :

• There is a gauge group $\prod_{\alpha=1}^n SU(k_\alpha)$.
• The coupling constant is given by the positions of the $NS5$ in the direction $6$, and the theta angle is given by the eleventh dimension $10$: with $s=(x^6 + i x^{10})/R$ we have $-i \tau_{\alpha} = s_{\alpha} - s_{\alpha -1}$.
• There are hypermultiplets transforming in the representations $(\mathbf{k}_{\alpha} ,\bar{\mathbf{k}}_{\alpha+1})$
• The masses of the hypers are given by the positions $v = x^4 + i x^5$ of the $D4$ in the directions $4,5$.

This is very interesting, but the story is even more beautiful once one realizes that both the $NS5$ and the $D4$ come from the same object in M-theory, namely the $M5$! Not only is this beautiful, but it will also solve a problem of the type $IIA$ description, by smoothing the singular junctions where the $D4$ end on the $NS5$. We will therefore consider an $M5$ that lives in the four space-time dimensions, and in addition in a two-dimensional Riemann surface $\Sigma$ inside the $\mathbb{R}^3 \times S^1$ parametrized by $(v,s)$. If locally $\Sigma$ is given by the equation $v = \mathrm{cst}$, then the ten-dimensional interpretation is a $D4$, and of the equation is $s = \mathrm{cst}$, we have an $NS5$, as illustrated in the table below.

\begin{equation}
\begin{array}{c|ccccccccccc}
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline
M5 & - & - & - & - & \Sigma & \Sigma & \Sigma & \cdot & \cdot & \cdot & \Sigma
\end{array}
\end{equation}

Let us now add one last element to our construction, a few $D6$ branes.
\begin{equation}
\begin{array}{c|cccccccccc}
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
NS5 & - & - & - & - & - & - & \mathrm{Re} \, s & \cdot & \cdot & \cdot \\
D4 & - & - & - & - & \mathrm{Re} \, v & \mathrm{Im} \, v & - & \cdot & \cdot & \cdot \\
D6 & - & - & - & - & \mathrm{Re} \, v & \mathrm{Im} \, v & \mathrm{Re} \, s & - & - & -
\end{array}
\end{equation}
More precisely, we put $d_{\alpha}$ $D6$ branes between the $NS5$ labeled $\alpha -1$ and $\alpha$. The gauge theory interpretation of the previous paragraph does not change, but we have now a different matter content, with additional hypermultiplets coming from the strings between the $D4$ and $D6$. Again, going to M-theory makes everything smooth, the singularity of the $D6$ core being replaced by the smooth and complete (multi-)Taub-NUT metric.

In the next article, I will explain how those constructions can be used to understand deep concepts of the field theories, both in four and higher dimensions.

#### References

• The classical book by C. Johnson, D-branes.
• For the M-wave, see this paper by Chu and Isono.
• The brane scan at nCatLab.
• E. Witten, Solutions Of Four-Dimensional Field Theories Via M Theory.