M-theory, that is a theory describing dynamics of 2 and 5 dimensional membranes, is believed to be a certain generalisation of 11 dimensional supergravity. It is often said that 11D SURGA is a low energy limit of M-theory. Basically, this is the only information known about M-theory to the moment. So why the fundamental objects of M-theory are known to be M2 and M5 branes? In one of the previous posts it has been figured out that this can be related to 11-dimensional superalgebra and Fierz identities. However, this does not explain why M5 branes are called magnetic. Here we will approach from the side of the supergravity in 11 dimensions and see that its fundamental gauge fields interact with electric and magnetic currents represented by 3- and 6-forms.
Start first with the usual electrodynamics in four dimensions, described by a field strength $F_{\mu\nu}$ and the corresponding gauge potential $A_\mu$
\begin{equation}\label{F}
F_{\mu\nu}=\partial_{\m}A_{\n}-\partial_{\nu}A_{\mu}.
\end{equation}The latter interacts with a point-like particle, or better to say with its charge, via the corresponding current and the full action reads
\begin{equation}\begin{aligned}
S&=\int d^4 x F^{\mu\nu}F_{\mu\nu}+\int d\tau j^\mu A_\mu\\
&=\int \star F \wedge F+\int_\gamma q_e dx^\mu A_\mu =\int \star F \wedge F+q_e \int \pi \wedge A,
\end{aligned}
\end{equation}where $\pi \sim \star j$ is a closed 3-form $d \pi=0$, which defines the worldline $\gamma$ of the particle. At this point one may call the charge $q_e$ either electric or magnetic, and we will choose the former.
Hence, equations of motion of electromagnetic field interacting with electric current are\begin{equation}
\begin{aligned}
d \star F= q_e \p,\\
d F=0,
\end{aligned}
\end{equation}where the second is the Bianchi identities which have to be imposed to keep $F=dA$, which is just \eqref{F}. Now, one notices that free EM EOM's, i.e. $\p=0$, are invariant under S-duality, that is just $U(1)$ acting on $F+i \star F$. To keep such duality properties for interacting EOM's it is tempting to add another charge density $\hat\p$ that will be associated to magnetic charge\begin{equation}
\begin{aligned}
d \star F=q_e \p,\\
d F=q_m \hat\p.
\end{aligned}
\end{equation}Indeed, setting $\p=0$ one obtains div$\vec{B}=q_m$ for the magnetic charge. Now the S-duality rotates magnetic and electric charges into each other acting on the combination $q_e+i q_m$.
To make the symmetry even more transparent, let us set $\p=0$ and introduce a field $G=\star F$, that renders the EOM's into\begin{equation}
\begin{aligned}
d G=0,\\
d \star G=q_m \hat\p.
\end{aligned}
\end{equation}That have precisely the same form as the EOM's for electric charge. Looking at the first line one concludes that EOM for the field $F$ turns into Bianchi identity for its magnetic dual $G$. This identity implies $G=d\tilde{A}$, with some magnetic gauge potential $\tilde{A}$.
This dual potential is again a 1-form, hence it interacts with a point particle, the magnetic monopole. This follows from the fact, that we are working in 4 dimensions and hence dual of the 2-form $F$ is again a 2-form $G$. In $D$ dimensions the dual would be a $D-2$-form and the corresponding magnetic monopole will be a $D-4$-dimensional object.
Finally, we are switching to dualization of a 2-dimensional object (the M2-brane) living in 11 dimensions. It interact with the 3-form $C_3=C_{\m\n\r}dx^\m \wedge dx^\n \wedge dx^\r$, which has field strength $F_4=dC_3$. Equations of motion for such field strength will be\begin{equation}
\begin{aligned}
dF_4=0,\\
d \star F_4=q \p,
\end{aligned}
\end{equation}which are however not completely true. The problem here is that the RHS of the above is not quite as it is written as it contains some terms quadratic in the gauge potential $C_3$ coming from Chern-Simons like terms $F_4 \wedge F_4 \wedge C_3$ in the full action. However, the claim is that the same procedure as described below will be equally applicable to the full story. For more technical details the reader is referred to the wonderful description of dualization of tensor fields in supergravity by Cremmer, Julia, Lu and Pope.
Here I will just count the dimensions and rank of the corresponding forms. Hence, M2-brane interacts with the 3-form, whose field strength is a 4-form $F_4$. As before, one may write the EOM's as Bianchi identities for a Hodge-dual 7-form $G_7$, whose solution is $G_7=d\tilde{C}_6$. Such 6-form interacts with a 5-dimensional object, that is supposed to be the desired M5-brane.
Hence, calling the initial M2-brane electric, as the one we started with, its dual M5-brane will be magnetic. In the same sense as the point-like magnetic monopole in 4 dimensions is dual to the point-like electric charge.
The M2 and M5 branes of M-theory gives rise to the whole spectrum of NS-NS and R-R objects of Type IIA string theory: string, NS5-brane, D-branes. Interestingly, there are many more branes in M-theory, than these two. First, one recalls the Kaluza-Klein monopole, which will be described in a further post. Second, one discovers the so-called exotic branes, similar to those NS-NS exotic branes of string theory. These also have their one gauge potentials and can be related to each other by similar electric-magnetic dualities, or U-duality, more generally speaking.
Start first with the usual electrodynamics in four dimensions, described by a field strength $F_{\mu\nu}$ and the corresponding gauge potential $A_\mu$
\begin{equation}\label{F}
F_{\mu\nu}=\partial_{\m}A_{\n}-\partial_{\nu}A_{\mu}.
\end{equation}The latter interacts with a point-like particle, or better to say with its charge, via the corresponding current and the full action reads
\begin{equation}\begin{aligned}
S&=\int d^4 x F^{\mu\nu}F_{\mu\nu}+\int d\tau j^\mu A_\mu\\
&=\int \star F \wedge F+\int_\gamma q_e dx^\mu A_\mu =\int \star F \wedge F+q_e \int \pi \wedge A,
\end{aligned}
\end{equation}where $\pi \sim \star j$ is a closed 3-form $d \pi=0$, which defines the worldline $\gamma$ of the particle. At this point one may call the charge $q_e$ either electric or magnetic, and we will choose the former.
Hence, equations of motion of electromagnetic field interacting with electric current are\begin{equation}
\begin{aligned}
d \star F= q_e \p,\\
d F=0,
\end{aligned}
\end{equation}where the second is the Bianchi identities which have to be imposed to keep $F=dA$, which is just \eqref{F}. Now, one notices that free EM EOM's, i.e. $\p=0$, are invariant under S-duality, that is just $U(1)$ acting on $F+i \star F$. To keep such duality properties for interacting EOM's it is tempting to add another charge density $\hat\p$ that will be associated to magnetic charge\begin{equation}
\begin{aligned}
d \star F=q_e \p,\\
d F=q_m \hat\p.
\end{aligned}
\end{equation}Indeed, setting $\p=0$ one obtains div$\vec{B}=q_m$ for the magnetic charge. Now the S-duality rotates magnetic and electric charges into each other acting on the combination $q_e+i q_m$.
To make the symmetry even more transparent, let us set $\p=0$ and introduce a field $G=\star F$, that renders the EOM's into\begin{equation}
\begin{aligned}
d G=0,\\
d \star G=q_m \hat\p.
\end{aligned}
\end{equation}That have precisely the same form as the EOM's for electric charge. Looking at the first line one concludes that EOM for the field $F$ turns into Bianchi identity for its magnetic dual $G$. This identity implies $G=d\tilde{A}$, with some magnetic gauge potential $\tilde{A}$.
This dual potential is again a 1-form, hence it interacts with a point particle, the magnetic monopole. This follows from the fact, that we are working in 4 dimensions and hence dual of the 2-form $F$ is again a 2-form $G$. In $D$ dimensions the dual would be a $D-2$-form and the corresponding magnetic monopole will be a $D-4$-dimensional object.
Finally, we are switching to dualization of a 2-dimensional object (the M2-brane) living in 11 dimensions. It interact with the 3-form $C_3=C_{\m\n\r}dx^\m \wedge dx^\n \wedge dx^\r$, which has field strength $F_4=dC_3$. Equations of motion for such field strength will be\begin{equation}
\begin{aligned}
dF_4=0,\\
d \star F_4=q \p,
\end{aligned}
\end{equation}which are however not completely true. The problem here is that the RHS of the above is not quite as it is written as it contains some terms quadratic in the gauge potential $C_3$ coming from Chern-Simons like terms $F_4 \wedge F_4 \wedge C_3$ in the full action. However, the claim is that the same procedure as described below will be equally applicable to the full story. For more technical details the reader is referred to the wonderful description of dualization of tensor fields in supergravity by Cremmer, Julia, Lu and Pope.
Here I will just count the dimensions and rank of the corresponding forms. Hence, M2-brane interacts with the 3-form, whose field strength is a 4-form $F_4$. As before, one may write the EOM's as Bianchi identities for a Hodge-dual 7-form $G_7$, whose solution is $G_7=d\tilde{C}_6$. Such 6-form interacts with a 5-dimensional object, that is supposed to be the desired M5-brane.
Hence, calling the initial M2-brane electric, as the one we started with, its dual M5-brane will be magnetic. In the same sense as the point-like magnetic monopole in 4 dimensions is dual to the point-like electric charge.
The M2 and M5 branes of M-theory gives rise to the whole spectrum of NS-NS and R-R objects of Type IIA string theory: string, NS5-brane, D-branes. Interestingly, there are many more branes in M-theory, than these two. First, one recalls the Kaluza-Klein monopole, which will be described in a further post. Second, one discovers the so-called exotic branes, similar to those NS-NS exotic branes of string theory. These also have their one gauge potentials and can be related to each other by similar electric-magnetic dualities, or U-duality, more generally speaking.
