What exactly does it mean to wrap a D-brane or a M-brane in a Riemann surface $\Sigma_g$?

Question

What exactly does it mean to wrap a D-brane or an M-brane in a Riemann surface $\Sigma_g$ ($g$ is the genous)? Is there some mathematical statement? And why do we get various supersymmetric gauge theories out of this?

Is there any relation to compactifications? Is there a good place for a read on the "wrapping" of branes on Riemann surfaces?

P.S. As far as the references are concerned I would appreciate something "easier" than Nunez's and Maldacena's paper for example (http://arxiv.org/abs/hep-th/0007018). Thanks!

This post imported from StackExchange Physics at 2015-02-23 17:11 (UTC), posted by SE-user Marion

Comments

@Urs Schreiber maybe you can help out with this? A very interesting question that currently is at the center of research in susy gauge theories. 

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@conformal_gk, to notify a user you need to eliminate all the spaces in his/her username, in this case you need to type "at"UrsSchreiber (replace "at" by @)

Answer 1

The simplest classical picture of a brane is a manifold in space-time, representing the trajectory of a spatially extended dynamical object in string theory. What is generally called a p-brane is extended in p spatial dimensions and so its "trajectory", generally called worldvolume, is a (p+1)-dimensional manifold in spacetime. In superstring theory, the spacetime is 10 dimensional and the data of a p-brane is the data of a (p+1) dimensional submanifold of this spacetime. The simplest textbooks examples take as spacetime $\mathbb{R}^{1,9}$ and flat p-branes i.e. $\mathbb{R}^{1,p}$. But one can imagine more complicated situations. For example, to construct realistic 4-dimensional theories, one can take as spacetime $\mathbb{R}^{1,3} \times X$ where $X$ is a compact six dimensional manifold. In this case, one can consider for example p-branes of the form $\mathbb{R}^{1,3} \times Y$ where $Y$ is a submanifold of $X$. One generally says that the p-brane is wrapped around $Y$. For example, if $X$ contains a Riemann surface, one can take $Y$ to be this Riemann surface and so the p-brane is wrapped on the Riemann surface $Y$.

To summarize: a p-brane is wrapped on a manifold $Y$ if its worlvolume is a (p+1)-manifold which is a product of a flat space with $Y$.

In what preceeds, I have only considered the worldvolume manifold underlying the brane. In fact, a brane is a much richer object because it is a dynamical object in the theory. For example, for D-branes in string theory, strings can end on the D-branes and in particular we have open strings with both ends on the D-brane. The massless spectrum of these open strings contain a gauge boson and so the effective theory living on a D-brane is a gauge theory. More precisely, it is a $U(1)$ gauge theory for a unique D-brane and a $U(N)$ gauge theory if one has $N$ D-branes stacked together (in the simplest cases, say in type II superstring without spacetime singularities, without B-field...). Due to spacetime supersymmetry in string theory, it is in general a supersymmetric gauge theory. The precise number of supersymmetries preserved in the gauge theory depends precisely on the way the brane is wrapped. If the brane is wrapped on a manifold $Y$, then taking the limit when $Y$ becomes small, one obtains an effective theory in the remaining non-compact dimensions of the brane. Thus starting with a gauge theory living on a flat brane, wrapping this brane around a non-trivial manifold gives you a way to construct a new gauge theory with less spacetime dimensions and in general less supersymmetries.

In general, to preserve supersymmetries by wrapping branes on a manifold $Y$, $Y$ has to satisfy strong conditions which are mathematically very interesting (these conditions have a BPS form and are related to the theory of calibrated submanifolds. Depending on the precise context, some examples are complex submanifolds, special Lagrangian submanifolds...)

EDIT: here are some explicit examples.

1) Start with $N$ parallel coincident M5 branes. The low energy theory living on this stack of brane is a $N=(2,0)$ six dimensional gauge theory of gauge group $U(N)$. If you wrap these branes on a 2-torus, the resulting theory at low-energy in the remaining four non-compact dimensions is $N=4$ super Yang-Mills of gauge group $U(N)$. If instead of a 2-torus, one chooses a more complicated Riemann surface, one obtains a four dimensional N=2 gauge theory whose details depend on the Riemann surface. These $N=2$ 4d theories are called of class S and most of them have no lagrangian description.

2)Again $N$ $M5$ branes but in a slightly different context (we write 6=4+2 rather 6=2+4). Assume that M-theory is compactified on a Calabi-Yau 3-fold $X$. Then wrapping the M5-branes around a four dimensional submanifold of $X$ (more precisely a complex surface in $X$), gives at low energy for the remaining two non-compact dimensions a $N=(4,0)$ 2d gauge theory, which was for example used for black holes entropy calculations.

3) Consider $N$ parallel coincident D3 branes. The low energy theory living on this stack of brane is a $N=4$ super Yang-Mills of gauge group $U(N)$. Assume that type IIB string theory is compactified on a Calabi-Yau 3-fold $X$. Then, you can wrap the branes on a point in $X$, which gives something non-trivial if this point is singular in $X$, for example if $X$ is a cone over a 5-manifold $Y$ and if we put the branes at the tip of the cones. The resulting gauge theory in the non-compact four dimensions have $N=2$ or $N=1$ supersymmetries depending on the details. An AdS/CFT argument shows that these four dimensional gauge theories are dual to type IIB string theory on $AdS^5 \times Y$.

4) Consider type IIA string theory compactified on a Calabi-Yau 3-fold $X$. One can wrap $N$ D6 branes around a 3-manifold in $X$ (more precisely a special Lagrangian submanifold of $X$) and the low-energy gauge theory living in the remaining four non-compact dimensions has $N=1$ supersymmetries...

5) Consider type IIB gauge theories compactified on a complex surface $X$ resolution of an ADE singularity. One can wrap $N$ D7-branes around the non-trivial 2-cycles in $X$ and the low-energy gauge theory living in the remaining six non-compact dimensions has $N=(1,0)$ supersymmetries...

6)...

Comments

Would you be able to give some specific gauge theory examples? 

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Thanks for your really good answer. I would give a second + but it's not possible. Would you be able to give some references for the points 1 and 5 (i.e. introductory or review or lectures if possible and if they exist)? Thanks a lot.

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Unfortunately, I don't know review/lectures. They probably exist but it is just that I never looked for them. The references I know are original papers. For 1), a great part of the recent activity comes from the work of Gaiotto, see for example http://arxiv.org/abs/0904.2715 For 5), they are probably many things in the F-theory litterature but what I had in mind is http://arxiv.org/abs/1312.5746