# wrapping M5-branes on a Riemann surface

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AdS/CFT gives us a way to use geometry to study field theory! I am trying to wrap M5-branes on a Riemann surface $\Sigma_{g}$. In my problem, for a Riemann surface in 11d, the normal bundle is max $SO(5)$. Here is my question: How do we put $SO(2)$ in $SO(5)$?

Urs Schreiber suggests the following mathematically precise interpretation of the question, which probably addresses the concerns of those who commented or who had voted to put the OP on hold:

There is a famous construction of (N=2)-supersymmetric 4-dimensional Yang-Mills field theories and their Seiberg-Witten theory from the N=(2,0)-superconformal 6-dimensional field theory on the worldvolume of M5-branes: by Kaluza-Klein-compactifying the latter on a Riemann surface. This construction was revived more recently in 2009 by the influential article

On page 22 of this article, around the displayed formula (2.27), the author mentions that the Kaluza-Klein compactification of the 6d theory on a Riemann surface involves a “well known twisting procedure” of the holonomy of the Riemann surface by choosing an SO(2)-subgroup of the SO(5) group that is the “R-symmetry” group of the 6-dimensional supersymmetric field theory (the group under which its supercharges transform).

Question: What is this “well known twisting procedure” exactly, and how does it work? Of course I know how to find $SO(2)$-subgroups of $SO(5)$, but what does such a choice amount to in the context of the construction of an N=2, D=4 SYM from the 6d-field theory on the 5-brane? Where is this twisting procedure discussed in the literature?

This post imported from StackExchange MathOverflow at 2014-12-24 12:03 (UTC), posted by SE-user Irina
retagged Dec 24, 2014
Irina: It is very much unclear what are you asking: (1) What do you mean by "max SO(5)"? (2) Are you interested in describing all embeddings $SO(2)\to SO(5)$? (There will be countably many, even up to conjugation, given by continuous monomorphisms $S^1\to S^1\times S^1$, the latter is the maximal torus in $SO(5)$.) Or you just want to have some examples of such embeddings? (math.stackexchange would be more suitable for the latter question.) (3) How does the question about these embeddings relate to $M5$ branes, whatever they are?

This post imported from StackExchange MathOverflow at 2014-12-24 12:03 (UTC), posted by SE-user Misha
p-branes are the fundamental objects in String/M-Theory (1-branes are strings), and a quick google search will explain them to you fairly nicely. Here "M5-brane wrapped on Riemann surface" is to consider the world-volume (trajectory of the brane) as $\mathbb{R}^{3,1}\times\Sigma_g$ (but the first factor should really be an Anti-de Sitter space). Anyway this background information is irrelevant for his "actual" question, embedding $SO(2)$ into $SO(5)$, which I don't know how to reconcile with the weird statement "normal bundle is max $SO(5)$".

This post imported from StackExchange MathOverflow at 2014-12-24 12:03 (UTC), posted by SE-user Chris Gerig
@ChrisGerig: Do you see any relation between the M5-branes and representations $SO(2)\to SO(5)$? (This may help to explain what can one possibly ask here.) BTW, "Irina" is (presumably) she, not he, as it is a common Russian female name.

This post imported from StackExchange MathOverflow at 2014-12-24 12:03 (UTC), posted by SE-user Misha
Not really; there are $SO(5)$-gauge transformations on the normal bundle of this (6-dimensional) world-volume in $\mathbb{R}^{11}$, but I don't see what this has to do with $SO(2)$.

This post imported from StackExchange MathOverflow at 2014-12-24 12:03 (UTC), posted by SE-user Chris Gerig
Just for completeness: the question is specifically about the construction of super Yang-Mills theory in d=4 by KK reduction on Riemann surfaces of the 6d (2,0)-SCFT on M5-branes as pioneered by Vafa et al and Witten and then more recently taken to further depth by Gaiotto, leading to what is now called the "AGT correspondence" which relates Liouville theory on that Riemann surface to the partition function of the 4d SYM theory. For commented pointers to the literature see here ncatlab.org/nlab/show/… .

This post imported from StackExchange MathOverflow at 2014-12-24 12:03 (UTC), posted by SE-user Urs Schreiber
The twist can be summarized as follows: the (2,0) theory has 5 scalar fields (which can be seen as arising from the 5 transverse directions to an M5 brane), acted on by the R-symmetry group SO(5). In order to define the theory on a general Riemann surface (times flat spacetime), you need to specify how these fields should transform: three will continue to be scalars, but two transform as a section of the cotangent bundle to the Riemann surface. This is why you need an embedding of SO(2) into SO(5) (just rotating the first two variables).

This post imported from StackExchange MathOverflow at 2014-12-24 12:03 (UTC), posted by SE-user David Ben-Zvi
For references try Witten's long paper on knot homology - the section on fivebranes discusses a twist of the (2,0) theory which is holomorphic in two dimensions and topological in four. Or try the fundamental series of papers of Gaiotto-Moore-Neitzke, this construction is essential to their work (and they're my source).

This post imported from StackExchange MathOverflow at 2014-12-24 12:03 (UTC), posted by SE-user David Ben-Zvi

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Now that the question is open again (now in my paraphrasing), maybe I'll repost my reply from the nForum with some brief comments thrown in:

That the 6-dimensional (2,0)-superconformal QFT on the worldvolume of the M5-brane yields N=2 D=4 super Yang-Mills theory under Kaluza-Klein compactification on a 2d Riemann surface was known since about the mid 90s. Edward Witten had famously advertized this in the Proceedings to Graeme Segal's 60th birthday conference that by this construction the remaining invariance under Moebius transformations of that compactification manifold geometrically explains the "Montonen-Olive"/"electric-magnetic" S-duality invariance of (super) Yang-Mills theory.

Later he realized further compactification of this down to 2-dimensions as a geometric realization of geometric Langlands duality. In the course of this the N=2 D=4 super Yang-Mills theory is "topologically twisted" in a way analogous to the well-known twisting of N=4 SYM that goes back to the work that won him the Fields medal. The twisting of the N=2 theory then also showed up in the more recent work by Gaiotto et al. that lead to the AGT correspondence.

While the details for the topological twisting of the N=2 supersymmetric field theory are a tad more involved than those of the N=4 theory, the basic idea is the same: one picks an embedding of the spacetime rotation symmetry into the R-symmetry group (the one under which the supercharges transform) and then asks for a linear combination $Q$ of the supercharges that is held fixed by the resulting external+internal symmetry transformations. The cohomology of this $Q$ is then seen to pick inside the quantum observables of the origional super gauge field theory those of a topological field theory. That is the topologically twisted theory.

Pointers to more details on this topological twisting that the above questing is after are collected here:

http://ncatlab.org/nlab/show/topologically+twisted+D=4+super+Yang-Mills+theory

Pointers specifically concerning the twistied Kaluza-Klein compactification of the M5-brane on a Riemann surface to a topologically twisted N=2 super Yang-Mills theory are here:

http://ncatlab.org/nlab/show/N=D2+D=4+super+Yang-Mills+theory#ReferencesConstructionFrom5Branes

This post imported from StackExchange MathOverflow at 2014-12-24 12:04 (UTC), posted by SE-user Urs Schreiber
answered Jun 30, 2013 by (5,785 points)

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