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Supergroups from Hanany-Witten-like constructions

One month ago, Cumrun Vafa released a paper articulating some provoking statements about the holographic duals of gauge theories with a gauge supergroup, \(U(m|n)\).

Today, we may read a 143-page-long (!) paper by Edward Witten and Victor Mikhaylov,

Branes and Supergroups,
which says some very interesting and very different things about gauge theories based on these \(U(m|n)\) supergroups. They may deduce some new statements about dualities of various Chern-Simons theories and similar beasts.




Their starting point is the Hanany-Witten setup. In type IIB string theory, you may consider parallel D3-branes (with a 4-dimensional world volume) that may terminate somewhere. These D3-branes are allowed to end on NS5-branes. The "end point" of these D3-branes, or the intersection, has a lower dimensionality by one, of course (like every boundary), so it hosts a 3-dimensional gauge theory.




Note that type IIB string theory boasts the \(SL(2,\ZZ)\) S-duality which exchanges NS5-branes with D5-branes, among other things, which is why the NS5-branes in type IIB carry a 5+1-dimensional gauge theory just like the D5-branes. The NS5-branes' world volume is non-chiral although the 10D spacetime dynamics is chiral. (In type IIA, all the properties are reversed: there is no S-duality, NS5-branes have a chiral \((2,0)\) theory on them, and the 10D spacetime dynamics is non-chiral.)

However, Witten and Mikhaylov also consider semi-infinite D3-branes that are stretched from the NS5-brane in the opposite direction than the original D3-branes. What happens if these half-D3-branes in two different semi-infinite directions are combined? The authors argue that the gauge symmetry inside the 3-dimensional boundary is generalized from \(U(m)\) to \(U(m|n)\) or, if you place an orientifold O3-plane into this intersection as well, to \(OSp(m|2n)\).

These 3-dimensional Chern-Simons-like theories are treated in various ways. They construct a magnetic dual and deduce various dualities i.e. equivalences. Some of those are known, some of them are new. And they study how line and surface operators transform under these dualities.

This line of thought is an example of the power of the string-theoretical reasoning that "geometrizes" certain things and allows one to deduce various equivalences and other properties that would look incomprehensible – and in practice, could be undiscoverable – without string theory.

The (Witten-)Chern-Simons theories don't belong to the list of interests of typical particle phenomenologists. They are more mathematical in character, especially due to their relationship to knot theory or things like topological quantum computation (let me mention Alexei Kitaev, a winner of the Milner prize, as a key researcher). These 3D theories are rather special and it may be wise for those who study them – either for purely mathematical or applied (quantum computation...) reasons – to know their properties and the reasons behind their properties in some details.

It just happens that many of these properties may only be understood (as of today) if these theories are derived as a limit of some construction within string theory. In this sense, string theory tells us where these theories come from (or may come from), who are their ancestors, and what are their family relationships, and how these family relationships impact particular properties of these theories. In other words, string theory puts all the good ideas in their proper context.

It is the only known – and probably the only allowed by mathematics – intellectual framework that goes beyond the rules of quantum field theory (write down a Lagrangian or other rules for a particular QFT, and purely technically study it, while not being distracted by any other QFTs). One could say that string theory plays the analogous role to the role that Darwin's theory of evolution plays in biology.

By the way, a co-father of string theory Holger Nielsen has a paper with Masao Ninomiya that elaborates on their new type of string field theory where everything is made out of scalar fields. I don't understand it well enough but it looks that they are constructing the strings from string bits (an even number of string bits for closed strings) so it's not new, after all. Charles Thorn would have worked on string bits for a long time, and my and DVV's matrix string theory really shown that a string-bit-like description is actually a non-perturbatively exact formulation of string theory; it is also true for "string bits" that emerge in the BMN \(pp\)-wave limit of the AdS/CFT. Correct me if I completely misunderstood their (Nielsen-Ninomiya) point.

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