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- TL;DR Summary
- Possible phase of M-theory in which gravity is part of a higher gauge symmetry

(2,0) theory is a six-dimensional field theory that has achieved a mild degree of notoriety. "(2,0)" is a notation for degree of supersymmetry, in dimensions where supersymmetries can have a "chirality" (left-handedness vs right-handedness); in this case, it means that this is a field theory possessing two supersymmetries with the same chirality, and none with the opposite chirality.

(2,0) theory is the worldvolume theory of a 5-brane from M-theory. It's how the world looks from inside the 5-brane (which from outside inhabits an 11-dimensional space-time). The other known stable brane from M-theory, the M2-brane, also has a worldvolume theory, "ABJM theory", a three-dimensional field theory with N=6 supersymmetry.

To be more precise, there is a whole family of (2,0) theories, classified by their symmetry group, just as e.g. Yang-Mills theory can have various gauge groups. These correspond to different numbers of coincident M5-branes, in particular 11-dimensional backgrounds. Nonetheless, one still speaks of "(2,0) theory" in the way that one speaks of "Yang-Mills theory" or "gauge theory", as a theoretical template shared by all members of a class of field theories.

Part of the interest of (2,0) theory is that it is thought to explain some of the properties of theories in lower dimensions. For example, Seiberg and Witten studied four-dimensional theories with N=2 supersymmetry, and were able to derive various properties such as particle masses from a two-dimensional object called "the Seiberg-Witten curve". This curve is now understood simply as the shape of two of the dimensions of a (2,0) theory - when (2,0) theory is compactified on the Seiberg-Witten curve, you get the corresponding four-dimensional theory. (At least, I think that's how it works.)

Another part of the interest of (2,0) theory is that we don't have full equations for it. We know the degrees of freedom, but not how they interact. The theory is studied heuristically, or indirectly via its conjectured relationships to other theories. One may read that it is a "higher gauge theory" whose symmetry group is a "2-group" rather than an ordinary Lie group; and that it is a "non-lagrangian" theory which is strongly coupled however you look at it, so the usual perturbative approaches won't work - leaving me to wonder what

Finally, it's of interest because understanding it and defining it is a step towards understanding and defining M-theory itself, which is similarly known only in a rather fragmentary way.

OK, so there's lots of unknowns here, but I thought I had a reasonable grasp of what is known about (2,0) theory. M-theory has 2-branes and 5-branes; (2,0) theory is the worldvolume theory of the 5-branes, so it describes things like virtual 2-branes budding from the 5-branes; it's a window on how M-theory works.

But yesterday I learned about a proposal that there is also a (4,0) theory in six dimensions

"Gravity, Duality, and Conformal Symmetry" by Chris Hull (2022)

and that it might be a "double copy" of the (2,0) theory.

The double copy refers to the phenomenon that one may in various ways "square a gauge theory" and obtain a theory with gravity. It was first discovered in string theories where gauge bosons are open strings and gravitons are closed strings, and one has "an intuitive picture of breaking up a closed string into two open strings glued together by phase factors"... quote from:

"Perturbative Gravity and Gauge Theory Relations - A Review" by Thomas Sondergaard (2011)

Double copy relations can also be proved working within field theory (see the 2011 paper), and have also been generalized in many ways.

Still, the idea of a double copy of (2,0) theory spoiled the harmony of my simple understanding above, potentially complicating things in an unknown way. So perhaps it's worth mentioning why one would believe that a (4,0) theory exists at all. One way to get there is Nahm's classification of supersymmetry representations in higher dimensions.

Years before anyone thought of M-branes, Nahm discovered that a (2,0) superconformal multiplet is possible in six dimensions, but no one thought there was actually a six-dimensional field theory containing that set of fields. Then, decades later, a (2,0) theory was found lurking in M-theory. Nahm's classification also includes a possible (4,0) superconformal multiplet in six dimensions, so what if there's a (4,0) theory too?

Chris Hull's proposal is that (4,0) theory arises as a strong coupling limit of five-dimensional N=8 supergravity. (This is reminiscent of how Witten originally obtained 11-dimensional M-theory, as the strong coupling limit of a 10-dimensional string theory.) The resulting theory, however, would be peculiar, since there are no gravitons in Nahm's (4,0) supermultiplet. Instead (quoting Hull 2022), "it has an exotic fourth-rank tensor gauge field ... with the algebraic properties of the Riemann tensor".

Perhaps what this suggests, is that the algebraic unification of gravity and ordinary gauge theory, occurs somewhere in the higher gauge theory on which M-theory is probably based. At the fundamental level, there is only higher gauge theory; in the compactifications that are best-studied, higher gauge theory has separated into gravity and ordinary gauge theory; but (4,0) theory would be a six-dimensional phase of M-theory, in which gravity has not yet emerged from higher gauge symmetry. That only happens when you take one further step, down to five dimensions.

My problem is that this does not seem to be comprehensible in terms of the simple visualizations I possess, like "2-branes interacting with 5-branes" or "two open string glued to make a closed string". Perhaps one could work one's way upwards as follows: Ordinary gauge theory, "squared", produces a gravity theory. Ordinary gauge theory is (2,0) theory compactified. Ordinary gravity is (4,0) theory compactified. So work with the stringy explanation of "gravity equals gauge theory squared", and lift those strings up into the M-world of 2-branes and 5-branes, and hopefully insight will come - eventually.

(2,0) theory is the worldvolume theory of a 5-brane from M-theory. It's how the world looks from inside the 5-brane (which from outside inhabits an 11-dimensional space-time). The other known stable brane from M-theory, the M2-brane, also has a worldvolume theory, "ABJM theory", a three-dimensional field theory with N=6 supersymmetry.

To be more precise, there is a whole family of (2,0) theories, classified by their symmetry group, just as e.g. Yang-Mills theory can have various gauge groups. These correspond to different numbers of coincident M5-branes, in particular 11-dimensional backgrounds. Nonetheless, one still speaks of "(2,0) theory" in the way that one speaks of "Yang-Mills theory" or "gauge theory", as a theoretical template shared by all members of a class of field theories.

Part of the interest of (2,0) theory is that it is thought to explain some of the properties of theories in lower dimensions. For example, Seiberg and Witten studied four-dimensional theories with N=2 supersymmetry, and were able to derive various properties such as particle masses from a two-dimensional object called "the Seiberg-Witten curve". This curve is now understood simply as the shape of two of the dimensions of a (2,0) theory - when (2,0) theory is compactified on the Seiberg-Witten curve, you get the corresponding four-dimensional theory. (At least, I think that's how it works.)

Another part of the interest of (2,0) theory is that we don't have full equations for it. We know the degrees of freedom, but not how they interact. The theory is studied heuristically, or indirectly via its conjectured relationships to other theories. One may read that it is a "higher gauge theory" whose symmetry group is a "2-group" rather than an ordinary Lie group; and that it is a "non-lagrangian" theory which is strongly coupled however you look at it, so the usual perturbative approaches won't work - leaving me to wonder what

*will*work. It's a conformal theory, so maybe one of the approaches peculiar to conformal field theories will work, like operator product expansions or the "conformal bootstrap".Finally, it's of interest because understanding it and defining it is a step towards understanding and defining M-theory itself, which is similarly known only in a rather fragmentary way.

OK, so there's lots of unknowns here, but I thought I had a reasonable grasp of what is known about (2,0) theory. M-theory has 2-branes and 5-branes; (2,0) theory is the worldvolume theory of the 5-branes, so it describes things like virtual 2-branes budding from the 5-branes; it's a window on how M-theory works.

But yesterday I learned about a proposal that there is also a (4,0) theory in six dimensions

"Gravity, Duality, and Conformal Symmetry" by Chris Hull (2022)

and that it might be a "double copy" of the (2,0) theory.

The double copy refers to the phenomenon that one may in various ways "square a gauge theory" and obtain a theory with gravity. It was first discovered in string theories where gauge bosons are open strings and gravitons are closed strings, and one has "an intuitive picture of breaking up a closed string into two open strings glued together by phase factors"... quote from:

"Perturbative Gravity and Gauge Theory Relations - A Review" by Thomas Sondergaard (2011)

Double copy relations can also be proved working within field theory (see the 2011 paper), and have also been generalized in many ways.

Still, the idea of a double copy of (2,0) theory spoiled the harmony of my simple understanding above, potentially complicating things in an unknown way. So perhaps it's worth mentioning why one would believe that a (4,0) theory exists at all. One way to get there is Nahm's classification of supersymmetry representations in higher dimensions.

Years before anyone thought of M-branes, Nahm discovered that a (2,0) superconformal multiplet is possible in six dimensions, but no one thought there was actually a six-dimensional field theory containing that set of fields. Then, decades later, a (2,0) theory was found lurking in M-theory. Nahm's classification also includes a possible (4,0) superconformal multiplet in six dimensions, so what if there's a (4,0) theory too?

Chris Hull's proposal is that (4,0) theory arises as a strong coupling limit of five-dimensional N=8 supergravity. (This is reminiscent of how Witten originally obtained 11-dimensional M-theory, as the strong coupling limit of a 10-dimensional string theory.) The resulting theory, however, would be peculiar, since there are no gravitons in Nahm's (4,0) supermultiplet. Instead (quoting Hull 2022), "it has an exotic fourth-rank tensor gauge field ... with the algebraic properties of the Riemann tensor".

Perhaps what this suggests, is that the algebraic unification of gravity and ordinary gauge theory, occurs somewhere in the higher gauge theory on which M-theory is probably based. At the fundamental level, there is only higher gauge theory; in the compactifications that are best-studied, higher gauge theory has separated into gravity and ordinary gauge theory; but (4,0) theory would be a six-dimensional phase of M-theory, in which gravity has not yet emerged from higher gauge symmetry. That only happens when you take one further step, down to five dimensions.

My problem is that this does not seem to be comprehensible in terms of the simple visualizations I possess, like "2-branes interacting with 5-branes" or "two open string glued to make a closed string". Perhaps one could work one's way upwards as follows: Ordinary gauge theory, "squared", produces a gravity theory. Ordinary gauge theory is (2,0) theory compactified. Ordinary gravity is (4,0) theory compactified. So work with the stringy explanation of "gravity equals gauge theory squared", and lift those strings up into the M-world of 2-branes and 5-branes, and hopefully insight will come - eventually.