Mathieu Moonshine & Monster Group: Uncovering Connections

[mitchell porter]

In mathematics, "moonshine" refers to a set of notorious coincidences connecting quantities from the representation theory of "sporadic" finite groups, and quantities from certain special complex functions. Best known is the moonshine connected with the "Monster group", but recently there has also been moonshine for the much smaller Mathieu group M₂₄, and now there has been further arcane progress regarding this Mathieu moonshine, as reported in Terry Gannon's "Much ado about Mathieu".

This is all another link in the chain connecting exceptional mathematical objects (like the group E₈) with physical "theories of everything". String theory is being used to explain moonshine, which in turn suggests that the fundamental definition of string theory may have something to do with exceptional symmetries. The Monster group gets all the press as the most exceptional of the exceptionals, because it's the biggest, but according to Gannon's introduction it's M₂₄ that is the "most remarkable" - in the opinion of John Conway, one of the great contemporary mathematicians - because of its centrality in the web of relations connecting the exceptionals.

M₂₄ also has natural connections to symmetries in 24 dimensions, the number of dimensions perpendicular to the worldsheet of the bosonic string (which contains 1 space and 1 time dimension, bringing us to the well-known 26 dimensions). I am rather inclined to think that the superstring is just a sector of the bosonic string, and so it's easy to suppose that "Mathieu moonshine" corresponds to something universal in string theory, perhaps related to the universal AdS3 factor of spatial geometry near a string.

 

[tom.stoer]

In mathematics, "moonshine" refers to a set of notorious coincidences connecting quantities from the representation theory of "sporadic" finite groups, and quantities from certain special complex functions. Best known is the moonshine connected with the "Monster group", but recently there has also been moonshine for the much smaller Mathieu group M₂₄, and now there has been further arcane progress regarding this Mathieu moonshine, as reported in Terry Gannon's "Much ado about Mathieu".

Interesting!

... that the fundamental definition of string theory may have something to do with exceptional symmetries.

What is this 'fundamental definition'? I haven't seen any. Is this another conjecture or speculation?

 

[mitchell porter]

What is this 'fundamental definition'?

The same one that everyone is still seeking; something yet to be found. The step beyond "perturbative string theory + duality tricks". It looks like this Mathieu moonshine stuff needs to be taken into account, along with other developments like the use of E₁₀ and E₁₁ (which are infinite-dimensional Kac-Moody algebras), "doubled field theory", the twistor convergence... If a person could hold all of that in their mind at once, and if they knew enough future math to see it all as an inevitable whole, then maybe they could see what string theory is really about.

 

[tom.stoer]

I agree, there seem to be deep relations between mathematical structures.

The physical questions are these: Does this point towards a fundamental definition of string theory in the physical sense? Does this mean that there is one underlying mathematical structure which serves as a physical theory and defines some 'fundamental degrees of freedom' from which other approaches can be derived? Is there any mathematical principle from which answers to these physical question can be derived?

Let's make an example: think about the space of all gauge theories including the SM; and think about effective field theories like chiral perturbation theory, heavy baryons, ... I would say that - given this theory space - there is no mathematical principle from which QCD + GSW as the fundamental theory can be inferred. But there are experiments which tell what the fundamental d.o.f. are.

In the same sense it could very well be that we cannot decide which mathematical structure is 'fundamental' b/c this is not a mathematical question.

That's why sometimes I do speculate whether it's the other way round: not string theory is the fundamental theory from which gauge and gravity can be derived, but string theories are effective theories for a certain class of gauge/gravity theories.

 

[mitchell porter]

it could very well be that we cannot decide which mathematical structure is 'fundamental' b/c this is not a mathematical question. (...) sometimes I do speculate whether it's the other way round: not string theory is the fundamental theory from which gauge and gravity can be derived, but string theories are effective theories for a certain class of gauge/gravity theories.

We may also need progress on "pre-string" issues like the status of real-time evolution in a theory defined by an S-matrix.

 

[arivero]

We may also need progress on "pre-string" issues like the status of real-time evolution in a theory defined by an S-matrix.

Well, the only practical problem of asymptotical states is not the lack of a real-time evolution, but the problem of building quarks as asymptotical

 

[tom.stoer]

I agree; the whole S-matrix program is a dead end when dealing with elementary d.o.f. which cannot be isolated; the S-matrix is certainly not a fundamental approach

 

[arivero]

I agree; the whole S-matrix program is a dead end when dealing with elementary d.o.f. which cannot be isolated; the S-matrix is certainly not a fundamental approach

I wonder what does it happen in string theory, about oriented and non oriented string theories. Do both of them possesses asyptotic states and thus a S-matrix theory? Or only oriented strings havs such?

 

[mitchell porter]

String theory is an outgrowth of the S-matrix program and it is perturbatively defined as an S-matrix. For a QFT, there are finite-time formalisms like Keldysh, but the string S-matrix is defined using vertex insertions in Riemann surfaces - the vertex insertion is an asymptotic state, and the Riemann surface is a conformal compactification of the infinite worldsheet running between the asymptotic states. Finite-time evolution might come from Riemann surfaces with finite-size holes (B. Sathiapalan) or perhaps from Keldysh formalism applied to string field theory. Either way it seems a bit unfinished.

What to do about time in string theory is an instance of the general problem of "how to think about events deep in the bulk, in a holographic theory".

 

[S.Daedalus]

I am rather inclined to think that the superstring is just a sector of the bosonic string,

Sorry for being a bit off-topic, but I don't believe I've encountered this before... How does it work? Is this a widespread view?

 

[mitchell porter]

Is this a widespread view?

No, but see "Cosmological unification of string theories", in which a superstring state evolves into a bosonic string state. The authors say they are embedding bosonic string theory into superstring theory, rather than vice versa.

 

[S.Daedalus]

Thanks!