
#1
Nov2512, 07:54 PM

P: 748

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 M_{24}, 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 E_{8}) 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 M_{24} 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. M_{24} 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 wellknown 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. 



#2
Nov2612, 01:36 AM

Sci Advisor
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#3
Nov2612, 05:00 PM

P: 748





#4
Nov2712, 12:35 AM

Sci Advisor
P: 5,307

M is for Moonshine
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. 



#5
Nov2812, 08:10 PM

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#6
Dec312, 12:23 PM

PF Gold
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#7
Dec312, 03:53 PM

Sci Advisor
P: 5,307

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




#8
Dec312, 07:22 PM

PF Gold
P: 2,884





#9
Dec412, 12:35 AM

P: 748

String theory is an outgrowth of the Smatrix program and it is perturbatively defined as an Smatrix. For a QFT, there are finitetime formalisms like Keldysh, but the string Smatrix 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. Finitetime evolution might come from Riemann surfaces with finitesize 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". 



#10
Dec412, 01:28 PM

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#11
Dec512, 02:57 AM

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#12
Dec512, 04:06 AM

P: 210

Thanks!



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