What is the Mathematical Foundation of Super String and M Theory?

[Jeebus]

I am curious on this subject and have been reading and reading, but I would like to know what goes behind Super String in math world terms? Like how does it function and operate? Could someone please tell me how the math works and functions as the property of Super String and maybe M Theory, too? I would greatly appreciate it. Just the math, that's it. I would like to know what some of the math involved in Super Strings or M Theory is exactly.

Thanks.

 

[jeff]

I think the easier question to answer is what kind of math won't be needed by strings, the answer being very possibly none.

 

[Jeebus]

I think the easier question to answer is what kind of math won't be needed by strings, the answer being very possibly none.

So what's the equation to Super String thus far? It can't be just mumbo jumbo, topology and other discrete mathematics have to be combined, so what do they got so far? That's probably a better question from the first.

Thanks for the reply by the way.

 

[Stingray]

I think the easier question to answer is what kind of math won't be needed by strings, the answer being very possibly none.

Hopefully that's not true, but it certainly needs a lot.

Jeebus, there is no "equation of super string." Its a collection of ideas. There are mathematical models of this or that, different methods developed etc, certain theorems proven within various contexts, but nothing so easily written as F=ma. You can write down actions I suppose, but I don't think that's what you're asking. Please be more specific.

 

[selfAdjoint]

Take the worldsheet - the spacetime being of the string through time. Being a surface you can put complex coordinates on it, and you find when you do that, because of the way tha acion behaves, that the physics expressed in those coordinate is invariant under conformal transformations, which means, from the textbooks of complex variable, that your worldsheet is a Riemann Surface, with all the mathematics thereunto appertaining. And so on and so on.

I can almost believe Jeff's hyperbole. I just bought a book on spectral sequences - a very boring tool of the algbraic topologist^*. And there in an advanced chapter what do I see? BRS quantization (as they call it) of strings! No math is safe! String physics eats it all up!


^*Not boring for me though. On page 10 I find: "This will turn out to be a subtle matter, though it has rich rewards." Sursum corda

 

[Pinball900]

There has to be certain mathematical equations to or squence of equations to help provide proof to the theory or there would be no theory.

 

[selfAdjoint]

Indeed, but the point of our answers is that after you define the action and quantize it (the start of any advanced quantum theory), the types of math used proliferate beyond simple explanations. I find that Kaku's book Hyperspace and Green's Elegant Universe describe some of the math in words pretty well - Kaku does modular functions and Greene does Calabi-Yau manifolds.

But even graduate students in intro string theory see the mathematical horizon receding faster than they can catch up. Which means in the end that the field breaks up into narrow specializations, which people can learn well enough to do research in them.

 

[matt grime]

There has to be certain mathematical equations to or squence of equations to help provide proof to the theory or there would be no theory.

What makes you say that? Look up Atiyah's topological quantum field theory.

A string is an object in a (suitably chosen) triangulated category, and a brane is a morphism.

Examples of other bizarre models (I use bizarre in the sense of 'to someone not well versed in higher mathematics) are cobordisms as evolution of time.

 

[Jeebus]

Hi,

Matt I have a question. On arxiv's page there is numerous pdf's with super string equations on them, I don't understand what they are trying to get at. So, here is just one link: http://arxiv.org/PS_cache/hep-th/pdf/0107/0107247.pdf

This criteria is over my head and I would like someone like you, selfAdjoint, jeff, etc, to show me what is going on here.

Thanks.

 

[Javier]

First of all, don't look at that paper...not everything on the preprint arxiv is worth reading (in fact, relatively little of it is). Try Polchinski's lectures: hep-th/9411028. Start with section 2 (skip the conformal field theory stuff...this is unecessary for a first understanding of the subject). If you have no idea what's going on after looking through this, you'll have to go back to quantum field theory, and if that's not understandable, go back further. The quantization of 1d extended objects is the first thing you learn...this requires an appreciation of techniques from QFT since there are a number of quantization methods you learn there: light-cone, canonical, covariant, BRST, etc. All of these methods emphasize different aspects of string quantization.
String theory, as someone mentioned before, is not a set of equations, but a set of ideas, some of which can be encoded and summarized in equations and principles as in the case of the action coupled with the path integral's use of the action (the modified "least action principle").
The perturbative basics (where you learn how to do string perturbation calculations using Feynman sum over histories) are really not the interesting part of string theory, and in research you don't sit around calculating string interaction amplitudes much (unless the quantum corrections have non-trivial and interesting consequences). A lot of interesting stuff is in understanding the spacetimes on which the strings propagate, the relationships among the various string theories, and the non-perturbative feature of string theory.
One of the main tools in all of this is differential geometry. Lie (and discrete) groups are essential. Classical field backgrounds (including spacetime) may have non-trivial topology, like in the case of monopoles and instantons, and so understanding things like cohomology and homotopy are very useful. In doing non-perturbative analysis of field theories arising from string theories, algebraic geometry is useful. And so on...
However, I'll offer this bit of advice: quantum field theories are the low energy theories of string theories, so if you do not have an appreciation for quantum field theory (and after that supersymmetric Yang-Mills and supergravity in particular), then you won't get much out of string theory other than some superficial things like "particles are actually strings vibrating in different ways" and "there are membranes on which string can live", and so on. Not very enlightening. In fact, most of the work in string theory is actually done using field theory techniques. For example, do a search on hep-th for E. Witten for "all years" and look at the titles in the list you get.

 

[matt grime]

Whilst there may be equations one can write down in string theory, equations do not define a theory. For instance, a simple theory is the theory of groups. We have a set of axioms, anything satisfying those axioms is a group, no equations in sight. One may then write down equations about certian kinds of things in the theory (a finite group acting on a set has the orbit stabilizer equation/theorem).

 

[sol2]

Revelations about the Metric

https://www.physicsforums.com/showpost.php?p=186137&postcount=10

Javier,

I appreciate the descriptions of all the maths involved.

From a beginners standpoint the dynamical nature of the Planck brane, would seem to be very alluring, especially, once it is understood what quantum geometry is? The math is transferred to a whole new arena, and now we have encapsulated Reinmannian geometry?

How would the standard model then arise from these perspectives and you have a interesting geometrical perspective? Please feel free to correct.