The road to AdS/CFT-Correspondence

[Hymne]

Hello there!
I´m currently writing a bachealor thesis and I need some advice from the pro's.
I have a good basis in QM and classical field theory and have started reading QFT. My question is if you think it is possible for me to reach Ads/CFT in three months, with a very good motivation?
Also what should I focus on if I want to reach this subject?
I have basically skipped the canonical quantization of field approach to QFT and is instead reading paths integral formalism with Srednicki and A. Zee. Is this the best way, or do you advice something else?

Please help me with a 'map' for getting an understanding of Ads/CFT.

Best Regards!
Hymne

 

[atyy]

I'm not a pro, but have found these readable.
http://arxiv.org/abs/0712.0689
http://www-library.desy.de/preparch/desy/proc/proc02-02/Proceedings/pl.6/deboer_pr.pdf
http://arxiv.org/abs/0909.0518

The QFT's energy scale corresponds to a dimension in the gravity dual. The energy scale is set up by Wilsonian renormalization.
http://arxiv.org/abs/0909.0859

QFT has a mathematical similarity with classical statistical physics. I think Wilsonian renormalization is easier to understand starting from the latter subject, where it was used to explain why, near a critical point, different materials behave in ways that depend only on their symmetries. This concept is explained in L7 of http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/

 

[mitchell porter]

I think it's possible but very difficult. You will need to understand basic QFT, quantization of gauge fields, CFTs, and supersymmetry.

That will prepare you to understand, on one side, N=4 super-Yang-Mills theory, which is the boundary CFT in the original version of the correspondence, and on the other side, Type IIB superstring theory with D-branes, which is the theory in the AdS space. String theory is also based on path integrals and CFTs - the "world-sheet" of the string is a 2-dimensional CFT.

But before you get to AdS/CFT, you have to understand that other stuff. Of course you can look ahead to papers on AdS/CFT, and that might help you in deciding where to focus, but the immediate challenge will just be to learn the precursors.

I haven't seen Srednicki and Zee. But for basic QFT, you need to understand how to get "particles" from quantized field modes, and then how to get Feynman diagrams from the perturbation expansion of the S-matrix for weakly interacting fields. You need to be able to switch between the sum-over-histories picture, the state-vector picture of ordinary QM, and the algebra of the perturbation expansion as figured out by Freeman Dyson. I think ideally you'd first do this for a bosonic scalar field theory, to get some intuition and some experience with the formal integrals behind the Feynman diagrams. Then you'd have to plunge into QED, and the new complexities like regularization and renormalization, fermions, and the beginnings of gauge theory. Given your three-month deadline, even with the luckiest decisions about where to focus, I doubt that you have time to become competent with the calculations - there are just too many technical details to learn. So mostly you will have to settle for understanding how it all works.

For gauge theory, the advanced concept that you want is "Faddeev-Popov quantization" or "BRST quantization". Because of the gauge symmetry, formally different histories are physically identical, so you need a way to avoid counting the same physical history twice in the path integral. In BRST you add extra fields, "ghost fields", which vary in a way that counters the redundancy.

Regarding CFTs and renormalization... The modern understanding of renormalization is that you are working with a low-energy approximation to the full field theory, and renormalizability means you can ignore the high-energy behavior even though you don't know what it is. The infinite counterterms which get added to cancel the divergent integrals are just a way of saying "in the real theory, those diagrams get canceled by other diagrams; I don't know the details, but I don't need to know the details". The advanced concept here is "renormalization group flow", and a point of view according to which all QFTs are CFTs with some extra source terms added.

Supersymmetry is less conceptually exotic than all of this - it's just a symmetry connecting fermion fields and boson fields - but it has its own specific algebraic complications, especially "extended supersymmetry" for N>1.

Another semi-technical thing which is relevant for getting to AdS/CFT is the "planar expansion" of QCD for large N (where this N is the number of "colors" in QCD, not the number of "copies of supersymmetry"... this is why the super-N sometimes gets written in a cursive script, to distinguish it). Planar means you only consider Feynman diagrams which can be drawn on the plane without the lines crossing.

When you get closer to learning about CFT, you could try the references in the answers here:

http://physics.stackexchange.com/questions/4743/superconformal-algebra

But first you need to get your basic QFT intuition working.

 

[Hymne]

Thank you so much, both Mitchell and atyy! I always get suprised by the helpfulness of the users at this site.

Indeed seems the road to be quite long and as you say difficult for 3 months..
Anyway do I need a topic related to the knowledge I will build up in QFT. Do you have any nice, but easier, suggestions that usually catch the stundents interest?


Btw, thanks for the arxiv links! I´m sure they will come to be handy the next year or so.

 

[Kevin_Axion]

Here are good lectures: http://www.perimeterscholars.org/253.html.

 

[Hymne]

****ing amazing

 

[Kevin_Axion]

Just note that in the 2010-2011 sessions, nothing below Cosmology is posted because those courses don't start until tomorrow. You can look at the 2009-2010 sessions though, they're up. The only thing is the 2009-2010 sessions don't have CFT, and Beyond the Standard Model.

 

[crackjack]

Susskind's "An Introduction to Black Holes, Information And The String Theory Revolution: The Holographic Universe" might help in the big picture (but not the specific holography of AdS5-CFT4)

 

[mitchell porter]

http://ocw.mit.edu/courses/physics/8-821-string-theory-fall-2008/lecture-notes/" are a very compact, high-speed tour of the basics of the subject. But they presuppose all that other material like ordinary QFT.

At the other extreme, in December a set of 23 papers on all aspects of "integrability" in AdS/CFT were posted to the arxiv, along with an http://arxiv.org/abs/1012.3982" - it has a link and a summary for each paper. So that's a book length review of part of the subject - though it doesn't touch on other related topics, like the twistor renaissance. It is truly amazing how much has come from this one equivalence, and I think Maldacena will get the Nobel for it one day - though not until we learn to apply these perspectives to QFTs in the real world.