Wednesday, March 30, 2005

Wijnholt's topological holography

It's a sunny Wednesday and Natalia Saulina convinced me to attend a seminar of Martijn Wijnholt, a Harvard alumnus, at the M.I.T. He's been working on it with Leonardo Rastelli, also at Princeton. Their work, a topological simplified version of the AdS/CFT correspondence, relates two theories:
  • the B-model on "H3 x S3" where the hyperboloid "H3" is a Euclidean version of "AdS3" (as a reader reminded me - thanks - the obvious answer why "H3" and not "AdS3" is that one needs an even number of time dimensions (0) for the complex structure to exist, so that we can twist the theory)
  • a subset of the symmetric orbifold CFT, living on the boundary
Martijn motivated this research by the question whether we can check the AdS/CFT correspondence at the level of stringy loops. This has formally been achieved because in their simplified context, the loop amplitudes are uniquely determined by some recursion relations - the same recursion relations on both sides. Explicit checks have not been done.

Unfortunately, Martijn did not really have time to discuss the second theory, namely the boundary CFT, and he focused on the bulk description. However, there were some interesting points already in the bulk description. The most important one is that he argued that the twisted WZW model is equivalent to
  • the bosonic (p,1) minimal model
where "p" is the flux through the "S^3". This equivalence of CFTs has been established kind of exactly although it was a bit confusing what the "(p,1)" model really was - in some sense, it does not have any operators for "q=1" - see e.g. (15.3.16) in Polchinski's book - unless you couple it to the two-dimensional gravity. (A discussion whether the gravity was needed and whether it was topological gravity or not followed - Ami Hanany, Hong Liu, me etc.) I did not quite understand where his operators eventually came from, but Martijn finally compared some operators - tachyonic ones - with the operators describing the "H3 x S3" geometry.

Martijn also realized that Vafa et al. argued that the minimal models were equivalent to a Calabi-Yau space, and he conjectured that "H3 x S3" is T-dual to the Calabi-Yau space where the required T-duality acts on two circles. I am slightly skeptical about this claim.


  1. Word up! Time to get smurfy, yah! This blog is so deep my smurf barely reaches the bottom! You must be a giant smurfin' brain!!!

  2. Dear papa smurf,

    your official name on this blog is "tatka Smoula". Thanks for your understanding.

    All the best

  3. Perhaps they need the euclidean H_3 instead of AdS_3 because they need to define complex structure on the target to perform the twist?

  4. Oh yes, sure, you're right, that was a part of the answer. Sorry for my cryptic remark. The next question of mine was whether it's OK to have an imaginary NS-NS flux over the H3 in topological string theory?