An exact AdS/CFT duality
Matthias Gaberdiel ETH Zürich
DIAS 8 July 2020
Based mainly on work with Rajesh Gopakumar and Lorenz Eberhardt
Much recent progress in string theory has been related to AdS/CFT duality [Maldacena ’97, ...]
superstrings on SU(N) super Yang-Mills = AdS S5 theory in 4 dimensions 5 ⇥ 4d non-abelian gauge theory similar to that appearing in the standard model of particle physics. Relation of parameters
The relation between the parameters of the two theories is
4 4 R 2 R 2 = N gstring = g = g N = l YM l YM Pl ⇥ s ⇥
AdS radius in AdS radius in Planck units string units ‘t Hooft parameter Strong weak duality
For example, in the large N limit of gauge theory at large ‘t Hooft coupling
4 4 R 2 R 2 = N gstring = g = g N = l YM l YM Pl ⇥ s ⇥ large small large supergravity (point particle) approximation is good for AdS description. This is interesting since it gives insights into strongly coupled gauge theories using supergravity methods. Strategy for a proof
There is very good evidence that at least the original duality is correct. However, we have not yet succeeded in proving it (and we therefore also do not know to which extent it generalises).
In order to make progress in this direction, we will consider
‣ Tensionless regime
‣ Lower dimensional version (AdS3) Weakly coupled gauge theory
The tensionless regime arises in another corner of parameter space where the gauge theory is weakly coupled
4 4 R 2 R 2 = N gstring = g = g N = l YM l YM Pl ⇥ s ⇥
large small
[Sundborg ’01] [Witten ’01] l `tensionless strings’ s ⇥ [Sezgin,Sundell ’01] Tensionless limit
In tensionless limit all string excitations become massless:
mass mass
spin spin
leading Regge trajectory Higher spin theory
Resulting theory has an infinite number of massless higher spin fields, which generate a very large gauge symmetry.
effective description in terms of Vasiliev Higher Spin Theory.
maximally symmetric/unbroken phase of string theory Higher spin CFT duality
Proposal for higher spin subsector: [Klebanov-Polyakov ’02] [Sezgin-Sundell ’02]
higher spin theory 3d O(N) vector model on AdS4 in large N limit
Many precision tests of this proposal have since been performed, in particular by [Giombi, Yin ’09-’10] [Maldacena, Zhiboedov ’11-’12], … Lower dimensions
The other key ingredient is to consider the lower-dimensional case, in particular, string theory on AdS3.
The advantage of going to the 3d case is that
‣ Solvable world-sheet theory for strings on AdS3 exists [sl(2,R) WZW model] [Maldacena, (Son), Ooguri ’00 - ’01]
‣ Much better control over 2d CFTs AdS3
It has long been suspected that the CFT dual of string theory on 3 4
AdS
Sym ( 4) ( 4)N /S
However, it was not known what precise string background is being described by the symmetric orbifold theory itself. see however [Larsen, Martinec ’99]
Conversely, it was not known what the precise CFT dual of the explicitly solvable world-sheet theory for strings in terms of an sl(2,R) WZW model is. [Maldacena, (Son), Ooguri ’00 & ‘01] AdS3
In fact, the only consensus was that the actual symmetric orbifold theory cannot be dual to the WZW model… AdS3
Long andIn fact, short the strings only consensus was that the actual symmetric orbifold theory cannot be dual to the WZW model…
Short string solution Long string solution The basic reason for this is that the WZW model describes the t background with pure NS-NS flux, which is known to have long string solutions.
I These long strings live close to the boundary[Seiberg, and Witten give ’99], rise to[Maldacena, a Ooguri ’00] continuum of excitations that are not present in the symmetric orbifold theory. So the spectrum cannot possibly match.
13 AdS3
Long andIn fact, short the strings only consensus was that the actual symmetric orbifold theory cannot be dual to the WZW model…
Short string solution Long string solution These long strings give rise to a continuum of t excitations that are not present in the actual symmetric orbifold theory.
I These long strings live close to the boundary[Seiberg, and Witten give ’99], rise to[Maldacena, a Ooguri ’00] continuum of excitations that are not present in the symmetric orbifold theory. So the spectrum cannot possibly match.
13 Joining forces…
Use insights from the HS/CFT duality to make progress with identifying the worldsheet description of symmetric orbifold. 3d version of Klebanov-Polyakov duality is:
AdS3: 2d CFT: higher spin theory N,k minimal models with a complex Win large N ‘t Hooft limit scalar of mass M with coupling
[MRG,Gopakumar ’10] Susy version
Just as in original Klebanov-Polyakov duality, this is purely bosonic. However, in order to establish relation to string theory, one can consider susy generalisation.
For susy case, the 2d CFT is the so-called Wolf space coset su(N + 2) so(4N + 4) k 1 u(1) , su(N) u(1)
In the large k limit, the Wolf space coset becomes a subsector of [MRG, Gopakumar ’14]
3 4 AdS3 S T hs theory based on ⇥ ⇥ string theory shs2[ ]