Free Yang-Mills 近似を用いた AdS/CFT 対応の解析 (An analysis of AdS/CFT using the free-field approximation) 高柳 匡 ( 京大理学部 ) Tadashi Takayanagi, Kyoto U. based on hep-th/ [JHEP 0701 (2007) 090] hep-th/ with 西岡 辰磨 ( Tatsuma Nishioka) KEK 理論研究会 2007

① Introduction AdS/CFT has been played a crucial role to understand the non-perturbative properties of gauge and gravity theory. This is owing to its S-duality nature: IIB String on AdS5×S5 4D N=4 SU(N) YM near horizon of N D3-branes SCFT R: AdS radius λ=Ng YM 2 [J.Maldacena 98’]

In the large radius limit, we find IIB SUGRA Strongly coupled N=4 SYM Easy ! An interesting theory In the small radius limit, we obtain IIB string Free N=4 SYM at the zero radius What is this? (Unknown…) Very easy !

Setup: AdS/CFT correspondence in Poincare Coordinate

To check the AdS/CFT duality, we need to compare the supergravity (or semi-classical string) results with those of strongly coupled Yang-Mills. Recently this has been achieved for several quantities in N=4 SYM by using the spin-chain descriptions. However, if we want to consider AdS/CFT(QFT) for more general examples without conformal invariance and SUSY, we cannot resort such a `integrable’ method.

Now we would like to assume the zero-th order approximation and to compare the following two theories. IIB SUGRA Free Yang-Mills in various backgrounds Naively, this crude approximation does not seem to work. However, there are several physical quantities which we can compare semi-quantitatively between these two theories.

A famous example will be the (thermodynamical) entropy. We can compute the entropy in free N=4 SYM by counting the number of bosons and fermions In the gravity dual description, it is given by the Bekenstein-Hawking entropy of AdS Schwarzschild BH

Therefore we find the semi-quantitative agreement [Gubser-Klebanov-Peet 96’] up to 30 %. In this talk we would like to check the AdS/CFT in various backgrounds by confirming such a semi-quantitative agreement. Our Examples: AdS bubbles, Sasaki-Einstein Mfds.

② AdS bubbles and Closed String Tachyon (2-1) AdS bubbles Compactify a space coordinate x i in AdS space and impose the anti-periodic boundary condition for fermions. Closed string tachyons in IR region Anti-periodic Nishioka-Takayanagi hep-th/

The end point of closed string tachyon condensation is conjectured to be the AdS bubbles (AdS solitons). [Horowitz-Silverstein 06’] Closed string tachyon Capped off ! Anti-Periodic

Explicit metric AdS Schwarz-Schild ( ⇔ Finite temperature SYM) AdS Bubble (AdS Soliton): Double Wick Rotation χ ｒ r=r 0

The dual gauge theory is the 4D N=4 SYM compactified on a circle with the anti-periodic boundary condition for all fermions (i.e. thermal circle). The supersymmetry is completely broken. Also there exists a mass gap in the IR. ~ a confinement from the viewpoint of 3D SYM. [Witten 98’] Since this system is at zero temperature, the thermal entropy is zero. So we compare the following quantities: (i) Casimiar Energy = ADM energy [Horowitz-Myers 98] (ii) Entanglement Entropy = Area of minimal surface [Ryu-Takayanagi 06]

(i) Casimir Eenergy Free SYM side Gravity side Thus we again find Summing over KK modes

Free field computation of Casimir Energy Ex. a massless scalar

(ii) Entanglement Entropy Free SYM side We divide the space into two parts A and B. Then the the total Hilbert space becomes factorized We define the reduced density matrix for A by taking trace over the Hilbert space of B. Now the entanglement entropy is defined by the von Neumann entropy A B

In the simplest case of the division by a straight line, we can calculate the entropy exactly. Consider the quantity This is the same as the partition function on We analytically continue the integer n to n=1/N<1. Then we get the orbifold A B

The entanglement entropy is found to be Note: this is essentially the same as the open string vacuum amplitude for a fractional D2-brane on C/Z N.

After the summation of KK modes, we obtain We find from the area law formula Thus we can conclude that Gravity side

Holographic Computation (1) Divide the space N is into A and B. (2) Extend their boundary to the entire AdS space. This defines a d dimensional surface. (3)Pick up a minimal area surface and call this. (4)The E.E. is given by naively applying the Bekenstein-Hawking formula as if were an event horizon. [Ryu-Takayanagi 06’] Explicit proof via GKP-Witten relation: Fursaev 06’

In our case, the minimal surface looks like r r=r 0

It is clear that the entropy decreases compared with the supersymmetric AdS5 background. Our conjecture The entanglement entropy always decreases under the closed string tachyon condensation. Note: We neglect the radiations produced during the tachyon condensation as we do so for the Sen’s conjecture about the open string tachyon condensation.

(2-2) Twisted AdS bubbles We would like to generalize the above discussions to the twisted AdS bubbles, dual to the N=4 4D Yang-Mills with twisted boundary conditions: Supersymmetries are broken except ζ ＝ 1. The dual metric can be obtained from the double Wick rotation of the rotating 3-brane solution. When ζ ＝ 0, the background becomes the AdS bubble. At ζ ＝ 1, it coincides with the supersymmetric AdS5.

The metric of the twisted AdS bubble

The smoothness of the metric at the tip of the bubble requires the twisted periodicity We will again have a closed string tachyon from the string wound around the twisted circle. Closed string tachyon condensation on a twisted circle (or Melvin background). [Review: Headrick-Minwalla-Takayanagi 04’]

The result of Casimir Energy=ADM mass Energy Twist parameter SUSY Free Yang-Mills AdS gravity Cf. Y.Hikida hep-th/ : C 2 /Z N

The result of the entanglement entropy This is a new quantitative evidence of AdS/CFT in a slightly susy breaking background. Twist parameter Entropy Free Yang-Mills AdS side (Strongly coupled YM) Supersymmetric Point

③ Free Fields vs. Sasaki-Einstein As we have seen, the free Yang-Mills approximation of the entropy and Casimir energy to the SYM semi-qualitatively agrees with the AdS gravity results. This suggests that the degree of freedom of free Yang-Mills is not so different from that of strongly coupled Yang-Mills. [ A comment] This semi-quantitative agreements are very non-trivial and may be special to QFTs which have their AdS duals.

We would like to test this speculation for infinitely many examples of N=1 SCFTs which are dual to toric Sasaki- Einstein manifolds X 5. [Examples of X 5 ] (i) [Klebanov-Witten 98’] (ii) [Gauntlett-Martelli-Sparks-Waldram 04’] (iii) [Cvetic-Lu-Page-Pope 05’] Below we will assume X 5 is a toric manifold. Infinitely Many examples

We compare the thermal entropy in free Yang-Mills with the one in the strongly coupled YM. The latter can be found as the black hole entropy where a is the central charge of 4D N=1 SCFT. It is related to the volume of the dual Sasaki-Einstein mfd via

Now we define the ratio of the entropy This index f can be found purely from the toric data of the (CY3 cone over) Sasaki-Einstein mfds, employing the Z-minimization [Martelli-Sparks-Yau] method. Properties (1) f=1 for any orbifolds of C 3 (2) f remains the same after orbifolding X 5 X 5 /Z n

Explicit Example (1):Ypq X=q/p f(x) 8/9 < f < T 1,1 S 5 /Z 2

Explicit Example (2):Lpqr x=p/q y=r/q f(x,y) 8/9 < f < Note the constraints: 0<x<y<1 y<(x+1)/2

Other examples (3) Xpq: [Hanany-Kazakopoulos-Wecht 05’] (4) Zpq: [Oota-Yasui 06’] (5) Symmetric Pentagon (6)Regular Polygon 8/9 < f < /9 < f < /9 < f < /9 < f <

Del Pezzo Surfaces etc.

In this way, the ratio f takes values within a narrow range. In other words, we can say that the N=1 SCFTs (quiver gauge theories) which have AdS duals are rather special kinds of super Yang-Mills: The degrees of freedom do not depend on the coupling constant so much. Notice also that 4f/3 is always greater than 1, which means the strongly coupled SYM has a smaller degree of freedom.

④ Conclusions and Discussions In various example, we have seen that several physical quantities agree semi-quantitatively between free Yang- Mills and IIB supergravity. It has not been known when a given CFT has its AdS dual. Therefore it would be useful to examine many examples even statistically and see if there exist any common physical properties. It would be very nice if the narrow range of the index f offers us a criterion of the existence of AdS dual.