Exact spectrum of	tachyons in AdS/CFT
Ryo Suzuki(Marie-Curie fellow, Mathematical Institute, Oxford)
March 2014
Based on arXiv:1312.3900 (to appear in JHEP) in collaboration with 	Zoltán Bajnok, Nadav Drukker, Árpád Hegedűs, Raphael Nepomechie, László Palla, Christoph Sieg

Tachyon and instability
Lagrangian density of a complex-scalar QFT
The 1st derivative defines the vacuum, the 2nd the mass

When the mass is pure imaginary,	

the corresponding particle is called a tachyon,	

and the extremum is unstable2Brane-antibrane system
D-brane & D-antibrane (D-D) system in the flat spacetime	

is an example of unstable state in string theory

D-brane & D-antibrane and open strings in between	

in the curved spacetime (AdS5 x S5) are less well-understood3

AdS/CFT correspondence
AdS5 x S5 is the primary example of AdS/CFTSuperstring theory 	

Stack of	N D3-branes
on AdS5 x S5

N=4 4dim SU(N)	

super Yang-Mills4

AdS/CFT correspondence
The energy of an open string in AdS5 x S5 ending on a pair of	“giant-graviton” D-D branes

should be dual to the dimension of a determinant-like operator	

in 4D SU(N) N=4 super Yang-Mills theory

Hope: demonstrate this duality using integrability[Balasubramanian, Berkooz, Naqvi, Strassler (2001)] [Balasubramanian, Huang, Levi, Naqvi (2002)]

5

Integrability Methods
The spectral problem at large N is now “solvable” through	(Asymptotic/Thermodynamic) Bethe Ansatz
We want to solve TBA; i.e. obtain ETBA(l)Example: the exact dimension of Konishi operator
DD

5.4
5.2
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8

D7Ê
Ê6ÊÊ5ÊÊÊ

5101520l

Green: SYM, weak 5-loopÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ ÊÊÊ

1210
8
6

50

100

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Blue: TBA, numerics

l

500

1000

1500

2000

l

Red: String, strong 1-loop[Gromov, Kazakov,Vieira (2009)] [Frolov (2010)] and others6To doWe propose BTBA equations	

(Boundary Thermodynamic Bethe Ansatz)	

and solve them numericallyHowever,7

Integrability vs Instability
Can we apply integrability methods to unstable systems?
Problem 1	

• Even in AdS5 x S5, there are unstable strings with complex energy	whereas the energy is always real in integrability methods
Problem 2	

• TBA energy may diverge in the system with closed string tachyons	

eg. non-susy TsT-transformed AdS5 x S5 vs g-deformed SYMWhat happens in the open string sector?
cf. similar question [Rastelli, Pomoni] 1002.0006

We encounter apparent singularities8Plan of Talk
✓ Introduction	

• Determinants and giant-gravitons	

• BTBA equations and energy bound	• Summary and outlook9Determinants and giant-gravitons

Spherical Maximal Giant Gravitons (SMGG’s)

SMGG’s are classified by the choice:[McGreevy, Susskind, Toumbas (2000)]11Giant graviton is determinant
SMGG’s are dual to determinants

Open strings on the Y=0 brane are dual to det-like operator

A pair of open strings on Y=0 and Ybar=0 should be dual to:[Balasubramanian, Berkooz, Naqvi, Strassler (2001)] [Balasubramanian, Huang, Levi, Naqvi (2002)]
12

GG as boundary condition
GG is a boundary condition for an asymptotic open spin chain

Dilatation operator of N=4 SYM = Spin chain Hamiltonian• Ground state• One-particle state• Two-particle state13

GG as boundary condition
GG is a boundary condition for an asymptotic open spin chain

The Y=0 preserves the symmetry psu(1|2)2 	

which determines the reflection matrix, 	

a solution of the boundary Yang-Baxter relation[Hofman, Maldacena (2007)] [Chen, Correa (2007)]

14

The Yq=0 brane
New reflection amplitudes can be found by rotating RY• N=4 SYM:
• Integrable system:•Rq still preserves integrability	(solutions of boundary Yang-Baxter relation)	

!

• q = p/2 corresponds to the Ybar=0 brane15

YbarY determinant-like operator
A pair of open strings on Y=0 and Ybar=0 should be dual to:

In the ’t Hooft limit, its dimension should take the factorized form
The simplest case is

The energy of a corresponding open string should be16

Caveat!
Actually the representative state is not a dilatation eigenstate

An example of potential mixings

We must use the true eigenstate before computing wrapping diagram• Nevertheless, the wrapping computation seem to be insensitive to the details	

• The classification of dilatation eigenstate at finite N is difficult particularly	

when the length of operator exceeds N(cf. to classify all representations of non-semi-simple Brauer algebra)17

Wrapping diagram
After a lot of tree-level contractions between Y-Y, we obtain

Spacetime structure	(amputated)
this is same as the so-called zig-zag diagram
cf. [Brown, Schnetz] arXiv:1208.1890, [Schnetz] arXiv:1210.5376

18

Wrapping diagram
After a lot of tree-level contractions between Y-Y, we obtain

The result isAgree with the boundary Lüscher formula for L = J >1
cf. [Brown, Schnetz] arXiv:1208.1890, [Schnetz] arXiv:1210.5376

19

BTBA equations 	

and energy boundExact dimension/energy
Begin with the equivalence of Euclidean worldsheet partition functions[Zamolodchikov (1990)]	

[Arutyunov, Frolov (2007)]
RLL

R

In Hamiltonian formalism,
Take the large R limit,

The “mirror” free energy can be computed by the “mirror”	

asymptotic Bethe Ansatz equations in the thermodynamic limit
Thermodynamic Bethe Ansatz equations (TBA)21

Mirror trick with boundary
A simple generalization is to change boundary conditions
RLL

RGround-state BTBA

Exact ground-state energy22

Summary of YbarY energy
YbarY BTBA:(∞ nonlinear integral equations can be solved by numerical iteration)

BTBA energy:
Our BTBA describes D of the determinant-like operator:

all wrapping corrections, negative values

Interestingly, there exists a lower bound for the (B)TBA energy23

YQ(v) at large v
BTBA equation for YQ in the large v limit24

YQ(v) at large v
BTBA equation for YQ in the large v limit

However, the integrals in BTBA energy diverges if YQ(v) ~ 1/v

The BTBA energy cannot be negative and large25

YQ(v) at large Q
BTBA equation for YQ in the large Q limit

However, the sum in BTBA energy diverges if YQ(v) ~ 1/Q

The BTBA energy cannot be negative and large26

Closer look at the boundThe stronger bound is

It is impossible to saturate this lower bound.
Suppose EBTBA = 1 - Lthen BTBA dictates YQ(v) ~ 1/Q
This implies EBTBA diverges, which is a contradiction
A sign of divergences can also be seen at numerical analysis	

(ie. indeed TBA energy seems to “hit” the bound)27

Numerical Results

Solid: BTBA solution, Dashed: Lüscher formula, Dotted: Lower bound28

Numerical Results

Solid: BTBA solution, Dashed: Lüscher formula, Dotted: Lower bound29

Phase diagram

under the assumption that the L = 1 energy diverges at g = 030

Summary and outlook

Summary• Studied the spectrum of determinant-like operators	

dual to open strings ending on giant gravitons	

• Wrapping corrections from N=4 SYM agree with the Lüscher formula	• Proposed and solved BTBA equations for Y=0 & Ybar=0	

• Found the lower-bound for the (B)TBA energy

Future works• Beyond the critical coupling? Compare with string theory?	

• How to compute the dimension of the L=1 state?	

• AdS/CFT for unstable systems?

Thank you for attention	Спасибо за внимание