Finite N Index and Angular Momentum Bound from Gravity

“KEK Theory Workshop 2007”

Yu Nakayama, 13th. Mar. 2007.

(University of Tokyo)

Based on hep-th/0701208

0. Introduction Classification of (S)CFT

2 dimension CFT (BPZ…) Central charge Character

2 Dimension SCFT Witten index Elliptic genus

Witten index Central charge (a-theorem, a-maximization) Character? Index for 4-dimensional SCFT Geometrical classification via AdS-CFT?

Similar classification exists for 4-dimensional SCFT?

Witten index for supersymmetric field theory

Witten Index on R4 (or T3 ×R) captures vacuum structure of the supersymmetric (field) theories

Bose-Fermi cancellation Only vacuum (H=0) states contribute Does not depend on

Many applications Study on vacuum structure Implication for SUSY breaking Derivation of index theorem (geometry)

The index for 4d SCFT

Consider SCFT on S3 × R. The index (Romelsberger, Kin

ney et al) can be defined by a similar manner.

Properties Only short multiplets (Δ=0) states contribute Does not depend on β No dep on continuous deformation of SCFT The index is unique (KMMR) Captures a lot more information of SCFT!

AdS-CFT @ Finite N

Index can be studied in the strongly coupled regime

AdS/CFT duality Large N limit SUGRA approximation

Excellent agreement N=4 SYM (KMMR) Orbifolds and conifold (Nakayama)

Finite N case? 1/N ~ gs

Quantized string coupling? What is the fundamental degrees of freedom?

Index does not depend on the coupling constant

Finite N Index and Angular Momentum BoundFinite N Index and Angular Momentum Bound from Gravity

Yu Nakayama

Index for N=4 SYM (gYM = 0)

Only states with will contribute.

Contribution to Index

Chiral LH multiplets and LH semi-long multiplets contribute to the Index

Chiral LH multiplet

LH semi-long multiplet

Computation of index from matrix model (AMMPR)

Strategy to determine Seff

Count Δ=0 single letter states Integrate over U Or direct path integral

Path integral on S3 ×R reduces to a matrix integral over the holonomy (Polyakov loop)

Large N Limit vs Finite N

Introduce eigenvalue density evaluate saddle point Saddle point is trivial leading contribution is just

Gaussian fluctuation

Finite N seems difficult. Even for SU(2), we have to evaluate

Explicit integration is possible in the large N limit

Maximal Angular Momentum Limit

We take Only states with will contribute.

Why do we call maximal angular momentum limit? The limit prevents us from taking too large j1 with fixed

j2.

Not protected by any BPS algebra!!

We propose a new limit, where the matrix integral is feasible

Index in maximal angular momentum limit

For SU(2), we have

Similarly, they are trivial for SU(N). Agrees with gravity (large N limit). No finite N corrections

Index is trivial nontrivially! No finite N corrections!

Partition function

For SU(2)

For SU(3)

For SU(∞)

Partition function does have finite N corrections in the maximal angular momentum limit

Does not agree with gravity computation

Partition function is nontrivial with finite N corrections

Maximal Angular Momentum Limit from GravityFinite N Index and Angular Momentum Bound from Gravity

Yu Nakayama

Physical meaning of angular momentum bound?

No consistent interacting theory with (finitely many) massless particles spin > 2. Gives the maximal angular momentum bound for dua

l CFTs.

Highest weight state should satisfy j1 1, j≦ 2 1.≦ Only decoupled free DOF contributes to the index in t

his limit. Any CFTs with dual gravity description (e.g. any Sasa

ki-Einstein) should satisfy this bound. Again there is no general proof from field theoy. Nontr

ivial bound!

SUGRA admits only massless particle spin up to 2!

Contribution from BH?

Asymptotically AdS (extremal = BPS) Black holes have charge

They do not satisfy maximal angular momentum bound.

consistent with our results They are not exhaustive?

In high energy regime, black holes may contribute to the index

Summary and Outlook

Finite N Index and Angular Momentum Bound from Gravity

Yu Nakayama

Summary and Outlook Counting states (index) for finite N gauge

theory is of great significance.Basic building blocks for nonperturbative string

theoryNature of quantum gravity

Difficult problem in general. Maximal Angular Momentum Limit was

proposed. No finite N corrections for index in this limit. Finite N corrections for full index?