Download Finite N Index

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
Similar classification exists for 4-dimensional SCFT?
 Witten
index
 Central charge (a-theorem, a-maximization)
 Character?
 Index for 4-dimensional SCFT
 Geometrical classification via AdS-CFT?
Witten index for supersymmetric field theory

Witten Index on R4 (or T3 ×R) captures vacuum
structure of the supersymmetric (field) theories
 Bose-Fermi


Only vacuum (H=0) states contribute
Does not depend on
 Many



cancellation
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,
Kinney 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 does not depend on the coupling constant


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?
Finite N Index and
Angular Momentum
Bound
Finite 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)
Path integral on S3 ×R reduces to a matrix integral over
the holonomy (Polyakov loop)

Strategy to determine Seff



Count Δ=0 single letter states
Integrate over U
Or direct path integral
Large N Limit vs Finite N
Explicit integration is possible in the large N limit

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
Maximal Angular Momentum Limit
We propose a new limit, where the matrix integral is feasible

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!!
Index in maximal angular momentum limit
Index is trivial nontrivially! No finite N corrections!

For SU(2), we have
Similarly, they are trivial for SU(N).
 Agrees with gravity (large N limit).
 No finite N corrections

Partition function
Partition function is nontrivial with finite N corrections

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

Maximal Angular
Momentum Limit from
Gravity
Finite N Index and Angular Momentum Bound
from Gravity
Yu Nakayama
Physical meaning of angular momentum bound?
SUGRA admits only massless particle spin up to 2!

No consistent interacting theory with (finitely
many) massless particles spin > 2.
 Gives the maximal angular momentum bound for
dual CFTs.

Highest weight state should satisfy j1 ≦ 1, j2 ≦1.
 Only
decoupled free DOF contributes to the index in
this limit.
 Any CFTs with dual gravity description (e.g. any
Sasaki-Einstein) should satisfy this bound.
 Again there is no general proof from field theoy.
Nontrivial bound!
Contribution from BH?
In high energy regime, black holes may contribute to the index

Asymptotically AdS (extremal = BPS) Black
holes have charge
They do not satisfy maximal angular momentum
bound.
 consistent with our results
 They are not exhaustive?

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 theory
 Nature 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?
