Upload
arch
View
43
Download
0
Tags:
Embed Size (px)
DESCRIPTION
Finite N Index and Angular Momentum Bound from Gravity. “KEK Theory Workshop 2007” Yu Nakayama , 13 th . 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 - PowerPoint PPT Presentation
Citation preview
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?