1. Correlators of Matrix Models on Homogeneous Spaces Yoshihisa Kitazawa Theory Division KEK, Japan

2. Ｉｎｔｒｏｄｕｃｔｉｏｎ Matrix models are a promising candidate for a nonperturabative formulation of supertsring theory It is possible to construct a compact fuzzy homogeneous space G/H in matrix models by a group theoretic construction They share many common features with de Sitter space (S 4 )

3. In such a construction, we obtain non- commutative (NC) gauge theory on G/H The gauge invariant operators in NC gauge theory are the Wilson lines The correlators of them are different from those in conformal field theory They may shed light on the origins of Newton’s law and the energy density fluctuations in the early universe.

4. Supergravity multiplets We consider plane wave type Wilson loop  r exp (ik  A  ) Tr  k     exp (ik  A  ) Unbroken SUSY: k     with k 2 =0  supergravity multiplets  k  Q   k   Q  Q   k  Q   Q   k 

5. Vertex Operators We can construct them by SUSY Tr exp (ikA) Tr exp (ikA)[A ,A ] Str exp(ikA)[A ,A  ][A,A  ] =∫d  Tr exp(i  kA)[A ,A  ]exp(i(1-  )kA)[A,A  ] Tr exp(ikA)[A ,A ]    Tr exp (ikA) 

6. NC gauge theory on G/H We obtain NC gauge theory on G/H by expanding IIB matrix model as A  = p  +a   where p  denotes a backgroud representing G/H We naturally obtain Wilson line operators of NC gauge theory from the vertex operators in IIB matrix model

7. Dilaton vetex operator V(k)=Tr exp (ikA) Chiral operators in CFT tr Z(x) j Z = A 8 +iA 9  r exp(ikp) + Tr exp(ikp)  ika + Str exp(ikp)(ika) 2  V(k)V(-k)> The two point correlator: <k  k ∫d  Tr exp(i  kp)a  exp(i(1-  )kp)a k  k  ∫d  Tr exp(-i  kp)a  exp(-i(1-  )kp)a  

8. k 4 ∫d  ∫d  ∫d 4 q exp(i(  -  )k^q)/q 2 (q+k) 2 = k 4 ∫d4q (1/k^q) 2  sin2(k^q/2) 1/q 2 (q+k) 2 = k 4 log(l 2 /(k 2 ) 2 )= k 4 ∫ k l/k dq /q e i(1-  kp eikpeikp e ikqp e -i  kp e -ikqp e -i(1-a)kp

9. Graviton vertex operators Str exp(ikA) [A ,A  ][A,A  ] =∫d  Tr exp(  ikA)     ]exp((1-  ) ikA)      ∫d  Tr exp(  ikA) [A ,p  ]exp((1-  ) ikA) [A,p  ] The two point correlator is  d  d  d  q exp(i(  -  )k^q) =  d  q  (1/k^q)^2 sin 2 (k^q/2) ~ l 4 /k 2 In conformal field theory k 4 log(l 2 /k 2 )

10. In order to make sense of highly divergent correlators,we regularize them on compact fuzzy G/H It is because UV cut-off  p and IR cutoff  p are related by the NC relationship  p  p ~ l p + =p 1 +ip 2 S 2 Y j = y j (p + ) j On G/H, the eigen-matrices of the Casimir Operators are the polynomials of p

11. Graviton vertex operators on G/H On G/H it is natural to consider spherical Harmonic than the planewave We insert Y j = y j (A + ) j into the Wilson lines  V(j) =  Str Y j [A ,A  ][A,A  ] = (1/j) y j  i  tr (A + ) j-i [A ,A  ] (A + ) i [A,A  ] ~ (1/j) y j  i  tr (p + ) j-i [a ,p  ] (p + ) i [a,p  ]

12. The two point function =  a,,b,,I,,k (1/j) 2 y j 2 Tr (p + ) j-i Y a (p + ) i Y b Tr Y + b (p - ) k Y + a (p - ) j-k =  i,,,k (1/j) 2 y j 2 Tr (p + ) i (p - ) k Tr (p + ) j - i (p - ) j - k = (1/j 2 )  (y j / y j-i y i ) 2 ~ N(1/j 2 )

13. We make use of  a Y a ij Y a kl =  il  jk

14. CP 2 Example CP 2 can be constructed by 8 matrices p  which are the generators of SU(3) in the (p,0) representation p  2 =p(p+3)/3N=(p+1)(p+2)/2 Around the state where p 8 2 =p 2 /3, (p 4,p 5,p 6,p 7 ) can be regarded as local coordinates Y j = y j (p+) j p + =(p 4 +ip 5 )/√2 [p ,[p , Y j ]] = j(j+2) Y j 0 < j < p+1

15. We determine y i such that Tr Y + Y = 1 Semicalssically we can represent p + =rz 1 /(1+z 1 *z 1 ) r 2 =N Tr (p + ) j (p-) j ~ r 2j+2  d 4 z(z 1 *z 1 ) 2j /(1+z 1 *z 1 +z 2 *z 2 ) 2j+3 ~ Nr 2j 2 (j!) 2 /(2J+2)! = 1/y j 2 = (1/j 2 )  i  (yj / yj-iyi) 2 = N √   /2j 2  j/i(j-i)) 3/2 ~ √  3/2  N/j 2

16. Conclusions The correlation functions on compact fuzzy homogeneous space may shed light on the origins of Newton’s law and origin of CMB fluctuations We have found that the two point functions of graviton vertex operators behave as 1/k 2 in contrast to k 4 log(l 2 /k 2 ) behavior in conformal field theory

17. The two point correlators of higher dimensional operators such as V j =Str Y j [A ,A ][A,A  ][A ,A  ] We can estimate in the same way as ~ N 2 / j 4 It might lead to confinement and mass gap  for these modes