Correlators of Matrix Models on Homogeneous Spaces

Correlators of Matrix Modelson Homogeneous Spaces Yoshihisa Kitazawa Theory Division KEK, Japan

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 withde Sitter space (S4)

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.

Supergravity multiplets We consider plane wave type Wilson loop Tr exp (ikmAm) Tr ekmGmy exp (ikmAm ) Unbroken SUSY: kmGme = 0 with k2=0 IIB supergravity multiplets |k>, Qa |k>,Qa Qb |k>, ..... , Qa1..... Qa8 |k>,

Vertex Operators We can construct them by SUSY Tr exp (ikA) Tr exp (ikA)y Tr exp (ikA)[Am,An] Tr exp(ikA)[Am,An]Gmy Str exp(ikA)[Am,Ar][An,Ar] =∫dbTr exp(ibkA)[Am,Ar]exp(i(1-b)kA)[An,Ar]

NC gauge theory on G/H We obtain NC gauge theory on G/H by expanding IIB matrix model as Am= pm+am where pm denotes a backgroud representing G/H We naturally obtain Wilson line operators of NC gauge theory from the vertex operators in IIB matrix model

Dilaton vetex operator Chiral operators in CFT tr Z(x) j Z = A8+iA9 V(k)=Tr exp (ikA) Tr exp(ikp) + Tr exp(ikp)ika + Str exp(ikp)(ika)2 The two point correlator: <V(k)V(-k)> <kmkn∫dbTr exp(ibkp)amexp(i(1-b)kp)an krks∫daTr exp(-iakp)arexp(-i(1-a)kp)as >

e-iakp e-i(1-a)kp ei(1-b)kp eibkp eikqp e-ikqp k4∫da ∫db ∫d4q exp(i(a-b)k^q)/q2(q+k)2 = k4∫d4q (1/k^q)2sin2(k^q/2) 1/q2(q+k)2 = k4∫kl/k dq /q = k4 log(l2/(k2)2)

Graviton vertex operators Strexp(ikA) [Am,Ar][An,Ar] =∫db Tr exp(bikA) [Am,Ar ]exp((1-b) ikA) [An,Ar] ~∫db Tr exp(bikA) [Am,pr ]exp((1-b) ikA) [An,pr] The two point correlator is ∫ da ∫db ∫d4q exp(i(a-b)k^q) = ∫d4q(1/k^q)^2 sin2(k^q/2) ~ l4/k2 In conformal field theory k4log(l2/k2)

In order to make sense of highly divergent correlators,we regularize them on compact fuzzy G/H It is because UV cut-off Dp and IR cutoff dp are related by the NC relationship Dpdp ~ l On G/H, the eigen-matrices of the Casimir Operators are the polynomials of p p+=p1+ip2 S2 Yj= yj(p+)j

Graviton vertex operators on G/H • On G/H it is natural to consider spherical Harmonic than the planewave • We insert Y j= yj(A+)jinto the Wilson lines V(j) =Str Yj[Am,Ar][An,Ar] = (1/j) yjSitr (A+)j-i [Am,Ar] (A+)i [An,Ar] ~ (1/j) yjSitr (p+)j-i [am,pr] (p+)i [an,pr]

The two point function < V(j) V(j)*> = Sa,,b,,I,,k (1/j)2yj2 Tr (p+)j-i Y a (p+)i Y b Tr Y+b (p-)k Y+a (p-)j-k =Si,,,k(1/j)2yj2 Tr (p+)i (p-)k Tr (p+)j-i (p-)j-k = (1/j2)S (yj /yj-iyi)2 ~ N(1/j2)

Wemake use of SaYaijYakl = dildjk

CP2 can be constructed by 8 matrices pm which are the generators of SU(3) in the (p,0) representation CP2 Example pm2=p(p+3)/3 N=(p+1)(p+2)/2 Around the state where p82=p2/3, (p4,p5,p6,p7) can be regarded as local coordinates Yj= yj(p+)j p+=(p4+ip5)/√2 [pm,[pm, Yj]] = j(j+2)Yj 0 < j < p+1

We determine yi such that Tr Y+Y = 1 Semicalssically we can represent p+=rz1/(1+z1*z1) r2=N Tr (p+)j(p-)j ~ r2j+2∫d4z(z1*z1)2j/(1+z1*z1+z2*z2)2j+3 ~ Nr2j2 (j!)2/(2J+2)! = 1/yj2 <V(j) V(j)*> = (1/j2) Si(yj / yj-iyi)2 = N √ p/2j2S (j/i(j-i))3/2 ~ √pz(3/2) N/j2

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/k2 in contrast to k4log(l2/k2) behavior in conformal field theory

The two point correlators of higher dimensional operators such as Vj=Str Yj [Am,An][An,Ar][Ar,As] We can estimate in the same way as <VjVj*> ~ N2 / j4 It might lead to confinement and mass gapfor these modes

Correlators of Matrix Models on Homogeneous Spaces