in collaboration with K.Furuta, M.Hanada and Y.Kimura. Hikaru Kawai (Kyoto Univ.) There are several types of matrix models, but here for the sake of

Transcription

1 Curved space-times and degrees of freedom in matrix models 1 Based on hep-th/ , hep-th/ and hep-th/ in collaboration with K.Furuta, M.Hanada and Y.Kimura. Hikaru Kawai (Kyoto Univ.) There are several types of matrix models, but here for the sake of concreteness we consider IIB matrix model. Abstract IIB matrix model is a candidate for the constructive definition of string theory. It is nothing but the large-n reduced model of 10D super Yang-Mills theory, 1 1 µ ν 2 1 µ µ S= Tr( [ A, A ] + ΨΓ [ A, Ψ]) 2 g 4 2 µ A ( = 1 ~ 10 µ ), Ψ (10D Majorana-Weyl) : N N N hermitian This theory seems to describe fluctuations around a flat space-time. The basic question is whether it can describe curved space-times, and it really contains invariance of the general relativity, diffeomophism and local Lorentz invariance, or not. Here, we will show that indeed it is the case, if we introduce a new interpretation for the matrices.

2 2 Matrix model contains space time. First of all let us see that the matrix model contains space-time in its degrees of freedom. IIB matrix model is formally obtained from 10D SU(N) super Yang Mills theory by a dimensional reduction to zero dimensions. Therefore, if we think naively, this reduction severely truncates the degrees of freedom. However it is not the case if N is infinitely large. In fact in this case, the matrix model contains the original 10D model. 10D super YM IIB matrix model 10D super YM dimensional reduction to 0D N

3 The reason is as follows: 3 The dynamical variablesa µ s are N N matrices, but let us treat them in a little abstract manner. That is, we regard N N matrices as linear transformations on some linear space V : N A µ EndV ( ), V C. So, if i N is infinitely large, V is an infinite dimensional vector space, and we can take various expressions for V. First let us assume V is the space of n-component complex scalar fields on 10D space-time time: i 10 n V = { ϕ ( x) : R C }. Then a matrix, a linear transformation on V, is nothing i j but an integration kernel, or a bilocal field κ ( x, y) : T EndV ( ), n i 10 i j j ( Tϕ) ( x) = d y κ ( x, y) ϕ ( y) j= 1 = c ( x) ϕ ( x) + c ( x) ϕ ( x) + c ( x) ϕ ( x) + i j j µ i j j µν ij j (0) (1) µ (2) µ ν

4 At least formally, such bilocal field can be expanded as a differential operator of infinite order whose coefficients are n n matrices. In particular as a special value ofa µ, we can take the covariant derivative i j i j ( δ ) Aµ = idµ = i µ + iaµ ( x) EndV ( ), which means the whole space of gauge field configurations is embedded in the space of matrices EndV ( ). 4

5 Thus we have seen that the dynamical degrees of freedom of the original 10D theory are completely included in the matrix model if we take the large-n limit: IIB matrix model SU(n) 10D super YM. ( N ) Furthermore in this embedding the local gauge symmetry is realized as a part of the SU(N) symmetry of the matrix model: δa = i λ, A, λ End( V ) : 0-th order differential operator µ µ. 5

6 6 Large arge-n reduction In this sense, IIB matrix model has more dynamical degrees of freedom than 10D super Yang-Mills theory. However, it turns out that the difference between them is not so large. This fact is known as the large-n reduction. For example, it is known that (1) If we quench the diagonal elements of A µ, Aµ = Pµ + A ɶ µ, P : diagonal and fixed, Aɶ : offdiagonal, µ µ the matrix model is equivalent to the large-n Yang-Mills theory. (2) If we expand A µ around a noncommutative background A = pˆ + a ( xˆ ), µ µ µ p ˆµ, pˆ ν = ibµν, µ µν νλ λ xˆ = C pˆ, B C = δ, and constrain the fluctuation that ν µν µ aµ ( x) < ( x ), the matrix model is equivalent to Yang-Mills theory in a noncommutative space.

7 More presicely, 7 (1) quenched reduced model Parisi (1982) Gross, Kitazawa Bhanot, Heller, Neuberger Das, Wadia The path integral for the off-diagonal elements of A µ can be expressed by such Feynman diagrams as j k i And if the eigenvalues p µ i of. of P µ are uniformly distributed in the momentum space, the summation over the indices becomes the integration over loop momenta in the large-n limit: i, j, k ( i j, j k, k i) ( pi pj)( pj pk)( pi pk) 1 d k d k k k k k ( 1+ 2) Therefore the quenched reduced model becomes equivalent to the original 10D field theory in the large-n limit.

8 8 It is useful to compare this result with the expression in terms of the differential operators. If we express A µ as a differential operator (0) i j (1) ν i j (2) ν λ i j Aµ cµ ( x) cµ ( x) ν cµ ( x) ν λ = + + +, the background of the quenched reduced model is nothing but (0) Aµ i µ =. So, in the classical level, fluctuations from this background consist of infinitely many fields of various higher spins. Because diagonal elements are nothing but polynomials of i µ, the condition of quenching suppresses the zero mode of each field. However we still have very large degrees of freedom as far as we consider classical dynamics. On the other hand the quenched reduced model tells us that these degrees of freedom are combined to one vector field,, if we treat them quantum mechanically.

9 9 (2) twisted reduced model Gonzalez-Arroyo, Okawa (1983) Gonzalez-Arroyo, Korthals Altes Rediscovered in the context of NCFT. We pick up a classical solution of the equation of motion Aµ, Aµ, A ν = 0 that is given by the CCR with 10 / 2= 5 degrees of freedom, = ˆ 1 (0) Aµ pµ n p ˆ, ˆ µ pν = ibµν ( Bµν R ), 1 n : n n unit matrix. and expand the dynamical variables around it: A ˆ µ = Aµ + aµ, ψ = 0+ ψ. (0) ˆ Next we introduce the following correspondence between operators and functions: d k d k oˆ = oɶ ( k)exp( ik xˆ µ ) o( x) o( k)exp( ik x µ ) 10 µ = 10 µ (2 π ) ɶ (2 π ) B C = δ, xˆ = C pˆ. νλ λ µ µν µν µ ν (ô is the Weyl ordering ring of o( x ˆ).)

10 Then we can show the following mapping rules: 10 p ˆ, ˆ µ o i µ o, oo ˆ ˆ o * o, ( ) Tr o detb = 10 ˆ d xo( x) 5 ( 2π) Using them, we can rewrite the matrix model action as that of a field theory on the noncommutative space-time time: det B 1 1 tr, ψ * µ SIIB = d x F D 5 µν + ψγ µ ( π) Again we have seen that the matrix model is almost equivalent to the 10D field theory.

11 11 matrix model contains gravity So far, we have seen that IIB matrix model is slightly larger than the 10D super Yang-Mills theory. On the other hand it is known that IIB matrix model also contains gravity, if we consider a different kind of background. Ishibashi, Kitazawa, Tsuchiya and HK (1996) We consider the linear space V as before: i 10 n V = { ϕ ( x) : R C }, but this time we consider a background (0) A µ that is diagonal in that is diagonal in the x-space. In terms of integration kernel,, it is expressed as (0) µ A x (10) ( x y) ij µ = δ δ. Or in term of differential operator A x µ =. (0) ij µ δ

12 If we expand A µ around it A = A + A ɶ (0) µ µ µ, 12 the one-loop effective Lagrangian contains terms that correspond to the exchange of graviton and dilaton: 1 ( x y) Seff= d xd y const f x f x f y f y 8 µλ νλ µρ νρ { tr( ( ) ( )) tr( ( ) ( )) const tr( f ( x) f ( x)) tr( f ( y) f ( y)) + } µν µν λρ λρ x y

13 Question: 13 We have seen that Yang-Mills theory is embedded in IIB matrix model, and in this embedding the local gauge symmetry is a part of the SU(N) symmetry of the matrix model. Furthermore we have seen that IIB matrix model contains gravity. Can we find a good way of embedding gravity in IIB matrix model, in such a way that diffeomorphism and local Lorentz invariance become a part of the SU(N) symmetry of the matrix model?

14 More concretely, suppose we have a D-dimensional curved space M and the covariant derivative a ( a= 1 ~ D) on it. Find (i) a good space V and (ii) a good object ( a) which is equivalent to such that each component of linear transformation on V. Difficulty of, ( ) ( a a ) a 14 = is expressed as a The covariant derivative a ( a= 1 ~ D) maps a scalar field to a vector field, and a vector field to a tensor field, and so on. Therefore, it can not simply be regarded as a linear transformation on some space such as A = i. a The difficulty may become clearer, if we compare the product a a b and and AA a b. In a b, the spin connection contained in a mixes the index b. On the other hand AAsimply a b means the product of Aa and Aa = i a can not hold. A b. Therefore the equation like

15 15 Answer: Suppose = I is a D I M U is a D-dim Riemannian manifold with a spin structure,, and we have a transition function on each overlapping region ( ) ( ), tij x spin D x UI UJ. Here we assume all indices are Lorentz indices, and in particular, the covariant derivative is expressed as bc ( µ ωµ O ) µ a = ea + bc, where Obc is the Lorentz generator. is the Lorentz generator.

16 (i) V = C ( E prin ) (i) First we consider the principal spin( D ) bundle 16 E prin on on M associated with the spin structure.. In other words, we consider a direct product UI Spin( D) consider a direct product and glue them together by the following rule: For x U U, or I J for each patch U I, ( x, g ) U Spin( D) ( x, g ) U Spin( D) I I J J g = t ( x) g I IJ J Then we take the space of functions on E prin as as V. In other words, we regard matrices as bilocal fields or differential operators on E prin.

17 b 1 µ (ii) ( ) ( cd = R g e ) ˆ ) µ + ωµ c (ii) Le ( ) ( ) ( xo End( C ( E )) a a b d prin R g be the rep. matrix of the vector rep. of Spin( D ), b Let a ( ) and O ˆab be the be the rep. matrix of the vector rep. of be the infinitesimal left action on the function space on Spin( D ) : ( O ˆab is the derivative along the fiber.) ab ˆ ab ε Oabϕ ( g) = ϕ((1 ε τab ) g) ϕ( g). Then we define a differential operator on 1 µ ( ) ( cd ( xo ) ˆ ) µ ωµ E prin by b ( a) = R( a) g eb + cd. by 17

18 We can show that each component of ( a),( a= 1..10) is a globally defined differential operator on E prin, and thus a linear transformation on V. Therefore it can be expressed by a 18 matrix. (Proof Proof) [ I] b 1 [ I] ( ) = R g ( a) ( a) I b [ J] 1 c [ J] ( I ) ( ( )) 1 [ J] ( I ( )) = R g R t x b ( a) b IJ c = R g t x c ( a) IJ c = R t x g ( ( ) ) 1 1 [ J] c ( a) IJ I c 1 [ J] ( ) = R g = c ( a) J c ( a) Furthermore we can show that each component of ( a) is hermitian for the natural measure on E prin :. * D * ( f, h) = f h= d x g dg f ( x, g) h( x, g) E prin

19 19 Example S 2 with homogeneous metric i spin(2) = { e θ ;0 θ < 2 π} S Eprin S1bundleoverS 2 S3 (Hopf bundle) V = C ( E ) = { ϕ( z, θ ); S C } prin 3 1 z: the stereographic coordinate of S 2 = ( + ) = ( ) i {(1 + ) + θ }, 2 i {(1 + ) θ}, 2 2iθ e zz z z 2iθ e zz z z i + = (1 + zz ) z+ 2 i = (1 + zz ) z 2 where ±= 1± i2. Each of ( + ) and ( ) is a globally defined differential zo zo operator on E prin.

20 In this manner, S 2 is realized in terms of two matrices. This should be distinguished from the ordinary fuzzy sphere, which is obtained by embedding to the space of three matrices. 20 Similarly, any Riemannian manifold with dimension ion less than or equal to D can be coded in the space of D matrices. CY 3 T 10 space of 10 matrices S 10

21 What is the space V? 21 V = ( V V ), V : space of a field with rep. r. ( ) r: rep. of Spin(D) dr r r r ( dim. of r ) Since an element of V is a function on on M, it gives a function Spin( D) C. E prin, at each point at each point In general, the space of functions on a group Gforms a special representation called the regular representation, which is isomorphic to v ( v v ) reg r r r:rep. of G scalar, spinor, vector, etc. etc. where v r is a representation ofg, and d r is its dimension. d r,

22 A function G C can be expanded as f g i, j i, j ( ) = c< r> R< r> ( g) ri,, j i, j where R< r> ( g) is the rep. matrix for r., 22 The action of an element h of G on f is assumed to be f ( h g) = c R ( h g) = = 1 i, j i, j 1 < r> < r> ri,, j ri,, j ri,, j Therefore, c R ( h ) R ( g) i, j ik, 1 k, j < r> < r> < r> c R ( h ) R ( g). k, j ki, 1 i, j < r> < r> < r> f g f g f h g c c R h c 1 i, j i, j ki, 1 k, j ( ) ( ) = ( ) < r> < r> = < r> ( ) < r>.

23 The regular rep. has the following remarkable property: 23 v v v v, for any r. reg r reg reg More explicitly, this isomorphism is given by Then ( i) f g f g v v f g = R g f g i ( i) ij 1 j ( ) reg r ( ) < r> ( ) ( ). ( i f ) ( g ) transforms under h G as ( ) R g R h f h g R g h f h g f h g ij ( ) 1 ( jk k ) ( ) ( ik ) ( k i r r r ) ( ) ( < > < > = < > = ). Therefore we have a : V V T V V, where T is the tangent bundle, and the combined map is given by 1 µ ( ) ( cd ( xo ) ˆ ) µ ωµ b ( a) = R( a) g eb + cd.

24 24 New interpretation of IIB matrix model We now regard the matrices in IIB matrix model as linear transformations on C ( E prin ). Here we consider the classical EOM derived from the action 1 S = Tr( [ A, ] 2 ) a Ab + fermions 4. If we set the fermions to be zero, it becomes [ ] Aa Aa, Ab = 0. Now we can impose the following Ansatz A i a = ( a), because ( a) is a well defined linear transformation,, and we have, = 0 ( a) ( a) ( b).

25 Let s rewrite this equation in terms of the ordinary ry covariant derivative a. Formula 1 ( ) d 1 R ( g ) 1 ( ) d 1 ( ) = R g c ( a) ( b) ( a) c ( b) d = R g R g c ( a) ( b) c d 25 In the last expression, the Lorentz generator in acts c on the index d of d. Using this, we have 0 =,, ( a) ( a) ( b) [ ] 0 = a,, cd cd ca =, R O = ( R ) O R cd R = 0, R = 0 R = 0. a b a ab cd a ab cd ab c a ab ab ab

26 The Einstein equation follows from the EOM of IIB matrix model. If we start with IIB action with a mass term 2 1 ( 2 [ ] ) m 2 S = Tr Aa, Ab + Tr( Aa ) + fermions 4 2, we have the Einstein equation with a cosmological constant 2 Rab m δab =. 26

27 Diffeomorphism and local Lorentz invariance 27 We now show that the symmetries of the general relativity are realized as parts of the SU(N) symmetry of the matrix model. In the matrix model the infinitesimal SU(N) symmetry is given by [ ] δ A = Λ, A, Λ End( V ). a a If we interpret V as as C ( E prin ), we can take variousλ from End( C ( E prin )) (1) diffeomorphism. 1 { ( a) (, ), ( ) } ( ( )) a λ x g a End C Eprin λ ( x ) a Λ= 2 λ 1 ( ) ( x, g) = R g λ ( x) b ( a) ( a) b [, A ] δ Aa = Λ a correctly reproduces the diffeomorphism. (2) local Lorentz correctly reproduces the diffeomorphism. ab ( ) ˆ Λ= λ xo End ( C ( E )) ab [, ] δaa = Λ Aa is prin is the local Lorentz transformation.

28 28 Summary Any D-dimensional manifold can be embedded in D matrices. Accordingly, the matrices in IIB matrix model can be interpreted as differential operators on the principal bundle on any manifold of less than or equal to 10 dimensions. The classical EOM of IIB matrix model gives the Einstein equation. So far we have analyzed classical EOM. However, as we have seen for the flat space, it is expected that the degrees of freedom become much smaller in the quantized level. It is important to consider what remains in the quantized level. hep-th/ , hep-th/ In order to implement SUSY, we can consider super manifold instead of the ordinary manifold, and consider the matrices as differential operators on the associated principal bundle. Then V becomes a super vector space and the matrices should be regarded as super matrices, but the form of the action of IIB matrix model seems to work without any modification.