Perturbative Vacua from IIB Matrix Model

Physics Letters B 659 (2008) 712–717 www.elsevier.com/locate/physletb

Perturbative vacua from IIB matrix model

Hikaru Kawai a,b, Matsuo Sato a,∗

a Department of Physics, Kyoto University, Kyoto 606-8502, Japan b Theoretical Physics Laboratory, The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan Received 21 August 2007; received in revised form 23 October 2007; accepted 16 November 2007 Available online 19 November 2007 Editor: T. Yanagida

Abstract It has not been clarified whether a matrix model can describe various vacua of string theory. In this Letter, we show that the IIB matrix model includes type IIA string theory. In the naive large N limit of the IIB matrix model, configurations consisting of simultaneously diagonalizable matrices form a moduli space, although the unique vacuum would be determined by complicated dynamics. This moduli space should correspond to a part of perturbatively stable vacua of string theory. Actually, one point on the moduli space represents type IIA string theory. Instead of integrating over the moduli space in the path-integral, we can consider each of the simultaneously diagonalizable configurations as a background and set the fluctuations of the diagonal elements to zero. Such procedure is known as quenching in the context of the large N reduced models. By quenching the diagonal elements of the matrices to an appropriate configuration, we show that the quenched IIB matrix model is equivalent to the two-dimensional large N N = 8 super-Yang–Mills theory on a cylinder. This theory is nothing but matrix string theory and is known to be equivalent to type IIA string theory. As a result, we find the manner to take the large N limit in the IIB matrix model. © 2007 Elsevier B.V. All rights reserved.

1. Introduction space at least in the one-loop level. Here we discuss stabil- ity of such conﬁgurations. In a naive reduced model given by I =− 1 tr[A ,A ]2 (μ = 1,...,D), the one-loop effective The IIB matrix model is one of the proposals for non- 4g2 μ ν perturbative string theory [1]. In the original interpretation, it action for the diagonal elements pi is given by S = (D − μ naturally describes type IIB string theory. In [2,3], the light- j 2) log((pi − p )2). Therefore, if D>2 extended con- cone Hamiltonian for type IIB string ﬁeld theory is derived i

0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2007.11.021 H. Kawai, M. Sato / Physics Letters B 659 (2008) 712–717 713 timated as follows. In the last expression, for each pair of i 4 8 and j we have three choices 1, a tr(S(i,j)), b tr(S(i,j)), which carry 0, 8, 16 powers of ξ, respectively. Because we have 16N- N 16 (i) dimensional fermionic integral i=1 d ξ , the number of factors other than 1 should be less than or equal to 2N. There- i fore, the effective action for pμ is expressed as a sum of terms consisting of less than or equal to 2N factors: p(i) − p(j) exp −S(p) = f ρ ρ (i) − (j) 2 2 various terms ((pμ pμ ) ) (i) (j ) pρ − pρ × f ··· Fig. 1. Uniform distribution of the diagonal elements. ((p(i ) − p(j ))2)2 μ μ ∼ exp O(N) , the diagonal elements to uniformly distributed values [11–15]. where f , f , ... are polynomials. This should be compared If we consider Feynman diagrams, the diagonal elements be- to the effective action in the bosonic case, which is of order have as the momenta in the gauge theories, and sums over in- exp(O(N 2)). We see that supersymmetry reduces the attractive dices become integrals over the momenta in the large N limit. force by order 1/N at least in the one-loop level. If this is true to In this way, the quenched matrix models and the gauge theo- all orders, in the large N limit any of the simultaneously diago- ries are equivalent. Therefore, the IIB matrix model can pro- nalizable configurations is stable and represents an independent duce the maximally supersymmetric large N gauge theories by vacuum as in the case of a moduli of scalar fields in the ordinary quenching the diagonal elements of some matrices to uniformly field theory.1 Contributions from such vacua are approximately distributed values and those of the other matrices to zero. As of the same order in the path-integral of the IIB matrix model. a generalization, if we quench the diagonal elements to dis- Instead of integrating over the moduli space, we can consider crete values instead of continuous ones, the matrix model cor- each of the simultaneously diagonalizable configurations as a responds to a toroidally compactified gauge theory, because the background and set the fluctuations of the diagonal elements to discrete momenta are conjugate to compactified coordinates. zero. Such procedure is known as quenching in the context of In the following, we show that the IIB matrix model the large N reduced model. In this Letter, we show that type IIA quenched in an appropriate way gives the two-dimensional N = string emerges as a vacuum of the IIB matrix model, if we intro- 8 super-Yang–Mills theory on a cylinder, whose bosonic duce such interpretation. More precisely, we show that the IIB sector is given by matrix model quenched appropriately is equivalent to matrix ∞ L string theory, which gives type IIA string theory. The moduli N 1 2 1 I 2 S2D = dt dxtr (Fμν) + DμA space of the IIB matrix model also includes other perturbative λ 4 2 −∞ 0 vacua. For example, we can show that the IIB matrix model N = 1 2 quenched in another way gives four-dimensional 4 super- − AI , AJ , (2.1) Yang–Mills theory, thus produces type IIB string theory on the 4 5 AdS5 × S . where μ,ν = 1, 2 and I = 3,...,10. The organization of this Letter is as follows. In Section 2, We start with the IIB matrix model, whose bosonic part is we show that by introducing a proper quenching the IIB ma- given by trix model becomes equivalent to the two-dimensional N = 8 1 super-Yang–Mills theory on a cylinder, which is nothing but S =− AM ,AN 2 . 2 tr (2.2) matrix string theory. In Section 3, we find relations between the 4gIIB coupling constant gIIB of the IIB matrix model and the string M = Let pi (i 1,...,N)be the diagonal elements of the matrices coupling gs , through matrix string theory. As a result, we find M 1 10 A .Weregard(pi ,...,pi ) as a ten-dimensional vector for how we should take the large N limit in the IIB matrix model. each i. We assume such vectors distributed uniformly in In Section 4, we summarize and discuss our results. 2π Λ Λ p, n, 0,...,0 p ∈ R,n∈ Z, −

(2.2) is rewritten as N 2π 2 S =− tr AM ,AN 2 . (2.5) 4λ Λ

We then expand the matrices as

Aμ = pμ + aμ (μ = 1, 2),

AI = aI (I = 3,...,10), (2.6) where all the diagonal elements of aμ and aI are fixed to zero. Fig. 2. A planar contribution to the free energy. In order to obtain the Feynman rule for this action, we choose a 2 4 2 2 =[ μ ]= λ Λ N 2π gauge fixing condition as F(aμ) p ,aμ 0. Then, we have FM = C N 2π 2 1 [ μ ]2 N 2π λ Λ a gauge fixing term λ ( Λ ) 2 tr( p ,aμ ) and a ghost term N ( 2π )2 tr(c¯[pμ, [p +a ,c]]) in the Feynman gauge. The total N λ Λ μ μ × 1 1 action is given by i − j 2 j − k 2 i,j,k,l=1 (pμ pμ) (pμ pμ) 2 1 1 N 2π 1 i j 2 ij Mji i j 2 ij ji × , (2.10) S = p − p a a + p − p c¯ c (pk − pl )2 (pl − pi )2 λ Λ 2 μ μ M μ μ μ μ μ μ − k − i − j − k μij jk Mki where C is a combinatorial factor. In the large N limit, pμ pμ pμ pμ a aM a 1 N i i 1 1 2π ∞ N i=1 f(p ,n ) is replaced by Λ dq Λ L m=−∞ f(q,m) i i Λ i Λ 1 ij jk ij because p and n are uniformly distributed in −

1 f = C(Nλ)2 gauge [6]. The relation between XI and AI is given by the sec- Y L3 ∞ ond equation of (3.1c). When showing the equivalence of (2.1) AI I × dp dq dr 1 and (2.7), we have identified with a , which is nothing but 2π 2π 2π 2 + 2π 2 AI by (2.6). Thus we find that the I th components of Aphys de- l,m,n=−∞ p ( L l) M fined by × 1 1 q2 + ( 2π m)2 r2 + ( 2π n)2 phys = = L L L AM sAM ,s gsls, (3.5) 1 2π × . (2.13) represent the string coordinates. In other words, an operator as (p + q + r)2 + ( 2π (l + m + n))2 L phys I phys ik AI Therefore, we have tr P AM e (3.6) 2π 2 corresponds to the emission vertex of a state with momentum F = f . (2.14) I M Λ Y k , where P is an appropriate polynomial. Now we discuss the manner to take the large N limit. From 3. Type IIA string theory from IIB matrix model (2.4) and (3.4) we find that gIIB and L should be tuned in the large N limit as We have shown that the IIB matrix model quenched in a N = Λ λ proper way is equivalent to the two-dimensional 8 super- gIIB = , Yang–Mills theory on a cylinder, which can be regarded as √2π N matrix string theory. Therefore, type IIA string theory emerges ( 2π)3 N L = . (3.7) as a vacuum of the IIB matrix model. This vacuum is specified gs λ by two parameters Λ and L as in (2.3). In this section, we dis- Note that we have assumed that λ and Λ are kept finite (2.9).In cuss how the string coupling g is expressed in terms of g , s IIB this limit, the UV cut-off Λ (3.3) goes to infinity as Λ and L and how we should take a large N limit to obtain matrix string O(N1/2), which guarantees type IIA string theory is produced type IIA string theory. from the IIB matrix model. In order to compare (2.1) with matrix string theory, we in- The IIB matrix model has a freedom of redefining the overall troduce the following redefinition: scale of the matrices. That is, the form of the action (2.2) is 2π 2π unchanged under τ = t, σ = x, (3.1a) L L AM = κA M ,g= κ2g , (3.8) (2π)3N IIB IIB g2 = , s 2 (3.1b) phys λL where κ is a constant. Here we consider AM defined in (3.5) A˜ = L A I = L AI as fundamental variables. Then the parameters gIIB, Λ and L μ μ,X gsls , (3.1c) phys phys phys 2π 2π become gIIB , Λ and L whicharegivenby where l is the string scale. Then, (2.1) becomes the well-known s phys = 2 form of matrix string theory. gIIB s gIIB, phys = ∞ 2π Λ sΛ, − 1 1 2 ˜ 2 1 ˜ I 2 Lphys = s 1L. (3.9) S = dτ dσ tr g (Fμν) + DμX MS 2π 4 s 2l2 −∞ s 0 Here the second and third equations follow from the fact that −1 μ 1 1 I J 2 Λ and L specify the eigenvalue distributions of A , and thus − X ,X . μ 2 4 (3.2) scale in the same way as A . By substituting (3.7) to (3.9),we 4 gs ls find how the large N limit should be taken in order for Aphys to Note that because of the redefinition of the world-sheet coordi- I represent the string coordinates: nates (3.1a), the UV cut-off for this action is given by phys 1 = 2 2 L gIIB Cls N , (3.10a) Λmatrix string = Λ. (3.3) √ phys 1 2π Λ = 2πClsN 2 , (3.10b) From (3.1a), (3.1b) and (3.1c), we can obtain the relation phys = 2π between the IIB matrix model and type IIA string theory. First, L , (3.10c) gsls substituting (2.4) to (3.1b), gs is expressed in terms of the parameters of the matrix model, where C is defined by √ Λ 2π Λ C = √ . (3.11) gs = . (3.4) λ L gIIB 2 Using (2.4), we can rewrite C as √1 Λ , which indicates that Next, we relate the string coordinates to matrices. The diag- 2π N gIIB onal elements of XI are string coordinates in the light-cone C is invariant under the redefinition (3.8). 716 H. Kawai, M. Sato / Physics Letters B 659 (2008) 712–717

−1/2 −1/2 So far we have discussed the leading order in the large N if the large N limit is taken with gIIBN and ΛN be- limit. As we discussed in the introduction, although we can ing fixed. Here the freedom of overall rescaling of the matrices freely fix eigenvalues of the matrices by hand in this order, is fixed such that the matrices represent the string coordinates. 1/N corrections should determine their distribution dynami- Any perturbative string should emerge in the same limit, be- cally [10,16–19]. In this case, the square of the range of the cause the way of taking the large N limit is expected not to eigenvalue distribution Λ is expressed by a function f as depend on the vacuum. Furthermore, if we assume these relations still hold when the eigenvalue distribution is dynamically Λ2 = g f(N) (3.12) IIB determined, ls and Λ are expressed as because it is given by − 1 1 − 1 = 2 2 4 ls C (gIIB) N , 1 μ 2 √ 1 1 1 tr A = 2 2 4 N Λ 2πC (gIIB) N . (4.1) dA 1 tr(Aμ)2 exp(− 1 1 tr([AM ,AN ]2)) Let us discuss how interactions of type IIA superstring are N 4 g2 = IIB derived in our new interpretation of IIB matrix model. The au- dAexp(− 1 1 tr([AM ,AN ]2)) 4 g2 thors in [9] show that half-BPS classical solutions of matrix IIB 1 μ 2 − 1 [ M N ]2 string theory determine world-sheets with definite genera when dAN tr(A ) exp( 4 tr( A ,A )) = gIIB . (3.13) they derive the Green–Schwartz action. In this sense, the world- − 1 [ M N ]2 dAexp( 4 tr( A ,A )) sheet genus expansion is not directly related to the ordinary On the other hand, from (3.10a) and (3.10b) we obtain that 1/N expansion. Although correlation functions of Wilson loops factorize in the large N limit, strings are not simply represented 2 phys 1 phys = 2 Λ gIIB 2πCN , (3.14) by Wilson loops in our case. Therefore there is a possibility that interactions of strings can be reproduced only by planer di- which suggests that agrams. In order to examine this possibility, we need a more 1 f(N)= 2πCN 2 . (3.15) precise analysis on the string states, which we intend to report in future publications. However, the value of C is not determined in the leading order Space–times emerge in various manners in the IIB matrix of the large N limit, because we can give any values to λ and Λ. model. First, in the original picture, matrices appear as a reg- phys Finally, we discuss how ls and Λ are expressed in terms ularization of the Schild action of type IIB string theory [1], phys of gIIB and N.From(3.10a) we have and they represent the space–time coordinates. Second, in the interpretation we have introduced in this Letter, two matrices − 1 phys 1 − 1 = 2 2 4 ls C gIIB N . (3.16) correspond to conjugate momenta of the world-sheet coordi- Substituting (3.16) to (3.10b), we obtain nates, whereas the other eight correspond to the light-cone coordinates of type IIA string. Third, as shortly discussed in √ 1 phys 1 phys 1 Λ = 2πC 2 g 2 N 4 . (3.17) this Letter, the IIB matrix model produces the four-dimensional IIB large N N = 4 super-Yang–Mills theory, and thus produces 5 These results (3.16) and (3.17) are expected to hold for any vac- type IIB string theory on the AdS5 × S background through uum because the way of taking the large N limit should not the AdS/CFT correspondence [20]. In this case, six matrices depend on the vacuum. In fact, they are consistent with the re- correspond to the radial coordinate of the AdS5 and the coordi- sults of some other analyses [10]. nates of the S5, whereas the other four matrices correspond to the conjugate momenta of the angular coordinates of the AdS5. 4. Conclusion and discussion Fourth, the matrices can be regarded as the covariant deriva- tives on curved space–times [21]. It is interesting to study the In the IIB matrix model, simultaneously diagonalizable con- relations among these ways of representing space–times. figurations are stable and form a moduli space in the leading order of the large N limit. If we consider fluctuations around Acknowledgements each of them with the diagonal elements being quenched, we obtain a perturbative vacuum of string theory. This work is supported in part by a Grant-in-Aid for the Actually, type IIA string theory and type IIB string theory 21st Century COE “Center for Diversity and Universality in AdS × S5 emerge, if we consider fluctuations around ap- on 5 Physics” from the Ministry of Education, Culture, Sports, Sci- propriate configurations. Therefore, the moduli space should ence, and Technology (MEXT) of Japan. represent at least a part of perturbatively stable vacua of string theory. References We have given a detailed analysis on the case of type IIA string theory. We have shown how the string coupling gs and [1] N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, Nucl. Phys. B 498 the string scale ls are related to the coupling constant gIIB and (1997) 467, hep-th/9612115. the range of the eigenvalue distribution Λ in the IIB matrix [2] M. Fukuma, H. Kawai, Y. Kitazawa, A. Tsuchiya, Nucl. Phys. B 510 model. As a result, we find that type IIA string theory emerges (1998) 158, hep-th/9705128. H. Kawai, M. Sato / Physics Letters B 659 (2008) 712–717 717

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