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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<j μ μ figurations of the diagonal elements are unstable because they from Schwinger–Dyson equations for Wilson loops. On the collapse to a point. In a supersymmetric case, if one ignores the other hand, matrix string theory describes type IIA string the- diagonal elements of fermionic matrices, the one-loop effective ory [4–7]. In this theory, the diagonal elements of the eight action for the diagonal elements of bosonic matrices is given scalars form coordinates of the light-cone strings. This inter- by S = (D − 2 − d ) log((pi − pj )2) = 0, and there is pretation correctly reproduces the world-sheet action and the F i<j μ μ no force between them. However, one cannot ignore the diag- joining and splitting of type IIA strings [8,9]. However, it has onal elements of fermions when the dimensions of a theory is not been clarified whether a matrix model can produce two or less than one. In fact, the one-loop effective action for the diag- more perturbative string theories, although a non-perturbative onal elements of both the bosonic and fermionic matrices in the string theory should include all perturbative vacua. S4 S8 = (i,j) + (i,j) In the large N limit of the IIB matrix model, configura- IIB matrix model is given by S(p,ξ) i<j tr( 4 8 ), tions of simultaneously diagonalizable matrices form a moduli p(i)−p(j) where (S ) = (ξ¯ (i) − ξ¯ (j))Γ μρν (ξ (i) − ξ (j)) ρ ρ (i,j) μ,ν (i)− (j) 2 2 ((pλ pλ ) ) [10]. By integrating out ξ, we have a complicated interac- * Corresponding author. i − = 16 (i) − = tion among pμ exp( S(p)) i d ξ exp( S(p,ξ)) E-mail addresses: [email protected] (H. Kawai), N 16 (i) + 4 + 8 [email protected] (M. Sato). i=1 d ξ i<j(1 a tr(S(i,j)) b tr(S(i,j))), which is es- 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, − <p< , 2. Gauge theories from IIB matrix model L 2 2 Λ 2π Λ − < n< , (2.3) In general, zero-dimensional matrix models are obtained by 2 L 2 dimensional reduction of gauge theories. Such models can re- produce the gauge theories in the large N limit by quenching as Fig. 1 and set the fluctuations of the diagonal elements to zero. By introducing the ’t Hooft coupling 1 However, such configurations would become unstable if we take the 1/N 2 corrections into account. This would correspond to instability of perturbative 2π λ = Ng2 , (2.4) vacua of string theory when non-perturbative corrections are included.
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