Physics Letters B 579 (2004) 211–216 www.elsevier.com/locate/physletb

Note on a fermionic solution of the model and noncommutative

Yuuichirou Shibusa, Tsukasa Tada

Theoretical Physics Laboratory, The Institute of Physical and Chemical Research (RIKEN), Wako 2-1, Saitama 351-0198, Japan Received 21 October 2003; accepted 31 October 2003 Editor: M. Cveticˇ

Abstract We present a new fermionic solution of the supersymmetric matrix model. The solution satisfies the commutation and anticommutation relations for noncommutative superspace. Therefore the solution can be considered as an implementation of noncommutative superspace on the matrix model.  2003 Elsevier B.V. All rights reserved.

1. Introduction naturally lead to IIB matrix model [27]: 1 a b 1 ¯ a S = Tr [Xa,Xb] X ,X − θΓ [Xa,θ] , (1) Noncommutative space has been a fascinating arena 4 2 in the study of nonperturbative aspects of string the- ory [1–4]. There is now a surge of renewed interest in where maximal 32 are realized on noncommutativity, this time in superspace [5–20]. In the set of N × N matrices. In the above action (1), this Letter, we study noncommutative superspace as a the contractions of the upper and lower indices are fermionic background of a certain matrix model. performed over 10-dimensional flat Minkowski met- In the so-called reduced model [21], the informa- ric. However, thanks to the maximal , tion of space–time is subtly converted into large N the dynamics of space–time is contained in the vari- color degrees of freedom. This yields a wider perspec- ous configurations of the matrices. In fact, this model tive to the implementation of space–time in quantum exhibits nontrivial space–time backgrounds such as field theory and many-body systems [22–26]. The cru- D-branes as a solution to the equation of motion cial point is to respect the appropriate symmetries that from (1). would correspond to the symmetries of the resultant While one can investigate the IIB matrix model field theory. Following the same spirit, one would be by the expansion around the commutative (diagonal) background [28], it is also interesting to start with the following solution: E-mail addresses: [email protected] (Y. Shibusa), a b =− abI α = [email protected] (T. Tada). X ,X iC N ,θ0, (2)

0370-2693/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2003.10.110 212 Y. Shibusa, T. Tada / Physics Letters B 579 (2004) 211–216 where Cab is an antisymmetric constant. The result 2. Noncommutative superspace and the is noncommutative Yang–Mills theory [3]. Xaswhich four-dimensional matrix model satisfy Eq. (2) themselves serve as noncommutative coordinates. In the string picture, this corresponds For the sake of the recent interest, let us first con- to the appearance of noncommutative geometry in a sider the four-dimensional case. In four dimension, the constant B-field background [2]. following commutation and anticommutation relations One may regard the foregoing analysis as the for the supercoordinates has been studied in [11] representation of (local) Poincaré group and its certain α β αβ noncommutative version in the large-N matrix model. θ ,θ = C , (6) Since superspace is nothing but the coset space of a α αβ a ¯α˙ X ,θ = iC σ ˙ θ , (7) the super-Poincaré group divided by the Lorentz βα a b ¯ 2 ab group, it would be also natural to try to represent X ,X = (θ) C , (8) ˙ ˙ ˙ ˙ (noncommutative) superspace on the matrix model in θ¯α, θ¯β = θ¯α,θβ = θ¯α,Xa = 0, (9) a similar manner. In this Letter, we exhibit such an attempt. where Also, searching a solution of the matrix model that a,b = 1,...,4,α,α˙ = 1, 2, corresponds to the graviphoton background consid- ered in [6–8,11] would have its own prudence. Con- ab ≡ αβ − ab C C σ  αβ , (10) sidering that the noncommutative solution (2) corre- αβ ab αβ sponds to a B-field background in string theory, non- C = σ Cab. (11) commutative superspace realized in the matrix model We have followed the notation of [29]. should have some connection with the graviphoton To analyze these relations in the matrix model, we background. consider the following four-dimensional matrix model In the following, we present a solution with nonzero α matrices for the matrix model. Xa sandθ s 1 a b 2 a ¯ S = Tr X ,X + Cθσ [θ,Xa] . (12) in the solution satisfy the relations which are similar 4 to those recently studied in the context of noncommu- C tative superspace. It will be shown that the solution The constant is left undetermined for the moment. preserves half the supersymmetries contained in the We treat the model in the Euclidean signature while model and the Killing spinor is explicitly constructed. maintaining the notation of [29] just as has previously been done in [11]. This is achieved by defining σ 0 To illustrate the basic idea for our fermionic solu- ¯ tion, let us consider following fermionic matrices here as i times that of [29]. Then θ and θ become independent [30] and the equations of motion are as θ ∼ (Grassmann) ⊗ Q + (Grassmann) ⊗ P +···, follows: (3) Xb, Xa,X + C θ,σaθ¯ = 0, (13) b where Q and P are a pair of matrices such that a Xa, θσ = 0, (14) [Q, P ]=i. The anticommutation relations among α˙ a ¯ = them would be something like σ θ α,Xa 0. (15) {θ,θ}∼i(Grassmann)2 ⊗ I. (4) In particular, (14) can also be derived by multiply- a = ing σa on the both side of (7) since σβα˙ (σa)αγ˙ αβ Then, as we will next describe in detail, we find that −2βαα˙ γ˙ , which cancels with symmetric C .This there is a set of bosonic matrices of the following form suggests that the equation of motion of the matrix model (12) is compatible with the algebra of noncom- ∼ 2 ⊗ + 2 ⊗ +··· X (Grassmann) Q (Grassmann) P , mutative superspace (6)–(9). In the following, we will (5) show that it is actually the case by constructing an ex- which satisfy both (anti)commutation relations and the plicit solution of the equations of motions that also sat- equation of motion for the matrix model. isfies (6)–(9). Y. Shibusa, T. Tada / Physics Letters B 579 (2004) 211–216 213

It is worth mentioning in passing, though, that this It follows that the only nontrivial structure constants N = matrix model has the following enhanced ( 2) fAˆBˆ Cˆ among the generators we have introduced are supersymmetry transformations:   0 i 0000  −  δXa =−iξσaθ¯ + iθσaξ,¯ (16)  i 00000   α  000i 00 α = A[ ] ab + αI fAB0 =   . (24) δθ Xa,Xb ξσ ξ , (17)  00−i 000   ¯α˙ ab ¯ α˙ ¯ ˙α 0000· δθ = B[Xa,Xb] σ¯ ξ + ξ I. (18) 0000 · ¯ Since θ and θ are now being treated as independent, We will try to find a new solution in terms of the we have introduced here independent parameters A above introduced generators, that is to say, we give an and B. To satisfy the consistency of the algebra (16)– ansatz for the solution in the following form: (18) and ordinary SUSY algebra, it is necessary that A =−B. This further leads the commutator of two 2n a = aA A supersymmetries as follows Xsol X T , (25) = A 1 [ ]=I a ¯  +  a ¯ 2n 2n δ1,δ2 iξ1σ ξ2 iξ1σ ξ2 ∂a ˙ ˙ θ α = θ αAT A, θ¯α = θ¯αAT A. (26) + δ −2iAX ξ σ cξ¯ − (1 ↔ 2) sol sol U(N)gauge c 1 2 A=1 A=1 ¯ + eq. of motion for θ and θ, (19) The equation of motions provides the following con- in a shorthand notation. Using these relations, it is ditions for the ansatz: ˙ ˙ straightforward to show that the action (12) is invariant θ¯α 0 = θ¯α, (27) under (16)–(18) provided aA =− αA a ¯β˙0 X iθ σ αβ˙θ , (28) AC =−i. (20) Others = 0. (29) One can use the above relation (20) to determine C in These relations give a new solution with nonzero terms of A. fermionic matrices. Moreover, if we denote Now we proceed to find a solution of (13)–(15) αA βB ≡ αβ with nonzero fermion matrices. Let us denote U(N) θ θ fAB0 C , (30) ˆ generators as T A and choose a integer n which is large AB  ¯ enough, but much smaller than N so that N/n 1. Xsol, θsol and θsol obey the noncommutative su- The reason for introducing n will shortly become perspace commutative and anticommutative relations clear. We will focus our attention on the following (6)–(9) provided that (26)–(29) hold. In other words, special generators which have n by n block structure: we can reproduce the relations (6)–(9) in the matrix model (12). ˆ A = 0,A, A few brief comments follows. In the above analy- αβ A = 1, 2,...,2n, sis, one could say that C is rather defined by the left-hand side of (30). So we are not constructing a so- A = T Q1,P1,Q2,P2,...,Qn,Pn, (21) lution for a given Cαβ . There is a possibility of another solution for more general Cαβ that is discussed later. where Qk sandPksareN/n× N/n matrices and Qks Also Cαβ is not genuine c-number but bi-Grassmann and Pks satisfy in our case. Since the square of Grassmann number equals zero, it might cause a problem. However, note [Qj ,Pk]=iδjk. (22) that (Cαβ )k = 0 provided k  2n and one can take n We also denote the identity as T 0,thatis, as large as one wishes in the large-N limit. This is be- cause we have 2n sets of Grassmann variables due to 0 T = IN . (23) the n by n block structure of the matrices. Therefore, 214 Y. Shibusa, T. Tada / Physics Letters B 579 (2004) 211–216 with the relation (30), we can avoid the power of Cαβ The commutator of two supersymmetries yields from vanishing to a certain extent. [δ ,δ ]=I iξ¯ Γ aξ  − iξ¯ Γ aξ  ∂ Let us next discuss the symmetry of the solution. 1 2 2 1 1 2 a It turns out that this background Killing spinor can be ¯ a + δU(N)gauge 2ξ2Γ ξ1Xa defined by using a two-component spinor ξ˜ as follows: + eq. of motion for θ, (40) ˜ ¯ 2 ξ = ξ(θsol) , (31) ¯ in a shorthand. ¯ = ˜ ··· α1 β1 α2 β2 ξ ξα1α2 β1β2 α3α4 β3β4  θ ,θ θ ,θ Now, just as 4-dimensional case, let us adopt the sol sol sol sol × α3 β3 ··· following ansatz for θ, θsol,θsol , (32)  ξ = 0, (33) 2n  a = aA A ξ¯ = 0. (34) θsol θ T , (41) A=1 ¯ αA ξ in (32) contains all of the 4nθ sinLorentz aA invariant way hence multiplying any θ αA yields zero. where θ s are 10-dimensional spinors. Then the This solution is 1/2 BPS, thus there is N = 1SUSY anticommutator for θ becomes while the trivial (diagonal) background is supposed to θ α,θβ = θ αAθ βBf I have N = 2 SUSY. It is also interesting to note that AB0 AB n has to be finite in order for the above (32) to be ≡ αβ I a Killing spinor. Should we take the limit n →∞ C . (42) first, the Killing spinor (32) is ill-defined hence half In 10 dimension, the spinor structure of Cαβ yields the of the symmetries are broken. This case may rather following expansion, corresponds to N = 1/2 SUSY studied in [11]. αβ 1 a αβ γδ C = Γ C (CΓa)γδC 16 3. 10-dimensional case 1 a ···a αβ γδ + Γ 1 5 C (CΓa ···a )γδC , (43) 32 · 5! 5 1 It is also straightforward to generalize the above where C is the conjugation matrix in 10 dimen- solution to 10-dimensional case. The action for the sion. From (36), (37), we can derive the following con- 10-dimensional IIB matrix model or so-called IKKT ditions as 4-dimensional case: model [27] is defined by the following action: aA =¯ a A 1 a b 1 ¯ a X ηΓ θ , (44) S = Tr X ,X [Xa,Xb]− θΓ [Xa,θ] . (35) 4 2 C a αβ = Γ αβ C 0. (45) Here the indices a and b run from 0 to 9 and θ Here we need to introduce one new Grassmann pa- should be understood as Majorana–Weyl fermion in 10 rameter η since θ and θ¯ are not independent unlike dimension, though we work in Euclidean space–time four-dimensional case. Also note that Eq. (45) actu- signature in the following. The equations of motion ally gives a constraint among 2nθαAs. derived from (35) are This solution has the Killing spinor which is 16nth αβ b a 1 ¯ a order with respect to C . Thus we have N = 1su- 0 = X , X ,Xb + θ,Γ θ , (36) 2 persymmetry. The commutation and anticommutation a 0 = X ,Γaθ . (37) relations of the solution are = Enhanced N 2 supersymmetry transformations are θ α,θβ = Cαβ I, (46) given by Xa,θα = ηΓ¯ a CβαI, (47) a = ¯ a β δX iξΓ θ, (38) a b 1 a a a ab αβ i α  X ,X = ηΓ¯ 1 2 3 η CΓ C I. (48) δθα = Xa ,Xb Γ abξ + ξ αI. (39) 96 a3a2a1 αβ 2 Y. Shibusa, T. Tada / Physics Letters B 579 (2004) 211–216 215

a 4. Discussion Xsol also contains bi-Grassmann factor. This seems inevitable from the commutation relation (8) where the We have presented a fermionic solution of the right-hand side contains a product of Grassmann vari- matrix model and shown that the solution yields the ables. It would be interesting to pursue the physical same commutation and anticommutation relations as interpretation of this feature and the relation with the those of noncommutative superspace. Let us comment preceding work [31], which we leave for future inves- on several aspects of the solution. tigation. Firstly, it is worth mentioning that the present solution has a self dual structure. In fact, denoting the commutator of Xa as a field strength, Acknowledgements ab ≡ a b F X ,X (49) We would like to thank Y. Kimura for his collab- and with the definition for the dual field strength oration at the early stage of the present work. We are also indebted to M. Hayakawa, N. Ishibashi, S. Iso, ˜ ab = i ab cd 0123 = H. Kawai and F. Sugino for fruitful discussions. This (F) E cdF ,E i, (50) 2 work is supported in part by the Grants-in-Aid for Sci- the field strength for the solution Fsol satisfy the entific Research (13135223) of the Ministry of Ed- (anti)self dual condition, ucation, Culture, Sports, Science and Technology of ˜ Japan. Fsol =−Fsol. (51) ¯ 2 Since Fsol contains (θ) , the would be instanton index for the solution, Tr F F˜ is zero. It would be interesting References to clarify the relation between the present solution and the more general self dual solutions including the [1] A. Connes, M.R. Douglas, A. Schwarz, JHEP 9802 (1998) 003, hep-th/9711162. instanton solutions. [2] N. Seiberg, E. Witten, JHEP 9909 (1999) 032, hep-th/9908142. αβ Secondly, the parameter C for non(anti)commu- [3] H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, tativity in our solution is a product of Grassmann Nucl. Phys. B 565 (2000) 176, hep-th/9908141. variables. One may find this as an unsatisfying aspect [4] For a comprehensive review and further references, see, of the solution. Therefore it would be interesting to M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73 (2001) 977, hep-th/0106048. observe that there is another solution of (13), (14) [5] D. Klemm, S. Penati, L. Tamassia, Class. Quantum Grav. 20 and (15), (2003) 2905, hep-th/0104190. [6] H. Ooguri, C. Vafa, hep-th/0302109. 1 1 θ = Γ ⊗ I, (52) [7] H. Ooguri, C. Vafa, hep-th/0303063. [8] J. de Boer, P.A. Grassi, P. van Nieuwenhuizen, hep-th/ θ 2 = Γ 2 ⊗ I, (53) 0302078. θ¯1 = Γ 3 + iΓ 4 ⊗ I, (54) [9] H. Kawai, T. Kuroki, T. Morita, Nucl. Phys. B 664 (2003) 185, hep-th/0303210. ¯2 5 6 θ = Γ + iΓ ⊗ I, (55) [10] I. Chepelev, C. Ciocarlie, hep-th/0304118. Xa =−iθσaθ¯ ⊗ I, [11] N. Seiberg, JHEP 0306 (2003) 010, hep-th/0305248. (56) [12] R. Britto, B. Feng, S.J. Rey, hep-th/0306215; where Γ i are SO(6) gamma matrix. The above solu- R. Britto, B. Feng, S.J. Rey, hep-th/0307091. [13] N. Berkovits, N. Seiberg, hep-th/0306226. tion obeys Seiberg’s noncommutativity relations (6)– [14] M. Hatsuda, S. Iso, H. Umetsu, hep-th/0306251. αβ αβ (9). In particular, we can set C = δ in this case. [15] S. Terashima, J.T. Yee, hep-th/0306237. And there is no Killing vector like (32) hence we have [16] S. Ferrara, M.A. Lledo, O. Macia, hep-th/0307039. exactly N = 1/2 supersymmetry. However, those ma- [17] J.H. Park, hep-th/0307060. trices are not in the representation of u(n),which [18] M.T. Grisaru, S. Penati, A. Romagnoni, hep-th/0307099. [19] R. Britto, B. Feng, hep-th/0307165. means, unfortunately, we cannot derive the equations [20] A. Romagnoni, hep-th/0307209. of motions (13)–(15) from a (Hermitian) matrix model [21] T. Eguchi, H. Kawai, Phys. Rev. Lett. 48 (1982) 1063. action in the first place. [22] G. Parisi, Phys. Lett. B 112 (1982) 463. 216 Y. Shibusa, T. Tada / Physics Letters B 579 (2004) 211–216

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