JHEP11(1998)013 limit, the N November 17, 1998 Accepted: - D October 28, 1998, Received: p-branes, D-branes. We study the /axion configuration near D-instantons in type IIB [email protected] [email protected] HYPER VERSION Spinoza Institute, University ofLeuvenlaan Utrecht 4, 3584 CEE-mail: Utrecht, The Netherlands Department of Physics, UniversityBerkeley, of California California at 94720 Berkeley Theoretical Physics Group, Mail StopLawrence Berkeley 50A-5101 National Laboratory, Berkeley,E-mail: California 94720 coupling constant becomesthis zero configuration except is near analyzed. theis An origin. discussed. implication The of this of result to theKeywords: IIB Matrix Model Abstract: Kostas Skenderis . In theflat field in theory the limit, string the frame metric as near well the as instantons becomes in the Einstein frame. In the large Hirosi Ooguri On the field theory limit of JHEP11(1998)013 5 S × +1)- 5 p limit, the AdS , 7 N S × +2)-dimensional 4 p AdS is subtle and we will instantons. This case N i.e. + 1)-dimensional conformal field p 1)-branes, − -brane. Another is to use the collective p 1 ) times a compact space is equivalent to a ( -instantons 4 +2 matrix model 5 p D IIB AdS -brane, which in some limit is a ( p -branes in two different ways. One is to consider strings propagating p are shown to be equivalent to conformal field theory on M2, D3 and 4 S × 7 AdS In this paper, we study the region near ( string coupling constant becomesof zero the except conjecture near [4,comment the on 5]. origin. this case This The toward seems situation the in end in support of the this case paper. of finite coordinates of the is of particular interestconjectured since to the give a corresponding non-perturbativestudy 0-dimensional definition the gauge of the theory dilaton/axion type has fieldstheory IIB been configuration superstring limit near [4, of 5]. the Senthe We instantons. and string Seiberg frame If [6, as we 7], well take the as the metric in field near the the Einstein instantons frame. becomes Moreover, flat in in the large in the curved background generated by the theory. The equivalence of the two descriptions implies the correspondence. M5 branes, respectively. Thesehorizon correspondences region of were discovered by studying the near 2. Wick rotation and supersymmetry The solution is definedtation of in type the IIB Euclidean signature isour subtle, space. prescriptions we would Since for like the the to Wick Wick start our ro- rotation discussion and by supersymmetry stating in the Euclidean space. and 1. Introduction According to the AdS/CFT correspondence [1,Anti-de 2, 3], Sitter on Space ( ( 5. Comment on the Contents 1. Introduction2. Wick rotation and supersymmetry3. Instanton solution4. Field theory limit of 1 1 3 dimensional conformal field theory. In particular, string theory on JHEP11(1998)013 ) ,R as (2.3) (2.5) (2.2) (2.4) (2.8) (2.7) (2.1) (2.6) (2 a SL as 6) are , V 4 to be a function of = 0 is used in [10]. , and the axion λ λ (1). Following [8], we =2 φ /U p . ) 1 , τ 2 ,R ) µ . τ , , ∂ a (2 ! ν iλ 2 1 2 , iλ =0 SL iλ a∂ e , . − e µ ! µ , . τ , τe 2 ∂ λ µ P Σ ν τ φ ! i Σ 0 . µ ) -form fields, ∂ 2 =0 =0 P e ) in [9], whereas µ 1 2 φ µ ∂ p + + e i , i ∗ 2 iλ µ ∂ − 2 iλ a V V Σ − ) i P = iQ µ − + + ie 2 Σ = + 2 e i e ¯ β τe τ φ 0 ∂a + + 1 2 µ β φ∂ − µ + 2 − − + ( ν − V ∗ e µ

ν φ V V P Q V a µ 2 ∂ µ 2 P µ

φ∂ ∂ e

µ ∂ µ = α ∇ iτ → → − as V α P ∂ 2 − + +2 = 1 ( V τ µ µ Im log( a V − φ a 2 1 = =( P → Q αβ V αβ − √ ∆ ∆ µ i  = V µν P and − − = in (2.3) is related to the dilaton µ R µν φ τ D V = = R µ µ P Q (1) action as U is set equal to (1) gauge symmetry by setting the scalar field λ U (1) phase. It is convenient to parametrize the matrix U (1) gauge symmetry (2.2), they transform as is the Ricci tensor. Substituting (2.5) into these, the equations of motion U µν R The complex scalar field In the Minkowski signature space, the dilaton and axion fields of type IIB theory . For example, We can fixτ the Under the where Σ is a To write the type IIB supergravity equations of motion, we introduce two parametrize the upper half-plane, or the coset space where can be expressed in terms of and define the local The equations of motion (in the absence of the singlet currents, introduce the frame field, JHEP11(1998)013 (2.9) (3.1) (2.10) (2.12) (2.11) (2.13) are given µ ψ , ∗  ) µ as independent spinors. Q ∗  i 2 , ) . + 2 α µ and ) , ν and the gravitino ,  ∇ ia , ρ ∂a iλ , α∂ . . µ ( 2 µ , = φ γ  and the dilaton and the axion are − =( ∂ 2 ∗ iλ µ e µ φ. φ e α 2 ∗ =0 ) 2 =0 µ Q µ e 2 1 µν 2 iP e ∂ 2 ) δ i τ α , 2 → φ ) . 2 ∗ − µ µ − − τ  − ∂ 1 a − 2 = µ µ e τ = 2 φ ∂α 2 φ∂ ) 3 ∂ µ γ µ ( τ ν + τ µ ± ∗ µ φ 2 ∂ µν 1 ∂ ∇ 2 ∂φ − τ φ∂ i τ g 2 ( = e iP  , and we treat 1 µ µ , δψ 1 2 τ ) ∂ ∂ α +2 + + ia µ = = ( ( µ µ ∂ − α φ λ 2 1 ∂ 2 1 ( µ Q = µ ,δρ δρ i ∆ ∆ R 2 2 1 ∗ − ∂ = α δψ  µ = − = = = µν γ → µ ∗ L µ µ µ µ R a ∇ P P Q iP =( = µ δρ δψ The instanton is a solution in Euclidean signature space. The equations is this case are obtained from (2.8) by the substitution where The supersymmetry transformations of the dilatino by In addition, we Wick rotate the spinors as in [12, 13]. This yields The supersymmetry transformation rules thatWe make can the be substitution derived analogously [11, 12, 13]. The invariance of theinvariance Wick-rotated of action under the these originalcomplete rules discussion action directly of under follows Euclidean the from spinors the and transformation Wick rules rotation (2.10). we refer to For [11, a 12, more 13]. related as 3. Instanton solution As shown in [9],metric the (in Euclidean the equations of Einstein motion frame) (2.11) is have flat a solution where the These can be derived from the Lagrangian density JHEP11(1998)013 at s (3.7) (3.5) (3.3) (4.2) (3.6) (3.4) (3.2) (4.1) g → φ e as α . . , 0 → +const s 2 . τ +const 1 − as − , ,l  = unites of charge is given by ! 8 N , c 8 s , 1 r 8 s ··· . +const N 8 ! Nl r 8  s + Nl s ,τ :fixed 8 s 1+ c g 8 8 r =0 2 g 1 4 0 s Nl  r ) 4 r s 0 s c l φ g 1 s c s 0 limit instead of the field theory limit (4.1). They e g c N g 8 g − 0 1+ = ∆ = -dependence of the dilaton/axion fields in (3.7) c  → 1+ = s 2 ' YM c l s Nu r

g φ g is a numerical constant related to the volume of the -instantons ( α 2 YM e 1 1+ 0 , 0 − s c D =

c g s µν ,g α − δ g ' 7) [16]). Notice that the final configuration they obtain is singular 2 s r l φ = = = as − e p< = φ r α µν e u g = 2 τ . The equation (3.1) then determines the axion In this limit, the -branes ( c p 1 is given by is related to the instanton charge c ∞ = is the string length and behaves for large r s l α The near instanton configuration was also studied in [14] (see also [15] for a related observation). To summarize, the instanton solution with A spherically symmetric solution to (3.2) with the boundary condition 1 the constant infinity for some constant where Since disappears and we obtain unit 9-sphere. of the solutions. 4. Field theory limit of It was shown in [9] thatHere the we solution (3.7) will preserves study half the of the field maximal theory supersymmetry. limit [1, 6, 7] In the following, weare choose then the satisfied plus if sign the in dilaton the obeys right hand side. The equations (2.11) In that paper they considered the strict as the dilaton diverges inas the in limit. (4.2). In the field theory limit (4.1), the dilaton and axion remain finite showed that the configurationresult one holds obtains for in all this D limit preserves maximal supersymmetry (a similar JHEP11(1998)013 4 / 1 (1) ) (5.2) (4.4) (5.3) (4.3) (4.5) (5.1) U given N µ =0and 2 YM Q g ∗ µ limit of this P , N 2 τ µ ∂ , −  ] = . = 0 by fixing the ,ψ 1 0 . µ µ τ ) µ A 2 9 Q [ ∂ → , is a cut-off parameter in the µ Ω 8 s  Γ d − 2 ¯ 2 of supersymmetry is preserved ψ u u / µ 1 2 ∂ + 2 1) brane, it is natural to expect that ,l 2 ) ]+ − . ν ,where N du 2 1 ( . τ − ,A µ N ) 2 YM µ N 8 ∂ 2 :fixed g u log A ( 5 p 4 s 0 = ][ s i l hermitian matrices. The large 2 YM 2 N − c ν g g 3 s µ l 0 N = = c ,A δψ µ = × λ 2 1 τ A -brane [17], the open string dynamics on the brane [ N µ 1+ p 1 4 2 matrix model YM ∂ v u u t  4 s are = l ,g tr µ Therefore it is natural to set ψ IIB = 2 s r P l 2 s 1 2 2 YM g = ds = 0 has the maximal number of solutions. − u µ = 10) and for the instanton solution (3.7), the composite gauge field δψ 2 S + 1)-dimensional supersymmetric gauge theory [18] in the limit τ , ..., µ p ∂ = 0. On the other hand, − =1 ∗ µ = δρ ( 1 µ τ µ A ∂ This is the case even before the field theory limit (4.1) is taken. Let us compare their results with the field theory limit of the instanton studied in Let us turn to the dilatino variation in (2.12). Since 2 where by (2.13) is a pure gauge. Thus the equation gauge symmetry as matrix integral. this paper. Since thestring metric for frame the metric instanton for solution the is instanton flat solution in is the given Einstein by frame, the therefore, In this gauge, the gravitino variation in (2.12) becomes simply which is non-zero in the fieldeven theory limit. in the Thus, only field 1 theory limit. and that the string coupling constant is ( 5. Comment on the In the D-brane descriptionreduces of to the the ( Since Repeating the argument in [1] intype the case IIB of the string D( into the the flat 0-dimensional metric matrix (5.4) model and given the by dilaton the background action (4.2) is equivalent model has been proposedtheory. in There [4, they 5] found as that a the non-perturbative string definition length of of type the IIB type string IIB string is ( JHEP11(1998)013 φ e (5.4) (5.5) (1998) 253 2 . , N e symmetry in ) ]. 2 9 odinger Institute Ω d 2 u +˜ 2 ˜ u d ( ]. N hep-th/9802109 = 2 super Poincar´ 2 YM Gauge theory correlators from non- g . N 0 2 c ) q 8 N = Nu 2 YM (1998) 105 [ ! 6 g 2 9 analysis of (5.2), one also finds free type IIB ( hep-th/9711200 2 0 Ω u finite, these two view points are in complete c to be large, the size of this region expands. In d N ' N + B 428 N φ 2 4 e limit of superconformal field theories and supergravity, 2 YM u du g N

(1998) 231 [ 2 N with Phys. Lett. 2 YM g , 0 .Aswetake c The large ]. q →∞ /N = 0. From the large , one obtains type IIB string in the flat space (5.4) with vanishing 1 ' φ N 2 s e is reminiscent of the string length found in [5]. Anti-de Sitter space and holography, Adv. Theor. Math. Phys.  . Thus the metric in the string frame also becomes flat. Moreover the 2 s l ds 8 N →∞ /u u is the line element of the unit 9-sphere. In the limit (4.1), this becomes N 2 YM =1 g 9 u Ω q d hep-th/9802150 Adv. Theor. Math. Phys. critical string theory [ At the same time, the dilaton in the field theory limit is In the limit [1] J.M. Maldacena, [2] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, [3] E. Witten, the limit string coupling factor where ˜ agreement. In the field theory limit, the string coupling given by the dilaton field This appears to be differentin from [5]. the expression for Thesupersymmetry the non-constant string near dilaton coupling the constant (5.5) found pointed instanton, is out as also in we responsible [4] thatten saw for the dimensions. in the matrix (4.5). breaking model (5.2) of has On 1/2 the of other hand, it was References Acknowledgments We would like toand thank Juan Maldacena Michael for Greentheory comments for on division of the discussions. CERN earlier wherealso version We this of like also work this to was paper. thank thank initiated We Tom for the thank the Banks Aspen the hospitality. Center K.S. for would Physics and the Erwin Schr¨ is small for in Vienna where partH.O. of was supported this in work partAC03-76SF00098. was by The completed the research NSF for of grantfor their K.S. PHY-95-14797 Scientific is hospitality. and Research supported the (NWO). The by DOE the work grant DE- Netherlands of Organization Here strings [4]. It would be very interesting to clarify the situation at finite JHEP11(1998)013 , , B Nucl. Phys. (1995) , , 75 (1998) 51 Class. and (1996) 335 (1998) 127 2 (1997) 3577 , Phys. Lett. 79 , reduced model as B 460 supergravity B 533 N ]. -branes and M-branes p ]. =10 Phys. Rev. Lett. . (I) Euclidean formulation , 4 Alarge ) ,D 3 ]. Φ Nucl. Phys. Nucl. Phys. =2 String field theory from IIB matrix Phys. Rev. Lett. Brane intersections, anti-de Sit- , N , hep-th/9511080 , ]. ]. Instantons and seven-branes in type IIB hep-th/9612115 -branes p 7 ]. hep-th/9705128 On Euclidean spinors and Wick rotations (1996) 37 [ (1978) 294. ]. D-instantons and asymptotic geometries (1997) 467 [ B 370 Supersymmetry constraints on type IIB supergravity hep-th/97041189 hep-th/9803090 Supersymmetry enhancement of D and matrix theory, Adv. Theor. Math. Phys. (1998) 158 [ B 140 B 498 n T hep-th/9608174 ]. B 510 Phys. Lett. , (1997) 4934 [ ]. ]. ]. ]. (1998) 1801 [ . hep-th/9702057 Covariant field equations of chiral Nucl. Phys. 15 Dirichlet branes and Ramond-Ramond charges (1983) 269. A Wick rotation for spinor fields: the canonical approach Why is the matrix theory correct? Nucl. Phys. (1996) 29 [ Bound states of strings and , -branes on A possible constructive approach to (super D56 , 0 D Nucl. Phys. B 226 hep-th/9510017 , B 389 (1998) 369 [ hep-th/9709220 hep-th/9710009 hep-th/9803231 hep-th/9510135 of the model Lett. 433 [ superstring Quant. Grav. [ model Phys. superstring theory Phys. Rev. hep-th/9808061 [ 4724 [ ter spacetimes and dual superconformal theories [ [7] N. Seiberg, [4] N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, [8] J.H. Schwarz, [5] M. Fukuma, H. Kawai, Y. Kitazawa and A. Tsuchiya, [9]G.W.Gibbons,M.B.GreenandM.J.Perry, [6] A. Sen, [12] P. van Nieuwenhuizen and A. Waldron, [14] E. Bergshoeff and K. Behrndt, [13] A. Waldron, [15] R. Kallosh and J. Kumar, [10] M.B. Green and S. Sethi, [11] H. Nicolai, [17] J. Polchinski, [18] E. Witten, [16] H.J. Boonstra, B. Peeters and K. Skenderis,