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Interactions of type IIB D-branes from the D-instanton matrix model
NUCLEAR PHYbilCS
ELSEVIER
m l
Nuclear Physics B 511 (1998) 629-646
Interactions of type IIB D-branes from the D-instanton matrix model I. Chepelev a,1, A.A. Tseytlin b,2 a Institute of Theoretical and Experimental Physics, Moscow 117259, Russia b Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 23 May 1997; accepted 24 September 1997
Abstract We compute long-distance interaction potentials between certain 1/2 and 1/4 supersymmetric D-brane configurations of type liB theory, demonstrating detailed agreement between classical supergravity and one-loop instanton matrix model results. This confirms the interpretation of Dbranes as described by classical matrix model backgrounds as being 'populated' by a large number of D-instantons, i.e. as corresponding to non-marginal bound states of branes of lower dimensions. In the process, we establish a precise relation between matrix model expressions and non-abelian F 4 terms in the super Yang-Mills effective action. (~) 1998 Elsevier Science B.V. PACS: 11.25.Hf; 11.30.Pb Keywords: Yang-Mills theory; Matrix model; D-instanton
1. Introduction The aim of this paper is to discuss interactions between some D-branes in the type IIB matrix model o f Ref. [ 1] (see also Ref. [ 2 ] ) . Our approach will be that of Ref. [3], where the D = 10 U(N) super Yang-Mills theory reduced to a point was not related to the (Schild form of) type IIB string action as in Refs. [ 1,4] but was interpreted as the direct D-instanton counterpart of the D0-brane matrix model of Ref. [5]. The two matrix models can be put into correspondence using T-duality in the time direction. I E-mail: [email protected] 2 Also at Lebedev Physics Institute, Moscow. E-mail: [email protected] 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S 0 5 5 0 - 3 2 13 ( 9 7 ) 0 0 6 5 8 - 5
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L Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
The Dp-brane configurations in the instanton matrix model can be described [ 1,3,68] in a similar way as in the 0-brane matrix model [5,9,10]. As was pointed out in [3], they should be identified not with 'pure' type IIB D-branes but with D-branes 'populated' by a large number of D-instantons just like D-branes in the matrix model of Ref. [5] are 'populated' by a large number of 0-branes [11,12]. In what follows we shall confirm this interpretation by demonstrating that the corresponding long-distance interaction potentials computed in the matrix model and in supergravity are in precise agreement. The matrix model (SYM) result is the same as the short-distance limit of the one-loop open string theory amplitude while the supergravity result is the long-distance limit of the tree-level closed string theory potential. They agree in the N ---* c¢ limit in which the brane configurations become supersymmetric for the same reason as in the 0-brane matrix model [5,11]. The U(N) SYM theory reduced to a point describes a collection of N D-instantons [ 13,14]. When some of the ten euclidean dimensions are compactified on a torus T p+I, the classical backgrounds represented by constant abelian fluxes (JAm, An] = iFmn) correspond [ 3] to 1/2 supersymmetric non-marginal bound states of type IIB Dp-branes (i.e. 1 + i , 3 ÷ 1 + i , 5 + 3 + 1 + i ) wrapped over the dual torus 7~p+I. The configuration with self-dual strength [Am, An] represents the 1/4 supersymmetric marginal bound state of D3-brane and D-instantons which we shall denote as 311i [15,16]. There is a close T-duality relation to similar configurations in the 0-brane matrix model [9-12]. Indeed, the interaction potentials between such D-branes in the instanton matrix model computed below are direct counterparts of the corresponding results in M(atrix) theory found in Refs. [5,17,11,12] for interactions between 1/2 supersymmetric branes and in [ 18] for interactions involving 1/4 supersymmetric branes. We shall consider two examples: (i) interaction between D-instantons and 1/2 supersymmetric 'Dp-branes', i.e. nonmarginal p -4- (p - 2) ÷ . . . + 1 + i bound states; (ii) interaction between 'D-string', i.e. 1-4-i bound state, and 1/4 supersymmetric marginal 3-brane-instanton bound state 3[1i. In Section 2 we shall determine the corresponding closed string theory (supergravity) potentials using the classical probe method (see Ref. [18] and references therein). In Section 4 we shall reproduce the same expressions by a one-loop calculation in the instanton matrix model. In Section 3 we shall present some general results about oneloop effective action in D ~< 10 SYM theories and explain their relation to the matrix model computations of the leading terms in long-distance interaction potentials. One natural generalisation of the present work is to 1/8 supersymmetric bound states probed by D-instantons or other type IIB 'D-branes'. In particular, one may consider D-brane configurations corresponding to D = 5 black holes as in Refs. [ 19-23].
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2. Closed string theory (supergravity) description 2.1. D-instanton-'Dp-brane' interaction To determine the D-instanton-'Dp-brane' interaction potential we shall consider the latter, i.e. the p + (p - 2) + . . . + 1 + i bound state of type lib D-branes (p = - 1 , 1 , 3 , 5 ) as a probe moving in the classical D-instanton background. 3 This probe can be described, as in [ 18], by the standard Dp-brane action with a constant worldvolume gauge field background. The relevant terms in the euclidean Dp-brane action are ( m , n = 1 . . . . . p + 1; i,j =p + 2 . . . . . 10)
lp=-TP[S-xeV'~ / Ap+I
p+l
--J"
/det
(Gmn+GijO.,XiO.XJ+.'Fm.)
k
where .~mn - T-IF,,,n (in what follows Bin, = 0) and C2k is the RR 2k-form potential. We used the static gauge and took the target-space metric in the block-diagonal form. In general, the Dp-brane tension is [24]
Tp = np~'p = npg -1 (27r)(I-p)t2T (p+l)I2,
T =_ (2'rrd) - I .
(2.2)
(0.)
We shall assume that the euclidean world-volume of a type IIB Dp-brane is wrapped over a (rectangular) torus T p+I with volume Vt,+l = (2~-)p+IRI . . . Rt>+l and that there is a constant world-volume gauge field background
-fl
0
U,,,, =
'..
,
l~ ½(p+l).
(2.3)
o ft -fl 0 The Dp-brane with the flux (2.3) on its world-volume represents the non-marginal bound state (p + (p - 2) + . . . + 1 + i) of D-branes of dimensions p , p - 2 . . . . [ 15] (with branes of 'intermediate' dimensions being wrapped over different cycles of the torus). The total numbers of branes of each type are l
np_2=npZTrrEfkR,k-lR21< .....
k=l
1 n-i
= npV~,+l I I rfk k=l
2~r '
(2.4)
as can be read off from the Chern-Simons terms in the D-brane action (2.1) [25]. 3 By the potential we shall mean the interaction part of the euclidean action. The euclidean time coordinate may be assumed to belong to the internal p + l-dimensional toms. Alternatively, one may consider the pbranes discussed below as being ' ( p + 1)-instantons' 115], with the time coordinate being orthogonal to the internal toms.
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The D-instanton background 'smeared' in the directions of the toms T p+I (Xl = Yl . . . . . Xp+l =Yp+l) is [26] 4
ds21o= Hl_/?(dy~ + ... + dy2+l + dxidx,), e 4'=H-l,
Co = H - I - 1,
(2.5)
Q(Pl+1 ) H-l = 1 + - - ?.7-p '
r2 = x i x i,
We shall use the notation Qp~") for the coefficient in the harmonic function Qp~") Hp = 1 + r7_p_n
of the p-brane background which is smeared in n transverse toroidal directions. In general,
Qp = Npg( 2.rr) (5-p )/7~l.(p-7)/2( to6_p ) -1, QCpn)= Npg( 2"n') (5-p)/2T(P-7)/2(
(Ok_ 1 =
gn~o6_p_n)
27rk/2/F(k/2),
(2.6)
-1
= NpNpl+nQp+n(2~r)n/2Tn/2Vn 1,
(2.7)
where V, is the volume of the flat internal toms. Substituting the background (2.5) into the Dp-brane action (2.1) and ignoring the dependence of Xi on the world-volume coordinates Xm (so that the matrix under the square root in (2.1) becomes Hl__/~3mn+ .T'mn), we find
,/ H_, + f~ -
lp = -TpVp+, H-_ 1 1-[
l ] ( H - I - 1) 1--[ fk
k=l
•
(2.8)
k=l
Defining the 'interaction potential' V(r) (r 2 XiXi) as the deviation from the 'free' action of the non-marginal p + ... + i bound state, =
/
lp = I~°) - V=-TpVp+, H V/1 + f ~ - V,
(2.9)
k=l
we get for the leading long-distance term in V
V = r--~_pa(__Pl+l)rpVp+1 H ~ m= 1 +o
(')
.
~ k= 1
1
2(1
l
fk + f2) + k~ll ~/ 1 + f 2 "-" k
]
J
1
(2.10)
The coefficient here is 4 We use the symbol 'i' and subscript ' - 1 ' to denote D-instantonsand the correspondingquantities.
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Q(('I+I)TpVp+I _ = 23-t(3
- - 1) .vl~ l - 4 n p l V*'_ l ,
p = 2 / - 1.
(2.11)
In the limit of the large background field ~,,, (f~ >> 1), i.e. for large instanton 'occupation number' n-i (2.4), we find (we assume that l ~< 3 and set T = 1)5 l
V-
l
1 l2_l(3_l)!npN_lHf rS_2
m
m=l
!
2Zf~_ 4 -
f~-2
+...
(2.12)
k=l
For example, in the case of p = 1, i.e. the D-instanton-'D-string' interaction, ~) =
1 - r--6nl N-I f3 + ....
f ~ f{-l.
(2.13)
Note that the potential (2.12) vanishes for p = 3 and fl = f2. In this case the background field Urn,, is self-dual and the interaction between the D-instanton and the 3 + 1 + i nonmarginal bound state becomes essentially the same as the interaction between the Dinstanton and the 31]i marginal bound state 6 hut ' i - (3ili)' is a BPS configuration [ 15]. An analogous conclusion is reached in the T-dual case of the 0-brane-4 + 2 + 0 bound state interaction: when the magnetic flux on the 4-brane is self-dual, the 0 - ( 4 + 2 + 0) interaction is the same as the 0-(4[i0) one [18]. The expression (2.12) can he put in the following 'covariant' form:
Jr- . . . .
.f~mn ~ ( .)Unto) - 1 .
(2.14)
Since the D-instanton number in (2.4) is equal to
n_j = n p ( 2 7 r ) - t V z # ~ ,
(2.15)
we can represent (2.14) also as (3-l)!
~1.
~,
1
~
-
2
~(Yrm.Yrm.)]+
If'ink ~k. f'.sSr~m
(2.16)
where 9zt is the volume of the dual torus, ½tgzt =
= (27r) 2t.
(2.17)
5 Let us note that the subleading l / r 2(7-p) term in V (2.9) is proportional to (cf. Eq. (2.10)) 1 m= 1
k= 1
2(1+s
+
"=
fk
1
x/J+s
8
1 k= I
1
1 k= 1
'
The leading term in the large field (fk ~ cx~) expansion of this expression vanishes. 6 The D-instanton does not couple to the D-string charge; the contribution of the latter is in any case suppressed for large fk.
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The ~4 coefficient in this expression is exactly the same as the quartic term in the expansion of the Born-Infeld action x/det(6mn + 7"ran) or in the open string effective action. This can be seen directly from (2.8) by noting that the expression there is
H- I [~/det(amn + .T'mn) - ~ ] =~
(HZll[ J
det(6mn + , , _ l J , , ~ j
The reason for this non-trivial coincidence (note that J-re, is the inverse of the background field .T,,m in the probe action) will become clear below when we reproduce (2.16) from the matrix model.
2.2. Interaction of 'D-string' with 3-brane-instanton bound state To determine the interaction potential between the non-marginal bound state of the D-string and the D-instanton and the marginal bound state of the D3-brane and the D-instanton we shall consider 1 + i as a probe moving in the 311i background. As above, the action for the 1 + i probe will be the D-string action (2.1) with a constant flux (2.3) on a 2-torus representing the D-instanton charge. The 3LIi type IIB supergravity background [16] is T-dual to 4110 or 5111 solutions [27]. We shall assume that the 3-brane world-volume is wrapped around the 4-torus (in directions 1,2, 3, 4) and that the world-volume of the ( 1 + i)-brane probe is parallel to the (5, 6) directions, i.e. the world-volumes do not share common dimensions. 7 The corresponding metric, dilaton and RR scalar fields smeared in the (5,6) directions are [271
ds20 =
(a_ln3)
e ~ = H-l,
1/2 [ n 3 [ (dy2
+ . . . + dy2) + dy~ + dy~ + dxidxi],
Co = HT_I - 1,
H-l = 1 + r--5-,
(2.18)
H3 = 1 + r2 ,
where Q(pn) are given by (2.7) (C2 = 0; the value of the C4 background will not be important below). Ignoring the dependence on the derivatives of Xi we find for the 'D-string' probe action ll ( f ~ f l )
II = - T 1 / d2j [H-I x/H-1H3 + f 2 - ( H~_l - 1 ) f ] =-T1V2f[I+HSI(X/I+H_IH3f-2-I)]
=-T1V2x/I+ f Z - V .
(2.19)
The leading long-distance interaction term in V is 7Here the adequateinterpretationis that the time directionis orthogonalto both of the ( 1 + i) and (31]i) world-volumes [ 15].
L Chepelev,A.A. Tseytlin/NuclearPhysicsB 511 (1998)629-646 ~=2~T1V21~---~[Q~2) 1 +1 f2
Q[6) ( 1
f
635
]
2
(2.20) This expression is in direct T-duality correspondence with the static potential between the 2 + 0 and 4110 bound states in [ 18]. The potential (2.20) has the following large f (large instanton charge n-l of 1 + i) expansion, cf. Eq. (2.13): V
= ~--~th ( N 3 f - N-,Tr2V4-'f 3 - 1N3/3) + . . . .
f~
f-',
(2.21)
where V4 is the volume of the 4-torus. s V can be expressed in terms of n-l = nl (2~) -J ½ f =
n127rf'z-lf -1
(2.22)
as follows:
4~r--------~n_l
(4~.)2N_IV4f 4 - ½N3f4
+...
(2.23)
Since in the matrix model representation N3 will be the instanton number of a gauge field on the dual 4-torus, (2.23) will be, like (2.16), proportional to the integral of F 4 terms over the dual 6-torus ( N 3 f 4 will be a subleading correction).
3. One-loop effective action in D ~ 10 super Yang-Milis theory To put matrix model computations in a proper perspective, it is useful to give a summary of some general results about the one-loop effective action F ( A ) of maximally supersymmetric YM theory in D ~< 10 dimensions.
3.1. UV divergences and 'large mass' expansion In general, oo
F= 1 ~ c a l n d e t A ~ = --21 a
A
str~--,Cae_S& ' 2
(3.1)
a
where the sum over a runs over bosonic, background gauge ghost and fermionic contributions taken with appropriate relative coefficients (ca = 1 , - 2 , - ¼). Aa are second-order differential operators (-D2+,t=') depending on the background value of the gauge field and A ~ oo is the UV cutoff. The divergent part of F can be expressed in terms of the DeWitt-Seeley coefficients b,,, 8 The large f limit of the r~ subleading term in V is
l --~-fiTl V2Q3( 2 ) (2Q_( 6 )j +
Q~2))f_3
1. Chepelev,A.A. Tseytlin/NuclearPhysics B 511 (1998)629-646
636 (tr
l ~ .... e -s~ ) s ~ o - (4,n.)o/--------~Zs 2° f dDxbn(A),
(3.2)
n--O
i.e. (4~r)l 0/2f
F (oo) =
( d° x ---D-bO A° + --ff-S-~b2 A °-2 1 In A2 g~ ) + ~Ao-4 b4 + . . . + -2 (3.3)
where bn = Y'~'~acabn(Aa)" For pure Y M theory [28] b4 = / ( D
- 2 6 ) T r F 2 n (the
appearance o f the coefficient D - 26 can be understood from string theory [ 29 ] ), while for D = 10 S Y M theory and its reductions to lower d i m e n s i o n s [28] b0 = I)2 = b4 = b6 = 0,
(3.4)
so that S Y M theories in D ~< 7 are o n e - l o o p U V finite. At the same time, b8 and hi0 are, in general, n o n - v a n i s h i n g . In particular, in a constant abelian b a c k g r o u n d bl0 = 0 but b8 ~ F 4 4: 0, i m p l y i n g the presence o f a logarithmic divergence in D = 8 S Y M and a quadratic divergence in D = 10 theory [ 2 8 ] . The general n o n - a b e l i a n expressions for b8 and bl0 ( u p to F 5 terms) in S Y M theory were found in Ref. [29] (based on the results o f Ref. [30] ) 1
b 8 = 2 T r ( F m k F n k F m r F n r + "~FmkFnkFnrFmr -
!
"~FmkFmkFnrFnr -- l F m k F n r F m k F n r )
(3.5) hi0 = - ~ T r
1 (DqFmkFnkDqFmrFnr -+- ~DqFmkFnkDqFnrFmr
-1DqVmkFmkDqVnrFnr - 1DqFmkFnrDqFmkFnr) + O ( F 5 ) .
(3.6)
The trace Tr is in the adjoint representation 9 and we dropped the gauge-dependent terms O(DmFmk) which vanish on the equations o f motion. The reason w h y the structure o f b8 (i.e. o f the coefficient o f the quadratic divergence in D = 10 S Y M ) is the same as the one of the F 4 term in the open superstring effective action was explained in [ 2 9 ] . lo Let us n o w formally shift Aa by the same constant term M 2 and define the 'IRregularised' effective action FM
FM = -~ Z a
c~,lndet(Aa + M2) = - 2
e-SM2tr~-'Cae-S~" A_ 2
(3.7)
a
9For generators of SU(N) Tr(TaTb) = N6ab, tr(TaTb) = ½8ab and TrX2 = 2NtrX2, TrX4 = 2NtrX 4 + 6(trXZ) 2, X = XaTa (see Ref. [31]; similar expressions in Appendix B of [29] should be multiplied by a factor of 2). The same relations hold for a matrix X belonging to the U(N) algebra provided X in the r.h.s. is replaced by its traceless part X ~ X"= X - l t r XI. 10This term can be extracted from the d ~ 0 limit of the string one-loop effective action ( al-rF4 "--+AZF4) if one includes planar as well as non-planar (tr F 2 )2) contributions. The tree-level open string effective action contains a similar F 4 term (the kinematic factor in the tree-level and one-loop 4-vector amplitude is the same [33] ) but with tr instead of Tr [32].
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637
This modified one-loop effective action is finite in D ~< 7 and has the following large M expansion (we use (3.2)):
F M ~ --'~
dDxZ
( 4 ,n .) D / 2 M n _ D b n n--'O
1
[r(~--~) b
[
r( ~o~_.~D) MIO_ o
] (3.8)
bj0 + . . . .j
The explicit form of the leading term is
I'M
=
,
(3 - ~D). 3(~-b/-~M8-_ D
/
l dD xTr (Fn,kF,,kFmrF,~ + ~FmkFnkFnrFmr
-¼FmkFmkFnrFnr - -gFmkFnrFmkFnr) + 0
(,) ~
(3.9)
,
or, equivalently, in terms of the trace in the fundamental representation of U ( N ) ,
FM =
2(3 - ½D)! i 1. . . . 3(--~) -6/T-~-o a dO x (Ntr [PmkPnkPmrPnr -'~ ~FmkFnkFnrFmr 1. . . . I . . . . -- ~FmkFmkFnrFnr - gFmkFnrFmkFnr]
+3 [tr(FmkP, k) tr(FmrPnr) + ½tr(FmkFnr) tr(FnkFmr)
--¼tr(FmkPnr) tr(FmkPnr) -- ½tr(g,,kPm~)tr(Fnr~r)]) + 0
(l) ~
, (3.10)
where P,,,, =_ Fro, - -~trFmnI. This expansion is useful in discussions of long-distance interactions between Dp-branes where D = p + 1 and M is proportional to the separation b between the branes, i.e. M 2 = Tb 2 (expressions related to special cases of (3.9), (3.10) appeared in [1,6,34] and, in particular, in [20]; see also below). Note that the subleading O ( 1 / M 1°-0) correction determined by bl0 vanishes in the case of constant abelian backgrounds which describe, e.g., interactions between 1/2 supersymmetric non-marginal bound states of D-branes. The coefficient bl0 is, in general, non-vanishing for non-abelian background fields.
3.2. Constant abelian gauge field background The one-loop effective action of SYM theory in D dimensions can be computed explicitly for a constant abelian gauge field background (i.e. for Finn = FmnTt I belonging to the Cartan subalgebra of a compact semisimple Lie algebra) following Refs. [28,35]. The basis Tl (1 = 1 . . . . . r) in the Cartan subalgebra in the adjoint representation can be chosen as a set o f d x d diagonal matrices Tt = diag(O, . . . , O , "~l~(1) , --t'tl~(l) , . . . , o ~ q ) , _ _ o ( ~ q ) ) , where {o~ i) } are positive roots (i . . . 1,. . q, q = ½(d - r), ~-]~7=1°~l(i)Oil'(i) = ~llZ). Let us
I. Chepelev,A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
638
define F~mn (i) = F ml n a 1(i) and assume that all F mn (i) have 'block-diagonal' form (we choose the space-time dimension to be even, D = 2l) it
0 fl --fl i)
/
0
F~,I~ =
"..
.
(3.11)
_f(i)
"D/2
Then one finds the following general expression for FM in (3.7) ( V o --= fdOx): 2VD F u - - (4.n.) o/2
ds 0
e_M2s
~
Z
i=1
×
k=l
sinh ~i)s
4( IIc°shf?s- )1}
(3.12)
In what follows we shall consider the special case when the background is such that N of F(,~ are equal to the same Finn while the rest vanish, i.e. when ~i) = fk, i = 1 . . . . . N. The corresponding background field strength is given by diagonal matrices in the adjoint or fundamental representations, F~,adj) = diag(O . . . . . O, Finn, -Finn
. . . . .
Finn, -Fmn ),
Fn(fund)= ( F ~ nI O) In
0
where I a unit n × n matrix and N =
n(N - n). Then
OO
/ ' M :-
2NVD (4¢r)0/2
I
ds
e_M2s
0
x ~~=t ~ = ~s
Z ( c o s h 2 f k s - - l ) - - 4 ( I c o s "h fk=l ks--lk=l
"
(3.13)
This integral is UV convergent for D ~< 7 and logarithmically divergent for D = 8, implying also the presence of an O( F 4) quadratic UV divergence in D = 10 SYM theory. It is also IR divergent for certain f~i) and small enough M (which is a manifestation of the well-known tachyonic instability of the YM theory in a constant abelian background which is not cured by supersymmetry). 11 The expressions that follow hold also in the more general case if the parameters ~i) are simply replaced by (i) (separately for each i) according to the rules [ 35 ] ff]~-I Lorentz invariants constructed out of F.m 2h =
(f(ki))
l (_l
2 .......
~hl~(i)
17(i)
inlinlm2,.
,Fn(;?_ltnl , h :
l .....
l.
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639
For example, the standard (M = 0) one-loop effective action for maximally supersymmetric SU(2) YM theory in D = 4 in the background Finn = Finn ~3 2 (i.e. N = 1) is [28] 4V4 7ds fls f2s (4~) 2 J s 3 sinhfj----ssinhf2~ ( c ° s h f l s - c ° s h f 2 s ) 2
F -
(3.14)
0
For comparison with the matrix model expressions, it is useful to separate a factor A/" ~ v/-d~ Fro, in FM, representing it as (3.15)
I'M = NAf)/V, 0/2
A/" = (2~')-D/2VD I ~ f ~ = ( 2 7 r ) - D / 2 V o ~ ,
(3.16)
k=l
zx3 ]/V = - 2 f
J
D/ 2
__dse_M2s -[7-
0
1 .tx 2 sinh fks
s
k=l
FD/2 /D/2 )] × /~-'~ ( c o s h 2 f k s - 1 ) - 4 ( 1 - - [ c o s h f k s - 1 . L k=l
(3.17)
"- k=l
In the matrix model context .Af-1 will be an integer (or a rational number, cf. (2.15), (2.22)) and will be cancelled against a factor contained in N (see Section 4). Special cases of FM (3.15) or (up to an overall coefficient) )4; appeared in the discussions of interaction potentials between D-branes (see, e.g., Refs. [36,1,17,11,12] ). 12 The general expression (3.17) was given in [6], where it was describing the potential between the parallel 'Dp-brane' and anti-'Dp-brane'. The leading terms in the large M expansions of FM and ~A; can be found directly from (3.13), (3.17),
Z2f ) +° =
"
(3-sD).N -(~)-~i~-"
(3-½o),
~/~---- ~ T ~ M ~ I ~ ~
Vo
~
[
1
](
ZmkFnkFmrFnr--~(FmkZmk) 2 ~ - 0
~
1
)
,
(3.18)
1 IFmkfnkFmrfnr - "~(Zmkfmk l ) 21~- 0 ( ~ l ) . (3.19)
They have the expected F 4 structure (3.9). In fact, for the abelian background considered above one has from (3.5) 12 For example, for fl =
iv,
f2 . . . . . ft = 0, M = b we get from (3.17) the (light open string mode part of)
phase shift for the scattering of two 0-branes [ 3 6 1 , 6 = -iYV
= f ~ ~ e- l'2'~( sin vs) -
1(cos
2vs-4
cos v s + 3 ).
640
I. Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646 I" D/2 =2N[2~-_
bs=Tr[F4-4(F2)2]=2N[F4-¼(F2)2]
/ D/2 ~ 2-I ~ ,J ' f2Z) (3.20)
4. Matrix model (super Yang-Mills) description In this section we will demonstrate that the leading-order terms in the long-distance potentials between BPS bound states with 1/2 and 1/4 of supersymmetry (2.12) and (2.21) computed in Section 2 using classical closed string effective field theory methods are indeed reproduced by the instanton matrix model, i.e. by the corresponding one-loop SYM computations. The instanton matrix model is defined by the D = 10 U(N) SYM Lagrangian reduced to 0+0 dimensions (in this section we shall assume that T -1 = 2~-ctt = 1) L = ---tr 2gs
1) ,
[X~z,X~]2 +20ryu [O,X~
(4.1)
where the elements of N x N matrix O are 16-component real spinors and Yl0 -- I16x16. We shall consider the background gauge field .4~ = T (X1 . . . . . -~10), where the components X'i=
0)
g~2)
,
i = l . . . . . 8,10
(4.2)
correspond to the coordinates of the two BPS objects and
-'~9 -----(b0 :)
(4.3)
represents their separation b. The calculation of the SYM one-loop effective action in this background is similar to the one described in [ 18]. Let us define the operators H = (x~l)® I _
I (~ .,~2)*) 2 ,
Hij
--..~jl) @ I +
1 ® .~,}j2),,
(4.4)
where _~j') = [XT~'),X)')], XT~j2)= [X'~2),,~J 2)] a n d * is complex conjugation. The one-loop effective action is the sum of the bosonic, ghost and fermionic contributions,
W -~ WB -b WG + WF,
(4.5)
WB =lndet(H6u~ + 2 H u , ) , WF=-½1ndet ( H + Z
WG = -21ndet H ,
yiyjHij),]
where the operators act in the U(N) matrix index space, Lorentz vector space and Lorentz spinor space. In the case of the background
1. Chepelev,A.A. Tseytlin/NuclearPhysicsB 511 (1998)629-646 " 1 0 = ( icgr 0 ) 0 0
641
)(i = ( b o O ) * 0 '
'
the resulting expression for W in (4.5) becomes the same as the one found in the 0-brane matrix model [ 18] for the relative motion of two BPS objects along the direction i,. This may be viewed as a manifestation of T-duality in string theory or the Eguchi-Kawai reduction in large N SYM theory [37,9,1].
4.1. D-instanton-'Dp-brane' interaction A 'Dp-brane' wrapped over a torus 7~p+l is represented by the following classical solution of the instanton matrix model (m, n = 1. . . . . p + I = 2/):
2,,, = T-l(icgm + ftm)In_,xn_~,
[Xm,X,,] =iT-2Fmnln_,Xn_l,
(4.6)
(0¢, ;,/
where 0m act on functions on the torus and Fro, is a constant abelian field strength. This configuration corresponds [3] to the (p + (p - 2) + . . . + 1 + i) type liB bound state wrapped over the torus T p+I dual to 7~p+1. We shall choose F,,,, in the form
-f, 0
~m, =-- T-l~',,m =
"..
.
(4.7)
0
-f, This background field is (minus) the inverse of the one which appears in the T-dual string theory picture (2.3), i.e. 5~m,SCm,, = ~mm', or fk = f~-l. Let us explain the reason for this inverse identification between the fluxes in the matrix model and string theory descriptions (see Refs. [9,11] for discussions of T-dual type IIA cases). The U(N) SYM theory on Te+lrepresents np = N Dp-branes with euclidean world-volumes wrapped over the torus [ 13]. By T-duality [37], it is also describing h_j = N D-instantons on the dual toms 7~p+l. Turning on the background field (2.3) on T p+I we get a non-marginal bound state p + (p - 2) + . . . -I- i with the 'induced' instanton number (2.4), (2.15) equal to n - i = npVp+l(Tf/2rr) 2 (for simplicity here we set all fk to be equal to f ) . Since T-duality along all of the directions of the torus T p+I interchanges instantons with Dp-branes, the corresponding bound state wrapped over 7~p+1 contains hp = n - i Dp-branes and h - i = np instantons. If the background field fi',,,n that produces this charge distribution is (4.7), then h-1 = fiz,Vp+l (Tf/2rr)'@. As a result, np - - n - l ,
~t-i = np
(4.8)
implies 2 ~'p+'
= 1,
i.e. f f = 1,
(4.9)
L Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
642
where we used (2.17). The matrix model background describing the configuration of a 'Dp-brane' (p = 2l - 1) with the world-volume directions Xl . . . . . Xp+l and N-1 D-instantons located at the origin, which are separated from each other by a distance b in the 9th direction, is thus represented by 1)=ql,
1)=pl . . . . . . .
[qk,P,] =ifk~k,I, fk
2t-l = q l ,
"'2t = P l ,
=b,
= f ~-l ,
(4.10) (4.11)
with all other )~(#1) components being equal to zero and the N-1 x N-l matrix )~2) ( # = 1 . . . . . 10) having zero entries. The bosonic, ghost and fermionic contributions to the one-loop effective action W (4.5) in this background are O(3
O0
wB = N 1
b{~}6#~
- 2i~'u~
)
Wo=-2N-, Z
,
{.}=o
Trlnb~.},
{n}=0
(4.12)
WF=-IN-I Z
(b~.} ' )
Trln
+-~'~ij.~(j
l
b~n}~b 2 4- Zfk(2nk + 1),
,
k=l
{,,}=o
(4.13) where {n} --- {nl . . . . . nt}. The constant background field matrix ~u~ has (4.7) as non-zero entries. The resulting effective action W (4.5) is given by __
W= --2N_l
x
0
s
I ~(c°sh2fksk=,
e_bEs
[I 2sinh1 fks k=l
-
1 ) (- 4
tc°shfksII
~=l
-
1)1
"
(4.14)
This is equal to the one-loop effective action F M of the U(N) SYM theory on the dual torus ~p+l in a constant abelian background proportional to #ran and with an IR cutoff M = b (see (3.7), (3.13), (3.15)). The dimension of the fundamental representation of the YM gauge group is the total number of instantons N = N-1 + n-1 (with both N-1 and n-l assumed to be large). Indeed, let us set D = p + 1 = 21, f~ = fk, VD = V2I and
N = N-1 + n - l ,
N = n-lN-i
(4.15)
in the SYM expression (3.13) or (3.15), (3.17). The corresponding abelian U(N) SYM background in the fundamental representation is given by a diagonal N x N matrix
( ;ni0)
1. Chepelev,A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
643
where I is a unit n - i × n-1 matrix. In the adjoint representation it has N = n - l N _ l non-zero entries ( ~ m , , - 5 ~ , m ) (differences of the diagonal values of the Caftan subalgebra element in the fundamental representation). Equivalently, N = q ( N - l + n - I ) q( N _ j ) - q ( n - l ) = n-1 N - l , where q( N ) = ½N( N - 1 ) is the number of positive roots of U ( N ) . The resulting SYM effective action is thus given by (3.13). Since the factor Af in (3.16) on the dual torus is N" = ~ - l / h p = n p / n _ l , we conclude that for np = 1 (assumed in the derivation of (4.14)) the factor of N" = l / n _ l cancels out, i.e. F M
= .A/'NW = N - 1 W = W.
(4.16)
Retaining only the leading term in the large distance (b ---, oc) expansion of W, we find the same expression as in (3.18), (3.19)
i [ W-
bS_2t2-1(3 - l ) ! N - l 1-I f ~ l 2 k=l
+ o
"
k=l
(4.17) Remarkably, with b = r and fk = f k I this coincides with the long-distance interaction potential (2.12), (2.14), (2.16) found from supergravity in the limit of large instanton number n - i (large fk or small f/,). The coefficient of the subleading t,,o1_2~ term in (4.17) turns out to be zero. This is a consequence of the vanishing of the coefficient bl0 (3.6) in (3.8) in a constant abelian background. Note, however, that the powers of r--- b in the subleading terms in (4.14), (4.17) and in the supergravity expression (2.10) do not match in general. The same universal expressions (4.17) or (3.20) describe also interactions of T-dual configurations of branes in the 0-brane matrix model. For example, the scattering of the two 0-branes is represented by the (electric) background F01 = iv, N = no + No, N = noNo, i.e. l = 1, ffl = iv, 8 = - i W ~ ~ v 3. The case of a 0-brane scattering on a 2 + 0 brahe is represented by the N × N matrix
F,,,,, ~-(1)
( ~,l,)I,0 ×,,0 0 "~ \ 0 5ctl,2,)INo×NoJ ' -
Y'mn = fE.m for m, n = 2, 3, .¢~(2),,1.= iVEmn for m, n = 0, 1 ( 1 is dual to the direction of the 0-brahe motion, 2, 3 are dual to the directions of 2-brane) and we assume that the volume V2 of the two-toms (20,21) is chosen so that i v ~ = 2~-. In the adjoint representation this background is given by Finn = diag(0 . . . . . 0, Finn, -Finn . . . . . Fro., -Fro.) with the total number of non-zero entries 2N = 2n0N0 and Finn = ~,(~) - ~-(2) (which has a block-diagonal structure as the non-zero components of ~,I,l) and ~,(2) are orthogonal). As a result, here 1 = 2, f , = f , f2 = iv and thus 6 = - i W ~ ( 1 / v f r 4 ) ( f 4 + 2 v 2 f 2 + v 4) in agreement with Ref. [ 1 1 ]. where
nln
~
mn
644
1. Chepelev.A.A. Tseytlin/NuclearPhysicsB 511 (1998)629-646
4.2. Interaction of a 'D-string' with a 3-brane-instanton bound state The matrix model background corresponding to the configuration of the 'D-string' (1 + i) wrapped over a 2-torus in the (5,6) directions and the D3-brane-D-instanton bound state (311i) wrapped over a 4-torus in the (1,2, 3,4) directions, which are separated by a distance b in the 9-direction, is given by (a = 1 . . . . . 4; T = 1) ~{l) = T - I ( i 0 . +.~a) = p . ,
f(~2) =q,
£(62) =p,
)~l) = b ,
(4.18)
[q,p] =ifI,
where the U ( N _ t ) gauge potential Aa is representing the charge N3 instanton on the dual torus 7TM (see, e.g., Ref. [9] ), i.e. its field strength Gab = OaAb- ObAa-i[Aa, Ab] satisfies
Gab = *Gab,
167r 21
fd4x
tr(GabGab) = N3 •
(4.19)
~4
The bosonic, ghost and fermionic contributions to the effective action (4.5) in this background are WB = Z T r l n
[(b] + P2)6u~ - 2i5t'~] ,
n=0
WG : --2 Z T r l n (b ] + p 2 ) , n=0
W F = - - ~1 Z~T r l n
( pb~2 +
i t k) , b~2 =--b2 + f(2n-t-1). + ~ymk.Y'£,
(4.20)
n=0
where
"T'tmn= ( Gab ) 0.,~O 'afl
'
Jr~ = fe~.
(4.21)
The expression for W is computed in a similar way as in the T-dual case of the
(2 + 0)-(4110) configuration considered in Ref. [ 18]. The final result for the leading long-distance (b ~ co) term in W is
l ~ 2 [f f d4x tr(GabGab) - gaN-lf3] +O ( ~ ) W - 32~.2b f-4 = 12b2 ( N 3 f - I@~2V4N-lf 3) + O ( ~ - - ~ ) .
(4.22)
This becomes exactly the same as the supergravity result for the interaction potential (2.21) after we set b = r, f = f - l , nl = 1, use the relation (2.17), i.e. f,'4I~ = (2~) 4, and note that since it is assumed that N-I >> N3, the last term in (2.21) can be neglected. The expression (4.22) is equivalent to the leading-order O ( F 4) term in the U(n_l + N-1 ) SYM effective action (3.9) on the dual 6-torus ~2 x i~4 computed in the background
1. Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646 Fmn=a~.mn=(Gab 0 ) 0 .~aflIn_l xn_, "
645 (4.23)
Indeed, substituting ~m, (4.23) into (3.10), i.e. into b8 in (3.5), and observing that the G 4 terms cancel out (b8 vanishes on a self-dual gauge field background) one is left with the abelian ~ 4 term and the 'cross term' ,fi"2G2, i.e. bs ( . T ' ) = T r [ ~ 4 -
'~(.T'-2 _ ½.~-2G2]=2n_, I N _ i f 4 - f2tr(GabGab)] ,
(4.24)
where in the first expression .~" and G are the adjoint representation counterparts of the N x N matrices in the fundamental representation with non-vanishing n - i x n - l and N_l x N_l blocks (note that the spatial components of,f" and G are orthogonal). One can also derive this expression by formally representing Gab as an abelian matrix with two equal 2 x 2 blocks, i.e. Gab = geabIN_j xN_l for a, b = 1,2 and a, b = 3,4. Then one may apply formula (3.20) for the abelian background with fl = f , 1"2 = f3 = g. This gives bs = 2 n - l N _ l [ 2 ( f 4 + 2g 4) -- ( f 2 +2g2)2] = 2 n - l N - l ( f 4 - 4f2g2), i.e. the same result as in (4.24) since tr(GabG,,b) = 4 N - l g 2, (2~)-2f,'ag2N_l = N3 (cf. Eq. (2.4)). For nl = 1 one has n-1 = 2 7 r ~ - t f -1 (see Eq. (2.22)) and concludes that
FM -
'
2(4~)3M2~'2
/
d4x b8 + O
(,) ~
(4.25)
f4 in (3.9) is equal to W in (4.22) for b = M. This is also in agreement with the supergravity potential represented in the form (2.23). Thus we have found complete agreement between the one-loop matrix model and classical supergravity expressions for the leading-order long-distance interaction potentials.
Acknowledgements We are grateful to Yu. Makeenko for useful discussions. The work of I.C. was supported in part by CRDF grant 96-RP1-253. A.A.T. acknowledges the support of PPARC and the European Commission TMR programme grant ERBFMRX-CT96-0045.
References [11 N. lshibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B 498 (1997) 467, hep-th/9612115. 121 V. Periwal, Matrices on a point as the theory of everything, Phys. Rev. D 55 (1997) 1711, hepth/9611103. 131 A.A. Tseytlin, On non-abelian generalisation of Born-lnfeld action in string theory, Nucl. Phys. B 501 (1997) 41, hep-th/9701125. 141 M. Li, Strings from liB matrices, Nucl. Phys. B 499 (1997) 149, hep-th/9612222. [5] T. Banks, W. Fischler, S.H. Shenker and k Susskind, Phys. Rev. D 55 (1997) 5112, hep-th/9610043.
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[6] I. Chepelev, Y. Makeenko and K. Zarembo, Properties of D-branes in matrix model of liB superstring, Phys. Lett. B 400 (1997) 43, hep-th/9701151. [7] A. Fayyazuddin and D.J. Smith, p-brane solutions in IKKT liB matrix theory, hep-th/9701168. [8] A. Fayyazuddin, Y. Makeenko, E Olesen, DJ. Smith and K. Zarembo, Towards a non-perturbative formulation of IIB superstrings by matrix models, hep-tfi/9703038. [9] O.J. Ganor, S. Ramgoolam and W. Taylor, Branes, fluxes and duality in M(atrix) theory, Nucl. Phys. B 492 (1997) 191, hep-th/9611202. [10] T. Banks, N. Seiberg and S. Shenker, Branes from matrices, hep-th/9612157. [11] G. Lifschytz and S.D. Mathur, Supersymmetry and membrane interactions in M(atrix) theory, hepth/9612087. [12] G. Lifschytz, Four-brahe and six-brane interactions in M(atrix) theory, hep-th/9612223. [13] E. Witten, Nucl. Phys. B 460 (1995) 335, hep-th/9510135. [14] M.B. Green and M. Gutpede, Phys. Lett. B 398 (1997) 69, hep-th/9612127. [ 15] M.B. Green and M. Gutperle, Nucl. Phys. B 476 (1996) 484, hep-th/9604091. [16] A.A. Tseytlin, Phys. Rev. Lett. 78 (1997) 1864, hep-th/9612164. [ 17] O. Aharony and M. Berkooz, Membrane dynamics in M(atrix) theory, hep-th/9611215. [ 18] I. Chepelev and A.A. Tseytlin, Long-distance interactions of D-brane bound states and longitudinal 5-brahe in M(atrix) theory, hep-th/9704127. 119] M. Douglas, J. Polchinski and A. Strominger, Probing five-dimensional black holes with D-branes, hep-th/9703031. [20] J. Maldacena, Probing near extremal black holes with D-branes, hep-th/9705053. [21] M. Li and E. Martinec, Matrix black holes, hep-th/9703211; On the entropy of matrix black holes, hep-th/9704134. [221 R. Dijkgraaf, E. Verlinde and H. Verlinde, 5-D Black holes and matrix strings, hep-th/9704018. [23] E. Halyo, M(atrix) black holes in five dimensions, hep-th/9705107. [24] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724; TASI Lectures on D-Branes, hep-th/9611050. 125] M.R. Douglas, Branes within branes, hep-th/9512077. [26] G.W. Gibbons, M.B. Green and M.J. Perry, Phys. Lett. B 370 (1996) 37, hep-th/9511080. [27] A.A. Tseytlin, Nucl. Phys. B 475 (1996) 149, hep-th/9604035. [28] E.S. Fradkin and A.A. Tseytlin, Nucl. Phys. B 227 (1983) 252. [29] R.R. Metsaev and A.A. Tseytlin, Nucl. Phys. B 298 (1988) 109. [301 A.E.M. Van de Ven, Nucl. Phys. B 250 (1985) 593. [31] S. Okubo and J. Patera, Phys. Rev. D 31 (1985) 2669. 132] A.A. Tseytlin, Nucl. Phys. B 276 (1986) 391; D. Gross and E. Witten, Nucl. Phys. B 277 (1986) 1. [33] M.B. Green and J. Schwarz, Nucl. Phys. B 198 (1982) 441; M.B. Green, J. Schwarz and L. Brink, Nucl. Phys. B 198 (1982) 474. [34] D. Berenstein and R. Corrado, M(atrix) theory in various dimensions, Phys. Lett. B 406 (1997) 37, hep-th/9702108. [35] I.G. Avramidi, J. Math. Phys. 36 (1995) 1557, gr-qc/9403035; hep-th/9604160. 1361 M.R. Douglas, D. Kabat, P. Pouliot and S.H. Shenker, Nuel. Phys. B 485 (1997) 85, hep-th/9608024. 137] W. Taylor, Phys. Lett. B 394 (1997) 283, hep-th/9611042.
ELSEVIER
m l
Nuclear Physics B 511 (1998) 629-646
Interactions of type IIB D-branes from the D-instanton matrix model I. Chepelev a,1, A.A. Tseytlin b,2 a Institute of Theoretical and Experimental Physics, Moscow 117259, Russia b Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 23 May 1997; accepted 24 September 1997
Abstract We compute long-distance interaction potentials between certain 1/2 and 1/4 supersymmetric D-brane configurations of type liB theory, demonstrating detailed agreement between classical supergravity and one-loop instanton matrix model results. This confirms the interpretation of Dbranes as described by classical matrix model backgrounds as being 'populated' by a large number of D-instantons, i.e. as corresponding to non-marginal bound states of branes of lower dimensions. In the process, we establish a precise relation between matrix model expressions and non-abelian F 4 terms in the super Yang-Mills effective action. (~) 1998 Elsevier Science B.V. PACS: 11.25.Hf; 11.30.Pb Keywords: Yang-Mills theory; Matrix model; D-instanton
1. Introduction The aim of this paper is to discuss interactions between some D-branes in the type IIB matrix model o f Ref. [ 1] (see also Ref. [ 2 ] ) . Our approach will be that of Ref. [3], where the D = 10 U(N) super Yang-Mills theory reduced to a point was not related to the (Schild form of) type IIB string action as in Refs. [ 1,4] but was interpreted as the direct D-instanton counterpart of the D0-brane matrix model of Ref. [5]. The two matrix models can be put into correspondence using T-duality in the time direction. I E-mail: [email protected] 2 Also at Lebedev Physics Institute, Moscow. E-mail: [email protected] 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S 0 5 5 0 - 3 2 13 ( 9 7 ) 0 0 6 5 8 - 5
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L Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
The Dp-brane configurations in the instanton matrix model can be described [ 1,3,68] in a similar way as in the 0-brane matrix model [5,9,10]. As was pointed out in [3], they should be identified not with 'pure' type IIB D-branes but with D-branes 'populated' by a large number of D-instantons just like D-branes in the matrix model of Ref. [5] are 'populated' by a large number of 0-branes [11,12]. In what follows we shall confirm this interpretation by demonstrating that the corresponding long-distance interaction potentials computed in the matrix model and in supergravity are in precise agreement. The matrix model (SYM) result is the same as the short-distance limit of the one-loop open string theory amplitude while the supergravity result is the long-distance limit of the tree-level closed string theory potential. They agree in the N ---* c¢ limit in which the brane configurations become supersymmetric for the same reason as in the 0-brane matrix model [5,11]. The U(N) SYM theory reduced to a point describes a collection of N D-instantons [ 13,14]. When some of the ten euclidean dimensions are compactified on a torus T p+I, the classical backgrounds represented by constant abelian fluxes (JAm, An] = iFmn) correspond [ 3] to 1/2 supersymmetric non-marginal bound states of type IIB Dp-branes (i.e. 1 + i , 3 ÷ 1 + i , 5 + 3 + 1 + i ) wrapped over the dual torus 7~p+I. The configuration with self-dual strength [Am, An] represents the 1/4 supersymmetric marginal bound state of D3-brane and D-instantons which we shall denote as 311i [15,16]. There is a close T-duality relation to similar configurations in the 0-brane matrix model [9-12]. Indeed, the interaction potentials between such D-branes in the instanton matrix model computed below are direct counterparts of the corresponding results in M(atrix) theory found in Refs. [5,17,11,12] for interactions between 1/2 supersymmetric branes and in [ 18] for interactions involving 1/4 supersymmetric branes. We shall consider two examples: (i) interaction between D-instantons and 1/2 supersymmetric 'Dp-branes', i.e. nonmarginal p -4- (p - 2) ÷ . . . + 1 + i bound states; (ii) interaction between 'D-string', i.e. 1-4-i bound state, and 1/4 supersymmetric marginal 3-brane-instanton bound state 3[1i. In Section 2 we shall determine the corresponding closed string theory (supergravity) potentials using the classical probe method (see Ref. [18] and references therein). In Section 4 we shall reproduce the same expressions by a one-loop calculation in the instanton matrix model. In Section 3 we shall present some general results about oneloop effective action in D ~< 10 SYM theories and explain their relation to the matrix model computations of the leading terms in long-distance interaction potentials. One natural generalisation of the present work is to 1/8 supersymmetric bound states probed by D-instantons or other type IIB 'D-branes'. In particular, one may consider D-brane configurations corresponding to D = 5 black holes as in Refs. [ 19-23].
I. Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
631
2. Closed string theory (supergravity) description 2.1. D-instanton-'Dp-brane' interaction To determine the D-instanton-'Dp-brane' interaction potential we shall consider the latter, i.e. the p + (p - 2) + . . . + 1 + i bound state of type lib D-branes (p = - 1 , 1 , 3 , 5 ) as a probe moving in the classical D-instanton background. 3 This probe can be described, as in [ 18], by the standard Dp-brane action with a constant worldvolume gauge field background. The relevant terms in the euclidean Dp-brane action are ( m , n = 1 . . . . . p + 1; i,j =p + 2 . . . . . 10)
lp=-TP[S-xeV'~ / Ap+I
p+l
--J"
/det
(Gmn+GijO.,XiO.XJ+.'Fm.)
k
where .~mn - T-IF,,,n (in what follows Bin, = 0) and C2k is the RR 2k-form potential. We used the static gauge and took the target-space metric in the block-diagonal form. In general, the Dp-brane tension is [24]
Tp = np~'p = npg -1 (27r)(I-p)t2T (p+l)I2,
T =_ (2'rrd) - I .
(2.2)
(0.)
We shall assume that the euclidean world-volume of a type IIB Dp-brane is wrapped over a (rectangular) torus T p+I with volume Vt,+l = (2~-)p+IRI . . . Rt>+l and that there is a constant world-volume gauge field background
-fl
0
U,,,, =
'..
,
l~ ½(p+l).
(2.3)
o ft -fl 0 The Dp-brane with the flux (2.3) on its world-volume represents the non-marginal bound state (p + (p - 2) + . . . + 1 + i) of D-branes of dimensions p , p - 2 . . . . [ 15] (with branes of 'intermediate' dimensions being wrapped over different cycles of the torus). The total numbers of branes of each type are l
np_2=npZTrrEfkR,k-lR21< .....
k=l
1 n-i
= npV~,+l I I rfk k=l
2~r '
(2.4)
as can be read off from the Chern-Simons terms in the D-brane action (2.1) [25]. 3 By the potential we shall mean the interaction part of the euclidean action. The euclidean time coordinate may be assumed to belong to the internal p + l-dimensional toms. Alternatively, one may consider the pbranes discussed below as being ' ( p + 1)-instantons' 115], with the time coordinate being orthogonal to the internal toms.
632
L Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
The D-instanton background 'smeared' in the directions of the toms T p+I (Xl = Yl . . . . . Xp+l =Yp+l) is [26] 4
ds21o= Hl_/?(dy~ + ... + dy2+l + dxidx,), e 4'=H-l,
Co = H - I - 1,
(2.5)
Q(Pl+1 ) H-l = 1 + - - ?.7-p '
r2 = x i x i,
We shall use the notation Qp~") for the coefficient in the harmonic function Qp~") Hp = 1 + r7_p_n
of the p-brane background which is smeared in n transverse toroidal directions. In general,
Qp = Npg( 2.rr) (5-p )/7~l.(p-7)/2( to6_p ) -1, QCpn)= Npg( 2"n') (5-p)/2T(P-7)/2(
(Ok_ 1 =
gn~o6_p_n)
27rk/2/F(k/2),
(2.6)
-1
= NpNpl+nQp+n(2~r)n/2Tn/2Vn 1,
(2.7)
where V, is the volume of the flat internal toms. Substituting the background (2.5) into the Dp-brane action (2.1) and ignoring the dependence of Xi on the world-volume coordinates Xm (so that the matrix under the square root in (2.1) becomes Hl__/~3mn+ .T'mn), we find
,/ H_, + f~ -
lp = -TpVp+, H-_ 1 1-[
l ] ( H - I - 1) 1--[ fk
k=l
•
(2.8)
k=l
Defining the 'interaction potential' V(r) (r 2 XiXi) as the deviation from the 'free' action of the non-marginal p + ... + i bound state, =
/
lp = I~°) - V=-TpVp+, H V/1 + f ~ - V,
(2.9)
k=l
we get for the leading long-distance term in V
V = r--~_pa(__Pl+l)rpVp+1 H ~ m= 1 +o
(')
.
~ k= 1
1
2(1
l
fk + f2) + k~ll ~/ 1 + f 2 "-" k
]
J
1
(2.10)
The coefficient here is 4 We use the symbol 'i' and subscript ' - 1 ' to denote D-instantonsand the correspondingquantities.
633
I. Chepelev, A,A. Tseytlin/Nuclear Physics B 511 (1998) 6 2 9 - 6 4 6
Q(('I+I)TpVp+I _ = 23-t(3
- - 1) .vl~ l - 4 n p l V*'_ l ,
p = 2 / - 1.
(2.11)
In the limit of the large background field ~,,, (f~ >> 1), i.e. for large instanton 'occupation number' n-i (2.4), we find (we assume that l ~< 3 and set T = 1)5 l
V-
l
1 l2_l(3_l)!npN_lHf rS_2
m
m=l
!
2Zf~_ 4 -
f~-2
+...
(2.12)
k=l
For example, in the case of p = 1, i.e. the D-instanton-'D-string' interaction, ~) =
1 - r--6nl N-I f3 + ....
f ~ f{-l.
(2.13)
Note that the potential (2.12) vanishes for p = 3 and fl = f2. In this case the background field Urn,, is self-dual and the interaction between the D-instanton and the 3 + 1 + i nonmarginal bound state becomes essentially the same as the interaction between the Dinstanton and the 31]i marginal bound state 6 hut ' i - (3ili)' is a BPS configuration [ 15]. An analogous conclusion is reached in the T-dual case of the 0-brane-4 + 2 + 0 bound state interaction: when the magnetic flux on the 4-brane is self-dual, the 0 - ( 4 + 2 + 0) interaction is the same as the 0-(4[i0) one [18]. The expression (2.12) can he put in the following 'covariant' form:
Jr- . . . .
.f~mn ~ ( .)Unto) - 1 .
(2.14)
Since the D-instanton number in (2.4) is equal to
n_j = n p ( 2 7 r ) - t V z # ~ ,
(2.15)
we can represent (2.14) also as (3-l)!
~1.
~,
1
~
-
2
~(Yrm.Yrm.)]+
If'ink ~k. f'.sSr~m
(2.16)
where 9zt is the volume of the dual torus, ½tgzt =
= (27r) 2t.
(2.17)
5 Let us note that the subleading l / r 2(7-p) term in V (2.9) is proportional to (cf. Eq. (2.10)) 1 m= 1
k= 1
2(1+s
+
"=
fk
1
x/J+s
8
1 k= I
1
1 k= 1
'
The leading term in the large field (fk ~ cx~) expansion of this expression vanishes. 6 The D-instanton does not couple to the D-string charge; the contribution of the latter is in any case suppressed for large fk.
L Chepelev,A.A. Tseytlin/NuclearPhysicsB 511 (1998)629-646
634
The ~4 coefficient in this expression is exactly the same as the quartic term in the expansion of the Born-Infeld action x/det(6mn + 7"ran) or in the open string effective action. This can be seen directly from (2.8) by noting that the expression there is
H- I [~/det(amn + .T'mn) - ~ ] =~
(HZll[ J
det(6mn + , , _ l J , , ~ j
The reason for this non-trivial coincidence (note that J-re, is the inverse of the background field .T,,m in the probe action) will become clear below when we reproduce (2.16) from the matrix model.
2.2. Interaction of 'D-string' with 3-brane-instanton bound state To determine the interaction potential between the non-marginal bound state of the D-string and the D-instanton and the marginal bound state of the D3-brane and the D-instanton we shall consider 1 + i as a probe moving in the 311i background. As above, the action for the 1 + i probe will be the D-string action (2.1) with a constant flux (2.3) on a 2-torus representing the D-instanton charge. The 3LIi type IIB supergravity background [16] is T-dual to 4110 or 5111 solutions [27]. We shall assume that the 3-brane world-volume is wrapped around the 4-torus (in directions 1,2, 3, 4) and that the world-volume of the ( 1 + i)-brane probe is parallel to the (5, 6) directions, i.e. the world-volumes do not share common dimensions. 7 The corresponding metric, dilaton and RR scalar fields smeared in the (5,6) directions are [271
ds20 =
(a_ln3)
e ~ = H-l,
1/2 [ n 3 [ (dy2
+ . . . + dy2) + dy~ + dy~ + dxidxi],
Co = HT_I - 1,
H-l = 1 + r--5-,
(2.18)
H3 = 1 + r2 ,
where Q(pn) are given by (2.7) (C2 = 0; the value of the C4 background will not be important below). Ignoring the dependence on the derivatives of Xi we find for the 'D-string' probe action ll ( f ~ f l )
II = - T 1 / d2j [H-I x/H-1H3 + f 2 - ( H~_l - 1 ) f ] =-T1V2f[I+HSI(X/I+H_IH3f-2-I)]
=-T1V2x/I+ f Z - V .
(2.19)
The leading long-distance interaction term in V is 7Here the adequateinterpretationis that the time directionis orthogonalto both of the ( 1 + i) and (31]i) world-volumes [ 15].
L Chepelev,A.A. Tseytlin/NuclearPhysicsB 511 (1998)629-646 ~=2~T1V21~---~[Q~2) 1 +1 f2
Q[6) ( 1
f
635
]
2
(2.20) This expression is in direct T-duality correspondence with the static potential between the 2 + 0 and 4110 bound states in [ 18]. The potential (2.20) has the following large f (large instanton charge n-l of 1 + i) expansion, cf. Eq. (2.13): V
= ~--~th ( N 3 f - N-,Tr2V4-'f 3 - 1N3/3) + . . . .
f~
f-',
(2.21)
where V4 is the volume of the 4-torus. s V can be expressed in terms of n-l = nl (2~) -J ½ f =
n127rf'z-lf -1
(2.22)
as follows:
4~r--------~n_l
(4~.)2N_IV4f 4 - ½N3f4
+...
(2.23)
Since in the matrix model representation N3 will be the instanton number of a gauge field on the dual 4-torus, (2.23) will be, like (2.16), proportional to the integral of F 4 terms over the dual 6-torus ( N 3 f 4 will be a subleading correction).
3. One-loop effective action in D ~ 10 super Yang-Milis theory To put matrix model computations in a proper perspective, it is useful to give a summary of some general results about the one-loop effective action F ( A ) of maximally supersymmetric YM theory in D ~< 10 dimensions.
3.1. UV divergences and 'large mass' expansion In general, oo
F= 1 ~ c a l n d e t A ~ = --21 a
A
str~--,Cae_S& ' 2
(3.1)
a
where the sum over a runs over bosonic, background gauge ghost and fermionic contributions taken with appropriate relative coefficients (ca = 1 , - 2 , - ¼). Aa are second-order differential operators (-D2+,t=') depending on the background value of the gauge field and A ~ oo is the UV cutoff. The divergent part of F can be expressed in terms of the DeWitt-Seeley coefficients b,,, 8 The large f limit of the r~ subleading term in V is
l --~-fiTl V2Q3( 2 ) (2Q_( 6 )j +
Q~2))f_3
1. Chepelev,A.A. Tseytlin/NuclearPhysics B 511 (1998)629-646
636 (tr
l ~ .... e -s~ ) s ~ o - (4,n.)o/--------~Zs 2° f dDxbn(A),
(3.2)
n--O
i.e. (4~r)l 0/2f
F (oo) =
( d° x ---D-bO A° + --ff-S-~b2 A °-2 1 In A2 g~ ) + ~Ao-4 b4 + . . . + -2 (3.3)
where bn = Y'~'~acabn(Aa)" For pure Y M theory [28] b4 = / ( D
- 2 6 ) T r F 2 n (the
appearance o f the coefficient D - 26 can be understood from string theory [ 29 ] ), while for D = 10 S Y M theory and its reductions to lower d i m e n s i o n s [28] b0 = I)2 = b4 = b6 = 0,
(3.4)
so that S Y M theories in D ~< 7 are o n e - l o o p U V finite. At the same time, b8 and hi0 are, in general, n o n - v a n i s h i n g . In particular, in a constant abelian b a c k g r o u n d bl0 = 0 but b8 ~ F 4 4: 0, i m p l y i n g the presence o f a logarithmic divergence in D = 8 S Y M and a quadratic divergence in D = 10 theory [ 2 8 ] . The general n o n - a b e l i a n expressions for b8 and bl0 ( u p to F 5 terms) in S Y M theory were found in Ref. [29] (based on the results o f Ref. [30] ) 1
b 8 = 2 T r ( F m k F n k F m r F n r + "~FmkFnkFnrFmr -
!
"~FmkFmkFnrFnr -- l F m k F n r F m k F n r )
(3.5) hi0 = - ~ T r
1 (DqFmkFnkDqFmrFnr -+- ~DqFmkFnkDqFnrFmr
-1DqVmkFmkDqVnrFnr - 1DqFmkFnrDqFmkFnr) + O ( F 5 ) .
(3.6)
The trace Tr is in the adjoint representation 9 and we dropped the gauge-dependent terms O(DmFmk) which vanish on the equations o f motion. The reason w h y the structure o f b8 (i.e. o f the coefficient o f the quadratic divergence in D = 10 S Y M ) is the same as the one of the F 4 term in the open superstring effective action was explained in [ 2 9 ] . lo Let us n o w formally shift Aa by the same constant term M 2 and define the 'IRregularised' effective action FM
FM = -~ Z a
c~,lndet(Aa + M2) = - 2
e-SM2tr~-'Cae-S~" A_ 2
(3.7)
a
9For generators of SU(N) Tr(TaTb) = N6ab, tr(TaTb) = ½8ab and TrX2 = 2NtrX2, TrX4 = 2NtrX 4 + 6(trXZ) 2, X = XaTa (see Ref. [31]; similar expressions in Appendix B of [29] should be multiplied by a factor of 2). The same relations hold for a matrix X belonging to the U(N) algebra provided X in the r.h.s. is replaced by its traceless part X ~ X"= X - l t r XI. 10This term can be extracted from the d ~ 0 limit of the string one-loop effective action ( al-rF4 "--+AZF4) if one includes planar as well as non-planar (tr F 2 )2) contributions. The tree-level open string effective action contains a similar F 4 term (the kinematic factor in the tree-level and one-loop 4-vector amplitude is the same [33] ) but with tr instead of Tr [32].
L Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
637
This modified one-loop effective action is finite in D ~< 7 and has the following large M expansion (we use (3.2)):
F M ~ --'~
dDxZ
( 4 ,n .) D / 2 M n _ D b n n--'O
1
[r(~--~) b
[
r( ~o~_.~D) MIO_ o
] (3.8)
bj0 + . . . .j
The explicit form of the leading term is
I'M
=
,
(3 - ~D). 3(~-b/-~M8-_ D
/
l dD xTr (Fn,kF,,kFmrF,~ + ~FmkFnkFnrFmr
-¼FmkFmkFnrFnr - -gFmkFnrFmkFnr) + 0
(,) ~
(3.9)
,
or, equivalently, in terms of the trace in the fundamental representation of U ( N ) ,
FM =
2(3 - ½D)! i 1. . . . 3(--~) -6/T-~-o a dO x (Ntr [PmkPnkPmrPnr -'~ ~FmkFnkFnrFmr 1. . . . I . . . . -- ~FmkFmkFnrFnr - gFmkFnrFmkFnr]
+3 [tr(FmkP, k) tr(FmrPnr) + ½tr(FmkFnr) tr(FnkFmr)
--¼tr(FmkPnr) tr(FmkPnr) -- ½tr(g,,kPm~)tr(Fnr~r)]) + 0
(l) ~
, (3.10)
where P,,,, =_ Fro, - -~trFmnI. This expansion is useful in discussions of long-distance interactions between Dp-branes where D = p + 1 and M is proportional to the separation b between the branes, i.e. M 2 = Tb 2 (expressions related to special cases of (3.9), (3.10) appeared in [1,6,34] and, in particular, in [20]; see also below). Note that the subleading O ( 1 / M 1°-0) correction determined by bl0 vanishes in the case of constant abelian backgrounds which describe, e.g., interactions between 1/2 supersymmetric non-marginal bound states of D-branes. The coefficient bl0 is, in general, non-vanishing for non-abelian background fields.
3.2. Constant abelian gauge field background The one-loop effective action of SYM theory in D dimensions can be computed explicitly for a constant abelian gauge field background (i.e. for Finn = FmnTt I belonging to the Cartan subalgebra of a compact semisimple Lie algebra) following Refs. [28,35]. The basis Tl (1 = 1 . . . . . r) in the Cartan subalgebra in the adjoint representation can be chosen as a set o f d x d diagonal matrices Tt = diag(O, . . . , O , "~l~(1) , --t'tl~(l) , . . . , o ~ q ) , _ _ o ( ~ q ) ) , where {o~ i) } are positive roots (i . . . 1,. . q, q = ½(d - r), ~-]~7=1°~l(i)Oil'(i) = ~llZ). Let us
I. Chepelev,A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
638
define F~mn (i) = F ml n a 1(i) and assume that all F mn (i) have 'block-diagonal' form (we choose the space-time dimension to be even, D = 2l) it
0 fl --fl i)
/
0
F~,I~ =
"..
.
(3.11)
_f(i)
"D/2
Then one finds the following general expression for FM in (3.7) ( V o --= fdOx): 2VD F u - - (4.n.) o/2
ds 0
e_M2s
~
Z
i=1
×
k=l
sinh ~i)s
4( IIc°shf?s- )1}
(3.12)
In what follows we shall consider the special case when the background is such that N of F(,~ are equal to the same Finn while the rest vanish, i.e. when ~i) = fk, i = 1 . . . . . N. The corresponding background field strength is given by diagonal matrices in the adjoint or fundamental representations, F~,adj) = diag(O . . . . . O, Finn, -Finn
. . . . .
Finn, -Fmn ),
Fn(fund)= ( F ~ nI O) In
0
where I a unit n × n matrix and N =
n(N - n). Then
OO
/ ' M :-
2NVD (4¢r)0/2
I
ds
e_M2s
0
x ~~=t ~ = ~s
Z ( c o s h 2 f k s - - l ) - - 4 ( I c o s "h fk=l ks--lk=l
"
(3.13)
This integral is UV convergent for D ~< 7 and logarithmically divergent for D = 8, implying also the presence of an O( F 4) quadratic UV divergence in D = 10 SYM theory. It is also IR divergent for certain f~i) and small enough M (which is a manifestation of the well-known tachyonic instability of the YM theory in a constant abelian background which is not cured by supersymmetry). 11 The expressions that follow hold also in the more general case if the parameters ~i) are simply replaced by (i) (separately for each i) according to the rules [ 35 ] ff]~-I Lorentz invariants constructed out of F.m 2h =
(f(ki))
l (_l
2 .......
~hl~(i)
17(i)
inlinlm2,.
,Fn(;?_ltnl , h :
l .....
l.
1. Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
639
For example, the standard (M = 0) one-loop effective action for maximally supersymmetric SU(2) YM theory in D = 4 in the background Finn = Finn ~3 2 (i.e. N = 1) is [28] 4V4 7ds fls f2s (4~) 2 J s 3 sinhfj----ssinhf2~ ( c ° s h f l s - c ° s h f 2 s ) 2
F -
(3.14)
0
For comparison with the matrix model expressions, it is useful to separate a factor A/" ~ v/-d~ Fro, in FM, representing it as (3.15)
I'M = NAf)/V, 0/2
A/" = (2~')-D/2VD I ~ f ~ = ( 2 7 r ) - D / 2 V o ~ ,
(3.16)
k=l
zx3 ]/V = - 2 f
J
D/ 2
__dse_M2s -[7-
0
1 .tx 2 sinh fks
s
k=l
FD/2 /D/2 )] × /~-'~ ( c o s h 2 f k s - 1 ) - 4 ( 1 - - [ c o s h f k s - 1 . L k=l
(3.17)
"- k=l
In the matrix model context .Af-1 will be an integer (or a rational number, cf. (2.15), (2.22)) and will be cancelled against a factor contained in N (see Section 4). Special cases of FM (3.15) or (up to an overall coefficient) )4; appeared in the discussions of interaction potentials between D-branes (see, e.g., Refs. [36,1,17,11,12] ). 12 The general expression (3.17) was given in [6], where it was describing the potential between the parallel 'Dp-brane' and anti-'Dp-brane'. The leading terms in the large M expansions of FM and ~A; can be found directly from (3.13), (3.17),
Z2f ) +° =
"
(3-sD).N -(~)-~i~-"
(3-½o),
~/~---- ~ T ~ M ~ I ~ ~
Vo
~
[
1
](
ZmkFnkFmrFnr--~(FmkZmk) 2 ~ - 0
~
1
)
,
(3.18)
1 IFmkfnkFmrfnr - "~(Zmkfmk l ) 21~- 0 ( ~ l ) . (3.19)
They have the expected F 4 structure (3.9). In fact, for the abelian background considered above one has from (3.5) 12 For example, for fl =
iv,
f2 . . . . . ft = 0, M = b we get from (3.17) the (light open string mode part of)
phase shift for the scattering of two 0-branes [ 3 6 1 , 6 = -iYV
= f ~ ~ e- l'2'~( sin vs) -
1(cos
2vs-4
cos v s + 3 ).
640
I. Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646 I" D/2 =2N[2~-_
bs=Tr[F4-4(F2)2]=2N[F4-¼(F2)2]
/ D/2 ~ 2-I ~ ,J ' f2Z) (3.20)
4. Matrix model (super Yang-Mills) description In this section we will demonstrate that the leading-order terms in the long-distance potentials between BPS bound states with 1/2 and 1/4 of supersymmetry (2.12) and (2.21) computed in Section 2 using classical closed string effective field theory methods are indeed reproduced by the instanton matrix model, i.e. by the corresponding one-loop SYM computations. The instanton matrix model is defined by the D = 10 U(N) SYM Lagrangian reduced to 0+0 dimensions (in this section we shall assume that T -1 = 2~-ctt = 1) L = ---tr 2gs
1) ,
[X~z,X~]2 +20ryu [O,X~
(4.1)
where the elements of N x N matrix O are 16-component real spinors and Yl0 -- I16x16. We shall consider the background gauge field .4~ = T (X1 . . . . . -~10), where the components X'i=
0)
g~2)
,
i = l . . . . . 8,10
(4.2)
correspond to the coordinates of the two BPS objects and
-'~9 -----(b0 :)
(4.3)
represents their separation b. The calculation of the SYM one-loop effective action in this background is similar to the one described in [ 18]. Let us define the operators H = (x~l)® I _
I (~ .,~2)*) 2 ,
Hij
--..~jl) @ I +
1 ® .~,}j2),,
(4.4)
where _~j') = [XT~'),X)')], XT~j2)= [X'~2),,~J 2)] a n d * is complex conjugation. The one-loop effective action is the sum of the bosonic, ghost and fermionic contributions,
W -~ WB -b WG + WF,
(4.5)
WB =lndet(H6u~ + 2 H u , ) , WF=-½1ndet ( H + Z
WG = -21ndet H ,
yiyjHij),]
where the operators act in the U(N) matrix index space, Lorentz vector space and Lorentz spinor space. In the case of the background
1. Chepelev,A.A. Tseytlin/NuclearPhysicsB 511 (1998)629-646 " 1 0 = ( icgr 0 ) 0 0
641
)(i = ( b o O ) * 0 '
'
the resulting expression for W in (4.5) becomes the same as the one found in the 0-brane matrix model [ 18] for the relative motion of two BPS objects along the direction i,. This may be viewed as a manifestation of T-duality in string theory or the Eguchi-Kawai reduction in large N SYM theory [37,9,1].
4.1. D-instanton-'Dp-brane' interaction A 'Dp-brane' wrapped over a torus 7~p+l is represented by the following classical solution of the instanton matrix model (m, n = 1. . . . . p + I = 2/):
2,,, = T-l(icgm + ftm)In_,xn_~,
[Xm,X,,] =iT-2Fmnln_,Xn_l,
(4.6)
(0¢, ;,/
where 0m act on functions on the torus and Fro, is a constant abelian field strength. This configuration corresponds [3] to the (p + (p - 2) + . . . + 1 + i) type liB bound state wrapped over the torus T p+I dual to 7~p+1. We shall choose F,,,, in the form
-f, 0
~m, =-- T-l~',,m =
"..
.
(4.7)
0
-f, This background field is (minus) the inverse of the one which appears in the T-dual string theory picture (2.3), i.e. 5~m,SCm,, = ~mm', or fk = f~-l. Let us explain the reason for this inverse identification between the fluxes in the matrix model and string theory descriptions (see Refs. [9,11] for discussions of T-dual type IIA cases). The U(N) SYM theory on Te+lrepresents np = N Dp-branes with euclidean world-volumes wrapped over the torus [ 13]. By T-duality [37], it is also describing h_j = N D-instantons on the dual toms 7~p+l. Turning on the background field (2.3) on T p+I we get a non-marginal bound state p + (p - 2) + . . . -I- i with the 'induced' instanton number (2.4), (2.15) equal to n - i = npVp+l(Tf/2rr) 2 (for simplicity here we set all fk to be equal to f ) . Since T-duality along all of the directions of the torus T p+I interchanges instantons with Dp-branes, the corresponding bound state wrapped over 7~p+1 contains hp = n - i Dp-branes and h - i = np instantons. If the background field fi',,,n that produces this charge distribution is (4.7), then h-1 = fiz,Vp+l (Tf/2rr)'@. As a result, np - - n - l ,
~t-i = np
(4.8)
implies 2 ~'p+'
= 1,
i.e. f f = 1,
(4.9)
L Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
642
where we used (2.17). The matrix model background describing the configuration of a 'Dp-brane' (p = 2l - 1) with the world-volume directions Xl . . . . . Xp+l and N-1 D-instantons located at the origin, which are separated from each other by a distance b in the 9th direction, is thus represented by 1)=ql,
1)=pl . . . . . . .
[qk,P,] =ifk~k,I, fk
2t-l = q l ,
"'2t = P l ,
=b,
= f ~-l ,
(4.10) (4.11)
with all other )~(#1) components being equal to zero and the N-1 x N-l matrix )~2) ( # = 1 . . . . . 10) having zero entries. The bosonic, ghost and fermionic contributions to the one-loop effective action W (4.5) in this background are O(3
O0
wB = N 1
b{~}6#~
- 2i~'u~
)
Wo=-2N-, Z
,
{.}=o
Trlnb~.},
{n}=0
(4.12)
WF=-IN-I Z
(b~.} ' )
Trln
+-~'~ij.~(j
l
b~n}~b 2 4- Zfk(2nk + 1),
,
k=l
{,,}=o
(4.13) where {n} --- {nl . . . . . nt}. The constant background field matrix ~u~ has (4.7) as non-zero entries. The resulting effective action W (4.5) is given by __
W= --2N_l
x
0
s
I ~(c°sh2fksk=,
e_bEs
[I 2sinh1 fks k=l
-
1 ) (- 4
tc°shfksII
~=l
-
1)1
"
(4.14)
This is equal to the one-loop effective action F M of the U(N) SYM theory on the dual torus ~p+l in a constant abelian background proportional to #ran and with an IR cutoff M = b (see (3.7), (3.13), (3.15)). The dimension of the fundamental representation of the YM gauge group is the total number of instantons N = N-1 + n-1 (with both N-1 and n-l assumed to be large). Indeed, let us set D = p + 1 = 21, f~ = fk, VD = V2I and
N = N-1 + n - l ,
N = n-lN-i
(4.15)
in the SYM expression (3.13) or (3.15), (3.17). The corresponding abelian U(N) SYM background in the fundamental representation is given by a diagonal N x N matrix
( ;ni0)
1. Chepelev,A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646
643
where I is a unit n - i × n-1 matrix. In the adjoint representation it has N = n - l N _ l non-zero entries ( ~ m , , - 5 ~ , m ) (differences of the diagonal values of the Caftan subalgebra element in the fundamental representation). Equivalently, N = q ( N - l + n - I ) q( N _ j ) - q ( n - l ) = n-1 N - l , where q( N ) = ½N( N - 1 ) is the number of positive roots of U ( N ) . The resulting SYM effective action is thus given by (3.13). Since the factor Af in (3.16) on the dual torus is N" = ~ - l / h p = n p / n _ l , we conclude that for np = 1 (assumed in the derivation of (4.14)) the factor of N" = l / n _ l cancels out, i.e. F M
= .A/'NW = N - 1 W = W.
(4.16)
Retaining only the leading term in the large distance (b ---, oc) expansion of W, we find the same expression as in (3.18), (3.19)
i [ W-
bS_2t2-1(3 - l ) ! N - l 1-I f ~ l 2 k=l
+ o
"
k=l
(4.17) Remarkably, with b = r and fk = f k I this coincides with the long-distance interaction potential (2.12), (2.14), (2.16) found from supergravity in the limit of large instanton number n - i (large fk or small f/,). The coefficient of the subleading t,,o1_2~ term in (4.17) turns out to be zero. This is a consequence of the vanishing of the coefficient bl0 (3.6) in (3.8) in a constant abelian background. Note, however, that the powers of r--- b in the subleading terms in (4.14), (4.17) and in the supergravity expression (2.10) do not match in general. The same universal expressions (4.17) or (3.20) describe also interactions of T-dual configurations of branes in the 0-brane matrix model. For example, the scattering of the two 0-branes is represented by the (electric) background F01 = iv, N = no + No, N = noNo, i.e. l = 1, ffl = iv, 8 = - i W ~ ~ v 3. The case of a 0-brane scattering on a 2 + 0 brahe is represented by the N × N matrix
F,,,,, ~-(1)
( ~,l,)I,0 ×,,0 0 "~ \ 0 5ctl,2,)INo×NoJ ' -
Y'mn = fE.m for m, n = 2, 3, .¢~(2),,1.= iVEmn for m, n = 0, 1 ( 1 is dual to the direction of the 0-brahe motion, 2, 3 are dual to the directions of 2-brane) and we assume that the volume V2 of the two-toms (20,21) is chosen so that i v ~ = 2~-. In the adjoint representation this background is given by Finn = diag(0 . . . . . 0, Finn, -Finn . . . . . Fro., -Fro.) with the total number of non-zero entries 2N = 2n0N0 and Finn = ~,(~) - ~-(2) (which has a block-diagonal structure as the non-zero components of ~,I,l) and ~,(2) are orthogonal). As a result, here 1 = 2, f , = f , f2 = iv and thus 6 = - i W ~ ( 1 / v f r 4 ) ( f 4 + 2 v 2 f 2 + v 4) in agreement with Ref. [ 1 1 ]. where
nln
~
mn
644
1. Chepelev.A.A. Tseytlin/NuclearPhysicsB 511 (1998)629-646
4.2. Interaction of a 'D-string' with a 3-brane-instanton bound state The matrix model background corresponding to the configuration of the 'D-string' (1 + i) wrapped over a 2-torus in the (5,6) directions and the D3-brane-D-instanton bound state (311i) wrapped over a 4-torus in the (1,2, 3,4) directions, which are separated by a distance b in the 9-direction, is given by (a = 1 . . . . . 4; T = 1) ~{l) = T - I ( i 0 . +.~a) = p . ,
f(~2) =q,
£(62) =p,
)~l) = b ,
(4.18)
[q,p] =ifI,
where the U ( N _ t ) gauge potential Aa is representing the charge N3 instanton on the dual torus 7TM (see, e.g., Ref. [9] ), i.e. its field strength Gab = OaAb- ObAa-i[Aa, Ab] satisfies
Gab = *Gab,
167r 21
fd4x
tr(GabGab) = N3 •
(4.19)
~4
The bosonic, ghost and fermionic contributions to the effective action (4.5) in this background are WB = Z T r l n
[(b] + P2)6u~ - 2i5t'~] ,
n=0
WG : --2 Z T r l n (b ] + p 2 ) , n=0
W F = - - ~1 Z~T r l n
( pb~2 +
i t k) , b~2 =--b2 + f(2n-t-1). + ~ymk.Y'£,
(4.20)
n=0
where
"T'tmn= ( Gab ) 0.,~O 'afl
'
Jr~ = fe~.
(4.21)
The expression for W is computed in a similar way as in the T-dual case of the
(2 + 0)-(4110) configuration considered in Ref. [ 18]. The final result for the leading long-distance (b ~ co) term in W is
l ~ 2 [f f d4x tr(GabGab) - gaN-lf3] +O ( ~ ) W - 32~.2b f-4 = 12b2 ( N 3 f - I@~2V4N-lf 3) + O ( ~ - - ~ ) .
(4.22)
This becomes exactly the same as the supergravity result for the interaction potential (2.21) after we set b = r, f = f - l , nl = 1, use the relation (2.17), i.e. f,'4I~ = (2~) 4, and note that since it is assumed that N-I >> N3, the last term in (2.21) can be neglected. The expression (4.22) is equivalent to the leading-order O ( F 4) term in the U(n_l + N-1 ) SYM effective action (3.9) on the dual 6-torus ~2 x i~4 computed in the background
1. Chepelev, A.A. Tseytlin/Nuclear Physics B 511 (1998) 629-646 Fmn=a~.mn=(Gab 0 ) 0 .~aflIn_l xn_, "
645 (4.23)
Indeed, substituting ~m, (4.23) into (3.10), i.e. into b8 in (3.5), and observing that the G 4 terms cancel out (b8 vanishes on a self-dual gauge field background) one is left with the abelian ~ 4 term and the 'cross term' ,fi"2G2, i.e. bs ( . T ' ) = T r [ ~ 4 -
'~(.T'-2 _ ½.~-2G2]=2n_, I N _ i f 4 - f2tr(GabGab)] ,
(4.24)
where in the first expression .~" and G are the adjoint representation counterparts of the N x N matrices in the fundamental representation with non-vanishing n - i x n - l and N_l x N_l blocks (note that the spatial components of,f" and G are orthogonal). One can also derive this expression by formally representing Gab as an abelian matrix with two equal 2 x 2 blocks, i.e. Gab = geabIN_j xN_l for a, b = 1,2 and a, b = 3,4. Then one may apply formula (3.20) for the abelian background with fl = f , 1"2 = f3 = g. This gives bs = 2 n - l N _ l [ 2 ( f 4 + 2g 4) -- ( f 2 +2g2)2] = 2 n - l N - l ( f 4 - 4f2g2), i.e. the same result as in (4.24) since tr(GabG,,b) = 4 N - l g 2, (2~)-2f,'ag2N_l = N3 (cf. Eq. (2.4)). For nl = 1 one has n-1 = 2 7 r ~ - t f -1 (see Eq. (2.22)) and concludes that
FM -
'
2(4~)3M2~'2
/
d4x b8 + O
(,) ~
(4.25)
f4 in (3.9) is equal to W in (4.22) for b = M. This is also in agreement with the supergravity potential represented in the form (2.23). Thus we have found complete agreement between the one-loop matrix model and classical supergravity expressions for the leading-order long-distance interaction potentials.
Acknowledgements We are grateful to Yu. Makeenko for useful discussions. The work of I.C. was supported in part by CRDF grant 96-RP1-253. A.A.T. acknowledges the support of PPARC and the European Commission TMR programme grant ERBFMRX-CT96-0045.
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