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Nuclear Physics B 634 (2002) 3–22 www.elsevier.com/locate/npe

Calculation of nonperturbative terms in open stringmodels

Julie D. BlumPhysics Department, University of Texas at Austin, Austin, TX 78712, USA

Received 19 December 2001; accepted 9 April 2002

Abstract

Nonperturbative corrections in type II string theory corresponding to Riemann surfaces with oneboundary are calculated in several noncompact geometries of desingularized orbifolds. One of these

models has a complicated phase structure which is explored. A general condition for integrality of the numerical invariants is discussed. © 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

String theory has provided mathematicians with many interesting conjectural resultsthat would be in some cases significantly more difficult to obtain through traditionaltechniques. Calculations that count the number of maps from Riemann surfaces into

Calabi–Yau manifolds are one example. Recently, these calculations have been extendedto Riemann surfaces with boundaries. The addition of boundaries for a generic type IIstring theory reduces the supersymmetry to N = 1 in four dimensions, and the counting of maps corresponds to holomorphic terms in the field theory. Such calculations are possiblyrelevant to an extension of the Standard Model.

In the following we will calculate the nonperturbative terms in a type II stringtheory generated by Riemann surfaces with one boundary, “disk instantons”. Most of thetechniques that will be employed here are discussed in [1,2], and other places. In honorof human decency, no additional references will be mentioned. We will focus on threenoncompact models, the blowups of the Z2 × Z2, Z2 × Z4, and Z7 orbifolds of C3. Thethree models support the conjecture that numerical invariants related mathematically to theEuler characteristic of the moduli space of open string instantons and physically to the

E-mail address: [email protected] (J.D. Blum).

0550-3213/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0550-32 13(02 )00 286- 9

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4 J.D. Blum / Nuclear Physics B 634 (2002) 3–22

counting of domain walls are integral in phases where Kähler parameters have a geometricinterpretation. The second model has an intricate moduli space with many phases. We

explore the phase structure of this model and see how it reduces in various limits to simplermodels. The complications of this model necessitated an understanding of what conditionsensure integrality of the numerical invariants. We explain why the geometric phases alwaysgive integers and the conditions under which fractions are possible.

2. Z2 × Z2

2.1. Toric geometry

The toric geometry for this model can be described by a linear sigma model (two-dimensional abelian gauge theory with N = 2 supersymmetry). There are six chiral fieldscarrying the following charges under three U (1) gauge fields.

l1 = (1, 0, 0, 1, −1, −1)

l2 = (0, 1, 0, −1, 1, −1)

(2.1)l3 = (0, 0, 1, −1, −1, 1)

The Calabi–Yau condition requires j lj i = 0. From the set of charges, one derives the

D-terms in the gauge theory.

(2.2)

j

lj i |xj |

2 = ri .

These equations can be solved leading to the “toric diagram”, shown in Fig. 1.The diagram is a projection of three dimensions onto the plane, and the angles shown

are not meant to be accurate. Generically, the diagram represents a T3 fibration. It shrinks

Fig. 1. Toric diagram of Z2 × Z2 blowup.

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J.D. Blum / Nuclear Physics B 634 (2002) 3–22 5

Fig. 2. Toric diagram of Z2 × Z2 flopped phase: r3 → −r3.

to a two-torus along planes xi = 0, to a circle along lines xi = xj = 0, and to a pointat the intersection of three lines. From the diagram one sees that there are three two-spheres and six noncompact two-cycles. Since this diagram resembles the intersection of three conifolds, one would expect other phases that replace a P1 by an S3. One can easilyvisualize a flopped phase as where r3 → −r3 in Fig. 2.

The equation for the Z2 × Z2 singularity can be obtained from gauge invariantcombinations of chiral fields as w2 = xyz where w, x , y , and z are complex variables.

To show explicitly that there are three P1’s, one should solve the Picard–Fuchsequations. These equations determine the Kähler parameters for the P1’s corrected byworldsheet instantons or equivalently complex parameters in the local mirror geometry,i.e.,

γ i

Ω , where Ω is the holomorphic three-form and γ i is a three-cycle. For this modelthe equations are

(2.3)

Θ1(Θ1 − Θ2 − Θ3) − z1

Θ21 − (Θ2 − Θ3)2

γ i

Ω = 0

and cyclic permutations where Θi = zi ∂zi and zi = e−t i with t i = ri + i θ i , the initial

Kähler parameter. Here θ i is a Fayet–Iliopoulos parameter in the linear sigma model. Onetakes linear combinations of the

γ iΩ normalized appropriately to obtain t̂

i, the instanton

corrected Kähler parameters. For the calculation here we need the inverse solutions whichturn out to be

(2.4)z1 = q1(1 + q2q3)

(1 + q1q2)(1 + q1q3)

and permutations where qi = e−t̂ i . Note that in finding unique solutions for the Picard–Fuchs equations, we have frequently had to change to a different basis of P1’s. Thesimplicity of these solutions makes this model amenable to obtaining exact results withoutgreat labor.

2.2. The mirror and open string amplitudes

The equation for the mirror is readily derived from Re(yi ) = −|xi|2, the D-termequations, and x z =

i e

yi where x and z are complex variables. Setting y5 = u, y6 = v,

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and y4 = 0 to fix a constant solution of the D-term equations yields

(2.5)xz = P(u,v) = 1 + eu + ev + e−t 1+u+v + e−t 2+v−u + e−t 3+u−v.

A noncompact, supersymmetric Lagrangian three-cycle in the original manifold isdetermined by three additional constraints which in this case take the form

|x5|2 − |x4|

2 = c1, |x6|2 − |x4|

2 = c2,

(2.6)

i

Arg(xi ) = 0 and/or π.

Let us write the above in the form

(2.7)

j

l(1

)j p |xj |2|3-cycle = cp

and

(2.8)

j

l(2)i Imlog(xi |3-cycle)

= 0 and/or π

for later use. Here

i l(1)ip l

(2)i = 0 makes the cycle Lagrangian and

i l(1)ip = 0 is

necessary for supersymmetry of the Lagrangian cycle. Additionally,

i lij l

(2)i = 0 impliesthat (2.8) is gauge invariant. If the cycle does not intersect a compact or noncompacttwo-cycle in the base, both

i Arg(xi ) = 0 and

i Arg(xi ) = π are needed to give a

composite cycle without boundary. For this case any worldsheet disks that intersect a D-brane wrapped on the two cycles will be oppositely oriented with respect to the two cyclesand not make a contribution. If the cycle intersects the toric base, one can choose either

i Arg(xi ) = 0 or

i Arg(xi ) = π to get a cycle without boundary, and there generallywill be a nonvanishing contribution from disks wrapping part of a P1 and intersecting thecycle in a circle. Allowing the three-cycle to end on the P1 where x6 = x4 = 0 (whichwill be denoted as phase I), the classical limit in the mirror corresponds to v = iπ ,Re u = −c1 ≈

−r22 → −∞, and xz = 0. More generally, one can choose the mirror two-

cycle of the Lagrangian three-cycle to be parametrized by z with x = P (u, v) = 0, aRiemann surface which is the moduli space of this cycle. The coordinates u and v canbe considered as transverse coordinates to a two-cycle inside a Calabi–Yau manifold.

The disk amplitude F g=0,h=1 (g is the genus, h is the number of boundaries) canbe determined classically in the mirror as ∂uF 0,1 = v where classically v = 0 and uparametrizes the area of a disk. In the original manifold, these disks can be interpretedas domain walls (e.g., fourbranes wrapping a disk) ending on a sixbrane wrapped on thethree-cycle. The tension of these domain walls is corrected by an amount δu that must beadded to the classical area of a disk. The amplitude F 0,1 takes the form

(2.9)F 0,1 =

k,n, m

1n2

N k, m

i

q mi ni

eûkn,

where k, n, mi are integers, û is the instanton corrected domain wall tension, and N k, mcounts the number of domain walls wrapping the two-cycle parametrized by

i mi t̂ i

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with a boundary wrapping the S1 k times. The assumption is that one counts isolateddomain walls and that the N k, m should be integers. If one has a continuous family of

domain walls, fractions may be possible. In section three we will determine a criterionfor obtaining integers and present an argument for integrality in those cases. Undermirror symmetry a domain wall fourbrane becomes a domain wall fivebrane with tensiondetermined classically by û = u − δu where δu = 12πi

Cu

u dv and v → v + 2π i aroundthe one-cycle on the Riemann surface Cu. This tension corresponds to the difference in thesuperpotential F 0,1 as one crosses the domain wall. In the original manifold, the change inImv corresponded to the change in Wilson line as one crossed the domain wall.

There is an ambiguity in F 0,1 due to the possibility of redefining the disk coordinateû → û + nv̂ where n is an integer since v = 0 classically. One requires n to be an integer

so that eû

is invariant under û → û + 2π i . This ambiguity can sometimes be related tomoving the Lagrangian cycle to a different phase along the toric base, and the amplitudeF 0,1 is not invariant. If the toric base is modeled by type IIB fivebranes, the ambiguitycorresponds to an SL(2, Z) transformation of type IIB.

Proceeding with the calculation, the Riemann surface P (u, v) = 0 looks like Fig. 3where the legs extend to infinity.

There are nine phases, but the three compact P1’s and the six noncompact two-cyclesof the original manifold are related by symmetries reducing the number of inequivalentphases to two (Fig. 4).

For instance, the three inner phases are exchanged by exchanging z3 ↔ z2 ↔ z1 along

with v ↔ u ↔ −u, and phase II and phase III are exchanged under z1 ↔ z3, v ↔ −v.Phase II corresponds to c1 = c2 + r2, c2 0.

In phase I one obtains

v=iπ − ln 21 + z1e

u + z2e−u

+ ln

1 + eu

(2.10)+

1 + eu2

− 4z3(eu + z1e2u + z2)

.

Fig. 3. Riemann surface P(u,v) = 0 in mirror of Z2 × Z2 blowup.

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Fig. 4. Phases for noncompact three-cycle in Z2 × Z2 blowup.

Extracting the piece of v that is independent of eu gives δv, and one has

(2.11)δv = t 1 − t̂ 1

2 −

t 3 − t̂ 32

+ iπ.

To get δu, we exchange t 3 and t 2 so

(2.12)δu = t 1 − t̂ 12

− t 2 − t̂ 22

+ iπ.

In the above equation for v, we must substitute, v = v̂ + δv, u = û + δu, and zi (qi ).Note that not all of the classical symmetries of the superpotential are preserved by thecorrections, and one cannot determine the correction uniquely by symmetry. The result is

v̂ = − ln1 − q1e

û

1 − q2e−û

−

m,n,a,b,c,d,e

(−1)m+b+c

(2.13)× (2n + m − 1)!eû(2a+b+c)qa+d +e1 q

m+n−a−b−c+d 2 q

m+n−c+e3

n!a!(c − e)!e!(b − d)!(m − c)!d !(n − a − b)! .

The indicates that we omit terms independent of eû. Clearly the first term of v̂ has

the required form. The lowest order terms of the second summation can be examined byhand or calculated on a computer using Mathematica, and one does obtain N k, m that areintegers after integrating and comparing with (2.9). One can show explicitly that all N 1, mare integers. One finds

N 1,0,m,m = −

mn=1

(−1)n+m(n + m − 1)!m

n!2(m − n)! ,

N 1,1,m−1,m =

m−1n=0

(−1)n+m(n + m − 1)!

n!2(m − n − 1)! ,

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Table 1m3 = 5, k = 5

m2 m1 = 0 1 2 3 4 5

0 5 −14 14 −6 1 01 −126 350 −350 150 −25 12 756 −2100 2100 −900 150 −63 −1764 4900 −4900 2100 −350 144 1764 −4900 4900 −2100 350 −145 −635 1764 −1764 756 −126 5

Table 2m3 = 5, k = 6

m2 m1 = 0 1 2 3 4 5

0 42 −126 140 −70 15 −11 −630 1890 −2100 1050 −225 152 2940 8820 9800 −4900 1050 −703 −5880 17640 −19600 9800 −2100 1404 5292 −15876 17640 −8820 1890 −1265 −1764 5292 −5880 2940 −630 42

Table 3m

3 = 5, k = 7

m2 m1 = 0 1 2 3 4 5

0 198 −630 756 −420 105 −91 −2310 7350 −8820 4900 −1225 1052 9240 −29400 35280 −19600 4900 −4203 −16632 52920 −63504 35280 −8820 7564 13860 −44100 52920 −29400 7350 −6305 −4356 13860 −16632 9240 −2310 198

and

N 1,0,m−1,m = N 1,1,m,m =m

n=1

(−1)n+m(n + m − 1)!

n!(n − 1)!(m − n)! .

We have verified that all N k,m1,m2,m3 are integers for k 10, m1 1, m2 1,and m3 10.Rather than present this data which is very cumbersome, we give Tables 1–3 with k andm3 set to specific values.

The diagonal symmetry N k,m1−x,m2−y,m1+m2 = N k,m1+y,m2+x,m1+m2 is generated byd ↔ b − d in (2.13). The similar symmetry

N k,m1−x,m1+m3−k,m3−y = N k,m1+y,m1+m3−k,m3+x

is generated by e ↔ c − e.In phase II we do the coordinate transformation u → u = u − v + t 2, v → v = v .

Correspondingly, δu = δu − δv + δt 2 = t 3−t̂ 3

2 + t 2−t̂ 2

2 and δv = δv. Phase II is almost

equivalent to phase I. One obtains phase II from (2.13) by ignoring the first log term,

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exchanging q2 ↔ q3 and û ↔ v̂. The first term is similar to the inner phase of a conifoldso it looks like the S3 symmetry relating the inner and two outer phases is broken by the

finite P1’s.One can extract amplitudes in the flopped phase of Fig. 2 by taking q3 → q 3 = 1/q3,

q1 → q1q

3, q2 → q

2q

3, and exp û → q

3 exp û

in (2.13). As a check on our results, theconifold in the two inequivalent phases is retrieved in the limit q1, q3 → 0. Replacingthe P1 by an S3 in this limit via a conifold transition also yields the same result as thecalculation of the expectation value of a Wilson line in the Chern–Simons theory on S3. Itwould be interesting to extend such calculations to the case of multiple S3’s.

3. Z2×

Z4

3.1. Toric geometry

Let us move on to the Z2 × Z4 case. Here we increase the complexity of the calculation,but the results reduce precisely to the Z2 × Z2 case in a particular limit. We start with thefollowing set of charges under six U (1)’s for nine fields

l1 = (1, −1, 0, 0, 0, −1, 1, 0, 0),

l2 = (0, −1, 1, 0, 0, 1, −1, 0, 0),

l3 = (0, 1, 0, 0, 0, −1, −1, 0, 1),

l4 = (0, 0, 0, 0, 0, 1, −2, 1, 0),

l5 = (0, 0, 0, 1, 0, 0, −1, −1, 1),

(3.1)l6 = (0, 0, 0, −1, 1, 0, 1, −1, 0),

yielding

j lj i |xj |

2 = ri . Solving these equations leads to the toric diagram (Fig. 5) in oneparticular region of moduli space where r7 = r2 + r3 − r5 > 0, r8 = −2r2 − r3 + r4 +r5 > 0,and all of the ri are large. The four-cycle represented by the hexagon is a P2 blown up insuccession at three points (one obtains inequivalent four-cycles depending on how onedoes this). It is also equivalent to the F2 Hirzebruch surface blown up at two points. Bytaking r4 and r8 to infinity, the above diagram and the theory reduce to two decoupledZ2 × Z2 cases. There are possible flop transitions to other geometric phases for r1, r2,r3, r5, r6, and r7 but not for r4 and r8. Shrinking r8 to a negative value removes a P1 as|x3| > 0 everywhere, and we enter a nongeometric phase where a Kähler parameter loses itscorrespondence to a geometric P1. The result of flopping r7 is shown in Fig. 6. Two moreflops (r4 − r2 → r2 − r4, r3 → −r3) and taking all ri to infinity with r4 − r3 > 0 and finitereduces the diagram to the blowup of a Z3 orbifold (see Fig. 7). The flops r2 → −r2 andr7 → −r7 generate a P1 × P1, and one can take external Kähler parameters to infinity to

obtain this model. The equations for the Z2 × Z4 singularity are v2

= yu and v4

= wzu2

.Before solving the Picard–Fuchs equations, one must choose a basis that depends bothon the open as well as closed string phase. A priori there are many inequivalent basischoices, but in this model the requirement that the open string expansion converge in aparticular phase limits the choices. Requiring the expansions in the zi to converge in a

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Fig. 5. Toric diagram of Z2 × Z4 blowup.

Fig. 6. Toric diagram of Z2 × Z4 flopped phase: r7 → −r7.

neighborhood of the origin yields a unique solution for the particular phase. By uniquenessthe solution solves the equations in any basis obtained from the original basis by a linear

transformation in which the transformation matrix has no negative entries. The solutionmay not be unique in any particular basis. Also, even when the expansions are convergentin a particular basis, there may be a correction which is “nonperturbative” with respectto that basis and necessary for integrality. We have found a unique basis for this phase inwhich the above problem does not occur. We need to define r9 = r6 − r7 for this basis. The

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Fig. 7. Toric diagram of Z2 × Z4 flopped phase: r7 → −r7, r4 − r2 → r2 − r4, r3 → −r3.

solutions are as follows:

z1 = q1e

−N +T

1 + q1q3,

z2 = q2(1 + q1q3)e−N +M −T

z3=

q3eN −T

1 + q1q3 ,z7 = q7(1 + q2q3q9)e−

P +M −T ,

z8 = q8eN +P −2M

z9 = q9e

−M +2T

(1 + q2q3q9)2,

(3.2)∆ = (t 1 + t 2 + 2t 7 + 2t 8 + t 9)2 + instanton corrections,

where ∆ is a solution corresponding to a four-cycle,

T =

m,n,p,q,r,s

(−1)n+p+r+m

(n + m + p − r − 2q − 1)!zr

1zp

2 zm

3 zn

7 zs

8zq

9m!q!r!(m − p − r + s)!(s − n)!(p − m − r)!

× 1

(n − 2q)!(p − 2s + n − q)!,

N =r,n,s

(−1)s(2r − s − 1)!zr1zr2z

2n7 z

s8z

n9

n!r!(s − 2n)!(r + n − 2s)! ,

M = r,n,s

(−1)n+r (2s − n − r − 1)!zr1zr2z

2n7 z

s8z

n9

r!n!(s − 2r)!(s − 2n)! ,

and

P =r,n,s

(−1)s (2n − s − 1)!zr1zr2z

2n7 z

s8z

n9

n!r!(s − 2r)!(r + n − 2s)!

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and the zi ’s must be found as a function of the qi ’s perturbatively.Examining Fig. 5 one sees that there is a reflection symmetry about a line through the

equator of P1(r4) and P1(r8). The above solutions do not reflect this symmetry because anychoice of basis necessarily breaks this symmetry. In this case the selection of q3 breaks thesymmetry. We had previouslychosen the basis with q6 instead of q9 and found that the openstring expansion did not give integers without the term (1 + q2q3q9) = (1 + q2q3q6/q7),but the perturbative solution, t̂ 6, of the Picard–Fuchs equations does not converge if weinclude this term. The pieces of the solution involving negative powers solve the Picard–Fuchs equations by themselves so there are ambiguities of the solution in general. Clearly,the “nonperturbative” pieces are essential for integrality of the numerical invariants. Wealso note that this model is the only one treated so far where the above ambiguity involving

negative powers occurs. In different phases of the theory we need to resolve the Picard–Fuchs equations.Up to total order fourteen in the qi ’s (linear order in q9) we find that the coefficients

in the expansions of the inverse mirror map are integral. The expansions in this phase are(z1z3 =

q1q3(1+q1q3)2

)

z1 = q1 + q1q2q3 + q1q2q3q8 + q1q2q7q8 − 2q1q2q3q7q8 + · · · ,

z2 = q2 + q1q2q3 + q2q8 + q1q2q3q8 + · · · ,

z7 = q7 − q2q3q7 + q7q8 + q1q2q7q8 − 2q2q3q7q8 + q2q3q7q9

+ q2q3q7q8q9 + · · · ,z8 = q8 + q1q2q8 + · · · ,

z9 = q9 − 2q2q3q9 − q8q9 + q1q2q8q9 − 2q2q7q8q9

(3.3)+ 4q2q3q7q8q9 − 2q1q2q3q7q8q9 + · · · .

In the limit that z7 = z8 = z9 = 0, eN = 1 + q1q2, eT = 1 + q2q3, and the solutionsare precisely those of the Z2 × Z2 case. The limit z1 = z2 = z

−13 = z7 = z8 = z9 = 0

while z2z3z7z8 is finite yields the solution for the blowup of the Z3 orbifold. Takingz1 = z

−12 = z3 = z

−17 = z8 = z9 = 0 with z2z3 and z2z7z8 finite yields the P

1 × P1 case.

3.2. The mirror and open string calculations

Putting y2 = u, y6 = v, and y7 = 0 the equation for the mirror is

xz = P(u,v) = 1 + eu + ev + eu+v−t 1 + eu−v−t 2 + ev−u−t 3

(3.4)+ e−v−t 2−t 7−t 8 + e−2v+u−2t 2−t 8 + e−3v+u−3t 2−2t 7−2t 8−t 9 .

This choice is sensible for open string phases on the four-cycle. Notice again that thisequation reproduces the Z2 × Z2 case when z7 = z8 = z9 = 0. To obtain the standard

version of the Z3 case, take v

= v − u − t 3 and the previously discussed limit. The standardversion of P1 × P1 results from u = u − v − t 2 and the above limit. The Riemann surfaceP(u,v) = 0 can be visualized Fig. 8.

A noncompact, supersymmetric three-cycle intersecting the toric base of the originalmanifold is determined by |x2|2 −|x7|2 = c1, |x6|2 −|x7|2 = c2,and

i Arg(xi ) = 0. There

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Fig. 8. Riemann surface P(u,v) = 0 in mirror of Z2 × Z4 blowup.

Fig. 9. Phases for noncompact three-cycle in Z2 × Z4 blowup.

are sixteen phases for the Lagrangian three-cycle with symmetries relating the phases inthe lower half of Fig. 9 to those in the upper half. For instance, the phase with a three-cycleintersecting P1(r7) is equivalent to phase I under the obvious permutation of the ri ’s andv → −v. Also, phases II and III are equivalent. There are, accordingly, nine inequivalentphases.

Five of these phases can be parametrized by u = 0 after a change of coordinates while

the rest correspond to v = 0. The latter yield quartic equations which can be solved andexpanded, but the amount of computer time required seems prohibitively large. Fortunately,the u = 0 phases are quadratic. These are still quite complicated because the expansioninvolves seven variables. We will, thus, restrict ourselves to two of the u phases and onlytest the integer hypothesis at low order in the expansion. Phase I corresponds to c1 = 0 and

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0 < c2 < r2. In phase II c1 = −r3 and c2 0. In phase I we calculate the zero mode pieceof u to be

(3.5)δu = iπ + t 1 − t̂ 1

2 −

t 3 − t̂ 32

= iπ − T + N.

We can determine δv by expanding an adjacent phase in which u − v = 0, and we find

δv = iπ + t 1 − t̂ 1

4 −

3(t 2 − t̂ 2)

4 −

t 7 − t̂ 72

− t 8 − t̂ 8

2 −

t 9 − t̂ 94

(3.6)= iπ − T + ln(1 + q1q3).

Solving (3.4) for u and substituting instanton corrected variables, we can write the

expansion for û in this phase.

û =

n,p,q,r

(−1)q (n − 1)!e−NneMr eP q

p!q!r!(n − p − q − r)!

×ev̂(4p+2r+q−3n)qp1 q

3n−3p−2r−q2 q

2(n−p−r−q)7 q

2n−2p−2r−q8 q

n−p−r−q9

−

a,b,c,d,e,f,m,n,p

(−1)b+d (2n + m − 1)!

n!a!b!c!d !e!f !(n − a − b − c − d)!(p − f )!(m − p − e)!

× e−T (2n+m)eMc eNbeP d

ev̂(4a+3b+2c+d +m−2p−2n)q a+e1 q3n−3a−3b−2c−d +p+f 2

(3.7)× q n+e+f 3 q2(n−a−b−c−d)+p7 q

2n−2a−2b−2c−d +p8 q

n−a−b−c−d +f 9

.

One can easily verify that the above expansion reduces to (2.13) in the limit q7 = q8 =q9 = 0. Also, the limit q1 = q9 = 0 yields the F2 blown up at two points. One can alsoshow that the inner phase of the Z3 model with ambiguity n = 1 is achieved in the limitq1 = q2 = q

−13 = q7 = q8 = q9 = 0 with q2q3q7q8 finite and v̂ → v̂ + t 3. In choosing this

basis we have required that the expansion be convergent in a neighborhood of the originin both open and closed string variables. In phase I this requirement entails that negativewinding terms of the form (

i q

nii )e−

mv̂ have n2 m. In an earlier calculation we chose abasis not meeting this last requirement and many terms had fractional invariants. Note thatthe basis with q6 instead of q9 does meet this latter requirement but still has fractions dueto the nonperturbative piece. Up to linear order in q9, quadratic order in q7, cubic order inq1, q3, and q8, quartic order in q2 and sixth order in e±v̂ , all of the numerical invariants inthis phase are integral. We present in Table 4 terms corresponding to the homology classesthat failed to be integral in the badly chosen basis.

3.3. Numerical invariants

What is the meaning of the numerical invariants given in Table 4 and why do weanticipate that the numerical invariants of this phase are integers? We consider the modulispace of maps from a Riemann surface of genus zero with one boundary into the Calabi–Yau such that the relative homology class of the image is labeled by the wrapping number

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Table 4Numerical invariants

k m1 m2 m3 m7 m8 m9 N k, m k m1 m2 m3 m7 m8 m9 N k, m

2 1 3 2 2 2 1 −64 −2 1 3 0 2 2 1 02 0 4 2 2 2 1 −36 −2 2 4 0 2 2 1 02 0 4 2 2 3 1 −36 −2 0 3 1 2 2 1 13 0 3 3 2 2 1 −621 −2 0 4 2 2 2 1 43 1 4 3 2 2 1 −2214 −3 0 3 0 2 2 1 03 1 4 3 2 3 1 −2214 −3 1 4 0 2 2 1 03 3 3 3 2 2 1 72 −3 0 3 0 2 2 1 04 1 3 2 2 2 1 −128 −3 1 4 0 2 2 1 04 3 3 2 2 2 1 0 −4 0 4 0 2 2 1 04 0 4 2 2 2 1 −56 −4 0 6 2 4 4 1 0

4 0 4 2 2 3 1 −56 −4 0 6 2 4 5 1 06 1 3 2 2 2 1 −224 6 3 3 3 2 2 1 10266 3 3 2 2 2 1 0 6 1 4 3 2 2 1 −106566 0 4 2 2 2 1 −84 6 0 4 2 2 3 1 −846 0 3 3 2 3 1 −936 6 1 4 3 2 3 1 −10656

on each two-sphere of the basis and the winding number around a noncontractible circleon the Lagrangian three-cycle. These maps should be holomorphic in the interior of thedisk. The moduli space of these maps is generally noncompact, and one must add in

extra maps that may be singular to define a compact space. There may be disconnectedcomponents of the moduli space when there are homotopically inequivalent maps intosome relative homology class. One then defines a cohomology class analogous to the Eulerclass, and the numerical invariant is obtained by integrating this class over the modulispace. For the case of genus zero closed strings, one should fix three complex parameterscorresponding to SL(2, C) transformations of the complex plane while for disks one fixesthree real parameters corresponding to SL(2, R) transformations of the upper half plane. If the dimension of the moduli space is zero after this fixing, the maps are isolated and thenumerical invariants can be interpreted as counting curves. Otherwise, the integral over themoduli space could give fractions when there are orbifold singularities in the moduli space.

Of course, there are many technicalities needed to make the above discussion rigorous.A first principles calculation from the nonlinear sigma model point of view as described

above is generally difficult. In this paper our determination of the invariants has beenfacilitated by an equivalent calculation on the local mirror. The drawback is that one doesnot have a direct argument that the invariants should be integers. Another approach is tostart with the boundary linear sigma model that flows to the conformally invariant nonlinearsigma model at low energies. The relevant correlation functions that yield the numericalinvariants are in a topological sector of the theory. The corresponding correlators in atopologically twisted version of the linear sigma model are scale invariant so the two

calculations should be equivalent. The correlators are intersection forms on the modulispace of classical solutions with a given instanton number mi = 12π

Σ F i and winding

number kp = 12π

∂Σ Λp where F i is the gauge field strength which is integrated over

the two-dimensional worldsheet Σ with boundary ∂ Σ and Λp is a boundary field whichcouples to the theta angle. It would be interesting to calculate correlators in the boundary

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linear sigma model corresponding to the numerical invariants. This correspondencebetween the two theories has been verified in several examples for closed worldsheets.

Our aim here is to discuss the structure of the boundary linear sigma model moduli spacein order to determine a criterion for integrality of the intersection forms. Assuming thecorrespondence is valid, we can apply this criterion to the nonlinear sigma model. Note thatthe moduli spaces of the two theories are different, but the correlators of this topologicalsector are expected to be the same.

In the linear sigma model the moduli space can be described as a space of holomorphicsections xi of a line bundle over the upper half complex plane H of degree d i =

j (mj lij + kpl

(1)ip ) (cf. (2.7)). One usually calls this bundle O(d i ). If d i 0, one can

write a section as xi = d i

n=0 xinzn where z is a coordinate on H and the xin must be

chosen so that solutions of xi = 0 all lie in H . If d i

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In this phase v > 0 and all terms have k 0 so the floppedP1 is nongeometric. The calculation gives half-integer invariants for terms of the formq42 q

23 q

8e

2nv̂ . (r 8 = r8 − r2 + r4 + r6, n ∈ Z+) These results are puzzling. Although there

is a Z2 orbifold singularity on the “geometric” P1(r 8) in this phase, the presence of thedisk prevents the moduli space from being an orbifold. We believe the resolution of thisparadox is that there is a “nonperturbative” contribution corresponding to P (3.2). Therelative homology classes in question are present in Table 4. Depending on the choice of

coordinates, one can have square root branch cuts ( q 8

) in the nongeometric phase but one

can always find coordinates without branch cuts. The correction P is an expansion in theflopped Kähler parameter for which we need the analytic continuation to this phase. Thereare other terms in this phase in which the moduli space does contain orbifold singularities(x3 = x5 = 0, x2 /≡ 0), and we anticipate fractional invariants. We have not pursued thiscalculation further.

We have in this model a flop transition that augments the four-cycle from an F2 blownup at two points to one blown up at four points (Fig. 10).

We can flop r1 and r6 to obtain this phase. The conifold type transitions occur for thepart of the diagram that reduces to the Z2 × Z2 case. As in that case one can substitute S3’sfor P1’s. Our amplitudes F 0,1 should correspond to Chern–Simons theory in a complicatedbackground.

Fig. 10. Flopped phase of Z2 × Z4 blowup: r1 → −r1, r6 → −r6.

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4. Z7

The Z7 case is interesting because there are three adjacent four-cycles. However, thereare no geometric flops unlike the previous examples. The charges of six chiral fields underthree U (1)’s are

l1 = (0, 1, −3, 0, 1, 1),

l2 = (1, −2, 1, 0, 0, 0),

(4.1)l3 = (−2, 1, 0, 1, 0, 0).

Solving the equations j

lj i |xj |

2 = ri generates the toric diagram (Fig. 11) where r4 =

r1 + 3r2 and r5 = r1 + 3r2 + 5r3.The instanton corrected Kähler parameters are given by

z1 = q1e−3M +P ,

z2 = q2eM −2P +Q,

(4.2)z3 = q3e−2Q+P ,

where

M = n,m,p

(−1)n+m(3n − m − 1)!zn1 zm2 z

p

3

n!2

p!(m − 2p)!(n − 2m + p)!

,

P =

n,m,p

(−1)m+p(2n − m − p − 1)!zp1 zn2 z

m3

p!2(n − 3p)!m!(n − 2m)! ,

Fig. 11. Toric diagram of Z7 blowup.

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and

Q =

n,m,p

(−1)m

(2n − m − 1)!zp

1 zm

2 zn

3n!p!2(m − 3p)!(n − 2m + p)!

.

These expressions for qi must be inverted to obtain zi (qi ). The expansions are

z1 = q1 + 6q21 + q1q2 + 10q21 q2 + 4q

21 q

22 + q1q2q3 + 10q

21 q2q3

+ 8q21 q22 q3 + 4q

21 q

22 q

23 + · · · ,

z2 = q2 − 2q1q2 + 5q21 q2 − 2q22 + 6q1q

22 − 20q

21 q

22 + q2q3 − 2q1q2q3 + 5q

21 q2q3

− 3q2

2q3 + 8q1q2

2q3 − 26q2

1q2

2q3 − 2q2

2q2

3 + 6q1q2

2q2

3 − 20q2

1q2

2q2

3 + · · · ,

z3 = q3 + q2q3 − 2q1q2q3 + 5q21 q2q3 − 2q1q22 q3 − 2q

23 − 3q2q

23 + 6q1q2q

23

(4.3)− 15q21 q2q23 − 2q

22 q

23 + 12q1q

22 q

23 − 44q

21 q

22 q

23 + · · · .

There are also three more solutions to the Picard–Fuchs equations corresponding to four-cycles. The Z3 and Z5 cases can be obtained in the appropriate limit. To obtain Z3 we setz2 = z3 = 0.

The equation for the mirror can be written as

(4.4)xz = P(u,v) = 1 + eu + ev + e−u−v−t 1 + e2u−t 2 + e3u−2t 2−t 3 ,

where y2 = u, y5 = v, and y3 = 0. The Riemann surface that describes moduli of the three-cycle is a genus three surface with legs extending to infinity (Fig. 12).

Clearly, we recover the O(−3) → P2 (Z3 blowup) when z2 = z3 = 0. Letting ourLagrangian three-cycle be determined by |x2|2 − |x3|2 = c1, |x5|2 − |x3|2 = c2, and

i Arg(xi ) = 0; we find eight inequivalent phases taking into account the obviouslysymmetric phases (Fig. 13) .

Again some of the phases are described by quadratic equations, whereas others requirea quartic equation. Phase I has c2 = 0 and c1 ≈

r12 while in phase II, c1 = c2 ≈

−r22 . In

phase I

(4.5)δu = iπ + 1

7

−δt 1 + 4δt 2 + 2δt 3

= iπ − M + P

Fig. 12. Riemann surface P(u,v) = 0 in mirror of Z7 blowup.

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Fig. 13. Phases for noncompact three-cycle in Z7 blowup.

Table 5Phase I: m1 = 2, k = 3

m3 m2 = 0 1 2 3 4

0 −9 −26 −49 −76 −110

1 0 −26 −76 −144 −2322 0 0 −49 −144 −2843 0 0 0 −76 −2364 0 0 0 0 −118


m3 m2 = 0 1 2 3 4

0 −12 −36 −72 −116 −172

1 0 −36 −112 −224 −3722 0 0 −72 −224 −4603 0 0 0 −116 −3764 0 0 0 0 −180


m3 m2 = 0 1 2 3 4

0 −15 −48 −99 −166 −250

1 0 −48 −156 −324 −5522 0 0 −99 −324 −6843 0 0 0 −166 −5564 0 0 0 0 −258


m3 m2 = 0 1 2 3 4

1 0 −62 −208 −444 −772

2 0 0 −132 −444 −9603 0 0 0 −226 −7764 0 0 0 0 −355

and

(4.6)δv = iπ + 1

7

−3δt 1 − 2δt 2 − δt 3

= iπ − M.

In phase II δu is unchanged while δv

= δv − δu = −P . The results are shown in theTables 5–12.There are nongeometric phases obtained by flopping the Kähler parameters. One would

need to analytically continue the expansions (4.2) to calculate in these phases. Theboundary conditions on the disk prevent the open instanton moduli space from being

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Table 9Phase II: m1 = 2, k = −3

m3 m2 = 0 1 2 3 4

0 −3 −8 −19 −46 −1101 0 −8 −28 −84 −2322 0 0 −19 −84 −2843 0 0 0 −46 −2364 0 0 0 0 −118


m3 m2 = 0 1 2 3 4

0 −4 −12 −32 −76 −1721 0 −12 −48 −144 −3722 0 0 −32 −144 −4603 0 0 0 −76 −3764 0 0 0 0 −180


m3 m2 = 0 1 2 3 40 −5 −18 −49 −116 −2501 0 −18 −76 −224 −5522 0 0 −49 −224 −6843 0 0 0 −116 −6844 0 0 0 0 −258


m3 m2 = 0 1 2 3 40 −7 −26 −36 −166 −3471 0 −26 −112 −324 −7722 0 0 −36 −324 −9603 0 0 0 −166 −7764 0 0 0 0 −355

an orbifold even when the closed string instanton moduli space is an orbifold unless theboundary of the disk is fixed by the orbifold. In this case fractional invariants are possible.

Acknowledgements

I wish to acknowledge J. Distler and A. Iqbal for beneficial discussions and R. McNeesfor answering Mathematica related questions. This work was supported in part by NSFgrant PHY-0071512.

References

[1] M. Aganagic, A. Klemm, C. Vafa, hep-th/0105045.[2] D. Morrison, M. Plesser, Nucl. Phys. B 440 (1995), hep-th/9412236.

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Nuclear Physics B 634 (2002) 23–50 www.elsevier.com/locate/npe

Dielectric fundamental strings in matrix stringtheory

D. Brechera

, B. Janssena

, Y. Lozanob

a Centre for Particle Theory, Department of Mathematical Sciences, University of Durham, South Road,

Durham DH1 3LE, United Kingdomb Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo 18, 33007 Oviedo, Spain

Received 11 April 2002; received in revised form 25 April 2002; accepted 25 April 2002

Abstract

Matrix string theory is equivalent to type IIA superstring theory in the light-cone gauge, togetherwith extra degrees of freedom representing D-brane states. It is therefore the appropriate frameworkin which to study systems of multiple fundamental strings expanding into higher-dimensional D-branes. Starting from Matrix theory in a weakly curved background, we construct the linear couplingsof closed string fields to type IIA Matrix strings. As a check, we show that at weak coupling the result-ing action reproduces light-cone gauge string theory in a weakly curved background. Further dualitiesgive a type IIB Matrix string theory and a type IIA theory of Matrix strings with winding. We com-ment on the dielectric effect in each of these theories, giving some explicit solutions describing fun-damental strings expanding into various Dp-branes.© 2002 Elsevier Science B.V. All rights reserved.

PACS: 11.25.-w; 11.27.+d

Keywords: D-branes; M(atrix) theories

1. Introduction

The idea that a collection of branes can undergo an “expansion” into a single higher-dimensional D-brane under the influence of a background Ramond–Ramond (R–R)potential was first explored by Emparan [1]. Although his main concern was with thedescription of N fundamental strings expanding into a Dp-brane, he also realised thatD(p − 2)-branes could undergo such an expansion. Emparan’s analysis of this effect was

E-mail addresses: [email protected] (D. Brecher), [email protected] (B. Janssen),[email protected] (Y. Lozano).

0550-3213/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0550-32 13(02 )00 344- 9

http://www.elsevier.com/locate/npehttp://www.elsevier.com/locate/npe

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24 D. Brecher et al. / Nuclear Physics B 634 (2002) 23–50

entirely at the level of the abelian theory relevant to the description of the single Dp-brane. Switching on a background R–R (p

+2)-form field strength, he found solutions of

the combined Born–Infeld–Chern–Simons theory with topology M2 × S p−1, where Mndenotes an n-dimensional Minkowski space, and with N units of dissolved electric flux,corresponding to the fundamental strings. Similar solutions, with topology Mp−1 × S 2,and with N units of dissolved magnetic flux, corresponding to D(p − 2)-branes, are easilyfound.

It was some years later that a description from the point of view of the lower-dimensional D(p − 2)-branes was provided [2]. This involved an analysis of certaincouplings in the non-abelian Born–Infeld–Chern–Simons action proposed by Taylor andvan Raamsdonk [3] and Myers [2]. From this perspective, the expansion is due to the fact

that the transverse coordinates of the D(p − 2)-branes are matrix-valued. Myers’ originalanalysis was of a collection of N D0-branes, in the presence of a background R–R four-form field strength: they spontaneously expand into a D2-brane of topologyR×S 2NC, whereS 2NC denotes the non-commutative two-sphere. The D2-brane is uncharged with respect tothe four-form, but has a non-zero dipole moment. Hence the name of a “dielectric” D2-brane. Configurations for arbitrary p, and more general configurations involving fuzzycosets, were described in [4].

These two descriptions of the dielectric effect—abelian and non-abelian—are, of course, dual to one another. In the limit of large N , the non-commutative nature of thefuzzy two-sphere is lost, it becomes a smooth manifold, and the two descriptions do indeedagree.

In principle, all such dielectric branes should have a correspondingsupergravitysolutiondescribing the back-reaction of the brane on spacetime. Technical difficulties, however,have limited the analysis of such solutions to that of N D4-branes [5,6] or N F-strings [6]expanding into a D6-brane with topologyM5 ×S 2 andM2 ×S 5, respectively. In both cases,there is a stable and an unstable radius of the dielectric sphere, the form of the effectivepotential matching the abelian worldvolume analysis precisely [5,7].

There is thus much evidence, from both the worldvolume and supergravity perspectives,that the dielectric effect is not limited to D-branes, but that dielectric F-strings also exist.

One can then pose the following question: can the expansion of F-strings into a Dp-branebe described from the point of view of the strings themselves? It is the purpose of thispaper to answer precisely this question.1

Since, from the strings’ perspective, the dielectric effect should be due to matrix-valuedcoordinates, we are led to a consideration of Matrix string theory [9–11]. Starting from theMatrix theory action [12,13], one compactifies on a circle, reinterprets the resulting theoryas a (1+1)-dimensional super Yang–Mills theory on the dual circle [14], and then performsthe so-called 9-11 flip, which implements a further S- and T-duality [11]. It is easy to seethat the result is a (1 +1)-dimensional super Yang–Mills theory describing N fundamentalstrings in the type IIA theory. In the weak string coupling limit, one recovers N copies of

the Green–Schwarz action describing light-cone gauge string theory.

1 A recent attempt to answer this question from the perspective of Matrix string theory has already beenmade [8], although the results seem somewhat opaque to the authors and there is little overlap with this work.

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D. Brecher et al. / Nuclear Physics B 634 (2002) 23–50 25

To discuss possible dielectric solutions of Matrix string theory, we need to know how theR–R potentials couple to the worldvolume of the (Matrix) string. Since the Matrix theory

action in a weakly curved background is known [15,16], one can in principle run throughthe above chain of dualities to derive the Matrix string theory action in a correspondingweakly curved background. Indeed, Schiappa has already considered this computation,and has written down an obvious dielectric-like solution [17]. However, it is not at allclear that this solution corresponds to a dielectric string, for reasons explained later in thispaper. Moreover, we find couplings to various components of the R–R fields over and abovethose found by Schiappa [17] and, for this reason, we consider the problem of deriving theMatrix string theory action in a weakly curved backgroundfrom scratch. We should furthernote that one of us has also already derived couplings of the type IIA F-string to various

background R–R potentials [18].One might think that F-strings could expand into a Dp-brane with topology M2 × S p−1

for any value of p, but this is not the case. As mentioned above, one generically finds astable and an unstable spherical solution of the Dp-brane theory with dissolved electricflux. Expansion into a D2-brane, however, is atypical: only an unstable solution exists [1].Indeed, this observation is mirrored in the corresponding (albeit smeared) supergravitysolution [7]. One might suspect, therefore, that stable cylindrical D2-brane solutionsof the type IIA Matrix string theory do not exist, and we aim to address this issuehere.

A further T-duality takes us to a type IIB Matrix string theory which, as the S-dual of theD-string theory, describes N F-strings in the static gauge. Just as D-strings can expand intoD3-branes, so can F-strings and the dielectric solution presented in [2] is equally applicablehere.

In the following section, we consider Matrix theory in a weakly curved back-ground [15,16]. It is easier to consider the chain of dualities leading to Matrix stringtheory in ten- rather than eleven-dimensional language, so we choose to work with theD0-brane theory in a weakly curved background. This has been derived from the Matrixtheory results in [19]—we show explicitly that it reproduces the lowest-order expansion

of the non-abelian D0-brane theory of Taylor and van Raamsdonk [3] and Myers [2]. T-duality, taking us to the D-string theory, is considered in Section 3, and the 9-11 flip inSection 4. We show that the resulting linear action reproduces the light-cone gauge stringtheory action in a weakly curved background, thus lending some weight to our results.Indeed, we need the extra couplings, relative to the results of [17], for this to be the case.A further T-duality taking us to the type IIB theory is considered in Section 5 and, al-though we cannot perform the S-duality rigorously, we argue that the result is equivalentto the S-dual of the D-string theory. In Section 6, we apply once more a T-duality trans-formation, this time in a direction transverse to the IIB string, giving a theory of type IIA

F-strings with winding number rather than light-cone momentum. Some of the resultingcouplings of both the type IIA and the type IIB strings have already been considered byone of us [18]. Dielectric solutions are considered in Section 7, where we give some ex-plicit solutions describing the expansion of F-strings into Dp-branes for different p. Weconclude in Section 8.

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2. Matrix theory

The bosonic sector of the Matrix theory action in a flat background is [13]

(2.1)S flat = 1

R

dt Tr

1

2 Dt X

i Dt Xi + R

2

16π 2l6P

Xi , Xj

Xi , Xj

,

where lP denotes the eleven-dimensional Planck length, R is the radius of the eleventhdimension and Dt Xi = ∂t Xi + i[At , Xi ]. We will choose the gauge At = 0 throughoutthis paper at the expense of losing explicit gauge invariance. With the string couplingset through the relation R

=gs

√ α, this action is the non-relativistic limit of the non-

abelian Born–Infeld action describing N D0-branes, with mass quantized in units of 1/R:the dimensional reduction to one dimension of ten-dimensional Yang–Mills theory withg2YM = gs /(4π 2α3/2). The connections are more subtle, however (see, e.g., [20]). Afterall, when N is large, Matrix theory captures eleven-dimensional physics in the infinitemomentum frame [13]. For finite N , it is equivalent to a DLCQ or null compactification of M-theory [21–23], whereas the type IIA D0-brane theory comes about via a spacelikecompactification. The two actions are then related by an infinite boost in the eleventhdimension. At any rate, if we take T 0 = 1/R and R = gs

√ α, so that lP = g1/3s

√ α, then

we recover the Yang–Mills description of D0-branes.

2.1. Linear couplings in Matrix theory

Matrix theory in an arbitrary background is understood only rather poorly (althoughsee [24–26] for early work on this subject) and, in the above sense, this is related to thequestion as to what is the form of the non-abelian Born–Infeld theory in curved space.Kabat and Taylor have derived the linear Matrix theory couplings to bosonic backgroundfields [15] and Taylor and van Raamsdonk have extended these calculations to includefermionic backgrounds [16]. They have further derived the linear couplings of the D0-

brane [19] and Dp-brane [3] theories from Matrix theory. The results certainly seem toagree with the linear order expansion of the combined non-abelian Born–Infeld–Chern–Simons action proposed by Taylor and van Raamsdonk [3] and Myers [2], including theform of the overall symmetrized trace due to Tseytlin [27]. More precise expressions forthe Matrix theory couplings, to all orders in both derivatives of the background fields andthe fermionic coordinates, can be had by dimensional reduction of the eleven-dimensionalsupermembrane vertex operators constructed in [28,29].

In other words, one can write the Matrix theory action as

(2.2)S =

S flat

+S linear,

with S flat given by (2.1), and where the linear action has the form [15]

(2.3)S linear = 1

R

dt STr

1

2hABT

AB + AABCJ ABC + AABCDEF MABCDEF

,

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where A, B = 0, . . . , 10 and STr denotes the symmetrized trace.2 The eleven-dimensionalmetric, hAB , 3-form potential, AABC , and its 6-form dual, AABCDEF , couple to

the energy–momentum tensor, membrane current and 5-brane current, respectively.The form of these currents has been worked out explicitly, and they match eleven-dimensional supergravitypredictions [15]. There are also higher-order multipole couplingsto derivatives of the background fields, but we will not be concerned with them here. Onecan relate this linear action to that relevant to the description of D0-branes, via the infiniteboost mentioned above. The D0-brane currents, denoted I , are related in this manner to theeleven-dimensional currents T , J and M, as explained in [19].

The linear D0-brane action is then

S linear = 1

R dt STr1

2 hµν I µν

h + φI φ + bµνI µν

s + b̃µ1...µ6 I µ1...µ6

5

+ C(1)µ I µ0 + C(3)µνρI µνρ2 +

1

60 C (5)µ1...µ5 I

µ1...µ54

(2.4)+ 1336

C (7)µ1...µ7 I µ1...µ76

,

where µ, ν = 0, . . . , 9. The currents I h, I s , I 5 and I p couple respectively to the metric, hµν ,the Neveu–Schwarz–Neveu–Schwarz (NS–NS) 2-form potential b(2), its 6-form Hodgedual b̃(6), and the R–R (p + 1)-form potentials C (p+1). The potentials C (5) and C(7), the

Hodge duals of C(3)

and C(1)

, have been rescaled relative to [3,19]. Note that there shouldalso be couplings to C(9), but these are not determined by the analysis of [3,19]. Thecurrents appearing in (2.4) are given in terms of the dimensional reduction of the Born–Infeld field strength,

(2.5)F 0i = −F 0i = ∂t Xi ≡ Ẋi , F ij = F ij = R

2π l3P i

Xi , Xj

,

where i, j = 1, . . . , 9. Usually R is taken to be R = gs√

α, and so one has λ ≡ 2π α =2π l3P /R. After the 9-11 flip of Section 4, however, the role of the ninth and the eleventh

coordinate are interchanged, and this relation is no longer true. For this reason we definea general quantity β ≡ 2π l3P /R, the D0-brane theory and its duals being recovered upontaking β = λ.

Substituting for (2.5), the NS–NS currents are [19]:

(2.6)

I φ = 1− 1

2 F 0i F 0i + 1

4 F ij F ij − 1

8

F µνF νρF

ρσ F σ µ − 1

4

F µν F µν

2

= 1− 14β2

[X, X]2 − 18β4

[X, X]4 − 1

4

[X, X]22

− 1

2 Ẋ21+ 14 Ẋ2 − 14β2 [X, X]2,

2 We include both the overall factor of 1/R and the overall gauge trace in the action, rather than in the currents,for convenience.

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(2.7)I 00h = 1+ 1

2 F 0i F 0i + 1

4 F ij F ij = 1+ 1

2Ẋ2 − 1

4β2[X, X]2,

(2.8)

I 0ih = −F 0i1+ 12 F 0j F 0j + 14 F j k F j k− F ij F j kF 0k= Ẋi

1+ 1

2Ẋ2 − 1

4β2[X, X]2

− 1

β2

Xi , Xj

Xj , Xk

Ẋk,

(2.9)I ij h = F 0i F 0j + F ik F kj = Ẋi Ẋj − 1

β2

Xi , Xk

Xk , Xj

,

(2.10)I 0is = 1

2 F ij F 0j = − i

2β

Xi , Xj

Ẋj ,

(2.11)

I ij s = 12 F ij 1+ 1

4 F kl F kl − 1

2 F 0j F 0j + 1

2 F 0i F 0kF kj − 1

2 F 0j F 0kF ki

+ 12

F ik F kl F lj

= i2β

Xi , Xj

1− 1

2Ẋ2 − 1

4β2[X, X]2

− i2β

Ẋi

Xj , Xk

Ẋk − Ẋj Xi , XkẊk− i2β3

Xi , Xk

Xk , Xl

Xl , Xj

,

where we have defined

Ẋ2 ≡ Ẋi Ẋi ,[X, X]2 ≡ Xi , Xj Xi , Xj ,[X, X]4 ≡ Xi , Xj Xj , XkXk , XlXl , Xi.

The R–R currents are [19]

(2.12)I 00 = 1,

(2.13)I i

0 = −F 0i

= Ẋi

,

(2.14)I 0ij 2 = −1

6 F ij = − i

6β

Xi , Xj

,

(2.15)

I ijk2 =

1

6

F 0i F j k + F 0j F ki + F 0kF ij

= − i6β

Ẋi

Xj , Xk+ Ẋj Xk, Xi+ ẊkXi , Xj ,

(2.16)

I 0ijkl4 =

1

2 F ij F kl + F ik F lj + F il F j k

= − 12β2

Xi , Xj Xk , Xl+ Xi , XkXl , Xj + Xi , XlXj , Xk,(2.17)I ijklm4 = −

15

2 F 0[i F j kF lm] = − 15

2β2 Ẋ[iXj , XkXl , Xm],

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(2.18)I 0ijklmn6 = −F [ij F kl F mn] = i

β3

X[i , Xj

Xk , Xl

Xm, Xn]

,

(2.19)I ijklmnp6 = 7F 0[i F j kF lmF np] = 7iβ3 Ẋ[iXj , XkXl , XmXn, Xp].

Up to a couple of caveats, we show below that the above currents agree with theexpansion, to linear order in the background fields, of the multiple D0-brane theory of Taylor and van Raamsdonk [3] and Myers [2]. The first such caveat is that we must haveI

µ1...µ65 = 0 if the two actions are to match [2]. In other words, D0-branes cannot expand

into NS5-branes. We will, therefore, drop this term from now on. The second is that thecurrents I 00h and I

ij h derived from Matrix theory actually contain further terms [19], the

form of which seems less certain. Indeed, although the expressions (2.7) and (2.9) match

precisely the first-order expansion of the non-abelian Born–Infeld action, as given in [2,3],we have not been able to match these extra terms in I 00h and I

ij

h . We will, for this reason,ignore them.

2.2. Matrix theory vs. multiple D0-branes

We will now show that the above currents are recovered in the expansion to linear orderin the background fields of the multiple D0-brane theory.3 If we set the NS–NS fields tozero, the multiple D0-brane action is [2,3,27]

(2.20)S = S flat + S CS,(2.21)S flat = −T 0

dt STr

1 − Ẋi (Q−1)ij Ẋj

det(Qij )

,

(2.22)S CS = T 0

STr

P

ei(iXiX)/λ

C(n)

,

where T 0 = 1/R = 1/(gs√

α ) and

(2.23)Qij = δij + iλ

Xi , Xj

.

The pull-back is defined in terms of gauge covariant derivatives, such that

(2.24)P [ω] = ω0 + Dt Xi ωidt = ω0 + Ẋi ωidt ,with obvious generalizations to forms of higher degree. The interior multiplication is

(2.25)iXΣ

µ1...µp

= Xi Σ iµ1...µp ,giving rise to the relevant commutators in the action. The Chern–Simons action (2.22)becomes

S CS = T 0

STr

P

C(1) + i

λ(iXiX)C

(3) − 12λ2

(iXiX)2C(5) − i

6λ3 (iXiX)3C

(7)

(2.26)+ 124λ4

(iXiX )4C(9)

.

3 See also [30].

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It is easy to see that the D0-brane R–R currents, (2.12)–(2.19), of the previous subsection,can be written such that

(2.27)C(1)µ I µ0 dt = P

C(1)

,

(2.28)C(3)µνρI µνρ2 dt =

i

λP iXiXC(3)

,

(2.29)1

60 C (5)µ1...µ5 I

µ1...µ54 dt = −

1

2λ2 P

(iXiX)2C(5)

,

(2.30)1

336 C (7)µ1...µ7 I

µ1...µ76 dt = −

i

6λ3 P

(iXiX)3C(7)

,

which reproduce the Chern–Simons action (2.26). Since we have set the NS–NS 2-form to

zero, these linear couplings are actually exact. We should also note that one could derivethe correct Matrix theory coupling to C(9) in (2.4) by comparing with the Chern–Simonsaction (2.26).

The NS–NS fields are somewhat harder to deal with. To linear order in the backgroundfields, the non-abelian Born–Infeld action is

(2.31)S NBI = S NBI(φ = 0) − φS flat,with S flat as in (2.21). This gives the dilaton current:

(2.32)I φ= 1 − ˙XQ−1 ˙Xdet Q,

where ẊQ−1 Ẋ = Ẋi (Q−1)ij Ẋj . As shown already in [2,27], expanding the current (2.32)to the relevant order, gives the Matrix theory result (2.6).

The action containing the couplings to the metric and NS–NS 2-form is morecomplicated:

(2.33)

S NBI(φ = 0) = −T 0

dt STr

−P E00 + E0iQ−1 − δi k Ekj Ej 0detQi j ,where

(2.34)Qi j = δi j + i

λ

Xi , Xk

Ekj .

To linear order in the background fields, Eµν = ηµν + hµν + bµν , so that

(2.35)Qi j = Qi k

δk j +

Q−1k

li

λ

Xl , Xm

(hmj + bmj )

,

which gives, again to linear order,

(2.36)Q−1

i

j

= δi

k

− Q−1

i

li

λ Xl , Xm(hmk + bmk)Q

−1

k

j ,

so we have all the ingredients we need to derive the linear Born–Infeld couplings to themetric and NS–NS 2-form potential. We find

(2.37)I 00h =

det Q1 − ẊQ−1 Ẋ−1/2,

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(2.38)I 0ih =

det Q1 − ẊQ−1 Ẋ−1/2Q−1(ij)Ẋj ,

(2.39)

I ij h = 12 det Q 1 − ẊQ−1 Ẋ−1/2×

Ẋi

Q−1j

k Ẋk − i

λẊl

Q−1

lm

Xm, Xi

Q−1

j k Ẋk

− iλ

1 − ẊQ−1 ẊQ−1j kXk , Xi+ (i ↔ j ),

(2.40)I 0is = 1

2

det Q

1 − ẊQ−1 Ẋ−1/2Q−1[ij ] Ẋj ,

(2.41)

I

ij

s = 1

4 det Q 1 − ẊQ−1 Ẋ−1/2

×

Ẋi

Q−1j

k Ẋk − i

λẊl

Q−1

lm

Xm, Xi

Q−1

j k Ẋk

− iλ

1 − ẊQ−1 ẊQ−1j kXk , Xi− (i ↔ j ).

One can verify that the lowest order expansion of these currents reproduces the results(2.7)–(2.11).

3. T-duality: multiple D-strings

To construct the Matrix string theory of Dijkgraaf, Verlinde and Verlinde [11], onecompactifies Matrix theory on a circle in, say, the x9 direction. Taylor has shown thatthis is equivalent to a (1 + 1)-dimensional super Yang–Mills theory on the dual circle [14].Taking now i, j = 1, . . . , 8, and denoting the dual coordinate by x̂, the worldvolume fieldstransform as [14]

(3.1)F ij = F ij

= i

β Xi , Xj −→ 12π R9 d x̂i

β Xi , Xj ,(3.2)F 9i = F 9i =

i

β

X9, Xi

−→ 12π R9

d x̂

λ

βDx̂ X

i ,

(3.3)F 0i = −F 0i = Ẋi −→ 1

2π R9

d x̂ Ẋi ,

(3.4)F 09 = −F 09 = Ẋ9 −→ 1

2π

R9

d x̂ λ Ȧx̂ ,

where R9 = α/R9 is the radius of the dual circle. Of course, this is just T-duality appliedto Matrix theory. By construction, the multiple D0-brane action (2.20) considered in theprevious section is covariant under T-duality, so an application of T-duality to the D0-braneaction (2.4) derived from Matrix theory should reproduce the non-relativistic limit of themultiple D-string action.

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where F 09 = Ȧ. We should note that only half of the terms necessary to form the pullbackof C(8) are present in the linear action (3.9). The missing terms come from the C(9)

coupling in the D0-brane action (2.4), as mentioned above. At any rate, this is just theexpansion of the Chern–Simons action

(3.11)S CS = T 1

STr

P

ei(iXiX)/λ

C(n)

∧ eλF

,

as expected. It is computationally more involved to check the NS–NS couplings in (3.9)against the non-abelian Born–Infeld theory of D-strings, and we will not do this here.

4. Matrix string theory: multiple IIA F-strings

4.1. The 9-11 flip

Having constructed the (1 + 1)-dimensional theory of the D-string, we are now in aposition to perform the so-called 9-11 flip, a rotation

(4.1)x9 → x11, x11 → −x9,which will give us the type IIA Matrix string theory action in a linear background.Whereas Schiappa [17] considers the action of the 9-11 flip on the currents, and leavesthe background fields invariant, we take the view here that it is the currents which areinvariant under the 9-11 flip. After all, in the flat space case, the 9-11 flip does notchange the worldvolume fields [11]. Moreover, Schiappa has argued that the currents arein fact invariant under the S- and T-dualities [17]. We simply take R9 = gs

√ α, so thatR9 = √ α/gs and lP = g4/3s R9. We then define the dimensionless worldsheet coordinates

(4.2)σ = x̂R9 , τ = R

α t,

and perform the rescalings Xi

→

√ α Xi and gs

→gs /(2π ). The flat space action (3.5)

becomes the Matrix string action of [11]:

(4.3)S flat = 1

2π

dτ dσ Tr

1

2Ẋ2 − 1

2 DX2 + g

2s

2Ȧ2 + 1

4g2s[X, X]2

,

where −∞ < τ < ∞ and 0 σ < 2π . Weakly coupled string theory at gs = 0 is recoveredin the IR limit, so is described by strongly coupled (1 + 1)-dimensional Yang–Mills. Theconformal field theory which describes this IR limit is a sigma model on an orbifoldtarget space [11]. The matrix-valued coordinates must commute in this limit, so can besimultaneously diagonalised, the eigenvalues x i1, . . . , x

iN corresponding to the positions of

the N strings. Then the action (4.3) reduces to a sum of Green–Schwarz actions for light-cone gauge string theory:

(4.4)S flat = 1

2π

dτ dσ

N n=1

1

2 ẋ2n −

1

2 ∂ x2n

.

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The couplings to linear background fields considered herein should thus tell us somethingabout light-cone gauge string theory in weakly curved backgrounds.

The action of the 9-11 flip on the background fields is easy to derive. Consider, forexample, the type IIB metric fluctuation ha9. Its T-dual on the type IIA side is −ba9, or ineleven-dimensional language −Aa9 11. Under the 9-11 flip −Aa9 11 → Aa11 9 = −ba9. Inother words, ha9 → −ba9 under the 9-11 flip, which is just (linearized) S-duality followedby (linearized) T-duality in the x 9 direction. Arguing in a similar manner, one finds that thelinear background fields transform in the following way under the 9-11 flip:

(4.5)

hab → hab − ηab

φ − 12

h99

, ha9 → −ba9, h99 → −

φ + 1

2h99

,

φ → −φ + 1

2 h99, bab → −C(3)

ab9, ba9 → −C(1)

a ,

C(0) → −C(1)9 , C(2)ab → bab , C(2)a9 → −ha9,C

(4)a1...a4 → C(5)a1...a49, C

(4)abc9 → C(3)abc , C(6)a1...a6 → N (7)a1...a69,

C(6)a1...a59

→ b̃a1...a59, C(8)a1...a8 → −C(9)a1...a89, C(8)a1...a79

→ −C(7)a1...a7 .

Here, N (7)a1...a69 is the field that couples minimally to a type IIA Kaluza–Klein monopole

whose Taub-NUT direction is along x 9. It is easy to verify that the above transformations

are precisely what one would find by performing a linearized S-duality (in the string frame)(4.6)h → h − ηφ, φ → −φ, b2 → −C(2),(4.7)C(0) → −C(0), C(2) → b2, C(6) → b̃6, C(8) → −C(8),

followed by a linearized T-duality in the x9 direction, as in (3.7) and (3.8), plus thelinearized T-duality rules for the field b̃6 [31]:

(4.8)b̃a1...a6 → N (7)a1...a69, b̃a1...a59 → b̃a1...a59.Performing the transformations (4.5) on the linear action (3.9), and substituting for

(4.2), one finds

S IIAlinear = 1

2π

dτ dσ

α

R2 STr

1

2

hab − ηab

φ − 1

2 h99

I abh + 2ba9I a9s

+ 12

φ + 1

2 h99

I 99h −

1

2

φ − 3

2 h99

I φ + C(1)a I a9h − C(3)ab9I abs − C(1)9 I 90

− ha9I a0 + 3babI ab92 + C(3)abc I abc2 + 1

12 C

(5)a1...a49

I a1...a494 +

1

60b̃a1...a59I

a1...a54

(4.9)+ 148

N (7)a1...a69

I a1...a696 −

1

336 C (7)a1...a7 I

a1...a76

.

Writing the currents in the appendix in terms of the dimensionless quantities τ and σ , andafter rescaling Xi →

√ αXi and gs → gs /(2π ), this is the action describing Matrix string

theory in a weakly curved background. There is no need to write out the new couplingsin full, suffice it to say that each term of the form Ẋ, λDX/β , [X, X]/β and λ Ȧ appears

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multiplied by a factor of R/√

α, each [X, X] term appears multiplied by a factor of 1/gs ,and each

˙A term appears with a factor of gs . We should note that the couplings derived

by Schiappa [17] are as above, but without those couplings to fields with a component inthe x 9 direction. As we will see below, these latter are necessary to match with light-conegauge string theory.

4.2. Light-cone gauge string theory in a general background

To examine the action (4.9), let us consider the weakly coupled string theory, gs = 0.Just as in the flat case, the Born–Infeld field strength drops out entirely, and one is forcedonto the space of commuting matrices as above. Moreover, the couplings to all Ramond–

Ramond fields vanish, as one might expect, and we find

S IIAlinear = 1

2π

dτ dσ

N n=1

α

R2

1

2 h00 − h09 +

1

2 h99

+√

α

R

(h0i − h9i )ẋin + (b0i − b9i )∂ xin

+ 14

(h00 − h99)

ẋ2n + ∂x2n

+ 12

hij

ẋinẋj n − ∂xin∂xj n

+ bij ẋin∂xj n + b09ẋn · ∂xn+ R√

α1

2 h0i ẋ

inẋ2n + ∂x2n+ bi912 ∂ xinẋ2n + ∂x2n+ ẋin∂xn · ẋn

(4.10)+ R2

α

1

4 φ − 3

8 h99

(ẋn · ∂xn)2 + 1

4

ẋ2n + ∂x2n

2.

If we now define the (somewhat non-standard) light-cone coordinates

(4.11)X± = 1√ 2

X0 ∓ X9,

then we have

S IIAlinear = 1

2π

dτ dσ

N n=1

α

R2 h++ +

√ α

R

√ 2

h+i ẋ in + b+i ∂xin

+ 12

h+−

ẋ2n + ∂x2n+ 1

2 hij

ẋ inẋ

j n − ∂x in∂xj n

(4.12)+ bij ẋin∂xj n + b+−ẋn · ∂xn +O

R√

α

.

We wish to compare this action to light-cone gauge string theory in a weakly curvedbackground. To this end, consider

(4.13)S = − 14π α

dτ dσ

√ −γ γ αβ∂α xµ∂β xν Gµν (x) + αβ∂αxµ∂β xνBµν(x),

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where α, β denote the worldsheet coordinates τ and σ , γ αβ is the worldsheet metric and γ its determinant. We are always free to take the worldsheet to be flat, in which case

(4.14)S = − 14π α

dτ dσ

−ẋµẋν Gµν(x) + ∂xµ∂xνGµν(x) − 2ẋµ∂xν Bµν (x).Since the worldsheet energy–momentum tensor vanishes, this must be supplemented withthe constraint

(4.15)

ẋµ ± ∂xµẋν ± ∂xνGµν = 0.The light-cone gauge is defined by taking x+(τ,σ) = τ . Since we interested only inlinear backgrounds, we set Gµν = ηµν + hµν , where η+− = −1 and ηij = δij , andBµν

=bµν . As usual in a light-cone treatment of gravity [32], we can further use spacetime

diffeomorphisms to set G−− = 0 = G−i . The string action is thenS = 1

2π α

dτ dσ

−2ẋ− + ẋ2 − ∂x2 + 1

2 h++ + h+i ẋi + h+−ẋ−

+ 12

hij

ẋi ẋj − ∂xi ∂xj + b+−∂x− + b+i ∂xi + bij ẋi ∂xj (4.16)+ b−i

1

2

ẋ2 + ∂x2∂xi − ẋ · ∂xẋi.

At first sight, the linear couplings to the background fields do not seem to match thosein the action (4.12). However, the constraint (4.15) can be solved for x

− giving, again to

linear order in hµν ,

(4.17)

ẋ− = 12

ẋ2 + ∂x2+ 1

2 h++ + ẋi h+i +

1

2

ẋi ẋj + ∂x i ∂xj hij + 1

2

ẋ2 +∂x2h+−,

(4.18)∂x− = ẋ · ∂x + ∂xi h+i + ẋi ∂xj hij + ẋ · ∂xh+−.To linear order, then, we can replace ẋ− and ∂x− in the string action with (ẋ2 + ∂x2)/2and ẋ · ∂x, respectively. In that case, we find exact agreement with the Matrix string theoryresult (4.12). In other words, we have succeeded in reproducing the correct form of the

light-cone gauge type IIA string action in a weakly curved background.

4.3. Ramond–Ramond couplings

Turning to the R–R couplings in the action (4.9), let us consider the 3-form couplings,which could potentially give rise to D2-brane solutions. If we set the Born–Infeld field tozero, we find

(4.19)S C(3) = i

2πgs

dτ dσ STr

√ α

RC

(3)+ + Ẋi C(3)i

,

where we have defined(4.20)C(3)+ =

1√ 2

C(3)+ij

Xj , Xi

,

(4.21)C(3)i = 1

2 C

(3)ijk

Xk , Xj

+ C(3)+−j Xj , Xi.

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The couplings given in [17] differ considerably from the ones given here, since in [17]only the C(3)

0ij and C(3)

ijk

terms were considered. Yet from our analysis it is clear that the

other terms not only contribute, but are necessary in order to be able to write the C(3)

couplings in the light-cone gauge.A nice check of the 3-form couplings (4.19) is against the eleven-dimensional

supermembrane theory. The connection between the light-cone treatment of the eleven-dimensional supermembrane in flat space and Matrix theory is well known [33]. Onecan further argue that Matrix string theory is found by compactifying the light-conesupermembrane theory on a circle, the off-diagonal elements of the Matrix string theoryfields being related to the infinite tower of Kaluza–Klein modes along the compactdirection [34]. Given further that the light-cone supermembrane theory in an arbitrary

supergravity background is known [35–37], one could in principle derive Matrix stringtheory in an arbitrary background using these techniques. Indeed, the Lagrangian densityfor the light-cone supermembrane in a curved (but not entirely general) background hasbeen derived in [36] and it is easy to see that the methods of [34] would generate some of the couplings in (4.19) of our Matrix string theory action. More specifically, the couplingsto C (3)+ij and C

(3)ijk are reproduced although, since the gauge C

(3)+−j = 0 was chosen in [36],

the coupling to these components of C(3) cannot be checked. Of course, neither can thecouplings to the remaining R–R fields, since it is not known how or, indeed, if their eleven-dimensional counterparts couple to the supermembrane.

It can be seen from (4.9) that as regards the R–R 5-form potential, only the terms of theform C (5)a1...a49 contribute. It is easy to check that the terms involving C(5)a1...a5 would couple

to the current I 5, which was previously set to zero in order to match the action (2.4) withthe action of the multiple D0-brane theory of [2,3]. Setting the Born–Infeld vector to zero,the remaining 5-form R–R field couplings can be written as:

(4.22)S C(5) = i

4πgs

R√ α

dτ dσ STr

√ α

RC

(5)+− + Ẋi C(5)+i − Ẋi C(5)−i

,

where we have defined

(4.23)C(5)µν = Xk , Xj DXi C(5)ijkµν .

Similarly, the C (7) couplings only have contributions involving terms of the form C (7)a1...a7 ,

since terms of the form C (7)a1...a69 couple to currents for which the explicit expression is not

known, corresponding to N (7) couplings in the D0-brane action (2.4). The C(7)a1...a7 termscan be written as

(4.24)S C(7) = i

96πg3s

R2

α

dτ dσ STr

√ α

R

1√ 2

C(7)+ +

√ α

R

1√ 2

C(7)− + Ẋi C(7)i

,

with

(4.25)C(7)µ =

Xn, Xm

Xl , Xk

Xj , Xi

C(7)ijklmnµ .

Let us for the sake of completeness also consider the couplings to b̃(6) and N (7). Asfor C(5), only the terms with a 9-component appear in the action. The other components

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couple to currents for which the explicit expression is not known. The b̃(6) couplings canbe written as:

(4.26)S ̃b(6) = 1

16πg2s

R√ α

dτ dσ STr

√ α

Rb̃

(6)+− + Ẋi b̃(6)+i − Ẋi b̃(6)−i

,

where

(4.27)b̃(6)µν =

Xl , Xk

Xj , Xi

b̃(6)ijklµν ,

and the N (7) couplings are given by

(4.28)S N (7) = − 1

16πg2s

R2

α dτ dσ STr√

α

R N

(7)

+− − Ẋi

N

(7)

+i + Ẋi

N

(7)

−i ,where

(4.29)N (7)µν =

Xm, Xl

Xk, Xj

DXi N (7)ijklmµν .

Note that the couplings of the different fields occur at a different order of the expansionparameter R/

√ α.

5. T-duality once more: multiple IIB F-strings

To describe fundamental strings in the type IIB theory, we perform another T-duality inthe x9 direction, as in (3.7) and (3.8). As before, we assume that the worldvolume fields(the currents) do not change, so the flat action (4.3) is unchanged. The linear action (4.9)becomes

S IIBlinear = 1

2π

dτ dσ

α

R2 STr

1

2 (hab − φηab)I abh + C(2)a9 I a9h +

1

2 (φ − h99)I 99h

−

1

2

(φ

+h99)I φ

−C

(2)ab I

abs

−2ha9I

a9s

+ba9I

a0

−C(0)I 90

+C

(4)abc9I

abc2

+ 3babI ab92 + 1

12 C (4)a1...a4 I

a1...a494 +

1

60b̃a1...a59I

a1...a54 +

1

48b̃a1...a6 I

a1...a696

(5.1)− 1336

C(8)a1...a79

I a1...a76

,

which should describe strongly coupled Matrix strings in the IIB theory, with the currentsas in the appendix, up to the coordinate transformations (4.2), and the rescalings Xi →√

αXi and gs → gs /(2π ).Note that precisely the same action is obtained if one applies the S-duality rules (4.6)

and (4.7) to the D1-brane action (3.9). Although this might be expected from the S-dualityconnection between D- and F-strings in the type IIB theory, it is not clear a priori howsuch an S-duality should be done directly, since we are dealing with non-abelian fields; asin Yang–Mills theory, we cannot rigorously perform a worldvolume duality transformationto show the S-duality equivalence between the two.

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It is worth pointing out however that the S-duality transformation of the NS–NS 2-form:

(5.2)b(2) → −C(2), C(2) → b(2),implies that the dynamics will now be governed by open D-strings, given that the invariant2-form field strength that will couple in the worldvolume is constructed with C (2) insteadof b(2). Therefore the Born–Infeld field A should transform into a new worldvolume vectorfield A whose abelian component forms a gauge invariant field strength with the R–R 2-form. The right S-duality transformation rule should be then

(5.3)A → −A, A → A,in order to match (5.2). This cannot, however, be seen explicitly at the level of the linearized

actions that we have considered.The action above can be rewritten in a more convenient form, filling in the expressions

for the currents as given in Appendix A (upon the rescaling). In particular, for the Chern–Simons action we have:

S IIBCS = 1

2π

dτ dσ STr

P

α

R2 b(2) + gs

√ α

RC(0) ∧ F + i

√ α

gs R(iXiX)C

(4)

+ i(iXiX)b(2) ∧ F − 1

2g2s(iXiX)2b̃

(6) − R2gs

√ α

(iXiX)2C(4) ∧ F

(5.4)+ iR6g3s

√ α

(iXiX)3C(8) − iR2

6g2s α (iXiX)3b̃

(6) ∧ F .Again we note that only half of the terms necessary to form the pullback of C(8) are presentin the linear action. Some of the above couplings have been given before in [18].

6. T-duality along a transverse direction: multiple IIA strings with winding number

Let us now also describe type IIA fundamental strings with winding number. The IIAstrings that are described by Matrix string theory carry momentum p+. This is the chargewhich is related by the 9-11 flip to the number of D-particles. T-duality in the x 9 directiongives IIB strings with winding number, which we have just checked are S-dual to multipleD-strings. Strongly coupled IIA F-strings with winding number can then be obtained fromIIB strings by performing a T-duality transformation in a direction transverse to the IIBstrings. These strings with winding number have interesting dielectric properties that wewill discuss in the next section.

Calling z the T-duality direction and a = (0, i),wherenow i = 1, . . . , 7, the linear actionthat is obtained from (5.1) is given by:

S IIAlinear = 1

2π

dτ dσ

α

R2 STr

12

hab + ηab

φ − 1

2 hzz

I abh − bazI azh

− 12

1

2 hzz + φ

I zzh + C(3)a9zI azh −

1

2

h99 − φ +

1

2 hzz

I 99h − C(1)9 I z9h

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− 12

φ − 1

2 hzz + h99

I φ − C(3)abz I abs − 2C(1)a I azs − 2ha9I a9s − 2bz9I z9s

+ ba9I a0 + hz9I z0 − C(1)z I 90 + C(5)abc9zI abc2 − 3Cab9I abz2 + 3babI ab92− 6hazI az92 +

1

60 N

(7)a1...a59z

I a1...a54 +

1

12b̃a1...a4z9I

a1...a4z4

+ 112

C (5)a1...a4zI a1...a494 +

1

3 C

(3)abc I

abcz94 −

1

336 C

(9)a1...a79z

I a1...a76

(6.1)+ 148

C(7)a1...a69

I a1...a6z6 +

1

48 N (7)a1...a6zI

a1...a696 +

1

8b̃a1...a5zI

a1...a5z96

.

The direction z in which the T-duality is performed appears as an isometry direction in the

transverse space of the strings. We denote the corresponding Killing vector as(6.2)kµ = δµz , kµ = ηzµ + hzµ.

In a manner similar to the Kaluza–Klein monopole, the non-abelian strings do not see thisspecial direction, the embedding scalar Xz is not a degree of freedom of the strings, but istransformed under T-duality into a world volume scalar ω [31]. This worldvolume scalarforms an invariant field strength with baz (see [18] for the details), and can, therefore,be associated to fundamental strings wrapped around the isometry direction z, whichthemselves end on the Matrix strings.

The action (6.1) can be written in a covariant way as a gauged sigma model,

where gauge-covariant derivatives DαXµ are used to gauge away the embedding scalarcorresponding to the isometry direction [38]:

(6.3)Dα Xµ = DαXµ − kρ Dα Xρ kµ,with α = σ, τ . These gauge-covariant derivatives reduce to the standard covariantderivatives Dα Xµ for µ = z and are zero for µ = z. The pull-backs that appear in the actionof the F-strings with winding are constructed from these gauge covariant derivatives. Forexample,

P

b(2)

=bµνDX

µDXν dt dx

(6.4)=b09 + R√ α

b0i DXi + R√

αbi9 Ẋi +

R2

α bij Ẋi DXj dt dx.

Filling in the expressions for the currents as given in the appendix (upon the rescaling)we can write the Chern–Simons action as:

S linear = 1

2π

dτ dσ STr

P

−gs

√ α

RikC

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