M Atrix String Theory on K3

SLAC-PUB-7462

UTTG-14-97

April 1997

Matrix String Theory on K3

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Willy Fischler ,Arvind Ra jaraman

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Theory Group,Department of Physics, University of Texas, Austin,TX 78712.

Stanford Linear Accelerator Center, Stanford, CA 94309

Abstract

We conjecture that M-theory compacti ed on an ALE space or K3 is de-

scrib ed by 0-branes moving on the ALE space. We give evidence for this

by showing that if we compactify another circle, we recover string theory on

the ALE space. This guarantees that in the large N limit, the matrix mo del

correctly describ es the force lawbetween gravitons moving in an ALE back-

ground. We also show the app earance in Matrix theory of the dualityof

M-theory on K3 with the heterotic string on a three-torus.

Submitted to Nuclear Physics B.

e-mail address: [email protected]

Supp orted in part by the Department of Energy under contract no. DE-AC03-76SF00515.

I. INTRODUCTION

Matrix theory [1] has b een prop osed as a nonp erturbative formulation of M-theory.

The conjecture is that M-theory in the in nite momentum frame is describ ed by a Hamilto-

nian that consists of zero-branes and their interactions. These zero-branes are the carriers of

longitudinal momentum.We shall refer to these partons as D 0-branes . This conjecture has

passed many tests. The Hamiltonian contains 11-dimensional sup ergravitons and sup erme-

mbranes in its sp ectrum and repro duces their interactions [1{4]. It also p ossesses T-duality

up on compacti cation [5,6]. In addition, up on compactifying on a circle and shrinking the

circle to zero radius, Matrix theory has b een shown [7{ 10] to contain multistring states.

These strings interact according to the usual light-cone interactions and the prop er scaling

between the coupling constant and the radius is recovered [9].

It is imp ortant to consider compactifying the Matrix mo del on more complicated sur-

faces which preserve less sup ersymmetry. The natural rst step is to consider the simplest

Calabi-Yau space, the four-dimensional K 3 surface. It is a reasonable guess that M-theory

on K 3 in the in nite momentum frame is describ ed by the dynamics of D 0-branes moving

on K 3 [11].

However, in [12], the authors computed the force lawbetween two D 0-branes moving

on K 3. They found that the v part of the force was not the same as the force exp ected

from sup ergraviton exchange. This app eared to signal a p otential problem with this mo del.

We b elieve that this disagreement is not a fatal one. There is no nonrenormalization

theorem for the gravitational coupling constantin N = 2 theories. Therefore the coupling

constant could easily b e di erent at short and large distance scales. In any case, for the

connection to Matrix theory,we need to consider the interaction of b ound states of large

numb ers of D 0-branes since it is only in the large N limit that one recovers the force law

derived from sup ergravity.

In order to showhow the large N limit is involved in this context and to simplify the

discusssion, we will b egin by following the prop osal of [11] for a matrix mo del of M-theory 1

on an ALE surface. We will then compactify an additional circle. In the limit where this

circle is small, keeping the ALE size xed in string units, we will showhow to recover

typ e I IA strings propagating in an ALE background. String theory then guarantees that

the gravitational force law is exactly repro duced in Matrix theory without the need for

additional degrees of freedom b eyond the D 0-branes . Since the force lawbetween gravitons

is repro duced in light-cone string theory only when N !1, there is no con ict with the

results of [12].

Finally,we sub ject this mo del to another test . In the limit when the volume of the K 3

shrinks to zero, string duality predicts that we should get the dynamics of heterotic strings

on T .We show that this is indeed the case, thus providing another consistency checkof

this mo del.

I I. MATRIX STRING THEORY

We will review the calculation of [9], expressing it in terms that will b e suitable for us.

The b osonic part of the Matrix theory hamiltonian is

2 i j 2

H = R tr +[X ;X ] 1

On compacti cation on a circle of radius R , the Matrix mo del is describ ed by a 1+1

dimensional gauge theory [13] with the Hamiltonian

d 1

2 i j 2 2 2 i 2

H = E +[X ;X ] 2 tr + R DX +

i 9

R R

1=2

i i

Rescaling X ! R X ,we get

2 i j 2 2 i 2

E +[X ;X ] 3 H = d tr +DX +

For small R , the typ e I IA limit, we nd that the terms [X; X ] and E must b e set equal

to zero. The condition [X; X ] = 0 implies that we are on the mo duli space of the theory

where the X commute. The condition E = 0 implies that the gauge dynamics is such that 2

only gauge-singlet states survive. The ob jects which are gauge-invariant are the eigenvalues

of the matrices X . Accordingly, the small R dynamics is reduced to the dynamics of the

eigenvalues of X . This can b e describ ed as a theory with Z gauge invariance.

The X are functions of the co ordinate in the YM theory. Due to the residual Z

symmetry, they do not have to b e p erio dically identi ed, but rather only identi ed upto a

p ermutation of the eigenvalues. This e ectively multiplies the length of the string by N .

We can make strings of any longitudinal momentum by this pro cedure.

The world sheet Hamiltonian reduces to

2 i 2

H = d tr +@X 4

Thus the long strings which are pro duced have the dynamics of IIA strings.

Wenow turn to the problem we are interested in, which is the Matrix mo del compact-

i ed on K 3. We shall start with a simpler case, where wehave an ALE surface instead of

K 3. The conjecture is then that M-theory on an ALE surface is describ ed by the dynamics

of 0-branes on the ALE. We will start byworking in the orbifold limit of the ALE surface.

Co ordinates will b e chosen so that x ;x ;x ;x are the directions along the ALE surface.

6 7 8 9

The dynamics of 0-branes on an ALE space was studied in great detail in [15,16]. For

illustrative purp oses, we will take the example R =Z . In this case, each 0-brane has an

image. We therefore take N 0-branes and their images under Z moving on R . One then

quantizes the op en strings connecting these 0-branes and then imp oses invariance under Z .

There are twotyp es of strings connecting these branes, those with p olarizations along the

ALE surface whichwe will call ;a =6;7;8;9 and those with p olarizations p erp endicular

to the ALE surface whichwe call X ;i =0;1;2;3;4;5. Each of them is represented bya

matrix in U 2N . One then imp oses the Z pro jection on these states.

We can schematically write the Z pro jection as

i 1 i a 1 a

UX U = X U U = 5

where U is a unitary op erator representing the Z .

2 3

We can represent the solution of these equations by

1 1 0 0

a a i i

C D A B

a i

A A @ @

6 = X =

a a i i

D C B A

i i a a

where A ;B ;C ;D are N N matrices.

In order to derive the Hamiltonian describing the dynamics of zero-branes, it is useful to

start with a 6-dimensional theory since the zero-branes moveon R ALE. The resulting

theory, thoughtofasaN =1;d = 6 theory, has U N U N gauge symmetry, and there

are in addition to the vector multiplet , twohyp ermultiplets transforming in the N; N

representation. The theory on the world-volume of the zero-branes is then the dimensional

reduction of this gauge theory to 0+1 dimensions.

We are interested in the case where in addition, we compactify on a circle of radius R .

We can T-dualize along the circle in order to get a theory of 1-branes wrapp ed on the dual

circle. After doing this the Hamiltonian is [15,16]

2 2 2 2 2 i 2 2 a 2

H = tr R E + P + P + R DX + R D +

9 X 9 9

X X X

a b 2 a i 2 i j 2

7 j[ ; ]j +2 j[ ;X ]j + j[X ;X ]j

a;i i;j

a;b

where commutators are taken in the U 2N matrices.

1=2 1=2

i i a a

We rescale X ! R X ; ! R .The Hamiltonian is then

9 9

2 2 i 2 a 2 2

H = d tr P + P +DX +D + E

h i

X X X

a b 2 a i 2 i j 2

Trj[ ; ]j +2 Trj[ ;X ]j + Trj[X ;X ]j 8

a;i i;j

a;b

In the limit R ! 0, to analyse the light sp ectrum, wemust set E = 0 and also set the

p otential terms to zero. The vanishing of E again implies that the e ective theory will b e

describ ed in terms of gauge-singlet ob jects .

The vanishing of the p otential terms implies that we are on the mo duli space. In this

case, the mo duli space is more complicated; wehave b oth a Higgs and a Coulomb branch.

We will analyse them separately. 4

The Higgs branch is the branch where 6=0. Since denotes the motion within

the ALE space, this implies that the D-branes havemoved o the xed p oint. The

Z pro jection implies that the images must then b e at the same p oint in the transverse

space i.e. the D-brane and its image are at p ositions x ;x ;x ;x ;x ;x ;x ;x and

2 3 4 5 6 7 8 9

x ;x ;x ;x ;x ;x ;x ;x resp ectively.

2 3 4 5 6 7 8 9

In general, di erent pairs of D-branes do not have to b e at the same value of

x ;x ;x ;x . We can therefore separate the 2N D-branes into N pairs.

2 3 4 5

For each pair of D-branes, we need to nd a set of gauge-invariant co ordinates parametriz-

ing their p osition on the Higgs branch. Wemust rst imp ose the condition that the p otential

energy vanishes. Wemust then cho ose a gauge invariant set of co ordinates.

It was shown in [15,16] that after these op erations, wehave precisely four gauge-invariant

scalars Y left. One can then nd the dynamics for these scalars. It was shown in [15,16]

a b

that these scalars propagate according to the Lagrangian g @Y @Y , where g coincides

ab ab

ALE

with the ALE metric: g = g .

This is true for a general ALE space, i.e. after imp osing the vanishing of the p otential

terms and gauge invariance, wehave four gauge-invariant co ordinates whichhave a nontrivial

mo duli space metric, which is identical to the ALE metric.

Wenow turn to the transverse co ordinates. Each pair of D-branes was at a transverse

co ordinate x ;x ;x ;x . If we take N pairs of 0-branes, we will havea NN matrix

2 3 4 5

parametrizing these p ositions. The gauge invariant ob jects are the eigenvalues of these

matrices.

The dynamics of these eigenvalues will determine the propagation of the string in the

transverse co ordinates. Accordingly, one exp ects a trivial metric in the kinetic terms. We

have not b een able to prove this directly, but this seems very plausible for several reasons.

The eigenvalue dynamics is a 1+1 sigma mo del. It is well known that such a theory ows

to a Ricci at metric in the IR. Furthermore, at the classical level, the metric is trivial.

There are no one-lo op corrections [12]. Sup ersymmetry then implies that the at metric

is not corrected p erturbatively. It seems reasonable to supp ose that the non-p erturbative 5

corrections will still allow the sigma mo del to ow to a free theory in the IR. We shall

therefore make this assumption.

As b efore, wehavea Z gauge symmetry.We can therefore pro duce long strings by

identifying the matrices upto a p ermutation of eigenvalues when we go around the YM

circle. These strings will haveaworld-sheet Lagrangian

i 2 ALE a b

L =@X + g @Y @Y + f er mions 9

which is precisely that of a string propagating in an ALE background.

We turn now to the Coulomb branch. On this branch, the are set equal to zero,

which means that the 1-branes are stuck to the xed p oint. The X can have arbitrary

values. These states were shown [15,24,11] to corresp ond to the extra gauge b osons needed

to enhance the gauge symmetry to a nonab elian group.

Hence wehave obtained all the states of typ e I IA string theory in the background of

an ALE space. The consistency of string theory will then automatically provide the force

lawbetween two gravitons. Although this is a consistent theory for nite N [23], the full

Lorentz invariance is only recovered in the large N limit. One should therefore exp ect to

recover the force lawbetween two gravitons only in this limit..

One must also show that the structure of the vertex is of the same form as in string

theory.To do this requires a more detailed analysis of the sup erconformal eld theory.We

exp ect that sup ersymmetry is sucient to recover the vertex structure, but a demonstration

of this is desirable.

So far wehave only lo oked at the orbifold limit. The case where the orbifold singularityis

blown up was also considered in [15,16], where it was shown that the blowup corresp onded to

the addition of Fayet-Iliop oulous D-terms. Remarkably, the result is very similar; the metric

on the Higgs branch is altered so that it is still identical to the metric on the nonsingular

ALE space! Thus everything wehave said go es through unchanged. We still recover string

theory in an ALE background, and the force law is guaranteed to work in the large N limit.

When wehave the full K 3 instead of an ALE space, there are certain complications that 6

arise. The compacti cation of M-theory on K 3 seems to b e describ ed by a 4+1 dimensional

eld theory, like the case of compacti cation on T which is a 4+1 YM theory. These

theories are dicult to de ne.

Presently,we can only discuss M-theory near the orbifold p oints of K 3 where the dy-

namics is simple enough to enable us to make statements. At these p oints, the K 3 is at

everywhere except near the orbifold singularities, where the geometry is that of the ALE

space. Wehave already seen that 0-branes moving in an ALE space see the ALE space

geometry.Itisobvious that 0-branes in at space see at space geometry. Putting these

facts together, we see that, at least near the orbifold limit, the D-branes see the geometry

of K 3.

We can then go through the pro cedure of constucting strings on K 3 in exact parallel to

the discussion for the ALE space.The only di erence is that on the Higgs branch, the mo duli

space metric is the K 3 metric instead of the ALE metric. Thus the strings we construct will

have exactly the worldsheet action of strings in the background of K 3. This again shows

that the force lawbetween two gravitons will work out correctly in the large N limit.

I I I. M-THEORY-HETEROTIC DUALITY

In the limit where the volume of the K 3 shrinks to zero, we exp ect to recover heterotic

string theory on T [17]. Wenow showhow this o ccurs in Matrix theory.

The idea is very simple. We are lo oking at the dynamics of D 0-branes in the background

of a shrinking K 3. It is natural to T-dualize this theory so that it b ecomes a theory of 4-

branes wrapp ed on K 3. It turns out that these 4-branes are at strong coupling, so it is

natural to interpret this as a theory of M-theory 5-branes wrapp ed on K 3. The extended

dimension of the 5-branes is much longer than the K 3 dimensions. We can therefore treat

it as e ectively b eing a 1+1 dimensional theory, with a Lagrangian following from wrapping

a 5-brane on K 3. This is known to have the same world-sheet structure as a heterotic

string [18,19]. We will therefore recover the heterotic string theory in this limit, exactly as 7

exp ected.

Wewould like to see that the tension of the heterotic string obtained in this manner is

exactly the value exp ected from M-theory-heterotic duality.

To simplify the calculation, we rst consider toroidal compacti cation along the lines

of [20], where M-theory is compacti ed on a four-torus with sides L ;i =1;2;3;4. This is

represented in Matrix theory by a 4+1 dimensional Yang-Mills theory on a four torus of

3 6

l l

2 2 6

11 11

radii =2 ;i =1;2;3;4, with a coupling constant g =2 .

RL RL L L L

i 1 2 3 4

It was shown in [14] that the coupling constant b ecomes an extra dimension. Thus we

reach a 5+1 dimensional theory. This is interpreted as the theory on the world-volume of

;i =1;2;3;4, and N M-theory 5-branes wrapp ed on a 5-torus with sides =2

=2 .

RL L L L

1 2 3 4

This 5+1 dimensional theory is still somewhat mysterious, but we can still extract some

information. In particular, if we shrink four out of the ve dimensions to zero size, we reach

a 1+1 dimensional theory describing a string. We can do this in veways, thus obtaining

ve strings. Four of these corresp ond to membranes wrapp ed on the four circles of the T .

The fth is the 5-brane wrapp ed on T . There is a symmetry among these ve strings. This

is related to the symmetry used in [21,22] to give a construction of the ve-brane wrapp ed

on T .

We already know the tensions of the four strings corresp onding to the four membranes.

The string tensions corresp onding to these strings are given by T = ;i =1;2;3;4: The

1 L L L L

1 2 3 4

symmetry then tells us that the fth string will have tension T = = , whichis

R 2 l

the right tension for a ve-brane wrapp ed on T .

We can p erform exactly the same manipulations for M-theory on a K 3ofvolume vl .

This, according to our conjecture, is a theory of D 0-branes moving on this K 3. After T-

12 8

l l

11 11

. duality [25], this is describ ed by a 4+1 dimensional theory on a K 3ofvolume =

4 4

R v

R vl

This is equivalent to a 5+1 dimensional theory, whichweinterpret as the world-volume of N

1 1 5

M-theory 5-branes wrapp ed on K 3 S . The length of the S is, as b efore, =2 .

In the limit v ! 0, this is a 1+1 dimensional theory where the 5-branes are wrapp ed on a 8

small K 3 and a large S .

Although we do not know the world-volume theory of the 5-brane, it is very reasonable

to assume that the dynamics will force us onto the mo duli space of the theory. This means

that the N 5-branes will b e separated from each other in the transverse space. Each 5-brane

is wrapp ed on a very small K 3, leading to a 1+1 dimensional theory which is known to b e

a theory which describ es the worldsheet action of the heterotic string [18,19]. As usual we

havea Z symmetry, and we can go through the usual pro cedure of making long strings.

1 v

The string scale of the heterotic string is, as b efore, T = , whichisthe =

R 2 l

correct value predicted from string duality.

It would b e interesting to see the connection of this construction to the matrix mo del

prop osed for heterotic strings [10].

IV. ACKNOWLEDGEMENTS

We are grateful to E.Halyo and esp ecially to L.Susskind for crucial discussions.

W. F. thanks the ITP at Stanford for its hospitality and was supp orted in part by the

Rob ert WelchFoundation and NSF Grant PHY-9511632. A.R. was supp orted in part by

the Department of Energy under contract no. DE-AC03-76SF0051 5. 9

REFERENCES

[1] T. Banks,W. Fischler,S. Shenker and L. Susskind,hep-th/9610043.

[2] M.R.Douglas, D.Kabat, P.Pouliot and S.Shenker, hep-th/9608024.

[3] G. Lifschytz, hep-th/9612223.

O. Aharony and M. Berko oz, hep-th/9611215.

G. Lifschytz and S. Mathur, hep-th/9612087.

[4] J. Polchinski and P.Pouliot, hep-th/9704029 .

[5] L. Susskind,hep-th/9611164.

[6] O.J. Ganor, S. Ramgo olam and W.Taylor,hep-th/9611202.

[7] L. Motl,hep-th/9701025.

[8] T. Banks and N. Seib erg,hep-th/9702187.

[9] R. Dijkgraaf, E. Verlinde and H. Verlinde, hep-th/9703030.

[10] S.Kachru and E.Silverstein, hep-th/9612162 .

T. Banks and L.Motl, hep-th/9703218.

D.Lowe, hep-th/9704041.

N.Kim and S.J. Rey, hep-th/9701139 .

[11] M.R. Douglas, hep-th/9612126.

[12] M.R. Douglas, H. Ooguri and S. Shenker, hep-th/9702203.

[13] W. Taylor,hep-th/9611042.

[14] M. Rozali,hep-th/9702136.

[15] M.R. Douglas and G. Mo ore, hep-th/9603167.

[16] C.V. Johnson and R. Myers, hep-th/9610140. 10

[17] E. Witten, Nuc. Phys. B443 1995 85.

[18] J. Harvey and A. Strominger, Nuc.Phys. B449 1995 535, hep-th/9504047.

[19] S. Cherkis and J. Schwarz, hep-th/9703062.

[20] W. Fischler, E. Halyo, A. Ra jaraman and L. Susskind, hep-th/9703102.

[21] E. Halyo, hep-th/9704086.

[22] M. Berko oz and M. Rozali, hep-th/9704089 .

[23] L. Susskind, hep-th/9704080.

[24] J. Polchinski, hep-th/9606165.

[25] K. Hori and Y.Oz, hep-th/9702173 . 11