SLAC-PUB-7462 UTTG-14-97 April 1997 Matrix String Theory on K3 } |yz Willy Fischler ,Arvind Ra jaraman } 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] y e-mail address: [email protected] z 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 4 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 3 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 11 i On compacti cation on a circle of radius R , the Matrix mo del is describ ed by a 1+1 9 dimensional gauge theory [13] with the Hamiltonian Z 2 d 1 2 i j 2 2 2 i 2 H = E +[X ;X ] 2 tr + R DX + i 9 2 R R 9 9 1=2 i i Rescaling X ! R X ,we get 9 Z 2 1 2 i j 2 2 i 2 E +[X ;X ] 3 H = d tr +DX + i 3 R 9 For small R , the typ e I IA limit, we nd that the terms [X; X ] and E must b e set equal 9 to zero. The condition [X; X ] = 0 implies that we are on the mo duli space of the theory i 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 i of the matrices X . Accordingly, the small R dynamics is reduced to the dynamics of the 9 i eigenvalues of X . This can b e describ ed as a theory with Z gauge invariance. N i The X are functions of the co ordinate in the YM theory. Due to the residual Z N 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 Z 2 2 i 2 H = d tr +@X 4 i 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 4 illustrative purp oses, we will take the example R =Z . In this case, each 0-brane has an 2 4 image. We therefore take N 0-branes and their images under Z moving on R . One then 2 quantizes the op en strings connecting these 0-branes and then imp oses invariance under Z . 2 There are twotyp es of strings connecting these branes, those with p olarizations along the a ALE surface whichwe will call ;a =6;7;8;9 and those with p olarizations p erp endicular i 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. 2 We can schematically write the Z pro jection as 2 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 6 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 . 9 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] Z d 2 2 2 2 2 i 2 2 a 2 H = tr R E + P + P + R DX + R D + 9 X 9 9 R 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 Z 1 2 2 i 2 a 2 2 H = d tr P + P +DX +D + E X 3 R 9 h i X X X 1 a b 2 a i 2 i j 2 Trj[ ; ]j +2 Trj[ ;X ]j + Trj[X ;X ]j 8 3 R 9 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 9 p otential terms to zero.