hep-th/9709118 IASSNS-HEP-97/99, LBNL-40723, RU-97-72 SLAC-PUB-7648, UCB-PTH-97/45 Matrix Description of 1; 0 Theories in Six Dimensions 1 2 3 4 Ofer Aharony , Micha Berkooz , Shamit Kachru , and Eva Silverstein 1 DepartmentofPhysics and Astronomy Rutgers University Piscataway, NJ 08855-0849, USA 2 Scho ol of Natural Sciences Institute for Advanced Study Princeton, NJ 08540, USA 3 DepartmentofPhysics University of California at Berkeley Berkeley, CA 94720, USA hep-th/9709118 16 Sep 1997 4 Stanford Linear Accelerator Center Stanford University Stanford, CA 94309, USA We prop ose descriptions of interacting 1; 0 sup ersymmetric theories without gravityin six dimensions in the in nite momentum frame. They are based on the large N limit of quantum mechanics or 1 + 1 dimensional eld theories with SON gauge group and four sup ercharges. We argue that this formulation allows for a concrete description of the chirality-changing phase transitions which connect 1; 0 theories with di erentnumb ers of tensor multiplets. Septemb er 1997 1. Intro duction In the past few years, large classes of interacting sup erconformal eld theories with between 4 and 16 sup ercharges have b een discovered in three, four, ve and six space-time dimensions. Most of these theories are not describ ed byweakly-coupled Lagrangians, and there is not even a known Lagrangian which ows to them in many cases. Therefore, we require a di erent approach to analyze them. This is an interesting abstract problem in itself, and it is rendered more urgentby the many applications these theories haveinM theory. In the matrix formulation of M theory [1] these theories are relevant for compacti- cations on four dimensional spaces [2-4]. These theories also arise in the study of certain black holes in string theory [5], and it has b een suggested that an improved understanding of some of these theories may lead to progress in solving large N nonsup ersymmetric QCD [6]. The xed p oints with 8 or fewer sup ercharges are imp ortant in the problem of unifying M-theory vacua, since they are crucial in connecting vacua with di erent sp ectra of chiral elds [7-10]. Fixed p oint theories with 2; 0 sup ersymmetry in six dimensions [11,12] were recently studied in a matrix mo del formulation in [13]. The purp ose of this pap er is to move on to theories with 1; 0 sup ersymmetry in six dimensions. We will formulate a matrix description of these theories and follow the chirality-changing phase transitions of [7,8] in this language. We b egin in section 2 with the de nition of the theory. In section 3 we analyze deformations away from the xed p oint, where we can see the low-energy sp ectrum in the spacetime theory, and observe the chirality-changing phase transition. We discuss various interesting issues, whichwe are not able to fully resolve, concerning the matrix description of these deformations. Section 4 contains the 1 + 1 dimensional generalization of the quantum mechanical theory, which corresp onds to a six dimensional \little string" theory in spacetime. As this pap er was b eing completed, similar results were indep endently obtained in [14]. 2. The Quantum Mechanical De nition of the Fixed Point Theory We will study here the simplest example of a xed p oint with 1; 0 sup ersymmetry, which is the low energy theory of a small instanton in the E E heterotic string. In M 8 8 theory this is describ ed bya vebrane at the end of the world ninebrane [7,8]. This theory 4 has a Coulomb branch of the form IR=ZZ times a decoupled IR factor, on which the low 2 energy sp ectrum consists of a tensor multiplet and a hyp ermultiplet. The scalars in these 1 multiplets lab el the transverse p osition of the vebrane in M theory. The scalar in the ten- sor multiplet parametrizes the distance b etween the vebrane and the ninebrane, and when its exp ectation value vanishes the low-energy theory is sup erconformal. Another branch coming out of the sup erconformal p oint is the Higgs branch, corresp onding to enlarging the size of the instanton. On this branch, the low-energy theory has 30 hyp ermultiplets, 1 which are in the 56+1+1 representation of the E symmetry left unbroken by the 7 2 instanton. Wewould like to prop ose an in nite momentum frame quantum mechanical description of this theory, which repro duces this mo duli space and low-energy sp ectrum. In particular, we will consider in this framework the chirality-changing phase transitions of [7,8]. There is an obvious generalization of this theory to k coincident vebranes or small instantons, which will also b e discussed. The arguments used in [13] for the construction of 2; 0 theories in six dimensions can also b e used for the construction of theories with 1; 0 sup ersymmetry.To get a light-cone 1 description of this system, we start with M theory on S =ZZ [15] with k vebranes, and 2 compactify a longitudinal direction of the vebranes on a circle of radius R. The theory 1 0 then b ecomes the typ e I IA string theory on S =ZZ a.k.a. typ e I , with 8 D8-branes at 2 each orientifold xed p oint [16] and k D4-branes. In the next subsection we will discuss the full matrix description of this theory.We will intro duce the degrees of freedom of the matrix description of this system, their interactions, and their representations under the various symmetries. In x2.2 we will consider the limit M !1in spacetime, and determine what remains of the degrees of freedom in the p matrix description in this limit. This surviving quantum mechanics is our formulation of the 1; 0 SCFT. 2.1. Heterotic Fivebranes in Matrix Theory 0 The ab ovetyp e I system is equivalent to the E E heterotic theory on a circle, 8 8 with a Wilson line A breaking the gauge symmetry to SO16 SO16 and k vebranes E wrapp ed around the circle. Let the radius of this circle in the E E theory b e denoted 8 8 r . This vacuum is related by T-duality [17] to the SO32 heterotic string on a circle of E radius r =1=4r , with a Wilson line A breaking the gauge group to SO16 SO16. S E S The winding number n of the SO32 theory maps to the D0-brane number N in the typ e S 0 I description. The SO17; 1 T-duality transformation maps this to a linear combination of momentum, winding, and E E lattice quantum numb ers in the E E theory: 8 8 8 8 2 N = n $ 2m A n 2A P ; 2:1 S E E E E E 2 where m ;n and P are the momentum, winding, and E E lattice quantum numb ers E E E 8 8 in the E E theory. 8 8 For the in nite momentum frame description we are interested in states with large momentum m around the circle in the E E theory. From 2.1 we see that this E 8 8 corresp onds to large D0-brane number N = n , though the two quantum numb ers are not S exactly the same. Let us now describ e the quantum mechanics of the D0-branes in this theory, near one of the orientifolds. This quantum mechanics without the D4-branes was studied in [18-20]. It is an SON gauge theory with 8 sup ersymmetries, containing 16 fermions in the fundamental representation which arise from the 0-8 strings. Adding the D4-branes longitudinal vebranes [21] is done simply by adding the 0-4 strings. These are k \hyp er- multiplets" in the fundamental representation, and there is an Spk USp2k global symmetry corresp onding to these. For N = 1 this theory was describ ed in [22] see also [23]. Altogether we are left with four linearly realized sup ersymmetries, which is the correct numb er for a lightcone description of a spacetime theory with 8 sup ersymmetries. The global symmetry of the quantum mechanics is SO4 SO4 SO16 Spk; 2:2 ? k where SO4 corresp onds to the rotation symmetry transverse to the 4-branes but inside ? the 8-branes, SO4 corresp onds to the rotations inside the 4-branes, SO16 is the k gauge symmetry on the 8-branes and Spk is the gauge symmetry of the 4-branes. The 4 sup ersymmetry generators transform in the f2; 12;111grepresentation of this group, so that two of its SU 2 factors are in fact R-symmetries. The representations of the elds under the SON gauge symmetry and the global symmetries are given in the following table : SON SO4 SO4 SO16 Spk ? k 0 0 states : A ;X NN 1=2 1;1 1;1 1 1 0 9 NN 1=2 2;1 2;1 1 1 L NN 1=2 1;2 1;2 1 1 L X NN + 1=2 2;2 1;1 1 1 k NN + 1=2 1;2 2;1 1 1 R 2:3 X NN + 1=2 1;1 2;2 1 1 ? NN + 1=2 2;1 1;2 1 1 R 0 4 states : v N 2; 1 1; 1 1 2k N 1; 1 2; 1 1 2k R N 1; 1 1; 2 1 2k L 0 8 states : N 1; 1 1; 1 16 1: L 3 Here X gives the p ositions of the zero branes along the fourbranes, X gives the p ositions ? k 1 p erp endicular to the fourbranes, and X gives the p ositions in the S =ZZ direction.