hep-th/9709118 16 Sep 1997 yin b ers limit vit um U-97-72 N 4 tn erstein gauge group and N a Silv SO y ter , and Ev eley 3 0 Theories 0 theories with di eren ; ws for a concrete description of the hru ; y y ysics They are based on the large ersit C-PUB-7648, UCB-PTH-97/45 ersit anced Study tofPh hep-th/9709118 IASSNS-HEP-97/99, LBNL-40723, R SLA ysics and Astronom ulation allo 0 sup ersymmetric theories without gra ; h connect 1 , Shamit Kac , CA 94720, USA 2 , NJ 08855-0849, USA y tum frame. tofPh a ooz eley y of California at Berk w Rutgers Univ Stanford Univ ho ol of Natural Sciences Departmen 3 Sc Berk Stanford, CA 94309, USA ersit Princeton, NJ 08540, USA 2 Institute for Adv teracting 1 in Six Dimensions ha Berk Piscata Stanford Linear Accelerator Cen Univ 4 Departmen 1 , Mic 1 y e argue that this form hanics or 1 + 1 dimensional eld theories with Matrix Description of 1 ultiplets. harges. W hanging phase transitions whic Ofer Aharon tum mec b er 1997 y-c e prop ose descriptions of in hiralit four sup erc c of quan of tensor m Septem W six dimensions in the in nite momen
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 SO 16 SO 16 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 SO 32 heterotic string on a circle of
E
radius r =1=4r , with a Wilson line A breaking the gauge group to SO 16 SO 16 .
S E S
The winding number n of the SO 32 theory maps to the D0-brane number N in the typ e
S
0
I description. The SO 17; 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 SO N 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 Sp k USp 2k 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
SO 4 SO 4 SO 16 Sp k ; 2:2
?
k
where SO 4 corresp onds to the rotation symmetry transverse to the 4-branes but inside
?
the 8-branes , SO 4 corresp onds to the rotations inside the 4-branes, SO 16 is the
k
gauge symmetry on the 8-branes and Sp k is the gauge symmetry of the 4-branes. The
4 sup ersymmetry generators transform in the f 2; 1 2;1 11grepresentation of this
group, so that two of its SU 2 factors are in fact R-symmetries. The representations of
the elds under the SO N gauge symmetry and the global symmetries are given in the
following table :
SO N SO 4 SO 4 SO 16 Sp k
?
k
0 0 states : A ;X N N 1 =2 1;1 1;1 1 1
0 9
N N 1 =2 2;1 2;1 1 1
L
N N 1 =2 1;2 1;2 1 1
L
X N N + 1 =2 2;2 1;1 1 1
k
N N + 1 =2 1;2 2;1 1 1
R
2:3
X N N + 1 =2 1;1 2;2 1 1
?
N N + 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. In
9 2
addition wehave scalars v in the fundamental representation. The fermions which are all
real are denoted with subscripts R or L, according to their chirality in the corresp onding
1 + 1 dimensional theory of 1-branes, vebranes and ninebranes, which is related to the
quantum mechanics we describ e by a T duality in the x direction. That theory has 0; 4
9
sup ersymmetry, and we will discuss it further in section 4. Sup ersymmetry pairs the right
moving fermions with the b osons app earing directly ab ove them in the table, and with
L
the gauge eld.
The mo duli of the spacetime theory are parameters in the quantum mechanics. These
mo duli are the scalars in the theory of the 4-branes and the 8-branes, which are in the
following representations :
SO N SO 4 SO 4 SO 16 Sp k
?
k
4
1 1; 1 2; 2 1 2k 2k 1 =2 4 4 states : X
?
4
2:4
X 1 1; 1 1; 1 1 2k 2k + 1 =2
9
4 8 states : H 1 1; 1 2; 1 16 2k
8
8 8 states : X 1 1; 1 1; 1 120 1:
9
Most of the interactions of this system may b e easily derived from those of the 0-
brane/4-brane system, which is the dimensional reduction of a 6D 1; 0 theory, and from
those of the 0-brane/8-brane system [18-20]. Among the terms app earing in the Lagrangian
1
are terms of the following schematic form :
8 4 4 4
2 2
X X + X X + X X + v X X +
L 9 L L 9 L R 9 R 9
9 9 9 9
4 4
2 2 2 2 2
+ [X ;X ]+v +[X ;X ] + v + v + + v X X X X
? ? R L L R R ? L ?
k k
? ?
2 2
[X ; ]+ [X ; ]+ [X ; ]+ [X ; ]+[X ;X ] + Hv + H :
L R L R L ? R L ? R ? R L
k k k
2:5
The singlet comp onents of the fermions are completely decoupled, and their shifts gen-
R
erate four non-linearly realized sup ersymmetries, completing the 8 spacetime sup ersymme-
tries. Quantization of the zero mo des of these elds will multiply the representation of each
state we get by f 1; 2 1;1 g+f 1;1 2;1 g which is the content of a half-hyp ermultiplet
in spacetime .
In the quantum mechanics describing the sup erconformal p oint in space-time, all the
parameters 2.4 vanish. Then, the quantum mechanical theory has a Coulomb branch
1
This formula do es not include the p owers of the gauge coupling g , whichmay b e put in
QM
on dimensional grounds. 4
in the usual sense of a Born-Opp enheimer approximation in which X 6= 0 and v =0.
?
In the matrix mo del interpretation, graviton states live here, as well as E gauge b osons
8
lo calized near X = 0 [19,20]. In addition, there is a Higgs branch in which X =0.Itis
9 ?
parametrized by exp ectation values of X and v , and has real dimension
k
N N +1 N N 1
dim M =4Nk +4 4 =4N k+1 : 2:6
H
2 2
2.2. Decoupling Gravity: Formulation of the 1; 0 SCFT
As in the case of the 2; 0 theories discussed in [13], the gauge coupling in the quantum
mechanics is related to the eleven dimensional Planck mass M and the compacti cation
p
6 3 2
M R .Thus, taking M !1 corresp onds to the g !1 or IR radius R by g
p QM
p
QM
limit of the quantum mechanics, where we exp ect the conformal theory in spacetime to
decouple from gravity,aswell as from the E gauge b osons whose gauge coupling go es to
8
zero in this limit.
As in [13], the presence of the Higgs branch with no apparent spacetime interpreta-
tion is what signals the presence of the nontrivial conformal theory in spacetime. In the
limit g !1, some degrees of freedom b ecome in nitely massive on the interior of the
QM
Higgs branch. In other words, the Coulomb and Higgs branches of the quantum mechanics
decouple. Integrating out the 0-4 states leads to an in nite tub e on the Coulomb branch,
so the origin is in nitely far away on that branch where the gravitons and the E gauge
8
b osons live . The degrees of freedom from 2.3 that are lifted in the interior of the Higgs
branch v 6=0, X 6= 0 are :
k
A ;X ; ;2N N 1 of the elds v and X and their sup erpartners ; ;
0 9 L R R
k
2:7
X ; ;2N N + 1 of the elds and :
? R L L
We are left with :
4N k + 1 of the elds v and X and their sup erpartners ; ;
R R
k
2:8
;4N k 1 of the elds and :
L L L
It is the g !1theory of these degrees of freedom that constitutes the matrix formula-
QM
tion of the spacetime SCFT. Note that, unlike in [13], even for k = 1 there is a non-trivial
Higgs branch here. This corresp onds to the non-trivial SCFT in spacetime which exists
even in this case.
The classical Higgs branch of the quantum mechanics is the mo duli space of Sp k
instantons [24]. There is no non-renormalization theorem for the mo duli space in this case. 5
In the quantum mechanics there could b e lo op corrections say,involving the s to the
L
metric of this space. However, there is some xed p ointgoverning the Higgs branch in the
infrared g !1 limit. We conjecture that, for N !1, it correctly describ es the
QM
1; 0 sup erconformal theories in the in nite momentum frame. In fact, it follows from the
results of [25] that the corresp onding 1 + 1 dimensional 0; 4 sigma mo del, with target
space M and with the additional left-moving fermion multiplets, is nite. This is not
H
to say that the infrared physics will necessarily b e transparent in terms of the degrees
of freedom 2.8 . The IR theory mayhave complicated interactions, arising for instance,
from the gauge constraint the A equation of motion in the original gauge theory we
0
start from [20].
Note that since N here is not the same as the spacetime momentum m , the nite N
E
theory do es not directly give us a discrete light-cone description of the 1; 0 sup erconformal
theories, as suggested in [26]. Presumably, as in [27], the nite N theory is a discrete light-
cone quantization of these theories compacti ed on a light-like circle with a Wilson lo op
breaking the E symmetry to SO 16 . In the quantum mechanics only an SO 16 subgroup
8
of the E global symmetry of the 1; 0 sup erconformal theory in spacetime is visible. As
8
0
in [19], the full E representations get lled out as the typ e I coupling go es to in nity and
8
0
states of energy 1= come down.
I
3. Low Energy States Away from the Fixed Point
As discussed in the intro duction, the six dimensional 1; 0 theories playavery inter-
esting role in giving chirality-changing phase transitions. Within Lagrangian eld theory
there is no way to lift chiral matter, so it is interesting to consider how this o ccurs in
our formulation. Let us p erturb the spacetime theory away from the conformal p oint,
going into its Higgs or Coulomb branches. Along these branches the low energy theory in
spacetime is free, and we should b e able to nd the correct low energy sp ectrum in our
quantum mechanical description. We will see how the quantum numb ers for these states
arise in this section.
It is not clear to us that the deformed theory can b e describ ed using only the degrees
of freedom 2.8 that were involved in formulating the critical theory. In principle, there
are twoways to analyze the theory away from the conformal p oint. We could either
p erform the p erturbation in the full gauge theory and then take the g !1limit
QM
while keeping the p erturbation parameters nite , or work directly in the theory which
describ es the Higgs branch of the quantum mechanics in the g !1 limit, and analyze
QM
the p erturbations in that mo del. As we will discuss in some detail b elow, wehave diculty
nding the correct spacetime sp ectrum using the second approach. 6
Weinterpret this diculty as resulting from the fact that this approach do es not
include information ab out states lo calized at the singularities at the b oundaries of the
Higgs branch. These quantum-mechanical variables, though decoupled from the interior of
the Higgs branch at the conformal p oint, may still b e imp ortant after we deform the theory
away from the conformal p oint. Because of the tub e metric, the singularities of the Higgs
branch still decouple from the graviton/gauge b oson states which live on the Coulomb
branch. After turning on the deformations 2.4 , the quantum mechanical Higgs branchis
generically lifted, and the wave functions of all states are concentrated near the origin
of the Higgs branch. Thus, it is not a surprise that the degrees of freedom related to the
singularities in the Higgs branch are required to describ e the states after the deformation.
It would b e interesting to understand b etter the role of the singularities at the b oundaries
of the Higgs branch, b oth in this theory and in the 2; 0 theories describ ed in [13].
3.1. The Coulomb Branch
First, let us discuss the Coulomb branch of the spacetime theory this is not to b e
confused with the Coulomb branch of the quantum mechanics . On this branch the ve-
branes are generically all separated from each other and from the ninebrane. There is
a tensor multiplet and a hyp ermultiplet forming a tensor multiplet of 2; 0 sup ersym-
metry living on each vebrane. For simplicity, let us fo cus on the case k = 1 the other
cases generically give k copies of this . Moving into the Coulomb branchaway from the
4
critical p oint is done by turning on X , and we exp ect to nd the vebrane states lo-
9
4
calized in the mo duli space near X = X sp eci cally, when half of the eigenvalues
9
9
4
of X are equal to one of the eigenvalues of X . In this region the 0-8 strings are
9
9
all massive and the SO N gauge theory is broken by the VEV of X to U N=2 here
9
we take N to b e even . In the IR, the theory reduces exactly to the quantum mechan-
ics of D0-branes and D4-branes with 8 sup ersymmetries discussed in [13], whichisa
sup ersymmetric quantum mechanics on the mo duli space of U k instantons. In b oth
cases the spacetime sp ectrum should include a tensor multiplet and a hyp ermultiplet for
2
k =1 . Thus, we should nd 16 ground states of this theory, which should b e in the
f 1; 3 1;1 g+f 1;1 1;1 g+f 1;1 2;2 g+f 1;2 1;2 g+f 1;2 2;1 grepresen-
tation of the SO 4 SO 4 global symmetry. This representation arises by quantizing
?
k
the fermion zero mo des of the U N=2 -singlet comp onents of the fermions and
L R
app earing in table 2.3 .
2
In the 2; 0 case it was not clear if a k = 1 theory whichwas decoupled from the Coulomb
branch existed or not [28], but here we are reaching this theory by a p erturbation from a theory
that was already decoupled from gravity, so there is no problem. 7
Note that in formulating the critical theory for k =1,we discarded b ecause it
L
b ecame in nitely massive on the interior of the Higgs branch 2.7 . But, as just noted,
quantizing its zero mo des gives the correct degeneracy and quantum numb ers to describ e
the tensor multiplet on the spacetime Coulomb branch. This is the rst dicultywe nd
in attempting to describ e the deformations away from the critical theory using only the
degrees of freedom involved in formulating the xed p oint itself.
In general, there is a corresp ondence b etween ground states of the sup ersymmetric
quantum mechanics on a space X and cohomology classes of X .Thus, we exp ect the mo des
of the tensor multiplet, which should exist for anyinteger value of momentum around the
circle in the E E theory i.e. for all even values of N in the original SO N theory , to
8 8
corresp ond to cohomology classes of the mo duli space of our theory. In the case k = 1 and
e
for non-zero X , this space is simply the mo duli space M U 1 of N = N=2 instantons
9
e
N
in a U 1 gauge group. Since these instantons are necessarily all of zero size, this space is
just
e
4N
M U 1 = IR =S : 3:1
e e
N N
e
For N =1,wehave simply a 0-brane/4-brane system, and the required state is just
the b ound state of [29]. Indeed, this state b ecomes completely lo calized on the 4-brane in
the M !1 limit.
p
e
For higher values of N , it is not apriori clear which cohomology should b e used,
since the states are all asso ciated with the orbifold singularities of the mo duli space.
It is natural to conjecture that the quantum mechanical ground states are given by the
orbifold cohomology of this space [30] this is more justi ed in the 1 + 1 dimensional theory
describ ed in section 4, but our theory is just a dimensional reduction of that theory . This
e
gives a state for every partition of N [31], in agreement with our exp ectation of nding
a single state for anyinteger value of momentum of the tensor multiplet. Quantizing the
zero mo des of and then gives this state the Lorentz quantum numb ers of a tensor
L R
multiplet and a hyp ermultiplet in the 1; 0 spacetime theory. These states are examples
of states living at the singularities of the Higgs branch, as discussed ab ove.
3.2. The Higgs Branch
The other branch of the spacetime theory is the Higgs branch, in which the vebranes
in spacetime turn into large E instantons. We are only interested in the regime in which
8
a quantum eld theory description, decoupled from gravity, remains valid. Let us de-
e
note by H the canonically normalized dimension 2 scalar eld in spacetime whose VEV
parameterizes the Higgs branch. The eld theory regime is
2
e
H< p 8 e On dimensional grounds, H is related to the scale size of the instanton/ vebrane by 3 e H = M : 3:3 p Thus, the eld theory regime is < p where the vebrane is still thin in Planck units. In the regime >l , the vebrane b ecomes p thick, gravity fails to decouple, and the matrix description necessarily involves the degrees of freedom 2.7 as well as 2.8 . In the eld theory regime 3.4 , as discussed ab ove, one might hop e to describ e the theory using only the degrees of freedom 2.8 . However, as with the spacetime Coulomb branch, we will encounter diculties in realizing this. We will analyze here only the case where the instantons are all emb edded in a single SU 2 . In this case, the E gauge symmetry in spacetime is broken to E , and its SO 16 8 7 subgroup which app ears explicitly in the quantum mechanics is broken to SO 12 SU 2 . In the quantum mechanics, wegointo this branchby turning on the parameters 4 . Note that turning on only corresp onding to the 4-8 strings H and to the 4-4 strings X ? the 4-8 strings when the instantons are all in the same SU 2 still leaves all but one of the vebranes/instantons at zero-size, so we still have a non-trivial conformal theory for k>1. In the quantum mechanics we see that not all of the Higgs branch is lifted in that 4 gives a case. In contrast, from 2.5 we can easily see that turning on b oth H and X ? mass to all the elds v and , and to 4k of the elds and . The rst 12 comp onents R L L of in the fundamental representation of the unbroken SO 12 and of SO N remain L massless, as do 4 combinations of and again, in the fundamental of SO N . L L Naively, when we turn on H the elds v and b ecome massive, and there is no R longer an in nite tub e in the Coulomb branch of the gauge theory, so gravity do es not seem to decouple from our theory.However, as discussed ab ove, we should b e careful in e howwe normalize H . In spacetime, wewant H to remain nite as M go es to in nity. p This corresp onds to having a nite H in the theory describing the Higgs branch in the g !1limit. In this limit, even for nite H there is still an in nite tub e in the Coulomb QM branch, and gravity still decouples from the Higgs branch of the 6D SCFT. For simplicity,we will analyze here only the case k = 1, where the combinations that 3 remain massless are exactly the 4 elds . The hyp ermultiplet H which obtains a VEV L on the Higgs branch is using 2.4 charged under SU 2 SO 16 Sp 1 , where SU 2 R R 3 Since to get a free low-energy theory in spacetime for k>1we are forced to turn on 4-4 strings, the general case decomp oses in the IR into k copies of this case living at di erentvalues 4 of X , corresp onding to the eigenvalues of X . ? ? 9 is the rst SU 2 factor in SO 4 which is identi ed with the SU 2 symmetry of the ? R 0 spacetime theory . Giving it a VEV breaks this symmetry to SU 2 SO 12 SU 2 , R 0 where SU 2 is a subgroup of SO 16 and SU 2 , and the last SU 2 is a subgroup of R R SO 16 and Sp 1 but note that away from the small instanton p oint this is a p erturbative gauge symmetry from the heterotic p oint of view . The fermions in the fundamental representation of SO N which remain massless are the , in the 1; 12; 1 representation, L and , in the 1; 1; 2 representation and in the 2 of the other SU 2 factor in SO 4 . L ? Since the v elds are all massive, the Higgs branch of the theory after the p erturbation 4N is given simply by the space of X s, whichisIR =S . What states do we exp ect to N k nd in this case? The massless states of the spacetime theory on the Higgs branch are 30 hyp ermultiplets. One of these hyp ermultiplets, which corresp onds to the transverse p osition of the instanton / vebrane and is free everywhere in the mo duli space is in the f 1; 1 2;2 g+f 1;2 1;2 grepresentation of the SO 4 SO 4 global symmetry ? k 0 where SO 4 now includes the new SU 2 group instead of the old SU 2 . The other ? R R hyp ermultiplets are all in the 2 f 1; 1 2;1 g+f 1;2 1;1 g representation, and in the 1 56 + 1 representation of the unbroken E gauge group in spacetime. This representation 7 2 1 1 decomp oses into a 32; 1 + 12; 2 + 1;1 oftheSO 12 SU 2 that we see in the 2 2 quantum mechanics. According to 2.1 , the momentum mo des of the rst representation should app ear for o dd values of N , while all the others should app ear for even values of N . Of course, this do es not mean that the momentum quantum numb er in the E E 8 8 string theory dep ends on the representation: from 2.1 one sees that for N = n o dd, P S E must b e a spinorial representation of SO 16 SO 16 but m can be odd or even. E As in the 0-8 system [18-20], we exp ect the structure of the ground states for o dd even values of N to b e the same as for N =1 N = 2 , with the only change b eing in the structure of the wave functions for the 0-8 b ound states. T duality and S duality relate our system to a heterotic SO 32 string theory with some non-trivial instanton bundle, and there we can show that the appropriate states exist the calculation is essentially as in [32], and the presence of torsion do es not change the results in this case [33] . 4 Let us analyze rst the case N = 1. In this case, the mo duli space is just IR ,sowe have only the ground state. The 12 remaining fermions are completely free in this case L since the gauge symmetry is just O 1 ZZ , so they have zero mo des. On the other 2 hand, as explained in [22], the s are sections of the SU 2 instanton bundle that lives in L the X directions. In this background they do not contribute any additional degeneracy. ? 0 Quantization of the zero mo des gives states in the 32 + 32 representation of the SO 12 L group. As in [19,20], imp osing a ZZ O 1 gauge constraint removes half of these states 2 and leaves us just with a 32. Adding the zero mo des turns each of these states into R 10 a half-hyp ermultiplet in spacetime, so we get exactly the exp ected sp ectrum of states for this value of N . In fact, for N =1 we can nd the right states also if wework only with the degrees of freedom 2.8 involved in the critical theory. Then v and its sup erpartners, as well as four of the elds , are lifted by H , and quantizing the zero mo des of the remaining fermions L 1 and provides us with the required 32 hyp ermultiplets after taking into account L R 2 the ZZ constraint . 2 For N = 2, the situation is more complicated as it was also in the 0-8 case , since the interactions b etween the elds play an imp ortant role in constructing the states. To realize these states in our formulation, we turn the op erators including and into creation L L and annihilation op erators as in [20] . We exp ect the ground states in the quantum mechanics to b e the same as those in the corresp onding 1 + 1 dimensional sigma mo del, where a level-matching constraint will force us to havetwo or oscillators in the L L sector where they are anti-p erio dic and no states will arise from other sectors . In the quantum mechanics, there will b e a gauge constraint analogous to the level-matching constraint of the heterotic string which will force the total charge of a state under the SO 2 gauge symmetry to b e equal to one [20]. We exp ect to nd ground states of the form j0i where and are now creation op erators , multiplied by an appropriate L L wave function which turns this state into an SO 4 SO 4 -singlet. These states will b e ? k in the 12; 2 representation of the SO 12 SU 2 global symmetry corresp onding to the remaining spacetime gauge symmetry, and again the zero mo des will turn them into R half-hyp ermultiplets. The 29th and 30th hyp ermultiplets will arise from states involving two s contracted to form a singlet of the SU 2 gauge symmetry , again with an L appropriate wave function for the rest of the elds. It would b e interesting to p erform the Born-Opp enheimer calculations explicitly, and see that exactly states of this form arise. The IR theory of the degrees of freedom 2.8 is complicated in this case, and wehave not b een able to nd these states directly by deforming that theory. Presumably, this is again a result of the theory at the singularities of the Higgs branch mixing with the theory describing the interior of the Higgs branchaswe deform away from the conformal p oint. 4. String Theories for string Theories The Higgs branch of the quantum mechanics formulated ab ove is exp ected to de- scrib e the 1; 0 sup erconformal theory in spacetime. In [13], a similar quantum mechanics describ ed the 2; 0 sup erconformal theories in spacetime. The corresp onding 1 + 1 di- mensional theory which gives the quantum mechanics up on dimensional reduction was 11 conjectured [13,28] to corresp ond to the \little string" theory of the typ e I IA NS vebrane [34,35], which reduces at low energies to the sup erconformal theory. Similarly,we exp ect the 1 + 1 dimensional theory with 0; 4 sup ersymmetry to describ e the \little string" theory of the heterotic E E vebrane, de ned by the limit g ! 0 in that theory [34]. 8 8 s The eld content and interactions of this theory are the same as those describ ed ab ove, with X now b ecoming part of the 1 + 1 dimensional gauge eld. The only di erence is that 9 there are now32chiral fermions , since we can no longer ignore the states of the \other L wall" these states are also required for anomaly cancellation . As in the Matrix theory descriptions of the heterotic string [36-38], half of these fermions have p erio dic b oundary conditions and the other half haveanti-p erio dic b oundary conditions. The X p ositions of 9 the D0-branes turn into the Wilson lo op around the circle, and half of the fermions are L massless when the value of this Wilson lo op corresp onds to the D0-branes b eing at eachof 4 the twowalls. However, this theory should still have a parameter X , corresp onding to 9 4 the X p osition of the vebranes , and the fermions as well as their b osonic partners 9 4 should only b e massless when the Wilson lo op is equal to the eigenvalues of X . This is 9 realized in the 1 + 1 dimensional eld theory byhaving the b oundary conditions for the 4 elds around the circle twisted by an arbitrary X matrix in the adjoint representation 9 of Sp k , namely 4 x +2 = exp X x : 4:1 9 The v shave similar b oundary conditions. Note that such b oundary conditions are not p ossible for the elds since a p otential would b e generated if their b oundary condition L were di erent [40]. We conjecture that the Higgs branch of this theory, in the g !1and large N YM limits, gives an in nite momentum frame description of the \little string" theory of the heterotic E E vebrane at zero coupling. Atlow energies this theory go es over to the 8 8 quantum mechanics of the previous sections, as exp ected. Note that the spacetime theory in this case includes two strings even for a single vebrane, coming from the membranes stretched b etween the vebrane and the two end of the world ninebranes. The sum of the 2 tensions of these two strings is the heterotic string tension M , but their ratio dep ends on s 4 the parameter X describ ed ab ove. 9 Acknowledgments Wewould like to thank T. Banks, J. de Bo er, G. Mo ore and N. Seib erg for useful discussions. O.A. and E.S. would like to thank the Asp en Center for Physics for hospitality 4 As noted in [39], this parameter actually lives in the \Coxeter blo ck" of Sp k . 12 during the course of this work. The work of O.A. is supp orted by DOE grant DE-FG02- 96ER40559. The work of M.B. is supp orted by NSF grant NSF PHY-9513835. The work of S.K. is supp orted by NSF grant PHY-95-14797 and a DOE Outstanding Junior Investigator Award. The work of E.S. is supp orted by the DOE under contract DE-AC03-76SF00515. 13 References [1] T. Banks, W. 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