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hep-th/9710009
IASSNS-HEP-97/108
Why is the Matrix Mo del Correct?
1
Nathan Seib erg
School of Natural Sciences
Institute for Advanced Study
Princeton, NJ 08540, USA
We consider the compacti cation of M theory on a light-like circle as a limit
of a compacti cation on a small spatial circle b o osted by a large amount.
Assuming that the compacti cation on a small spatial circle is weakly cou-
pled typ e IIA theory, we derive Susskind's conjecture that M theory com-
pacti ed on a light-like circle is given by the nite N version of the Matrix
mo del of Banks, Fischler, Shenker and Susskind. This p oint of view pro-
vides a uniform derivation of the Matrix mo del for M theory compacti ed
p
on a transverse torus T for p = 0; :::; 5 and clari es the diculties for larger
values of p.
10/97
1 seib [email protected]
Ab out a year ago Banks, Fischler, Shenker and Susskind (BFSS) [1] prop osed an
amazingly simple conjecture relating M theory in the in nite momentum frame to a certain
p
quantum mechanical system. The extension to compacti cations on tori T for p =1; :::; 5
was worked out in [1-5]. This prop osal was based on the compacti cation of M theory on
N
around that circle. In the a spatial circle of radius R in a sector with momentum P =
s
R
s
limit of small R M theory b ecomes the typ e IIA string theory and the lowest excitations
s
in the sector with momentum P are N D0-branes [6]. When the D0-brane velo cities are
small and the string interactions are weak the D0-branes are describ ed [7] by the minimal
sup ersymmetric Yang-Mills theory with sixteen sup ercharges (SYM). When the velo cities
or the string coupling are not small, this minimal sup ersymmetric theory is corrected by
higher dimension op erators. The suggestion of BFSS was that M theory in the in nite
momentum frame in uncompacti ed space is obtained by considering the minimal SYM
p
quantum mechanical system in the limit N; R ;P ! 1. For compacti cation on T the
s
prop osal of [1,2] is to consider Dp-branes, which are describ ed by SYM in p + 1 dimensions,
and again to truncate to the minimal theory. This prop osal raised a few questions:
1. Why is this prop osal correct?
2. Why is the theory with small R related to the theory with large R ?
s s
3. More sp eci cally, the minimal sup ersymmetric theory is corrected by higher dimension
op erators, which are imp ortant when R and the velo cities are not small. Whyisthe
s
extrap olation from the minimal theory, which is valid at small R , correct for large
s
R ?
s
4. Furthermore, for p 4 the minimal theory is not renormalizable and hence it is ill
de ned. Then, higher dimension op erators must b e included in the description. They
re ect the fact that the theory must b e emb edded in a larger theory with more degrees
of freedom. This theory for p = 4; 5 was found in [3-5]. The pro cedure to nd these
extensions of the minimal theories did not app ear systematic. What is then the rule
to construct the theory in di erent backgrounds?
Susskind noted that the nite N Matrix mo del enjoys some of the prop erties exp ected
to hold only in the large N limit and suggested that it is also physically meaningful [8].
He suggested that the matrix mo del describ es M theory compacti ed on a light-like circle
N
+
. Such a compacti cation on a light-like circle with of radius R with momentum P =
R
nite momentum is known as the discrete light-cone and the quantization of this theory is
known as the discrete light-cone quantization (DLCQ). With a light-like circle the value
of R can b e changed by a b o ost. Therefore, the uncompacti ed theory cannot b e obtained 1
+
by simply taking R to in nity. Instead, it is obtained by taking R; N ! 1 holding P
xed.
In this note we will relate these two approaches to the Matrix theory. In the pro cess
of doing so, we will derive the Matrix mo del and will answer the questions ab ove. We
will also present a uniform derivation of the Matrix mo del for M theory on a compacti ed
transverse space.
We start by reviewing some trivial facts ab out relativistic kinematics. A compacti -
cation on a light-like circle corresp onds to the identi cation
R
p
x x
2
; (1) +
R
t t
p
2
10
where x is a spatial co ordinate, e.g. x . We consider it as the limit of a compacti cation
on a space-like circle which is almost light-like
q
2
2
R
R
R
s
2
p p
+ R
+
x x x
s
2
2 2R
+ + (2)
R
R
t t t
p
p
2
2
with R R . The light-like circle (1) is obtained from (2) as R ! 0. This compacti ca-
s s
tion is related by a large b o ost with
2
R R
s
p
(3) = 1
2
2 2
R
R +2R
s
to a spatial compacti cation on
x x R
s
+ : (4)
t t 0
A longitudinal b o ost of the light-like circle (1) rescales the value of R . It also rescales
the value of the light-cone energy P . Therefore P is prop ortional to R . For small R
s
the value of P in the system with the almost light-like circle (2) is also prop ortional to
R (an exception to that o ccurs when P = 0 for the light-like circle; then P can be
non-zero for the almost light-like circle). The b o ost (3) rescales P to be indep endent of
R and of order R (if originally P =0, the resulting P after the b o ost can be smaller
s
than order R ).
s
Following [9] (as referred to in [10]) wenow consider M theory compacti ed on a light-
like circle (1) as the R ! 0 limit of the compacti cation on an almost light-like circle (2)
s 2
or as the limit of the b o osted circle (4). This way the DLCQ of M theory discussed in [8] is
related to the compacti cation on a small spatial circle as in [1]. For small R the theory
s
3
2
compacti ed on (4) is weakly coupled string theory with string coupling g = (R M ) ,
s s P
2 3
(M is the Planck mass). For xed energies and xed M , and string scale M = R M
P P s
s
P
the limit R ! 0 yields a complicated theory with vanishing string scale.
s
However, as we said ab ove, starting with P of order one, the e ect of the b o ost is to
2 2
is inserted on dimensional grounds). This is exactly (M reduce P to b e of order R M
s
P P
the range of energies in the discussion of [11], which was one of the motivations for the
Matrix mo del of BFSS [1]. In order to fo cus on the mo des with such values of P we
rescale the parameters of the theory. We do that by replacing the original M theory, which
f
is compacti ed on a light-like circle of radius R by another M theory, referred to as M
f
theory with Planck scale M compacti ed on a spatial circle of radius R . The transverse
P s
f
geometry of the original M theory is replaced by that of the M theory. For example, for
e
a compacti cation on a transverse torus with radii R the other theory has radii R .
i i
The relations b etween the parameters of these two theories are obtained by combining
2
f f
the limit R ! 0 with M !1 holding P R M xed. Therefore we identify
s P s
P
2 2
f
R M = RM ; (5)
s
P P
which is nite in the limit. Since the b o ost do es not a ect the transverse directions, we
identify
f e
M R = M R (6)
P i P i
and keep it xed. In this limit the energies are nite and we nd string theory with string
coupling and string scale
3
3 3
2
4
f
2 4
(RM ) =R eg =(R M )
s
s s P
P
1
3
3 2 2
2
f f
2 (7)
= R M = R ) M (RM
s
s
P s P
:
For R ! 0 with nite M and R we recover weakly coupled string theory with large
s P
string tension. This theory is very simple and is at the ro ot of the simpli cation of the
Matrix mo del.
N
+
A sector with P = in the original M theory is mapp ed to a sector of momentum
R
N
f
P = in the new M theory. In terms of this latter theory it includes N D0-branes.
R
s 3
Therefore, the original M theory is mapp ed to the theory of D0-branes. These D0-branes
1
2
e
move in a small transverse space of size R R ! 0 (it is small even relative to the
s
i
1
4
e f
string length R M R ! 0).
s
i s
We conclude that M theory with Planck scale M compacti ed on a light-like circle
P
N
+
f f
of radius R and momentum P = is the same as M theory with Planck scale M
P
R
compacti ed on a spatial circle (4) of radius R with N D0-branes in the limit
s
R ! 0
s
f
M !1
P
(8)
2 2
f
R M =RM = xed
s
P P
f e
M R = M R = xed :
P i P i
Here R should be understo o d as generic parameters in the transverse metric { not only
i
radii in a toroidal compacti cation. Clearly, the general discussion applies to curved space
with any number of unbroken sup ercharges.
p
For a compacti cation on T we can use T duality to map the system of N D0-branes
to N Dp-branes on a torus with larger radii
1 1
= = : (9)
i
3
2
e f
R RM
i
R M
P
i
s
Note that are nite when R ! 0. The string coupling after this T duality transforma-
i s
tion is
Y Y
0 p p 3 3 6
f f
ge = ge M = M R M : (10)
s i i
s s s P
The low energy dynamics of these N Dp-branes are controlled by p + 1 dimensional SYM
[7] with gauge coupling
0
Y
ge
s
3 6 2
= R M ; (11) g =
i
P YM
p 3
f
M
s
which also has a nite R ! 0 limit. Even though the low energy dynamics of these
s
Dp-branes is nite in this limit, we should explore the b ehavior at higher energies. We will
do that shortly.
To summarize, wehave mapp ed the original M theory problem with a light-like circle
N
+
to a problem of N of radius R and parameters M and R in the sector with P =
P i
R 4
Dp-branes wrapping a torus in string theory. The radii of the torus, the string coupling
and string scale are
1
=
i
3
R RM
i
P
Y
0 p 3 3 6
(12)
f
eg = M R M
i
s s P
1
3
2 2
2
f
2
M = R (RM )
s
s P
and R ! 0. Exactly this limit was analyzed recently in [12].
s
Let us analyze this limit for various values of p. For p = 0 the T duality which
we p erformed is not necessary. The theory is that of D0-branes with vanishing string
coupling and in nite string tension. Since the gauge coupling g is nite, the theory is
YM
not trivial. The relevant degrees of freedom are strings stretched between the D0-branes.
The in nite string scale decouples all the oscillators on the strings. Therefore the full
theory is the minimal SYM theory. Note that closed strings or gravitons in the bulk of
space time decouple b oth b ecause the string scale b ecomes large and b ecause the string
coupling vanishes. This is exactly the nite N version of the Matrix mo del of BFSS [1].
For p = 1; 2; 3 we recover the SYM prescription of [1,2]. Again, the in nite string
scale decouples the oscillators on the strings which are stretched b etween the N Dp-branes.
Therefore the Lagrangian is that of the minimal SYM theory without higher order cor-
0
rections. For p = 3 the string coupling ge do es not vanish, but there are still no higher
s
dimension op erators in the 3 + 1 dimensional SYM, since they are all suppressed byinverse
f
powers of the string scale M .
s
For p = 4 several new complications arise. First, the low energy SYM theory is not
renormalizable and therefore cannot give a complete description of the theory. It breaks
1
down at energies of order , where new degrees of freedom must b e added. Second, the
2
g
YM
string coupling also diverges in our limit. Therefore, in order to analyze the system we
f
need to study the strong coupling limit of the M theory, which is an eleven dimensional
theory. In this limit the D4-branes b ecome 5-branes wrapping the eleventh dimension.
Using (12) we nd that this eleven dimensional theory is compacti ed on a circle of nite
radius
0
Y
eg
s
3 6
= = R M ; (13)
5 i
P
f
M
s
but its eleven dimensional Planck scale diverges
f
1
M
s
6
R !1: (14)
s
1
0
3
(ge )
s 5
Since the eleven dimensional Planck scale is in nite, the mo des in the bulk of space-time
decouple, and the theory on the brane is a 5+1 dimensional theory. This non-trivial theory,
known as the (2; 0) eld theory, was rst found in [13,14]. The new degrees of freedom,
1
, can now be interpreted as asso ciated with which we had to add at energy of order
2
g
YM
momentum mo des around the circle (13). These are related to instantons in the SYM
f
theory, which are D0-branes in the M theory. We have thus derived the prop osal of [3,4]
4
to use this theory as a Matrix theory for M theory on T .
A similar analysis applies to p = 5. Here we study the strong coupling limit of D5-
branes in IIB string theory. Using S duality of this theory we map it to NS5-branes in
weakly coupled IIB theory. We thus recover the prop osal of [4,5] for the description of M
5
theory on T in terms of a new theory obtained by studying NS5-branes in typ e I I theory.
This theory, which can be called a non-critical string theory, is not a lo cal quantum eld
theory. In addition to the ve sides of the torus (9) it is characterized by the string
i
Q
0 2 3 6
slop e = g = R M . The SYM description breaks down at energies of order
i
YM P
1
, where new degrees of freedom are added. These degrees of freedom are the strings in
g
YM
f
the theory. In terms of the M theory and its typ e I IB string theory, these are D1-branes.
For p = 6 the situation is more complicated [15]. Here we are led to consider the
1
strong coupling limit of D6-branes . As for p = 4, this limit is describ ed by an eleven
dimensional theory. However, here the eleven dimensional Planck scale
f
M 1
s
= (15)
Q
1 1
2
0
3 3
(ge ) RM ( )
i
s
P
remains nite, but the radius of the eleventh dimension diverges
0
1
eg
s
2
R !1: (16)
s
f
M
s
Since the radius diverges, the D6-branes, which are Kaluza-Klein monop oles asso ciated
with the eleventh dimension, expand and b ecome an A singularity. The gauge coupling
N 1
of the asso ciated SYM theory is given by the eleven dimensional Planck scale, which
remains nite. Since the eleven dimensional Planck scale is nite, there is no reason
to assume that the gravitons in the bulk of the ALE space with an A singularity
N 1
f
decouple from it (see the discussion in [16-18]). From the M theory p oint of view (b efore
the T duality transformation) these graviton mo des can be identi ed as Kaluza-Klein
1
We thank E. Witten for very helpful discussions on this limit. 6
2 6 2
monop oles , which wrap the small T . Their energy is of order R . After the b o ost (3)
s
their energy is of order R and it vanishes as R ! 0. Therefore, the DLCQ theory has
s s
an in nite number of new massless mo des.
N
6 +
We conclude that M theory on T and a light-like circle with momentum P = is
R
6
the same as M theory with Planck scale (15) compacti ed on T with an A singularity
N 1
in the non-compact spatial directions. The eleven dimensional gravitons propagate in the
entire space. There is also an SU (N ) SYM in 6+1 dimensions describing the interactions
of gluons at the singularity. Comparing with the situation for lower values of p,we seem to
miss a decoupled 6+1 dimensional U (1) multiplet. However, considering the limit which
leads to this con guration carefully,we see that the U (1) multiplet exists. It is \smeared"
over the four dimensional non-compact space.
Unfortunately, this result is not satisfying. The Matrix theory o ered a simple de-
scription of M theory, which can lead to useful computations. Here we see that for p =6,
it go es over to a situation which is apparently as complicated as the underlying M theory.
After the completion of this work we received a pap er [19] which partially overlaps
with ours. We also learned that J. Polchinski had indep endently reached some of these
conclusions.
Acknowledgements
Wehave b ene ted from discussions with O. Aharony, T. Banks, J. Schwarz, S. Shenker,
L. Susskind and E. Witten. This work is supp orted in part by #DE-FG02-90ER40542.
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