BPS Quantization of the Five-Brane

Home , Robbert Dijkgraaf

April 1996

hep-th/9604055

cern-th/96-93

pupt-1603

BPS Quantization of the Five-Brane

Robbert Dijkgraaf

Department of Mathematics

University of Amsterdam, 1018 TV Amsterdam

Erik Verlinde

TH-Division, CERN, CH-1211 Geneva 23

and

Institute for Theoretical Physics

University of Utrecht, 3508 TA Utrecht

and

Herman Verlinde

Institute for Theoretical Physics

University of Amsterdam, 1018 XE Amsterdam

and

Joseph Henry Laboratories

Princeton University, Princeton, NJ 08544

Abstract

We give a uni ed description of all BPS states of M-theory compacti ed on T in terms

of the ve-brane. We compute the mass sp ectrum and degeneracies and nd that the

SO(5; 5; Z) U-duality symmetry naturally arises as a T-duality by assuming that the

world-volume theory of the ve-brane itself is describ ed by a string theory. We also

1 4

consider the compacti cation on S =Z T , and give a new explanation for its corre-

sp ondence with heterotic string theory by exhibiting its dual equivalence to M-theory on

K 3 S .

1. Introduction

There is bynow considerable evidence that the various dual relations b etween di erent

string theories can eventually b e understo o d in terms of an underlying \M-theory", whose

low energy e ective action is given by eleven dimensional sup ergravity [1,2,3,4,5,6,7,

8, 9], or its twelve-dimensional variant [10, 11]. It is not yet known what M-theory lo oks

like, but it is reasonable to exp ect that just like string theory it has some formulation in

terms of uctuating extended ob jects, b eing the membane or its dual, the ve-brane. The

fundamental strings that we know in ten dimensions or less are obtained by dimensional

reductions from the membrane or the ve-brane. From this p oint of view these branes

must be considered just as fundamental as strings [4]. In particular, one exp ects that

up on quantization their sp ectrum will take the form of a tower of states in an analogous

fashion as for strings.

Unlike string theory, M-theory do es not have a p erturbative coupling constant, since

there is no dilaton- eld in 11-dimensional sup ergravity. The dual relationship between

membranes and ve-branes must therefore be di erent from more standard weak-strong

coupling dualities, suchasbetween strings and ve-branes in d = 10. In particular, there

is the logical p ossibility of a double corresp ondence, in which the dual membrane may in

fact also be viewed as a particular limiting con guration of the ve-brane itself. Hence

M-theory could in a sense be self-dual. This p ossibility is supp orted by the fact that,

b esides to the dual six-form C of eleven-dimensional sup ergravity, the ve-brane also

couples directly to the three-form C itself, via an interaction of the typ e

C ^ T ; (1.1)

3 3

where T is a self-dual three-form eld strength that lives on the world-volume. By

allowing this eld T to have non-trivial uxes through the three-cycles on the world-

brane, the ve-brane can thus in principle carry all membrane quantum numb ers. These

con gurations are therefore naturally interpreted as b ound states between the two typ es

of branes.

One of the aims of the eleven-dimensional p oint of view is to shed light on the mysteri-

ous U-duality symmetry of string theory [1]. The most convincing evidence for U-duality

so far has b een obtained by considering the sp ectrum of BPS states. These studies nec-

essarily involve D-branes that describ e the states charged with resp ect to the RR gauge

elds [12]. Indeed, there have b een convincing results in D-brane analysis, in particular

in the form of degeneracy formulas, that supp ort the symmetry under certain U-duality

transformations [13, 14]. However, the formalism quickly b ecomes rather involved, since

in general one has to take into account D-branes of various dimensions and also b ound

states of fundamental strings and D-branes [15]. 2

As we just argued, the eleven-dimensional ve-brane is a natural candidate to give

a more uni ed treatment of all BPS states in string theory. In fact, one could hop e

that, in an appropriate covariant quantization, U-duality b ecomes a manifest symmetry

of the ve-brane. Unfortunately, there are various diculties in extending the covariant

formalism from strings to higher-dimensional extended ob jects [16]. In this pap er we

will make a rst step in developing a formalism for describing the BPS con gurations

of the ve-brane for compacti cations down to six dimensions, which indeed exhibits

the maximal symmetry. An imp ortant ingredient in this formalism is the idea that the

relevant degrees of freedom on the ve-brane are formed by the ground states of a string

theory living on the world-volume itself. In fact, in compacti cations on a 5-torus the

U-duality group SO(5; 5; Z) can then b e identi ed with the T-duality group of this string.

In six-dimensional compacti cations only the ground states of this string give rise to

space-time BPS states. The string excitations app ear only after further compacti cation

down to ve and four dimensions, where one can consider BPS representations that are

annihilated by 1/8 instead of 1/4 of the sup ercharges. The structure of the resulting BPS

sp ectrum has b een describ ed in our previous pap er [17]. In this pap er we will restrict

ourselves to the six-dimensional theory.

Outline of the paper

In section 2 we will start with a detailed description of the BPS sp ectrum of M-theory

in six dimensions. First we derive the BPS mass formula from the space-time sup ersym-

metry algebra. Then, by comparing with the known result of the BPS sp ectrum of typ e

IIA strings, we prop ose an explicit formula for all the degeneracies that is manifestly

invariant under the complete U-duality group SO(5; 5; Z). In section 3 we analyze the

zero-mo de structure of the ve-brane and show that the central charges corresp ond to

particular uxes through homology cycles on the ve-brane. We use this in section 4 to

construct the complete space-time sup ersymmetry algebra, including the central charges,

as op erators in the Hilb ert space of the ve-brane. In section 5 we rst argue that U-

duality implies that the world-brane theory must contain string degrees of freedom. We

then describ e a p ossible light-cone formulation of this six-dimensional string theory. The

low-energy elds will corresp ond to the collective mo des of the ve-brane. Finally, in

section 6we set out to calculate the BPS sp ectrum of the ve-brane, rst by considering

the linearized quantum uctuations and then by including the winding quantum numb ers.

We also discuss the relation with more conventional D-brane counting. In section 7 we

1 4 1

consider M-theory compacti cations on the orbifolds S K 3 and T S =Z . Here we

nd a new derivation of the corresp ondence with heterotic string theory and demonstrate

the dual equivalence of these two compacti cations. 3

2. The BPS Mass Formula and U-duality

Before we turn to our discussion of the ve-brane theory, let us rst give a description

of the BPS states and the mass formula from the six dimensional space-time p oint of view.

The maximally extended six-dimensional N = (4; 4) sup ersymmetry algebra is given by

[18]

n o

a b ab

Q ;Q = ! p= ;

n o

a ab

Q ; Q = Z ; (2.1)

where a; b =1;:::;4 are SO(5) spinor indices and ! is an anti-symmetric matrix, that

will b e used to raise and lower indices. (We refer to the App endix for our conventions on

spinors and gamma-matrices.) The algebra contains 16 central charges that are combined

in the 4 4 matrix Z and transform as a bi-spinor under the SO(5)SO(5) R-symmetry.

It further satis es the reality condition Z = !Z ! . Wenow wish to obtain a convenient

expression of the masses of BPS states in terms of the matrix Z .

In general, BPS states form short multiplets of the sup ersymmetry algebra which

are annihilated by a subset of the sup ersymmetry generators. We will see that in six

dimensions the generic BPS state is annihilated by 1/4 of the 32 sup ercharges. For a

given multiplet the condition can be written as

" Q )jBPSi =0: (2.2) (" Q +

b a

Since this condition holds with xed "; " for all states in the BPS multiplet, we can

take the commutator with the sup ercharges, and derive the following conditions on the

sup ersymmetry parameters

a ab

p=" + Z " = 0;

p=" + Z " = 0: (2.3)

2 2

Combining these equations with the mass shell condition p + m = 0, one deduces that

BPS

2 y y

m coincides with the highest eigenvalue of the hermitean matrices ZZ and Z Z , with

BPS

" and " b eing the corresp onding eigenvectors,

y a b 2 a

(ZZ ) " = m " ;

BPS

2 y b

" = m " : (2.4) (Z Z )

b a a

BPS

This determines the BPS masses completely in terms of the central Z . 4

The number of states within a BPS sup ermultiplet are determined by the number of

. In fact, one can show that for a given eigenvectors with the highest eigenvalue m

BPS

(non-zero) value of m there are always two indep endent eigenvectors " and ". This can

BPS

y y

be seen, for example, by decomp osing the matrices ZZ and Z Z in terms of hermitean

gamma-matrices , m =1;:::;5as

y 2 m

ZZ = m 1 +2K ;

y 2 m

Z Z = m 1+2K : (2.5)

These relations, which de ne m and the 5-vectors K and K , directly follow from the

L R

" are now replaced reality condition on Z . The eigenvalue equation for the spinors " and

by the Dirac-like conditions

(K jK j)" = 0;

L L

(K jK j)" = 0; (2.6)

R R

whichhave indeed (generically) two indep endent solutions. Thus the BPS condition (2.2)

can b e imp osed for 8 of the 32 sup ercharges. Consequently, a BPS sup ermultiplet contains

3 2

(16) states: this is in between the size of a massless multiplet, which has (16) states,

and that of a generic massive sup ermultiplet with (16) states.

The ten comp onents (K ;K ) can b e understo o d as follows. Wehave seen that the 16

L R

comp onents of the central charge can naturally b e combined into a spinor Z of SO(5; 5).

Out of two such spinors we can construct in the usual way a ten-dimensional vector with

1 1

m m y m m y

comp onents K = tr( ZZ ) and K = tr( Z Z ) as intro duced in (2.5). In fact,

8 8

L R

these quantities form a nul l vector (K ;K ) of SO(5; 5), since one easily veri es that

L R

jK j = jK j. We can express the BPS masses in terms of this vector, by combining the

L R

results (2.4) and (2.5). We nd

2 2

m = m +2jK j; (2.7)

L;R

BPS

2 y

where m = tr(ZZ ). The ab ove BPS mass formula is invariant under an SO(5) SO(5)

symmetry, which acts on Z on the left and on the right resp ectively.

U-duality invariant multiplicities of BPS states

Charge quantization implies that the central charge Z is a linear combination of

integral charges. The expression of Z in terms of the integers dep ends on the exp ec-

tation values of the 25 scalar elds of the 6-dimensional N = (4; 4) sup ergravity the-

ory. From the p oint of view of eleven dimensions these scalars represent the metric G

mn 5

and three-form C on the internal manifold T , and parametrize the coset manifold

mnk

M = SO(5; 5)=S O (5) SO(5). A convenient way to parametrize this coset is to replace

the 3-form C on T by its Ho dge-dual B = C . The parametrization of M in terms of

3 3

G and B is then familiar from toroidal compacti cations in string theory. In nitesi-

mn mn

mal variations of G and B are represented via the action of the spinor representation of

SO(5; 5) on Z . Concretely,

ab mn mn ab

Z =(G + B )( Z ) ; (2.8)

m n

where are hermitean gamma-matrices of SO(5). The U-duality group is now de ned

as those SO(5; 5) rotations that map the lattice of integral charges on to itself. Thus the

U-duality group can be identi ed with SO(5; 5; Z).

The 16 charges contained in Z can be interpreted in various ways dep ending on the

starting p oint that one cho oses. In this pap er we will be interested in the BPS states

that come from the ve-brane in 11 dimensions. From the p oint of view of the ve-brane

it is natural to break the SO(5; 5; Z) to a SL(5; Z) subgroup, b ecause this represents

the mapping-class group of the ve-torus T . The 16 charges split up in 5 Kaluza-Klein

m mn

momenta r , 10 charges s that couple to the gauge- elds C that come from the

3-form, and one single charge q that represents the winding number of the ve-brane.

We can work out the BPS mass formula in terms of these charges q , r and s . To

simplify the formula we consider the sp ecial case where the scalar elds asso ciated with

three-form C are put to zero, and the volume of T is put equal to one. Note that

mnk

these restrictions are consistent with the SL(5) symmetry. For this situation the central

charge Z takes the form

m mn

Z = q 1 + r + s ; (2.9)

ab ab m;ab mn

where the Dirac-matrices satisfy f ; g =2G . (Note that in our notation 1 = ! .)

m n mn ab ab

Inserting this expression in to the BPS mass formula gives

2 2

m n mn

G K K + G W W ; (2.10) m = m +2

mn m n

BPS

with

2 2 m n mk nr

m = q + G r r + G G s s ;

mn mn kr

mnk l r m m

s s ; (2.11) K = qr +

nk lr

W = s r :

m mn 6

m mn m

= K G W . Here we have written K

L;R

The BPS sp ectrum has the following interpretation in terms of the toroidal compact-

i cation of the typ e IIA string. Because the string coupling constant coincides with one

of the metric-comp onents, say G , string p erturbation theory breaks the SO(5; 5; Z)U-

duality to a manifest SO(4; 4; Z) T-duality. Accordingly, the 16 charges split up in an

SO(4; 4) vector of NS-charges, b eing the 4 momenta and 4 string winding numb ers n

(= r ), and m (= s ), and an 8 dimensional spinor that combines the RR-charges q ,

i i5

s and r (= r ) of the 0-branes, 2-branes and 4-branes. U-duality relates RR-solitons to

ij 5

the p erturbative string states, and can thus be used to predict the multiplicities of the

solitonic BPS states from the known sp ectrum of string BPS states. This fact has b een

exploited by Sen and Vafa [14, 13] to give a non-trivial check on U-dualityby repro ducing

the exp ected multiplicities from the D-brane description of the RR-solitons [12]. The

multiplicities of the string BPS states, i.e. with vanishing RR-charges, is given by d(N )

where N = n m and

X Y

1+t

N 2

: (2.12) d(N )t = (16)

1 t

This formula also describ es the degeneracies of the RR-solitons where N = qr + s s~

represents the self-intersection number of the D-branes. In fact, with the help of our

analysis, it is not dicult to obtain a generalized formula that satis es all requirements:

we nd that the degeneracies are given by the same numb ers d(N ), but where N is now

given by the greatest common divisor of the ten integers K and W ,

N = gcd(K ;W ): (2.13)

Indeed, one easily veri es that for the string BPS states all integers K and W vanish

i 5

except W = n m , while for the RR-solitons the only non-vanishing comp onent is K =

5 i

qr + s s~ . Furthermore, the formula is clearly invariant under the U-duality group

SO(5; 5;Z). It can be shown that this is the unique degeneracy formula with all these

prop erties.

In the remainder of this pap er we will present evidence that al l BPS states can be

obtained from the ve-brane. To this end we will prop ose a concrete quantum description

of the ve-brane dynamics that repro duces the complete BPS sp ectrum, including the

ab ove degeneracy formula.

5;5

This follows from the fact that anytwo primitivenull vectors v; w 2 can b e rotated into each

? 4;4

other byaSO(5; 5; Z) transformation. This is implied by the isomorphism v =hv i , i.e. the little

group of v is always SO(4; 4; Z). We thank G. Mo ore and C. Vafa for discussions on this p oint. 7

3. Charges and Fluxes on the Five-Brane

In this section we consider the compacti cation of the ve-brane coupled to eleven-

dimensional sup ergravity to six dimensions on a ve-torus T . From its description as a

soliton it is known that, after appropriate gauge- xing, the ve-brane is describ ed by an

e ective world-brane theory consisting of ve scalars, an anti-symmetric tensor with self-

dual eld strength T = dU and 4 chiral fermions [19]. These elds, which parametrize the

collective mo des of the ve-brane solution, form a tensor multiplet of the chiral N =(4;0)

sup ersymmetry on the world-brane.

The ve-brane couples directly to the six-form C , the metric and the three-form C ,

6 3

and hence after dimensional reduction it is charged with resp ect to all the corresp onding

m mn

16 gauge elds that we denote as A; A and A . To make this concrete, we consider

a ve-brane with top ology of T R where R represents the world-brane time . First

we consider the coupling to A , which is deduced from the term (1.1) by taking C =

m n m mn

A ^ dX ^ dX , where X are the emb edding co ordinates of the ve-brane, and A is

5 m n

constant along T . Now we use that dX ^ dX represents a closed two-form, and hence

de nes a dual three-cycle T . In this way we nd that the charge s with resp ect to

the gauge elds A is given by the ux of the self-dual three-form eld strength T = dU

s = dU (3.1)

through the 10 three-cycles T on the ve-brane. A ve-brane for which these charges

are non-zero, is actually a b ound state of a ve-brane with a number of membranes:

the quantum numb ers s count the number of membranes that are wrapp ed around

2 3

the 2-cycle T that is dual to T . This is similar to Witten's description of b ound

mn mn

states of (p; q )-strings [15]. Notice that in this case that the zero-mo de of the canonical

momentum is also quantized, but these are not indep endent b ecause the self-duality

condition implies that = dU .

Our aim in this pap er is to arrive at a U-dualityinvariant description of the ve-brane.

Wehave shown that under the U-duality group SO(5; 5; Z) the charges s are part of an

irreducible 16-dimensional spinor representation together with the Kaluza-Klein momenta

r of the ve-brane and its winding numer q around the ve-torus. In fact, the spinor

representation of SO(5; 5) is naturally identi ed with the odd (co)homology of the ve-

5 m

torus T . This observation motivates us to try write the other charges q and r as uxes of

5- and 1-form eld strengths through the 5-cycle and 1-cycles on T . Indeed, the winding

number of the ve-brane around the internal T can be written as

q = dV ; (3.2)

T 8

where dV is interpreted as the pull-back of the constant volume element on the T -

manifold. We now would like to turn the 4-form p otential V into an indep endent eld

that is part of the world-brane theory.

As mentioned, the e ective action on the world-brane contains as b osonic elds, b esides

the tensor eld U , ve scalars. In the following we will interpret four of these as b eing

the transversal co ordinates in space-time. This leaves us with one additional scalar Y . In

particular, since the world-brane is 6-dimensional, we can dualize it and obtain a four-form

V . We will now identify this eld with the eld V in (3.2). Its ve-form eld strength

W = dV is normalized such that the ux q is integer, but for the rest it can take any

value including zero. Now, formally we can go back to the description in terms of the

dual scalar eld Y by taking W to be an indep endent ve-form and intro ducing Y as

the Lagrange multiplier that imp oses the Bianchi identity dW =0. The fact that W has

integral uxes implies that Y must be a p erio dic eld, i.e. Y Y + r with r integer.

Notice that on-shell dY = S= V = , and so dY is the canonical momentum for V .

The 5 remaining charges r can now be identi ed with the integer winding numb ers of

Y around the 5 one-cycles T :

r = dY: (3.3)

It is straightforward (see also the previous fo otnote) to show that these op erators are

the generators of translations along the internal directions on the 5-torus, and so they

indeed represent the Kaluza-Klein momenta of the ve-brane. In this way all the 16

charges (q; r ;s ) have b een written as uxes through the odd homology cycles on the

ve-brane, and so, in view of our previous remark, are naturally identi ed with a 16

comp onent SO(5; 5; Z) spinor.

To make this more manifest it is convenient to combine the elds Y , U and V , and

their eld strengths dY; dU and dV using SO(5) gamma-matrices as

m mn

Y = Y 1 + V + U ;

ab ab m mn

ab ab

m mn

(rY ) = dV 1 +(dY ) +(dU ) ; (3.4)

ab ab m mn

ab ab

where r = @ . The eld Y is not an unconstrained eld, since it would describ e

ab m ab

to o many degrees of freedom. Namely, we still have to imp ose the condition that dU

is self-dual and dY is the dual of dV . This can be done in a rather elegant way in a

In fact, in a light-cone formalism for extended ob jects one naturally obtains a residual gauge

symmetry under volume preserving di eomorphisms [20]. For our ve-brane these can be used to

eliminate all dep endence on the ve compact emb edding co ordinates X except the volume-form

m n k l p m

dV = dX ^ dX ^ dX ^ dX ^ dX . Hence, the ve spatial comp onents V are basically the

mnk lp

emb edding co ordinates X . Notice also that the gauge-transformation V ! V + d corresp onds to a

volume preserving di eomorphism on the ve-brane, since it leaves dV invariant. 9

y ab ab ab

canonical formalism be imp osing the constraint = rY , where is the canonical

Y Y

conjugate momentum of Y . This constraint reduces the number of on-shell degrees of

freedom from 8 to 4.

The advantage of the notation (3.4) is that it makes the action of the U-duality group

more manifest: the action of SO(5) SO(5) is from the left and right resp ectively, while

the other generators of SO(5; 5) act as in (2.8). The results of this section can now be

summarized by the statement that the central charge Z coincides with the zero-mo de

part of (rY ) .

4. Space-time Supersymmetry

Our aim in this section is to investigate the BPS sp ectrum from the view p oint of

the world-brane theory. At present we do not know a consistent quantum theory for

ve-branes that is derived from a covariant world-volume action. Fortunately, our only

aim is to study the quantum states of the ve-brane that are part of the space-time

BPS sp ectrum, and, as we will see, for this we do not need to consider the full ve-

brane dynamics. Furthermore, even without using the details of the world-brane theory

one can already say a lot ab out its quantum prop erties just on the basis of symmetry

considerations and other general principles. Our only assumption is that the ve-brane

theory p ermits a light-cone gauge, so that there are 4 transversal co ordinates X in the

6 uncompacti ed dimension. In the following sections this assumption will be justi ed

by the fact that from this starting p oint we are able to derive a Lorentz invariant BPS

sp ectrum.

In the light-cone gauge the SO(5; 1) space-time Lorentz group is broken to the SO(4)

subgroup of transversal rotations. On the world-brane this group plays the role of an

R-symmetry. We organize the elds accordingly using SO(4) representations with ; _

indicating the two chirality spinors. In addition our elds may carry one or two spinor

indices of the SO(5) group of spatial rotations on the ve-brane. These are denoted by

a, b, etc. In this notation we have the following elds on the ve-brane

_ _

X ; ; ;Y ; (4.1)

b a

where Y is the eld that weintro duced in (3.4). Notice that each of these elds has four

on-shell comp onents. These elds represent the collective mo des of the ve-brane and

This generalizes the condition of self-dualitytointeracting theories, since we do not need to assume

that the eld is describ ed by a free action. 10

their zero-mo des will be used to construct the space-time sup ersymmetry algebra. More

_ _

b ab

precisely, each of these elds has a canonical conjugate momentum ; ; and .

Their zero-mo des

_ _

;Z (4.2) p ;S ;S

a b

enter in the N =(4;4) space-time sup ersymmetry algebra resp ectively as the transversal

momentum, part of the space-time sup ercharges (namely those that are broken by the

ve-brane) and the central charge.

The world-brane theory carries a chiral N =(4;0) sup ersymmetry that is generated by

a set of sup ercharges G and G . These sup ercharges satisfy in general the commutator

a a

algebra

fG ;G g = 2 (1 H + (P + W ));

ab m m

a b ab

_ _ m

fG ;G g = 2 (1 H + (P W )): (4.3)

ab m m

a ab

Here H is the Hamiltonian on the ve-brane, and P are world-brane momentum op-

erators that generate translations in the ve spatial directions. It will sometimes be

convenient to combine them into the matrix

P = 1 H + P : (4.4)

ab ab m

L in string theory, and as is clear from this The op erators P play the same role as L

0 m 0

analogy, will have to annihilate the physical states in the sp ectrum of the ve-brane

P jphys i =0: (4.5)

The op erator W that app ear in the sup ersymmetry algebra (4.3) represents a p ossi-

ble vector-like central charge. In terms of the uxes of dY and T = dU it receives a

contribution given by the top ological charge

W = dY ^ T: (4.6)

In order to have a realization of space-time sup ersymmetry (without space-time vector

central charges) we will also have to put W to zero on physical states

W jphys i =0: (4.7)

If we do not intro duce extra degrees of freedom, this condition implies the relation K =

+ +

K . Finally, from the light-cone condition x = p ,we nd that we have to imp ose the

condition

(H p p )jphys i =0: (4.8) 11

We will now turn to a discussion of the space-time sup ersymmetry. To reduce the

number of equations somewhat, we will concentrate rst on the \left-movers" G , and

discuss the dotted \right-moving" comp onents G afterwards. The world-brane sup er-

charges represent the unbroken part of the full N = (4; 4) space-time sup ersymmetry

algebra. The other generators must be identi ed with the zero-mo des S and S of the

a a

conjugate momenta of the world-brane fermions, since these are the Goldstone mo des

asso ciated with the broken sup ersymmetry. Under the world-brane sup ersymmetry these

zero-mo des transform into the zero-mo des of the b osonic elds,

fG ;S g = Z ;

fG ;S g = ! p= : (4.9)

To get the complete set of relations one needs to use the world-brane sup ersymmetry

algebra (4.3) together with the conditions (4.8) and (4.5). This gives on physical states

fG ;G g = 2p p ! ;

a b

fS ;S g = ! ; (4.10)

and similarly for the dotted comp onents. The space-time sup ersymmetry generators are

therefore

p p

_ a

+ +

2p S ;G = p ); (4.11) Q =(

a a

where on the left-hand side denotes a chiral four-comp onent SO(5; 1) space-time spinor

index.

We can now discuss the space-time BPS states from the world-brane p oint of view.

The value of the BPS mass is determined by the central charge Z , and so we know that

for a BPS state we should nd that

2 2

H jBPSi = (p + m )jBPSi (4.12)

BPS

where m is given in (2.7). We will now prove that this is in fact the lowest eigenvalue of

BPS

the Hamiltonian H in the sector with a given central charge Z . For this purp ose, let us

^ ^

intro duce new op erators G and P which are de ned as the non-zero mo de contributions

of the resp ective op erators. They satisfy the relations

y b

G + G = Z S S ; + p=

a a a b

2 y

1 1

P = p + (Z Z ) + P : (4.13)

ab ab ab ab

2 2

Using the world-brane and space-time sup ersymmetry algebra we derive that the anti-

commutator of two of these op erators is

y 2 y

^ ^ ^ ^

fG ; G (G G) g =2P p (Z Z ) : (4.14)

ab ab ab ab

a b i

2 12

Next, by taking the trace with the non-negative de nite matrix = (1 + K ), where

K is a unit vector in the direction of K = K = K , one deduces that

L R

2 2 y

1 1 1

^ ^ ^

H + K P = p + m + tr( G G): (4.15)

2 2 2

BPS

The last op erator on the right-hand-side is clearly p ositive de nite. Furthermore, since

BPS states are physical, they have to be annihilated by the translation generators P .

Combining these two facts gives the statement we wanted to prove. It also tells us that

BPS states are annihilated by half of the op erators G

" G jBPSi =0: (4.16)

In this sense, space-time BPS states are also BPS states from the world-brane p oint of

view.

5. Strings on the Five-Brane

The U-duality group acts on the shap e and size of the internal 5 torus, and in particular

contains transformations that map large to small volumes. If we require that the ve-

brane theory is invariant under such transformations, it is clearly necessary to include

short distance degrees of freedom. Furthermore, these extra degrees of freedom need to

b ehave the same for very small box sizes as momentum mo des do for large box sizes.

This suggests that we can p ossibly restore the full U-duality invariance by replacing the

world-brane theory by a string theory. Another indep endent indication that the ve-brane

world-volume theory may in fact contain string-like excitations is the p ossible o ccurrence

of vector-like central charges in the sup ersymmetry algebra, since only one-dimensional

extended ob jects can carry such charges.

One easily sees that BPS states necessarily corresp ond to the string ground states.

These states are annihilated by 1/4 of the space-time sup ercharges, which implies, as we

have seen in the previous section, that they are annihilated by 1/2 of ve-brane sup er-

charges. If we rep eat this pro cedure once more, we conclude that in terms of the string,

BPS states are annihilated by all of the string sup ercharges. Although we only use the

string ground states in this pap er (see however [17]) we will now make some remarks

concerning the formulation of this six-dimensional string theory.

We are lo oking for a string mo del whose ground states represent the massless tensor

multiplet describing the collective mo des of the ve-brane. Sp eci cally,we exp ect ground 13

_ _

states of the form j i that describ e the four scalars X = X , states j bi and ja i

that describ e the4world-brane fermions and , and nally we need states jabi that

corresp ond to the fth scalar Y and the 3 helicity states of the tensor eld U . Here the

indices a; b =1;2now lab el chiral SO(4) spinors.

This structure arises naturally in the following mo del. We will assume that the world-

sheet theory of this string theory can be formulated in a light-cone gauge, and so one

aa_

z ) together with fermionic exp ects to have 4 transversal b osonic co ordinate elds x (z;

partners (z ); (z ). The world-sheet theory has 4 left-moving and 4 right-moving

H H

a_ _

a _ ab_ ab_

sup ercharges F = @x and F = @x which corresp ond to the unbroken part

b b

of the world-brane sup ersymmetry and satisfy

_ _

b a_b a _

g =2 L ; (5.1) fF ; F

where L is the left-moving world-sheet Hamiltonian. The ground states must form a

multiplet of the zero-mo de algebra f ; g = . This gives 2 left-moving b osonic

a b

ground states j i and 2 fermionic states jai. By taking the tensor pro duct with the

right-moving vacua one obtains in total 16 ground states

j ; k ija; k i j ; k i jb; k i ; (5.2)

L L R R

just as we wanted. Here we also to ok into account the momenta (k ;k ), which form a

L R

lattice, since we have assumed that the ve-brane has the top ology of T . Since the

5;5

light-cone co ordinates parametrize a cylinder S R, the standard light-cone formalism

has to b e slightly mo di ed, as wehave to takeinto account that the string can also wind

around this circle. Sp eci cally, we must imp ose the mass-shell condition

+ 2 5 2

1 1

L = k k = k (k ) ; (5.3)

2 2

L L L

so that the level matching condition b etween the left-moving and right-moving sectors of

the string b ecomes

5 2 5 2 5 5

1 1

L L = (k ) (k ) = k w : (5.4)

0 0

2 2

L R

5 1

with w the winding number around the S . Level matching implies that for the ground

states jk j = jk j.

L R

Though the string theory we want needs to contain the tensor eld U in its sp ectrum, note that

it should not carry any charge with resp ect to it, since this would violate charge conservation on the

ve-brane. Rather, the U - eld should couple via its eld strength T , like an RR-vertex op erator. The

string describ ed in this section should therefore b e distinguished from the self-dual string considered in

[21]. We thank M. Becker, J. Polchinski, and A. Strominger for discussions on this p oint. 14

The world-sheet sup ersymmetry transformation relates the fermion zero-mo des and

the transversal momentum k of the string

a _ b ab

fF ; g = k : (5.5)

Hence one can construct a (4,0) sup ersymmetry on the 5+1-dimensional target space of

the string, i.e. on the world-volume of the ve-brane, as follows

p p

a a _ a

+ +

; F = ); (5.6) G =( 2k k

L L

where on the left-hand side a denotes a chiral SO(5,1) spinor index. These generate the

algebra

a b 0 ab m ab

^ ^

fG ; G g =2 (k 1 + k ) (5.7)

and similar for the right-moving generators.

We now prop ose that the world-brane dynamics of a single ve-brane is describ ed by

the second quantization of this string theory. The most imp ortant consequence for our

study of the BPS states is that the world-brane sup ersymmetry algebra gets mo di ed.

Namely, as we have just shown, the anti-commutator of the left-moving sup ercharges

pro duces the left-moving momentum k , while the right-movers give k . Hence the string

L R

states form representations of the N =(4;0) sup ersymmetry algebra

a b ab m ab

fG ;G g = 2 (H 1 + P );

_ _

a _ b _ ab m ab

fG ;G g = 2 (H1 + P ): (5.8)

m m

The op erators H , P and P act on multi-string states that form the Hilb ert space of

L R

the ve-brane. The ve-brane Hamiltonian H measures the energy of the collection of

m m

strings, while (P + P ) measure the total momentum. But we see that the algebra also

L R

M m m

naturally contains a vector central charge W = (P P ), which measures the sum

L R

of the string winding numb ers around the 5 indep endent one-cycles on the world-brane.

We are interested in BPS states, which are annihilated by eight of the world-brane

sup ersymmetry generators. Such states are obtained by combining individual string states

jbps i that satisfy

" G jbps i =0; (5.9)

where " is constrained by

k " =0; (5.10)

y a

As explained in section 4, the appropriate second-quantized generators G also contain a zero-mo de

contribution in addition to the part G which is expressed in terms of the string creation and annihilation

op erators. Wehave silently added the zero-mo de contribution here. 15

and similarly for the opp osite chirality. In terms of the world-sheet generators this con-

dition reads

a ab

F jbpsi = k jbpsi: (5.11)

This condition tells us that the string must be in the left-moving ground state. The full

BPS condition thus implies that the individual strings must be in their ground state.

6. U-duality Invariant BPS Spectrum of the Five-Brane

We will now discuss the BPS quantization of the ve-brane. A natural starting p oint

is the low-energy e ective eld theory on the world-volume, which is appropriate for

describing the world-brane dynamics at large volume. Our hyp othesis is that the e ective

theory in this regime takes the most simple form consistent with all the symmetries of

the ve-brane. As we have argued, this minimal requirement is ful lled by the eld

theory consisting of the single tensor multiplet describ ed in sections 3 and 4. These elds

describ e the ground states of the strings, except that at small volumes we have to take

into account the winding con gurations of the underlying string theory. We will further

make the assumption that for the purp ose of constructing the BPS sp ectrum it is allowed

to treat the ve-brane uctuations in a linear approximation, that is, we will use free eld

theory.

Since we will work towards a Hamiltonian formalism, we will from the b eginning

distinguish the time-co ordinate from the ve space-like co ordinates . We can further

pick a Coulomb gauge for the two-form eld U by demanding U =0 and @ U =0.

m0 mn

It is useful to intro duce the matrix-valued derivative

D = 1 @ + @ (6.1)

ab ab 0 m

and the world-volume bi-spinor eld Y intro duced in equation (3.4). Its equation of

motion then takes the form

b b

D (DY ) = D (DY ) =0; (6.2)

a bc a

where D acts on Y via matrix multiplication. These two equations reduce the number of

indep endent on-shell comp onents of Y to four. The free eld equations of motion of the

other elds are in this notation

D (DX ) = 0;

(D ) = 0: (6.3)

a 16

The free eld Hamiltonian and momentum op erators on the world-brane take the quadratic

form

i h

b a b a b _ ab 5 ac y b ac i

+ (D ) + (D ) (6.4) P = d (DY ) (DY ) +(DX ) (DX )

i c

_ c

To describ e the quantum uctuations of the ve-brane, we expand the various elds in

plane waves that propagate on the world-volume. Since we assume that the world-volume

has the top ology of T R, we can lab el these waves with 5 integral momenta k . For

example, the expansion of Y is

I ik I ik

(DY ) = Z + (a (k )u (k )e + a (k ) u (k )e ); (6.5)

ab ab I

ab ab

where I runs from 1 to 4, and k runs over the ve-dimensional momentum lattice of T .

Together with the four other b osonic elds X and fermionic elds and ,we obtain

a a

creation and annihilation mo des a (k ), (k ), where the indices I and A now b oth run

I A

from 1to8.

We could now imp ose the BPS conditions in the light-cone formalism that we have

b een using as describ ed in section 4. However, as we have seen, the e ective eld theory

approach is only sucient (and consistent) in the sp ecial case that the central charge

m m m

satis es the condition W = (K K )=0. This condition, which breaks U-duality,

L R

can be understo o d as the absence of vector central charges on the world-brane. In terms

of the string theory this corresp onds to considering the sector with zero total winding

numb er. Instead of working out this case in detail, we will immediately give a manifestly

U-dualityinvariant derivation of the BPS sp ectrum. The eld theory limit can simply b e

obtained afterwards by simply putting W to zero.

Indeed, it is clear from the discussion in section 5 that a eld theory description is

incomplete and that at small volumes of T extra degrees of freedom must be included

in the e ective description. In particular, the presence of string winding states will imply

that the mo des of the quantum elds can carry b esides the momentum k also a winding

number w. To put this idea into e ect, we note that the equation of motion (6.2) of the

eld DY in momentum space can indeed b e generalized to an SO(5; 5) invariant equation

by intro ducing a left- and a right-momentum vector, as follows

ab I

(jk j1 k ) u (k ;k ) = 0;

L L R

ab y I

(jkj1 k ) u (k ;k ) = 0: (6.6)

R L R

I I

Here u (k ;k ) denotes the generalization of the bi-spinors u (k ) in the mo de expansion

ab ab

L R

(6.5) of DY . Via the action of SO(5; 5) on the spinor indices of DY ,we deduce that the

ab 17

2 2

=0. In a k pair of momenta k and k combine into an SO(5; 5) null-vector, with k

L R

R L

similar way, we can argue that all other elds must also dep end on two momenta instead

of one.

I A

We are thus led to consider a Fock space with stringy oscillators a (k ;k ); (k ;k )

L R L R

5;5

where (k ;k ) 2 and jk j = jk j = jk j. The generalized number op erator is

L R L R

y y

I A

N = a (k ;k )a (k ;k )+ (k ;k ) (k ;k ): (6.7)

k ;k

I A

L R L R L R L R L R

We can now write the Hamiltonian and momentum op erators on the ve-brane (see equa-

tion (6.4)) in terms of the contributions of the zero-mo des and these particular string

mo des as

2 2

1 1

H = m + p + jk jN ;

k ;k

0 i

L R

2 2

k ;k

L R

m m

P = K + k N : (6.8)

k ;k

L L R

L L

k ;k

L R

1 1

2 m y m m y m

= where m = = with K trZ Z and K tr( Z Z ). Similarly, we de ne P

4 8

R L R

m y

tr( ZZ ).

Note that we now have indep endent left-moving and right-moving momentum op era-

tors P and P on the ve-brane. In order to b e able to realize the space-time sup ersym-

L R

metry algebra, wehave to imp ose on physical states the conditions that b oth momentum

op erators vanish

m m

P jphys i = P jphys i =0: (6.9)

L R

These equations tell us that the sum of the individual left-moving or right-moving string

momenta k ;k have to cancel the contribution K ;K of the zero-mo de uxes on the

L R L R

ve-brane. We see in particular that, in the case that K 6= K , this necessitates the

L R

inclusion of string winding mo des.

We will now imp ose the BPS condition. Hereto we can use the result demonstrated at

the end of section 4, that BPS states saturate the lower b ound (4.15) for the \light-cone

Hamiltonian" H + K P for a given value of the central charge Z . In more detail we

L;R

pro ceed as follows. The (mass) of these ve-brane states is measured by the op erator

2 2

+ 2jk jN : (6.10) m = m

k ;k

L R

k ;k

L R

As we have explained in detail in section 2, the BPS condition implies that m must be

2 2 y y

equal to m = m +2jK j which is the highest eigenvalue of Z Z or ZZ . Using the

L;R

BPS

physical state condition P = 0, the BPS mass formula may be rewritten as

2 2

1 1

m + K P = m + jK j; (6.11)

L L L

2 2 18

where we intro duced the unit vector K in the direction of K . Inserting the mo de-

L L

expansions of P and m shows that the zero-mo de part of the left-hand side is already

equal to the right-hand-side. The remaining oscillator part must therefore add up to zero

(jk j + K k )N =0: (6.12)

k ;k

L L L R

k ;k

L R

The left-hand side is a non-negative expression and can vanish only if the excitations have

momentum k directed in the opp osite direction of K . Similarly, one obtains an analogous

L L

result for the right-moving momenta k . So, we conclude that the BPS sp ectrum for given

5;5

central charge vector K =(K ;K ) 2 is obtained by acting with only those oscillators

L R

a and for which the momentum vector k = (k ;k ) p oints in the direction opp osite

k k

L R

to K .

5;5

To count the number of BPS states for given vector K 2 , let [K ] be the largest

p ositive integer so that K=[K ] is still an integral vector. In other words, K is [K ] times

a primitive vector, and the [K ] can be de ned as the greatest common divisor of the ten

m m

. With this notation, the allowed momenta of the oscillators must be ;K integers K

R L

of the form k = nK =[K ], for some p ositive integer n. Let N denote the o ccupation

n n

number of these mo des. The level matching conditions (6.9) together with (6.8) now

reduce to the simple combinatorial relation

nN =[K]: (6.13)

Since there are 8 b osonic and 8 fermionic mo des that contribute at each oscillator level,

we obtain the result that we announced at the end of section 2. The number of BPS

states is given by d([K ]) with

Y X

1+t

N 2

: (6.14) d(N )t = (16)

1 t

We would like to p oint out that the ab ove result has a rather striking interpretation.

We see that in the BPS limit the excitations of the ve-brane are constrained to lie in

a single space-like direction, which is determined by the value of the central charge. So,

e ectively the six-dimensional world-volume reduces to a world-sheet and the BPS ve-

brane b ehaves like a string, in fact a chiral typ e I I string. The U-duality group SO(5; 5; Z)

acts on the momentum vector K and so p ermutes the various string-like excitations of

the ve-brane.

If wewant to relate this eleven-dimensional p oint of view to the ten-dimensional typ e

I I string, wehave to single out a particular direction on the world-brane. As we discussed 19

b efore, this breaks the U-duality group to the little group SO(4; 4; Z) and corresp onds

to distinguishing NS-NS and R-R typ e charges. The string-like excitations in this xed

direction then corresp ond directly to the BPS states of the fundamental typ e I I string.

Comparison with D-branes

It might be illustrative to compare the counting of BPS states in this pap er with the

more conventional counting using D-branes [12]. It is esp ecially instructive to see how the

string degrees of freedom of the ve-brane, which from our p oint of view were a crucial

ingredient in a complete description of the BPS sp ectrum, manifest themselves in the

world-volume theories of the D-branes. In particular wewould like to see to which extent

string-like excitations o ccur for the Dirichlet four-brane in typ e IIA sup erstring theory,

which can be considered as a simultaneous dimensional reduction to ten dimensions of

the ve-brane in M-theory.

So let us reconsider the compacti cation of the typ e IIA sup erstring on a four-torus.

As we mentioned in section 2, the 16 quantum numb ers now decomp ose in the 8 NS-NS

and 8 R-R charges

odd 4

(p; w ) 2 H (T );

ev en 4

(q ;q ;q ) 2 H (T ) (6.15)

0 2 4

Here p ;w are the momenta and winding number of the fundamental NS string, and q

i p

(p =0;2;4) is the Dirichlet p-brane charge. In terms of these charges the 10 comp onents

of the vector (K ;W ) of equation (2.11) are given by

q ^ q K = q q +

2 2 4 0

i i i

K = q p +(q ^w)

4 2

W = p w

5 i

W = q w +(q p) (6.16)

i 0 i 2 i

Our analysis and U-duality predict that the degeneracy is a function of the g.c.d. of these

10 integers.

Let us now lo ok at con gurations in which we can make a meaningful comparison

with standard D-brane computations to count b ound state degeneracies. We will be

particularly interested in con gurations with a single 4-brane, no 2-brane (for simplicity)

and an arbitrary number of 0-branes, i.e. we put q = 1 and q = 0. In that case the

4 2

degeneracy is given by d(N ) with

i i i

N = gcd(q ;p ;p w; q w )=gcd(q ;p ): (6.17)

0 0 0 20

Can we make a macroscopic BPS string-state on the four-brane which has a non-zero

winding number W ? Such an ob ject will manifest itself, in the large volume, as a long

string-like ob ject. In our philosophy such a state is a coherent sum of BPS ve-brane

strings with paralel momenta and winding numb ers.

If we cho ose for simplicity the total momentum K to b e zero, we see from the ab ove

expressions that it is indeed quite easy to make such an ob ject. For q = 1 and p = 0 the

i i

winding number W simply equals the winding number w of the fundamental NS string.

Furthermore, the degeneracy of such a state is given by the usual numb er of string ground

states d(0) = 2 . So in this case, the string BPS state is nothing but the fundamental

closed string b ound with a zero-brane to the four-brane.

In fact, this p oint of view can b e easily generalized to a con guration with an arbitrary

number q of zero-branes, where the degeneracy is predicted to b e d(q ). There is a simple

0 0

explanation of this counting using the so-called \necklace" mo del. In this case we have

q zero-brane \b eads" that are stringed together with a NS string and b ound to the four-

brane. These zero-branes will cut the string in q pieces, each of which has the usual 8 + 8

ground states. However, the zero-branes can cluster together and form b ound states.

Therefore, to compute the total number of BPS states we have to sum over partitions

giving the usual degeneracy formula

Y X

1+t

N 8

(6.18) d(N )t =2

1 t

7. Heterotic String Theory from the Five-Brane.

The ab ove results can be generalized in a straightforward fashion to other internal

5 1

manifolds than T . As an explicit example, we will now brie y discuss the cases of S K 3

4 1

and T S =Z . Previous studies of M-theory compacti ed on these manifolds [6, 7, 22]

have shown that in b oth cases the theory b ecomes equivalent to the heterotic string

compacti ed on T (provided that the Z action in the latter case is de ned appropriately).

Here we con rm this by means of an explicit study of the ve-brane BPS sp ectrum. In

particular we will nd that the two di erent compacti cations are T-dual from the p oint

of view of the world-volume string theory.

It will be useful in the following to think ab out K 3 as (a resolution of ) the orbifold

T =Z , so that b oth typ es of compacti cation manifolds are obtained as Z -orbifolds of

2 2

5 4 1

T , with the two Z 's acting on T and S resp ectively. These two transformations are

2 21

ab ab

represented on the uxes Z of DY as follows

Z ! Z (7.1)

5 5

and

Z ! Z (7.2)

5 5

resp ectively. To b e able to construct the appropriate orbifolds we will rst have to extend

these Z -actions to a transformation on the complete elds, and furthermore, check that

the resulting transformations are indeed symmetries of the ve-brane theory. Notice that

i 5

under b oth Z actions the quantities K and W (with i =1;:::;4) are o dd, while K and

2 i

W are invariant. This suggest that the Z action must b e accompanied by a re ection of

5 2

the world-volume of the ve-brane. More precisely, we nd that the unique Z -symmetry

of the world-volume action that incorp orates the ab ove transformation on the uxes is

given by the orientifold transformation

DY ( ) ! DY (~) ;

5 5

DX ( ) ! DX (~); (7.3)

( ) ! (~);

i i 5 5

where ~ = for i = 1;:::;4, and ~ = . Without this co ordinate re ection this

transformation would not be a symmetry. For example, it would not leave the de nition

of the translation op erators P invariant.

So for b oth typ es of orbifolds, the world-volume co ordinates lie on the same orbifold

1 4 1

S T =Z , which we may also think of as S K 3. The two theories di er, however,

via their b oundary condition on the eld Y . In b oth cases, the world-volume theory has

a chiral N = 2 global sup ersymmetry and the R-symmetry group is reduced to SU (2),

which commutes with the holonomy group SU (2) of K 3. The 6-dimensional space-time

theory has an unbroken N =2 sup ersymmetry with central charge (a; b =1;2)

ab a

Q g = Z : (7.4) fQ ;

As b efore, this symmetry is realized on the ve-brane via the world-volume sup ersymmetry

a a

generators G and fermion zero-mo des S .

It is now straightforward to see that the two typ es of theories are related via the

U -duality (or T -duality) transformation that interchanges the momentum and winding

mo des of the string theory on the ve-brane. To see this wehave to treat the two orbifolds

1 5

separately. For the K 3 S compacti cation, of the uxes intro duced ab ove for T , only 22

the 8 Z -invariant uxes q , r and s with i; j =1;:::4 survive. In addition, there are 16

2 5 ij

extra uxes

s = dU; (7.5)

S S

2 4

where the S are the two-spheres surrounding the 16 xed p oints on T =Z . The intersec-

tion form on the total set of uxes has signature (4,20), and the integers therefore lab el a

4;20

vector (p ;p ) in the lattice . In terms of the quadratic quantities K ;W the only

m m

L R

non-vanishing comp onent is given by

2 2

(p p ): (7.6) K =

L R

4 1

In the dual compacti cation on T S =Z , on the other hand, we are left with the uxes

r and s . These are complemented by 16 extra uxes s~ , which again combine with the

i i5 I

4;20

other uxes into a vector (p ~ ; p~ ) 2 . In this case we nd only a non-zero contribution

L R

2 2

(~p p~ ): (7.7) W =

L R

As we will now show, these results imply that after imp osing the level matching constraints

5 5

P = P = 0 and the BPS condition, only strings on the ve-brane with either pure

L R

momentum or pure winding number in the 5-direction will contribute in BPS states. In

particular, the T-duality map on this S will interchange the momentum and winding

mo des and thus the two compacti cations.

In b oth cases, the dep endence of the central charge Z on the integer uxes is parametrized

4;20

by means of the action of SO(4; 20) on the lattice . The central charge only dep ends

on the 4 `left-moving' comp onents p (or p~ ) via

L L

ab i ab

Z = p : (7.8)

BPS states again satisfy a condition of the form (" Q + " Q )jBPSi =0. In this case the

a a

y 2 y 2

central charge matrix Z satis es Z Z = p 1, and the eigenvalue equation Z Z" = m "

L BPS

has therefore two indep endent solutions, with eigenvalue

m = jp j: (7.9)

BPS L

The resulting BPS multiplets are 16 dimensional, which equals the dimension of the

massless representations of the N =(2;2) space-time sup ersymmetry.

This `left-right' sux will turn out to corresp ond to the twochiral sectors of the space-time heterotic

string, and should not b e confused with the lab el distinguishing left or right-moving string mo des on the

ve-brane. 23

To obtain the multiplicities of the BPS states it is again necessary to intro duce a mo de

expansion of the low-energy string elds on the world-volume. The mo des are lab eled by

5 5 1

integral momentum k = k and winding number w = w in the S direction together with

a quantum numb er that lab els the eigenmo des with eigenvalue h of the Laplacian on K 3.

The Hamiltonian takes the form

2 2 2

1 1 1

2 2

p + p + p + k + w + hN (7.10) H =

k;w;h

2 2 2

L R

k;w;h

5 1

Of the translation op erators only the comp onent P in the direction of the S survives.

For the S K 3 compacti cation it takes the form

2 2 5

1 1

p p + kN : (7.11) P =

k;w;h

2 2

L R

k;w;h

In a similar way as b efore, we deduce from combining the BPS mass-shell condition

1 1

5 2 2

with the constraint P = 0, that charged BPS states can only oscillate H = p + m

2 2

BPS

1 5

in the left-moving S direction, i.e. opp osite to K , while the winding number must vanish

w =0. The total number of oscillators is constrained by the level matching condition

2 2

1 1

p p + kN =0: (7.12)

k;0;0

2 2

L R

4 1

In the dual compacti cation T S =Z the only mo di cation is the interchange of the

momentum lab el k and the winding lab el w . The total winding number of the string is

related to the uxes via the constraint W = 0, which reduces for BPS states to

2 2

1 1

p~ p~ + wN =0: (7.13)

0;w ;0

2 2

L R

This gives a concrete description of the BPS states in terms of the underlying string

mo des.

The number of oscillators that contribute to these expressions is determined by the

number of harmonic zero-mo des with h = 0 of the various elds on the K3 manifold.

There are 5 b osonic oscillators a with i = 1;:::5 corresp onding to the constant zero-

mo des of the co ordinate elds X and Y on K3. The 19 anti-self-dual harmonic 2-forms

on K3 lead to 19 additional left-moving mo des a with i = 6;:::;24. So in the end we

are left with 24 left-moving b osonic oscillator mo des a , which can be recognized as the

4 y

left-moving sector of the heterotic string compacti ed on T to 6 dimensions. Since also

The 3 self-dual two-forms on K 3 and the chiral covariantly constant spinor on K 3 give rise to

1 i

oscillator mo des which are right-moving on S , and form, together with the right-moving X and Y

mo des, the chiral world-sheet content of the sup erstring. These mo des are however all eliminated via the

BPS restriction. 24

4;20

the uxes combine into a vector (p ;p )on , from this p oint on the counting of BPS

L R

states exactly parallels that of the heterotic string.

8. Concluding Remarks

In this pap er we have presented a detailed analysis of the sp ectrum of BPS states

of the ve-brane theory in eleven dimensions. We motivated our prop osed quantization

pro cedure by using hints obtained from known results ab out BPS states in six-dimensional

string theory, such as the symmetry under U-duality. Although our formulation was not

manifestly Lorentz invariant in six dimensions, the nal result for the sp ectrum of BPS

states is in fact invariant under the full Lorentz group SO(5; 1). This is a consequence

of the fact that the BPS restriction e ectively reduces the ve-brane dynamics to that

of critical typ e II sup erstring theory. Hence we can use the standard derivation to show

that our result for the complete BPS sp ectrum is indeed fully covariant.

Our derivation should be compared with the analysis of D-brane states [12]. In prin-

ciple, it should be p ossible to make a concrete identi cation between sp eci c ve-brane

excitations and con gurations of D-branes and fundamental strings. Our construction of

the BPS sp ectrum in terms of a Fock space in section 6 indeed matches the description

of multiple D-brane con gurations as given in [13]. Via this corresp ondence our results

on the BPS sp ectrum also give useful information ab out the b ound states of strings and

D-branes [15].

Wehave concentrated on compacti cations to 6 dimensions. But it is straightforward

to extend our metho ds to compacti cations to other dimensions. In particular, for di-

mensions higher than 6 one can deduce the BPS sp ectra by simply putting some of the

16 charges equal to zero. Thus, the results presented in this pap er imply that at least for

compacti cations to six dimensions and higher, all BPS states can be obtained from the

ve-brane of M-theory. In particular, all the states that one would naively asso ciate with

the two-brane have now all b ecome part of the sp ectrum of ve-brane states.

This can b e nicely illustrated in the sp eci c example of the compacti cation on T

S =Z discussed in section 7 by considering the decompacti cation limit for large volume

of the T . Here one obtains the ten-dimensional heterotic string states as particular

excitations of the ve-brane. This should be compared to the observations made in [6]

that from the eleven-dimensional p oint of view the heterotic string naturally arises from

the two-brane wrapp ed around the S =Z . Our result however suggest that one should

b e able to think of the two-brane as a limiting con guration of the ve-brane, in this case

with the world-brane top ology of K 3 S . This could give an alternative explanation of

the E E gauge symmetry.

8 8 25

To extend our formalism to dimensions ve and lower, wehave to consider BPS states

that are annihilated by 1/8 instead of 1/4 of the space-time sup ercharges. This can be

achieved by including vector central charges (b oth in space-time and on the world-volume)

that mo dify the level matching conditions. In particular in ve dimensions this extends

the ux sectors to all 27 charges, that transform in the U-duality group E (Z) [17]. Hence

the ve-brane also provides a uni ed description of all BPS states of 5 dimensional string

theory. It would b e interesting to see to which extent this approach can b e used to obtain

the complete string BPS sp ectrum in four dimensions.

Acknowledgements

We would like to thank M. Becker, S. Ferrara, C. Kounnas, W. Lerche, R. Minasian,

G. Mo ore, J. Polchinski, A. Strominger, C. Vafa, B. de Wit and G. Zwart for interesting

discussions and helpful comments. This research is partly supp orted by a Pionier Fel-

lowship of NWO,aFellowship of the Royal Dutch Academy of Sciences (K.N.A.W.), the

Packard Foundation and the A.P. Sloan Foundation. 26

Appendix

Our conventions are the following. We denote the SO(5) gamma-matrices as

m b

with m =1;:::;5 and a; b =1;:::;b. They satisfy

f ; g =2 ; (A.1)

m n mn

and

y T T

= ; = ! ! (A.2)

m m

with ! = ! the symplectic form that gives the isomorphism SO(5) Sp(4). The other

indep endent elements in the Cli ord algebra are of the form

= [ ; ]; = : (A.3)

mn m n mn

We can use the matrices 1; ; to expand a general matrix Z that satis es the reality

m mn

condition

Z = !Z; ! (A.4)

m mn

Z = a1 + b + c : (A.5)

m mn

Note that

y m mn

Z = a1 + b c ; (A.6)

m mn

so, if in addition Z is hermitian, it is a linear combination of the identity and .

In the text, we will use the form ! to lower and raise indices of these matrices.

Note that the matrices are nowantisymmetric. In particular we will use the notation

1 = ! .

ab ab

A matrix Z satisfying (A.4) forms a 16-dimensional Ma jorana-Weyl spinor represen-

tation of the group SO(5; 5). In fact the SO(5; 5) gamma matrices can be written as

= 1 i ; (A.7)

m 2

= 1 ; (A.8)

m 1

and satisfy

L L R R L R

f ; g =2 ; f ; g = 2 ; f ; g =0: (A.9)

mn mn

m n m n m n

(11)

In the chiral representation S with = 1 1 = 1, the generators of the

SO(5) SO(5) subgroup are given by 1; 1 whereas the o -diagonal generators

mn mn

are of the form .

m n 27

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