M-THEORY and N=2 STRINGS 1. Introduction

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M-THEORY AND N=2 STRINGS

EMIL MARTINEC

Enrico Fermi Institute and Dept. of Physics

5640 South El lis Ave., Chicago, IL 60637-1433

Abstract. N=2 heterotic strings may provide a windowinto the physics of

M-theory radically di erent than that found via the other sup ersymmetric

string theories. In addition to their sup ersymmetric structure, these strings

carry a four-dimensional self-dual structure, and app ear to be completely

integrable systems with a stringy density of states. These lectures give an

overview of N=2 heterotic strings, as well as a brief discussion of p ossible

applications of b oth ordinary and heterotic N=2 strings to D-branes and

matrix theory.

1. Intro duction

A few years ago, if asked to describ e string theory, the average practi-

tioner would have classi ed its di erent manifestations according to their

various worldsheet gauge principles. On the 1+1 dimensional worldsheet,

there can be (p; q ) sup ersymmetries that square to translations along the

(left,right)-handed light cone; one says that the worldsheet has (p; q ) gauged

sup ersymmetry. The b osonic string has no sup ersymmetry; p = q = 0. The

sup ersymmetric string theories have, say, q = 1. Thus typ e I IA/B string

theory has (1,1) sup ersymmetry. The typ e I/IA strings are the orbifold of

these byworldsheet parity, and the heterotic strings are in the class (0,1).

Remarkably, we now understand that all the sup ersymmetric string theo-

ries { typ e I IA/B, typ e I, and heterotic { app ear to describ e asymptotic

expansions of a single nonp erturbative master theory: M-theory. This the-

ory has many miraculous duality prop erties that are now only b eginning to

b e unravelled; other lecturers at this scho ol will review the current state of

a airs. In these lectures, I will giveanoverview of a relatively unexplored

corner of string theory, namely the N=2 strings [1, 2, 3] (more sp eci cally

strings with (2,2) or (2,1) gauged worldsheet sup ersymmetry).

2 EMIL MARTINEC

The driving force b ehind the recent uni cation has b een the recognition

of the fundamental imp ortance of the spacetime (as opp osed to worldsheet)

sup ersymmetry algebra. The small (BPS) representations of the sup ersym-

metry algebra form a quasi-top ological sector of the theory. By tracking

this BPS sp ectrum across mo duli space, one can deduce the interconnec-

tions of the various string limits. The issue we wish to address is the role

of the heterotic (2,1) string, which is also a stringy realization of spacetime

sup ersymmetry and therefore ought to play a role somehow. As we will see

[4, 5], many of the basic ob jects of M-theory are realized in the (2,1) string.

Self-duality and integrability are further features of the (2,1) string, arising

from the chiral sector with N=2 worldsheet sup ersymmetry.

We will see that the chiral critical dimensions of the (2,1) string are

d = 4 (2 space, 2 time) for the N=2 sector, and d = 12 (10 space, 2 time) for

the N=1 sector. To see the relation to M-theory, consider standard heterotic

target space geometry. Here this is a 2+2 dimensional base manifold for

the dimensions common to b oth chiralities, with the additional left-movers

parametrizing an eight-dimensional torus of stringy dimensions b ered over it (see gure 1).

T 8

M2,2

Figure 1. (2,1) heterotic geometry.

The geometrical elds are the graviton h, antisymmetric tensor eld

b, gravitino , and the gauge connection " on the b er. These elds are

further restricted by the extra constraints of N=2 lo cal sup ersymmetry on

the worldsheet, giving rise to prep otentials

h + b ! I @ a

a a

" ! I @ '

! I @ : (1)

M-Theory and N=2 Strings 3

Here I is an almost complex structure on the base space, which is required

by the lo cal N=2 worldsheet sup ersymmetry. Under linearized gauge trans-

formations, these restricted elds transform as

(h + b)=@ ! a

" = @ ! '

= @ ! : (2)

In other words, the remnant elds are Nambu-Goldstone elds of sp on-

taneously broken symmetries (spacetime antisymmetric tensor eld gauge

transformations, translations, sup ersymmetries) on a D-brane. More pre-

cisely, one has in a complex co ordinate basis " = i(@ @ )' +(@ +@) =

@(+i')+@( i'), where the (real) gauge symmetry is =.However,

one can go to a holomorphic basis by complexifying the gauge group G; then

' is a co ordinate on G =G whose dynamics is determined by holomorphic

gauge symmetry much as in the 2d WZW mo del [6, 7, 8]. This virtually

guarantees us a connection to brane physics, since brane dynamics is almost

by de nition given by the nonlinear Lagrangian of the spacetime symme-

tries broken by the brane. A picture of this asp ect of (2,1) string theory is

shown in gure 2. The (2,1) string worldsheet maps into the worldvolume

of a D-brane, which is itself emb edded in spacetime. The quanta of this

brane are the (2,1) strings themselves. Since the transverse uctuations of

the brane can b e traced to those of the b er connection " @', while the

longitudinal directions are those of the base space, we see that spacetime

itself is the total space of the heterotic geometry of gure 1.

Figure 2. Chain of brane embeddings implied by (2,1) string states.

The dimension of the brane is determined by the numb er of indep endent

ik x

momentum comp onents in string vertex op erators O e . When the (2,1)

2;2

string target space is IR T , this kinematics is 1+1 or 2+1 dimensional;

4 EMIL MARTINEC

when the spatial dimensions are further compacti ed, the kinematics is

more or less 9+1 dimensional { the target is a kind of ninebrane.

As mentioned ab ove, an additional geometric structure is self-duality.

The almost complex structure I is one of a triplet of such structures pre-

served by the target space geometry (these almost complex structures are

not integrable, since (@ @ )I = db = torsion). Thus one can bring to b ear

the machinery of twistor theory to characterize the classical solution space.

One of the characteristic prop erties of self-dual gravity is the symmetry

group of area preserving di eomorphisms SDi SU (1), suggesting a

connection to matrix theory [9].

There are indeed a number of intriguing analogies among the matrix

mo del of M-theory, (2,1) strings, and other matrix mo dels:

1. The (BFSS) matrix mo del of M-theory [9] realizes the membrane as a

collective phenomenon of D0-brane sup ergravitons, i.e. as a state in the

large N collective eld theory. In a sense, the BFSS matrix mo del gives

a map from a single noncommutative torus into `spacetime'. In the

graviton limit, the matrices approximately commute; in the membrane

limit, the commutators are large. Similarly, the (2,1) string describ es

a map from a single brane into spacetime.

2. Other examples where large N collective eld theory generates string

theory as an asymptotic expansion around a particular master eld

include:

(a) The 1+1d noncritical string based on the original matrix mo del

of [10].

(b) 2d Yang-Mills [11].

(d) 2+2d self-dual gravity [13],[1]. This connection will be outlined

in section 2 b elow.

3. (2,1) string p erturbation theory is an expansion around the in nite

tension limit; T g . In other words, it is an expansion in low

brane

str

energy relative to the brane tension. This is similar to 1+1d noncritical

string theory, where the e ective expansion parameter is the energy

of excitations relative to some scale (in that case the cosmological

constant of Liouville theory, or equivalently the fermi energy of the

matrix partons).

Thus one sees common themes cropping up in diverse settings.

The fo cus of these lectures will be the structure of N=2 strings, and

their p ossible relation to matrix dynamics. We b egin in section 2 with the

somewhat simpler (2,2) string, in order to intro duce the novel features of

lo cal N=2 worldsheet sup ersymmetry and its asso ciated self-duality struc-

ture. Included are an overview of the relation b etween the self-dual gravity

M-Theory and N=2 Strings 5

of the (2,2) string and SDi dynamics; as well as an aside on the D-brane

sp ectrum of (2,2) strings, which leads to yet another interesting connection

to matrix theory. Section 3 intro duces (2,1) strings { their worldsheet gauge

algebra, sp ectrum, and simple compacti cations. The target space e ective

action for (2,1) strings is derived in section 4; parallels with matrix theory

are explored in section 5.

2. (2,2) Strings

The gauged N=2 sup ersymmetry on the string worldsheet consists of cur-

rents T (the stress tensor), G (the two sup ercurrents), and J (a U(1)

R-symmetry current). Their algebra is schematically

G [T; G ]

[T; J ] J

[G ;G ] T + J (3)

[J;G ] G :

In the conformal gauge, the contributions to the Virasoro central charge

from the Faddeev-Pop ov ghosts is 26+211 2=6, so that the critical

dimension is d =2=3 c = 4. A free eld representation of the ab ove

crit matter

currents is

T = @X @X @

J = I ; =0; :::; 3

G = ( I ) @X = G G : (4)

N=1 2

Here, I is a self-dual tensor which acts as a complex structure. Preserv-

ing this structure under analytic continuation requires the signature to b e

Euclidean (4+0) or ultrahyp erb olic (2+2). Even though the inde nite sig-

nature case has more than one negative metric string co ordinate, there are

no negative metric states. Each gauge invariance removes two elds' worth

of degrees of freedom; in fact, the two b osonic (T; J ) and two fermionic

(G ) gauge symmetries kill al l the oscillator mo des of the string.

One of the powerful consequences of this fact is the triviality of the

string S-matrix. On the one hand, there can b e no Regge b ehavior in scat-

tering amplitudes, since the sequence of p oles in the S-matrix are asso ciated

with physical oscillator excitations of the string, of which there are none.

On the other hand, the covariant formalism generates a Koba-Nielsen (inte-

gral) representation for the S-matrix amplitude, which exhibits such p oles.

The tension between these two prop erties is resolved by the vanishing of

any amplitude b eyond the three-p oint function (the three-p oint function is

6 EMIL MARTINEC

protected b ecause it do es not involve Koba-Nielsen integrals). The single

nontrivial S-matrix elementis

hV(1)V (2)V (3)i =(k I k ) (k I k ) : (5)

1 2 1 2 r

Since the only center of mass of the string can be excited, it is p ossible

to nd a representative of the string vertex op erator which is a simple

exp onential

ik x

V = e (6)

gh gh

apart from ghost (measure) factors (see App endix A for a brief sum-

mary). A BRST-equivalent representative (or `picture') of the vertex rep-

resents it as an integral over (2,2) sup erspace,

Z Z

ik X ik X 2 2

D X )(k I D X )e ; e = d d j (k I V = d d

N=1 N=1 N=1

(7)

i.e. from the N=1 p oint of view the single physical state is a graviton

whose only physical p olarization is a uctuation in the Kahler p otential {

g = @ @ K in complex co ordinates.

ij j

Another imp ortant feature of the (2,2) string is the absence of tar-

get space sup ersymmetry. Consider a Wilson line of the U(1) R-current,

exp[i J ]. Parallel transp ort of a complex fermion around the p oint

z picks up a phase exp[i ]; however J is gauged, and so the phase cannot

havephysical signi cance. In particular, there is no physical distinction b e-

tween p erio dic (NS) and antip erio dic (R) b oundary conditions, hence no

target space fermions and no target space sup ersymmetry.

The e ective action which generates the S-matrix (5) is

1 1

S = d x[ @K @K + K@@K ^ @@K ] ; (8)

2 2 3

str

from which one obtains the equation of motion

I ^ @ @K + @@K ^ @@K =0 ; (9)

known as the Plebanski equation. This is the equation that governs the dy-

namics of self-dual gravity. It is a straighforward exercise in 2+2 kinematics

to show that the quartic S-matrix computed from the action (8) vanishes

[1].

An alternate route to the Plebanski equation pro ceeds from the gener-

alized b eta-function of the background sigma mo del for string propagation.

Similar to the 1+1d noncritical string, where the `bulk' S-matrix is trivial, the e ec-

tive action is cubic (see b elow).

M-Theory and N=2 Strings 7

In (2,2) sup ersymmetry, there are two kinds of scalar sup er elds: chiral

and twisted chiral [14]. The simplest situation has the sigma mo del back-

ground describ ed entirely in terms of chiral sup er elds; then the target

space holonomy lies in U(d). The sigma mo del b eta function equations are

the conditions for SU(d) holonomy

R [] = 2r r (10)

H is the connection with torsion H ). These equations ( = f g

maybeintegrated in complex co ordinates to yield

log det[g ] = 2 + f (x)+f(x) : (11)

A further analysis of the conditions for (2,2) sup ersymmetry in the sigma

mo del coupled to worldsheet gravity [15] shows that the dilaton is lo-

cally the real part of a holomorphic function; thus local ly one can cho ose

co ordinates such that the dynamical equation is

det[g ]=1 : (12)

Some solutions of these equations have b een describ ed in [15]. The sin-

gle physical degree of freedom of the N=2 string is the center of mass

mo de K (x; x ) describing uctuations of the Kahler p otential g = I +

ij ij

@ @ K , where I is the background Kahler form. Expanding (12) in this way

yields the Plebanski equation (9). Note that although K contains the eld-

theoretic degrees of freedom of the string, there may b e additional mo duli

in the global mo des of the metric, antisymmetric tensor, and dilaton { for

instance the action (8) on K3 dep ends implicitly on the full 80 mo duli of

the conformal eld theory, as well as the string coupling = e . These

are analogues of the sp ecial states of 2d noncritical string theory [16] that

exist only for discrete momenta.

Backgrounds involving one chiral and one twisted chiral sup er eld must

havetwo commuting complex structures [14] and are thus essentially trivial

[17] { the quasi-Kahler p otential K describing the background geometry

satis es a free eld equation. When spacetime has a translational Killing

vector eld with compact orbits, these backgrounds are related to the ab ove

self-dual geometry by T-duality [14, 15]. Thus one might call the theory

discrib ed by (8) the N=2B theory, and the theory describ ed by the free

quasi-Kahler p otential K the N=2A string.

There are also interesting solutions which fall outside the class describ ed

by constant dilaton, for example the NS instanton [18],[15]

2 2

e = e +

H = r

G = e ; (13)

8 EMIL MARTINEC

whichisthe counterpart of the NS ve-brane in the usual 10d theory (an

instanton is the magnetic dual to a string in 4d). In particular, this ob ject

is not that same as the D-instanton encountered b elow.

The Ricci- at four dimensional geometry required by the N=2 string

admits a hyp erKahler structure, and the metric is self-dual. It is easy to see

that the N=2 lo cal sup ersymmetry generates N=4 global sup ersymmetry

in the critical dimension d = 4. The canonical normalization of the U(1)

crit

R-current

(14) J (z )J (w )

(z w )

implies that the exp onential elds J (z )=exp[i 2 J ], together with

J (z ), form an SU(2) current algebra; furthermore, J G (z w ) G are

two additional sup ersymmetry currents. These enlarge the N=2 currents (3)

to the N=4 current multiplet. Since any solution to the background eld

equations of motion preserves an SU(2) symmetry in the tangent space

(in Euclidean signature; SL(2,R) in ultrahyp erb olic signature), the target

space is hyp erKahler. There is a two-sphere's worth of choices of which U(1)

inside this global SU(2) is the gauged lo cal R-symmetry.Parametrizing this

CP = SU (2)=U (1) of self-dual geometry. choice is the twistor variable 2

For more on the relation of twistor geometry to the (2,2) string, see for

instance [1].

This self-duality structure leads to an in nite number of conservation

laws, and it is likely that this integrability underlies the triviality of the

S-matrix seen ab ove. Another way to exhibit these conservation laws re-

veals a connection to the area-preserving di eomorphism group SDi [13].

Consider the zero-curvature condition in two dimensions

[@ + A ;@ + A ]=0 ; (15)

t t x x

where is an arbitrary (sp ectral) parameter, and A ( =1;2) is an SDi -

valued connection. One can represent SDi as the algebra of canonical

transformations of a one-dimensional phase space parametrized by co ordi-

nates (p; q ). Then (15) at order implies

A (x; t; p; q ) = (@ @ )@ +(@ @ )@

t t p q t q p

A (x; t; p; q ) = (@ @ )@ +(@ @ )@ ;

x x p q x q p

(16)

the o( ) equations then read (in an appropriate choice of co ordinates)

(@ @ )(@ @ ) (@ @ )(@ @ )=1 ; (17)

t p x q t q x p

i.e. the Plebanski equation. The symmetry group of (15) is the lo op group

of SDi ; the self-dual gravity equations are essentially the same as the 2

M-Theory and N=2 Strings 9

two-dimensional chiral mo del equations for the group SDi , where the

four-dimensional spacetime is parametrized by(x; t; p; q ).

2.1. OPEN STRINGS, D-BRANES, AND MATRIX THEORY

The connection of the target space dynamics of the (2,2) string to SDi

leads to an intriguing toy version of the circle of ideas underlying ma-

trix theory. N=2 strings have very little dynamics; the N=(2,2) string has

no physical transverse excitations. The center-of-mass mo de of the closed

string describ es uctuations of the Kahler p otential of the self-dual metric;

the corresp onding mo de of the op en string describ es uctuations of a self-

dual Yang-Mills eld. The op en string S-matrix is again essentially trivial,

vanishing b eyond the three-p oint function. In sucha simpli ed dynamical

setting, one might exp ect to b e able to make exact statements ab out N=2

string D-brane dynamics as well, p erhaps even nonp erturbatively. The D-

branes should strongly a ect the strong-coupling b ehavior of the theory,

since they b ecome light in this regime.

In a sense, in N=2 strings self-duality plays a role similar to the BPS

condition. There is no BPS structure per se, due to the absence of space-

time sup ersymmetry; the almost top ological nature of the dynamics makes

up for this, however. This is b ecause spacetime sup ersymmetry of the usual

sup erstring is intimately related to the sp ectral ow that analytically re-

lates the NS and R sectors in worldsheet N=2 sup ersymmetry [19]. In the

N=2 string, this sp ectral ow is gauged, so in a sense one has only the

`BPS' sector . The sp ectral ow also generates the global SU(2) symme-

try whose preservation is tantamount to the hyp erKahler condition on the

target space, and hence self-duality.

The fascinating prop osal of matrix theory [9] casts the full light-cone

gauge dynamics of M-theory in terms of the quantum mechanics of the BPS

sector of zero-branes. Only the unexcited op en strings stretching between

the zero-branes are kept, yet one recaptures a remarkable amount of the

complete theory in this way. One might hop e that N=2 string D-branes

serve as a toy mo del of this dynamics, since N=2 lo cal sup ersymmetry

p ermits only such states in the physical state space of op en strings { all

op en string oscillations are unphysical.

App endix B is devoted to a cursory explanation of the op en string

sector of (2,2) strings. The upshot is that there are D-branes, just as in the

(1,1) and (0,0) strings. The latter lead to conventional (sup er)Yang-Mills

dynamics on the brane, dimensionally reduced from 10 or 26 dimensions.

Thus it is not surprising that, since the (2,2) op en string sector describ es

More precisely, the sort of BPS states that preserve the self-dual sup ercharge, i.e.

Q and not Q

10 EMIL MARTINEC

four dimensional self-dual gauge theory, the D-branes are governed by the

dimensional reduction of the self-duality equation:

worldvolume integrable

equation

dimension system

4 SDYM F + F =0

3 Bogomolny F + D =0

ij k k

2 Hitchin F +[X; X ]=0

i i j k

1 Nahm D X + [X ;X ]=0

0 ADHM [X ;X ]+ [X ;X ]=0

Wehave denoted the branes by their total worldvolume dimensions rather

than the space dimension, since that notation is a bit ambiguous in signa-

ture 2+2. Curiously, for worldvolume dimension equal to two (D-strings),

the equation is very similar to (15). In fact one can approximate matrix

theory rather closely, in the following sense. Consider the limit of N co-

incident D-strings, so that the [X; X ] term in the Hitchin equation drops

out. then the equation governing the D-string is a zero-curvature condi-

tion for its two-dimensional gauge eld. In the limit N !1, one has the

equations of motion of the two dimensional chiral mo del with gauge group

SU(1) SDi ; this can be shown to be equivalent to the self-duality

equations (c.f. the last of refs. [13]). Thus one has a situation where one

can recover the low-energy theory corresp onding to the N=2 string from the

large-N limit of one of its D-branes. The principal di erence is the lackofa

connection to the in nite momentum frame, and the corresp onding app ear-

ance of an extra dimension at strong coupling. Also, the transverse space to

the D-branes do es not app ear to b e related to the extra dimensions arising

from the matrix phase space; one starts from a two-dimensional ob ject in

a four-dimensional space whose transverse p osition is nondynamical, and

grows a di erenttwo dimensions by taking the large N limit. It may b e that

the appropriate starting p oint is a six-dimensional theory [20]. Perhaps the

(0+2)-brane is close to an analogue of the D-particle in the construction of

[9] since it spans b oth time directions of 2+2 spacetime. It would indeed

be intriguing if one could realize self-dual gravity in terms of D-brane `con-

stituents'; p erhaps this would provide an interpretation of the `entropy' of

the nuts and b olts of compacti ed self-dual solutions [21].

Indep endent of matrix theoretic ideas, the D-branes are the light ob jects

of the theory at strong N=2 string coupling; they, together with the NS

instanton (13), should dominate the nonp erturbative b ehavior of self-dual

gravity.

M-Theory and N=2 Strings 11

The N=2 string D-brane system is also a convenient lab oratory for the

exploration of the Nahm transform; T-duality relates the di erent branes

up on, for instance, compacti cation on T (mirror symmetry plays an anal-

ogous role on K3). Another intriguing p oint is that the mo duli space of k

instantons in SU(N ) gauge theory has b een prop osed as the b osonic part of

the con guration space of matrix noncritical strings [22]. N=2 strings gen-

erate exactly this mo duli space as the space of physical deformations, with

the D-instantons regulating the singularity at zero instanton scale size; and

manifestly incorp orate T-duality.Thus there might b e a close relationship

to the dynamics of this so-far mysterious 6d noncritical string theory to

quantum mechanics on the mo duli space of N=2 strings.

3. (2,1) Strings

Heterotic (2,1) strings [2] combine the self-dual, integrable structure of

(2,2) strings with the spacetime sup ersymmetry present in strings with

N=1 gauged worldsheet sup ersymmetry. A free eld representation for the

N=1 currents is

1 1

@x @x @ @y @y @ T =

2 2

G = @x + @y : (18)

a a 3

Here y , a = 1; :::; 8 is a chiral b oson and a Ma jorana-Weyl fermion.

Using these currents as gauge constraints in conformal gauge is not sucient

to remove b oth of the two timelike oscillator mo de towers of x and ; one

needs another b osonic and fermionic gauge current. The simplest choice [2]

is a U(1) sup ercurrent

J = v @x + v @y

int

= v + v : (19)

int

In order to gauge this current, it must b e anomaly free, which forces v =

2 2

(v; v )tobeanull vector: v + v = 0. The ghosts for J , have c=3;

int

therefore, they increase the critical dimension bytwo, so d = 12 = 10+2.

crit

One app ears to have a `sup erstring in d=12', but not really; the additional

gauge constraints pro ject momenta and p olarizations to be orthogonal to

2;2

v . If x 2 IR , then the (y; ) system must form a c = 12 holomorphic

sup erconformal eld theory (just as the internal sector of the (1,0) string

9;1

in IR must form a c = 16 conformal eld theory). The unique choice

Our conventions are as follows: the N=1 chiral sector will b e the left-moving oscilla-

tions; the N=2 chiral sector will b e right-movers. The indices are =0;1;2;3 for x 2

2;2 a 8

IR ; a =1; :::; 8 for y 2 T ; often we will use a combined index M =(; a)=0; :::; 11

when dealing with purely left-moving quantities.

12 EMIL MARTINEC

preserving spacetime sup ersymmetry, mo dular invariance, etc., is for y to

live on the Cartan torus of E .

Massless vertex op erators are BRST equivalentto

(N=2)

grav

(N =1) ik x

V = ( )( (k) )e

(N=2)

g aug e

(N =1) a ik x

V = ( )( (k) )e

(N =2)

(N =1) ik x

V = ( )( u (k )S )e : (20)

As in the (2,2) case, an equivalent representative of the graviton is

Z Z

ik X a 2

e = d d j I k (D X DX )exp[ik X +ik Y ] ; V = d d

N=1 N=1 a

(21)

so that the graviton/antisymmetric tensor has a sp ecial p olarization h +

b I @ , verifying the claim (1) of the intro duction. Similarly, the

a a

gauge eld " I @ ' and gravitino I @ . If the target space is

2;2

IR T , only the states (20) satisfy level matching

int

L = k k =0

L = k k +N =0 (22)

0 M

In this case the only physical states are at the massless level, N = 0,

much like the (2,2) string; with no momentum in the internal T directions,

k =0.

The Virasoro/null current constraints imp ose restrictions on the p olar-

izations and momenta:

Virasoro k =0 + k NS

ku= =0 R

Null current v =0 + v NS

v= u =0 R.

These constraints reduce the 12 p olarizations of the NS vector to 8 trans-

verse, and the 32 comp onents of the Ma jorana-Weyl R sector spinor to 8

physical states, as exp ected. The spacetime interpretation of these physical

states dep ends on the sp eci c orientation of the null vector v , see gure 3.

M-Theory and N=2 Strings 13

8 TT8

v t vt time time vs space space RR2,2 2,2

(a) (b)

Figure 3. Choices of nul l constraint on the left-movers.

2;2

In gure 3a, the null vector is oriented entirely within the IR base

space. The kinematics consists of 1+1 dimensional momenta k (recall that

the level matching constraint eliminates any momentum comp onents in the

E directions), with the physical p olarizations consisting of a 1+1 gauge

eld a , 8 scalars ' , and 8 fermions . The sp ectrum is that of the typ e

I IB D-string! On the other hand, if the null vector has its time comp onent

2;2

in IR and its space comp onentinT (see gure 3b), then the kinematics is

2+1 dimensional, and the physical p olarizations are the 2+1 gauge p otential

a , 7 scalars ' , and 8 fermions . The sp ectrum in this case is that of

the typ e I IA D2-brane.

One always has sixteen sup ersymmetries

(N =1)

S ; v= Q =0 ; (23) Q =

g h;R

just as a typ e I I D-brane breaking half the sup ersymmetries. The algebra of

these sup ercharges is (on-shell, since all considerations are mo dulo BRST

equivalence of the (2,1) string)

fQ ;Q g =( ) P v : (24)

M N

If our interpretation is correct, one would exp ect there to b e an additional

sixteen sup ersymmetries which are sp ontaneously broken by the brane.

These sup ersymmetries would be nonlinearly realized in the worldvolume

theory, and hence not visible in the single-string Hilb ert space. One might

see them in a careful study of the vertex algebra. In any event, the (2,1)

string would app ear to describ e D1- or D2-branes in static gauge, stretched

14 EMIL MARTINEC

2;2

across the noncompact spatial directions of IR . The transverse directions

app ear to b e compacti ed on a torus, probably the E Cartan torus.

3.1. TOROIDAL COMPACTIFICATION OF (2,1) STRINGS

Qualitatively new features arise up on further compacti cation of the (2,1)

string, since the level-matching constraints (22) prove to be much less

restrictive. In fact, the physical sp ectrum consists of a stringy tower of

`Dabholkar-Harvey' states { ground states on the right-moving N=2 side,

oscillator excitation at arbitrary level on the left-moving N=1 side, with

momentum and winding to comp ensate. Note that since it is the world-

sheet chirality that carries spacetime sup ersymmetry that must b e excited,

states with N 6= 0 break target space sup ersymmetry; i.e. these states are

not BPS-saturated.

2 3

Consider then the further compacti cation of the spatial x ;x co ordi-

2;2 2

2 8

nates of IR , so that the target is IR (time) T T . More generally,

int

one could consider an arbitrary spatial torus corresp onding to a p oint in

10;2

the Narain mo duli space N , but a pro duct torus will suce for illustra-

tive purp oses. The (2,1) string will in general have b oth momentum and

winding

i i

m R n

; i =2;3 : (25) p =

`;r

R 2

Reexamining the level-matching constraints

2 2 2 2

0 = p p + p + p

0 1 2;r 3;r

2 2 2 2 2

0 = p p + p + p + p~ +2N ; (26)

0 1 2;` 3;`

one sees that many more states are nowavailable { one need only satisfy the

mass shell condition, the level-matching constraint, and the null constraint

(as well as the highest weight condition under the current algebra):

i 2 2 2

1 1

m R ) + p~ +N 0 = p p + +(

0 1

2 2

i i 2

0 = 2m n +~p +2N

0 = v p : (27)

Note that the constraints now allow the internal momenta p~ 8 to b e arbi-

trarily excited, so that the kinematics is more or less ten dimensional. These

states are similar to the p erturbative BPS states of the typ e I I sup erstring;

for instance, the level density grows exp onentially. Even with this change

of circumstances, one can check that the four-p oint function continues to

M-Theory and N=2 Strings 15

vanish, indicating that the trivialityof the S-matrix continues to hold. T-

duality of the (2,1) string seems to imply that there is a minimum `size' to

(2;1)

the target circle(s) over which the target D-brane is stretched: R ` .

min

str

3.2. HETEROTIC/TYPE I CONSTRUCTION

It is now well-understo o d that the heterotic and typ e IA string theories

are particular asymptotic limits in the mo duli space of Z orientifolds of

M-theory, see gure 4.

E8 x E8 heterotic 11d sugra on R/Z_2

R11

SO(32) heterotic/type I type IA

1 1

Figure 4. Moduli space of S S =Z compacti cations of M-theory.

The simplest situation has 16 ninebranes at each of the two orientifold

planes to cancel anomalies; the low-energy dynamics consists of 11d sup er-

gravity in the bulk, and 10d SYM on these walls. Since the (2,1) string

seems only to describ e a brane in spacetime, to realize these vacua as a

(2,1) string background one might lo ok for a wrapp ed membrane/string that sees this structure, see gure 5.

x_perp // x_2

x_3 //

x_3=0 x_3= π R_3

Figure 5. The heterotic membrane geometry generated by the (2,1) string.

16 EMIL MARTINEC

In other words, one lo oks to describ e a cylindrical, op en membrane

stretched b etween the orientifold planes x =0;R . Its b oundaries should

3 3

have 1+1 dimensional elds describing rank eight current algebra, such that

one obtains a heterotic string when R R ; on the other hand, when

2 3

R R , these elds will be frozen, leaving the nite number of Chan-

2 3

Paton lab els of typ e IA theory. The orientifold symmetry acts as x !x ,

3 3

A ! A . The sign ip of the three-form gauge eld indicates

MNP MNP

orientation reversal of the M-theory membrane worldvolume. Since one ap-

p ears to be in static gauge, the orbifold x !x of the (2,1) string will

3 3

accomplish b oth the orbifold of spacetime and the orientation reversal of

the brane worldvolume. One simply wants to nd a consistent sup ersym-

metric orbifold of the (2,1) string with this op eration as part of the orbifold

group. A straightforward chain of logic leads to a unique answer satisfying

the orbifold level-matching constraints, preserving half the sixteen space-

time sup ersymmetries, and treating all the internal co ordinates y on the

same fo oting. Right-moving worldsheet sup ersymmetry is preserved only

if, in addition to x !x , one simultaneously ips ! .Now con-

3 3 3 3

0 3 1 2

sider the right-moving U(1) R-current, e.g. J = + ; there are two

choices for the Z to have a well-de ned action on the gauge algebra: (1)

! , which one can show do es not lead to a spacetime sup ersym-

0 0

metric solution; and (2) ! , which gives the twisted N=2 algebra

1 1

[23]

J !J ; G $G : (28)

Pro ceeding along these lines, one nds a unique Z satisfying the ab ove

requirements:

(x ;x ;y ; ::; y ) !(x ;x ;y ; ::; y )

1 3 1 8 1 3 1 8

( ; ; ; :::; )!( ; ; ; :::; )

1 3 1 8 1 3 1 8

( ; ) !( ; ) ,

1 3 1 3

with all other co ordinates invariant.

Vertex op erators describ e physical uctuations of the target brane, and

have the form O cos[k x ] if the p olarization op erator O is Z even, and

3 2

O sin[k x ]ifO is Z o dd. At the massless level, one nds Neumann b ound-

3 2

ary conditions (cos [k x ]) for p olarizations along (0; 2; 4; :::; 11), and Dirich-

let b oundary conditions (sin[k x ]) for p olarizations along (1; 3). Thus, in

space there are nine Neumann and one Dirichlet b oundary condition; in

time, one Dirichlet and one Neumann. Since we wish to avoid thinking

ab out what a Dirichlet b oundary condition means in physical time, we

place the time comp onent of the vector v (de ning the null pro jection J )

in the x direction; thus its space comp onent must also be one of the Z

1 2

twisted directions (2; 4; :::; 11). Cho osing the latter in one of the internal di-

rections, one nds now an op en membrane stretched b etween eight-branes

M-Theory and N=2 Strings 17

in a typ e I IA description; that is, when the target (x ;x ) torus orbifold is a

2 3

large cylinder, in the low-energy limit one has a 2+1 target space gauge eld

coupled to 7 scalars with Neumann b oundary conditions. This is precisely

the bulk dynamics of an op en membrane stretched b etween D8-branes. One

can lift to this e ective theory to eleven dimensions by dualizing the vec-

tor to another scalar (which one easily checks also has Neumann b oundary

conditions) to nd an op en membrane stretched between orientifold nine-

branes.

The b oundary dynamics comes from the twisted sectors of the (2,1)

string orbifold, as these states are pinned to the xed p oints x =0;R

3 3

which are the orientifold planes. Since the orbifold is an asymmetric one, the

numb er of states at each of the two xed p oints x =0;R is the square

3 3

2 = 16. ro ot of the numb er of xed p oints of the internal T =Z orbifold,

Consistency of the op erator algebra shows that these states app ear in the

Ramond sector, and an analysis of the BRST constraints shows that they

are chiral in the target space. Thus there are sixteen massless fermion elds

living on each b oundary of the op en target membrane, which describ e a rank

eight current algebra, and are inert under spacetime sup ersymmetry. Half

the sup ersymmetry charges (23) are broken by the orbifold. Note that this

is exactly the sp ectrum found on the D8-brane b oundaries of the matrix

theory description of the heterotic string, a fact that will b e imp ortant for

us b elow.

There is a very similar orbifold [5], also breaking half the sup ersymme-

try, obtained if one relaxes the condition that all internal co ordinates are

a a

treated identically. Twisting half rather than all the y ,

(x ;x ;y ; ::; y ) !(x ;x ;y ; ::; y )

1 3 1 4 1 3 1 4

( ; ; ; :::; )!( ; ; ; :::; )

1 3 1 4 1 3 1 4

( ; ) !( ; ) ,

1 3 1 3

is also a consistent asymmetric orbifold. The twisted sector ground states

transform as a hyp ermultiplet under the eight remaining sup ersymmetries.

Finally, it should b e noted that wehave only describ ed the low-energy

sp ectrum; there is again a stringy tower of left-moving states just as in the

untwisted, toroidally compacti ed (2,1) string.

4. S-matrix, e ective action and geometry of (2,1) strings

The tree-level, three-p oint S-matrix factorizes b etween left- and right-movers,

hence we can immediately deduce the answer by combining (2,2) string

result (5) with that of the sup erstring; for three massless metric p erturba-

tions, the result is

h V (1)V (2)V (3)i =(k I k ) ( k ) + cyclic ; (29)

1 3 r 1 2 1 3

18 EMIL MARTINEC

leading to the cubic e ective Lagrangian

(3)

1 1

F F F F (F F )(F F ) (30) L =

2 8

with F = @ . Curiously, this is a term in the expansion of the Born-

( )

Infeld Lagrangian around the background F = I . One can incorp orate

the e ects of the internal scalars ' simply by letting the p olarization indices

in (30) run over 10d (or 12d, since we are imp osing the null constraints by

hand), while keeping the kinematics 1+1 or 2+1 dimensional. Regarding

the fermions, their S-matrix is

hV (1)V (2)V (3) + V (1)V (2)V (3)i =(k I k ) [u ( + )u ] ;

' 1 3 r 1 a 3

(31)

where are twelve dimensional gamma matrices. One obtains the e ec-

tive Lagrangian

(3)

a a

L =( @ )I F +( @ )I @ ' : (32)

2;2

4 8

These are the only nonzero S-matrices in IR T , but this do es not mean

that we have found the full e ective action. It happ ens that the iteration

of these cubic eld-theoretic vertices generates a four-p oint S-matrix, such

2;2

that kinematics in IR allows the p ole terms to cancel among s, t, and u

channels, leaving an explicit four-p oint contact term. For the full S-matrix

to vanish, one must add a cancelling quartic term in the e ective action.

Continued iteration yields terms to all orders in small uctuations. This

situation contrasts with the (2,2) string, where the symmetry is powerful

enough to cause the p otential four-p oint contact term to vanish, and allows

the e ective action to cubic order to in fact b e the exact theory.

The simplest way to extract the full answer is to realize that the iteration

n n th

of the cubic terms will generate contact terms of the form k () at n

order in uctuations, while uctuations in the geometrical elds h; b; " are

all of the form @ () due to the (2,1) sup ersymmetry constraints. Hence

the e ective action found from the b eta function will come entirely from

0 2

one lo op (higher lo ops will come with extra factors of k , and hence more

powers of momenta than small uctuations). The b eta function equations

(10) can b e integrated once to give

I =0 (33)

with the e ective action

grav

d x det [ + F ] ; (34) S =

ij ij

str

Hence the analogy with the 1+1d noncritical string made in the intro duction, wherein the `bulk' S-matrix is also trivial.

M-Theory and N=2 Strings 19

at least for the gravity sector. The gauge elds may b e added via an analysis

of sigma mo del anomalies [24],[8]; the result is

4 0 a a

d x det S = [ + F + @ ' @ ' ] : (35)

ij ij j

str

It has b een checked [24] that this action repro duces all cubic and induced

quartic vertices involving only b osons, and that the S-matrix vanishes at

quartic order. The action is remarkably similar to the static gauge e ective

action for D-branes. There are two signi cant di erences, however: (1) For

2;2

8 0 2

target space IR T , it is exact to all orders in = ` , presumably

str

related to the fact that there are no oscillator excitations of the underly-

ing (2,1) string; and (2) the dynamics is integrable. The vanishing of the

S-matrix in the fully compacti ed case indicates that this integrability con-

tinues to hold even when the kinematics is essentially ten-dimensional. It

would b e very interesting to work out the generalization of (35) in this case.

It would also b e interesting to explore the p ossibility of nontrivial solutions

along the lines of (13).

The terms in the e ective action involving fermions should in principle

b e determined by sup ersymmetry.However, it is dicult to compute quan-

tities involving nontrivial Ramond vertex backgrounds (there is no sigma-

mo del approach). A b etter understanding of the spacetime sup ersymmetry

current algebra and its anomaly structure would be helpful, since it was

essentially this structure which enabled us to determine the dep endence of

the e ective action on the scalars ' . As mentioned in the intro duction, the

physical elds on the target brane are all Nambu-Goldstone mo des of the

spacetime symmetries sp ontaneously broken by the brane, and therefore the

e ective action is exp ected to b e largely determined by the various broken

and unbroken symmetries. Because the b osonic structure is so similar to

the Dirac-Born-Infeld/Nambu-Goto action, it is natural to guess that the

full answer including fermions has a structure similar to that of Dp-branes

in static gauge

a a (p) p+1

+ F + @ @ (36) S = d det

a a a a

2 ( + @ )@ +( @ )( @ ) :

The sorts of terms app earing in the expansion of this action are compatible

with with the cubic S-matrix (32), but the full structure is far from under-

stood. This fermionic completion of of the action (35) represents a rather

nontrivial coupling of SDYM to self-dual gravity with torsion, with an ad-

ditional fermionic symmetry whose geometry ought to b e quite intriguing.

20 EMIL MARTINEC

5. Connections between (2,1) strings and matrix theory

In this nal section of these lectures, I would like to present some evidence

for the idea that (2,1) strings are closely related to the formulation of M-

theory on T as a matrix mo del.

A striking feature of sup ergravity U-duality is the app earance of ex-

ceptional groups in less than six noncompact spacetime dimensions. Naive

extrap olation leads to a duality group E (Z) in two dimensions [25].

9(9)

However, the solutions to low-energy sup ergravity which exhibit the vari-

ous BPS charges p ermuted under duality, b ecome more and more singular

the lower the dimension. For instance, in 2+1 dimensions, n fundamen-

tal strings wound around a given cycle of the internal T has a dilaton

background of the form [26]

e = e 8G ` ln j~r ~r j ; (37)

N i

str

i=1

one can intro duce an arbitrary number of such sources, but the dynamics

b ecomes strongly coupled near the core of each one. Presumably the cor-

rect solution receives mo di cations due to the nonp erturbative states that

b ecome light there (c.f. [27]). This is supp orted by the observation that, in

the IIB theory, one can wrap a D7-brane around the T ; these are in the

same U-dualitymultiplet as the fundamental string, yet they have a rather

di erent classical solution { the core is nonsingular (from the viewp ointof

F-theory), but one cannot intro duce more than 24 sources. In 1+1 dimen-

sions, the situation is even worse [28]; any source generates a resp onse in

the dilaton of the form

Z Z

(x )

e = e T ; (38)

p ositive stress-energy of the source forces a singularity in the dilaton at

nite x , so again information of a more nonp erturbative nature is needed.

Finally, if one considers the velo city-dep endent forces b etween these ob jects

in low dimensions, eventually they b ecome con ning even b etween the BPS

states. For instance, in matrix theory the velo city dep endent force b etween

10d;1

d 4 7d

T two matrix partons on IR is v =r , b ecoming con ning b elow

3+1 dimensions.

It is not clear what to make of these observations. Two p ossibilities are

Sup ergravity makes sense as a low-energy theory, but that the branes

that have b een our guide to de ning matrix theory do not make sense

M-Theory and N=2 Strings 21

{ or at least cannot be present in the arbitrary numb ers needed to

de ne a large-N continuum limit.

Matrix theory exists in all dimensions; however, the con ning p otential

between matrix partons means that there will b e no mo duli space for

the gauge dynamics de ning matrix theory in low dimensions, therefore

no low-energy approximation that one could call spacetime, therefore

no sup ergravity.

The connection b etween (2,1) strings and brane dynamics cries out for some

connection to M-theory. Could (2,1) strings describ e the `low-dimensional

phase' of matrix M-theory conjectured ab ove? There are a numb er of rea-

sons to think so:

(2;1)

1. At least eight co ordinates are compact on the scale ` .

str

2. Massless physical states of the fully compacti ed (2,1) string have in-

teractions similar to what one would exp ect from SYM with gauge

group SDi .

3. The Z orbifold that gives the heterotic string acts as exp ected in

matrix theory [29].

4. In the fully compacti ed (2,1) string, the decompacti cation limits are

analogous to the limits which yield matrix I IA/B strings.

Consider again the vertex op erators of the fully compacti ed (2,1) string

V = (ghosts)( (p ) ) exp[ip x + ip x ] A (x ;x )

NS M r r M r

` ` ` `

V = (ghosts)( u (p )S ) exp[ip x + ip x ] (x ;x ) : (39)

R A r r A r

` ` ` `

The three-p oint function of, for instance, the b osons (29) generates the

cubic coupling

(3) M N

L = g @ A fA ; A g ; (40)

str M N

where fF; Gg = (@ F =@ x )I (@ G=@ x ) is a Poisson bracket induced by

r r

the complex structure of the right-moving N=2 sup ersymmetry. This cu-

bic vertex has the same structure as Yang-Mills with a gauge group of

symplectic di eomorphisms. In this interpretation, we wish to regard x as

transverse `spacetime' co ordinates, and x as phase space co ordinates on

the Lie algebra of SDi a la matrix theory. In fact, the left- and right-

moving degrees of freedom are not indep endent; they are coupled through

10;2

the level-matching constraints (26). Moreover, the lattice of momenta

10 2

is generically not factorized as . Nevertheless, there may b e a sense

in which a structure of SDi gauge theory is present o -shell [30]. It is

Also the branes have a strong back-reaction on spacetime, p erhaps precluding the

existence of the null Killing vector needed to de ne the light cone gauge.

6 9

In matrix theory on T , the theory cannot quite b e SYM even at low energies, due to anomalies.

22 EMIL MARTINEC

p ossible that this gluing of gauge and spacetime uctuations is an artifact

of the expansion ab out a particular classical solution, much as a monop ole

ties rotations in space and isospin.

To see the action of SDi , consider the right-moving chiral part of the

vertex op erators, taking sets which are mutually lo cal in their OPE's; this

means

p ; i =1;2;3 : p = p =0 : (41)

i i

i=1

The generic construction of such sets is as follows: Taketwo orthogonal null

2 2

vectors n ;n (i.e. n = n = n n =0, and n 6= n ). This is

(1) (2) (1) (2) (1) (2)

(1) (2)

p ossible in signature 2+2. The n span a self-dual nul l plane N .Now fo cus

(a)

on momenta p = an + bn which are arbitrary linear combinations of the

(1) (2)

H R

two basis vectors. The chiral vertex op erators J = dz d exp[ip x ]

p r

satisfy the algebra

[J ;J ]=pI qJ (42)

p q p+q

mo dulo BRST equivalence. Momenta p 62 N do not have mutually lo cal

OPE's, and hence no well-de ned algebra structure. These on-shell p 2N

are completely determined by their spatial comp onents; the null condition

forces the length of the spacelike part to equal that of the timelike part,

so that the triple (p; q ; p q ) form congruent triangles in the temp oral

x -x and spatial x -x planes (an overall relative orientation of these two

0 1 2 3

triangles is determined by the choice of N ). The upshot is that the algebra

(42) is an algebra of two-dimensional and not four-dimensional symplectic

di eomorphisms, i.e. SDi SU(1).

Thus the structure of the fully compacti ed (2,1) string strongly resem-

bles that of matrix theory on T , although there are profound di erences

due to the constraints relating left- and right-moving co ordinates (x ;x ).

Further supp ort for this idea is provided by the fact that the decompacti -

cation limit of the (2,1) string is describ ed by a D-brane style Lagrangian for

either strings or membranes (dep ending on the orientation of the null vector

2;2

v ); in fact it is precisely the D-string action if the null vector v 2 IR . D-

string dynamics is precisely what one nds in matrix theory if one shrinks

a compacti ed circle to a radius R ` ; then the dual circle on which

the SYM dynamics takes place lives on a circle of radius ` =R R ,so

that the IR dynamics b ecomes 1+1d SYM in the limit [22].

This choice is not unique; in fact the set of self-dual null planes is parametrized by

a co ordinate 2 IRP which is the twistor parameter.

The left-right constraints are most powerful here, since they restrict all momenta

to p = p .Itwould seem then that this limit describ es only a single D-string and not

N !1 of them. It may b e that this freezing of U(N) down to U(1) is an artifact of the

string p erturbation expansion, which takes place ab out a particular classical solution;

p erhaps this symmetry is broken in the particular vacuum seen by the (2,1) string.

M-Theory and N=2 Strings 23

The heterotic/typ e I construction in section 3.2 follows precisely the

pattern one would exp ect of matrix theory on T . In matrix theory, the

d d 1

~ ~

heterotic/typ e I theory on T arises from a SYM orientifold on T S ;

the orientifold Z acts simultaneously on the torus and acts as the involu-

tion that sends SU (N ) ! SO(N ). One must also add by hand 32 fermionic

matter multiplets in order to cancel gauge anomalies; these represent the

couplings of the zero-brane/sup ergravitons to the nine-branes of the back-

ground spacetime. All of this structure has a direct parallel in the (2,1)

string. The orbifold group of section 3.2 acts on the left-moving `spacetime'

co ordinates x by re ecting nine spatial co ordinates (of which one is re-

moved by null pro jection) while leaving one untouched { in other words,

8 1

~ ~

as though the parameter space were T =Z S . Simultaneously, it acts

to ip the right-moving U(1) R-current J !J.Thus the SDi algebra

(42) will undergo a Z quotient which one can p erhaps think of as SO(1).

Finally, one nds the requisite 32 fermionic states in the twisted sector

(and here they are not simply put in by hand). Decompacti cation of a

circle again resembles the construction of the matrix heterotic string [29],

although again the left-right gluing hides the gauge group structure in this

limit.

The alternate orbifold presented at the end of section 3.2 seems to be

4 5

~ ~

a T =Z T orbifold, but rather than sixteen hyp ermultiplets app earing

in the twisted sector (as exp ected in matrix theory on this space), we nd

only one.

At this p oint several remarks are in order. First, the (2,1) string app ears

to lack the parameters needed to explore the full 128-parameter mo duli

(2;1)

space of matrix theory on T . Eight circles are xed to the scale ` , so

str

it is not clear exactly how to connect the construction to the rest of M-

theory. This aim might be furthered by understanding how to construct

U-duality multiplets in the context of (2,1) string theory. The p erturba-

tive string states will be the momentum states of the SYM theory under

the presentinterpretation (which are longitudinally wrapp ed membranes in

matrix theory). The duality group is probably E (Z) [25] and do es not

9(9)

act entirely within the p erturbative theory. In fact, one exp ects [32] states

whose masses scale as arbitrarily high powers of 1=g . Since g / g ,

YM str

we are not going to nd these easily.

App endices

This corrects an erroneous interpretation in [31], where an attempt was made to

connect the duality group to the symmetries of the internal E torus. 8

24 EMIL MARTINEC

A. Technology of lo cal N=2 worldsheet sup ersymmetry

The ghost op erators in sup erstring vertices provide the appropriate sup er-

conformally covariant geometrical measure for their integration over the

worldsheet (for details, see [33, 34]). The sup erghosts have a number of

di erent `pictures', or BRST representatives, b ecause the splitting of the

sup ersymmetry ghosts into creation and annihilation op erators is ambigu-

ous. For the N=1 sup erconformal algebra, there is a single pair of such

ghosts , ; for N=2 one has two { , for the two sup ersymmetries

G . The ghost currentmay b e b osonized as @ = . The exp onen-

tials exp[ ]interp olate b etween the various vacua; are free elds.

In the NS sector there are two canonical pictures for a given chiral

sector of the usual N=1 sup erstring; the N=2 string doubles this. Consider

a sup er eld, with expansion in comp onents

(1) (1)

(0) (2)

O = O + O + O + O (43)

+ +

under the right-moving N=2 sup ersymmetry.For example, the exp onential

exp[ik X ] expands as

ik X ik x

e = e [1 + i k (1l+I) +i k (1l I ) (44)

+ ( k I @x (k )(k I )) ] : (45)

BRST invariantvertex op erators are then

(0) ik x

V = e e

(1;) ik x

V = e [k (1l I ) ]e

(2) ik x

V = [k I @x (k )(k I )]e : (46)

Any combination of these with () ghost charges adding up to 2 (to

cancel the background charge on the sphere) will yield the same on-shell

S-matrix amplitude (5). The N=1 structure is as usual [33] in terms of the

b osonization @ = , with additional ghost factors in the Ramond sector

from the spin eld for the sup ercurrent , equation (19). These are again

constructed in the standard way: the corresp onding spin 1/2 ghosts ,~ are

b osonized via @ = ~, and the standard picture for the Ramond sector

1 1

(N =1)

2 2

. The NS vertex op erators in vertices involves a factor = e

g h;R

equations (6),(20) have b een written in the zero picture for b oth left- and

right-movers.

For the twisted N=2 algebra, it is more convenient to b osonize the twist eigenstates

rather than the charge eigenstates.

+ +

M-Theory and N=2 Strings 25

B. (2,2) op en strings and D-branes

The op en string sector describ es self-dual Yang-Mills, in a formulation due

to Yang [1],[35]. The equations of motion are

(2;0) (0;2)

F = F =0

k^F = 0 (47)

The single physical string mo de is again the center-of-mass, which is an

adjoint scalar ' related to the gauge eld by

' ' ' '

A =(De )e ; A =(De )e ; (48)

with D a background covariant derivative. This ansatz automatically solves

the (2,0) and (0,2) comp onents of (47), and converts the (1,1) part into a

wave equation for '. In the mo dern interpretation of gauge charge, the

Chan-Paton indices are carried by branes lling spacetime, whose number

may be determined by closed string tadp ole cancellation (of course, such

restrictions are not active at op en string tree level, where we can formally

consider any classical Chan-Paton group). A calculation of the tadp ole [35]

indicates a gauge group G=SO(2); however, it should b e noted that there

are many massless tadp oles in N=2 string lo op amplitudes, whose inter-

pretation is currently unclear, rendering the determination of G somewhat

ambiguous.

B.1. BOUNDARY CONDITIONS AND D-BRANES

The op en string b oundary conditions of N=2 sup erconformal eld theory

have b een investigated by [36]. They fall into two classes:

A typ e : J = J ; G = G

B typ e : J =+J ; G =G ; (49)

where d is the complex dimension of the target space and p +1 is the

brane dimension . These two typ es of b oundary condition are related to

the twotyp es of N=2 sup er eld [14]: If a chiral eld ob eys typ e B b oundary

conditions, then its T-dual is a twisted chiral sup er eld satisfying typ e A

b oundary conditions, and vice-versa. Also, for a single free sup er eld, typ e

B b oundary conditions arise when the scalar comp onents are either b oth

Neumann or b oth Dirichlet, whereas typ e A has one Dirichlet and one

Neumann b oundary condition.

Unfortunately, this convention is not particularly well-adapted to N=2 strings with

signature 2+2; nevertheless we will adhere to it, using the notation (s+t)-brane for a

brane with s space and t time dimensions when necessary.

26 EMIL MARTINEC

The dynamics of typ e II branes in ten dimensions is, at long wave-

lengths, the dimensional reduction of the Yang-Mills theory governing 10d

op en strings, the dynamics of the Dirichlet co ordinates b eing \frozen" on

the brane. Corresp ondingly, one exp ects the dynamics of N=2 string D-

branes to be governed by the dimensional reduction of the self-dual equa-

tions (47). For instance, the D-string equations are the Hitchin equations

X =0 ; F +[X; X ]=0 : (50) D X = D

A A A

Here the dynamical eld is a 2d scalar ' in the adjointofU(N ) determin-

ing the reduced gauge eld (48). The covariantly constant (and therefore

nondynamical) Higgs eld X describ es the transverse p ositions of the N D-

strings. When N D-strings coincide, these equations b ecome the 2d chiral

mo del equations. Similarly, one can show that the D-particle is describ ed

by the Nahm equations

i ij k j k

DX = [X ;X ] ; (51)

and the D2-brane by the Bogomolny equations (for recent related work, see

[37])

F = D X : (52)

ij k k

There are other p ossible reductions, for instance imp osing Dirichlet b ound-

ary conditions along a null direction, or wrapping branes around homology

cycles of nontrivial self-dual four-manifolds such as K3; we will not dis-

cuss them here. Note that the equations (50)-(52) are all in the family of

Uhlenbeck-Yau typ e equations describing BPS con gurations of D-branes

(c.f. [38]), another indication of the resemblance of the dynamics to the

BPS sector of sup erstrings.

The D-instanton is also of some interest, in that its amplitudes may serve

to de ne an o -shell continuation of N=2 string dynamics, and thus an o -

shell quantum theory of self-dual gravity and Yang-Mills. These amplitudes

are in some resp ects similar to those of macroscopic lo ops in noncritical

string theory [39].

One may also consider comp osite systems of p- and p -branes. For exam-

ple D-instantons in the op en string theory are comp osites of a (0+0)-brane

and a (4+0)-brane which regularize an ab elian instanton . The 0-4 strings

should regulate the mo duli space of ADHM instantons (although this needs

to b e checked).

Assuming the tadp ole cancellation giving gauge group SO(2); otherwise { for instance

at op en string tree level{itmay b e consistent to consider multiple (4+0)-branes and

thus nonab elian (e.g. SU(N ) self-dual YM gauge group. Then the D-instantons carry

4+0

fundamental representation gauge quantum numb ers.

M-Theory and N=2 Strings 27

B.2. COUPLING D-BRANES TO SELF-DUAL GRAVITY

The SDYM op en string dynamics couples to the self-dual gravity of N=2

closed strings, since as usual op en string lo ops factorize on closed string ex-

change. Moreover, p erforming a Kahler gauge transformation K =(X)+

(X ) on the worldsheet action

Z I

2 2 2 2

S = d zd d K(X; X )+ dsd '(X; X ) ; (53)

one sees that can b e pushed onto gauge transformations of the U(1) part

of the prep otential ' which enco des the self-dual U(N)=SU(N)U(1) op en

string gauge background (48).

Marcus has shown [35] that the leading correction to the closed string

equation of motion (12) is

ij 4

Tr[F F ]+O(g ) ; (54) det [g ]=1

ij ij

where g and are the op en and closed string couplings, related by

hg . Using the op en string equation of motion k ^ F = 0, one may rewrite

this in a way that manifests the ab ove gauge symmetry:

1 1

2 2

F ) ^ (k + gN F )] = ^ (55) Tr[(k + gN

where is the holomorphic two-form.

It seems, however, that D-branes will not act as linearized sources for

static gravitational and/or axion-dilaton elds. A static string solution, as

for instance that of the fundamental string [26] is not self-dual (as one may

see for instance by the set of sup ersymmetries it preserves).

The D-branes of ordinary 10d string theory carry charges under the

antisymmetric tensor elds of the R-R sector. In the N=2 string these

elds are gauge equivalent to NS-NS elds by sp ectral ow in the N=2

U(1); e ectively, there are no R-R gauge elds, and the D-branes serveas

sources for the NS sector elds (as one sees by inserting closed string vertex

op erators on the disk). There app ear to b e no gauge charges carried by N=2

string D-branes, apart from those of the self-dual gravitational eld.

Acknowledgements

It is my great pleasure to thank Martin O'Loughlin and esp ecially David

Kutasov for the intense collab oration which led to the ideas presented

here. Thanks also to Tom Banks, Je Harvey, Chris Hull, Greg Mo ore,

and Stephen Shenker for fruitful discussions. The author is supp orted in

part by DOE grant DE-FG02-90ER-40560.

28 EMIL MARTINEC

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