Page 1

arXiv:hep-th/9303057v2 31 Mar 1993

NSF-ITP-93-29

TIFR-TH-93-07

hep-th/9303057

March, 1993

MAGNETIC MONOPOLES, BOGOMOL’NYI BOUND

AND SL(2,Z) INVARIANCE IN STRING THEORY

Ashoke Sen⋆

Institute for Theoretical Physics, University of

California, Santa Barbara, CA 93106, U.S.A.

and

Tata Institute of Fundamental Research, Homi

Bhabha Road, Bombay 400005, India†

ABSTRACT

We show that in heterotic string theory compactified on a six dimensional

torus, the lower bound (Bogomol’nyi bound) on the dyon mass is invariant under

the SL(2,Z) transformation that interchanges strong and weak coupling limits of

the theory.Elementary string excitations are also shown to satisfy this lower

bound. Finally, we identify specific monopole solutions that are related via the

strong-weak coupling duality transformation to some of the elementary particles

saturating the Bogomol’nyi bound, and these monopoles are shown to have the

same mass and degeneracy of states as the corresponding elementary particles.

⋆ e-mail addresses: SEN@TIFRVAX.BITNET, SEN@SBITP.UCSB.EDU

† Permanent address.

1

Page 2

Introduction

Following earlier ideas [1 − 10] we have proposed recently [11] that heterotic

string theory compactified on a six dimensional torus may have an SL(2,Z) sym-

metry that exchanges electric and magnetic fields, and also the strong and weak

coupling limits of the string theory. Existence of this symmetry demands that the

theory must necessarily contain magnetically charged particles. Allowed values of

electric and magnetic charges in this theory that are consistent with Dirac quan-

tization condition were found, and the set of these allowed values was shown to

be invariant under SL(2,Z) transformation [12]. This, however, does not establish

that states whose quantum numbers are related by SL(2,Z) transformation have

identical masses, − a necessary condition for SL(2,Z) invariance of the theory. This

is the problem that we try to address in this paper.

Elementary string excitations carry only electric charge, and their masses are

well known in the weak coupling limit of the theory. SL(2,Z) transform of these

states carry both electric and magnetic charges in general, and must arise as soliton

solutions in this theory. Thus in order to establish the SL(2,Z) invariance of the

mass spectrum, we must compare the elementary particle masses at weak coupling

to the soliton masses at strong coupling. In a generic theory, calculating soliton

masses at strong coupling would have been an impossible task; however, since the

theory under consideration has N = 4 supersymmetry, one can derive some results

about the soliton masses in this theory that are not expected to receive any quan-

tum corrections [13]. In particular, for a soliton carrying a given amount of electric

and magnetic charges, one can derive a lower bound (known as the Bogomol’nyi

bound) for the mass of the soliton. The bound is saturated for supersymmetric

solitons, and the masses of such solitons are expected not to receive any quantum

corrections. Thus one can compare these exact mass formulae as well as the lower

bound on the soliton masses with the masses of the elementary string excitations

and ask if they agree with the postulate of SL(2,Z) invariance of the theory. Al-

though this would not prove that SL(2,Z) is a symmetry of the theory, this would

provide a stringent test of this symmetry.

2

Page 3

In this paper we show first that the Bogomol’nyi bound is invariant under

SL(2,Z) transformation, and second, that the masses of the elementary string ex-

citations also satisfy the Bogomol’nyi bound, with a subset of them saturating

the bound. This implies that the elementary string excitations saturating the Bo-

gomol’nyi bound, and the supersymmetric solitons whose quantum numbers are

related to those of these elementary particles by SL(2,Z) transformation, have the

same mass. We also identify the specific soliton solutions that are related by an

SL(2,Z) transformation to some of the elementary string excitations saturating the

Bogomol’nyi bound.

Some other aspects of SL(2,Z) invariance have been discussed in ref.[14].

Review

The low energy effective action describing ten dimensional heterotic string

theory is given by

S =

1

32π

−1

?

d10x

?

−detG(10)

G(10)NN′

S

S

e−Φ(10)?

G(10)TT′

S

R(10)

S

+ G(10)MN

S

∂MΦ(10)∂NΦ(10)

12G(10)MM′

S

H(10)

MNTH(10)

M′N′T′−1

8G(10)MM′

S

G(10)NN′

S

F(10)I

MNF(10)I

(1)

M′N′

?

where

F(10)I

MN= ∂MA(10)I

N

− ∂NA(10)I

M

(2)

and

H(10)

MNT=

?

∂MB(10)

NT−1

4A(10)I

M

F(10)I

NT

+ cyclic permutations of M,N,T

?

(3)

Here Φ(10)is the dilaton field, G(10)

denote the rank two antisymmetric tensor field, and A(10)I

fields. The superscript(10)indicates that we are dealing with ten dimensional

SMNdenote ten dimensional σ-model metric, B(10)

MN

M

denote 16 U(1) gauge

fields, the indices M,N,T are ten dimensional Lorentz indices and run from 0 to 9,

and the indices I denote 16 dimensional gauge indices and run from 1 to 16. Note

3

Page 4

that we have included only the abelian gauge fields in the effective action. For a

generic toroidal compactification to four dimensions, all the non-abelian symmetry

is spontaneously broken, and only the U(1) gauge fields remain massless [15] [16].

We now compactify the theory on a 6 dimensional torus. Let us denote by

m,n (1 ≤ m,n ≤ 6) the six internal directions, and by µ,ν (µ,ν = 0,7,8,9) the

four uncompactified directions. In terms of the ten dimensional fields, we define

the four dimensional fields as follows:⋆

ˆGmn= G(10)

Smn,

ˆBmn= B(10)

mn,

ˆAI

m= A(10)I

m

,Φ = Φ(10)−1

ˆAI

µ,

2lndetˆG

Am

µ=1

2

=1

ˆGmnG(10)

Snµ,AI+12

µ

= −

1

2√2A(10)I

µ

+

1

√2

mAm

Am+6

µ

2B(10)

Sµν− G(10)

1 ≤ m,n ≤ 6,

mµ −ˆBmnAn

SmµG(10)

µ+

1

2√2

ˆAI

mAI+12

µ

GSµν= G(10)

SnνˆGmn,Bµν= B(10)

µν − 4B(10)

mnAm

µAn

ν− 2(Am

µAm+6

ν

− µ ↔ ν)

1 ≤ I ≤ 16

(4)

whereˆGmndenotes the inverse matrix ofˆGmn. The field strengths associated with

the four dimensional gauge fields and the anti-symmetric tensor field are defined

as

Fα

µν= ∂µAα

ν− ∂νAα

µ,1 ≤ α ≤ 28(5)

and

Hµνρ=

?

∂µBνρ+ 2Aα

µLαβFβ

νρ+ cyclic permutations of µ,ν,ρ

?

(6)

where L denotes the 28 × 28 matrix,

L =

0I6

0

I6

00

00−I16

(7)

The Einstein metric in 4 dimensions is obtained from the metric GSµνthrough the

⋆ In writing down these relations, we have made a change of normalization from the one used

in ref.[11] to the one used in ref.[12].

4

Page 5

rescaling,

Gµν= e−ΦGSµν

(8)

From now on, we shall choose the convention that all four dimensional indices will

be raised and lowered with the Einstein metric. With this convention, we define

the dual field strength

˜Fαµν=1

2(√−detG)−1ǫµνρσFα

ρσ

(9)

The equations of motion of the anti-symmetric tensor field allow us to define

a scalar field Ψ through the equation:

Hµνρ= −(√−detG)−1e2Φǫµνρσ∂σΨ(10)

We can combine the fields Φ and Ψ into a single complex scalar field λ:

λ = Ψ + ie−Φ≡ λ1+ iλ2

(11)

Finally, all information about the scalar fieldsˆGmn,ˆBmnandˆAI

mmay be included

in a single 28 dimensional matrix M satisfying,

MT= M,MTLM = L(12)

M is defined as,

M =

P

QT

RT

QR

SU

UT

V

(13)

where,

Pmn=ˆGmn,Qm

n=ˆGmp(ˆBpn+1

4

ˆAI

pˆAI

n),RmI=

1

√2

ˆGmpˆAI

p,

Smn=(ˆGmp−ˆBmp+1

UI

4

ˆAI

mˆAI

p)ˆGpq(ˆGqn+ˆBqn+1

4

ˆAJ

qˆAJ

n)

m=1

√2(ˆGmp−ˆBmp+1

4

ˆAJ

mˆAJ

p)ˆGpqˆAI

q,VIJ= δIJ+1

2

ˆAI

pˆGpqˆAJ

q

(14)

andTdenotes the transpose of a matrix. The equations of motion derived from the

5