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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.
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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.
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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
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of states as the elementary string excitations. Furthermore, we have shown that
both, the dyon solutions, and the elementary excitations in string theory satisfy a
lower bound to their masses, and this lower bound is invariant under the SL(2,Z)
transformation. These results provide a further support to the conjecture that
SL(2,Z) might be an exact symmetry of heterotic string theory compactified on a
six dimensional torus.
Acknowledgements: This work was supported in part by the National Science
Foundation grant no. PHY89-04035.
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