a r X i v : h e p  t h / 9 5 1 2 1 2 9 v 3 9 F e b 1 9 9 6
HUBEP95/33CERNTH/95341SNUTP95/095hepth/9512129
BPS Spectra and NonPerturbative Gravitational Couplings in
N
= 2
,
4
Supersymmetric String TheoriesGabriel Lopes Cardoso
a
, Gottfried Curio
b
, Dieter L¨ust
b
, Thomas Mohaupt
b
andSooJong Rey
c
1
a
Theory Division, CERN, CH1211 Geneva 23, Switzerland
b
HumboldtUniversit¨ at zu Berlin, Institut f¨ ur Physik D10115 Berlin, Germany
c
Department of Physics, Seoul National University, Seoul 151742 Korea
ABSTRACT
We study the BPS spectrum in
D
= 4
,N
= 4 heterotic string compactiﬁcations, with some emphasis on intermediate
N
= 4 BPS states. Theseintermediate states, which can become short in
N
= 2 compactiﬁcations, arecrucial for establishing an
S
−
T
exchange symmetry in
N
= 2 compactiﬁcations. We discuss the implications of a possible
S
−
T
exchange symmetry forthe
N
= 2 BPS spectrum. Then we present the exact result for the 1loopcorrections to gravitational couplings in one of the heterotic
N
= 2 modelsrecently discussed by Harvey and Moore. We conjecture this model to havean
S
−
T
exchange symmetry. This exchange symmetry can then be usedto evaluate nonperturbative corrections to gravitational couplings in some of the nonperturbative regions (chambers) in this particular model and also inother heterotic models.December 1995
1
email: cardoso@surya15.cern.ch, curio@qft2.physik.huberlin.de, luest@qft1.physik.huberlin.de,mohaupt@qft2.physik.huberlin.de, sjrey@phyb.snu.ac.kr
1 Introduction
Recently, some major progress has been obtained in the understanding of nonperturbative dynamics in ﬁeld theories and string theories with extended supersymmetry[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. One important feature of these theories is the existenceof BPS states. These BPS states play an important role in understanding duality symmetries and nonperturbative eﬀects in string theory in various dimensions. They are,for instance, essential to the resolution of the conifold singularity in type II string theory[13]. BPS states also play a central role in 1loop threshold corrections to gauge andgravitational couplings in
N
= 2 heterotic string compactiﬁcations, as shown recently in[14].In the context of
D
= 4
,N
= 4 compactiﬁcations, BPS states also play a crucial role intests [15] of the conjectured strong/weak coupling
SL
(2
,
Z
)
S
duality [16, 17, 18] in toroidalcompactiﬁcations of the heterotic string. Moreover, the conjectured string/string/stringtriality [19] interchanges the BPS spectrum of the heterotic theory with the BPS spectrumof the type II theory. In an
N
= 4 theory, BPS states can either fall into short or intointermediate multiplets. In going from the heterotic to the type IIA side, for example,the four dimensional axion/dilaton ﬁeld
S
gets interchanged with the complex K¨ahlermodulus
T
of the 2torus on which the type IIA theory has been compactiﬁed on [20, 21].Thus, it is under the exchange of
S
and
T
that the BPS spectrum of the heteroticand the type IIA string gets mapped into each other. The BPS mass spectrum of theheterotic(type IIA) string is, however, not symmetric under this exchange of
S
and
T
.This is due to the fact that BPS masses in
D
= 4
,N
= 4 compactiﬁcations are givenby the maximum of the 2 central charges

Z
1

2
and

Z
2

2
of the
N
= 4 supersymmetryalgebra [22].On the other hand, states, which from the
N
= 4 point of view are intermediate, areactually short from the
N
= 2 point of view. This then leads to the possibility thatthe BPS spectrum of certain
N
= 2 heterotic compactiﬁcations is actually symmetricunder the exchange of
S
and
T
. If such symmetry exists a lot of information aboutthe BPS spectrum at strong coupling can be obtained, in particular about those BPSstates which can become massless at speciﬁc points in the moduli space. Assuming thatthe contributions to the associated gravitational couplings are due to BPS states only(as was shown to be the case at 1loop for some classes of compactiﬁcations in [14]), itfollows that these gravitational couplings should also exhibit such an
S
↔
T
exchangesymmetry. The evaluation of nonperturbative corrections to gravitational couplingsis, however, very diﬃcult. The existence of an exchange symmetry
S
↔
T
is extremly1
helpful in that it allows for the evaluation of nonperturbative corrections to gravitationalcouplings in some of the nonperturbative regions (chambers) in moduli space. This isachieved by taking the known result for the 1loop correction in some perturbative region(chamber) of moduli space and applying the exchange symmetry to it. Three exampleswill be discussed in this paper, namely the 2 parameter model
P
1
,
1
,
2
,
2
,
6
(12) of [23], the3 parameter model
P
1
,
1
,
2
,
8
,
12
(24) [7, 12] (for these two models an exchange symmetry
S
↔
T
has been observed in [9]) and the
s
= 0 model of [14] (for this example weconjecture that there too is such an exchange symmetry).The paper is organised as follows. In section 2 we introduce orbits for short and intermediate multiplets in
D
= 4
,N
= 4 heterotic string compactiﬁcations and we show how theyget mapped into each other under string/string/string triality. In section 3 we discussBPS states in the context of
D
= 4
,N
= 2 heterotic string compactiﬁcations and showthat states, which from the
N
= 4 point of view are intermediate, actually play an important role in the correct evaluation of nonperturbative eﬀects such as nonperturbativemonodromies. We also discuss exchange symmetries of the type
S
↔
T
in the 2 and 3parameter models
P
1
,
1
,
2
,
2
,
6
(12) and
P
1
,
1
,
2
,
8
,
12
(24). In section 4 we introduce an
N
= 4free energy as a sum over
N
= 4 BPS states and suggest that it should be identiﬁed withthe partition function of topologically twisted
N
= 4 string compactiﬁcations. In section5 we introduce an
N
= 2 free energy as a sum over
N
= 2 BPS states and argue thatit should be identiﬁed with the heterotic holomorphic gravitational function
F
grav
. Wediscuss 1loop corrections to the gravitational coupling and compute them exactly in the
s
= 0 model of [14]. We then argue that this model possesses an
S
↔
T
exchange symmetry and use it to compute nonperturbative corrections to the gravitational coupling insome nonperturbative regions of moduli space. We also discuss the 2 parameter model
P
1
,
1
,
2
,
2
,
6
(12) of [23] and compute the associated holomorphic gravitational coupling in thedecompactiﬁcation limit
T
→ ∞
. Finally, appendices A and B contain a more detaileddiscussion of some of the issues discussed in section 2.
2 The
N
= 4
BPS spectrum2.1 The truncation of the mass formula
In this section we recall the BPS mass formulae for fourdimensional string theories with
N
= 4 spacetime supersymmetry [17, 19]. Speciﬁcally, we ﬁrst consider the heteroticstring compactiﬁed on a sixdimensional torus. In
N
= 4 supersymmetry, there are ingeneral two central charges
Z
1
and
Z
2
. There exist two kinds of massive BPS multiplets,2
namely ﬁrst the short multiplets which saturate two BPS bounds (the associated solitonbackground solutions preserve 1
/
2 of the supersymmetries in
N
= 4), i.e.
m
2
S
=

Z
1

2
=

Z
2

2
; (2.1)the short vector multiplets contain maximal spin one. Second there are the intermediate multiplets which saturate only one BPS bound and contain maximal spin 3
/
2 (theassociated solitonic backgrounds preserve only one supersymmetry in
N
= 4), i.e.
m
2
I
= Max(

Z
1

2
,

Z
2

2
)
.
(2.2)The BPS masses are functions of the moduli parameter as well as functions of the dilatonaxion ﬁeld
S
=
4
πg
2
−
i
θ
2
π
=
e
−
φ
−
ia
. Speciﬁcally, the two central charges
Z
1
,
2
have thefollowing form [24, 19]

Z
1
,
2

2
=
Q
2
+
P
2
±
2
Q
2
P
2
−
(
Q
·
P
)
2
,
(2.3)where
Q
and
P
are the (6dimensional) electric and magnetic charge vectors which dependon the moduli and on
φ,a
. One sees that for short vector multiplets, for with

Z
1

=

Z
2

,the square root term in (2.3) must be absent, which is satisﬁed for parallel electric andmagnetic charge vectors. In this case the BPS masses agree with the formula of Schwarzand Sen [17].In a general compactiﬁcation on a sixdimensional torus
T
6
the moduli ﬁelds locallyparametrize a homogeneous coset space
SO
(6
,
22)
/
(
SO
(6)
×
SO
(22)). In terms of thesemoduli ﬁelds, the two central charges are then given
2
by [19]

Z
1
,
2

2
= 116
γ
T
M
(
M
+
L
)
γ
±
(
γ
T
ǫγ
)
ab
(
γ
T
ǫγ
)
cd
(
M
+
L
)
ac
(
M
+
L
)
bd
(2.4)where
γ
T
= (
α,β
). Let us from now on restrict the discussion by considering only an
SO
(2
,
2) subspace which corresponds to two complex moduli ﬁelds
T
and
U
. This meansthat we will only consider the moduli degrees of freedom of a twodimensional twotorus
T
2
. (
Q
and
P
are now twodimensional vectors.) Then, converting to a basis where
L
has diagonal form, ˇ
L
=
T
−
1
LT,
ˇ
M
=
T
−
1
MT,
ˇ
M
+ ˇ
L
= 2
φφ
T
=
ϕϕ
†
+ ¯
ϕϕ
T
, the twocentral charges can be written as

Z
1
,
2

2
= 116
ˇ
γ
T
M
(
ϕϕ
†
+ ¯
ϕϕ
T
)ˇ
γ
±
2
(ˇ
γ
T
ǫ
ˇ
γ
)
ab
(ˇ
γ
T
ǫ
ˇ
γ
)
cd
R
ac
R
bd
(2.5)where ˇ
γ
T
= (ˇ
α,
ˇ
β
) = (
T
−
1
α,T
−
1
β
) and where
R
ac
=
12
(
ϕϕ
†
+ ¯
ϕϕ
T
)
ac
. Using that(ˇ
γ
T
ǫ
ˇ
γ
)
ab
= ˇ
α
a
ˇ
β
b
−
ˇ
α
b
ˇ
β
a
it follows that

Z
1
,
2

2
= 116
ˇ
γ
T
M
(
ϕϕ
†
+ ¯
ϕϕ
T
)ˇ
γ
±
4
i
ˇ
α
T
I
ˇ
β
2
We are using the notation of [18, 19].
3
= 14(
S
+ ¯
S
)
ˇ
α
T
R
ˇ
α
+
S
¯
S
ˇ
β
T
R
ˇ
β
+
i
(
S
−
¯
S
)ˇ
α
T
R
ˇ
β
±
i
(
S
+ ¯
S
)ˇ
α
T
I
ˇ
β
) (2.6)where
I
=
12
(
ϕϕ
†
−
¯
ϕϕ
T
). The central charges

Z
1
,
2

2
can ﬁnally also be rewritten into

Z
1
,
2

2
= 14(
S
+ ¯
S
)
ˇ
α
T
R
ˇ
α
+
S
¯
S
ˇ
β
T
R
ˇ
β
±
i
(
S
−
¯
S
)ˇ
α
T
R
ˇ
β
±
i
(
S
+ ¯
S
)ˇ
α
T
I
ˇ
β
= 14(
S
+ ¯
S
)(
T
+ ¯
T
)(
U
+ ¯
U
)
M
1
,
2

2
M
1
=
ˇ
M
I
+
iS
ˇ
N
I
ˇ
P
I
M
2
=
ˇ
M
I
−
i
¯
S
ˇ
N
I
ˇ
P
I
(2.7)whereˇ
P
0
=
T
+
U ,
ˇ
P
1
=
i
(1 +
TU
)ˇ
P
2
=
T
−
U ,
ˇ
P
3
=
−
i
(1
−
TU
) (2.8)and where ˇ
M
= ˇ
α,
ˇ
N
= ˇ
β
. Here, the ˇ
M
I
(
I
= 0
,...,
3) are the integer electric chargequantum numbers of the Abelian gauge group
U
(1)
4
and the ˇ
N
I
are the correspondinginteger magnetic quantum numbers.Note that

Z
2

2
can be obtained from

Z
1

2
by
S
↔
¯
S,
ˇ
N
I
→ −
ˇ
N
I
. This amounts tocomplex conjugating ˇ
M
I
+
iS
ˇ
N
I
.Finally, rotating the ˇ
P
I
into ˆ
P
= (1
,
−
TU,iT,iU
)
T
ˇ
P
=
A
ˆ
P , A
=
i
0 0
−
1
−
11
−
1 0 00 0
−
1 1
−
1
−
1 0 0
(2.9)gives that

Z
1
,
2

2
= 14(
S
+ ¯
S
)(
T
+ ¯
T
)(
U
+ ¯
U
)
M
1
,
2

2
M
1
=
ˆ
M
I
+
iS
ˆ
N
I
ˆ
P
I
M
2
=
ˆ
M
I
−
i
¯
S
ˆ
N
I
ˆ
P
I
(2.10)where ˆ
M
=
A
T
ˇ
M,
ˆ
N
=
A
T
ˇ
N
.4