The degeneracy of quarter BPS dyons in N=4 supersymmetric string theories is
known to jump across walls of marginal stability on which a quarter BPS dyon
can decay into a pair of half BPS dyons. We show that as long as the electric
and magnetic charges of the original dyon are primitive elements of the charge
lattice, the subspaces of the moduli space on which a quarter BPS dyon becomes
marginally unstable against decay into a pair of quarter BPS dyons or a half
BPS dyon and a quarter BPS dyon are of codimension two or more. As a result any
pair of generic points in the moduli space can be connected by a path avoiding
these subspaces and there is no jump in the spectrum associated with these
subspaces.Comment: LaTeX file, 9 pages; v2: a minor logical error corrected with no
  change in the result

A. Dabholkar

A. Sen

Ashoke Sen

B. Bates

D. Gaiotto

D. Shih

D.P. Jatkar

F. Denef

G. Lopes Cardoso

J.R. David

M. Cheng

M.C.N. Cheng

N. Banerjee

English

arXiv

The degeneracy of quarter BPS dyons in N=4 supersymmetric string theories is known to jump across walls of marginal stability on which a quarter BPS dyon can decay into a pair of half BPS dyons. We show that as long as the electric and magnetic charges of the original dyon are primitive elements of the charge lattice, the subspaces of the moduli space on which a quarter BPS dyon becomes marginally unstable against decay into a pair of quarter BPS dyons or a half BPS dyon and a quarter BPS dyon are of codimension two or more. As a result any pair of generic points in the moduli space can be connected by a path avoiding these subspaces and there is no jump in the spectrum associated with these subspaces

Sen, Ashoke

Indian Academy of Sciences

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Rare Decay Modes of Quarter BPS Dyons
Ashoke Sen
Harish-Chandra Research Institute
Chhatnag Road, Jhusi, Allahabad 211019, INDIA
E-mail: sen@mri.ernet.in, ashokesen1999@gmail.com
Abstract
The degeneracy of quarter BPS dyons in N=4 supersymmetric string theories is known to
jump across walls of marginal stability on which a quarter BPS dyon can decay into a pair
of half BPS dyons. We show that as long as the electric and magnetic charges of the original
dyon are primitive elements of the charge lattice, the subspaces of the moduli space on which a
quarter BPS dyon becomes marginally unstable against decay into a pair of quarter BPS dyons
or a half BPS dyon and a quarter BPS dyon are of codimension two or more. As a result any
pair of generic points in the moduli space can be connected by a path avoiding these subspaces
and there is no jump in the spectrum associated with these subspaces.
1
We now have a good understanding of the exact spectrum of a class of quarter BPS dyons
in N = 4 supersymmetric string theories [1–16]. It is also known that as we cross various
walls of marginal stability associated with the possible decay of the dyon into a pair of half
BPS states, the degeneracy changes by a certain amount that is exactly computable [12, 13].
Furthermore in the gravity description this jump can be accounted for by the (dis)appearance
of two centered small black holes [17–22] as in the asymptotic moduli space we cross walls
of marginal stability [13, 15, 16]. This raises the question: why aren’t there similar effects
associated with the decay of a quarter BPS dyon into a pair of quarter BPS dyons, or into a
quarter BPS dyon and a half BPS dyon? In this note we shall show that such decays take place
on subspaces of codimension higher than one as long as electric and magnetic charge vectors
of the original dyon are primitive elements of the charge lattice. Hence we can move from any
generic point in the moduli space to another generic point in the moduli space without ever
passing through these subspaces, and there is no effect of the type discussed in [12, 13, 15, 16]
associated with these decays.1
We denote by r the total number of U(1) gauge fields in the model, by ~Q and ~P the r
dimensional electric and the magnetic charge vectors, by τ = a + iS the axion-dilaton moduli
field parametrizing the upper half plane and by M the r×r matrix valued scalar field satisfying
MLMT = L, MT = M, L =
(
I6
−Ir−6
)
(1)
where Ik denotes k×k identity matrix. We shall use the subscript ∞ to denote the asymptotic
values of various scalar fields. Let us now introduce the SO(6, r − 6) matrix Ω∞ via the
relations2
M∞ = Ω∞Ω
T
∞, Ω∞LΩ
T
∞ = L , (2)
and define
QR =
1
2
(Ir + L)Ω
T
∞ Q, PR =
1
2
(Ir + L)Ω
T
∞ P . (3)
The vectors ~QR and ~PR lie in the six dimentional subspace spanned by the eigenvectors of L
with eigenvalue 1. In terms of ~QR and ~PR the BPS mass formula of [25,26] takes the form [12],
m( ~Q, ~P ) =
√
2 f( ~QR, ~PR; a∞, S∞) , (4)
1Different approaches to this problem have been advocated in [23, 24].
2Since (2) is invariant under a right multiplication of Ω∞ by an SO(6) × SO(r − 6) matrix that preserves
both the identity matrix and L, (2) does not fix Ω∞ completely in terms of M∞. This problem may be avoided
by choosing a suitable ‘gauge condition’ on Ω∞ so that there is one to one correspondence between M∞ and
Ω∞.
2
where
f( ~QR, ~PR; a∞, S∞) =
√
1
S∞
( ~QR − a∞ ~PR)2 + S∞ ~P 2R + 2
[
~Q2R
~P 2R − ( ~QR · ~PR)2
]
1/2
. (5)
The inner products of ~QR and ~PR are calculated with the identity matrix.
Let us now consider a possible marginal decay ( ~Q, ~P ) → ( ~Q1, ~P1) + ( ~Q− ~Q1, ~P − ~P1). This
requires adjusting the moduli such that
f( ~QR, ~PR; a∞, S∞) = f( ~Q1R, ~P1R; a∞, S∞) + f( ~QR − ~Q1R, ~PR − ~P1R; a∞, S∞) . (6)
For fixed ~Q, ~P and M∞, ~QR and ~PR span a two dimensional subspace of the six dimensional
space on which L has eigenvalue +1. Let us denote by ~Q1R‖ and ~P1R‖ the projection of ~Q1R
and ~P1R along this two dimensional subspace. Then we have the following inequalities:
f( ~QR, ~PR; a∞, S∞) ≤ f( ~Q1R‖, ~P1R‖; a∞, S∞) + f( ~QR − ~Q1R‖, ~PR − ~P1R‖; a∞, S∞) , (7)
f( ~Q1R‖, ~P1R‖; a∞, S∞) ≤ f( ~Q1R, ~P1R; a∞, S∞) , (8)
f( ~QR − ~Q1R‖, ~PR − ~P1R‖; a∞, S∞) ≤ f( ~QR − ~Q1R, ~PR − ~P1R; a∞, S∞) . (9)
The inequality (7) is proved by defining
~a =
~QR − a∞ ~PR√
S∞
, ~b = ~PR
√
S∞ , ~a1 =
~Q1R‖ − a∞ ~P1R‖√
S∞
, ~b1 = ~P1R‖
√
S∞ , (10)
and using the inequality:
√
~a2 +~b2 + 2|~a ×~b| ≤
√
~a2
1
+~b2
1
+ 2|~a1 ×~b1| +
√
(~a − ~a1)2 + (~b −~b1)2 + 2|(~a − ~a1) × (~b −~b1)|
(11)
for any set of vectors ~a, ~b, ~a1, ~b1 lying in a two dimensional plane. (11) can be easily proven
with the help of triangle inequality if we note that
√
~a2 +~b2 + 2|~a ×~b| can be interpreted as
|~a+ǫ~b| where ǫ is the π/2 rotation matrix in the plane of ~a and ~b, with the sign of ǫ chosen such
that aT ǫb > 0. Requiring the inequality (11) to be saturated gives one equation and several
inequalities among the components of ~a, ~b, ~a1 and ~b1:
~a1 + ǫ~b1 = λ(~a + ǫ~b) with 0 ≤ λ ≤ 1 , aT1 ǫb1 ≥ 0, (a − a1)T ǫ(b − b1) ≥ 0 . (12)
Using (10) we can translate these conditions into one constraint equation and some inequalities
involving the variables (a∞, S∞, M∞, ~Q, ~P ).
3
The inequality (8) follows from the observations that
| ~Q1R‖ − τ̄∞ ~P1R‖|2 ≤ | ~Q1R − τ̄∞ ~P1R|2,
√
~Q2
1R‖
~P 2
1R‖ − ( ~Q1R‖ · ~P1R‖)2 ≤
√
~Q2
1R
~P 2
1R − ( ~Q1R · ~P1R)2 . (13)
The first of these is obvious since the (complex) vector on the left hand side is a projection
of the vector on the right hand side along the plane spanned by ~QR and ~PR. The second one
follows from the fact that the right hand side of the inequality represents the area of a triangle
formed by the vectors ~Q1R and ~P1R and the left hand side represents the area of the projection
of this triangle in the plane spanned by ~QR and ~PR. Both inequalities are saturated when
~Q1R and ~P1R lie in the plane spanned by ~QR and ~PR. The inequality (9) can be proved in an
identical manner and is also saturated when ~Q1R and ~P1R lie in the plane spanned by ~QR and
~PR. This requires adjusting Ω∞ or equivalently M∞ appropriately.
Now in order to satisfy the condition for marginal stability (6) we must saturate all the
three inequalities (7)-(9). This would require adjusting moduli M∞ to make ( ~Q1R, ~P1R) lie in
the plane of ( ~QR, ~PR), and additional adjustment of (a∞, S∞) to saturate the inequality (7).
Thus we have a surface of codimension two or more, and we can go from any generic point
in the moduli space to another generic point in the moduli space without ever encountering
this subspace of marginal stability. This shows that there is no discontinuous change in the
spectrum associated with these subspaces.
It is instructive to compare this with the condition for marginal stability of half BPS dyons
in the N = 2 supersymmetric S-T-U model. In that case the BPS mass formula is identical to
the one given in (4), (5), but M and L are 4 × 4 matrices and L has two eigenvalues +1 and
two eigenvalues −1. As a result the vectors ~QR, ~PR, ~Q1R, ~P1R all lie inside a two dimensional
subspace spanned by the eigenvectors of L with eigenvalue +1, and the inequalities (8), (9)
are automatically saturated. Thus we only need to saturate the inequality (7). This gives one
condition on the asymptotic moduli, producing a codimension one surface.
There is a special case where our argument fails for the N = 4 supersymmetric theory.
If the full r dimensional charge vectors ~Q1 and ~P1 happen to lie in the plane spanned by ~Q
and ~P then ~Q1R and ~P1R automatically lie in the plane of ~QR and ~PR and we do not get any
condition on the moduli M∞ from (8), (9). This would require ~Q1 and ~P1 to be of the form:
~Q1 = α~Q + β ~P , ~P1 = γ ~Q + δ ~P . (14)
If we take ~Q and ~P to be primitive then charge quantization would require α, β, γ, δ to be
integers. Furthermore in the ZZN orbifold models γ must be integer multiples of N since in
4
one particular direction along the charge lattice Q is quantized in units of 1/N while P is
quantized in integer units [5, 8]. (14) now implies that
~Q1R = α~QR + β ~PR, ~P1R = γ ~QR + δ ~PR . (15)
We can substitute these into (7) and use the discussion below (11) to determine under what
condition the inequality might be saturated. We shall only consider the case when ~QR and ~PR
are not parallel, ı.e. ~Q2R
~P 2R − ( ~QR · ~PR)2 6= 0, since for generic ~Q and ~P aligning ~QR and ~PR
will impose more than one condition on Ω∞ and will produce a surface of codimension higher
than one. In this case the aT
1
ǫb1 ≥ 0, (a − a1)T ǫ(b − b1) ≥ 0 conditions give
αδ − βγ ≥ 0 , (1 − α)(1 − δ) − βγ ≥ 0 . (16)
On the other hand the (~a1 + ǫ~b1) = λ(~a + ǫ~b) condition gives
(α − λ − a∞γ)
√
~Q2R
~P 2R − ( ~QR · ~PR)2 + +S∞
(
γ ~QR · ~PR + (δ − λ)~P 2R
)
= 0 , (17)
(β − (δ − λ)a∞)
√
~Q2R
~P 2R − ( ~QR · ~PR)2 − S∞
(
γ ~Q2R + (δ − λ) ~QR · ~PR
)
= 0 , (18)
where λ is an arbitrary parameter with
0 ≤ λ ≤ 1 . (19)
We can solve for a∞ using (17) and substitute into (18) to get
(
βγ − αδ + λ(α + δ) − λ2
)
√
~Q2R
~P 2R − ( ~QR · ~PR)2 − S∞
{
(λ − δ)~PR − γ ~QR
}
2
= 0 . (20)
If γ = 0 then (18) contains additional information beyond what can be obtained from (20) and
(17).
We focus on the constraint (20). Since the second term proportional to S∞ is negative
semi-definite, if we can show that the first term is also negative definite then we would have
shown that the equation has no solution. For this analysis we shall make use of (16). Let us
consider the following cases separately:
• First consider the case when both the inequalities in (16) are saturated:
αδ − βγ = 0 , (1 − α)(1 − δ) − βγ = 0 . (21)
5
This correponds to the case when in each of the decay products the electric and the
magnetic charge vectors are parallel. Hence both the decay products are half BPS. (21)
gives α + δ = 1. Using this we can express the first term on the left hand side of (20) as
(
λ − λ2
)
√
~Q2R
~P 2R − ( ~QR · ~PR)2 . (22)
Since this is positive semi-definite in the range (19) this can cancel the second term in (20)
on a codimension 1 subspace in the (λ, S∞) space. Using (17) or (18) we can convert this
to a codimension 1 subspace in the (a∞, S∞) space, reproducing the marginal stability
walls studied in [12].
• Now consider the case where at least one of the decay products is quarter BPS. Without
any loss of generality we can take this to be the state carrying charges (α~Q+β ~P , γ ~Q+δ ~P ).
In this case the first inequality in (16) will be a strict inequality. Combining this with
the information that α, β, γ, δ are integers we get
αδ − βγ ≥ 1 , (1 − α)(1 − δ) − βγ ≥ 0 . (23)
Our goal is to use these results to analyze the first term on the left hand side of (20).
Due to (23),
(
βγ − αδ + λ(α + δ) − λ2
)
(24)
is negative or zero at λ = 0, 1. Thus in order for it to be positive in some range of value
between λ = 0 and λ = 1, it must have a maximum in this range, and its value at the
maximum must be positive. Now (24) has a maximum at
λ =
α + δ
2
, (25)
where it takes the value
1
4
(α + δ)2 − (αδ − βγ) . (26)
Eq.(25) shows that in order that the maximum lies in the range (0, 1) we must have
0 ≤ (α + δ) ≤ 2 . (27)
Using eqs.(23), (27) we see that (26), representing the maximum value of (24), must be
negative or zero. As a result there is no cancellation between the two terms in the left
6
hand side of (20). The only possibility is that both terms may vanish simultaneously.
Vanishing of the second term will require
γ = 0, λ = δ , (28)
while from (23), (25)-(27) we see that the vanishing of the first term would require
αδ − βγ = 1, α + δ = 2, λ = α + δ
2
. (29)
As a consequence of (28), (29) we get
α = δ = 1, γ = 0, λ = 1 . (30)
We have seen however that for γ = 0, (17), (18) may contain additional information
beyond the ones which have been already discussed. In particular substituting (30) into
(18) we get β = 0. This choice of (α, β, γ, δ) corresponds to the trivial case where the
final decay products have charges ( ~Q, ~P ) and (0, 0).
Similar analysis shows that the decay of a quarter BPS dyon into three or more quarter
or half BPS dyons occur on subspaces of codimension larger than one, since this would
require aligning multiple six dimensional vectors along a plane and/or aligning multiple
two dimensional vectors along a line. This completes our proof that the only possible
codimension one subspaces of marginal stability arise from the decay of a quarter BPS
dyon into a pair of half BPS dyons.
Before concluding this paper we would like to offer a physical explanation of why decay
of a quarter BPS state into quarter BPS states requires more constraint than decay into
half BPS states. This essentially arises from the fact that at a point of marginal stability
the supersymmetries of decay products must align. Since half-BPS states have more
supersymmetry than quarter BPS states, it is clearly easier to ensure that a pair of half
BPS states have one common supersymmetry than ensuring that a pair of quarter BPS
states have a common supersymmetry. Similar argument can be given for the decay of a
quarter BPS state into three or more states.
Acknowledgement: I would like to thank A. Mukherjee, S. Mukhi and R. Nigam for
pointing out a logical error in an earlier version of the draft.
7
References
[1] R. Dijkgraaf, E. P. Verlinde and H. L. Verlinde, “Counting dyons in N = 4 string theory,”
Nucl. Phys. B 484, 543 (1997) [arXiv:hep-th/9607026].
[2] G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, “Asymptotic degener-
acy of dyonic N = 4 string states and black hole entropy,” JHEP 0412, 075 (2004)
[arXiv:hep-th/0412287].
[3] D. Shih, A. Strominger and X. Yin, “Recounting dyons in N = 4 string theory,”
arXiv:hep-th/0505094.
[4] D. Gaiotto, “Re-recounting dyons in N = 4 string theory,” arXiv:hep-th/0506249.
[5] D. P. Jatkar and A. Sen, “Dyon spectrum in CHL models,” JHEP 0604, 018 (2006)
[arXiv:hep-th/0510147].
[6] J. R. David, D. P. Jatkar and A. Sen, “Product representation of dyon partition function
in CHL models,” JHEP 0606, 064 (2006) [arXiv:hep-th/0602254].
[7] A. Dabholkar and S. Nampuri, “Spectrum of dyons and black holes in CHL orbifolds using
Borcherds lift,” arXiv:hep-th/0603066.
[8] J. R. David and A. Sen, “CHL dyons and statistical entropy function from D1-D5 system,”
JHEP 0611, 072 (2006) [arXiv:hep-th/0605210].
[9] J. R. David, D. P. Jatkar and A. Sen, “Dyon spectrum in N = 4 supersymmetric type II
string theories,” arXiv:hep-th/0607155.
[10] J. R. David, D. P. Jatkar and A. Sen, “Dyon spectrum in generic N = 4 supersymmetric
Z(N) orbifolds,” arXiv:hep-th/0609109.
[11] A. Dabholkar and D. Gaiotto, “Spectrum of CHL dyons from genus-two partition func-
tion,” arXiv:hep-th/0612011.
[12] A. Sen, “Walls of marginal stability and dyon spectrum in N = 4 supersymmetric string
theories,” arXiv:hep-th/0702141.
8
[13] A. Dabholkar, D. Gaiotto and S. Nampuri, “Comments on the spectrum of CHL dyons,”
arXiv:hep-th/0702150.
[14] N. Banerjee, D. P. Jatkar and A. Sen, “Adding charges to N = 4 dyons,” arXiv:0705.1433
[hep-th].
[15] A. Sen, “Two Centered Black Holes and N=4 Dyon Spectrum,” arXiv:0705.3874 [hep-th].
[16] M. C. N. Cheng and E. Verlinde, “Dying Dyons Don’t Count,” arXiv:0706.2363 [hep-th].
[17] F. Denef, “Supergravity flows and D-brane stability,” JHEP 0008, 050 (2000)
[arXiv:hep-th/0005049].
[18] F. Denef, “On the correspondence between D-branes and stationary supergravity solutions
of type II Calabi-Yau compactifications”, arXiv:hep-th/0010222.
[19] F. Denef, B. R. Greene and M. Raugas, “Split attractor flows and the spectrum of BPS
D-branes on the quintic,” JHEP 0105, 012 (2001) [arXiv:hep-th/0101135].
[20] B. Bates and F. Denef, “Exact solutions for supersymmetric stationary black hole com-
posites,” arXiv:hep-th/0304094.
[21] F. Denef and G. W. Moore, “Split states, entropy enigmas, holes and halos,”
arXiv:hep-th/0702146.
[22] F. Denef and G. W. Moore, “How many black holes fit on the head of a pin?,”
arXiv:0705.2564 [hep-th].
[23] F. Denef, private communications.
[24] M. Cheng, private communications.
[25] M. Cvetic and D. Youm, “Dyonic BPS saturated black holes of heterotic string on a six
torus,” Phys. Rev. D 53, 584 (1996) [arXiv:hep-th/9507090].
[26] M. J. Duff, J. T. Liu and J. Rahmfeld, “Four-Dimensional String-String-String Triality,”
Nucl. Phys. B 459, 125 (1996) [arXiv:hep-th/9508094].
9


Rare decay modes of quarter BPS dyons

Publications of the IAS Fellows

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20
07
Rare Decay Modes of Quarter BPS Dyons
Ashoke Sen
Harish-Chandra Research Institute
Chhatnag Road, Jhusi, Allahabad 211019, INDIA
E-mail: sen@mri.ernet.in, ashokesen1999@gmail.com
Abstract
The degeneracy of quarter BPS dyons in N=4 supersymmetric string theories is known to
jump across walls of marginal stability on which a quarter BPS dyon can decay into a pair
of half BPS dyons. We show that as long as the electric and magnetic charges of the original
dyon are primitive elements of the charge lattice, the subspaces of the moduli space on which a
quarter BPS dyon becomes marginally unstable against decay into a pair of quarter BPS dyons
or a half BPS dyon and a quarter BPS dyon are of codimension two or more. As a result any
pair of generic points in the moduli space can be connected by a path avoiding these subspaces
and there is no jump in the spectrum associated with these subspaces.
1
We now have a good understanding of the exact spectrum of a class of quarter BPS dyons
in N = 4 supersymmetric string theories [1–16]. It is also known that as we cross various
walls of marginal stability associated with the possible decay of the dyon into a pair of half
BPS states, the degeneracy changes by a certain amount that is exactly computable [12, 13].
Furthermore in the gravity description this jump can be accounted for by the (dis)appearance
of two centered small black holes [17–22] as in the asymptotic moduli space we cross walls
of marginal stability [13, 15, 16]. This raises the question: why aren’t there similar effects
associated with the decay of a quarter BPS dyon into a pair of quarter BPS dyons, or into a
quarter BPS dyon and a half BPS dyon? In this note we shall show that such decays take place
on subspaces of codimension higher than one as long as electric and magnetic charge vectors
of the original dyon are primitive elements of the charge lattice. Hence we can move from any
generic point in the moduli space to another generic point in the moduli space without ever
passing through these subspaces, and there is no effect of the type discussed in [12, 13, 15, 16]
associated with these decays.1
We denote by r the total number of U(1) gauge fields in the model, by ~Q and ~P the r
dimensional electric and the magnetic charge vectors, by τ = a + iS the axion-dilaton moduli
field parametrizing the upper half plane and byM the r×r matrix valued scalar field satisfying
MLMT = L, MT =M, L =
(
I6
−Ir−6
)
(1)
where Ik denotes k×k identity matrix. We shall use the subscript ∞ to denote the asymptotic
values of various scalar fields. Let us now introduce the SO(6, r − 6) matrix Ω∞ via the
relations2
M∞ = Ω∞Ω
T
∞, Ω∞LΩ
T
∞ = L , (2)
and define
QR =
1
2
(Ir + L)Ω
T
∞Q, PR =
1
2
(Ir + L)Ω
T
∞ P . (3)
The vectors ~QR and ~PR lie in the six dimentional subspace spanned by the eigenvectors of L
with eigenvalue 1. In terms of ~QR and ~PR the BPS mass formula of [25,26] takes the form [12],
m( ~Q, ~P ) =
√
2 f( ~QR, ~PR; a∞, S∞) , (4)
1Different approaches to this problem have been advocated in [23, 24].
2Since (2) is invariant under a right multiplication of Ω∞ by an SO(6) × SO(r − 6) matrix that preserves
both the identity matrix and L, (2) does not fix Ω∞ completely in terms of M∞. This problem may be avoided
by choosing a suitable ‘gauge condition’ on Ω∞ so that there is one to one correspondence between M∞ and
Ω∞.
2
where
f( ~QR, ~PR; a∞, S∞) =
√
1
S∞
( ~QR − a∞ ~PR)2 + S∞ ~P 2R + 2
[
~Q2R
~P 2R − ( ~QR · ~PR)2
]
1/2
. (5)
The inner products of ~QR and ~PR are calculated with the identity matrix.
Let us now consider a possible marginal decay ( ~Q, ~P )→ ( ~Q1, ~P1)+ ( ~Q− ~Q1, ~P − ~P1). This
requires adjusting the moduli such that
f( ~QR, ~PR; a∞, S∞) = f( ~Q1R, ~P1R; a∞, S∞) + f( ~QR − ~Q1R, ~PR − ~P1R; a∞, S∞) . (6)
For fixed ~Q, ~P and M∞, ~QR and ~PR span a two dimensional subspace of the six dimensional
space on which L has eigenvalue +1. Let us denote by ~Q1R‖ and ~P1R‖ the projection of ~Q1R
and ~P1R along this two dimensional subspace. Then we have the following inequalities:
f( ~QR, ~PR; a∞, S∞) ≤ f( ~Q1R‖, ~P1R‖; a∞, S∞) + f( ~QR − ~Q1R‖, ~PR − ~P1R‖; a∞, S∞) , (7)
f( ~Q1R‖, ~P1R‖; a∞, S∞) ≤ f( ~Q1R, ~P1R; a∞, S∞) , (8)
f( ~QR − ~Q1R‖, ~PR − ~P1R‖; a∞, S∞) ≤ f( ~QR − ~Q1R, ~PR − ~P1R; a∞, S∞) . (9)
The inequality (7) is proved by defining
~a =
~QR − a∞ ~PR√
S∞
, ~b = ~PR
√
S∞ , ~a1 =
~Q1R‖ − a∞ ~P1R‖√
S∞
, ~b1 = ~P1R‖
√
S∞ , (10)
and using the inequality:√
~a2 +~b2 + 2|~a×~b| ≤
√
~a2
1
+~b2
1
+ 2|~a1 ×~b1|+
√
(~a− ~a1)2 + (~b−~b1)2 + 2|(~a− ~a1)× (~b−~b1)|
(11)
for any set of vectors ~a, ~b, ~a1, ~b1 lying in a two dimensional plane. (11) can be easily proven
with the help of triangle inequality if we note that
√
~a2 +~b2 + 2|~a×~b| can be interpreted as
|~a+ǫ~b| where ǫ is the π/2 rotation matrix in the plane of ~a and ~b, with the sign of ǫ chosen such
that aT ǫb > 0. Requiring the inequality (11) to be saturated gives one equation and several
inequalities among the components of ~a, ~b, ~a1 and ~b1:
~a1 + ǫ~b1 = λ(~a+ ǫ~b) with 0 ≤ λ ≤ 1 , aT1 ǫb1 ≥ 0, (a− a1)T ǫ(b− b1) ≥ 0 . (12)
Using (10) we can translate these conditions into one constraint equation and some inequalities
involving the variables (a∞, S∞,M∞, ~Q, ~P ).
3
The inequality (8) follows from the observations that
| ~Q1R‖ − τ¯∞ ~P1R‖|2 ≤ | ~Q1R − τ¯∞ ~P1R|2,√
~Q2
1R‖
~P 2
1R‖ − ( ~Q1R‖ · ~P1R‖)2 ≤
√
~Q2
1R
~P 2
1R − ( ~Q1R · ~P1R)2 . (13)
The first of these is obvious since the (complex) vector on the left hand side is a projection
of the vector on the right hand side along the plane spanned by ~QR and ~PR. The second one
follows from the fact that the right hand side of the inequality represents the area of a triangle
formed by the vectors ~Q1R and ~P1R and the left hand side represents the area of the projection
of this triangle in the plane spanned by ~QR and ~PR. Both inequalities are saturated when
~Q1R and ~P1R lie in the plane spanned by ~QR and ~PR. The inequality (9) can be proved in an
identical manner and is also saturated when ~Q1R and ~P1R lie in the plane spanned by ~QR and
~PR. This requires adjusting Ω∞ or equivalently M∞ appropriately.
Now in order to satisfy the condition for marginal stability (6) we must saturate all the
three inequalities (7)-(9). This would require adjusting moduli M∞ to make ( ~Q1R, ~P1R) lie in
the plane of ( ~QR, ~PR), and additional adjustment of (a∞, S∞) to saturate the inequality (7).
Thus we have a surface of codimension two or more, and we can go from any generic point
in the moduli space to another generic point in the moduli space without ever encountering
this subspace of marginal stability. This shows that there is no discontinuous change in the
spectrum associated with these subspaces.
It is instructive to compare this with the condition for marginal stability of half BPS dyons
in the N = 2 supersymmetric S-T-U model. In that case the BPS mass formula is identical to
the one given in (4), (5), but M and L are 4× 4 matrices and L has two eigenvalues +1 and
two eigenvalues −1. As a result the vectors ~QR, ~PR, ~Q1R, ~P1R all lie inside a two dimensional
subspace spanned by the eigenvectors of L with eigenvalue +1, and the inequalities (8), (9)
are automatically saturated. Thus we only need to saturate the inequality (7). This gives one
condition on the asymptotic moduli, producing a codimension one surface.
There is a special case where our argument fails for the N = 4 supersymmetric theory.
If the full r dimensional charge vectors ~Q1 and ~P1 happen to lie in the plane spanned by ~Q
and ~P then ~Q1R and ~P1R automatically lie in the plane of ~QR and ~PR and we do not get any
condition on the moduli M∞ from (8), (9). This would require ~Q1 and ~P1 to be of the form:
~Q1 = α~Q+ β ~P , ~P1 = γ ~Q+ δ ~P . (14)
If we take ~Q and ~P to be primitive then charge quantization would require α, β, γ, δ to be
integers. Furthermore in the ZZN orbifold models γ must be integer multiples of N since in
4
one particular direction along the charge lattice Q is quantized in units of 1/N while P is
quantized in integer units [5, 8]. (14) now implies that
~Q1R = α~QR + β ~PR, ~P1R = γ ~QR + δ ~PR . (15)
We can substitute these into (7) and use the discussion below (11) to determine under what
condition the inequality might be saturated. We shall only consider the case when ~QR and ~PR
are not parallel, ı.e. ~Q2R
~P 2R − ( ~QR · ~PR)2 6= 0, since for generic ~Q and ~P aligning ~QR and ~PR
will impose more than one condition on Ω∞ and will produce a surface of codimension higher
than one. In this case the aT
1
ǫb1 ≥ 0, (a− a1)T ǫ(b− b1) ≥ 0 conditions give
αδ − βγ ≥ 0 , (1− α)(1− δ)− βγ ≥ 0 . (16)
On the other hand the (~a1 + ǫ~b1) = λ(~a+ ǫ~b) condition gives
(α− λ− a∞γ)
√
~Q2R
~P 2R − ( ~QR · ~PR)2 ++S∞
(
γ ~QR · ~PR + (δ − λ)~P 2R
)
= 0 , (17)
(β − (δ − λ)a∞)
√
~Q2R
~P 2R − ( ~QR · ~PR)2 − S∞
(
γ ~Q2R + (δ − λ) ~QR · ~PR
)
= 0 , (18)
where λ is an arbitrary parameter with
0 ≤ λ ≤ 1 . (19)
We can solve for a∞ using (17) and substitute into (18) to get
(
βγ − αδ + λ(α + δ)− λ2)√ ~Q2R ~P 2R − ( ~QR · ~PR)2 − S∞ {(λ− δ)~PR − γ ~QR}2 = 0 . (20)
If γ = 0 then (18) contains additional information beyond what can be obtained from (20) and
(17).
We focus on the constraint (20). Since the second term proportional to S∞ is negative
semi-definite, if we can show that the first term is also negative definite then we would have
shown that the equation has no solution. For this analysis we shall make use of (16). Let us
consider the following cases separately:
• First consider the case when both the inequalities in (16) are saturated:
αδ − βγ = 0 , (1− α)(1− δ)− βγ = 0 . (21)
5
This correponds to the case when in each of the decay products the electric and the
magnetic charge vectors are parallel. Hence both the decay products are half BPS. (21)
gives α+ δ = 1. Using this we can express the first term on the left hand side of (20) as
(
λ− λ2)√ ~Q2R ~P 2R − ( ~QR · ~PR)2 . (22)
Since this is positive semi-definite in the range (19) this can cancel the second term in (20)
on a codimension 1 subspace in the (λ, S∞) space. Using (17) or (18) we can convert this
to a codimension 1 subspace in the (a∞, S∞) space, reproducing the marginal stability
walls studied in [12].
• Now consider the case where at least one of the decay products is quarter BPS. Without
any loss of generality we can take this to be the state carrying charges (α~Q+β ~P , γ ~Q+δ ~P ).
In this case the first inequality in (16) will be a strict inequality. Combining this with
the information that α, β, γ, δ are integers we get
αδ − βγ ≥ 1 , (1− α)(1− δ)− βγ ≥ 0 . (23)
Our goal is to use these results to analyze the first term on the left hand side of (20).
Due to (23), (
βγ − αδ + λ(α+ δ)− λ2) (24)
is negative or zero at λ = 0, 1. Thus in order for it to be positive in some range of value
between λ = 0 and λ = 1, it must have a maximum in this range, and its value at the
maximum must be positive. Now (24) has a maximum at
λ =
α + δ
2
, (25)
where it takes the value
1
4
(α + δ)2 − (αδ − βγ) . (26)
Eq.(25) shows that in order that the maximum lies in the range (0, 1) we must have
0 ≤ (α + δ) ≤ 2 . (27)
Using eqs.(23), (27) we see that (26), representing the maximum value of (24), must be
negative or zero. As a result there is no cancellation between the two terms in the left
6
hand side of (20). The only possibility is that both terms may vanish simultaneously.
Vanishing of the second term will require
γ = 0, λ = δ , (28)
while from (23), (25)-(27) we see that the vanishing of the first term would require
αδ − βγ = 1, α + δ = 2, λ = α + δ
2
. (29)
As a consequence of (28), (29) we get
α = δ = 1, γ = 0, λ = 1 . (30)
We have seen however that for γ = 0, (17), (18) may contain additional information
beyond the ones which have been already discussed. In particular substituting (30) into
(18) we get β = 0. This choice of (α, β, γ, δ) corresponds to the trivial case where the
final decay products have charges ( ~Q, ~P ) and (0, 0).
Similar analysis shows that the decay of a quarter BPS dyon into three or more quarter
or half BPS dyons occur on subspaces of codimension larger than one, since this would
require aligning multiple six dimensional vectors along a plane and/or aligning multiple
two dimensional vectors along a line. This completes our proof that the only possible
codimension one subspaces of marginal stability arise from the decay of a quarter BPS
dyon into a pair of half BPS dyons.
Before concluding this paper we would like to offer a physical explanation of why decay
of a quarter BPS state into quarter BPS states requires more constraint than decay into
half BPS states. This essentially arises from the fact that at a point of marginal stability
the supersymmetries of decay products must align. Since half-BPS states have more
supersymmetry than quarter BPS states, it is clearly easier to ensure that a pair of half
BPS states have one common supersymmetry than ensuring that a pair of quarter BPS
states have a common supersymmetry. Similar argument can be given for the decay of a
quarter BPS state into three or more states.
Acknowledgement: I would like to thank A. Mukherjee, S. Mukhi and R. Nigam for
pointing out a logical error in an earlier version of the draft.
7
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Rare Decay Modes of Quarter BPS Dyons

Rare Decay Modes of Quarter BPS Dyons

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