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The flat directions previously noticed for non-BPS attractors
extend to the whole • ˆ
flow. This is due to the existence of a nontrivial subgroup H
of the duality group
G which leaves non-BPS charge vectors invariant, while acting
non-trivially on the
scalars in G/H. We expect all these results to extend to all
symmetric supergravity theories by extension from the STU case. On the
other hand, the flat directions we see are intricately tied to the
symmetries of our scalar manifold so we don't expect to see such flat
directions occur naturally in non-symmetric = 2 theories. N The marginality
property of our non-BPS mass formula begs for an explanation. Such an
explanation might possibly be found by following up on the surprising
observation in [48] that the near horizon regions for our non-BPS black
holes can be lifted to super-symmetric five dimensional solutions, with
"supersymmetry without supersymmetry" as in [49, 50]. In any case, an
explanation is beyond the scope of this paper. One might also wonder if the
marginality property extends to non-BPS extremal black holes (with or
without wrapped D6-brane charge) in other = 2 theories, especially those N
with cubic prepotentials. Finally, it would be interesting to know how the
marginality ′ property survives α corrections to the action.
In this context, it would be particularly interesting to look at cases
where Γ is properly quantized but the subparts Γ are not.
µ Acknowledgments We would like to thank Oleg Lunin for early
discussions on this work. JS would like to thank the University of
California at Berkeley for hospitality during part of this work. The work
of EG is supported in part by the US DOE under contract No.
DE-AC03-76SF00098 and the Berkeley Center for Theoretical Physics. The work
of FL is supported by DoE under grant DE-FG02-95ER40899. The work of JS was
partially funded by DOE under the contract number DE-AC02-05CH11231. A.
Non-BPS D0-D4 with general moduli: derivation We consider a non-BPS system
with D0-brane charge Q < 0 and three D4-brane charges 0 i P > 0 for i
= 1, 2, 3. In order to be sufficiently general we include arbitrary complex
moduli fields z = x i y (j = 1, 2, 3) at the outset. j j j − The
Lagrangian (2.14) for the analogue mechanics problem is: 3 ′
′2 2 (y ) + (x ) 1 ′ 2U 2 i i + e V , (A.1) = (U ) + BH eff 2
4 y L i i=1 ( ( ( V (x , y ) = BH i i where the potential V in the case of
the STU-model is: BH 3 3 2 1 2 1 P x x +P x x +P x x 1 Q 1 2 1 3 2 3
− 2 i 2 0 y y y .JHEP01(2008)040 (( (A.2) (P ) y + 2 y y y 1 2 3 1 y
y y 1 2 3 +Q 0 y y y 1 2 3 3 2 1 2 1 2 1 3 2 3 ( (P x x +P x x +P x x ) + +
1 2 3 i 2 ( i=1 ( 3 2 k j 2 i j k (y ) (P x +P x ) ( j=k=i=1( (
(A.17) i j BH ∂φ 2 −
− k i<j i<j k ( ( ( ( ( The dynamical
equations still appear very complicated at this point but they are in fact
integrable. It would be interesting to explore the structure that makes
this possible but we will be content with simply finding the solutions,
proceeding as follows [40]. Suppose we could find solutions α
satisfying: i 2 M − − ′′ 2(α α )
2(α α ) 2α 0 j j i k k , i = j = k , e + e e α = i
2 ( ( ′ ′ ′ ′′ − ( ( 2 2
2(α +α α ) j i k α α [(α ) 2α ] +
= M e .
(A.18) i j i i 0 − − − ( ( i i<j i=j=k
( ( ( ( ( If this is possible then (A.12) reduces to: ′ −
− 3α α α i j k , j = k = i ,
(A.19) t = M (t + t ) e 0 j k i ( ( - and (A.14) reduces to:
′ β β = M e 1 + t t .
(A.20) 0 i j - i<j (
One can check that (A.19) and (A.20) ensure that the remaining equations
(A.13) and (A.7)
(A.15)JHEP01(2008)040 are satisfied, provided we take the same
sign in both equations. It is therefore sufficient to solve (A.18)-(A.20).
The first equation in (A.18) can be reorganized as:
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