Three-dimensional solutions of M-theory on $S^1/Z_2 \times T_7$

6 Pages·0.1 MB·English

by  Gabriel Lopes Cardoso

Three-dimensional solutions of M-theory
on S1/Z T
2 7
×
8
Gabriel Lopes Cardoso
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9
1
Institute for Theoretical Physics, Utrecht University
n
a NL-3508 TA Utrecht, The Netherlands
J

6
1

1 Abstract: We review various three-dimensional solutions in the low-energy de-
v scription of M-theory on S1/Z T . These solutions have an eleven dimensional
2 7
3 ×
interpretation in terms of intersecting M-branes.
1
1
1
0
8 1 Introduction
9
/
h
t Eleven dimensional supergravity, the low-energy effective theory of M-theory, has
-
p four 1/2-supersymmetric solutions: the M-2-brane, the M-5-brane, the M-wave and
e (in the S1 Kaluza-Klein vacuum) the M-monopole. We will refer to them collectively
h
: as M-branes. Intersections of these four basic configurations give rise to solutions
v
i preserving a smaller amount of supersymmetry (for a review see for instance [1, 2]
X
and references therein). In the following, we will review various three-dimensional
r
a solutions in the low-energy description of M-theory on S1/Z T . These solutions
2 7
×
have an eleven dimensional interpretation in terms of orthogonally intersecting M-
branes.
The eleven-dimensional space-timeline element describing asupersymmetric con-
figuration of orthogonally intersecting M-branes is given in terms of harmonic func-
tions which depend on some of the overall transverse directions [2, 3, 4]. If these
overall transverse directions do not include the eleven-th dimension x , then its
11
compactification on S1/Z T down to three dimensions is also a solution of the
2 7
×
low-energy effective field theory of ten-dimensional heterotic string theory compact-
ified on a seven-torus. Such a solution is determined in terms of harmonic functions
¯
H(z,z¯) = f(z) + f(z¯), where z and z¯ denote the spatial dimensions. The static
solutions presented in [5, 6, 7] are of this type. If, on the other hand, the overall
transverse directions include the eleven-th dimension x , then compactifying the
11
eleven-dimensional configuration on S1/Z T gives rise to a solution which is
2 7
×
determined in terms of harmonic functions H(x ,z,z¯). An example of such a super-
11
gravity solution has recently been given [8] in the context of M-theory compactified
1
on T . There, it was also shown that this solution corresponds to a BPS state with
8
mass that goes like 1/g3. Evidence for the existence of BPS states with masses that
go like 1/g3 or higher inverse powers of the string coupling constant g in M-theory
compactifications down to three (andlower) dimensions has been emerging in studies
of U-duality multiplets of M-theory on tori [9, 8].
If the harmonic functions H are taken to be independent of the eleven-th di-
mension, then the resulting three-dimensional solutions can be turned into finite
energy solutions [5, 7] by utilizing a mechanism first discussed in the context of four-
dimensionalstringycosmicstringsolutions[10]. Forinstance, thesolutionspresented
in [7] have, at spatial infinity (z ), an asymptotic behaviour corresponding to
→ ∞
f(z) lnz. At finite distance these asymptotic solutions become ill-defined and so
∝
need to be modified. The associated corrections are all encoded in f(z). They can
be determined by requiring the solutions to have finite energy. This requirement,
together with the appropriate asymptotic behaviour, determines f(z) to be given
by f(z) j−1(z). In section 2 we will review some of the three-dimensional solu-
∝
tions constructed in [7], namely those which have a ten-dimensional interpretation
in terms of a fundamental string, a wave and up to three orthogonally intersecting
NS 5-branes as well as up to three Kaluza–Klein monopoles. These solutions are la-
belled by an integer n with n = 1,2,3,4. The energy E carried by these (one-center)
solutions is given by E = 2n π (in units where 8πG = 1).
6 N
Another class of solutions constructed in [7]consists ofsolutions carrying energies
E = nπ, where n = 1,2,3,4. An example of a solution with E = π is given by a
6 6
wrapped M-monopole. In section 3 we briefly review this solution. We then compare
it to the solution recently discussed in [8], which can be constructed by considering
a periodic array of M-monopoles along a transverse direction and by identifying
this transverse direction with the eleven-th dimension. This latter solution is thus
specified by a harmonic function H(x ,z,z¯) on S1/Z R2.
11 2
×
2 A class of finite energy solutions
The effective low-energy field theory of theten-dimensional heterotic string compact-
ified on a seven-dimensional torus is obtained from reducing the ten-dimensional
N = 1 supergravity theory coupled to U(1)16 super Yang–Mills multiplets (at a
generic point in the moduli space). The massless ten-dimensional bosonic fields are
(10) (10) (10)I
the metric G , the antisymmetric tensor field B , the U(1) gauge fields A
MN MN M
and the scalar dilaton Φ(10) with (0 M,N 9, 1 I 16). The field strengths
≤ ≤ ≤ ≤
are F(10)I = ∂ A(10)I ∂ A(10)I and H(10) = (∂ B(10) 1A(10)IF(10)I)+ cyclic
MN M N − N M MNP M NP − 2 M NP
permutations of M,N,P.
The bosonic part of the ten-dimensional action is
d10x G(10)e−Φ(10)[ (10) +G(10)MN∂ Φ(10)∂ Φ(10)
M N
S ∝ − R
Z q
1 1
H(10) H(10)MNP F(10)IF(10)IMN]. (1)
−12 MNP − 4 MN
2
The reduction to three dimensions (see [5] and references therein) introduces the
graviton g , the dilaton φ Φ(10) ln√detG , with G the internal 7D metric,
µν mn mn
≡ −
30 U(1) gauge fields A(a) (A(1)m, A(2), A(3)I ) (a = 1,...,30, m = 1,...,7, I =
µ ≡ µ µm µ
1,...,16) , where A(1)m are the 7 Kaluza–Klein gauge fields coming from the
µ
reduction of G(10), A(2) B +B A(1)n+1aI A(3)I are the 7 gauge fields coming
MN µm ≡ µm mn µ 2 m µ
from the reduction of B(10) and A(3)I AI aI A(1)m are the 16 gauge fields from
MN µ ≡ µ − m µ
A(10)I. The field strengths F(a) are given by F(a) = ∂ A(a) ∂ A(a). Finally, B(10)
M µν µν µ ν − ν µ MN
induces the two-form field B with field strength H = ∂ B 1A(a)L F(b)+
µν µνρ µ νρ − 2 µ ab νρ
cyclic permutations.
The bosonic part of the three-dimensional action in the Einstein frame is then
1 1
= d3x√ g gµν∂µφ∂νφ e−4φgµµ′gνν′gρρ′HµνρHµ′ν′ρ′
S 4 − {R− − 12
Z
1 1
e−2φgµµ′gνν′F(a)(LML) F(b) + gµνTr (∂ ML∂ ML) , (2)
−4 µν ab µ′ν′ 8 µ ν }
where a = 1,...,30. Here M denotes a matrix comprising the scalar fields G , aI
mn m
and B [11, 5].
mn
We now construct static solutions by setting the associated three-dimensional
Killing spinor equations to zero. In doing so, we restrict ourselves to backgrounds
with H = 0 and aI = 0. It can be checked that the resulting solutions to
µνρ m
the Killing spinor equations satisfy the equations of motion derived from (2). The
associated three-dimensional Killing spinor equations in the Einstein frame are [6]
1
δχI = e−2φF(3)Iγµνε,
2 µν
1 1
δλ = e−φ∂ φ+lndetea γµ I ε+ e−2φ[ B F(1)n +F(2) ]γµνγ4 Σmε
−2 µ{ m} ⊗ 8 4 − mn µν µνm ⊗
1
+ e−φ∂ B γµ Σmnε ,
µ mn
4 ⊗
1 1
δψ = ∂ ε+ ω γαβε+ (e eν e eν)∂ φγαβε
µ µ 4 µαβ 4 µα β− µβ α ν
1 1
+ (en∂ e en∂ e )I Σabε e−φ[emF(2) e F(1)m]γνγ4 Σaε
8 a µ nb− b µ na 4 ⊗ −4 a µν(m) − ma µν ⊗
1 1
∂ B I Σmnε+ e−φB F(1)n γνγ4 Σmε ,
−8 µ mn 4 ⊗ 4 mn µν ⊗
1 1
δψ = e−φ(em∂ e +em∂ e )γµγ4 Σaε+ e−2φemB F(1)n γµνε
d −4 d µ ma a µ md ⊗ 8 d mn µν
1 1
+ e−φemen∂ B γµγ4 Σaε e−2φ[ e F(1)m +emF(2) ]γµνε , (3)
4 d a µ mn ⊗ − 8 md µν d µνm
where δψ emδψ denotes the variation of the internal gravitini. Here we have
d ≡ d m
performed a 3+7 split of the ten-dimensional gamma matrices [6].
In the following, we review a particular class of static solutions to the Killing
spinor equations (3) constructed in [6, 7]. The solutions in this class are labelled by
an integer n (n = 1,2,3,4). The associated space-time line element is given by
ds2 = dt2 +H2ndωdω¯ , H = f(ω)+f¯(ω¯) , (4)
−
3
where ω = alnz = a(lnr + iθ) = rˆ+ iθˆ. Here, a = n+1 α α , where α and
2π | i i+7| i
αi+7 denote two electric charges carried by each of the soluqtions in this class. The
associated field strengths of the three-dimensional gauge fields are [7]
∂ H ∂ H
F(1)i = η √Gii β , F(2) = η G β , β = rˆ,θˆ , η = . (5)
tβ αi H2 tβi − αi ii H2 αi ±
q
The three-dimensional dilaton field is given by e−φ = H. The i-th component of the
internal metric G reads G = αi . In addition, there are (depending on the
mn ii |αi+7|
integer n) various additional non-constant background fields G and B , which
mn mn
are also determined in terms of f(ω) [7].
Solving the Killing spinor equations (3) does not determine the form of f(ω). Its
form can be determined by demanding the solution to behave as f(ω) ω at
≈ 2(n+1)
spatial infinity [6] and by requiring the solution to have finite energy, as follows [7].
The energy carried by any of the solutions in this class is compute to be (in units
where 8πG = 1)
N
¯
¯ ˆ ˆ
∂ f∂ f ∂ f∂ f
ω ω¯ z z¯
E = i2n dωdω¯ = i2n dzdz¯ , (6)
(f +f¯)2 (fˆ+f¯ˆ)2
Z Z
where we have introduced fˆ = n+1f for later convenience. There is an elegant
π
mechanism [10]forrenderingtheintegral(6)finite. Letustakez tobethecoordinate
of a complex plane. Then there is a one-to-one map from a certain domain F on
ˆ
the f-plane (the so called ‘fundamental’ domain) to the z-plane. This map is known
ˆ
as the j-function, j(f) = z. By means of this map, the integral (6) can be pulled
back from the z-plane to the domain F (the z-plane covers F exactly once). Then,
by using integration by parts, this integral can be related to a line integral over the
boundary of F, which is evaluated to be [10]
2π π
E = 2n = 2n (7)
12 6
and, hence, is finite. By expanding j(fˆ) = e2πfˆ+ 744 + (e−2πfˆ) = z we recover
O
f(ω) ω at spatial infinity.
≈ 2(n+1)
We note that the solutions discussed above represent one-center solutions. They
ˆ
can be generalised to multi-center solutions via j(f(z)) = P(z)/Q(z), where P(z)
and Q(z) are polynomials in z with no common factors. These are the analogue of
the multi-string configurations discussed in [10].
¯
It can be checked that the curvature scalar ∂ f∂ f blows up at the special
ω ω¯
ˆ R ∝
point f = 1 (at this point, the j-function and its derivatives are given by j =
1728,j′ = 0), whereas it is well behaved at the point fˆ = eiπ/6 (at this point, the
j-function and its derivatives are given by j = j′ = j′′ = 0). It would be interesting
to understand the physics at this special point in moduli space further.
As mentioned in the introduction thethree-dimensional solutions reviewed inthis
section can be given an eleven dimensional interpretation [7] in terms of orthogonally
intersecting M-branes. The two gauge fields (5), in particular, arise from a wave and
from a M-2-brane in eleven dimensions, respectively.
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3 Wrapped M-monopoles
Another class of solutions constructed in [7] is the class of solutions carrying energy
E = nπ, where n = 1,2,3,4. An example of a solution carrying energy E = π is
6 6
obtained by compactifying the M-monopole in the following way. The M-monopole
in eleven dimensions is given by the metric [12]
ds2 = dt2 +Hdy2+H−1(dψ +A dy )2 +dx2 +...+dx2, i = 1,2,3, (8)
11 − i i i 1 6
with H = H(y ), F = ∂ A ∂ A = cε ∂ H, c = . Here, ψ denotes a periodic
i ij i j j i ijk k
− ±
variable. Identifying one of the x with the eleven-th cooordinate and compactifying
i
the remaining x on a five-torus yields the five-dimensional line element
i
ds2 = dt2 +Hdy2+H−1(dψ +A dy )2. (9)
5 − i i i
Now, consider the case that H only depends on two of the coordinates y , that is
i
H = H(z,z¯) with z = y + iy . Then we can set A = A = 0, and ∂ A =
2 3 2 3 2 1
c∂ H, ∂ A = c∂ H. The metric is now
3 3 1 2
−
ds2 = dt2 +Hdzdz¯+Hdy2+H−1(dψ +A dy )2
5 − 1 1 1
= g dxµdxν +G dxmdxn, (10)
µν mn
where the off-diagonal internal metric is given by
H +A2H−1 A H−1
G = 1 1 . (11)
mn A H−1 H−1
1 !
The resulting three-dimensional solution can be turned into a finite energy solution
with energy E = π by using the mechanism described in the previous section.
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Let us now compare this solution to the one obtained by identifying the eleven-th
dimension notwith oneof thex , but rather with oneofthe y [8]. Compactifying the
i i
eleven-dimensional line element (8) on a six-torus yields again the five-dimensional
line element (9). Compactifying over ψ yields the four-dimensional line element
ds2 = dt2 +Hdy2 (12)
4 − i
withinternalmetriccomponentG = H−1andmagneticgaugefieldF = cε ∂ H.
ψψ ij ijk k
The Bianchi identity of F yields ∆H = 0. Now, identifying one of the transverse
coordinates y with the eleven-th coordinate (say y = x ) and setting z = y +iy
i 1 11 2 3
yields the Laplacian as ∆ = ∂2 + 4∂ ∂ . Thus, H is now a harmonic function on
x11 z z¯
S1/Z R2. In [13] a solution to the Laplace equation on S1 R2 was constructed.
2
× ×
It can be readily adapted to the case at hand. We take the range of x to be
11
x [ 1, 1], with a peridic identification x x + 1 of the endpoints. The Z
11 ∈ −2 2 11 ∼ 11 2
symmetry acts as x x . A Z invariant solution to the Laplace equation on
11 11 2
→ −
S1/Z R2 is then given by
2
×
1 1
H(x ,z,z¯) = . (13)
11 nX∈Z|n− 21| − (x11 −(n− 21))2 +zz¯
 q 
5
The constant term in (13) is such that H is regular at z = 0 on the orbifold plane
x = 0. For z H, on the other hand, H reduces to H logzz¯ on the plane
11
| | → ∞ ≈
x = 0.
11
Acknowledgement
We would like to thank K. Behrndt, M. Bourdeau and E. Verlinde for useful discus-
sions.
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