BPS Membranes in Supergravity

A supergravity theory is a gauge theory of supersymmetry [van Nieuwenhuizen], [Duff-86]. Perhaps the most amazing aspect of these theories is the fact that they automatically include General Relativity in the bosonic sector. Before looking at the membranes of our title, let us examine supergravity theories in general. We will approach them from the point of view of 11 dimensional supergravity, since all other supergravity theories can be obtained from it through dimensional reduction, and it has the simplest super-multiplet.

The first thing to say about supergravity theories is that they can be approached as purely classical theories, with the proviso that there is no contradiction in the idea of a classical fermionic field. The common use of spinors in General Relativity is certainly consistent with that view, and we shall take the approach here that supergravity is a classical field theory, and that the membranes we will discuss are described by classical fields. We will use the traditional names for the fields but in no way imply that they are quantized.

In these theories, the metric is generally replaced by the vielbein as the fundamental field describing the geometry:

g_{μ ν} = η_{a b} e_μ^a e_ν^b

where we are using Greek letters to denote spacetime indices and Latin letters to denote tangent space indices. e is a map between spacetime and Minkowski Spacetime (which is everywhere equivalent to the tangent space); its inverse is

e^μ_a

If the metric is diagonal, the vielbein can be simply the square root of the metric, although this is not always the most useful form. In general when constructing vielbeins by hand, it is a good idea to check them, making sure that in addition to the above, they satisfy:

g_{μ ν} e^μ_a e^ν_b = η_{a b}
g^{μ ν} = η^{a b} e^μ_a e^ν_b and
e_μ^a e_ν^b g^{μ ν} = η^{a b}

The supersymmetric partner of the vielbein is Ψ_{μ α}, the spin 3/2 gravitino (we will use Greek letters from the beginning of the alphabet to denote spinor indices). Spinors are in general 2^{Floor D / 2} - dimensional, which means that the D = 11 gravitino spinor index α varies from 1 to 32.

Γ denote the gamma matrices, which satisfy the Clifford Algebra

{ Γ_{a α}^γ , Γ_{b γ}^β } = 2 η_{a b} δ_α^β

For n = Floor ( D / 2 ), the gamma matrices can be computed in pairs as the following tensor products of sigma matrices [Tanii]:

Γ_{2 i + 1} = σ₃ⁱ x σ₁ x I^{n - i - 1}
Γ_{2 i + 2} = σ₃ⁱ x σ₂ x I^{n - i - 1}

where the superscripts here indicate the number of factors for each sigma matrix in the tensor product and i runs from 0 to n - 1. These can be computed and checked using the following Mathematica code:

sigma = {{{0, 1}, {1, 0}}, {{0, -I}, {I, 0}}, {{1, 0}, {0, -1}}, IdentityMatrix[2]};
anticomm[ xx_List, yy_List] := Dot[xx, yy] + Dot[yy, xx];
recurprod[ mm_List, nn_] := If[ nn == 1, mm[[1]], Dot[ recurprod[mm, nn - 1], mm[[nn]]]];
gammamatrix[ t_, ss_] := Block[ {mm, ind, gmdim, fact, fact2, check, mtest, gammam}, (
mm = t + ss;
If[ Floor[mm / 2] == (mm / 2), fact = I^( Mod[mm, 4] / 2), fact = (-1)^((-3 - mm) / 4)];
gmdim = 2^Floor[mm / 2];
ind = Table[ Flatten[ {
Table[3, {jj, Floor[(ii - 1) / 2]}], 2 - Mod[ii, 2],
Table[4, {jj, Floor[mm / 2] - Floor[(ii - 1) / 2] - 1}]}],
{ii, 2 Floor[mm / 2]}];
If[ Floor[mm / 2] == (mm / 2), , AppendTo[ ind, Table[4, {ii, Floor[mm / 2]}]]];
gammam = Table[ Table[ Table [Product[
sigma[[ind[[kk, ll]],
1 + Mod[ Floor[(ii - 1) / (2^Floor[mm / 2 - ll])], 2],
1 + Mod[ Floor[(jj - 1) / (2^Floor[mm / 2 - ll])], 2]]],
{ll, mm / 2}], {jj, gmdim}], {ii, gmdim}], {kk, mm}];
If[ (Floor[mm / 2] == (mm / 2)), ,
If[ (mm - 1) > 0, gammam[[mm]] = fact * recurprod[ gammam, mm - 1],]];
If[ t == 1, gammam[[mm]] = gammam[[mm]] * I,];
check = Table[ Table[ anticomm[ gammam[[ii]], gammam[[jj]]], {jj, mm}], {ii, mm}];
Print["Anticommutator check: ", {Table[
check[[ii, ii]] == 2 IdentityMatrix[gmdim], {ii, mm - 1}],
check[[mm, mm]] == ((-1)^t) 2 IdentityMatrix[gmdim],
Table[ Table[
{ii, jj, check[[ii, jj]] == 0 IdentityMatrix[gmdim]},
{jj, ii + 1, mm}], {ii, mm - 1}]}];

fact2 = fact;
If[ t == 1, fact2 = I^(1 - Mod[Floor[mm / 2], 2]),];
mtest = fact2 * recurprod[ gammam, mm];
AppendTo[ gammam, mtest];
Print["Chiral operator check: ", Dot[mtest, mtest] == IdentityMatrix[gmdim]];
Return[gammam];)];

The gammamatrix function has two arguments: the number of timelike and the number of spacelike coordinates in the spacetime for which we are generating the matrices. The first parameter is generally either 0 or 1; we will see the utility of this in the following. The variable ind is a list specifying the order of the tensor products of the sigma matrices for each gamma matrix. The recurprod function recursively performs a matrix multiplication of all of the gamma matrices in order to construct the chiral operator Γ_D+1, with an overall numerical factor specified by the variable fact2. Γ_D+1² is equal to the Identity Matrix; in odd dimensions, Γ_D+1 is proportional to the Identity Matrix. Note that the gamma matrix spacetime index runs from 1 to D, with Γ_D being the timelike matrix. Dot is used for matrix multiplication.

The gamma matrices are frequently used in totally antisymmetrized products such as

Γ_{a b α}^β = Γ_{a α}^γ Γ_{b γ}^β - Γ_{b α}^γ Γ_{a γ}^β, etc.

These can be computed using the following Mathematica code as an example:

antigamma[ gammad_, nn_] := Block[ {dim, agamma, ind},(
dim = Length[gammad] - 1;
If[ nn == 2, agamma = Table[ Table[
Dot[ gammad[[jj]], gammad[[kk]]],
{kk, dim}], {jj, dim}] -
Outer[ Times, IdentityMatrix[dim], IdentityMatrix[ Length[ gammad[[1]]]]],];

If[ nn == 3, (
agamma = Table[ Table[ Table[0, {kk, dim}], {jj, dim}], {ii, dim}];
Do[ Do[ Do[(
ind = Permutations[ {ii, kk, ll}];
agamma[[ii, kk, ll]] = Dot[ gammad[[ii]], gammad[[kk]], gammad[[ll]]];
ind = Rest[ind];
Do[
agamma[[ind[[mm,1]], ind[[mm,2]], ind[[mm,3]]]] = Signature[ ind[[mm]]] agamma[[ii, kk, ll]],
{mm, Length[ind]}];),
{ll, kk + 1, dim}], {kk, ii + 1, dim - 1}], {ii, dim - 2}];),];

Return[agamma];)];

This algorithm makes use of the fact that due to the Clifford Algebra, products of different gamma matrices are automatically antisymmetric. Outer is used to create the block diagonal identity matrices which are subtracted out of the simple products in the 2 index case; the usual technique of using the antisymmetry to fill in the remaining matrices is used for the case of the 3 index product. This code assumes that the gamma matrices include the chiral operator as described above.

In order to be able to compute the covariant derivative of fermionic fields

δ_α^β d_μ + ω _μ^{a b} Γ_{a b α}^β / 4

we must define the spin connection ω _μ^{a b}, which can be computed from the Christoffel Connection and the vielbein

ω _μ^{a b} = - η^{b c} e^ν_c (d e_ν^a / d x^μ - Γ^ρ_{μ ν} e_ρ^a)

and which is antisymmetric in the upper indices.

Finally, we must define the gauge field strength tensor, which is self-dual, totally antisymmetric and in D = 11 has 4 indices:

f_{ν ρ σ τ}

BPS (Bogomolny, Prasad and Sommerfield) membranes are bosonic and preserve partial supersymmetry [Duff-95]. For consistency, if we are to be able to set the fermionic fields to zero, we must make sure that the variational equations for those fields also vanish. In D = 11, this means that the gravitino variations must be zero [Stelle]:

&delta Ψ_{μ α} = (δ_α^β d_μ + ω _μ^{a b} Γ_{a b} _α^β / 4 + f^{ν ρ σ τ} e_μ^a e_ν^b e_ρ^c e_σ^d e_τ^e Γ_{a b c d e}_α^β / 144 -
f_μ^{ν ρ σ} e_ν^a e_ρ^b e_σ^c Γ_{a b c}_α^β / 18) ε_β

where we have omitted terms proportional to the gravitino. The supersymmetry parameter ε is called a Killing Spinor, and is taken to be a tensor product of two "sub-spinors" of definite chirality relative to their respective spaces. These spaces correspond to the membrane and the submanifold orthogonal to it. If the gamma matrices for each subspace are denoted by γ and Γ, respectively, and the dimension of the γ subspace is n, the D = 11 gamma matrices have the form

{γ x I, γ_m+1 x Γ}

The following Mathematica code will compute such matrices:

crossmatrix[ mat1_List, mat2_List] := Block[ {row1, col1, row2, col2, crow, ccol, cross},(
row1 = Length[mat1];
col1 = Length[mat1[[1]]];
row2 = Length[mat2];
col2 = Length[mat2[[1]]];
crow = row1 * row2;
ccol = col1 * col2;
cross = Table[ Table[
mat1[[1 + Mod[ Floor[(ii - 1) / row2], row1], 1 + Mod[ Floor[(jj - 1) / col2], col1]]] * mat2[[1 + Mod[ (ii - 1), row2], 1 + Mod[(jj - 1), col2]]],
{jj, ccol}], {ii, crow}];
Return[cross];)];
splitgamma[ gamma1_List, gamma2_List] := Block[ {p1, p2, gd, sgamma},(
p1 = Length[gamma1] - 1;
p2 = Length[gamma2] - 1;
sgamma = Table[0, {ii, p1 + p2}];
If[ p1 == (2 Floor[p1 / 2]),(
gd = Length[ gamma2[[1]]];
Do[ sgamma[[ii]] = crossmatrix[ gamma1[[ii]], IdentityMatrix[gd]],
{ii, p1}];
Do[ sgamma[[ii + p1]] = crossmatrix[ gamma1[[p1 + 1]], gamma2[[ii]]],
{ii, p2}];),
(gd = Length[ gamma1[[1]]];
Do[ sgamma[[ii]] = crossmatrix[ gamma1[[ii]], gamma2[[p2 + 1]]],
{ii, p1}];
Do[ sgamma[[ii + p1]] = crossmatrix[ IdentityMatrix[gd], gamma2[[ii]]],
{ii, p2}];)];

AppendTo[ sgamma, recurprod[ sgamma, p1 + p2]];
Return[sgamma];)];

The crossmatrix function creates a block-diagonal matrix from its arguments, while splitgamma uses it to construct the block-diagonal gamma matrices. Here we see the utility of being able to create gamma matrices corresponding to a spacelike manifold. If we wish, for instance, to construct a set of gamma matrices which explicitly reflect an SO(7) X SO(3,1) symmetry, we might use the following:

gammap = gammamatrix[0, 7];
gammaq = gammamatrix[1, 3];
gamma11 = splitgamma[ gammap, gammaq];

crossmatrix can also be used to construct supersymmetry parameters which reflect the same symmetry:

parmp = Table[{Symbol["pp" <> ToString[i]]}, {i, 8}];
parmq = Table[{Symbol["pq" <> ToString[i]]}, {i, 4}];
parm11 = crossmatrix[ parmp, parmq];

Of course, the membranes must also satisfy the gauge field equations:

D_μ f^{μ ν ρ σ} = - ε^{ν ρ σ α β χ δ ε φ γ η} f_{α β χ δ} f_{ε φ γ η} / 576

and what are usually termed the graviton field equations:

R_{μ ν} - R g_{μ ν} / 2 = f_μ^{ρ σ τ} f_{ν ρ σ τ} / 3 - g_{μ ν} f² / 24

These latter are of course the supergravity generalization of Einstein's Equations. Mosty solutions use highly specific ansatze, ie.:

f^{μ ν ρ σ} = ε^{μ ν ρ σ} f ( r ),

where ε^{μ ν ρ σ} is the volume form for a sub-manifold and r is a radial coordinate on the complementary submanifold. But, one could start with a geometric solution and attempt to find a self-dual gauge field which satisfies the field equations.

We will also be interested in a D = 10 theory, called the IIB Supergravity Theory. It has a dilatino λ in addition to the gravitino, a dilaton φ in addition to the vielbein, and a gauge 3-form field H and a self-dual gauge 5-form field F in place of the 4-form gauge field. These fields all arise from the dimensional reduction. The dilatino variation equation is

δ λ_α = (Γ^a_α^β e^μ_a d_μ φ - e^{- φ / 2} Γ^{a b c}_α^β e^μ_a e^ν_b e^ρ_c H_{μ ν ρ} / 12) ε_β

once again omitting terms proportional to the dilatino and gravitino. The gravitino variation equation is now

δ Ψ_{μ α} = (δ_α^β d_μ + ω _μ^{a b} Γ_{a b} _α^β / 4 +
e^{- φ / 2} (H^{ν ρ σ} e_μ^a e_ν^b e_ρ^c e_σ^d Γ_{a b c d}_α^β - 9 H_μ^{ν ρ} e_ν^a e_ρ^b Γ_{a b}_α^β) / 96 +
i F^{ν ρ σ τ φ} e_ν^a e_ρ^b e_σ^c e_τ^d e_φ^e Γ_{a b c d e}_α^γ e_μ^f Γ_f_γ^β / 480) ε_β

The dilaton field equation is

D^μ d_μ φ = - e^{- φ} H^{μ ν ρ} H_{μ ν ρ} / 12

and the gauge field equations are

D^μ (e^{- φ} H_{μ ν ρ}) = 0

and

F_{ν ρ σ τ φ} = ε_{ν ρ σ τ φ}^{η ι θ κ λ} F_{η ι θ κ λ}

(the latter simply enforcing the self-duality condition). Finally, the graviton field equation for the IIB theory is

R_{μ ν} = d_μ φ d_ν φ / 2 + F_μ^{ρ σ τ λ} F_{ν ρ σ τ λ} / 6 + e^{- φ} (H_μ^{ρ σ} H_{ν ρ σ} - g_{μ ν} H^{μ ν ρ} H_{μ ν ρ} / 12) / 4

Curvature Invariants on Supergravity Membranes

In the following, we will be using the so-called "Einstein metric". The Weyl rescaling to the "string metric" is a conformal transformation (when well-defined, that is, when everwhere finite and nonzero) which means that the resulting metric is not in the same diffeomorphism class as the Einstein metric. We will examine three relatively simple BPS membranes:

the 1-brane in D = 10, using an SO(1,1) X SO(8) "split" and a positive chirality Killing Spinor:
ds² = r^9/2 / (k + r⁶)^3/4 ( dx² - dt²) + (k + r⁶)^1/4 / r^3/2 dy²
φ = ln (r⁶ / (k + r⁶)) / 2
H_{μ 0 1} = - d_μ (ε_{0 1} r⁶ / (k + r⁶))
F = 0
ε = (r⁶ / (k + r⁶))^3/16 {1, 0} X {0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0}
(where r² = Σ y_i²),

the 5-brane in D = 10, using an SO(5,1) X SO(4) split and a positive chirality Killing Spinor:
ds² = r^1/2 / (k + r²)^1/4 ( dx² - dt²) + (k + r²)^3/4 / r^3/2 dy²
φ = ln ((k + r²) / r²) / 2
H_{μ ν ρ} = 2 k ε_{μ ν ρ σ} y^σ / r⁴
F = 0
ε = (r² / (k + r²))^1/16 {1, 0, 0, 1, 0, 1, 1, 0} X {0, 1, 1, 0}

and the 5-brane in D = 11, using an SO(5,1) X SO(5) split and a negative chirality Killing Spinor:
ds² = r / (k + r³)^1/3 ( dx² - dt²) + (k + r³)^2/3 / r² dy²
f_{μ ν ρ σ} = 3 k ε_{μ ν ρ σ τ} y^τ / r⁵
ε = (r³ / (k + r³))^1/12 {0, 1, 1, 0, 1, 0, 0, 1} X {1, 0, 0, 1}

In each case, the y coordinates are orthogonal to the membrane, and the vielbein is simply the square root of the metric.

We compare the invariants we have chosen to examine for these membranes in such a way as to be able to compare the membranes to each other:

D p R

10 1 - 9 k² / (2 r^1/2 (k + r⁶)^9/4)

10 5 3 k² / (2 r^1/2 (k + r²)^11/4)

11 5 3 k² / (2 (k + r³)^8/3)

D	p	R
10	1	- 9 k² / (2 r^1/2 (k + r⁶)^9/4)
10	5	3 k² / (2 r^1/2 (k + r²)^11/4)
11	5	3 k² / (2 (k + r³)^8/3)

D p R^{a b} R_{a b}

10 1 2349 k⁴ / (4 r (k + r⁶)^9/2)

10 5 33 k⁴ / (4 r (k + r²)^11/2)

11 5 207 k⁴ / (4 (k + r³)^16/3)

D	p	R^{a b} R_{a b}
10	1	2349 k⁴ / (4 r (k + r⁶)^9/2)
10	5	33 k⁴ / (4 r (k + r²)^11/2)
11	5	207 k⁴ / (4 (k + r³)^16/3)

D p R^{a b c d} R_{a b c d}

10 1 27 k² (211 k² - 448 k r⁶ + 1792 r¹²) / (16 r (k + r⁶)^9/2)

10 5 3 k² (59 k² + 192 k r² + 384 r⁴) / (16 r (k + r²)^11/2)

11 5 9 k² (13 k² + 32 k r³ + 160 r⁶) / (4 (k + r³)^16/3)

D	p	R^{a b c d} R_{a b c d}
10	1	27 k² (211 k² - 448 k r⁶ + 1792 r¹²) / (16 r (k + r⁶)^9/2)
10	5	3 k² (59 k² + 192 k r² + 384 r⁴) / (16 r (k + r²)^11/2)
11	5	9 k² (13 k² + 32 k r³ + 160 r⁶) / (4 (k + r³)^16/3)

D p R^{a b} R^c_a R_{b c}

10 1 - 40095 k⁶ / (8 r^3/2 (k + r⁶)^27/4)

10 5 75 k⁶ / (8 r^3/2 (k + r²)^33/4)

11 5 675 k⁶ / (8 (k + r³)⁸)

D	p	R^{a b} R^c_a R_{b c}
10	1	- 40095 k⁶ / (8 r^3/2 (k + r⁶)^27/4)
10	5	75 k⁶ / (8 r^3/2 (k + r²)^33/4)
11	5	675 k⁶ / (8 (k + r³)⁸)

D p R^{a b c d} R_{a c} R_{b d}

10 1 729 k⁵ ( -25 k + 168 r⁶) / (8 r^3/2 (k + r⁶)^27/4)

10 5 3 k⁵ (37 k + 72 r²) / (8 r^3/2 (k + r²)^33/4)

11 5 27 k⁵ (25 k + 144 r³) / (8 (k + r³)⁸)

D	p	R^{a b c d} R_{a c} R_{b d}
10	1	729 k⁵ ( -25 k + 168 r⁶) / (8 r^3/2 (k + r⁶)^27/4)
10	5	3 k⁵ (37 k + 72 r²) / (8 r^3/2 (k + r²)^33/4)
11	5	27 k⁵ (25 k + 144 r³) / (8 (k + r³)⁸)

D p R^{a b c d} R^e_a R_{b c d e}

10 1 - 243 k⁴ (389 k² - 1736 k r⁶ + 3164 r¹²) / (64 r^3/2 (k + r⁶)^27/4)

10 5 3 k⁴ (169 k² + 552 k r² + 636 r⁴) / (64 r^3/2 (k + r²)^33/4)

11 5 27 k⁴ (19 k² + 96 k r³ + 80 r⁶) / (16 (k + r³)⁸)

D	p	R^{a b c d} R^e_a R_{b c d e}
10	1	- 243 k⁴ (389 k² - 1736 k r⁶ + 3164 r¹²) / (64 r^3/2 (k + r⁶)^27/4)
10	5	3 k⁴ (169 k² + 552 k r² + 636 r⁴) / (64 r^3/2 (k + r²)^33/4)
11	5	27 k⁴ (19 k² + 96 k r³ + 80 r⁶) / (16 (k + r³)⁸)

D p R^{a b c d} R^{e f}_{a b} R_{c d e f}

10 1 81 k³ ( -3757 k³ + 24864 k² r⁶ - 84504 k r¹² + 89600 r¹⁸) / (128 r^3/2 (k + r⁶)^27/4)

10 5 3 k³ (841 k³ + 4032 k² r² + 6408 k r⁴ + 768 r⁶) / (128 r^3/2 (k + r²)^33/4)

11 5 3 k³ (121 k³ + 816 k² r³ + 1152 k r⁶ - 320 r⁹) / (8 (k + r³)⁸)

D	p	R^{a b c d} R^{e f}_{a b} R_{c d e f}
10	1	81 k³ ( -3757 k³ + 24864 k² r⁶ - 84504 k r¹² + 89600 r¹⁸) / (128 r^3/2 (k + r⁶)^27/4)
10	5	3 k³ (841 k³ + 4032 k² r² + 6408 k r⁴ + 768 r⁶) / (128 r^3/2 (k + r²)^33/4)
11	5	3 k³ (121 k³ + 816 k² r³ + 1152 k r⁶ - 320 r⁹) / (8 (k + r³)⁸)

D p R^{a b c d} R^e_a^f_c R_{b e d f}

10 1 81 k³ ( -2747 k³ + 27552 k² r⁶ - 20328 k r¹² + 125440 r¹⁸) / (512 r^3/2 (k + r⁶)^27/4)

10 5 3 k³ (895 k³ + 4608 k² r² + 15480 k r⁴ + 20736 r⁶) / (512 r^3/2 (k + r²)^33/4)

11 5 3 k³ (221 k³ + 1776 k² r³ + 5328 k r⁶ + 14720 r⁹) / (32 (k + r³)⁸)

D	p	R^{a b c d} R^e_a^f_c R_{b e d f}
10	1	81 k³ ( -2747 k³ + 27552 k² r⁶ - 20328 k r¹² + 125440 r¹⁸) / (512 r^3/2 (k + r⁶)^27/4)
10	5	3 k³ (895 k³ + 4608 k² r² + 15480 k r⁴ + 20736 r⁶) / (512 r^3/2 (k + r²)^33/4)
11	5	3 k³ (221 k³ + 1776 k² r³ + 5328 k r⁶ + 14720 r⁹) / (32 (k + r³)⁸)

D p R^{a b c d} R^e_a^f_c R_{b f d e}

10 1 81 k³ (505 k³ + 1344 k² r⁶ + 32088 k r¹² + 17920 r¹⁸) / (256 r^3/2 (k + r⁶)^27/4)

10 5 9 k³ (9 k³ + 96 k² r² + 1512 k r⁴ + 3328 r⁶) / (256 r^3/2 (k + r²)^33/4)

11 5 3 k³ (25 k³ + 240 k² r³ + 1044 k r⁶ + 3760 r⁹) / (8 (k + r³)⁸)

D	p	R^{a b c d} R^e_a^f_c R_{b f d e}
10	1	81 k³ (505 k³ + 1344 k² r⁶ + 32088 k r¹² + 17920 r¹⁸) / (256 r^3/2 (k + r⁶)^27/4)
10	5	9 k³ (9 k³ + 96 k² r² + 1512 k r⁴ + 3328 r⁶) / (256 r^3/2 (k + r²)^33/4)
11	5	3 k³ (25 k³ + 240 k² r³ + 1044 k r⁶ + 3760 r⁹) / (8 (k + r³)⁸)

D p R^{a b; c} R_{a b; c}

10 1 81 k⁴ (283 k² + 3752 k r⁶ + 46480 r¹²) / (32 r^3/2 (k + r⁶)^27/4)

10 5 9 k⁴ (11 k² + 200 k r² + 1104 r⁴) / (32 r^3/2 (k + r²)^33/4)

11 5 3474 k⁴ r⁶ / (k + r³)⁸

D	p	R^{a b; c} R_{a b; c}
10	1	81 k⁴ (283 k² + 3752 k r⁶ + 46480 r¹²) / (32 r^3/2 (k + r⁶)^27/4)
10	5	9 k⁴ (11 k² + 200 k r² + 1104 r⁴) / (32 r^3/2 (k + r²)^33/4)
11	5	3474 k⁴ r⁶ / (k + r³)⁸

D p R^{a b; c} R_{a c; b}

10 1 81 k⁴ (541 k² + 10136 k r⁶ + 22960 r¹²) / (64 r^3/2 (k + r⁶)^27/4)

10 5 3 k⁴ (55 k² + 744 k r² + 1872 r⁴) / (64 r^3/2 (k + r²)^33/4)

11 5 1089 k⁴ r⁶ / (k + r³)⁸

D	p	R^{a b; c} R_{a c; b}
10	1	81 k⁴ (541 k² + 10136 k r⁶ + 22960 r¹²) / (64 r^3/2 (k + r⁶)^27/4)
10	5	3 k⁴ (55 k² + 744 k r² + 1872 r⁴) / (64 r^3/2 (k + r²)^33/4)
11	5	1089 k⁴ r⁶ / (k + r³)⁸

D p R^{a b}_{; a} R^c_{b; c}

10 1 81 k⁴ (k + 28 r⁶)² / (64 r^3/2 (k + r⁶)^27/4)

10 5 9 k⁴ (k + 12 r²)² / (64 r^3/2 (k + r²)^33/4)

11 5 36 k⁴ r⁶ / (k + r³)⁸

D	p	R^{a b}_{; a} R^c_{b; c}
10	1	81 k⁴ (k + 28 r⁶)² / (64 r^3/2 (k + r⁶)^27/4)
10	5	9 k⁴ (k + 12 r²)² / (64 r^3/2 (k + r²)^33/4)
11	5	36 k⁴ r⁶ / (k + r³)⁸

D p R^{a b c d; e} R_{a b c d; e}

10 1 27 k² (499 k⁴ + 8680 k³ r⁶ + 282576 k² r¹² - 401408 k r¹⁸ + 286720 r²⁴) / (32 r^3/2 (k + r⁶)^27/4)

10 5 3 k² (59 k⁴ + 840 k³ r² + 5904 k² r⁴ + 6144 k r⁶ + 18432 r⁸) / (32 r^3/2 (k + r²)^33/4)

11 5 72 k² r⁶ (56 k² - 100 k r³ + 175 r⁶) / (k + r³)⁸

D	p	R^{a b c d; e} R_{a b c d; e}
10	1	27 k² (499 k⁴ + 8680 k³ r⁶ + 282576 k² r¹² - 401408 k r¹⁸ + 286720 r²⁴) / (32 r^3/2 (k + r⁶)^27/4)
10	5	3 k² (59 k⁴ + 840 k³ r² + 5904 k² r⁴ + 6144 k r⁶ + 18432 r⁸) / (32 r^3/2 (k + r²)^33/4)
11	5	72 k² r⁶ (56 k² - 100 k r³ + 175 r⁶) / (k + r³)⁸

D p R^{a b c d}_{; a} R^e_{b c d; e}

10 1 81 k⁴ (25 k² - 2632 k r⁶ + 70000 r¹²) / (32 r^3/2 (k + r⁶)^27/4)

10 5 3 k⁴ (11 k² + 456 k r² + 4752 r⁴) / (32 r^3/2 (k + r²)^33/4)

11 5 4770 k⁴ r⁶ / (k + r³)⁸

D	p	R^{a b c d}_{; a} R^e_{b c d; e}
10	1	81 k⁴ (25 k² - 2632 k r⁶ + 70000 r¹²) / (32 r^3/2 (k + r⁶)^27/4)
10	5	3 k⁴ (11 k² + 456 k r² + 4752 r⁴) / (32 r^3/2 (k + r²)^33/4)
11	5	4770 k⁴ r⁶ / (k + r³)⁸

D p Euler class

10 1 229635 k⁵ (5 k + 8 r⁶)² (5 k³ + 24 k² r⁶ - 102 k r¹² - 256 r¹⁸) / (2097152 π⁵ r^5/2 (k + r⁶)^45/4)

10 5 - 135 k⁶ (8 k³ + 57 k² r² + 48 k r⁴ - 64 r⁶) / (1048576 π⁵ r^1/2 (k + r²)^55/4)

D	p	Euler class
10	1	229635 k⁵ (5 k + 8 r⁶)² (5 k³ + 24 k² r⁶ - 102 k r¹² - 256 r¹⁸) / (2097152 π⁵ r^5/2 (k + r⁶)^45/4)
10	5	- 135 k⁶ (8 k³ + 57 k² r² + 48 k r⁴ - 64 r⁶) / (1048576 π⁵ r^1/2 (k + r²)^55/4)

D p ε^{a b c ...} ε^{e f g}_... R_{b c e}^h R_{f g a h}

10 1 - 17010 k² (485 k² + 448 k r⁶ - 1792 r¹²) / (r (k + r⁶)^9/2)

10 5 1890 k² ( -29 k² + 192 k r² + 384 r⁴) / (r (k + r²)^11/2)

11 5 - 181440 k² (33 k² - 32 k r³ - 160 r⁶) / (k + r³)^16/3

D	p	ε^{a b c ...} ε^{e f g}_... R_{b c e}^h R_{f g a h}
10	1	- 17010 k² (485 k² + 448 k r⁶ - 1792 r¹²) / (r (k + r⁶)^9/2)
10	5	1890 k² ( -29 k² + 192 k r² + 384 r⁴) / (r (k + r²)^11/2)
11	5	- 181440 k² (33 k² - 32 k r³ - 160 r⁶) / (k + r³)^16/3

The D = 10 membranes have essential singularities at r=0, and the D=11 5-brane has one at r = -k^1/3. The singularity in the D=11 5-brane metric at r=0 appears to be a coordinate singularity.

For the D = 10 branes, we find that

2 R^{a b; c} R_{a b; c} - R^{a b c d}_{; a} R^e_{b c d; e} = R^{a b; c} R_{a c; b}

while for the D = 11 brane,

2 R^{a b; c} R_{a b; c} - R^{a b c d}_{; a} R^e_{b c d; e} = 2 R^{a b; c} R_{a c; b}

For all three membranes, we find that

ε^{a b c ...} ε^{e f g}_... R_{b c e}^h R_{f g a h} - 2 (D - 3)! R^{a b c d} R_{a b c d} ~ R^{a b} R_{a b}

The next section discusses several general results concerning warped product spacetimes.

Table of Contents

Index:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

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