# 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.

gμ ν = ηa b eμa eνb
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
eμa

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:

gμ ν eμa eνb = ηa b

gμ ν = ηa b eμa eνb and

eμa eνb gμ ν = ηa b

{ Γa αγ , Γb γβ } = 2 ηa b δαβ
For n = Floor ( D / 2 ), the gamma matrices can be computed in pairs as the following tensor products of sigma matrices [Tanii]:
Γ2 i + 1 = σ3i x σ1 x In - i - 1

Γ2 i + 2 = σ3i x σ2 x In - 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};

anticomm[ xx_List, yy_List] := Dot[xx, yy] + Dot[yy, xx];

recurprod[ mm_List, nn_] := If[ nn == 1, mm[], 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 ΓD+1, with an overall numerical factor specified by the variable fact2. ΓD+12 is equal to the Identity Matrix; in odd dimensions, ΓD+1 is proportional to the Identity Matrix. Note that the gamma matrix spacetime index runs from 1 to D, with ΓD being the timelike matrix. Dot is used for matrix multiplication.

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

Γa b αβ = Γa αγ Γb γβ - Γb αγ Γa γβ, etc.
These can be computed using the following Mathematica code as an example:
antigamma[ gammad_, nn_] := Block[ {dim, agamma, ind},(

If[ nn == 2, agamma = Table[ Table[

{kk, dim}], {jj, dim}] -

Outer[ Times, IdentityMatrix[dim], IdentityMatrix[ Length[ gammad[]]]],];
If[ nn == 3, (
agamma = Table[ Table[ Table[0, {kk, dim}], {jj, dim}], {ii, dim}];

Do[ Do[ Do[(

ind = Permutations[ {ii, kk, 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.
δαβ dμ + ω μa b Γa b αβ / 4
we must define the spin connection ω μa b, which can be computed from the Christoffel Connection and the vielbein
ω μa b = - ηb c eνc (d eνa / d xμ - Γρμ ν eρa)
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:

fν 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 Ψμ α = (δαβ dμ + ω μa b Γa b αβ / 4 + fν ρ σ τ eμa eνb eρc eσd eτe Γa b c d e αβ / 144 -
fμν ρ σ eνa eρb eσc Γa b c αβ / 18) εβ
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, γm+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[]];

row2 = Length[mat2];

col2 = Length[mat2[]];

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[]];

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[]];

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:
Dμ fμ ν ρ σ = - εν ρ σ α β χ δ ε φ γ η fα β χ δ fε φ γ η / 576
and what are usually termed the graviton field equations:
Rμ ν - R gμ ν / 2 = fμρ σ τ fν ρ σ τ / 3 - gμ ν f2 / 24
These latter are of course the supergravity generalization of
Einstein's Equations. Mosty solutions use highly specific ansatze, ie.:
fμ ν ρ σ = εμ ν ρ σ f ( r ),

where εμ 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.

δ λα = (Γaαβ eμa dμ φ - e - φ / 2 Γa b cαβ eμa eνb eρc Hμ ν ρ / 12) εβ
once again omitting terms proportional to the dilatino and gravitino. The gravitino variation equation is now
δ Ψμ α = (δαβ dμ + ω μa b Γa b αβ / 4 +
e - φ / 2 (Hν ρ σ eμa eνb eρc eσd Γa b c d αβ - 9 Hμν ρ eνa eρb Γa b αβ) / 96 +

i Fν ρ σ τ φ eνa eρb eσc eτd eφe Γa b c d e αγ eμf Γf γβ / 480) εβ

The dilaton field equation is
Dμ dμ φ = - e - φ Hμ ν ρ Hμ ν ρ / 12
and the gauge field equations are
Dμ (e - φ Hμ ν ρ) = 0
and
Fν ρ σ τ φ = εν ρ σ τ φη ι θ κ λ Fη ι θ κ λ
(the latter simply enforcing the self-duality condition). Finally, the graviton field equation for the IIB theory is
Rμ ν = dμ φ dν φ / 2 + Fμρ σ τ λ Fν ρ σ τ λ / 6 + e - φ (Hμρ σ Hν ρ σ - gμ ν Hμ ν ρ Hμ 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 = r9/2 / (k + r6)3/4 ( dx2 - dt2) + (k + r6)1/4 / r3/2 dy2

φ = ln (r6 / (k + r6)) / 2

Hμ 0 1 = - dμ0 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 yi2),
• the 5-brane in D = 10, using an SO(5,1) X SO(4) split and a positive chirality Killing Spinor:
ds2 = r1/2 / (k + r2)1/4 ( dx2 - dt2) + (k + r2)3/4 / r3/2 dy2

φ = ln ((k + r2) / r2) / 2

Hμ ν ρ = 2 k εμ ν ρ σ yσ / 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

fμ ν ρ σ = 3 k εμ ν ρ σ τ yτ / 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:

DpR
101- 9 k2 / (2 r1/2 (k + r6)9/4)
1053 k2 / (2 r1/2 (k + r2)11/4)
1153 k2 / (2 (k + r3)8/3)

DpRa b Ra b
1012349 k4 / (4 r (k + r6)9/2)
10533 k4 / (4 r (k + r2)11/2)
115207 k4 / (4 (k + r3)16/3)

DpRa b c d Ra b c d
10127 k2 (211 k2 - 448 k r6 + 1792 r12) / (16 r (k + r6)9/2)
1053 k2 (59 k2 + 192 k r2 + 384 r4) / (16 r (k + r2)11/2)
1159 k2 (13 k2 + 32 k r3 + 160 r6) / (4 (k + r3)16/3)

DpRa b Rca Rb c
101- 40095 k6 / (8 r3/2 (k + r6)27/4)
10575 k6 / (8 r3/2 (k + r2)33/4)
115675 k6 / (8 (k + r3)8)

DpRa b c d Ra c Rb d
101729 k5 ( -25 k + 168 r6) / (8 r3/2 (k + r6)27/4)
1053 k5 (37 k + 72 r2) / (8 r3/2 (k + r2)33/4)
11527 k5 (25 k + 144 r3) / (8 (k + r3)8)

DpRa b c d Rea Rb c d e
101- 243 k4 (389 k2 - 1736 k r6 + 3164 r12) / (64 r3/2 (k + r6)27/4)
1053 k4 (169 k2 + 552 k r2 + 636 r4) / (64 r3/2 (k + r2)33/4)
11527 k4 (19 k2 + 96 k r3 + 80 r6) / (16 (k + r3)8)

DpRa b c d Re fa b Rc d e f
10181 k3 ( -3757 k3 + 24864 k2 r6 - 84504 k r12 + 89600 r18) / (128 r3/2 (k + r6)27/4)
1053 k3 (841 k3 + 4032 k2 r2 + 6408 k r4 + 768 r6) / (128 r3/2 (k + r2)33/4)
1153 k3 (121 k3 + 816 k2 r3 + 1152 k r6 - 320 r9) / (8 (k + r3)8)

DpRa b c d Reafc Rb e d f
10181 k3 ( -2747 k3 + 27552 k2 r6 - 20328 k r12 + 125440 r18) / (512 r3/2 (k + r6)27/4)
1053 k3 (895 k3 + 4608 k2 r2 + 15480 k r4 + 20736 r6) / (512 r3/2 (k + r2)33/4)
1153 k3 (221 k3 + 1776 k2 r3 + 5328 k r6 + 14720 r9) / (32 (k + r3)8)

DpRa b c d Reafc Rb f d e
10181 k3 (505 k3 + 1344 k2 r6 + 32088 k r12 + 17920 r18) / (256 r3/2 (k + r6)27/4)
1059 k3 (9 k3 + 96 k2 r2 + 1512 k r4 + 3328 r6) / (256 r3/2 (k + r2)33/4)
1153 k3 (25 k3 + 240 k2 r3 + 1044 k r6 + 3760 r9) / (8 (k + r3)8)

DpRa b; c Ra b; c
10181 k4 (283 k2 + 3752 k r6 + 46480 r12) / (32 r3/2 (k + r6)27/4)
1059 k4 (11 k2 + 200 k r2 + 1104 r4) / (32 r3/2 (k + r2)33/4)
1153474 k4 r6 / (k + r3)8

DpRa b; c Ra c; b
10181 k4 (541 k2 + 10136 k r6 + 22960 r12) / (64 r3/2 (k + r6)27/4)
1053 k4 (55 k2 + 744 k r2 + 1872 r4) / (64 r3/2 (k + r2)33/4)
1151089 k4 r6 / (k + r3)8

DpRa b; a Rcb; c
10181 k4 (k + 28 r6)2 / (64 r3/2 (k + r6)27/4)
1059 k4 (k + 12 r2)2 / (64 r3/2 (k + r2)33/4)
11536 k4 r6 / (k + r3)8

DpRa b c d; e Ra b c d; e
10127 k2 (499 k4 + 8680 k3 r6 + 282576 k2 r12 - 401408 k r18 + 286720 r24) / (32 r3/2 (k + r6)27/4)
1053 k2 (59 k4 + 840 k3 r2 + 5904 k2 r4 + 6144 k r6 + 18432 r8) / (32 r3/2 (k + r2)33/4)
11572 k2 r6 (56 k2 - 100 k r3 + 175 r6) / (k + r3)8

DpRa b c d; a Reb c d; e
10181 k4 (25 k2 - 2632 k r6 + 70000 r12) / (32 r3/2 (k + r6)27/4)
1053 k4 (11 k2 + 456 k r2 + 4752 r4) / (32 r3/2 (k + r2)33/4)
1154770 k4 r6 / (k + r3)8

DpEuler class
101229635 k5 (5 k + 8 r6)2 (5 k3 + 24 k2 r6 - 102 k r12 - 256 r18) / (2097152 π5 r5/2 (k + r6)45/4)
105- 135 k6 (8 k3 + 57 k2 r2 + 48 k r4 - 64 r6) / (1048576 π5 r1/2 (k + r2)55/4)

Dpεa b c ... εe f g... Rb c eh Rf g a h
101- 17010 k2 (485 k2 + 448 k r6 - 1792 r12) / (r (k + r6)9/2)
1051890 k2 ( -29 k2 + 192 k r2 + 384 r4) / (r (k + r2)11/2)
115- 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 = -k1/3. 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 Ra b; c Ra b; c - Ra b c d; a Reb c d; e = Ra b; c Ra c; b
while for the D = 11 brane,
2 Ra b; c Ra b; c - Ra b c d; a Reb c d; e = 2 Ra b; c Ra c; b
For all three membranes, we find that
εa b c ... εe f g... Rb c eh Rf g a h - 2 (D - 3)! Ra b c d Ra b c d ~ Ra b Ra b
The next section discusses several general results concerning warped product spacetimes.