Matrix Theory of pp Waves

0.15 MB·

by  Jeremy Michelson

hep-th/0401050
UK/04-01
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MATRIX THEORY OF pp WAVES∗
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2

n JEREMY MICHELSON
a
J Department of Physics and Astronomy
University of Kentucky
8
Lexington, KY 40506 USA

1 E-mail: [email protected]
v
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0 TheMatrixTheorythathasbeenproposedforvariousppwavebackgroundsisdis-
1 cussed. Particularemphasisisontheexistenceofnovelnontrivialsupersymmetric
0 solutions of the Matrix Theory. These correspond to branes of various shapes
(ellipsoidal, paraboloidal, and possibly hyperboloidal) that are unexpected from
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previousstudiesofbranesinppwavegeometries.
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/
h
t
- 1. INTRODUCTION
p
e TheBMN1 MatrixTheory,whichdescribesthemaximallysupersymmetric
h
: pp wave of M-theory2, is a very nice arena for learning about M-theory. It
v
is simple enough to be reasonably tractable, yet describes a curved back-
Xi ground. Moreover, it has a dimensionless parameter µα′—the ratio, in
R
r string units, between the strength of the four-form flux and the DLCQ
a
radius—which, when large, permits a perturbative treatment of the Yang-
Mills theory3. Additionally, the four-form flux supports fuzzy sphere so-
lutions to the Matrix Theory1. The fuzzy spheres are the general vacua
of the Matrix Theory, which preserve all the SUSYs. These fuzzy spheres
have very interesting properties upon compactification4 of the theory to
a IIA Matrix String Theory. These results have already appeared in the
literature5,6 andspaceconstraintspreventusfromsummarizingthemhere.
TherearemanyppwavesolutionstoM-theory7,8,9,10andaMatrixThe-
ory has been proposed8,11 for each. The original argument noted that the
form reproduces BMN and has the right number of SUSYs8. A derivation
from membrane quantization has since been given11. One can also show,
beyondcountingfermionicgenerators,thatthefullSUSYalgebrasmatch12.
∗ToappearintheProceedingsofthe3rdInternationalSymposiumonQuantumTheory
andSymmetries(QTS3)10–14Sept2003—(cid:13)c WorldScientific.
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2
Having proposed a Matrix Theory for each M-theory pp wave, one
should now study them. The zeroth question to answer is: what are
the vacua which preserve all of the SUSYs? We will study this for three
ppwaves: the Penroselimit ofAdS S3 7,8,13,14; the T-dual4 ofthe maxi-
3
×
mallysupersymmetric ppwaveofthe IIB theory15; andthe 26supercharge
pp wave10. For the first two, we find, at infinite N, brane solutions which
do not follow from treating g as a superpotential16.a
++
2. MATRIX THEORIES
The general 11-dimensional pp wave considered is17,8,9
9
ds2 =2dx+dx− µ2(xi)2 (dx+)2+(dxi)2, F =dx+ Θ, (1)
−" i # ∧
i=1
X
withconstantµ ,Θ,andalsoΘisa“spatial”(supportinthexi-directions)b
i
three-form, satisfying the equation of motion (e.o.m.) Θ2 =12 µ2.
ijk i i
This geometry admits 16+2n Killing spinors. To present them—and
P
define n—set, (with all sums explicit here, and U 3ΓiΘΓi+Θ)
(i)
≡
1 1
Ω Θ(Γ+Γ−+1l), Ω 0, Ω ΓiU Γ+. (2)
+ ≡−12 − ≡ i ≡ 24 (i)
The solutions of the Killing spinor equation, ǫ= ǫ Ω ǫ=0, are
A A A
D ∇ −
ǫ(x,ǫ0)= 1+ xiΩi e−112(Γ+Γ−+1l)θx+ǫ0, (3)
h Xi i
where the constant spinor ǫ obeys17,8,9
0
U2 Γ+ǫ = 144µ2Γ+ǫ . (4)
(i) 0 − i 0
So there are 16 “standard” SUSYs which obey Γ+ǫ = 0 and an even
0
number, 2n, of “supernumerary”7 SUSYs obeying
U2 ǫ = 144µ2ǫ , (5)
(i) 0 − i 0
for each i. Equation (5) need not have any solutions, and never has 12 or
14solutions10. Existenceof“supernumerary”SUSYsguaranteesasolution
to the e.o.m.s, by summing Eq. (5) over i and tracing over Killing spinors.
aUnfortunatelythereisnoroomtolistthemyriadpapersonbranesinppwaves.
bTheindexi,j,... runsfrom1to9,andA,B,... runsoverallcoordinates. Γ-matrices
are defined with respect to the elfbein e−ˆ =dx−− 1 9 µ2(xi)2dx+, e+ˆ =dx+,
2 i=1 i
eˆı=dxi. TheFeynmanslash,e.g.Θ= 31!ΓijkΘijk,isusePdextensively.
3
In units for which the 11-dimensional Planck length, ℓ = 1, and em-
P
ploying Majorana so(9) spinors Ψ, the Matrix Theory action8,11 isc
S =R dτTr 1 (D Xi)2+ i ΨTD Ψ+ΨTΓi Xi,Ψ + 1 Xi,Xj 2
2R2 τ R τ 4
Z (cid:26)
(cid:2) (cid:3) (cid:2) (cid:3)
1 µ2 i i
i (Xi)2 ΨTΘΨ Θ XiXjXk . (6)
−2 R2 − 4R − 3R ijk )
i
X
The first line is the familiar BFSS Matrix Theory18 action; the second line
containsmass terms for the bosons andfermions,and the Myersterm19,20.
TheMatrixTheory(6)isinvariantunderthenonlinearlyrealizedSUSY
δΨ=e14Θτǫ+ corresponding to the 16 “standard” SUSYs of the pp wave.
Consider also the transformations,
δXi =iΨTΓiǫ(τ), δAτ =iRΨTǫ(τ), ǫ(τ) e−112Θτǫ0, (7a)
≡
1 1 i
δΨ= D XiΓiǫ(τ)+ XiΩˆ ǫ(τ)+ Xi,Xj Γijǫ(τ), (7b)
τ i
2R R 4
where Ωˆ 1 ΓiU and ǫ is constant (cf. Bo(cid:2)nelli21).(cid:3) These transfor-
i ≡ 24 (i) 0
mations preserve the action precisely when ǫ obeys (5). So the Matrix
0
quantum mechanics (6) preserves exactly the right number of SUSYs to
describe the pp wave (1)7,11. Indeed, the SUSY algebras match12.
3. THE AdS3 ×S3 pp WAVE
The Penrose limit of AdS S3 has been consideredin7,8,13,14. Lifting the
3
×
geometry supported with NS-NS flux to 11-dimensions,
4 9
ds2 =2dx+dx− µ2 (xa)2(dx+)2+ (dxi)2,
− (8)
a=1 i=1
X X
(4)F =2µdx+ dx1 dx2 dx9+2µdx+ dx3 dx4 dx9.
∧ ∧ ∧ ∧ ∧ ∧
TheM-theorydirectionisx9,andthedirectionsxa,a=1 4arethespace-
···
likeAdS S3directionstransversetothenullgeodesicofthePenroselimit.
3
×
Vanishing of the fermionic variation (7b) gives the conditions
X1,X2 = X3,X4 , X2,X3 = X1,X4 , X1,X3 = X2,X4 ,
−
µ µ
(cid:2) (cid:3) (cid:2) X2(cid:3),X9(cid:2) =i X(cid:3)1, (cid:2) X1,X(cid:3)9 =(cid:2) i X(cid:3)2, (cid:2) (cid:3)
R − R
µ µ
(cid:2)X4,X9(cid:3)=i X3, (cid:2)X3,X9(cid:3)= i X4.
R − R
(9)
(cid:2) (cid:3) (cid:2) (cid:3)
cDτXi=∂τXi+i Aτ,Xi and(ΨαΨβ)∗=Ψ∗βΨ∗α. RistheDLCQradius.
(cid:2) (cid:3)
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All other commutators vanish, and the vacua are static.
For finite N, all vacua preserving all eight linearly-realized SUSYs are
trivial: X1 =X2 =X3 =X4 =0. However, there is a family of solutions
for infinite N. Specifically, take X1,X2 and X3,X4 to form two noncom-
mutative planes, with equal noncommutativity parameter,
X1,X2 =iϑ= X3,X4 ; X1,X3 =0= X2,X3 , (10)
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
and take
µ
X9 =X9 (X1)2+(X2)2+(X3)2+(X4)2 , (11)
0 − 2Rϑ
(cid:2) (cid:3)
where X9 commutes with all the matrices. Then it is straightforward to
0
see that equations (9) are satisfied. This solution describes a longitudinal
fivebrane of a (fuzzy) paraboloidal shape (11). The fivebrane wraps the
entire “AdS S3 directions” but it extends into the M-theory direction
3
×
through the paraboloid equation.
4. THE 26 SUPERCHARGE MATRIX THEORY VACUA
The pp wave with 26 supercharges was presented in10. Supersymmetric
vacua obey (I,J, =1...7, I′′,J′′, =1,2,3, with the conventions10)
··· ···
µ µ
D XI =0,I =1...7, D X8 = X9, D X9 = X8,
τ τ τ
2 −2
X4 =X5 =X6 =X7 =0, XI′′,XJ′′ =−iRµǫI′′J′′K′′XK′′,
X8,XI′′ =h0= X9,XiI′′ .
h i h i (12)
That is, states preserving all the SUSYs are fuzzy spheres at the origin of
the (4,5,6,7)-hyperplane,and orbit in the (8,9)-plane with frequency µ.
2
Recall10 that the 26 supercharge pp wave can be compactified to a 26
supercharge IIA pp wave. Such a reduction occurs along a Killing vector
which rotates in the (8,9)-plane. Thus, in the IIA theory, the fully super-
symmetric solutions (12) are static. This is reminsicent of the maximally
supersymmetric pp wave, for which5 the orbiting fuzzy spheres broke the
same half of the (supernumerary) SUSYs as were broken by the reduc-
tion to the IIA theory, yielding fully (24 supercharge) supersymmetric IIA
fuzzy spheres located at any static value of X8 . Here, the solutions have
IIA
identical form, but with two additional supercharges.
5
5. IIB T-dual Matrix Theory Vacua
The T-dual, lifted to M-theory, of the IIB maximally supersymmetric
ppwaveis4 (the coordinateshavebeenreindexedrelativetothe reference)
6 9
ds2 =2dx+dx− 4µ2 (xI)2+16µ2(x7)2 (dx+)2+ (dxi)2, (13a)
−" #
I=1 i=1
X X
F = 4µdx+ dx7 dx8 dx9+8µdx+ dx5 dx6 dx7. (13b)
− ∧ ∧ ∧ ∧ ∧ ∧
The linear SUSYs of the Matrix Theory are parametrized by Γ5689ǫ =ǫ .
0 0
Fully supersymmetric vacua must be static, at X1...4 =0 and satisfy
µ i i
2 X7+ X8,X9 X5,X6 =0, (14a)
− R 2 − 2
µ µ
X6,X7 =2i(cid:2) X5, (cid:3)X7,X(cid:2) 5 =2i(cid:3) X6, (14b)
R R
X5(cid:2),X8 =(cid:3) X6,X9 , (cid:2) X6,X(cid:3)8 = X5,X9 , (14c)
−
X7,X8 =0(cid:2)= X7,(cid:3)X9 .(cid:2)These e(cid:3)quat(cid:2)ions are(cid:3)sim(cid:2)ilar to (cid:3)those for non-
maximally SUSic vacua of the BMN theory22. Solutions to equation (14)
(cid:2) (cid:3) (cid:2) (cid:3)
satisfy the e.o.m.s.
ForfiniteN,the onlysolutionsarefuzzy ellipsoids,( Ja,Jb =iǫ Jc)
abc
µ µ (cid:2) µ (cid:3)
X5 =2√2 J1, X6 =2√2 J2, X7 =2 J3. (15)
R R R
Constantvalues ofX8,9, thatarediagonalandproportionalto the identity
in each irreducible SU(2) block, give the positions of the fuzzy ellipsoids.
At strictly infinite N there are many more solutions. For example,
µ µ 1
X5,6 =2√2 J1,2 1l, X7 =2 J3 ϑ1l 1l, (16a)
R ⊗ R − 8 ⊗
(cid:18) (cid:19)
X8 =1l xˆ8, X9 =1l xˆ9, (16b)
⊗ ⊗
where xˆ8,xˆ9 = iϑ is a c-number. That is, X5,6,7 form a fuzzy ellipsoid
and X8 and X9 parameterize a noncommutative plane orthogonal to the
(cid:2) (cid:3)
ellipsoid and translated, along X7, by an amount µ ϑ from the origin.
−4R
This is the longitudinal M5-brane, formed as a stack of M2-branes18,23,
with oneM2-braneblownupinto a fuzzy ellipsoid. Thus,the solution(16)
describes a longitudinal M5-brane of topology R1,3 S2.
×
One can also find a solution for every complex simple Lie algebra of
rank 2. If the algebra is not simply-laced—i.e. except for su(3)—the al-
gebra gives two inequivalent solutions. Explicitly, if the two roots of the
6
algebraareα =(a , b )andα =(0,a ),withCartansubalgebrah ,h ,
1 1 1 2 2 1 2
−
([h ,α ]=a α , α ,α† =a h , etc.), then
1 1 1 1 2 2 2 2
h i
2µ 2µ 2µ
X7 = h , X5 = (e +e†), X6 = (e e†),
a R 1 a R 1 1 a Ri 1− 1
1 1 1
b µ b µ
X8 = 2i 1 (e +e†), X9 = 2 1 (e e†),
− sa21a2R 2 2 − sa21a2R 2− 2
(17)
is a solution. Demanding hermitian matrices then restricts to a particular
noncompact realification of each algebra. The resulting realifications are
su(2,1), sp(1,1)=so(1,4), sp(2,R)=so(2,3) and g . Two inequivalent
∼ ∼ 2(2)
solutions are obtained from the last algebra. As these are all noncompact,
theirnontrivialunitaryrepresentations(hermitianmatrices)areallinfinite
dimensional.
For example, explicit24 su(2,1) solutions can be written in terms of
threecommutingsetsofannihilationoperatorsa ,a ,b,andtheirhermitian
1 2
conjugate creation operators,
2 µ 2 µ
X5 =2 a†a +a†a , X6 = i2 a†a a†a ,
3R 1 2 2 1 − 3R 1 2− 2 1
r r
2 µ (cid:16) (cid:17) 2 µ (cid:16) (cid:17)
X8 = a b+a†b† , X9 = i a b a†b† ,
√3R 1 1 − √3R 1 − 1
(cid:16) (cid:17) (cid:16) (cid:17)
2 µ
X7 = a†a 2a†a bb† .
3R 1 1− 2 2−
(cid:16) (cid:17) (18)
The quadratic Casimir of su(2,1) takes the form
Csu(2,1) = 3R2 (X8)2+(X9)2 + 3R2 (X5)2+(X6)2+2(X7)2
2 −4 µ2 8 µ2
9 R4(cid:0) X8,X9 2+ 9(cid:1) R4 X(cid:0)5,X8 2+ X5,X9 2 . ((cid:1)19)
− 64 µ4 32 µ4
(cid:2) (cid:3) (cid:16)(cid:2) (cid:3) (cid:2) (cid:3) (cid:17)
Thefirstlinesuggestsahyperboloidalinterpretationtothesolution,though
this needs refinement.
Compactifying the Matrix Theory along X8 gives12, at least for weak
stringcoupling,the Green-Schwarzactionfor the IIB stringin the pp wave
background. Theradiusofcompactificationmatchesprecisely. Thisdemon-
strationrequiresafieldredefinitionintheMatrixStringTheorythatisthe
inverseofthecoordinatetransformationthatmakestheisometrymanifest4
on the IIB side. Unfortunately, there is no space to give details here.
7
Acknowledgments
This workwas supported in partby Department ofEnergy contract#DE-
FG01-00ER45832.
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