On the transverse M5-branes in matrix theory

0.12 MB·

by  Yuhma Asano

DIAS-STP-16-14,UTHEP-701, YITP-16-149
On the transverse M5-branes in matrix theory
Yuhma Asano,1 Goro Ishiki,2,3 Shinji Shimasaki,4 and Seiji Terashima5
1School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland
2Center for Integrated Research in Fundamental Science and Engineering (CiRfSE),
University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
3Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
4Research and Education Center for Natural Sciences, Keio University,
Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan
5Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
(Dated: 25 Jan 2017)
It has been a long-standing problem how the transverse M5-branes are described in the matrix-
modelformulationsofM-theory. WeconsiderthisproblemforM-theoryonthemaximallysupersym-
7
1 metricpp-wavegeometry,whichadmitstransversesphericalM5-braneswithzerolight-coneenergy.
0 Byusingthelocalization,wedirectlyanalyzethestrongcouplingregionofthecorrespondingmatrix
2 theorycalledtheplanewavematrixmodel(PWMM).UndertheassumptionthattheSO(6)scalars
inPWMMbecomemutuallycommutinginthestrongcouplingregion,weshowthattheeigenvalue
n
densityoftheSO(6)scalarsinPWMMexactlyagreeswiththeshapeofthesphericalM5-branesin
a
the decoupling limit. This result gives a strong evidence that the transverse M5-branes are indeed
J
contained in thematrix theory and thetheory realizes a second quantization of theM-theory.
5
2

Introduction: A nonperturbative formulation of M- M-theory was conjectured in [6]. In particular, vacua
]
h theory in the light-cone frame is conjectured to be given corresponding to the spherical transverse M5-brane and
t by the matrix theory [1]. The matrix theory is expected itsmultiplegeneralizationwerespecified. Thisconjecture
-
p to achieve second quantization of M-theory, in which all wastestedforthecaseofasingleM5-branebycomparing
e
fundamental objects in M-theory are described in terms the BPS protected mass spectra of PWMM and those of
h
oftheinternaldegreesoffreedomofmatrices. Ithasbeen the M5-brane.
[
shownthatthereexistmatrixconfigurationscorrespond- In this Letter, we explicitly show that the spheri-

1
ingtovariousobjectsinM-theorysuchassupergravitons, cal M5-brane emerges in the strong coupling regime of
v
M2-branes and longitudinal M5-branes [1–4]. PWMM as the eigenvalue density of the SO(6) scalars.
0
4 On the other hand, the description of transverse M5- We apply the localizationmethod [7] to PWMM and re-
1 branes has not been fully understood. The charge of duce the partition function to a simple matrix integral.
7 the transverse M5-branes is known to be absent in the By evaluating the matrix integral in the strong coupling
0
supersymmetryalgebraofthematrixtheory,andhenceit limit, we find that the eigenvalue density of the SO(6)
.
1 seemstobeimpossibletoconstructmatrixconfigurations scalar matrices forms a five-dimensional spherical shell
0 for transverse M5-branes with nonvanishing charges. and its radius exactly agrees with that of the spherical
7
M5-brane in the M-theory on the pp-wave background.
1 TheabsenceoftheM5-branecharge,however,doesnot
SphericalM5-brane on thepp-wave background:
: prohibit the presence of M5-branes with compact world-
v We firstreviewthe sphericaltransverseM5-braneonthe
i volume,whichhavezeronetcharge. Itshouldbeclarified
X pp-wave background. The maximally supersymmetric
whethersuchcompacttransverseM5-branesareincluded
pp-wavesolution of 11-dimensionalsupergravityis given
r in the matrix theory.
a by
The plane wave matrix model (PWMM) provides a
very nice arena to understand this problem. PWMM 9
ds2 =g dxµdxν = 2dx+dx−+ dxAdxA
is the matrix theory for M-theory on the maximally su- µν −
persymmetricpp-wavebackgroundofthe11-dimensional AX=1
supergravity [5]. On this background, M-theory admits µ2 3 µ2 9
xixi+ xaxa dx+dx+,
a stable spherical transverse M5-brane with vanishing − 9 36 !
i=1 a=4
light-cone energy. In general, objects with zero light- X X
F =µ, (1)
cone energy in M-theory are mapped to vacuum states 123+
in the matrix theory. Thus, in finding the description of whereµisaconstantparametercorrespondingtotheflux
the spherical M5-branes, the target is restricted to the of the three form field. Throughout this Letter, we use
vacuum sector of PWMM. the notation such that µ,ν = +, ,1,2, ,9, A,B =
− ···
As we will see below, vacua of PWMM are given by 1,2, ,9, i,j =1,2,3 and a,b=4,5, ,9.
··· ···
fuzzysphereandarelabeledbythepartitionofN,where We consider a single M5-brane in this background[6].
N is the matrix size of PWMM. For each vacuum, the Let Xµ(σ) be embedding functions of the M5-brane,
correspondingobjectwith vanishing light-coneenergyin where σα(α = 0,1, ,5) are world-volume coordinates
···
2
on the M5-brane. The bosonic part of the M5-brane ac- M2-braneare mapped to N N Hermitian matrices YA
×
tion is given by as
µp+
S = T d6σ deth +T C , (2) XA(σ0,σ1,σ2) YA(t), (8)
M5 − M5 − αβ M5 6 → 12πNTM2
Z Z
p
and Poisson brackets and integrals on the spatial world-
where h is the induced metric h =
αβ αβ
g (X)∂ Xµ∂ Xν and C is the potential for the volume are mapped to commutators and traces of ma-
µν α β 6
trices, respectively [13]. The complicated factor in (8) is
magnetic flux dC = F . T is the M5-brane tension,
6 4 M5
∗ chosen so that the action (7) takes the simple form. In
which is written in terms of the 11-dimensional Planck
length l as T = 1 . equation(8),thetimecoordinatetisrelatedtoσ0 bythe
p M5 (2π)5l6p same rescaling factor. The coupling constant g2 in (7) is
By applying the standard procedure of the gauge fix-
related to the original parameters in the M-theory by
ing in the light-cone frame [8], we obtain the light-cone
Hamiltonian of the M5-brane as T2 12πN 3
g2 = M2 , (9)
V T2 2π µp+
H = d5σ 5 P2 + M5 XA1, ,XA5 2 (cid:18) (cid:19)
M5 2p+ A 5! { ··· }
Z p+ (cid:20) µ2 (cid:18) µ2 (cid:19) whereTM2 = (2π1)2l3p is the tensionofthe M2-brane. The
+ (Xi)2+ (Xa)2 covariantderivativeDin(7)isdefinedbyDYA =∂ YA
t
2V 9 36 −
5 (cid:18) (cid:19) i[A,YA]. This is introduced to take into account the
− µT6M! 5ǫa1a2···a6Xa1{Xa2,··· ,Xa6} , (3) cthoensgtarauignet-fioxfintgheinfothrmeli{gPhtA-c,oXnAe}fra=me0,[8w].hIicnhthaeriAses=in0
(cid:21)
gauge,theGausslawconstraintcorrectlyreproducesthis
where V = π3, p+ is the total light-cone momentum,
5 constraint.
PA are the conjugate momenta of the transverse modes Noticing that the potential for Yi forms a perfect
XA and the curly bracket is defined by f , ,f =
1 5 square, one can easily find the vacuum configuration of
ǫa1···a5(∂ f ) (∂ f ). { ··· }
a1 1 ··· a5 5 PWMM as
By noticing that the potential term for Xa in (3) can
be rewritten as a perfect square, one can easily find the Yi =2Li, Ya =0, A=0. (10)
vacuum configuration as
Here, Li are N-dimensional representation matrices of
PA =0, Xi =0, Xa =r xa, (4) the SU(2) generators. The representation can be re-
M5
ducible and one can make an irreducible decomposition,
where xa are the embedding functions of the unit 5-
sphere into R6 satisfying Λ
L = L[ns], (11)
i i
xaxa =1, {xa1,··· ,xa5}=ǫa1a2···a6xa6. (5) Ms=1
where L[n] stand for the generatorsin the n-dimensional
The constant rM5 is determined as irreducibile representation and Λ n = N. Thus,
s=1 s
µp+ 1/4 the vacua are labeled by the partition of N, ns ns
r = . (6) P { | ≥
M5 (cid:18)6π3TM5(cid:19) ns+T1h,e csonnsje=ctNur}e. on the spherical M5-brane: The
P
Thus, we find that the zero energy configuration is a vacuum of the form (10) is the fuzzy sphere configura-
spherical M5-brane with the radius given by (6). tion and hence it has a clear interpretation as a set of
Plane wave matrix model: The action of PWMM spherical M2-branes. Indeed, the M-theory on the pp-
is obtained by the matrix regularization of a single M2- wavebackgroundallowszeroenergysphericalM2-branes
brane action on the pp-wave background [5], which is as well. The commutative limit of the fuzzy sphere (10),
given by the 1+2 dimensional analogue of (2). The where ns become large, can naturally be identified with
bosonic part of the action of PWMM is given by those M2-branes in M-theory.
On the other hand, the correspondence between (11)
S = 1 dtTr 1(DYA)2 2Y2 1Y2 and the spherical M5-brane can be understood by in-
g2 2 − i − 2 a troducing a dual way of looking at the Young tableau
Z (cid:20)
1 of the vacuum. For the vacuum (11), let us consider
+ [YA,YB]2 iǫ Yi[Yj,Yk] . (7)
ijk the Young tableau corresponding to the partition n
4 − s
(cid:21) such that the length of the sth column is given by{n}.
s
In obtaining this action, we first apply the matrix regu- Let us denote by m the length of the kth row, where k
k
larization, where the embedding functions XA(σ) of the runs from 1 to max n . It was conjectured in [6] that
s
{ }
3
when m are large, the vacuum corresponds to multiple Spherical M5-branes from PWMM: Let us ana-
k
M5-branes, where the number of M5-branes is given by lyze PWMM in the decoupling limit of the M5-brane by
N :=max n andeachM5-branecarriesthelight-cone usingthelocalizationmethod. Weconsiderthefollowing
5 s
{ }
momentum proportional to m (k =1,2, ,N ). scalar field:
k 5
···
Inwhatfollows,wetestthisconjecturefocusingonthe
vacua such that the partition is of the form, φ(t)=Y3(t)+i(Y8(t)sin(t)+Y9(t)cos(t)). (16)
L =L[N5] 1 . (12) This field preserves1/4 of the whole supersymmetries in
i i ⊗ N2 PWMM and any expectation values made of only φ can
N and N satisfy N N = N and correspond to the becomputedbythelocalizationmethod[7]. Sinceweare
2 5 2 5
number ofM2-andM5-branes,respectively. Inorderto interested in PWMM around a specific vacuum (12), we
describe the M5-brane, we consider the limit, impose the boundary conditions such that all the fields
take the vacuum configurations at t . With this
→ ±∞
N2 , N5 : fixed. (13) boundary condition, the localization computation leads
→∞
to the following equality [9–11]:
With the above interpretation, this limit corresponds to
N M5-branes, each of which carries the light-cone mo-
5 Trf (φ(t )) = Trf (2L +iM) , (17)
I I I 3 MM
mentum with an equal amount. h i h i
I I
Y Y
In order to isolate the degrees of freedom of the M5-
branes in PWMM, the ’t Hooft coupling of PWMM where fI are arbitrary smooth functions of φ, 2L3 is the
should also be sent to infinity in taking the limit (13) vacuum configuration for Y3, and M is an N N Her-
×
[6]. This can be understood as follows. We first rewrite mitian matrix which commutes with all La(a = 1,2,3).
the metric (1) so that the compactified direction x− is For the vacuum given by (12), M takes the form, M =
orthogonalto the other directions as 1N5⊗M˜,whereM˜ isanN2×N2 Hermitianmatrix. The
expectation value in the left-hand side of (17) is
ds2 = µ2r2dx˜+dx˜++ 36 dx˜−dx˜−+r2dΩ5+ , taken with respecth·t·o·ithe original partition function of
− 36 µ2r2 2 ··· PWMM around (12), while that in the right-hand side,
(14) , is taken with respect to the matrix integral,
MM
h···i
xw˜+he=re xr+2 = √36xaxx−a, xi˜s−th=e xr−ad.iuTshoefpthhyesi5c-aslpchoemrepaacntd- Z = dqie−2gN25Piqi2N5−1N2−1 N2
− µ2r2
ification radius is then given by R˜ R/(µr), where R is Z Yi JY=0 jY=1 i=Yj+1
∼ (2J +2)2+(q q )2 (2J)2+(q q )2
the original compactification radius of M-theory. Upon i j i j
{ − }{ − }. (18)
compactification, transverse M5-branes in the M-theory × (2J +1)2+(qi qj)2 2
{ − }
become NS5-branes in the type IIA superstring theory.
The world-volume theory of the NS5-branes is known as Here, qi (i=1,2, ,N2) are the eigenvalues of M˜ [15].
···
the little string theory, which has a characteristic scale In the decoupling limit (15), the saddle point approx-
givenbythestringtension l−2. ForthesphericalNS5- imation becomes exact in evaluating the matrix inte-
∼ s
branewiththe radius(6), thetheoryis controlledbythe gral. We first introduce the eigenvalue density for qi
dimensionless combinationrM25/ls2. We keepthis ratio fi- by ρ(q) = N12 Ni=21δ(q − qi), which is normalized as
nite to obtain an interacting theory on the NS5-branes qm dqρ(q) = 1. Here, we assume that ρ(q) has a fi-
−qm P
while we send rM5 to infinity to make the bulk grav- Rnite support [−qm,qm]. In the large-λ limit, the saddle
ity decouple. By using (6) and the well known relation point equation of the partition function (18) is reduced
ls ∼ (lp3/R˜)1/2, we find that the decoupling limit of the to
NS5-brane is given by p+ with R4p+ fixed [14].
The M5-brane in 11 dimens→ion∞s is recovered by further β =πρ(q)+ 2N5q2 qm dq′ 2N5 ρ(q′),
taking R4p+ to be large. Finally, by combining this ob- λ −Z−qm (2N5)2+(q−q′)2
servation with (13), we find that the decoupling limit of (19)
the M5-brane is written in terms of the parameters of
PWMM as where β is the Lagrangemultiplier for the normalization
of ρ and we used the fact that q /N 1 in this limit.
m 5
λ ≫
The solution of (19) is given by
N , N : fixed, λ , 0, (15)
2 5
→∞ →∞ N →
2
wThhuerse, tλhe=degc2oNup2liinsgtlhimei’ttoHf othoeftMco5u-bprlainnge coofrPreWspMonMds. ρ(q)= 3π8λ4341 (cid:20)1− qqm22(cid:21)23 , qm =(8λ)14, β = 8λ12N12 5.
(20)
to the strong coupling limit in the ’t Hooft limit.
4
By using (17) and the solution for the eigenvalue den- Summary: In this Letter, we considered the ma-
sity (20), we can compute any operator made of φ. In trix theoretical description of the spherical transverse
particular, let us consider the resolvent of φ defined by M5-branes with vanishing light-cone energy in M-theory
Tr(z φ)−1. According to the result of the localization on the maximally supersymmetric pp-wave background.
−
(17), the expectation value of this operator is equal to Following the proposal in [6], we considered PWMM
that of Tr(z 2L iM)−1 in the matrix integral (18). expanded around the vacuum associated with the M5-
3
− −
Note that the support of the eigenvalue density of M branes. We applied the localization to this theory and
is much larger than that of L in the decoupling limit. obtained an eigenvalue integral. We then analyzed and
3
Thus,we find that, withthe suitable normalizationasin solved the eigenvalue integral in the decoupling limit of
(8), the spectrum of φ lies on the imaginary axis in this theM5-branes,whichcorrespondstothestrongcoupling
limit. This fact enables us to ignore the realpart of φ in limit of PWMM. Finally, under the assumption that the
the decoupling limit. Thus, by putting t = 0, we obtain SO(6) scalar fields become mutually commuting in the
the following equality in the decoupling limit: strongcouplingregion,wefoundthattheeigenvalueden-
sity of the SO(6) scalar fields in PWMM in this limit
1 qm
Tr(Y9)n(0) = dqρ(q)qn. (21) forms a 5-dimensional spherical shell and the radius of
Nh i Z−qm the spherical shell exactly agrees with that of the M5-
brane in M-theory. Thus, we concluded that the M5-
Namely, ρ(q) is identified with the eigenvalue density of
Ya in this limit. branein M-theoryis formedby the eigenvaluedensity of
the SO(6) scalar fields. We also computed the radius of
By using the SO(6) symmetry in PWMM, we then
the multiple M5-branes.
introduce the SO(6) symmetric up-lift of the eigenvalue
densityofasingleYa obtainedabove. We definethe up-
Acknowledgments: We thank H. Shimada for valu-
lifted density function ρ˜as a solution to the equations,
able discussions. The work of G. I. was supported, in
µp+ 2n qm part, by Program to Disseminate Tenure Tracking Sys-
Z d6xaρ˜(r)x29n =(cid:18)12πNTM2(cid:19) Z−qmdqρ(q)q2n (22) tSe.mw,aMs sEuXppTo,rJteadpabnyatnhde MbyEKXTA-KSEupNpHoIrt(e1d6KPr1o7g6r7a9m).foSr.
for any n. Here, r = x2 and ρ˜ is normalized as the Strategic Research Foundation at Private Universi-
a a
d6xaρ˜(r) = 1. Note that ρ˜depends only on r because ties Topological Science (Grant No. S1511006).
pP
of the SO(6) symmetry. The first factor on the right-
R
hand side of (22) just reflects the rescaling (8), so that
ρ˜(r)canbethoughtofasadensityfunctionintheoriginal
target space.
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5 0 (cid:18) 5 M5(cid:19)
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0
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thatthetransverseM5-braneisformedbytheeigenvalue
[13] We usethe mappingrules in [8].
density of the SO(6) scalars in PWMM.
[14] Note that we are interested in the case where lp and µ
For N > 1, r in (23) is interpreted as the radius of
5 0 are fixed.
themultiplesphericalM5-branes. TheN5-dependenceof [15] In the derivation of (17), possible instanton corrections
r coincideswiththe conjecturedformin[6]basedonan are ignored. However,thesecorrections areindeed negli-
0
observation on perturbative expansions in PWMM. gible in thedecoupling limit (15).