Continuum limit(s) of BMN matrix model: Where is the (nonabelian) gauge group?

Abstract

We discuss the continuum limits of Berenstein-Maldacena-Nastase matrix model. They give rise to Poisson bracket gauge field theories on the ordinary two sphere or on a set of two spheres with a gauge groups U(n) depending on the degeneracy of the classical solution about which the model is considered. We show that these models fail to be equivalent among each other in the continuum limit.

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arXiv:hep-th/0206239v5 24 Feb 2003
Continuum limits of
Berenstein–Maldacena–Nastase matrix theory:
Where is the (nonabelian) gauge group?
Corneliu Sochichiu∗
High Energy and Elementary Particle
Physics Division, University of Crete,
P.O. Box 2208, GR-710 03 Heraclion,
Crete, GREECE
and
Laboratori Nazionali di Frascati,
INFN, Via E. Fermi 40,
C.P. 13, 00044 Frascati
Italy
February 1, 2008
Abstract
We discuss the continuum limits of Berenstein-Maldacena-Nastase ma-
trix model. The special attention is paid to limits that give rise to Poisson
bracket gauge field theories with gauge groups U(n) on the ordinary two
sphere. The gauge group and the space depending on the degeneracy of
the classical solution about which the model is considered. We compare
these limits as well as different solutions in the framework of the same
limit model. We show that these models fail to be equivalent in the con-
tinuum limit, i.e. the continuum limit does not commute with dualities
of the matrix theory.
1Introduction
The correspondence between gauge fields and strings is a long standing problem
[1]. To establish such an equivalence a good understanding of both gauge and
string theories is needed, inclusively on the nonperturbative level. Some progress
in this direction was reached recent years. (For a comprehensive review see [2].)
On the other hand, hints for connection between nonperturbative string dy-
namics and matrix models which are dimensionally reduced Yang–Mills theories
where given by matrix theories [3, 4], describing branes.
∗On leave from: Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, 141980 Dubna, Moscow Reg., RUSSIA and Institutul de Fizic˘ a Aplicat˘ a AS ¸, str.
Academiei, nr. 5, Chi¸ sin˘ au, MD2028 MOLDOVA; e-mail: sochichi@lnf.infn.ti
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In recent papers [5]–[7], the correspondence between Yang–Mills theory and
string theory in pp-wave background [8, 9], was considered. The pp-wave back-
ground is a curved space corresponding to a plane parallel gravitational wave. In
such a space the string theory preserves many of its nice features, in particular
it is treatable and can be quantized [10].
The DLCQ compactification which in the flat space gives the BFSS model [3],
in the case of pp-wave leads to a modified matrix theory for zero-brane. The
modification with respect to the BFSS model results in the addition of mass
terms for all matrix fields and of Chern–Simons terms to some of them. The
result of this modification is that the stationary vacua of the model are given by
fuzzy sphere or by a set of fuzzy spheres. The perturbation theory around such
classical vacua was analyzed [11] together with the continuum limit (see also
[16]). Also, in [11], it was shown that BMN matrix theory arises as the result
of the world sheet quantization of membranes in pp-wave background described
by a Poisson bracket model. Like BFSS matrix theory the BMN matrix theory
also possesses a number of supersymmetric solutions [12]–[15].
Here we review the fact that in the scaling limit proposed by BMN one in fact
recovers the above Poisson bracket action of the commutative spherical brane.
Depending on what solution of the matrix model is chosen, one can get in the
above limit field models with different local gauge groups. In earlier papers [17]–
[21] we analyzed the equivalence relations arising in the N → ∞ limit of “flat”
matrix theories. In that cases, N → ∞ limit yielded noncommutative gauge
models in different dimensions and/or having different gauge groups. Now, the
situation is slightly different. Since the N → ∞ limit of BMN matrix theory
is combined also with the commutative limit, the scaling prescription leads to
a commutative, although exotic, field model on an ordinary sphere or set of
spheres. The equivalence of these different limits in this case is not clear a
priori. To find such an equivalence, if it exists, one has to identify the solutions
in the limit model, which correspond to different sets of spheres and analyze the
model in the vicinity of such solutions. We do this and find that, in fact, the
models are not equivalent. One can recover, starting with the U(1) model, the
spectrum corresponding to the Abelian subalgebra only of nonabelian models.
The plan of the paper is as follows.
classical solutions and continuum limit of the BMN matrix model. After that,
we consider the Poisson gauge model with the gauge group U(1) and find in this
model solutions which are commutative analogs of U(n) backgrounds of matrix
theory. We find that it is only maximal abelian subgroup U(1)nof U(n) which is
manifest while the remaining part is hidden in the large gauge transformations of
the original irreducible limit. The world sheet quantization makes these modes
to arise explicitly.
In the next section we review the
2Classical solutions and continuum limit
The BMN matrix model appears as the DLCQ quantization of zero brane in
the pp-wave background,
ds2= −4dx+dx−−
??µ
3
?2
x2
α+
?µ
6
?2
x2
µ
?
(dx+)2+ dx2
i, (1)
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were the early Greek indices run α,β = 1,2,3, late ones µ,ν = 4,...,9, while
the Latin ones span both of these sets, i.e. i = 1,2,...,9, xi = (xα,xµ). In
this approach, the sector of M-theory corresponding to the light cone momentum
2p+= −p−= N/R (R is the DLCQ radius) is described by the following matrix
action [6],
S =
?
dttr
?
1
2(2R)(D0φi)2+(2R)
1
2(2R)3
4
[φi,φj]2−
−
??µ
?2
φ2
α+
?µ
6
?2
φ2
µ
?
−µ
3iǫαβγφαφβφγ
?
+ fermions, (2)
where φiare N × N hermitian matrices and “fermions” denotes the fermionic
part of the action which is not written explicitly, since it is not important for
our further analysis. In (2) the indices run according to the same convention as
in eq. (1).
One can see, that there are no stable nontrivial vacua involving only fields
φµ(cfy. Ref. [16]), while one can build nontrivial vacuum solutions out of φα.
In what follows, we will consider the model about such configurations.
The φαdependent part of the action can be rewritten in the following form:
S0=
?
dttr
?
1
2(2R)(D0φα)2+(2R)
4
?
[φα,φβ] − iµ
6Rǫαβγφγ
?2?
. (3)
As it is not difficult to see from the form (3) of the action, the vacua of this
sector of the model are given by matrices satisfying su(2) algebra,
[φα,φβ] = iµ
6Rǫαβγφγ.(4)
The matrices φα, satisfying vacuum condition (4) can be split into blocks
corresponding to irreducible representations Rλof su(2) of spins jλ, having the
total dimensionality,
?
The cases of interest for us are when the solution is represented by n-times
degenerate irreducible representation of the spin j satisfying 2j = N/n − 1. In
particular, the simplest case is for n = 1 when one has a simple irreducible
representation with 2j + 1 = N. Although there are other interesting cases, in
what follows we concentrate mainly on the above ones.
So, let us consider a solution φα≡ Yα, which is n times irreducible represen-
tation. An arbitrary Hermitian n×n matrix can be uniquely expanded in terms
of n×n Hermitian matrices whose entries of symmetrised traceless polynomials
of Yα. This polynomials are noncommutative analogues of spherical functions
and treating them as such one has a map from the space of operators on N
dimensional space to the space of n × n matrix valued functions on a sphere of
the radius,
Y2
α=
6R
These functions are subject to the star product on fuzzy sphere whose exact
form we will not need1.
λ
(2jλ+ 1) = N. (5)
?µ
?2
j(j + 1).(6)
1For the details referring the fuzzy sphere star product we send readers to [16, 11].
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Thus, an arbitrary matrix configuration can be considered as a perturbation
of the background solution,
φα= Yα+ Aα,
φµ= φµ,
...,
(7a)
(7b)
(7c)
where Aαand φµare now fields on the fuzzy sphere and dots stay for fermions.
In this parametrization the action (3) is essentially one of Yang–Mills–Higgs
model on a fuzzy sphere.
Having in mind this map one can switch between different solutions and re-
expand as in (7) to obtain equivalence maps between models with different gauge
groups living on fuzzy spheres of different radii which are related as rn =const
(see [17]–[21]).
According to BMN prescription, as N goes to infinity the radius of the sphere
remans finite,
Y2
α→ r2
0=
?µp+
6n
?2
,N → ∞,(2R) ∼N
p+,(8)
while the background becomes commutative,
[Yα,Yβ] = in
Nr0ǫαβγYγ→ 0. (9)
Ploughing this into action one should be careful with divergent factors of (2R).
The contribution to the action will be given by the leading term in the expansion
of commutators,
[f,g] ≈ in
N{f,g} + 0(N−2),
where {,} is the Poisson bracket on the sphere which is given by,
(10)
{f,g} = ǫαβγYα∂βf∂γg.(11)
Also the trace is replaced by integration over the sphere according to,
4πn
N
tr →
?
dΩ. (12)
Then, the action becomes,
SN→∞=
?
dtdΩtru(n)
?1
2(D0φi)2+
1
4g2({φα,φβ} − rǫαβγφγ)2
?2
+
1
4g2{φµ,φν}2+1
2
?r
2g
φ2
µ
?
+ fermions, (13)
where φα = Yα+ Aα, g2= (p+)2. Integration is performed over time and a
sphere whose radius is r = (µp+)/6n, all fields are n dimensional Hermitian
matrices subject to u(n) trace.
The model possesses a Poisson bracket gauge symmetry,
φi→ φi+ {φi,u}, (14)
where u is an arbitrary hermitian n × n matrix valued function.
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3Commutative “dualities”
In the previous section we found that the N → ∞ limit of the BMN matrix
theory is sensitive to the background around which we are considering it. For any
finite N and finite noncommutativity they are just different parameterizations
of the same matrix model and, therefore, these models are all equivalent. This
may not hold true as N goes to infinity and the noncommutativity vanishes.
Let us try to check, however, at which extend this equivalence is still present in
the limiting model (13).
In order to do this consider the model (13) for n = 1 and r = r0 which is
obtained from the N → ∞ limit of the irreducible algebra. The gauge symmetry
here is just U(1) Poisson bracket gauge symmetry. Let us find static vacuum
solutions of this model most close to one producing the U(n) model.
The static vacua satisfy an equation analogous to (4) that the commutator
is replaced with the Poisson bracket,
{φα,φβ} = r0ǫαβγφγ. (15)
φα are functions on (ordinary) sphere of radius r0. Since φ2
with all φαthe solution is, in fact a map of two spheres: S2→ S2. Nontrivial
solutions are given, therefore, by the homotopically nontrivial maps.
π2(S2) = Z it is natural to identify the homotopy class of the solution with the
rank n of the gauge group U(n). Indeed, zero and one class solutions correspond
to φα= 0 and φα= Yα, respectively.
Let us find higher classes. In spherical coordinates,
αP.b.-commutes
Since
Y1= r0sinθcosϕ,
Y2= r0sinθsinϕ,
Y3= r0cosθ,
(16a)
(16b)
(16c)
the Poisson bracket is given by,
{f,g} =
1
sinθ(∂θf∂ϕg − ∂θg∂ϕf). (17)
On the other hand, the simplest way to get a map of n-th homotopically
class is to wrap along ϕ,
φ1≡ Y(n)
φ2≡ Y(n)
φ3≡ Y(n)
1
= rsinθcosnϕ,(18a)
2
= rsinθsinnϕ,(18b)
3
= Y3= cosθ.(18c)
Fortunately, we are lucky enough and the map as it appears in (18) satisfies
the vacuum condition (15) if the radius r is chosen to be r = r0/n. This relation
is encouraging since it is exactly the relation of the radii of the spheres on which
U(n) models live in the N → ∞ limit (cfy. (8)).
To proceed further we have to consider the functions (18) as new “coordi-
nates” by which we should substitute the old ones. Since Y(1)are wrapping n
times about Y(n)a generic function of Y(1)becomes a n-fold ambiguous func-
tion of Y(n). Locally, any function of Y(1)will become a set of n functions of
Y(n),
φ(Y(1)) ?→ φa(Y(n)),a = 1,...,n,(19)
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one for each sheet. (In general functions are mapped to sections of a nontrivial
n dimensional fibre bundle.)
Unfortunately, this is not in a total accordance with our expectations, since
in order to get U(n) gauge group the fields should map to n × n dimensional
matrices rather then to n component fields. In fact, the fields in new coordi-
nates represent the diagonal part of the expected matrices. Indeed, the gauge
transformation (14) splits in n U(1) parts (one U(1) for each sheet) which is an
indication that the gauge group is U(1)n.
Summarizing, it appears that the maximum we can get in the limiting model
is to map the U(1) model to a model were U(n) is truncated down to U(1)n.
4World volume quantization and restoration of
the whole U(n) group
Let us try to recover the remaining non-diagonal part of the desired U(n) sym-
metry group.
In fact, the symmetry can be restored upon the worldvolume quantization.
The idea can be illustrated by the following example. Consider a particle moving
in a space consisting of n sheets (branes). The position of the particle is given
by its (continuous) coordinate x and the number of the sheet a = 1,...,n.
Classically, the particle can move smoothly along x and jump through indices
a. Suppose the observer does not care about the sheet numbers. So far there is
no nonabelian symmetry in the system.
Now, consider the above model as being quantum. Since the particle can be
found on different branes, the wave function of the particle is a n dimensional
vector ψa(x). There are n2Hermitian operators describing the jumps of the
particle from a to b, which commute with x and p and which generate a U(n)
symmetry group.
Let us return now to our model. The gauge symmetry (14) is in fact only the
infinitesimal version of the whole gauge invariance. Eq. (14) can be integrated
to Hamiltonian flows to yield the finite gauge transformations. For example, a
rotation by ∆ϕ along the Y3axis can be formally written as,
φ(ϕ) ?→ φ(ϕ + ∆ϕ) = e{∆ϕY3,φ}(ϕ),(20)
where we symbolically denoted the exponentiated Poisson bracket,
e{A,B} = B +1
1!{A,B} +1
2!{A,{A,B}} + ...(21)
Thus the “large” rotations of ϕ by, say, 2πk where k < n is an integer, result
in cyclic jumping over k sheets. Y(n)are invariant under such transformations
since they are degenerate along the sheet numbers. The total number of inde-
pendent large rotations is precisely n2(including identical rotations). Thus the
nonabelian structure is hidden in large gauge transformations!
Consider now the world volume quantization. It results in the replacement
of the Poisson bracket algebra,
{Y(n)
α
,Y(n)
β
} = rǫαβγ,Y(n)
γ
,(22)
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by an operator one,
[Y(n)
α
,Y(n)
β
] = i?rǫαβγ,Y(n)
γ
, (23)
in such a way that it forms an irreducible representation modulo the action of
the sheet jump operators. It is n-tuple degenerate one and this degeneracy is
governed by U(n) gauge group. Now arbitrary operator about the background
Y(n)is represented by n-dimensional matrix valued noncommutative function
on the fuzzy sphere (23).
Let us note that the gauge group was restored at the moment when we
replaced the “classical” sheet number label by an operator.
5 Discussion
In this note we considered the properties of N → ∞ limit of the BMN matrix
theory.
Field theory models describing the fluctuations of the matrix theory in
N → ∞ limit depend on the classical background around which the fluctua-
tions are measured. Different backgrounds lead to models in different spaces or
having different gauge groups. In BMN matrix theory there is a class of classical
backgrounds given by fields satisfying su(2) algebra. In the case when such a
background is given by a n times degenerate irreducible representation of spin
j ∼ N/n, the continuum limit of the matrix model is described by a Poisson
gauge model with group U(n). Since at finite N the models with different gauge
groups appear as different parameterizations of the same matrix theory there
is an equivalence between them. As N goes to infinity, however, the models
become nonequivalent. Thus, we are able to map the Poisson gauge model with
group U(1) to at most the abelian sector of the U(n) model. Quantization of
the worldsheet seems to restore the whole gauge group.
The string interpretation of the above properties of the limit is as follows.
Continuum limit of the BMN matrix theory corresponds to zero slope or infinite
tension limit of the string theory. So, the transition from a one-brane configu-
ration, which corresponds to gauge group U(1), to a multi-brane configuration,
corresponding to group U(n), passes through intermediary configurations with
concentric branes of different radii. Infinite tension strings can not stretch be-
tween branes of different radii, but rather can begin and end on the same brane.
The last gives modes in the abelian sector of the gauge theory, while nonabelian
modes can not appear even when spherical branes become of the same radius.
Here we considered only very specific backgrounds given by irreducible repre-
sentations or product of identical irreducible representations. Another interest-
ing class is given by background with products of different irreducible represen-
tations. In the case of these solutions one can expect to have a Poisson gauge
model with spontaneously broken gauge group U(n) → U(n1) × ···×U(nk),
n1+ ··· + nk = n, where ni is the multiplicity of the i-th irreducible factor.
These can serve as subjects for a future study.
Unfortunately, there is no known direct relation between BMN matrix theory
and BMN super–Yang–Mills/PP (SYM/PP) string correspondence [5], except
that the respective matrix theory was designed to describe the brane dynamics
on the pp-wave. It would be interesting to know such a relation in case it exists,
but so far the SYM/PP correspondence is formulated in terms of perturbative
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string states and extending the analysis beyond the perturbative limits is a good
challenge.
Acknowledgments
I benefited from useful discussions with K. Selivanov, R. Woodard, K. Anag-
nostopoulos. I would like to thank Th.Tomaras for kind hospitality and interest
in my work.
This work was done during my visit to University of Crete supported by
Greek Nato fellowship program. I would like to thank Mr. N. Sinanis of Uni-
versity of Crete for various help during my stay in Crete.
This work was also supported by RFFI grant 02-01-00126a and the grant for
support of leading scientific schools 0015-96046 and INTAS grant 000262.
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