HU-EP-07/12
arXiv:0704.0929
Noncommutative Electromagnetism As A Large N Gauge Theory
Hyun Seok Yang ∗
Institut für Physik, Humboldt Universität zu Berlin
Newtonstraße 15, D-12489 Berlin, Germany
ABSTRACT
We map noncommutative (NC) U(1) gauge theory on IRdC × IR2nNC to U(N → ∞) Yang-Mills the-
ory on IRdC , where IR
C is a d-dimensional commutative spacetime while IR
NC is a 2n-dimensional NC
space. The resulting U(N) Yang-Mills theory on IRdC is equivalent to that obtained by the dimensional
reduction of (d+2n)-dimensional U(N) Yang-Mills theory onto IRdC . We show that the gauge-Higgs
system (Aµ,Φ
a) in the U(N → ∞) Yang-Mills theory on IRdC leads to an emergent geometry in the
(d+2n)-dimensional spacetime whose metric was determined by Ward a long time ago. In particular,
the 10-dimensional gravity for d = 4 and n = 3 corresponds to the emergent geometry arising from
the 4-dimensional N = 4 vector multiplet in the AdS/CFT duality. We further elucidate the emergent
gravity by showing that the gauge-Higgs system (Aµ,Φ
a) in half-BPS configurations describes self-
dual Einstein gravity.
PACS numbers: 11.10.Nx, 02.40.Gh, 04.50.+h
Keywords: Noncommutative Gauge Theory, Large N Gauge Theory, Emergent Gravity
October 31, 2018
∗hsyang@physik.hu-berlin.de
http://arxiv.org/abs/0704.0929v3
http://arxiv.org/abs/0704.0929
1 Introduction
A noncommutative (NC) spacetimeM is obtained by introducing a symplectic structure B = 1
Babdy
dyb and then by quantizing the spacetime with its Poisson structure θab ≡ (B−1)ab, treating it as a
quantum phase space. That is, for f, g ∈ C∞(M),
{f, g} = θab
⇒ −i[f̂ , ĝ]. (1.1)
According to the Weyl-Moyal map [1, 2], the NC algebra of operators is equivalent to the deformed
algebra of functions defined by the Moyal ⋆-product, i.e.,
f̂ · ĝ ∼= (f ⋆ g)(y) = exp
θab∂ya∂
f(y)g(z)
. (1.2)
Through the quantization rules (1.1) and (1.2), one can define NC IR2n by the following commutation
relation
[ya, yb]⋆ = iθ
ab. (1.3)
It is well-known [2, 3] that a NC field theory can be identified basically with a matrix model or a
large N field theory. This claim is based on the following fact. Let us consider a NC IR2 for simplicity,
[x, y] = iθ, (1.4)
although the same argument equally holds for a NC IR2n as it will be shown later. After scaling the
coordinates x →
θx, y →
θy, the NC plane (1.4) becomes the Heisenberg algebra of harmonic
oscillator
[a, a†] = 1. (1.5)
It is a well-known fact from quantum mechanics that the representation space of NC IR2 is given by
an infinite-dimensional, separable Hilbert space H = {|n〉, n = 0, 1, · · · } which is orthonormal, i.e.,
〈n|m〉 = δnm and complete, i.e.,
n=0 |n〉〈n| = 1. Therefore a scalar field φ̂ ∈ Aθ on the NC plane
(1.4) can be expanded in terms of the complete operator basis
Aθ = {|m〉〈n|, n,m = 0, 1, · · · }, (1.6)
that is,
φ̂(x, y) =
Mmn|m〉〈n|. (1.7)
One can regard Mmn in (1.7) as components of an N × N matrix M in the N → ∞ limit. More
generally one may replace NC IR2 by a Riemann surface Σg of genus g which can be quantized via
deformation quantization [4]. For a compact Riemann surface Σg with finite area A(Σg), the matrix
representation can be finite-dimensional, e.g., for a fuzzy sphere [5]. In this case, A(Σg) ∼ θN but
we simply take the limit N → ∞. We then arrive at the well-known relation:
Scalar field on NC IR2 (or Σg) ⇐⇒ N ×N matrix at N → ∞. (1.8)
If φ̂ is a real scalar field, then M should be a Hermitean matrix. We will see that the above relation
(1.8) has far-reaching applications to string theory.
The matrix representation (1.7) clarifies why NC U(1) gauge theory is a large N gauge theory.
An important point is that the NC gauge symmetry acts as a unitary transformation on H for a field
φ̂ ∈ Aθ in the adjoint representation of U(1) gauge group
φ̂ → Uφ̂ U †. (1.9)
This NC gauge symmetry Ucpt(H) is so large that Ucpt(H) ⊃ U(N) (N → ∞) [6, 7], which is
rather obvious in the matrix basis (1.6). Therefore the NC gauge theory is essentially a large N gauge
theory. It becomes more precise on a NC torus through the Morita equivalence where NC U(1) gauge
theory with rational θ = M/N is equivalent to an ordinary U(N) gauge theory [8]. For this reason,
it is not so surprising that NC electromagnetism shares essential properties appearing in a large N
gauge theory such as SU(N → ∞) Yang-Mills theory or matrix models.
It is well-known [9] that 1/N expansion of any large N gauge theory using the double line for-
malism reveals a picture of a topological expansion in terms of surfaces of different genus, which
can be interpreted in terms of closed string variables as the genus expansion of string amplitudes. It
has been underlain the idea that large N gauge theories have a dual description in terms of gravita-
tional theories in higher dimensions. For example, BFSS matrix model [10], IKKT matrix model [11]
and AdS/CFT duality [12]. From the perspective (1.8), the 1/N expansion corresponds to the NC
deformation in terms of θ/A(Σg).
All these arguments imply that there exists a solid map between a NC gauge theory and a large N
gauge theory. In this work we will find a sound realization of this idea. It turns out that the emergent
gravity recently found in [13, 14, 15, 16] can be elegantly understood in this framework. Therefore
the correspondence between NC field theory and gravity [3] is certainly akin to the gauge/gravity
duality in large N limit [10, 11, 12].
This paper is organized as follows. In Section 2 we map NC U(1) gauge theory on IRdC × IR2nNC to
U(N → ∞) Yang-Mills theory on IRdC , where IRdC is a d-dimensional commutative spacetime while
IR2nNC is a 2n-dimensional NC space. The resulting U(N) Yang-Mills theory on IR
C is equivalent to
that obtained by the dimensional reduction of (d + 2n)-dimensional U(N) Yang-Mills theory onto
IRdC . In Section 3, we show that the gauge-Higgs system (Aµ,Φ
a) in the U(N → ∞) Yang-Mills
theory on IRdC leads to an emergent geometry in the (d + 2n)-dimensional spacetime whose metric
was determined by Ward [17] a long time ago. In particular, the 10-dimensional gravity for d = 4 and
n = 3 corresponds to the emergent geometry arising from the 4-dimensional N = 4 vector multiplet
in the AdS/CFT duality [12]. We further elucidate the emergent gravity in Section 4 by showing that
the gauge-Higgs system (Aµ,Φ
a) in half-BPS configurations describes self-dual Einstein gravity. A
notable point is that the emergent geometry arising from the gauge-Higgs system (Aµ,Φ
a) is closely
related to the bubbling geometry in AdS space found in [18]. Finally, in Section 5, we discuss several
interesting issues that naturally arise from our construction.
2 A Large N Gauge Theory From NC U(1) Gauge Theory
We will consider a NC U(1) gauge theory on IRD = IRdC × IR2nNC , where D-dimensional coordinates
XM (M = 1, · · · , D) are decomposed into d-dimensional commutative ones, denoted as zµ (µ =
1, · · · , d) and 2n-dimensional NC ones, denoted as ya (a = 1, · · · , 2n), satisfying the relation (1.3).
We assume the metric on IRD = IRdC × IR2nNC as the following form 1
ds2 = GMNdXMdXN
= gµνdz
µdzν +Gabdy
adyb. (2.1)
The action for D-dimensional NC U(1) gauge theory is given by
4g2YM
detGGMPGNQ(FMN + ΦMN ) ⋆ (FPQ + ΦPQ), (2.2)
where the NC field strength FMN is defined by
FMN = ∂MAN − ∂NAM − i[AM , AN ]⋆. (2.3)
The constant two-form Φ will be taken either 0 or −B = −1
Babdy
a ∧ dyb with rank(B) = 2n.
Here we will use the background independent prescription [8, 19] where the open string metric
Gab, the noncommutativity θ
ab and the open string coupling Gs are determined by
θab =
, Gab = −κ2
, Gs = gs
det′(κBg−1), (2.4)
with κ ≡ 2πα′. The closed string metric gab in Eq.(2.4) is independent of gµν in Eq.(2.1) and det′
denotes a determinant taken along NC directions only in IR2nNC . In terms of these parameters, the
couplings are related by
, (2.5)
det′G
gs|Pfθ|
. (2.6)
An important fact is that translations in NC directions are basically gauge transformations, i.e.,
eik·y ⋆ f(z, y) ⋆ e−ik·y = f(z, y+ θ · k) for any f(z, y) ∈ C∞(M). This means that translations along
NC directions act as inner derivations of the NC algebra Aθ:
[ya, f ]⋆ = iθ
ab∂bf. (2.7)
1 Here we can take the d-dimensional spacetime metric gµν with either Lorentzian or Euclidean signature since the
signature is inconsequential in our most discussions. But we implicitly assume the Euclidean signature for some other
discussions.
Using this relation, each component of FMN can be written as the following forms
Fµν = i[Dµ, Dµ]⋆, (2.8)
Fµa = θ
[Dµ, x
b]⋆ = −Faµ, (2.9)
Fab = −iθ−1ac θ−1bd
[xc, xd]⋆ − iθcd
, (2.10)
where the covariant derivative Dµ and the covariant coordinate x
a are, respectively, defined by
Dµ ≡ ∂µ − iAµ, (2.11)
xa ≡ ya + θabAb. (2.12)
Collecting all these facts, one gets the following expression for the action (2.2) with Φ = −B 2
(2πκ)
detgµνTrH
gµλgνσFµν ⋆ Fλσ +
gµνgabDµΦ
a ⋆ DνΦ
gacgbd[Φ
a,Φb]⋆ ⋆ [Φ
c,Φd]⋆
, (2.13)
where we defined adjoint scalar fields Φa ≡ xa/κ of mass dimension and
TrH ≡
(2π)n|Pfθ| . (2.14)
Note that the number of the adjoint scalar fields is equal to the rank of θab. The resulting action (2.13)
is not new but rather well-known in NC field theory, e.g., see [19, 20].
The NC algebra (1.3) is equivalent to the Heisenberg algebra of an n-dimensional harmonic os-
cillator in a frame where θab has a canonical form:
[ai, a
j ] = δij , (i, j = 1, · · · , n). (2.15)
The NC space (1.3) is therefore represented by the infinite-dimensional Hilbert space H = {|~m〉 ≡
|m1, · · · , mn〉;mi = 0, 1, · · · , N → ∞ for i = 1, · · · , n} whose set of eigenvalues forms an n-
dimensional positive integer lattice. A set of operators in H
Aθ = {|~m〉〈~n|;mi, ni = 0, 1, · · · , N → ∞ for i = 1, · · · , n} (2.16)
can be identified with the generators of a complete operator basis and so any NC field φ(z, y) ∈ Aθ
can be expanded in the basis (2.16) as follows,
φ(z, y) =
~m,~n
Ω~m,~n (z)|~m〉〈~n|. (2.17)
2If Φ = 0 in Eq.(2.2), the only change in Eq.(2.13) is [Φa,Φb] → [Φa,Φb]− i
Now we use the ‘Cantor diagonal method’ to put the n-dimensional positive integer lattice in H
into a one-to-one correspondence with the infinite set of natural numbers (i.e., 1-dimensional positive
integer lattice): |~m〉 ↔ |i〉, i = 1, · · · , N → ∞. In this one-dimensional basis, Eq.(2.17) is relabeled
as the following form
φ(z, y) =
Ωij (z)|i〉〈j|. (2.18)
Following the motivation discussed in the Introduction, we regard Ωij(z) in (2.18) as components of
an N ×N matrix Ω in the N → ∞ limit, which also depend on zµ, the coordinates of IRdC . If the field
φ(z, y) is real which is the case for the gauge-Higgs system (Aµ,Φ
a) in the action (2.13), the matrix
Ω should be Hermitean, but not necessarily traceless. So the N × N matrix Ω(z) can be regarded as
a field in U(N → ∞) gauge theory on d-dimensional commutative space IRdC , where TrH in (2.14)
is identified with the matrix trace over the basis (2.18). All the dependence on NC coordinates is now
encoded into N ×N matrices and the noncommutativity in terms of star product is transferred to the
matrix product.
Adopting the matrix representation (2.18), the D-dimensional NC U(1) gauge theory (2.2) is
mapped to the U(N → ∞) Yang-Mills theory on d-dimensional commutative space IRdC . One can
see that the resulting U(N) Yang-Mills theory on IRdC in Eq.(2.13) is equivalent to that obtained by
the dimensional reduction of (d + 2n)-dimensional U(N) Yang-Mills theory onto IRdC . It might be
emphasized that the map between the D-dimensional NC U(1) gauge theory and the d-dimensional
U(N → ∞) Yang-Mills theory is “exact” and thus the two theories should describe a completely
equivalent physics. For example, we can recover the D-dimensional NC U(1) gauge theory on IRdC ×
IR2nNC from the d-dimensional U(N → ∞) Yang-Mills theory on IRdC by recalling that the number of
adjoint Higgs fields in the U(N) Yang-Mills theory is equal to the dimension of the extra NC space
IR2nNC and by applying the dictionary in Eqs.(2.8)-(2.10).
One can introduce linear algebraic conditions of D-dimensional field strengths FMN as a higher
dimensional analogue of 4-dimensional self-duality equations such that the Yang-Mills equations in
the action (2.2) follow automatically. These are of the following type [21, 22]
TMNPQFPQ = λFMN (2.19)
with a constant 4-form tensor TMNPQ. The relation (2.19) clearly implies via the Bianchi identity
D[MFPQ] = 0 that the Yang-Mills equations are satisfied provided λ is nonzero. For D > 4, the 4-
form tensor TMNPQ cannot be invariant under SO(D) transformations and the equation (2.19) breaks
the rotational symmetry to a subgroup H ⊂ SO(D). Thus the resulting first order equations can be
classified by the unbroken symmetry H under which TMNPQ remain invariant [21, 22]. It was also
shown [23] that the first order linear equations above are closely related to supersymmetric states, i.e.,
BPS states in higher dimensional Yang-Mills theories.
The equivalence between D- and d-dimensional gauge theories can be effectively used to clas-
sify classical solutions in the d-dimensional U(N) Yang-Mills theory (2.13). The group theoretical
classification [21], integrability condition [22] and BPS states [23] for the D-dimensional first-order
equations (2.19) can be directly translated into the properties of the gauge-Higgs system (Aµ,Φ
a) in
the d-dimensional U(N) gauge theory (2.13). These classifications will also be useful to classify the
geometries emerging from the gauge-Higgs system (Aµ,Φ
a) in the U(N → ∞) Yang-Mills theory
(2.13), which will be discussed in the next section. Unfortunately, the D = 10 case is missing in
[21, 22, 23] which is the most interesting case (d = 4 and n = 3) related to the AdS/CFT duality.
3 Emergent Geometry From NC Gauge Theory
Let us first recapitulate the result in [17]. It turns out that the Ward’s construction perfectly fits with
the emergent geometry arising from the gauge-Higgs system (Aµ,Φ
a) in the U(N → ∞) Yang-Mills
theory (2.13). Suppose that we have gauge fields on IRdC taking values in the Lie algebra of volume-
preserving vector fields on an m-dimensional manifold M [24, 25]. In other words, the gauge group
G = SDiff(M). The gauge covariant derivative is given by Eq.(2.11), but the Aµ(z) are now vector
fields on M , also depending on zµ ∈ IRdC . The other ingredient in [17] consists of m Higgs fields
Φa(z) ∈ sdiff(M), the Lie algebra of SDiff(M), for a = 1, · · · , m. The idea [24, 25] is to specify
f−1(D1, · · · , Dd,Φ1, · · · ,Φm) (3.1)
forms an orthonormal frame and hence defines a metric on IRdC ×M with a volume form ν = ddz∧ω.
Here f is a scalar, a conformal factor, defined by
f 2 = ω(Φ1, · · · ,Φm). (3.2)
The result in [24, 25] immediately implies that the gauge-Higgs system (Aµ,Φ
a) leads to a metric
on the (d + m)-dimensional space IRdC × M . A local coordinate expression for this metric is easily
obtained from Eq.(3.1). Let ya be local coordinates on M . So Aµ(z) and Φa(z) have the form
Aµ(z) = A
µ(z, y)
, Φa(z) = Φ
a(z, y)
, (3.3)
where the y-dependence, originally hidden in the Lie algebra of G = SDiff(M), now explicitly
appears in the coefficients Aaµ and Φ
a. Let V
b denote the inverse of the m×m matrix Φba, and let Aa
denote the 1-form Aaµdz
µ. Then the metric is [17]
ds2 = f 2δµνdz
µdzν + f 2δabV
d (dy
c −Ac)(dyd −Ad). (3.4)
It will be shown later that the choice of the volume form ω for the conformal factor (3.2) corresponds
to that of a particular conformally flat background although we mostly assume a flat volume form,
i.e., ω ∼ dy1 ∧ · · · ∧ dy2n, unless explicitly specified.
The gauge and Higgs fields in Eq.(3.3) are not arbitrary but must be subject to the Yang-Mills
equations, for example, derived from the action (2.13), which are, in most cases, not completely
integrable. Hence to completely determine the geometric structure emerging from the gauge-Higgs
system (Aµ,Φ
a) is as much difficult as solving the Einstein equations in general. But the self-dual
Yang-Mills equations in four dimensions or Eq.(2.19) in general are, in some sense, “completely
solvable”. Thus the metric (3.4) for these cases might be completely determined. Let us discuss two
notable examples. See [17] for more examples describing 4-dimensional self-dual Einstein gravity.
• Case d = 0, m = 4: This case was dealt with in detail in [25, 26, 27]. It was proved that the
self-dual Einstein equations are equivalent to the self-duality equations
[Φa,Φb] = ±
εabcd[Φc,Φd] (3.5)
on the four Higgs fields Φa. Furthermore reinterpreting n of the Φa’s as Dµ leads to the case d =
n, m = 4− n. In Section 5, we will discuss the physical meaning about the interpretation Φa 7→ Dµ.
• Case d = 3, m = 1: Here M is one-dimensional, so the Lie algebra of vector fields on M is the
Virasoro algebra. Thus Aµ and Φ are now real-valued vector fields on M which must be independent
of y to preserve the volume form ν = d3z ∧ dy [27]. The metric (3.4) reduces to
ds2 = Φd~z · d~z + Φ−1(dy −Aµdzµ)2 (3.6)
and has a Killing vector ∂/∂y. In this case, the self-duality equations (3.5) reduce to the Abelian
Bogomol’nyi equations, ∇× ~A = ∇Φ, and the metric (3.6) describes a gravitational instanton [28].
Recently we showed in [15, 16] for the d = 0 and m = 4 case that self-dual electromagnetism in
NC spacetime is equivalent to self-dual Einstein gravity and the metric is precisely given by Eq.(3.4).
A key observation [16] was that the self-dual system (3.5) defined by vector fields on M can be
derived from the action (2.2) or (2.13) for slowly varying fields, where all ⋆-commutators between
NC fields are approximated by the Poisson bracket (1.1). An important point in NC geometry is
that the adjoint action of (covariant) coordinates with respect to star product can be identified with
(generalized) vector fields on some (curved) manifold [15, 16], as the trivial case was already used
in Eq.(2.7). In the end, a D-dimensional manifold described by the metric (3.4) corresponds to an
emergent geometry arising from the gauge-Higgs system in Eq.(3.3). Now we will show in a general
context how the nontrivial geometry (3.4) emerges from the gauge-Higgs system (Aµ,Φ
a) in the
action (2.13).
Let us collectively denote the covariant derivatives Dµ in (2.11) and the Higgs fields Da ≡
−iκBabΦb = −i(Babyb +Aa) in (2.12) as DA(z, y). Therefore DA(z, y) transform covariantly under
NC U(1) gauge transformations
DA(z, y) → g(z, y) ⋆ DA(z, y) ⋆ g−1(z, y). (3.7)
Define the adjoint action of DA(z, y) with respect to star product acting on any NC field f(z, y) ∈ Aθ:
adDA[f ] ≡ [DA, f ]⋆. (3.8)
Then it is easy to see [16] that the above adjoint action satisfies the Leibniz rule and the Jocobi
identity, i.e.,
[DA, f ⋆ g]⋆ = [DA, f ]⋆ ⋆ g + f ⋆ [DA, g]⋆, (3.9)
[DA, [DB, f ]⋆]⋆ − [DB, [DA, f ]⋆]⋆ = [[DA, DB]⋆, f ]⋆. (3.10)
These properties imply that adDA can be identified with ‘generalized’ vector fields or Lie deriva-
tives acting on the algebra Aθ, which can be viewed as a gauge covariant generalization of the inner
derivation (2.7). Note that the generalized vector field in Eq.(3.8) is a kind of general higher or-
der differential operators in [29]. Indeed it turns out that they constitute a generalization of volume
preserving diffeomorphisms to ⋆-differential operators acting on Aθ (see Eqs.(4.1) and (4.2) in [7]).
In particular, the generalized vector fields in Eq.(3.8) reduce to usual vector fields in the commu-
tative, i.e. O(θ), limit:
adDA [f ] = iθ
ab∂DA
+ · · · = i{DA, f}+O(θ3)
≡ V aA(z, y)∂af(z, y) +O(θ3) (3.11)
where we defined [∂µ, f ]⋆ = ∂µf . Note that the vector fields VA(z, y) = V
A(z, y)∂a are exactly of
the same form as Eq.(3.3) and belong to the Lie algebra of volume preserving diffeomorphisms, as
precisely required in the Ward construction (3.1), since they are all divergence free, i.e., ∂aV
A = 0.
Thus the vector fields f−1VA(z, y) for A = 1, · · · , D can be identified with the orthonormal frame
(3.1) defining the metric (3.4). It should be emphasized that the emergent gravity (3.4) arises from a
general, not necessarily self-dual, gauge-Higgs system (Aµ,Φ
a) in the action (2.13).
Note that
[DA, DB]⋆ = −i(FAB − BAB) (3.12)
where the NC field strength FAB is given by Eq.(2.3). Then the Jacobi identity (3.10) leads to the
following identity for a constant BAB
ad[DA,DB]⋆ = −i adFAB = [adDA, adDB ]⋆. (3.13)
The inner derivation (3.11) in commutative limit is reduced to the well-known map C∞(M) →
TM : f 7→ Xf between the Poisson algebra (C∞(M), {·, ·}) and vector fields in TM defined by
Xf(g) = {g, f} for any smooth function g ∈ C∞(M). The Jacobi identity for the Poisson algebra
(C∞(M), {·, ·}) then leads to the Lie algebra homomorphism
X{f,g} = −[Xf , Xg] (3.14)
where the right-hand side is defined by the Lie bracket between Hamiltonian vector fields. One can
check by identifying f = DA and g = DB that the Lie algebra homomorphism (3.14) correspond
to the commutative limit of the Jacobi identity (3.10). That is, one can deduce from Eq.(3.14) the
following identity
XFAB = −[VA, VB] (3.15)
using the relation {DA, DB} = −FAB +BAB and XDA = iVA.
Using the homomorphism (3.15), one can translate the generalized self-duality equation (2.19)
into the structure equation between vector fields
TABCDFCD = λFAB ⇔
TABCD[VC , VD] = λ[VA, VB]. (3.16)
Therefore a D-dimensional NC gauge field configuration satisfying the first-order system defined by
the left-hand side of Eq.(3.16) is isomorphic to a D-dimensional emergent geometry defined by the
right-hand side of Eq.(3.16) whose metric is given by Eq.(3.4). For example, in four dimensions
where TABCD = εABCD and λ = ±1, the right-hand side of Eq.(3.16) is precisely equal to Eq.(3.5)
describing gravitational instantons [24, 25, 26, 27]. This proves, as first shown in [15, 16], that self-
dual NC electromagnetism is equivalent to self-dual Einstein gravity. Note that the Einstein gravity
described by the metric (3.4) arises from the commutative, i.e., O(θ) limit. Therefore it is natural to
expect that the higher order differential operators in Eq.(3.11), e.g. O(θ3), give rise to higher order
gravity [16]. We will further discuss the derivative correction in Section 5.
The 10-dimensional metric (3.4) for d = 4 and n = 3 (m = 6) is particularly interesting since it
corresponds to an emergent geometry arising from the 4-dimensional N = 4 vector multiplet in the
AdS/CFT duality. Note that the gravity in the AdS/CFT duality is an emergent phenomenon arising
from particle interactions in a gravityless, lower-dimensional spacetime. As a famous example, the
type IIB supergravity (or more generally the type IIB superstring theory) on AdS5 × S5 is emergent
from the 4-dimensional N = 4 supersymmetric U(N) Yang-Mills theory [12].3 In our construction,
N × N matrices are mapped to vector fields on some manifold M , so the vector fields in Eq.(3.3)
correspond to master fields of large N matrices [30], in other words, (Aµ,Φ
a) ∼ N2. According to
the AdS/CFT duality, we thus expect that the metric (3.4) describes a deformed geometry induced
by excitations of the gauge and Higgs fields in the action (2.13). For example, we may look for
1/2 BPS geometries in type IIB supergravity that arise from chiral primaries of N = 4 super Yang-
Mills theory [18]. Recently this kind of BPS geometries, the so-called bubbling geometry in AdS
space, with a particular isometry was completely determined in [18], where the AdS5 × S5 geometry
emerges from the simplest and most symmetric configuration. In next section we will illustrate such
kind of bubbling geometry described by the metric (3.4) by considering self-dual configurations in
the gauge-Higgs system.
3The overall U(1) = U(N)/SU(N) factor actually corresponds to the overall position of D3-branes and may be
ignored when considering dynamics on the branes, thereby leaving only an SU(N) gauge symmetry.
4 Self-dual Einstein Gravity From Large N Gauge Theory
In the previous section we showed that the Ward’s metric (3.4) naturally emerges from the D-dimensional
NC U(1) gauge fields AM on IR
C × IR2nNC or equivalently the gauge-Higgs system (Aµ,Φa) in d-
dimensional U(N) gauge theory on IRdC . So, if an explicit solution for AM or (Aµ,Φ
a) is known, the
corresponding metric (3.4) is, in principle, exactly determined. However, it is extremely difficult to
get a general solution by solving the equations of motion for the action (2.2) or (2.13). Instead we
may try to solve a more simpler system such as the first-order equations (2.19), which are morally
believed to be ‘exactly solvable’ in most cases. In this section we will further elucidate the emer-
gent gravity arising from gauge fields by showing that the gauge-Higgs system (Aµ,Φ
a) in half-BPS
configurations describes self-dual Einstein gravity. Since the case for D = 4 and n = 2 has been
extensively discussed in [14, 15, 16], we will consider the other cases for D ≥ 4. For simplicity,
the metrics in the action (2.13) are supposed to be the form already used in Eq.(3.4); gµν = δµν and
gab = δab.
Note that the action (2.2) or (2.13) contains a background B, due to a uniform condensation of
gauge fields in a vacuum. But we will require a rapid fall-off of fluctuating fields around the back-
ground at infinity in IRD as usual.4 Our boundary condition is FMN → 0 at infinity. Eq.(2.10) then
requires that [xa, xb]⋆ → iθab at |y| → ∞. Thus the coordinates ya in (2.12) are vacuum expectation
values of xa characterizing the uniform condensation of gauge fields [16]. This condensation of the
B-fields endows the ⋆-algebra Aθ with a remarkable property that translations act as an inner auto-
morphism of the algebra Aθ as shown in Eq.(2.7). But the gauge symmetry on NC spacetime requires
the covariant coordinates xa in Eq.(2.12) instead of ya [31]. The inner derivation adDa in Eq.(3.8) is
then a ‘dual element’ related to the coordinate xa. This is also true for the covariant derivatives Dµ
in (2.11) since they are related to Da = −iBabxb by the ‘matrix T -duality’; Da 7→ Dµ, as will be
explained in Section 5.
It is very instructive to take an analogy with quantum mechanics. Quantum mechanical time
evolution in Heisenberg picture is defined as an inner automorphism of the Weyl algebra obtained
from a quantum phase space
f(t) = eiHtf(0)e−iHt
and its evolution equation is of the form (3.8)
df(t)
= i[H, f(t)].
Here we liberally interpret DA(z, y) in Eq.(3.8) as ‘multi-Hamiltonians’ determining the spacetime
evolution in IRD. Then it is quite natural to interpret Eq.(3.8) as a spacetime evolution equation
determined by the “covariant Hamiltonians” DA(z, y).
4In the matrix representation (2.18), this means that matrix components Ωij(z) for the fluctuations are rapidly vanish-
ing for i, j = N → ∞ as well as for |z| → ∞, since roughly N ∼ ~y · ~y.
Let us be more precise about the meaning of the spacetime evolution. If the Hamiltonian is slightly
deformed, H → H + δH , the time evolution of a system is correspondingly changed. Likewise, the
fluctuation of gauge fields AM or (Aµ,Φ
a) around the background specified by ya’s changes DA(z, y),
which in turn induces a deformation of the background spacetime according to Eq.(3.11). This is
precisely the picture about the emergent geometry in [15, 16] and also a dependable interpretation of
the Ward’s geometry (3.4). A consistent picture related to the AdS/CFT duality was also observed in
the last of Section 3.
For the above reason, all equations in the following will be understood as inner derivations acting
on Aθ like as (3.8). The adjoint action defined in this way naturally removes a contribution from
the background in the action (2.2) or (2.13) [15]. For example, the first equation in (4.1) can be
consistent only in this way since the left hand side goes to zero at infinity but the right hand side
becomes ∼ θ/κ2. It might be remarked that this is the way to define the equations of motion in
the background independent formulation [8, 19] and thus it should be equivalent to the usual NC
prescription with ΦMN = 0.
4.1 D = 4 and n = 1
NC instanton solutions in this case were constructed in [32]. As was proved in Eq.(3.16), NC U(1)
instantons are in general equivalent to gravitational instantons. We thus expect that the NC self-
duality equations for D = 4 and n = 1 are mapped to self-dual Einstein equations. We will show
that the gauge-Higgs system (Aµ,Φ
a) in this case is mapped to two-dimensional U(∞) chiral model,
whose equations of motion are equivalent to the Plebański form of the self-dual Einstein equations
[33, 17, 34].
We showed in Section 2 that 4-dimensional NC U(1) gauge theory on IR2C × IR2NC is mapped
to 2-dimensional U(N → ∞) gauge theory with the action (2.13). The 4-dimensional self-duality
equations now become the U(N → ∞) Hitchin equations on IR2C :
Fµν = ±
εµν [Φ,Φ
†], DµΦ = ±iεµνDνΦ, (4.1)
where Φ = Φ1 + iΦ2. Note that the above equations also arise as zero-energy solutions in U(N)
Chern-Simons gauge theory coupled to a nonrelativistic complex scalar field in the adjoint represen-
tation [35]. It was shown in [36] that the self-dual system in Eq.(4.1) is completely solvable in terms
of Uhlenbeck’s uniton method. A NC generalization of Eq.(4.1), the Hitchin’s equations on IR2NC ,
was also considered in [37] with very parallel results to the commutative case. We will briefly discuss
the NC Hitchin’s equations in Section 5.
The equations (4.1) for the self-dual case (with + sign) can be elegantly combined into a zero-
curvature condition [35, 36] for the new connections defined by5
A+ = A+ + Φ, A− = A− − Φ† (4.2)
5Here we will relax the reality condition of the fields (Aµ,Φ
a) and complexify them.
with A± = A1 ± iA2:
F+− = ∂+A− − ∂−A+ − i[A+,A−] = 0 (4.3)
where ∂± = ∂1 ± i∂2. Thus the new gauge fields should be a pure gauge, that is, A± = ig−1∂±g for
some g ∈ GL(N, IC). Thus we can choose them to be zero, viz.
A+ = −Φ, A− = Φ†. (4.4)
Then the self-dual equations (4.1) reduce to
† + ∂−Φ + 2i[Φ,Φ
†] = 0, (4.5)
† − ∂−Φ = 0. (4.6)
Introducing another gauge fields C+ = −2Φ and C− = 2Φ†, Eq.(4.5) also becomes the zero-curvature
condition, hence C± are a pure gauge or
Φ = − i
h−1∂+h, Φ
h−1∂−h. (4.7)
A group element h(z) defines a map from IR2C to GL(N, IC) group, which is contractible to the map
from IR2C to U(N) ⊂ GL(N, IC). Then Eq.(4.6) implies that h(z) satisfies the equation in the two-
dimensional U(N) chiral model [35, 36]
−1∂−h) + ∂−(h
−1∂+h) = 0. (4.8)
Eq.(4.8) is the equation of motion derived from the two-dimensional U(N) chiral model governed
by the following Euclidean action
d2zTr ∂µh
−1∂νhδ
µν . (4.9)
A remarkable (mysterious) fact has been known [33, 17, 34] that in the N → ∞ limit the chiral
model (4.9) describes a self-dual spacetime whose equation of motion takes the Plebański form of
self-dual Einstein equations [38]. Thus, including the case of D = 4 and n = 2 in [14, 15, 16], we
have confirmed Eq.(3.16) stating that the 4-dimensional self-dual system in the action (2.2) or (2.13)
in general describes the self-dual Einstein gravity where self-dual metrics are given by Eq.(3.4).
4.2 D = 6 and n = 1
Our current work has been particularly motivated by this case since it was already shown in [39] that
SU(N) Yang-Mills instantons in the N → ∞ limit are gravitational instantons too. Since NC U(1)
instantons are also gravitational instantons as we showed before, it implies that there should be a close
relationship between SU(N) Yang-Mills instantons and NC U(1) instantons. A basic observation was
the relation (1.8), which leads to the sound realization in Eq.(2.13). But we will simply follow the
argument in [39] for the gauge group G = U(N); in the meantime, we will confirm the results for the
emergent geometry from NC gauge fields.
Let us look at the instanton solution in U(N) Yang-Mills theory. The self-duality equation is
given by
Fµν = ±
εµναβFαβ (4.10)
where the field strength is defined by
Fµν = ∂µAν − ∂νAµ − i[Aµ, Aν ]. (4.11)
In terms of the complex coordinates and the complex gauge fields defined by
(x2 + ix1), z2 =
(x4 + ix3),
Az1 = A2 − iA1, Az2 = A4 − iA3,
Eq.(4.10) can be written as
Fz1z2 = 0 = Fz̄1z̄2 , (4.12)
Fz1z̄1 ∓ Fz2z̄2 = 0. (4.13)
Now let us consider the anti-self-dual (ASD) case. We first notice that Fz1z2 = 0 implies that there
exists a u(N)-valued function g such that Aza = ig
−1∂zag (a = 1, 2). Therefore one can choose a
gauge
Aza = 0. (4.14)
Under the gauge (4.14), the ASD equations lead to
∂z̄1Az̄2 − ∂z̄2Az̄1 − i[Az̄1 , Az̄2 ] = 0, (4.15)
∂z1Az̄1 + ∂z2Az̄2 = 0. (4.16)
First notice a close similarity with Eqs.(4.5) and (4.6). Eq.(4.16) can be solved by introducing a
u(N)-valued function Φ such that
Az̄1 = −∂z2Φ, Az̄2 = ∂z1Φ. (4.17)
Substituting (4.17) into (4.15) one finally gets
(∂z1∂z̄1 + ∂z2∂z̄2)Φ− i[∂z1Φ, ∂z2Φ] = 0. (4.18)
Adopting the correspondence (1.8), we now regard Φ ∈ u(N)⊗C∞(IR4) in Eq.(4.18) as a smooth
function on IR4 × Σg, i.e., Φ = Φ(xµ, p, q) where (p, q) are local coordinates of a two-dimensional
Riemann surface Σg. Moreover, a Lie algebra commutator is replaced by the Poisson bracket (1.1)
{f, g} = ∂f
that is,
[Φ1,Φ2] → i{Φ1,Φ2}, (4.19)
where we absorbed θ into the coordinates (p, q). After all, the ASD Yang-Mills equation (4.18) in
the large N limit is equivalent to a single nonlinear equation in six dimensions parameterized by
(xµ, p, q):
(∂z1∂z̄1 + ∂z2∂z̄2)Φ + {∂z1Φ, ∂z2Φ} = 0. (4.20)
Since Eq.(4.20) is similar to the well-known second heavenly equation [38], it was called in [39] as
the six dimensional version of the second heavenly equation.
Starting from U(N) Yang-Mills instantons in four dimensions, we arrived at the nonlinear dif-
ferential equation for a single function in six dimensions. It is important to notice that the resulting
six-dimensional theory is a NC field theory since the Riemann surface Σg carries a symplectic struc-
ture inherited from the u(N) Lie algebra through Eq.(4.19) and it can be quantized in general via
deformation quantization [4]. Since the function Φ in (4.20) is a master field of N ×N matrices [30],
so Φ ∼ N2, the AdS/CFT duality [12] implies that the master field Φ describes a six-dimensional
emergent geometry induced by Yang-Mills instantons.
To see the emergent geometry, consider an appropriate symmetry reduction of Eq.(4.20) to show
that it describes self-dual gravity in four dimensions. There are many reductions from six to four
dimensional subspace leading to self-dual four-manifolds [39]. A common feature is that the four
dimensional subspace necessarily contains the NC Riemann surface Σg. We will show later how the
symmetry reduction naturally arises from the BPS condition in six dimensions. As a specific example,
we assume the following symmetry,
∂z1Φ = ∂z̄1Φ, ∂z2Φ = ∂z̄2Φ, (4.21)
Φ(z1, z2, z̄1, z̄2, p, q) = Λ(z1 + z̄1 ≡ x, z2 + z̄2 ≡ y, p, q). (4.22)
Then Eq.(4.20) is precisely equal to the Husain’s equation [34] which is the reduction of self-dual
Einstein equations to the sdiff(Σg) chiral field equations in two dimensions:
Λxx + Λyy + ΛxqΛyp − ΛxpΛyq = 0. (4.23)
Note that we already encountered in Section 4.1 the two-dimensional sdiff(Σg) chiral field equations
since sdiff(Σg) ∼= u(N) according to the correspondence (1.8). We showed in [16] that Eq.(4.23)
can be transformed to the first heavenly equation [38] which is a governing equation of self-dual
Einstein gravity. In the end we conclude that self-dual U(N) Yang-Mills theory in the large N limit
is equivalent to self-dual Einstein gravity.
Now it is easy to see that the self-dual Einstein equation (4.23) is coming from a 1/2 BPS equation
in six dimensions (see Eq.(34) in [23]) defined by the first-order equation (2.19). According to our
construction, the six-dimensional NC U(1) gauge theory (2.2) is equivalent to the four-dimensional
U(N) gauge theory (2.13). Therefore six-dimensional BPS equations can be equivalently described
by the gauge-Higgs system (Aµ,Φ
a) in the action (2.13). Let us newly denote the NC coordinates
y1, y2 and commutative ones z3, z4 as uα, α = 1, 2, 3, 4 while z1, z2 as vA, A = 1, 2. The 1/2 BPS
equations, Eq.(34), in [23] can then be written as the following form
Fαβ = ±
εαβγδFγδ, (4.24)
FαA = FAB = 0. (4.25)
Using Eqs.(2.8)-(2.10), the above equations can be rewritten in terms of (Aµ,Φ
a) where the constant
term in (2.10) can simply be dropped for the reason explained before.
FAB = 0 in Eq.(4.25) can be solved by AB = 0 (B = 1, 2) and then FαA = 0 demand that the
gauge fields Aα should not depend on v
A. Thereby Eq.(4.24) precisely reduces to the self-duality
equation (4.1) for D = 4 and n = 1. The symmetry reduction considered above is now understood
as the condition (4.25); in specific, the coordinates vA correspond to i(z1 − z̄1) and i(z2 − z̄2) for
the reduction (4.22). However there are many different choices taking a four-dimensional subspace
in Eq.(4.24) which are related by SO(6) rotations [23]. Unless vA ∈ (y1, y2), that is, Eq.(4.24)
becomes commutative Abelian equations in which there is no non-singular solution, Eqs.(4.24) and
(4.25) reduce to four-dimensional self-dual Einstein equations, as was shown in [39]. The above BPS
equations also clarify why the two-dimensional chiral equations in Section 4.1 reappear in Eq.(4.23).
4.3 D = 8 and D = 10
The analysis for the first-order system (2.19) becomes much more complicated in higher dimensions.
The unbroken supersymmetries in D = 8 have been analyzed in [23]. Because the integrable structure
of Einstein equations in higher dimensions is little known, it is difficult to precisely identify governing
geometrical structures emergent from the gauge theory (2.2) or (2.13) even for BPS states. Neverthe-
less some BPS configurations can be easily implemented as follows. As we did in Eqs.(4.24)-(4.25),
one can imbed the 4-dimensional self-dual system for n = 1 or n = 2 into eight or ten dimensions.
The simplest case is that the metric (3.4) becomes (locally) a product manifold M4×X where M4 is
a self-dual (hyper-Kähler) four-manifold. For example, we can consider an eight-dimensional config-
uration where (A1, A2,Φ
3,Φ4) depend only on (z1, z2, y3, y4) coordinates while (Φ1,Φ2, A3, A4) do
only on (y1, y2, z3, z4) in a B-field background with θ12 6= 0 and θ34 6= 0, only non-vanishing com-
ponents. There are many similar configurations. We will not exhaust them, instead we will consider
the simplest cases which already have some relevance to other works.
The simplest BPS state in D = 8 is the case with n = 2 in the action (2.13); see Eq.(55) in [23].
The equations are of the form
Fµν = ±
εµνλσFλσ, (4.26)
[Φa,Φb] = ±1
εabcd[Φ
c,Φd], (4.27)
a = 0. (4.28)
A solution of Eq.(4.28) is given by Aµ = Aµ(z) and Φ
a = Φa(y). Then Eq.(4.26) becomes com-
mutative Abelian equations which allow no non-singular solutions, while (4.27) reduces to Eq.(3.5)
describing 4-dimensional self-dual manifolds [15]. Thus the metric (3.4) in this case leads to a half-
BPS geometry IR4×M4. Since we don’t need instanton solutions in Eq.(4.26), we may freely replace
IR4 by 4-dimensional Minkowski space IR1,3 (see the footnote 1).
The above system was considered in [40] in the context of D3-D7 brane inflationary model. The
model consists of a D3-brane parallel to a D7-brane at some distance in the presence of Fab =
(B + F )ab on the worldvolume of the D7-brane, but transverse to the D3-brane. The F -field plays
the role of the Fayet-Illiopoulos term from the viewpoint of the D3-brane worldvolume field theory.
Because of spontaneously broken supersymmetry in de Sitter valley the D3-brane is attracted towards
the D7-brane and eventually it is dissolved into the D7-brane as a NC instanton. The system ends in
a supersymmetric Higgs phase with a smooth instanton moduli space. An interesting point in [40] is
that there is a relation between cosmological constant in spacetime and noncommutativity in internal
space. Our above result adds a geometrical picture that the internal space after tachyon condensation
is developed to a gravitational instanton, e.g., an ALE space or K3.
Another interesting point, not mentioned in [40], is that it effectively realizes the dynamical com-
pactification of extra dimensions suggested in [41]. Since the D3-brane is an instanton inside the
D7-brane, particles living in the D3-brane are trapped in the core of the instanton with size ∼ θ2
where the noncommutativity scale θ is believed to be roughly Planck scale. Since the instanton
(D3-brane) results in a spontaneous breaking of translation symmetry and supersymmetry partially,
Goldstone excitations corresponding to the broken bosonic and fermionic generators are zero-modes
trapped in the core of the instanton. “Quarks” and “leptons” might be identified with these fermionic
zero-modes [41].
We argued in the last of Section 3 that the 10-dimensional metric (3.4) for d = 4 and n = 3
reasonably corresponds to an emergent geometry arising from the 4-dimensional N = 4 supersym-
metric U(N) Yang-Mills theory. Especially it may be closely related to the bubbling geometry in AdS
space found by Lin, Lunin and Maldacena (LLM) [18]. One may notice that the LLM geometry is
a bubbling geometry deformed from the AdS5 × S5 background which can be regarded as a vacuum
manifold emerging from the self-dual RR five-form background, while the Ward’s geometry (3.4) is
defined in a 2-form B-field background and becomes (conformally) flat if all fluctuations are turned
off, say, (Aµ,Φ
a) → (0, ya/κ). But it turns out that the LLM geometry is a special case of the Ward’s
geometry (3.4).
To see this, recall that the AdS5 × S5 background is conformally flat, i.e.,
ds2 =
(ηµνdz
µdzν + dyadya) =
(ηµνdz
µdzν + dρ2) + L2dΩ25 (4.29)
where ρ2 =
a=1 y
aya and dΩ25 is the spherically symmetric metric on S
5. It is then easy to see that
the metric (4.29) is exactly the vacuum geometry of Eq.(3.4) when the volume form ω in Eq.(3.2) is
given by
dy1 ∧ · · · ∧ dy6
. (4.30)
Therefore it is obvious that the Ward’s metric (3.4) with the volume form (4.30) describes a bubbling
geometry which approaches to the AdS5 × S5 space at infinity where fluctuations are vanishing,
namely, (Aµ,Φ
a) → (0, ya/κ). Note that the flat spacetime IR1,9 is coming from the volume form
ω = dy1 ∧ · · · ∧ dy6, so Eq.(4.30) should correspond to some nontrivial soliton background from the
gauge theory point of view. We will discuss in Section 5 a possible origin of the volume form (4.30).
Now let us briefly summarize half-BPS geometries of type IIB string theory corresponding to
the chiral primaries of N = 4 super Yang-Mills theory [18]. These BPS states are giant graviton
branes which wrap an S3 in AdS5 or an S̃
3 in S5. Thus the geometry induced (or back-reacted)
by the giant gravitons preserves SO(4) × SO(4) × R isometry. It turns out that the solution is
completely determined by a single function which is specified with two types of boundary conditions
on a particular plane corresponding to either of two different spheres shrinking on the plane in a
smooth fashion. The LLM solutions are thus in one-to-one correspondence with various 2-colorings
of a 2-plane, usually referred to as ‘droplets’ and the geometry depends on the shape of the droplets.
The droplet describing gravity solutions turns out to be the same droplet in the phase space describing
free fermions for the half-BPS states.
The solutions can be analytically continued to those with SO(2, 2) × SO(4) × U(1) symmetry
[18], so the solutions have the AdS3 × S3 factor rather than S3 × S̃3. After an analytic continuation,
a underlying 4-dimensional geometry M4 attains a nice geometrical structure at asymptotic region,
where AdS3 × S3 → IR1,5 and M4 reduces to a hyper-Kähler geometry. But it loses the nice picture
in terms of fermion droplet since the solution is now specified by one type of boundary condition. It
is interesting to notice that the asymptotic bubbling geometry for the type IIB case is the Gibbons-
Hawking metric [28] and the real heaven metric [42] for the M theory case, which are all solutions of
NC electromagnetism [15, 16].
It is quite demanding to completely determine general half-BPS geometries emerging from the
gauge-Higgs system in the action (2.13). Hence we will look at only an asymptotic geometry (or a
local geometry) which is relatively easy to identify. For the purpose, we consider the n = 3 case on
4-dimensional Minkowski space IR1,3. It is simple to mimic the previous half-BPS configurations in
D = 6, 8 with trivial extra Higgs fields. Then the resulting metric (3.4) will be locally of the form
M4 × IR1,5 akin to the asymptotic bubbling geometry. However M4 can be a general hyper-Kähler
manifold. Therefore the solutions we get will be more general, whose explicit form will depend on
underlying Killing symmetries and boundary conditions. For example, the type IIB case is given by a
hyper-Kähler geometry with one translational Killing vector (Gibbons-Hawking) while the M theory
case is with one rotational Killing vector (real heaven) [43]. Therefore we may get in general bubbling
geometries in the M theory as well as the type IIB string theory.
5 Discussion
We showed reasonable evidences that the 10-dimensional metric (3.4) for d = 4 and n = 3 describes
the emergent geometry arising from the 4-dimensional N = 4 supersymmetric U(N) Yang-Mills
theory and thus might explain the AdS/CFT duality [12]. An important point in this context is that
the volume form (4.30) is required to describe the AdS5 × S5 background. What is the origin of this
nontrivial volume form ? In other words, how to realize the self-dual RR five-form background from
the gauge theory point of view ?
To get some hint about the question, first note that the AdS5 × S5 geometry emerges from multi-
instanton collective coordinates which dominates the path integral in a large N limit [44]. The factor
d4zdρρ−5 appears in bosonic collective coordinate integration (with zµ the instanton 4-positions)
which agrees with the volume form of the conformally invariant space AdS5, where instanton size
corresponds to the radial coordinate ρ in Eq.(4.29). Another point is that the AdS5 × S5 space cor-
responds to the LLM geometry for the simplest and most symmetric configuration which reduces to
the usual Gibbons-Hakwing metric (3.6) at asymptotic regions [18]. This result is consistent with
the picture in Section 4.2 that U(N) instantons at large N limit are indeed gravitational instantons.
It is then tempted to speculate that the AdS5 × S5 geometry would be emerging from a maximally
supersymmetric instanton solution of Eq.(2.19) in D = 10. It should be an interesting future work.
In addition, we would like to point out that an AdSp × Sq background arises from Eq.(3.4) in the
same way as Eq.(4.30) by choosing the volume form ω as follows
dy1 ∧ · · · ∧ dyq+1
(5.1)
with ρ2 =
a=1 y
aya and (Aµ,Φ
a) = (0, ya/κ). A particularly interesting case is d = 2 and n = 2
for which the volume form (5.1) leads to the AdS3 × S3 background and the action (2.13) describes
matrix strings [45, 46]. We believe that the metric (3.4) with ω = dy1 ∧ · · · ∧ dy4/ρ2 describes a
bubbling geometry emerging from the matrix strings.
One might already notice a subtle difference between the matrix action (2.13) and the Ward’s
metric (3.4). According to our construction in Section 2, the number of the Higgs fields Φa is even
while the Ward construction has no such restriction. But it was shown in [15, 16] that the Gibbons-
Hawking metric (3.6) for the d = 3 and m = 1 case also arises from the d = 0 and m = 4. It implies
that we can replace some transverse scalars by gauge fields and vice versa. Recalling that the fields
in the action (2.13) are all N × N matrices, of course, N → ∞, it is precisely ‘matrix T -duality’
exchanging transverse scalars and gauge fields associated with a compact direction in p-brane and
(p+ 1)-brane worldvolume theories through (see Eq.(154) in [46])
Φa ↔ iDµ = i(∂µ − iAµ). (5.2)
With this identification, the d-dimensional U(N) gauge theory (2.13) can be obtained by applying the
d-fold ‘matrix T -duality’ (5.2) to the 0-dimensional IKKT matrix model [11, 20]
S = −2πκ
gMPgNQ[Φ
M ,ΦN ][ΦP ,ΦQ]
. (5.3)
However, the T -duality (5.2) gives rise to qualitatively radical changes in worldvolume theory.
First it changes the dimensionality of the theory and thus it affects its renormalizability (see Sec. VI
in [46] and references therein for this issue in Matrix theory). For example, the action (2.13) for
d > 4 is not renormalizable since the coupling constant g2YM ∼ gsκ
2 ∼ gsm4−ds has negative mass
dimension in this case. Second it also changes a behavior of the emergent metric (3.4). But these
changes are rather consistent with the fact that under the T -duality (5.2) a Dp-brane is transformed
into a D(p+ 1)-brane and vice versa.
Our construction in Section 2 raises a bizarre question about the renormalization property of NC
field theory. If we look at the action (2.2), the theory superficially seems to be non-renormalizable for
D > 4 since the coupling constant (2.5) has negative mass dimension. But this non-renormalizability
appears as a fake if we use the matrix representation (2.18) together with the redefinition of variables
in Eq.(2.4). The resulting coupling constant, denoted as gd, in the final action (2.13) depends only on
the dimension of commutative spacetime rather than the entire spacetime. Since the resulting U(N)
theory is in the limit N → ∞, while the ’t Hooft coupling λ ≡ g2dN is kept fixed, planar diagrams
dominate in this limit [9]. Since the dependence of NC coordinates in the action (2.2) has been
encoded into the matrix degrees of freedom, one may suspect that the divergence of the original theory
might appear as a divergence of perturbation series as a whole in the action (2.13). The convergence
aspect of the planar perturbation theory concerns Np(n), the number of planar diagrams in nth order
in λ. It was shown in [47] that Np(n) behaves asymptotically as
Np(n)
n→∞∼ cn, c = constant. (5.4)
Therefore the planar theory (unlike the full theory) for d ≤ 4 has a formally convergent perturbation
series, provided the ultraviolet and infrared divergences of individual diagrams are cut off [47]. It will
be interesting to carefully examine the renormalization property of NC field theories along this line.
We showed in Section 3 that the Ward metric (3.4) is emerging from commutative, i.e., O(θ),
limit. Since the vector fields in Eq.(3.11) are in general higher order differential operators acting on
Aθ, we thus expect that they actually define a ‘generalized gravity’ beyond Einstein gravity, e.g., the
NC gravity [29] or the NC unimodular gravity [48].6 It was shown in [16] that the leading derivative
6The latter seems to be quite relevant to our emergent gravity since the vector fields VM in Eq.(3.11) always belong to
the volume preserving diffeomorphisms, which is a generic property of vector fields defined in NC spacetime. It should
be interesting to more clarify the relation between the NC unimodular gravity [48] and the emergent gravity.
corrections in NC gauge theory start with four derivatives, which was conjectured to give rise to higher
order gravity. As was explicitly checked for the self-dual case, Einstein gravity maybe emerges from
NC gauge fields in commutative limit, which then implies that the leading derivative corrections give
rise to higher order terms with four more derivatives compared to the Einstein gravity. This means
that the higher order gravity starts from the second order corrections in θ with higher derivatives, that
is, no first order correction in θ to the Einstein gravity. Interestingly this result is consistent with those
in [29] and also in [49] calculated from the context of NC gravity.
It was shown in Section 4.1 that the self-duality system for the D = 4 and n = 1 case is mapped
to the two-dimensional U(∞) chiral model (4.9) which is remarkably equivalent to self-dual Einstein
gravity [33, 17, 34]. But this case should not be much different from the D = 4 and n = 2 case
in [15, 16] since they equally describe the self-dual Einstein gravity. Indeed we can make them
bear a close resemblance each other. For the purpose, let us consider a four-dimensional NC space
IR2NC × IR2NC . We can choose the matrix representation (2.18) only for the second factor, i.e.,
φ(y1, y2, y3, y4) =
Ωij (y
1, y2)|i〉〈j|. (5.5)
As a result, the action (2.13) now becomes two-dimensional U(N) gauge theory on IR2NC . The self-
dual equations in Eq.(4.1) in this case are given by the NC Hitchin equations, now defined on IR2NC
instead of IR2C . The NC Hitchin equations have been considered by K. Lee in [37] with very parallel
results with the commutative case (4.1). It is interesting that there exist two different realizations for
self-dual Einstein gravity, whose relationship should be more closely understood.
Finally it will be interesting to consider a compact NC space instead of IR2nNC , for instance, a
NC 2n-torus T2nNC . Since the module over a NC torus is still infinite dimensional [8], the matrix
representation (2.18) is also infinite dimensional. Thus we expect that our construction in Section 2
and 3 can be applied even to the NC torus without many essential changes.
Acknowledgments
After posting this paper to the arXiv, we were informed of related works on YM-Higgs BPS configu-
rations on NC spaces [50] and on the relation between a large N gauge theory, a Moyal deformation
and a self-dual gravity [51] by O. Lechtenfeld and C. Castro, respectively. We thank them for the
references. This work was supported by the Alexander von Humboldt Foundation.
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	Introduction
	A Large N Gauge Theory From NC U(1) Gauge Theory
	Emergent Geometry From NC Gauge Theory
	Self-dual Einstein Gravity From Large N Gauge Theory
	D=4 and n=1
	D=6 and n=1
	D=8 and D=10
	Discussion

  We map noncommutative (NC) U(1) gauge theory on R^d_C X R^{2n}_{NC} to U(N ->
\infty) Yang-Mills theory on R^d_C, where R^d_C is a d-dimensional commutative
spacetime while R^{2n}_{NC} is a 2n-dimensional NC space. The resulting U(N)
Yang-Mills theory on R^d_C is equivalent to that obtained by the dimensional
reduction of (d+2n)-dimensional U(N) Yang-Mills theory onto R^d_C. We show that
the gauge-Higgs system (A_\mu,\Phi^a) in the U(N -> \infty) Yang-Mills theory
on R^d_C leads to an emergent geometry in the (d+2n)-dimensional spacetime
whose metric was determined by Ward a long time ago. In particular, the
10-dimensional gravity for d=4 and n=3 corresponds to the emergent geometry
arising from the 4-dimensional N=4 vector multiplet in the AdS/CFT duality. We
further elucidate the emergent gravity by showing that the gauge-Higgs system
(A_\mu,\Phi^a) in half-BPS configurations describes self-dual Einstein gravity.

Introduction
A noncommutative (NC) spacetimeM is obtained by introducing a symplectic structure B = 1
Babdy
dyb and then by quantizing the spacetime with its Poisson structure θab ≡ (B−1)ab, treating it as a
quantum phase space. That is, for f, g ∈ C∞(M),
{f, g} = θab
⇒ −i[f̂ , ĝ]. (1.1)
According to the Weyl-Moyal map [1, 2], the NC algebra of operators is equivalent to the deformed
algebra of functions defined by the Moyal ⋆-product, i.e.,
f̂ · ĝ ∼= (f ⋆ g)(y) = exp
θab∂ya∂
f(y)g(z)
. (1.2)
Through the quantization rules (1.1) and (1.2), one can define NC IR2n by the following commutation
relation
[ya, yb]⋆ = iθ
ab. (1.3)
It is well-known [2, 3] that a NC field theory can be identified basically with a matrix model or a
large N field theory. This claim is based on the following fact. Let us consider a NC IR2 for simplicity,
[x, y] = iθ, (1.4)
although the same argument equally holds for a NC IR2n as it will be shown later. After scaling the
coordinates x →
θx, y →
θy, the NC plane (1.4) becomes the Heisenberg algebra of harmonic
oscillator
[a, a†] = 1. (1.5)
It is a well-known fact from quantum mechanics that the representation space of NC IR2 is given by
an infinite-dimensional, separable Hilbert space H = {|n〉, n = 0, 1, · · · } which is orthonormal, i.e.,
〈n|m〉 = δnm and complete, i.e.,
n=0 |n〉〈n| = 1. Therefore a scalar field φ̂ ∈ Aθ on the NC plane
(1.4) can be expanded in terms of the complete operator basis
Aθ = {|m〉〈n|, n,m = 0, 1, · · · }, (1.6)
that is,
φ̂(x, y) =
Mmn|m〉〈n|. (1.7)
One can regard Mmn in (1.7) as components of an N × N matrix M in the N → ∞ limit. More
generally one may replace NC IR2 by a Riemann surface Σg of genus g which can be quantized via
deformation quantization [4]. For a compact Riemann surface Σg with finite area A(Σg), the matrix
representation can be finite-dimensional, e.g., for a fuzzy sphere [5]. In this case, A(Σg) ∼ θN but
we simply take the limit N → ∞. We then arrive at the well-known relation:
Scalar field on NC IR2 (or Σg) ⇐⇒ N ×N matrix at N → ∞. (1.8)
If φ̂ is a real scalar field, then M should be a Hermitean matrix. We will see that the above relation
(1.8) has far-reaching applications to string theory.
The matrix representation (1.7) clarifies why NC U(1) gauge theory is a large N gauge theory.
An important point is that the NC gauge symmetry acts as a unitary transformation on H for a field
φ̂ ∈ Aθ in the adjoint representation of U(1) gauge group
φ̂ → Uφ̂ U †. (1.9)
This NC gauge symmetry Ucpt(H) is so large that Ucpt(H) ⊃ U(N) (N → ∞) [6, 7], which is
rather obvious in the matrix basis (1.6). Therefore the NC gauge theory is essentially a large N gauge
theory. It becomes more precise on a NC torus through the Morita equivalence where NC U(1) gauge
theory with rational θ = M/N is equivalent to an ordinary U(N) gauge theory [8]. For this reason,
it is not so surprising that NC electromagnetism shares essential properties appearing in a large N
gauge theory such as SU(N → ∞) Yang-Mills theory or matrix models.
It is well-known [9] that 1/N expansion of any large N gauge theory using the double line for-
malism reveals a picture of a topological expansion in terms of surfaces of different genus, which
can be interpreted in terms of closed string variables as the genus expansion of string amplitudes. It
has been underlain the idea that large N gauge theories have a dual description in terms of gravita-
tional theories in higher dimensions. For example, BFSS matrix model [10], IKKT matrix model [11]
and AdS/CFT duality [12]. From the perspective (1.8), the 1/N expansion corresponds to the NC
deformation in terms of θ/A(Σg).
All these arguments imply that there exists a solid map between a NC gauge theory and a large N
gauge theory. In this work we will find a sound realization of this idea. It turns out that the emergent
gravity recently found in [13, 14, 15, 16] can be elegantly understood in this framework. Therefore
the correspondence between NC field theory and gravity [3] is certainly akin to the gauge/gravity
duality in large N limit [10, 11, 12].
This paper is organized as follows. In Section 2 we map NC U(1) gauge theory on IRdC × IR2nNC to
U(N → ∞) Yang-Mills theory on IRdC , where IRdC is a d-dimensional commutative spacetime while
IR2nNC is a 2n-dimensional NC space. The resulting U(N) Yang-Mills theory on IR
C is equivalent to
that obtained by the dimensional reduction of (d + 2n)-dimensional U(N) Yang-Mills theory onto
IRdC . In Section 3, we show that the gauge-Higgs system (Aµ,Φ
a) in the U(N → ∞) Yang-Mills
theory on IRdC leads to an emergent geometry in the (d + 2n)-dimensional spacetime whose metric
was determined by Ward [17] a long time ago. In particular, the 10-dimensional gravity for d = 4 and
n = 3 corresponds to the emergent geometry arising from the 4-dimensional N = 4 vector multiplet
in the AdS/CFT duality [12]. We further elucidate the emergent gravity in Section 4 by showing that
the gauge-Higgs system (Aµ,Φ
a) in half-BPS configurations describes self-dual Einstein gravity. A
notable point is that the emergent geometry arising from the gauge-Higgs system (Aµ,Φ
a) is closely
related to the bubbling geometry in AdS space found in [18]. Finally, in Section 5, we discuss several
interesting issues that naturally arise from our construction.
2 A Large N Gauge Theory From NC U(1) Gauge Theory
We will consider a NC U(1) gauge theory on IRD = IRdC × IR2nNC , where D-dimensional coordinates
XM (M = 1, · · · , D) are decomposed into d-dimensional commutative ones, denoted as zµ (µ =
1, · · · , d) and 2n-dimensional NC ones, denoted as ya (a = 1, · · · , 2n), satisfying the relation (1.3).
We assume the metric on IRD = IRdC × IR2nNC as the following form 1
ds2 = GMNdXMdXN
= gµνdz
µdzν +Gabdy
adyb. (2.1)
The action for D-dimensional NC U(1) gauge theory is given by
4g2YM
detGGMPGNQ(FMN + ΦMN ) ⋆ (FPQ + ΦPQ), (2.2)
where the NC field strength FMN is defined by
FMN = ∂MAN − ∂NAM − i[AM , AN ]⋆. (2.3)
The constant two-form Φ will be taken either 0 or −B = −1
Babdy
a ∧ dyb with rank(B) = 2n.
Here we will use the background independent prescription [8, 19] where the open string metric
Gab, the noncommutativity θ
ab and the open string coupling Gs are determined by
θab =
, Gab = −κ2
, Gs = gs
det′(κBg−1), (2.4)
with κ ≡ 2πα′. The closed string metric gab in Eq.(2.4) is independent of gµν in Eq.(2.1) and det′
denotes a determinant taken along NC directions only in IR2nNC . In terms of these parameters, the
couplings are related by
, (2.5)
det′G
gs|Pfθ|
. (2.6)
An important fact is that translations in NC directions are basically gauge transformations, i.e.,
eik·y ⋆ f(z, y) ⋆ e−ik·y = f(z, y+ θ · k) for any f(z, y) ∈ C∞(M). This means that translations along
NC directions act as inner derivations of the NC algebra Aθ:
[ya, f ]⋆ = iθ
ab∂bf. (2.7)
1 Here we can take the d-dimensional spacetime metric gµν with either Lorentzian or Euclidean signature since the
signature is inconsequential in our most discussions. But we implicitly assume the Euclidean signature for some other
discussions.
Using this relation, each component of FMN can be written as the following forms
Fµν = i[Dµ, Dµ]⋆, (2.8)
Fµa = θ
[Dµ, x
b]⋆ = −Faµ, (2.9)
Fab = −iθ−1ac θ−1bd
[xc, xd]⋆ − iθcd
, (2.10)
where the covariant derivative Dµ and the covariant coordinate x
a are, respectively, defined by
Dµ ≡ ∂µ − iAµ, (2.11)
xa ≡ ya + θabAb. (2.12)
Collecting all these facts, one gets the following expression for the action (2.2) with Φ = −B 2
(2πκ)
detgµνTrH
gµλgνσFµν ⋆ Fλσ +
gµνgabDµΦ
a ⋆ DνΦ
gacgbd[Φ
a,Φb]⋆ ⋆ [Φ
c,Φd]⋆
, (2.13)
where we defined adjoint scalar fields Φa ≡ xa/κ of mass dimension and
TrH ≡
(2π)n|Pfθ| . (2.14)
Note that the number of the adjoint scalar fields is equal to the rank of θab. The resulting action (2.13)
is not new but rather well-known in NC field theory, e.g., see [19, 20].
The NC algebra (1.3) is equivalent to the Heisenberg algebra of an n-dimensional harmonic os-
cillator in a frame where θab has a canonical form:
[ai, a
j ] = δij , (i, j = 1, · · · , n). (2.15)
The NC space (1.3) is therefore represented by the infinite-dimensional Hilbert space H = {|~m〉 ≡
|m1, · · · , mn〉;mi = 0, 1, · · · , N → ∞ for i = 1, · · · , n} whose set of eigenvalues forms an n-
dimensional positive integer lattice. A set of operators in H
Aθ = {|~m〉〈~n|;mi, ni = 0, 1, · · · , N → ∞ for i = 1, · · · , n} (2.16)
can be identified with the generators of a complete operator basis and so any NC field φ(z, y) ∈ Aθ
can be expanded in the basis (2.16) as follows,
φ(z, y) =
~m,~n
Ω~m,~n (z)|~m〉〈~n|. (2.17)
2If Φ = 0 in Eq.(2.2), the only change in Eq.(2.13) is [Φa,Φb] → [Φa,Φb]− i
Now we use the ‘Cantor diagonal method’ to put the n-dimensional positive integer lattice in H
into a one-to-one correspondence with the infinite set of natural numbers (i.e., 1-dimensional positive
integer lattice): |~m〉 ↔ |i〉, i = 1, · · · , N → ∞. In this one-dimensional basis, Eq.(2.17) is relabeled
as the following form
φ(z, y) =
Ωij (z)|i〉〈j|. (2.18)
Following the motivation discussed in the Introduction, we regard Ωij(z) in (2.18) as components of
an N ×N matrix Ω in the N → ∞ limit, which also depend on zµ, the coordinates of IRdC . If the field
φ(z, y) is real which is the case for the gauge-Higgs system (Aµ,Φ
a) in the action (2.13), the matrix
Ω should be Hermitean, but not necessarily traceless. So the N × N matrix Ω(z) can be regarded as
a field in U(N → ∞) gauge theory on d-dimensional commutative space IRdC , where TrH in (2.14)
is identified with the matrix trace over the basis (2.18). All the dependence on NC coordinates is now
encoded into N ×N matrices and the noncommutativity in terms of star product is transferred to the
matrix product.
Adopting the matrix representation (2.18), the D-dimensional NC U(1) gauge theory (2.2) is
mapped to the U(N → ∞) Yang-Mills theory on d-dimensional commutative space IRdC . One can
see that the resulting U(N) Yang-Mills theory on IRdC in Eq.(2.13) is equivalent to that obtained by
the dimensional reduction of (d + 2n)-dimensional U(N) Yang-Mills theory onto IRdC . It might be
emphasized that the map between the D-dimensional NC U(1) gauge theory and the d-dimensional
U(N → ∞) Yang-Mills theory is “exact” and thus the two theories should describe a completely
equivalent physics. For example, we can recover the D-dimensional NC U(1) gauge theory on IRdC ×
IR2nNC from the d-dimensional U(N → ∞) Yang-Mills theory on IRdC by recalling that the number of
adjoint Higgs fields in the U(N) Yang-Mills theory is equal to the dimension of the extra NC space
IR2nNC and by applying the dictionary in Eqs.(2.8)-(2.10).
One can introduce linear algebraic conditions of D-dimensional field strengths FMN as a higher
dimensional analogue of 4-dimensional self-duality equations such that the Yang-Mills equations in
the action (2.2) follow automatically. These are of the following type [21, 22]
TMNPQFPQ = λFMN (2.19)
with a constant 4-form tensor TMNPQ. The relation (2.19) clearly implies via the Bianchi identity
D[MFPQ] = 0 that the Yang-Mills equations are satisfied provided λ is nonzero. For D > 4, the 4-
form tensor TMNPQ cannot be invariant under SO(D) transformations and the equation (2.19) breaks
the rotational symmetry to a subgroup H ⊂ SO(D). Thus the resulting first order equations can be
classified by the unbroken symmetry H under which TMNPQ remain invariant [21, 22]. It was also
shown [23] that the first order linear equations above are closely related to supersymmetric states, i.e.,
BPS states in higher dimensional Yang-Mills theories.
The equivalence between D- and d-dimensional gauge theories can be effectively used to clas-
sify classical solutions in the d-dimensional U(N) Yang-Mills theory (2.13). The group theoretical
classification [21], integrability condition [22] and BPS states [23] for the D-dimensional first-order
equations (2.19) can be directly translated into the properties of the gauge-Higgs system (Aµ,Φ
a) in
the d-dimensional U(N) gauge theory (2.13). These classifications will also be useful to classify the
geometries emerging from the gauge-Higgs system (Aµ,Φ
a) in the U(N → ∞) Yang-Mills theory
(2.13), which will be discussed in the next section. Unfortunately, the D = 10 case is missing in
[21, 22, 23] which is the most interesting case (d = 4 and n = 3) related to the AdS/CFT duality.
3 Emergent Geometry From NC Gauge Theory
Let us first recapitulate the result in [17]. It turns out that the Ward’s construction perfectly fits with
the emergent geometry arising from the gauge-Higgs system (Aµ,Φ
a) in the U(N → ∞) Yang-Mills
theory (2.13). Suppose that we have gauge fields on IRdC taking values in the Lie algebra of volume-
preserving vector fields on an m-dimensional manifold M [24, 25]. In other words, the gauge group
G = SDiff(M). The gauge covariant derivative is given by Eq.(2.11), but the Aµ(z) are now vector
fields on M , also depending on zµ ∈ IRdC . The other ingredient in [17] consists of m Higgs fields
Φa(z) ∈ sdiff(M), the Lie algebra of SDiff(M), for a = 1, · · · , m. The idea [24, 25] is to specify
f−1(D1, · · · , Dd,Φ1, · · · ,Φm) (3.1)
forms an orthonormal frame and hence defines a metric on IRdC ×M with a volume form ν = ddz∧ω.
Here f is a scalar, a conformal factor, defined by
f 2 = ω(Φ1, · · · ,Φm). (3.2)
The result in [24, 25] immediately implies that the gauge-Higgs system (Aµ,Φ
a) leads to a metric
on the (d + m)-dimensional space IRdC × M . A local coordinate expression for this metric is easily
obtained from Eq.(3.1). Let ya be local coordinates on M . So Aµ(z) and Φa(z) have the form
Aµ(z) = A
µ(z, y)
, Φa(z) = Φ
a(z, y)
, (3.3)
where the y-dependence, originally hidden in the Lie algebra of G = SDiff(M), now explicitly
appears in the coefficients Aaµ and Φ
a. Let V
b denote the inverse of the m×m matrix Φba, and let Aa
denote the 1-form Aaµdz
µ. Then the metric is [17]
ds2 = f 2δµνdz
µdzν + f 2δabV
d (dy
c −Ac)(dyd −Ad). (3.4)
It will be shown later that the choice of the volume form ω for the conformal factor (3.2) corresponds
to that of a particular conformally flat background although we mostly assume a flat volume form,
i.e., ω ∼ dy1 ∧ · · · ∧ dy2n, unless explicitly specified.
The gauge and Higgs fields in Eq.(3.3) are not arbitrary but must be subject to the Yang-Mills
equations, for example, derived from the action (2.13), which are, in most cases, not completely
integrable. Hence to completely determine the geometric structure emerging from the gauge-Higgs
system (Aµ,Φ
a) is as much difficult as solving the Einstein equations in general. But the self-dual
Yang-Mills equations in four dimensions or Eq.(2.19) in general are, in some sense, “completely
solvable”. Thus the metric (3.4) for these cases might be completely determined. Let us discuss two
notable examples. See [17] for more examples describing 4-dimensional self-dual Einstein gravity.
• Case d = 0, m = 4: This case was dealt with in detail in [25, 26, 27]. It was proved that the
self-dual Einstein equations are equivalent to the self-duality equations
[Φa,Φb] = ±
εabcd[Φc,Φd] (3.5)
on the four Higgs fields Φa. Furthermore reinterpreting n of the Φa’s as Dµ leads to the case d =
n, m = 4− n. In Section 5, we will discuss the physical meaning about the interpretation Φa 7→ Dµ.
• Case d = 3, m = 1: Here M is one-dimensional, so the Lie algebra of vector fields on M is the
Virasoro algebra. Thus Aµ and Φ are now real-valued vector fields on M which must be independent
of y to preserve the volume form ν = d3z ∧ dy [27]. The metric (3.4) reduces to
ds2 = Φd~z · d~z + Φ−1(dy −Aµdzµ)2 (3.6)
and has a Killing vector ∂/∂y. In this case, the self-duality equations (3.5) reduce to the Abelian
Bogomol’nyi equations, ∇× ~A = ∇Φ, and the metric (3.6) describes a gravitational instanton [28].
Recently we showed in [15, 16] for the d = 0 and m = 4 case that self-dual electromagnetism in
NC spacetime is equivalent to self-dual Einstein gravity and the metric is precisely given by Eq.(3.4).
A key observation [16] was that the self-dual system (3.5) defined by vector fields on M can be
derived from the action (2.2) or (2.13) for slowly varying fields, where all ⋆-commutators between
NC fields are approximated by the Poisson bracket (1.1). An important point in NC geometry is
that the adjoint action of (covariant) coordinates with respect to star product can be identified with
(generalized) vector fields on some (curved) manifold [15, 16], as the trivial case was already used
in Eq.(2.7). In the end, a D-dimensional manifold described by the metric (3.4) corresponds to an
emergent geometry arising from the gauge-Higgs system in Eq.(3.3). Now we will show in a general
context how the nontrivial geometry (3.4) emerges from the gauge-Higgs system (Aµ,Φ
a) in the
action (2.13).
Let us collectively denote the covariant derivatives Dµ in (2.11) and the Higgs fields Da ≡
−iκBabΦb = −i(Babyb +Aa) in (2.12) as DA(z, y). Therefore DA(z, y) transform covariantly under
NC U(1) gauge transformations
DA(z, y) → g(z, y) ⋆ DA(z, y) ⋆ g−1(z, y). (3.7)
Define the adjoint action of DA(z, y) with respect to star product acting on any NC field f(z, y) ∈ Aθ:
adDA[f ] ≡ [DA, f ]⋆. (3.8)
Then it is easy to see [16] that the above adjoint action satisfies the Leibniz rule and the Jocobi
identity, i.e.,
[DA, f ⋆ g]⋆ = [DA, f ]⋆ ⋆ g + f ⋆ [DA, g]⋆, (3.9)
[DA, [DB, f ]⋆]⋆ − [DB, [DA, f ]⋆]⋆ = [[DA, DB]⋆, f ]⋆. (3.10)
These properties imply that adDA can be identified with ‘generalized’ vector fields or Lie deriva-
tives acting on the algebra Aθ, which can be viewed as a gauge covariant generalization of the inner
derivation (2.7). Note that the generalized vector field in Eq.(3.8) is a kind of general higher or-
der differential operators in [29]. Indeed it turns out that they constitute a generalization of volume
preserving diffeomorphisms to ⋆-differential operators acting on Aθ (see Eqs.(4.1) and (4.2) in [7]).
In particular, the generalized vector fields in Eq.(3.8) reduce to usual vector fields in the commu-
tative, i.e. O(θ), limit:
adDA [f ] = iθ
ab∂DA
+ · · · = i{DA, f}+O(θ3)
≡ V aA(z, y)∂af(z, y) +O(θ3) (3.11)
where we defined [∂µ, f ]⋆ = ∂µf . Note that the vector fields VA(z, y) = V
A(z, y)∂a are exactly of
the same form as Eq.(3.3) and belong to the Lie algebra of volume preserving diffeomorphisms, as
precisely required in the Ward construction (3.1), since they are all divergence free, i.e., ∂aV
A = 0.
Thus the vector fields f−1VA(z, y) for A = 1, · · · , D can be identified with the orthonormal frame
(3.1) defining the metric (3.4). It should be emphasized that the emergent gravity (3.4) arises from a
general, not necessarily self-dual, gauge-Higgs system (Aµ,Φ
a) in the action (2.13).
Note that
[DA, DB]⋆ = −i(FAB − BAB) (3.12)
where the NC field strength FAB is given by Eq.(2.3). Then the Jacobi identity (3.10) leads to the
following identity for a constant BAB
ad[DA,DB]⋆ = −i adFAB = [adDA, adDB ]⋆. (3.13)
The inner derivation (3.11) in commutative limit is reduced to the well-known map C∞(M) →
TM : f 7→ Xf between the Poisson algebra (C∞(M), {·, ·}) and vector fields in TM defined by
Xf(g) = {g, f} for any smooth function g ∈ C∞(M). The Jacobi identity for the Poisson algebra
(C∞(M), {·, ·}) then leads to the Lie algebra homomorphism
X{f,g} = −[Xf , Xg] (3.14)
where the right-hand side is defined by the Lie bracket between Hamiltonian vector fields. One can
check by identifying f = DA and g = DB that the Lie algebra homomorphism (3.14) correspond
to the commutative limit of the Jacobi identity (3.10). That is, one can deduce from Eq.(3.14) the
following identity
XFAB = −[VA, VB] (3.15)
using the relation {DA, DB} = −FAB +BAB and XDA = iVA.
Using the homomorphism (3.15), one can translate the generalized self-duality equation (2.19)
into the structure equation between vector fields
TABCDFCD = λFAB ⇔
TABCD[VC , VD] = λ[VA, VB]. (3.16)
Therefore a D-dimensional NC gauge field configuration satisfying the first-order system defined by
the left-hand side of Eq.(3.16) is isomorphic to a D-dimensional emergent geometry defined by the
right-hand side of Eq.(3.16) whose metric is given by Eq.(3.4). For example, in four dimensions
where TABCD = εABCD and λ = ±1, the right-hand side of Eq.(3.16) is precisely equal to Eq.(3.5)
describing gravitational instantons [24, 25, 26, 27]. This proves, as first shown in [15, 16], that self-
dual NC electromagnetism is equivalent to self-dual Einstein gravity. Note that the Einstein gravity
described by the metric (3.4) arises from the commutative, i.e., O(θ) limit. Therefore it is natural to
expect that the higher order differential operators in Eq.(3.11), e.g. O(θ3), give rise to higher order
gravity [16]. We will further discuss the derivative correction in Section 5.
The 10-dimensional metric (3.4) for d = 4 and n = 3 (m = 6) is particularly interesting since it
corresponds to an emergent geometry arising from the 4-dimensional N = 4 vector multiplet in the
AdS/CFT duality. Note that the gravity in the AdS/CFT duality is an emergent phenomenon arising
from particle interactions in a gravityless, lower-dimensional spacetime. As a famous example, the
type IIB supergravity (or more generally the type IIB superstring theory) on AdS5 × S5 is emergent
from the 4-dimensional N = 4 supersymmetric U(N) Yang-Mills theory [12].3 In our construction,
N × N matrices are mapped to vector fields on some manifold M , so the vector fields in Eq.(3.3)
correspond to master fields of large N matrices [30], in other words, (Aµ,Φ
a) ∼ N2. According to
the AdS/CFT duality, we thus expect that the metric (3.4) describes a deformed geometry induced
by excitations of the gauge and Higgs fields in the action (2.13). For example, we may look for
1/2 BPS geometries in type IIB supergravity that arise from chiral primaries of N = 4 super Yang-
Mills theory [18]. Recently this kind of BPS geometries, the so-called bubbling geometry in AdS
space, with a particular isometry was completely determined in [18], where the AdS5 × S5 geometry
emerges from the simplest and most symmetric configuration. In next section we will illustrate such
kind of bubbling geometry described by the metric (3.4) by considering self-dual configurations in
the gauge-Higgs system.
3The overall U(1) = U(N)/SU(N) factor actually corresponds to the overall position of D3-branes and may be
ignored when considering dynamics on the branes, thereby leaving only an SU(N) gauge symmetry.
4 Self-dual Einstein Gravity From Large N Gauge Theory
In the previous section we showed that the Ward’s metric (3.4) naturally emerges from the D-dimensional
NC U(1) gauge fields AM on IR
C × IR2nNC or equivalently the gauge-Higgs system (Aµ,Φa) in d-
dimensional U(N) gauge theory on IRdC . So, if an explicit solution for AM or (Aµ,Φ
a) is known, the
corresponding metric (3.4) is, in principle, exactly determined. However, it is extremely difficult to
get a general solution by solving the equations of motion for the action (2.2) or (2.13). Instead we
may try to solve a more simpler system such as the first-order equations (2.19), which are morally
believed to be ‘exactly solvable’ in most cases. In this section we will further elucidate the emer-
gent gravity arising from gauge fields by showing that the gauge-Higgs system (Aµ,Φ
a) in half-BPS
configurations describes self-dual Einstein gravity. Since the case for D = 4 and n = 2 has been
extensively discussed in [14, 15, 16], we will consider the other cases for D ≥ 4. For simplicity,
the metrics in the action (2.13) are supposed to be the form already used in Eq.(3.4); gµν = δµν and
gab = δab.
Note that the action (2.2) or (2.13) contains a background B, due to a uniform condensation of
gauge fields in a vacuum. But we will require a rapid fall-off of fluctuating fields around the back-
ground at infinity in IRD as usual.4 Our boundary condition is FMN → 0 at infinity. Eq.(2.10) then
requires that [xa, xb]⋆ → iθab at |y| → ∞. Thus the coordinates ya in (2.12) are vacuum expectation
values of xa characterizing the uniform condensation of gauge fields [16]. This condensation of the
B-fields endows the ⋆-algebra Aθ with a remarkable property that translations act as an inner auto-
morphism of the algebra Aθ as shown in Eq.(2.7). But the gauge symmetry on NC spacetime requires
the covariant coordinates xa in Eq.(2.12) instead of ya [31]. The inner derivation adDa in Eq.(3.8) is
then a ‘dual element’ related to the coordinate xa. This is also true for the covariant derivatives Dµ
in (2.11) since they are related to Da = −iBabxb by the ‘matrix T -duality’; Da 7→ Dµ, as will be
explained in Section 5.
It is very instructive to take an analogy with quantum mechanics. Quantum mechanical time
evolution in Heisenberg picture is defined as an inner automorphism of the Weyl algebra obtained
from a quantum phase space
f(t) = eiHtf(0)e−iHt
and its evolution equation is of the form (3.8)
df(t)
= i[H, f(t)].
Here we liberally interpret DA(z, y) in Eq.(3.8) as ‘multi-Hamiltonians’ determining the spacetime
evolution in IRD. Then it is quite natural to interpret Eq.(3.8) as a spacetime evolution equation
determined by the “covariant Hamiltonians” DA(z, y).
4In the matrix representation (2.18), this means that matrix components Ωij(z) for the fluctuations are rapidly vanish-
ing for i, j = N → ∞ as well as for |z| → ∞, since roughly N ∼ ~y · ~y.
Let us be more precise about the meaning of the spacetime evolution. If the Hamiltonian is slightly
deformed, H → H + δH , the time evolution of a system is correspondingly changed. Likewise, the
fluctuation of gauge fields AM or (Aµ,Φ
a) around the background specified by ya’s changes DA(z, y),
which in turn induces a deformation of the background spacetime according to Eq.(3.11). This is
precisely the picture about the emergent geometry in [15, 16] and also a dependable interpretation of
the Ward’s geometry (3.4). A consistent picture related to the AdS/CFT duality was also observed in
the last of Section 3.
For the above reason, all equations in the following will be understood as inner derivations acting
on Aθ like as (3.8). The adjoint action defined in this way naturally removes a contribution from
the background in the action (2.2) or (2.13) [15]. For example, the first equation in (4.1) can be
consistent only in this way since the left hand side goes to zero at infinity but the right hand side
becomes ∼ θ/κ2. It might be remarked that this is the way to define the equations of motion in
the background independent formulation [8, 19] and thus it should be equivalent to the usual NC
prescription with ΦMN = 0.
4.1 D = 4 and n = 1
NC instanton solutions in this case were constructed in [32]. As was proved in Eq.(3.16), NC U(1)
instantons are in general equivalent to gravitational instantons. We thus expect that the NC self-
duality equations for D = 4 and n = 1 are mapped to self-dual Einstein equations. We will show
that the gauge-Higgs system (Aµ,Φ
a) in this case is mapped to two-dimensional U(∞) chiral model,
whose equations of motion are equivalent to the Plebański form of the self-dual Einstein equations
[33, 17, 34].
We showed in Section 2 that 4-dimensional NC U(1) gauge theory on IR2C × IR2NC is mapped
to 2-dimensional U(N → ∞) gauge theory with the action (2.13). The 4-dimensional self-duality
equations now become the U(N → ∞) Hitchin equations on IR2C :
Fµν = ±
εµν [Φ,Φ
†], DµΦ = ±iεµνDνΦ, (4.1)
where Φ = Φ1 + iΦ2. Note that the above equations also arise as zero-energy solutions in U(N)
Chern-Simons gauge theory coupled to a nonrelativistic complex scalar field in the adjoint represen-
tation [35]. It was shown in [36] that the self-dual system in Eq.(4.1) is completely solvable in terms
of Uhlenbeck’s uniton method. A NC generalization of Eq.(4.1), the Hitchin’s equations on IR2NC ,
was also considered in [37] with very parallel results to the commutative case. We will briefly discuss
the NC Hitchin’s equations in Section 5.
The equations (4.1) for the self-dual case (with + sign) can be elegantly combined into a zero-
curvature condition [35, 36] for the new connections defined by5
A+ = A+ + Φ, A− = A− − Φ† (4.2)
5Here we will relax the reality condition of the fields (Aµ,Φ
a) and complexify them.
with A± = A1 ± iA2:
F+− = ∂+A− − ∂−A+ − i[A+,A−] = 0 (4.3)
where ∂± = ∂1 ± i∂2. Thus the new gauge fields should be a pure gauge, that is, A± = ig−1∂±g for
some g ∈ GL(N, IC). Thus we can choose them to be zero, viz.
A+ = −Φ, A− = Φ†. (4.4)
Then the self-dual equations (4.1) reduce to
† + ∂−Φ + 2i[Φ,Φ
†] = 0, (4.5)
† − ∂−Φ = 0. (4.6)
Introducing another gauge fields C+ = −2Φ and C− = 2Φ†, Eq.(4.5) also becomes the zero-curvature
condition, hence C± are a pure gauge or
Φ = − i
h−1∂+h, Φ
h−1∂−h. (4.7)
A group element h(z) defines a map from IR2C to GL(N, IC) group, which is contractible to the map
from IR2C to U(N) ⊂ GL(N, IC). Then Eq.(4.6) implies that h(z) satisfies the equation in the two-
dimensional U(N) chiral model [35, 36]
−1∂−h) + ∂−(h
−1∂+h) = 0. (4.8)
Eq.(4.8) is the equation of motion derived from the two-dimensional U(N) chiral model governed
by the following Euclidean action
d2zTr ∂µh
−1∂νhδ
µν . (4.9)
A remarkable (mysterious) fact has been known [33, 17, 34] that in the N → ∞ limit the chiral
model (4.9) describes a self-dual spacetime whose equation of motion takes the Plebański form of
self-dual Einstein equations [38]. Thus, including the case of D = 4 and n = 2 in [14, 15, 16], we
have confirmed Eq.(3.16) stating that the 4-dimensional self-dual system in the action (2.2) or (2.13)
in general describes the self-dual Einstein gravity where self-dual metrics are given by Eq.(3.4).
4.2 D = 6 and n = 1
Our current work has been particularly motivated by this case since it was already shown in [39] that
SU(N) Yang-Mills instantons in the N → ∞ limit are gravitational instantons too. Since NC U(1)
instantons are also gravitational instantons as we showed before, it implies that there should be a close
relationship between SU(N) Yang-Mills instantons and NC U(1) instantons. A basic observation was
the relation (1.8), which leads to the sound realization in Eq.(2.13). But we will simply follow the
argument in [39] for the gauge group G = U(N); in the meantime, we will confirm the results for the
emergent geometry from NC gauge fields.
Let us look at the instanton solution in U(N) Yang-Mills theory. The self-duality equation is
given by
Fµν = ±
εµναβFαβ (4.10)
where the field strength is defined by
Fµν = ∂µAν − ∂νAµ − i[Aµ, Aν ]. (4.11)
In terms of the complex coordinates and the complex gauge fields defined by
(x2 + ix1), z2 =
(x4 + ix3),
Az1 = A2 − iA1, Az2 = A4 − iA3,
Eq.(4.10) can be written as
Fz1z2 = 0 = Fz̄1z̄2 , (4.12)
Fz1z̄1 ∓ Fz2z̄2 = 0. (4.13)
Now let us consider the anti-self-dual (ASD) case. We first notice that Fz1z2 = 0 implies that there
exists a u(N)-valued function g such that Aza = ig
−1∂zag (a = 1, 2). Therefore one can choose a
gauge
Aza = 0. (4.14)
Under the gauge (4.14), the ASD equations lead to
∂z̄1Az̄2 − ∂z̄2Az̄1 − i[Az̄1 , Az̄2 ] = 0, (4.15)
∂z1Az̄1 + ∂z2Az̄2 = 0. (4.16)
First notice a close similarity with Eqs.(4.5) and (4.6). Eq.(4.16) can be solved by introducing a
u(N)-valued function Φ such that
Az̄1 = −∂z2Φ, Az̄2 = ∂z1Φ. (4.17)
Substituting (4.17) into (4.15) one finally gets
(∂z1∂z̄1 + ∂z2∂z̄2)Φ− i[∂z1Φ, ∂z2Φ] = 0. (4.18)
Adopting the correspondence (1.8), we now regard Φ ∈ u(N)⊗C∞(IR4) in Eq.(4.18) as a smooth
function on IR4 × Σg, i.e., Φ = Φ(xµ, p, q) where (p, q) are local coordinates of a two-dimensional
Riemann surface Σg. Moreover, a Lie algebra commutator is replaced by the Poisson bracket (1.1)
{f, g} = ∂f
that is,
[Φ1,Φ2] → i{Φ1,Φ2}, (4.19)
where we absorbed θ into the coordinates (p, q). After all, the ASD Yang-Mills equation (4.18) in
the large N limit is equivalent to a single nonlinear equation in six dimensions parameterized by
(xµ, p, q):
(∂z1∂z̄1 + ∂z2∂z̄2)Φ + {∂z1Φ, ∂z2Φ} = 0. (4.20)
Since Eq.(4.20) is similar to the well-known second heavenly equation [38], it was called in [39] as
the six dimensional version of the second heavenly equation.
Starting from U(N) Yang-Mills instantons in four dimensions, we arrived at the nonlinear dif-
ferential equation for a single function in six dimensions. It is important to notice that the resulting
six-dimensional theory is a NC field theory since the Riemann surface Σg carries a symplectic struc-
ture inherited from the u(N) Lie algebra through Eq.(4.19) and it can be quantized in general via
deformation quantization [4]. Since the function Φ in (4.20) is a master field of N ×N matrices [30],
so Φ ∼ N2, the AdS/CFT duality [12] implies that the master field Φ describes a six-dimensional
emergent geometry induced by Yang-Mills instantons.
To see the emergent geometry, consider an appropriate symmetry reduction of Eq.(4.20) to show
that it describes self-dual gravity in four dimensions. There are many reductions from six to four
dimensional subspace leading to self-dual four-manifolds [39]. A common feature is that the four
dimensional subspace necessarily contains the NC Riemann surface Σg. We will show later how the
symmetry reduction naturally arises from the BPS condition in six dimensions. As a specific example,
we assume the following symmetry,
∂z1Φ = ∂z̄1Φ, ∂z2Φ = ∂z̄2Φ, (4.21)
Φ(z1, z2, z̄1, z̄2, p, q) = Λ(z1 + z̄1 ≡ x, z2 + z̄2 ≡ y, p, q). (4.22)
Then Eq.(4.20) is precisely equal to the Husain’s equation [34] which is the reduction of self-dual
Einstein equations to the sdiff(Σg) chiral field equations in two dimensions:
Λxx + Λyy + ΛxqΛyp − ΛxpΛyq = 0. (4.23)
Note that we already encountered in Section 4.1 the two-dimensional sdiff(Σg) chiral field equations
since sdiff(Σg) ∼= u(N) according to the correspondence (1.8). We showed in [16] that Eq.(4.23)
can be transformed to the first heavenly equation [38] which is a governing equation of self-dual
Einstein gravity. In the end we conclude that self-dual U(N) Yang-Mills theory in the large N limit
is equivalent to self-dual Einstein gravity.
Now it is easy to see that the self-dual Einstein equation (4.23) is coming from a 1/2 BPS equation
in six dimensions (see Eq.(34) in [23]) defined by the first-order equation (2.19). According to our
construction, the six-dimensional NC U(1) gauge theory (2.2) is equivalent to the four-dimensional
U(N) gauge theory (2.13). Therefore six-dimensional BPS equations can be equivalently described
by the gauge-Higgs system (Aµ,Φ
a) in the action (2.13). Let us newly denote the NC coordinates
y1, y2 and commutative ones z3, z4 as uα, α = 1, 2, 3, 4 while z1, z2 as vA, A = 1, 2. The 1/2 BPS
equations, Eq.(34), in [23] can then be written as the following form
Fαβ = ±
εαβγδFγδ, (4.24)
FαA = FAB = 0. (4.25)
Using Eqs.(2.8)-(2.10), the above equations can be rewritten in terms of (Aµ,Φ
a) where the constant
term in (2.10) can simply be dropped for the reason explained before.
FAB = 0 in Eq.(4.25) can be solved by AB = 0 (B = 1, 2) and then FαA = 0 demand that the
gauge fields Aα should not depend on v
A. Thereby Eq.(4.24) precisely reduces to the self-duality
equation (4.1) for D = 4 and n = 1. The symmetry reduction considered above is now understood
as the condition (4.25); in specific, the coordinates vA correspond to i(z1 − z̄1) and i(z2 − z̄2) for
the reduction (4.22). However there are many different choices taking a four-dimensional subspace
in Eq.(4.24) which are related by SO(6) rotations [23]. Unless vA ∈ (y1, y2), that is, Eq.(4.24)
becomes commutative Abelian equations in which there is no non-singular solution, Eqs.(4.24) and
(4.25) reduce to four-dimensional self-dual Einstein equations, as was shown in [39]. The above BPS
equations also clarify why the two-dimensional chiral equations in Section 4.1 reappear in Eq.(4.23).
4.3 D = 8 and D = 10
The analysis for the first-order system (2.19) becomes much more complicated in higher dimensions.
The unbroken supersymmetries in D = 8 have been analyzed in [23]. Because the integrable structure
of Einstein equations in higher dimensions is little known, it is difficult to precisely identify governing
geometrical structures emergent from the gauge theory (2.2) or (2.13) even for BPS states. Neverthe-
less some BPS configurations can be easily implemented as follows. As we did in Eqs.(4.24)-(4.25),
one can imbed the 4-dimensional self-dual system for n = 1 or n = 2 into eight or ten dimensions.
The simplest case is that the metric (3.4) becomes (locally) a product manifold M4×X where M4 is
a self-dual (hyper-Kähler) four-manifold. For example, we can consider an eight-dimensional config-
uration where (A1, A2,Φ
3,Φ4) depend only on (z1, z2, y3, y4) coordinates while (Φ1,Φ2, A3, A4) do
only on (y1, y2, z3, z4) in a B-field background with θ12 6= 0 and θ34 6= 0, only non-vanishing com-
ponents. There are many similar configurations. We will not exhaust them, instead we will consider
the simplest cases which already have some relevance to other works.
The simplest BPS state in D = 8 is the case with n = 2 in the action (2.13); see Eq.(55) in [23].
The equations are of the form
Fµν = ±
εµνλσFλσ, (4.26)
[Φa,Φb] = ±1
εabcd[Φ
c,Φd], (4.27)
a = 0. (4.28)
A solution of Eq.(4.28) is given by Aµ = Aµ(z) and Φ
a = Φa(y). Then Eq.(4.26) becomes com-
mutative Abelian equations which allow no non-singular solutions, while (4.27) reduces to Eq.(3.5)
describing 4-dimensional self-dual manifolds [15]. Thus the metric (3.4) in this case leads to a half-
BPS geometry IR4×M4. Since we don’t need instanton solutions in Eq.(4.26), we may freely replace
IR4 by 4-dimensional Minkowski space IR1,3 (see the footnote 1).
The above system was considered in [40] in the context of D3-D7 brane inflationary model. The
model consists of a D3-brane parallel to a D7-brane at some distance in the presence of Fab =
(B + F )ab on the worldvolume of the D7-brane, but transverse to the D3-brane. The F -field plays
the role of the Fayet-Illiopoulos term from the viewpoint of the D3-brane worldvolume field theory.
Because of spontaneously broken supersymmetry in de Sitter valley the D3-brane is attracted towards
the D7-brane and eventually it is dissolved into the D7-brane as a NC instanton. The system ends in
a supersymmetric Higgs phase with a smooth instanton moduli space. An interesting point in [40] is
that there is a relation between cosmological constant in spacetime and noncommutativity in internal
space. Our above result adds a geometrical picture that the internal space after tachyon condensation
is developed to a gravitational instanton, e.g., an ALE space or K3.
Another interesting point, not mentioned in [40], is that it effectively realizes the dynamical com-
pactification of extra dimensions suggested in [41]. Since the D3-brane is an instanton inside the
D7-brane, particles living in the D3-brane are trapped in the core of the instanton with size ∼ θ2
where the noncommutativity scale θ is believed to be roughly Planck scale. Since the instanton
(D3-brane) results in a spontaneous breaking of translation symmetry and supersymmetry partially,
Goldstone excitations corresponding to the broken bosonic and fermionic generators are zero-modes
trapped in the core of the instanton. “Quarks” and “leptons” might be identified with these fermionic
zero-modes [41].
We argued in the last of Section 3 that the 10-dimensional metric (3.4) for d = 4 and n = 3
reasonably corresponds to an emergent geometry arising from the 4-dimensional N = 4 supersym-
metric U(N) Yang-Mills theory. Especially it may be closely related to the bubbling geometry in AdS
space found by Lin, Lunin and Maldacena (LLM) [18]. One may notice that the LLM geometry is
a bubbling geometry deformed from the AdS5 × S5 background which can be regarded as a vacuum
manifold emerging from the self-dual RR five-form background, while the Ward’s geometry (3.4) is
defined in a 2-form B-field background and becomes (conformally) flat if all fluctuations are turned
off, say, (Aµ,Φ
a) → (0, ya/κ). But it turns out that the LLM geometry is a special case of the Ward’s
geometry (3.4).
To see this, recall that the AdS5 × S5 background is conformally flat, i.e.,
ds2 =
(ηµνdz
µdzν + dyadya) =
(ηµνdz
µdzν + dρ2) + L2dΩ25 (4.29)
where ρ2 =
a=1 y
aya and dΩ25 is the spherically symmetric metric on S
5. It is then easy to see that
the metric (4.29) is exactly the vacuum geometry of Eq.(3.4) when the volume form ω in Eq.(3.2) is
given by
dy1 ∧ · · · ∧ dy6
. (4.30)
Therefore it is obvious that the Ward’s metric (3.4) with the volume form (4.30) describes a bubbling
geometry which approaches to the AdS5 × S5 space at infinity where fluctuations are vanishing,
namely, (Aµ,Φ
a) → (0, ya/κ). Note that the flat spacetime IR1,9 is coming from the volume form
ω = dy1 ∧ · · · ∧ dy6, so Eq.(4.30) should correspond to some nontrivial soliton background from the
gauge theory point of view. We will discuss in Section 5 a possible origin of the volume form (4.30).
Now let us briefly summarize half-BPS geometries of type IIB string theory corresponding to
the chiral primaries of N = 4 super Yang-Mills theory [18]. These BPS states are giant graviton
branes which wrap an S3 in AdS5 or an S̃
3 in S5. Thus the geometry induced (or back-reacted)
by the giant gravitons preserves SO(4) × SO(4) × R isometry. It turns out that the solution is
completely determined by a single function which is specified with two types of boundary conditions
on a particular plane corresponding to either of two different spheres shrinking on the plane in a
smooth fashion. The LLM solutions are thus in one-to-one correspondence with various 2-colorings
of a 2-plane, usually referred to as ‘droplets’ and the geometry depends on the shape of the droplets.
The droplet describing gravity solutions turns out to be the same droplet in the phase space describing
free fermions for the half-BPS states.
The solutions can be analytically continued to those with SO(2, 2) × SO(4) × U(1) symmetry
[18], so the solutions have the AdS3 × S3 factor rather than S3 × S̃3. After an analytic continuation,
a underlying 4-dimensional geometry M4 attains a nice geometrical structure at asymptotic region,
where AdS3 × S3 → IR1,5 and M4 reduces to a hyper-Kähler geometry. But it loses the nice picture
in terms of fermion droplet since the solution is now specified by one type of boundary condition. It
is interesting to notice that the asymptotic bubbling geometry for the type IIB case is the Gibbons-
Hawking metric [28] and the real heaven metric [42] for the M theory case, which are all solutions of
NC electromagnetism [15, 16].
It is quite demanding to completely determine general half-BPS geometries emerging from the
gauge-Higgs system in the action (2.13). Hence we will look at only an asymptotic geometry (or a
local geometry) which is relatively easy to identify. For the purpose, we consider the n = 3 case on
4-dimensional Minkowski space IR1,3. It is simple to mimic the previous half-BPS configurations in
D = 6, 8 with trivial extra Higgs fields. Then the resulting metric (3.4) will be locally of the form
M4 × IR1,5 akin to the asymptotic bubbling geometry. However M4 can be a general hyper-Kähler
manifold. Therefore the solutions we get will be more general, whose explicit form will depend on
underlying Killing symmetries and boundary conditions. For example, the type IIB case is given by a
hyper-Kähler geometry with one translational Killing vector (Gibbons-Hawking) while the M theory
case is with one rotational Killing vector (real heaven) [43]. Therefore we may get in general bubbling
geometries in the M theory as well as the type IIB string theory.
5 Discussion
We showed reasonable evidences that the 10-dimensional metric (3.4) for d = 4 and n = 3 describes
the emergent geometry arising from the 4-dimensional N = 4 supersymmetric U(N) Yang-Mills
theory and thus might explain the AdS/CFT duality [12]. An important point in this context is that
the volume form (4.30) is required to describe the AdS5 × S5 background. What is the origin of this
nontrivial volume form ? In other words, how to realize the self-dual RR five-form background from
the gauge theory point of view ?
To get some hint about the question, first note that the AdS5 × S5 geometry emerges from multi-
instanton collective coordinates which dominates the path integral in a large N limit [44]. The factor
d4zdρρ−5 appears in bosonic collective coordinate integration (with zµ the instanton 4-positions)
which agrees with the volume form of the conformally invariant space AdS5, where instanton size
corresponds to the radial coordinate ρ in Eq.(4.29). Another point is that the AdS5 × S5 space cor-
responds to the LLM geometry for the simplest and most symmetric configuration which reduces to
the usual Gibbons-Hakwing metric (3.6) at asymptotic regions [18]. This result is consistent with
the picture in Section 4.2 that U(N) instantons at large N limit are indeed gravitational instantons.
It is then tempted to speculate that the AdS5 × S5 geometry would be emerging from a maximally
supersymmetric instanton solution of Eq.(2.19) in D = 10. It should be an interesting future work.
In addition, we would like to point out that an AdSp × Sq background arises from Eq.(3.4) in the
same way as Eq.(4.30) by choosing the volume form ω as follows
dy1 ∧ · · · ∧ dyq+1
(5.1)
with ρ2 =
a=1 y
aya and (Aµ,Φ
a) = (0, ya/κ). A particularly interesting case is d = 2 and n = 2
for which the volume form (5.1) leads to the AdS3 × S3 background and the action (2.13) describes
matrix strings [45, 46]. We believe that the metric (3.4) with ω = dy1 ∧ · · · ∧ dy4/ρ2 describes a
bubbling geometry emerging from the matrix strings.
One might already notice a subtle difference between the matrix action (2.13) and the Ward’s
metric (3.4). According to our construction in Section 2, the number of the Higgs fields Φa is even
while the Ward construction has no such restriction. But it was shown in [15, 16] that the Gibbons-
Hawking metric (3.6) for the d = 3 and m = 1 case also arises from the d = 0 and m = 4. It implies
that we can replace some transverse scalars by gauge fields and vice versa. Recalling that the fields
in the action (2.13) are all N × N matrices, of course, N → ∞, it is precisely ‘matrix T -duality’
exchanging transverse scalars and gauge fields associated with a compact direction in p-brane and
(p+ 1)-brane worldvolume theories through (see Eq.(154) in [46])
Φa ↔ iDµ = i(∂µ − iAµ). (5.2)
With this identification, the d-dimensional U(N) gauge theory (2.13) can be obtained by applying the
d-fold ‘matrix T -duality’ (5.2) to the 0-dimensional IKKT matrix model [11, 20]
S = −2πκ
gMPgNQ[Φ
M ,ΦN ][ΦP ,ΦQ]
. (5.3)
However, the T -duality (5.2) gives rise to qualitatively radical changes in worldvolume theory.
First it changes the dimensionality of the theory and thus it affects its renormalizability (see Sec. VI
in [46] and references therein for this issue in Matrix theory). For example, the action (2.13) for
d > 4 is not renormalizable since the coupling constant g2YM ∼ gsκ
2 ∼ gsm4−ds has negative mass
dimension in this case. Second it also changes a behavior of the emergent metric (3.4). But these
changes are rather consistent with the fact that under the T -duality (5.2) a Dp-brane is transformed
into a D(p+ 1)-brane and vice versa.
Our construction in Section 2 raises a bizarre question about the renormalization property of NC
field theory. If we look at the action (2.2), the theory superficially seems to be non-renormalizable for
D > 4 since the coupling constant (2.5) has negative mass dimension. But this non-renormalizability
appears as a fake if we use the matrix representation (2.18) together with the redefinition of variables
in Eq.(2.4). The resulting coupling constant, denoted as gd, in the final action (2.13) depends only on
the dimension of commutative spacetime rather than the entire spacetime. Since the resulting U(N)
theory is in the limit N → ∞, while the ’t Hooft coupling λ ≡ g2dN is kept fixed, planar diagrams
dominate in this limit [9]. Since the dependence of NC coordinates in the action (2.2) has been
encoded into the matrix degrees of freedom, one may suspect that the divergence of the original theory
might appear as a divergence of perturbation series as a whole in the action (2.13). The convergence
aspect of the planar perturbation theory concerns Np(n), the number of planar diagrams in nth order
in λ. It was shown in [47] that Np(n) behaves asymptotically as
Np(n)
n→∞∼ cn, c = constant. (5.4)
Therefore the planar theory (unlike the full theory) for d ≤ 4 has a formally convergent perturbation
series, provided the ultraviolet and infrared divergences of individual diagrams are cut off [47]. It will
be interesting to carefully examine the renormalization property of NC field theories along this line.
We showed in Section 3 that the Ward metric (3.4) is emerging from commutative, i.e., O(θ),
limit. Since the vector fields in Eq.(3.11) are in general higher order differential operators acting on
Aθ, we thus expect that they actually define a ‘generalized gravity’ beyond Einstein gravity, e.g., the
NC gravity [29] or the NC unimodular gravity [48].6 It was shown in [16] that the leading derivative
6The latter seems to be quite relevant to our emergent gravity since the vector fields VM in Eq.(3.11) always belong to
the volume preserving diffeomorphisms, which is a generic property of vector fields defined in NC spacetime. It should
be interesting to more clarify the relation between the NC unimodular gravity [48] and the emergent gravity.
corrections in NC gauge theory start with four derivatives, which was conjectured to give rise to higher
order gravity. As was explicitly checked for the self-dual case, Einstein gravity maybe emerges from
NC gauge fields in commutative limit, which then implies that the leading derivative corrections give
rise to higher order terms with four more derivatives compared to the Einstein gravity. This means
that the higher order gravity starts from the second order corrections in θ with higher derivatives, that
is, no first order correction in θ to the Einstein gravity. Interestingly this result is consistent with those
in [29] and also in [49] calculated from the context of NC gravity.
It was shown in Section 4.1 that the self-duality system for the D = 4 and n = 1 case is mapped
to the two-dimensional U(∞) chiral model (4.9) which is remarkably equivalent to self-dual Einstein
gravity [33, 17, 34]. But this case should not be much different from the D = 4 and n = 2 case
in [15, 16] since they equally describe the self-dual Einstein gravity. Indeed we can make them
bear a close resemblance each other. For the purpose, let us consider a four-dimensional NC space
IR2NC × IR2NC . We can choose the matrix representation (2.18) only for the second factor, i.e.,
φ(y1, y2, y3, y4) =
Ωij (y
1, y2)|i〉〈j|. (5.5)
As a result, the action (2.13) now becomes two-dimensional U(N) gauge theory on IR2NC . The self-
dual equations in Eq.(4.1) in this case are given by the NC Hitchin equations, now defined on IR2NC
instead of IR2C . The NC Hitchin equations have been considered by K. Lee in [37] with very parallel
results with the commutative case (4.1). It is interesting that there exist two different realizations for
self-dual Einstein gravity, whose relationship should be more closely understood.
Finally it will be interesting to consider a compact NC space instead of IR2nNC , for instance, a
NC 2n-torus T2nNC . Since the module over a NC torus is still infinite dimensional [8], the matrix
representation (2.18) is also infinite dimensional. Thus we expect that our construction in Section 2
and 3 can be applied even to the NC torus without many essential changes.
Acknowledgments
After posting this paper to the arXiv, we were informed of related works on YM-Higgs BPS configu-
rations on NC spaces [50] and on the relation between a large N gauge theory, a Moyal deformation
and a self-dual gravity [51] by O. Lechtenfeld and C. Castro, respectively. We thank them for the
references. This work was supported by the Alexander von Humboldt Foundation.
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	Introduction
	A Large N Gauge Theory From NC U(1) Gauge Theory
	Emergent Geometry From NC Gauge Theory
	Self-dual Einstein Gravity From Large N Gauge Theory
	D=4 and n=1
	D=6 and n=1
	D=8 and D=10
	Discussion
