Non-commutativity and Open Strings Dynamics in Melvin Universes
MAD-TH-07-02
Non-commutativity and Open Strings Dynamics
in Melvin Universes
Danny Dhokarh, Akikazu Hashimoto, and Sheikh Shajidul Haque
Department of Physics, University of Wisconsin, Madison, WI 53706
Abstract
We compute the Moyal phase factor for open strings ending on D3-branes wrapping a
NSNS Melvin universe in a decoupling limit explicitly using world sheet formalism in
cylindrical coordinates.
http://arxiv.org/abs/0704.1124v2
Melvin Universe is an exact axially symmetric solution of Einstein gravity in a background
with magnetic flux [1]. It arises naturally as a Kaluza-Klein reduction of twisted flat space
ds2 = −dt2 + d~x2 + dr2 + r2(dϕ+ ηdz)2 + dz2 , (1)
along the coordinate z. The twist is parameterized by variable η. The fact that z ∼ z+2πR
is periodic makes the twist deformation physical.
Melvin universes has a natural embedding in string theory [2–4]. Simply embed (1) in
11-dimensional supergravity. Reducing along z gives rise to a background in type IIA string
theory with a background of magnetic RR 2-form field strength.
Along similar lines, one can embed (1) in type IIA supergravity and T-dualize along z.
This gives rise to a background in type IIB string theory
ds2 = −dt2 + d~x2 + dr2 +
r2dϕ2
1 + η2r2
1 + η2r2
1 + η2r2
dϕ ∧ dz̃
1 + η2r2
z̃ = z̃ + 2πR̃, R̃ =
, (2)
with an axially symmetric magnetic NSNS 3-form field strength in the background. String
theories in backgrounds like (2) are very special in that the world sheet theory is exactly
solvable [5–10]. Quantization of open strings in Melvin backgrounds have also been studied
and was shown to be exactly solvable [11, 12] as well.
Embedding D-branes in Melvin universes can give rise to interesting field theories in
the decoupling limit. A D3-brane extended along t, z̃, and two of the ~x coordinates gives
rise to a non-local field theory known as the “dipole” theory [13, 14]. Orienting the D3-
brane to be extended along the t, r, ϕ, and z̃ coordinates, on the other hand, gives rise
to a non-commutative gauge theory with a non-constant non-commutativity parameter1
[16,17]. These are field theories, whose Lagrangian [17] is expressed most naturally using the
deformation quantization formula of Kontsevich2 [19]. Field theories arising as a decoupling
limits of various orientations of D-branes in Melvin and related closed string backgrounds
along these lines3 were tabulated and classified in Table 1 of [16].4
1The first explicit construction of models of this type is [15].
2General construction of non-commutative field theory on curved space-time with non-constant non-
commutativity parameter, arising from D-branes in non-vanishing H field background, and their relation to
the deformation quantization formula of Kontsevich, was first discussed in [18].
3The S-dual NCOS theories with non-constant non-commutativity parameter was studied in [20, 21].
4More recently, a novel non-local field theory, not included in the classification of [16], was discovered
[22, 23].
To show that the decoupled field theory is a non-commutative field theory, the authors
of [16] presented the following arguments:
• The application of Seiberg-Witten formula5 [24]
)µν = [(g +B)µν ]
−1 (3)
to the closed string background (2) gives the following open string metric and the
non-commutativity parameter
Gµνdx
ν = −dt2 + dr2 + r2dϕ2 + dz2
θϕz = 2πα′η (4)
which are finite if α′ is scaled to zero keeping ∆ = α′η fixed.
• Solution of the classical equations of motion of an open string traveling freely on the
D3-brane with angular momentum J has a dipole structure whose size is given by [16]
L = θϕzJ . (5)
Another suggestive argument is the similarity between α′ → 0 limit of critical string
theory and the boundary Poisson sigma-model [25] as was pointed out, e.g., in [26]. As was
emphasized in [26], however, the two theories are not to be understood as being equivalent
or continuously connected. This apparent similarity therefore does not constitute a proof
that the decoupled theory is a non-commutative field theory.
A physical criteria for non-commutativity is the Moyal-like phase factor in scattering
amplitudes. Scattering amplitudes of open strings ending on a D-brane can be computed
along the lines reviewed in [27]. In the case of the constant non-commutativity parameter,
one can show very explicitly that
1x(τ1)eip
2x(τ2) . . . eip
nx(τn)〉G,θ = e
θijpm
ǫ(τn−τm)〈eip
1x(τ1)eip
2x(τ2) . . . eip
nx(τn)〉G,θ=0
which implies that the scattering amplitudes receive corrections in the form of the Moyal
phase factor [24, 28, 29]. The goal of this article is to derive the analogous statement (60)
for the model of [16, 17]. Once (60) is established in polar coordinates, the connection
to Kontsevich formula follows from performing a change of coordinates to the rectangular
coordinate system and a non-local field redefinition as is described in [17, 30].
5The normalization of B field is such that BHashimoto−Thomas = 2πα
BSeiberg−Witten.
A useful first step in this exercise is to reproduce the master relation (6) in a slightly
different formalism than what was used in [24]. Let us begin by constructing the closed
string background as follows. Start with flat space
ds2 = dy′2 + dz̃2 , (7)
where y and z̃ are compactified with period L = 2πR. Then,
I T-dualize along the z direction so that the metric becomes
ds2 = dy′2 + dz2 . (8)
II Twist by shifting the coordinates y′ = y + ηz
ds2 = (dy + ηdz)2 + dz2 . (9)
III T-dualize on z so that
1 + η2
(dy2 + dz̃2), B =
1 + η2
dy ∧ dz̃ . (10)
The open string metric associated to this background is
Gµνdx
ν = dy2 + dz̃2, θyz̃ = 2π∆2 (11)
if we scale
∆2 = α′η . (12)
The transformation of the coordinates and the orientation of the branes are illustrated in
figure 1. This sequence of dualities was referred to as the “Melvin shift twist” in [16].
The approach of [24] was to work directly in the duality frame III, but one can just as
easily work in a framework which makes the T-duality between duality frame II and III
manifest, by working with a sigma model of the form
dσ1dσ2
∂ay∂by + 2η∂ayVb + (1 + η
2)VaVb
+ iǫab∂az̃Vb
where we have chosen to work in conformal gauge in Eucledian signature. This action utilizes
the Bushar’s formulation of T-duality [31]. To see this more explicitly, consider integrating
out the field z̃. This imposes the constraint
dV = 0 → Va = ∂az (14)
Figure 1: In I and II, the thick line denotes a D2-brane, and the dotted line is the minimum
energy configuration of the open strings ending on the D2-branes. The I and II are related
by coordinate transformation y′ = y + ηz. III is the T-dual of II, and the shaded region in
III denotes a D3-brane.
which brings the action (13) into the form
dσ1dσ2
∂ay∂by + 2η∂ay∂bz + (1 + η
2)∂az∂bz
which is the sigma model for the duality frame II. On the other hand, integrating out V
first gives rise to a sigma model of the form
dσ1dσ2
1 + η2
δab (∂ay∂by + ∂az̃∂bz̃) + i
1 + η2
ǫab∂ay∂bz̃
which is the string action for the duality frame III.
In extracting non-commutative gauge theory as a decoupling limit, we are interested in
embedding a D-brane extended along the y and z̃ coordinates in the duality frame III. We
must therefore take the sigma model to be defined on a Riemann surface with one boundary,
which we take to be the upper half plane. It is also necessary to impose the appropriate
boundary condition for all of the world sheet fields. We impose the boundary condition
which is free along the y direction and Dirichlet along the z direction:
∂ny(σ, σ̄) + ηVn(σ, σ̄)|∂Σ = 0 , (17)
Vt|∂Σ = ∂tz|∂Σ = 0 . (18)
Using the equation of motion from the variation of Va field
η∂by + (1 + η
2)Vb + iǫab∂az̃ = 0 (19)
and (18), we infer
∂nz̃ − iη∂ty = 0 . (20)
The boundary conditions (17) and (20) are precisely the boundary condition imposed in the
analysis of [24].
In order to complete the derivation of (6), we add a source term
e−Ssource =
eikyny(σn,σ̄n)+ikznz̃(σn,σ̄n) = e
n(ikyny(σn,σ̄n)+ikznz̃(σn,σ̄n)) (21)
to the action (13). Integrating out the V fields and bringing the sigma model (13) into duality
frame III would lead to identical computation as what was described in [24] to derive (6). We
will show below that the same conclusion can be reached using a slightly different argument
which turns out to easily generalize to the case of Melvin deformed theories [16, 17].
The approach we take here is to go to the duality frame I. This brings the sigma model
(13) to a simpler form
dσ1dσ2
δab (∂ay
′ + ∂az∂bz)
. (22)
The z̃ field in the vertex operator now plays the role of a disorder operator of the dual
field z. It has the effect of shifting the Dirichlet boundary condition, incorporating the fact
that strings are stretched along the z direction in frames I and II. Also, the fact that the
periodicity in (y′, z) coordinate system are twisted
(y′, z) = (y′ + ηLn, z + Ln) (23)
requires an insertion of a disorder operator for the y′(σ, σ̄) field as well. We therefore find
that the source term has the form
e−Ssource =
eikyny
′(σn,σ̄n)+iηkznỹ
′(σn,σ̄n)−iηkynz(σn,σ̄n)+ikznz̃(σn) . (24)
The boundary condition is now simply Neumann for y′
′(σ, σ̄) = 0|
∂Σ , (25)
and Dirichlet for z
∂tz(σ, σ̄) = 0|∂Σ . (26)
In this form, y′ and the z sector decouple, allowing their correlators to be computed sepa-
rately. In order to compute the correlation functions involving order and disorder operators
with boundary conditions (25) and (26), it is convenient to decompose the fields into holo-
morphic and anti holomorphic parts
y′(σ, σ̄) = y′(σ) + ȳ′(σ̄) , ỹ′(σ, σ̄) = y′(σ)− ȳ′(σ̄) , (27)
z(σ, σ̄) = z(σ) + z̄(σ̄) , z̃′(σ, σ̄) = z(σ)− z̄(σ̄) . (28)
Their correlation functions are given by
〈y′(σ1)y
′(σ2)〉 = −
α′ log(σ1 − σ2) (29)
〈ȳ′(σ1)ȳ
′(σ2)〉 = −
α′ log(σ̄1 − σ̄2) (30)
〈ȳ′(σ̄1)y
′(σ2)〉 = −
α′ log(σ̄1 − σ2) (31)
〈z(σ1)z(σ2)〉 = −
α′ log(σ1 − σ2) (32)
〈z̄(σ̄1)z̄(σ̄2)〉 = −
α′ log(σ̄1 − σ̄2) (33)
〈z̄(σ̄1)z(σ2)〉 =
α′ log(σ̄1 − σ2), (34)
from which we infer
〈y′(σ1, σ̄1)y
′(σ2, σ̄2)〉 = −
α′(log(σ1 − σ2) + log(σ1 − σ̄2) + log(σ̄1 − σ2) + log(σ̄1− σ̄2)) (35)
〈ỹ′(σ1, σ̄1)y
′(σ2, σ̄2)〉 = −
α′(log(σ1 −σ2) + log(σ1 − σ̄2)− log(σ̄1 −σ2)− log(σ̄1 − σ̄2)) (36)
〈ỹ′(σ1, σ̄1)ỹ
′(σ2, σ̄2)〉 = −
α′(log(σ1 −σ2)− log(σ1 − σ̄2)− log(σ̄1− σ2)+ log(σ̄1 − σ̄2)) (37)
〈z̃(σ1, σ̄1)z̃(σ2, σ̄2)〉 = −
α′(log(σ1 − σ2) + log(σ1 − σ̄2) + log(σ̄1 − σ2) + log(σ̄1 − σ̄2)) (38)
〈z(σ1, σ̄1)z̃(σ2, σ̄2)〉 = −
′(log(σ1 − σ2) + log(σ1 − σ̄2)− log(σ̄1 − σ2)− log(σ̄1 − σ̄2)) (39)
〈z(σ1, σ̄1)z(σ2, σ̄2)〉 = −
α′(log(σ1−σ2)− log(σ1− σ̄2)− log(σ̄1−σ2)+ log(σ̄1− σ̄2)) . (40)
In terms of these correlation functions, one can easily show that
〈O(σ1, σ̄1)O(σ2, σ̄2)〉 (41)
′(ky1ky2 + kz1kz2)(log(σ1 − σ2) + log(σ1 − σ̄2) + log(σ̄1 − σ2) + log(σ̄1 − σ̄2))
−ηα′(ky1kz2 − ky2kz1)(log(σ1 − σ̄2)− log(σ̄1 − σ2))
η2α′(ky1ky2 + kz1kz2)(log(σ1 − σ2)− log(σ1 − σ̄2)− log(σ̄1 − σ2) + log(σ̄1 − σ̄2))
On(σn, σ̄n) = ikyny
′(σn, σ̄n) + iηkznỹ
′(σn, σ̄n)− iηkynz(σn, σ̄n) + ikznz̃(σn, σ̄n) . (42)
When these operators are pushed toward the boundary
σ → τ + 0+i , (43)
the correlation function (41) reduces to
〈O(τ1)O(τ2)〉 = 2α
′(ky1ky2 + kz1kz2) log(τ1 − τ2)− πiηα
′(ky1kz2 − ky2kz1)ǫ(τ2 − τ1) (44)
where ǫ(τ), following the notation of [24], is a function which takes the values ±1 depending
on the sign of τ . The term of order η2 vanishes in this limit. From these results, we conclude
eOn(τn)〉 = e
m<n〈Om(τm)On(τn)〉 (45)
from which the main statement (6) follows immediately.
Finally, let us discuss the generalization of (6) to D3-brane embedded into Melvin universe
background (2) along the lines of [16, 17]. We will consider the simplest case of embedding
(2) into bosonic string theory. For the Melvin universe background (2), it is convenient to
prepare a vertex operator that corresponds to tachyons in cylindrical basis
V (ν,m,~k) =
dk1 dk2 δ(ν
2 − k21 − k
imθeik1x1(σ,σ̄)+k2x2(σ,σ̄)+
~k~x(σ,σ̄)
~k~x(σ,σ̄)Jν(r(σ, σ̄))e
imϕ(σ,σ̄) (46)
where
r2 = x21 + x
2, ϕ = arg(x1 + ix2), θ = arg(k1 + ik2) . (47)
As long as ~k2 + ν2 are taken to satisfy the on-shell condition of the tachyon, (46) is linear
combination of operators of boundary conformal dimension 1, and must itself be an operator
of boundary conformal dimension one. Such construction of vertex operator as a linear
superposition is similar in spirit to what was considered in [32, 33].
dσ1dσ2
∂ar∂br + r
∂aϕ∂bϕ+ 2ηr
∂aϕVb + (1 + η
2)VaVb
+ iǫab∂az̃Vb
on the upper half plane. Integrating out z̃ brings this action to the form appropriate for the
analogue of II
dσ1dσ2
∂ar∂br + r
2∂aϕ∂bϕ+ 2ηr
2∂aϕ∂bz + (1 + η
2r2)∂az∂bz
. (49)
The vertex operators can be represented as a source term
e−Ssource =
Jvn(r(σn, σ̄n))e
imnϕ(σn,σ̄n)+ikznz̃(σn,σ̄n) (50)
where z̃ is a disorder operator. Now, if we let
ϕ′(σ, σ̄) = ϕ(σ, σ̄) + ηz(σ, σ̄) , (51)
the action becomes
dσ1dσ2
∂ar∂br + r
′ + ∂az∂bz
e−Ssource =
Jvn(r(σn, σ̄n))e
On (53)
On = imnϕ
′(σn, σ̄n) + iηkznϕ̃
′(σn, σ̄n)− iηmnz(σn, σ̄n) + ikznz̃(σn, σ̄n) (54)
where
ϕ̃′(σ, σ̄) (55)
is the disorder field for ϕ′ satisfying the relation
∂aϕ̃′ = iǫabr2∂bϕ
′ (56)
which follows naturally from the Busher rule applied to the ϕ fields.
This time, the problem is slightly complicated by the fact that (r, ϕ′) sector is interacting.
It is still the case that (ϕ′, z) sector, for some fixed r(σ, σ̄), is non-interacting. We will exploit
this fact and do the path integral in the order where we integrate over ϕ′ and z first. The
two point function of ϕ′ formally has the form
〈ϕ′(σ1, σ̄1)ϕ
′(σ2, σ̄2)〉 = (∂r
2(σ, σ̄)∂)−1 . (57)
Then, it follows that
〈ϕ′(σ1, σ̄1)∂
aϕ̃′(σ2, σ̄2)〉 = iǫ
ab(∂b)−1 (58)
from which it follows
〈ϕ̃′(σ1, σ̄1)ϕ
′(σ2, σ̄2)〉 = −
α′(log(σ1−σ2)+ log(σ1− σ̄2)− log(σ̄1−σ2)− log(σ̄1− σ̄2)) (59)
in complete analogy with (36). The correlator (59) tells us that while the field-field correlator
〈ϕ′ϕ′〉 is complicated and r dependent, the field/disorder field correlator 〈ϕ̃′ϕ′〉 stays simple
and topological.
We can then proceed to compute the analogue of (44) and (45) for the operator (54) in
the (ϕ′, z) sector. While we do not explicitly compute the 〈ϕ̃′ϕ̃′〉 correlator which appear at
order η2 in (44), it is clear that the boundary condition forces this term to vanish as was the
case in the earlier example. The term of order η in the exponential can be made to take the
Moyal-like form
(2π∆)(makzb−kzamb)ǫ(τb−τa) (60)
which is finite in the scaling limit α′ → 0 with
keeping ∆ finite. This is precisely the scaling considered in [16, 17]. The dependence on
r(σ, σ̄) drops out for this term of order η, allowing us to further path integrate over this
field trivially, with the only effect of η being the overall phase factor (60). This establishes
that the decoupled theory of D-branes in Melvin universes considered in [16,17] has an effec-
tive dynamics which includes the Moyal-like phase factor involving the angular momentum
quantum number m and the momentum kz. In Cartesian coordinates, this Moyal phase cor-
responds to a position dependent non-commutativity [16,17]. This analysis extends straight
forwardly to other simple models of position dependent non-commutativity, such as6 the
“Melvin Null Twist” [15] and “Null Melvin Twist” [34]. It would be interesting to extend
this analysis to superstrings and to consider the scattering of states other than the open
string tachyon.
Acknowledgements
We would like to thank I. Ellwood and O. Ganor for discussions. This work was supported
in part by the DOE grant DE-FG02-95ER40896 and funds from the University of Wisconsin.
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