Non-commutativity from the double sigma model

0.16 MB·English

by  Dimitri Polyakov

CTP-SCU/2015001
Non-commutativity from the double sigma model
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Dimitri Polyakov, Peng Wang, Houwen Wu and Haitang Yang
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h Center for theoretical physics
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- Sichuan University
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Chengdu, 610064, China
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polyakov, pengw, [email protected], [email protected]
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5 Abstract
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. Weshowhownon-commutativity arisesfromcommutativity inthedoublesigmamodel. Wedemon-
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strate that this model is intrinsically non-commutative by calculating the propagators. In the simplest
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5 phase configuration, there are two dual copies of commutative theories. In general rotated frames, one
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gets a non-commutative theory and a commutative partner. Thus a non-vanishing B also leads to a
:
v commutative theory. Our results imply that O(D,D) symmetry unifies not only the big and small
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X torus physics, but also the commutative and non-commutative theories. The physical interpretations
r of themetricandotherparameters inthedoublesigmamodelarecompletely dictated by theboundary
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conditions. The open-closed relation is also an O(D,D) rotation and naturally leads to the Seiberg-
Witten map. Moreover, after applying asecond dual rotation, we identify the description parameter in
the Seiberg-Witten map as an O(D,D) group parameter and all theories are non-commutative under
this composite rotation. As a bonus, the propagators of general frames in double sigma model for open
string are also presented.
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Contents
1 Introduction 2
2 Non-commutativity in string theory and the Seiberg-Witten map 4
3 O(D,D) rotations of the propagators 6
4 The general descriptions of Seiberg-Witten map 12
1 Introduction
Symmetries and dualities play a central role in modern physics. There are some evidences that various
dualities in string theory may emerge from symmetry breaking in some select backgrounds. In string
theory,therearethreewellknowndualities: T-dualityrelateslargeandsmalldistancephysics;S-duality
connects weak and strong couplings; and U-duality unifies the T-duality and S-duality. M-theory is
dictated by these three types of dualities. There is another significant equivalence in string theory,
the Seiberg-Witten map [1] which seems to disconnect from known dualities or symmetries. The
Seiberg-Witten map relies on the non-commutativity from string theory and the open-closed relation.
In the language of string theory, commutativity is a natural description for closed string and non-
commutativity is natural for open string. The Seiberg-Witten map is a map between commutative
variables gˆ ,gˆ ,Bˆ of closed strings and non-commutative variables (g ,g ,B ) of open strings.
s ij ij s ij ij
TheDirac(cid:16)-Born-Infeld(cid:17)(DBI)action,obtainedfromcalculatingtheone-loopbetafunctionofworldsheet
action, is originally expressed with commutative variables gˆ ,gˆ ,Bˆ . Utilizing the Seiberg-Witten
s ij ij
map, one can derive a non-commutative version of the DBI(cid:16)action. The(cid:17)refore, the Seiberg-Witten map
unifies the non-commutative gauge theories and commutative gauge theories.
In this work, we present solid evidences to show that the Seiberg-Witten map can be described
by a pure coordinate transformation of O(D,D) group. The relationship between non-commutative
property and T-duality was first proposed by Connes, Douglas and Schwarz [2], where they showed
that the Matrix model on tori with an anti-symmetric field produces non-commutativity in a natural
way. Non-commutativities from open string ending on D-branes are first addressed in [3]. In [4] ,
Maharana and Pal made a nice try to construct a relationship between O(D,D) transformations and
the Seiberg-Witten map. They introduced T-dual coordinates in an ad hoc way and found some links
between O(D,D) and the Seiberg-Witten map. But since their setup is not a consistent O(D,D)
covariant theory, the underlying connection of O(D,D) and the Seiberg-Witten map was not revealed.
2
The O(D,D) symmetry is a continuous symmetry for non-compact background, where D is a
number of spacetime dimensions. If we compactify d = D −n dimensions, the continuous O(D,D)
breaks into an O(n,n)×O(d,d;Z) group. The O(d,d;Z) group is known as T-duality group in the
compactified background. Moreover, it is well known that the solution space of the closed string low
energy effective action possesses O(n,n) symmetry for certain backgrounds. These features makes it
tempting to construct theories having O(D,D) invariance at the very beginning. The double sigma
model was proposed by Tseytlin [5] and developed in [6, 7] to fulfill this purpose. Recent developments
of double field theory [8, 9, 10, 11] have the similar inspirations. In the double sigma action, the
O(D,D) invariance is manifested by introducing another set of target space coordinates. In the same
pattern of Polyakov action, openstringandclosedstring sharethesamedouble sigmaaction. Whether
the theory represents open or closed string is determined by the boundary conditions. In this paper,
we will address the open string scenario. Surprisingly, after calculating the propagators, we find that
the double sigma model intrinsically has both commutativity and non-commutativity. Starting from
the simplest phase configuration, there are two commutative theories, either of which can represent
our ordinary picture. However, the hidden mixed propagators between the dual theories exhibit non-
commutative property, which implies that the ordinary coordinate X is non-commutative to its dual
X˜. With this property, it is not hard to see that after an O(D,D) rotation, a non-commutative theory
emerges. Our calculation shows that this non-commutative theory is completely identical to that in
[1], [12, 13]. Itturns out ingeneral phase frame, besidesthe non-commutative theory, we have another
commutative companion even ifthenon-commutative parameter isnonvanishing. The Neven-Schwartz
B field is simply a group parameter in this theory. Another very interesting observation is that, unlike
the situation in the Polyakov action, the physical interpretations of the metric and other parameters in
the double sigma action are completely dictated by the boundary conditions. The parameters between
closed and open strings are linked by an O(D,D) transformation. With all these results in mind,
we conclude that the double sigma action unifies both commutativity and non-commutativity and
Seiberg-Witten map can bedescribed by a subset of O(D,D) group. It is quite inspiring to notice that
O(D,D)group notonlyincludes dualtheories of smallandlarge tori, butalsounifies thecommutative
and non-commutative theories. The unification of the commutative and non-commutative theories is
realized by the intrinsic non-commutative property of the action, observation of commutativity or
non-commutativity depends on which frame is taken.
It is remarkable that after performing another dual rotation with a group parameter C, we find
that the description parameter Φ in the Seiberg-Witten map is not necessary any more and can be
removed. For fixed closed string parameters, the corresponding open string parameters are not unique.
The arbitrariness of the non-commutative parameter θ is accounted by the the alterable open string
metric g, which in turn derived from the arbitrariness of the group parameter C. Therefore, C replaces
the role of Φ to make the non-commutative theory is optional for different metric and * product
parameter θ. Another feature of the composite rotation is that both the ordinary theory and its dual
become non-commutative.
Thereminderofthispaperisoutlinedasfollows. Insection2,webrieflyreviewthenon-commutativity
3
in string theory and the Seiberg-Witten map. By using the double sigma model, we derive the non-
commutative theory from a commutative one through O(D,D) coordinate transformations in section
3. We address the general description under a composite rotation in section 4.
2 Non-commutativity in string theory and the Seiberg-Witten map
This short review is based on Seiberg and Witten’s work [1], and Mukhi’s talk [14]. We begin with
Euclidean worldsheet action
1
S = g ∂ Xi∂αXj −iǫαβB ∂ Xi∂ Xj , (2.1)
ij α ij α β
2ˆ
Σ
(cid:16) (cid:17)
where ǫ is an anti-symmetric matrix, g is the spacetime metric. The worldsheet equation of motion is
given by
g ∂ ∂αXj = 0. (2.2)
ij α
The open string boundary condition is
g ∂ Xj +iB ∂ Xi = 0. (2.3)
ij σ ij τ ∂Σ
To calculate the correlation function, we map the disc to(cid:12)the upper half z plane by z = τ +iσ. The
(cid:12)
propagator is
Xi(z,z¯)Xj z′,z¯′ = −α′ gijlog z−z′ −gijlog z−z¯′
′
(cid:10) (cid:0) (cid:1)(cid:11) +Gi(cid:2)jlog z−(cid:12) z¯′ 2+(cid:12) 1 θij(cid:12)log z−(cid:12)z¯ +Dij , (2.4)
(cid:12) (cid:12) 2πα′ (cid:12) z¯−(cid:12)z′
(cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
where
ij
1 1
G = g−Bg−1B , θij = − B , (2.5)
ij ij g−B g+B
(cid:18) (cid:19)
(cid:0) (cid:1)
and can be grouped as
1 1
+θ = . (2.6)
G g+B
Now, consider the propagator on the boundary and one gets
i
Xi(τ)Xj τ′ = −α′ 2Gijlog τ −τ′ + θijε τ −τ′ , (2.7)
2
(cid:20) (cid:21)
where ε(τ −τ′) = +1(cid:10)when τ −(cid:0)τ′ (cid:1)>(cid:11)0, and ε(τ −τ′)(cid:12)(cid:12)= −1 w(cid:12)(cid:12) hen τ −(cid:0)τ′ < 0.(cid:1)Then, Gij is effectively
identified as the open string metric. The antisymmetric quantity θij introduces non-commutativity
4
into the theory. To see the map between commutative and non-commutative gauge theories, we turn
to the DBI action
1
SDBI = dDx det(g+B+F), (2.8)
g ˆ
s
p
where g is the closed string coupling and F = ∂ A −∂ A is the gauge field strength. For simplicity,
s ij i j j i
the field strength F is assumed to be constant. Expanding this action in order of F, one gets the
Maxwell equation where S ∼ F Fij. Applying the open-closed string relation between the closed
ij
string side(g,B) and theopen´stringside (G,θ), the DBIaction can berewritten in theterms of (G,θ)
as
1 det(1+θF) 1
SDBI = dDx det G+F . (2.9)
gs ˆ pdet(1+θG)s (cid:18) 1+θF(cid:19)
After redefining p
1
Fˆ ≡ F , G ≡ g det(1+θG). (2.10)
s s
1+θF
the DBI action becomes p
1 1
SDBI = dDx det G+Fˆ . (2.11)
G ˆ
s det 1−θFˆ r (cid:16) (cid:17)
r
(cid:16) (cid:17)
The coupling G can be seen as a new string coupling, and B field disappeared. Since the prefactor
s
det 1−θFˆ canbecanceledby openWilsonlineorunderstoodasanontrivialJacobian factorfrom
r
a local(cid:16)coordin(cid:17)ate transformation eliminating U(1) gauge fields [16] and does not affect our discussion,
the action is simplified as
1
SDBI = dDx det G+Fˆ . (2.12)
G ˆ
s r
(cid:16) (cid:17)
To understand the physical implication of Fˆ, we expand it in order of θ,
Fˆ = F −F θklF +O θ2 . (2.13)
ij ij ik lj
Since F = ∂ A −∂ A , if we redefine a new gauge potential(cid:0)Aˆ(cid:1)as
ij i j j i i
1
Aˆ ≡ A −θkl A ∂ A + A ∂ A +O θ2 . (2.14)
i i k l i k i l
2
(cid:18) (cid:19)
(cid:0) (cid:1)
The Fˆ can be rewritten as
Fˆ = ∂ Aˆ −∂ Aˆ +θkl∂ Aˆ ∂ Aˆ
ij i j j i k i l j
= ∂ Aˆ −∂ Aˆ + Aˆ ,Aˆ , (2.15)
i j j i i j
n o
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where the Poisson bracket Aˆ ,Aˆ can be lifted to a non-commutative commutator as −i Aˆ ,Aˆ
i j i j
∗
through the Moyal-Weyl prnoduct aos follows h i
{f,g} → −i(f ∗g−g∗f), (2.16)
with
iθij ∂ ∂
f(x)∗g(x)= e2 ∂xi∂yjf(x)g(y) . (2.17)
(cid:12)x=y
(cid:12)
Therefore, we have two copies of DBI actions, one is commutative(cid:12)
(cid:12)
1
SDBI = dDx det(g+B+F), (2.18)
g ˆ
s
p
the other is non-commutative
1
SDBI = dDx det G+Fˆ . (2.19)
G ˆ
s r
(cid:16) (cid:17)
where Fˆ = ∂ Aˆ − ∂ Aˆ − i Aˆ ,Aˆ . The relations between these two DBI actions, or relations
ij i j j i i j
∗
between commutative gauge thheories aind non-commutative gauge theories are
1
Fˆ = F ,
1+θF
1
Aˆ = A −θkl A ∂ A + A ∂ A +O θ2 . (2.20)
i i k l i k i l
2
(cid:18) (cid:19)
(cid:0) (cid:1)
3 O(D,D) rotations of the propagators
To understand how the non-commutativity arises from O(D,D) symmetry, we start from the simplest
phase configuration of the double sigma model. The action with Lorentz signature is given as follows
1
S = −4πα′ ˆ −∂1XMhMN∂1XN +∂1XMηMN∂0XN , (3.21)
Σ
(cid:0) (cid:1)
where ∂0 = ∂τ, ∂1 = ∂σ and
g 0 0 1 Xi
h = ij , η = , XM = . (3.22)
MN 0 gij MN 1 0 X˜
i
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Inthedoublesigmaaction,theD dimensionaltargetspacecoordinate Xi isdoubledto2D dimensional
XM Xi,X˜ , where M = 1,2,...,2D, while the worldsheet Σ is still two dimentional. Therefore,
i
hMN(cid:16)and ηM(cid:17)N are 2D×2D matrices. gij is the spacetime metric. It is important that this action has
an O(D,D) symmetry: invariant under rotation Ω satisfying ΩηΩT = η. Here for later convenience,
we use XM = Xi,X˜ in the action, different from the conventions in [5] . In this paper, we assume
i
the open string(cid:16)ending(cid:17)on spacefilling branes. The case for Dp−branes can be easily generalized. It is
worth noting that open and closed strings share the same action. The differences between open and
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closed strings are determined by the boundary conditions. In recent works, the double field theory
action of closed strings has been derived from this action [11] . The equation of motion (EOM) is
obtained by varying XM
∂1 hMN∂1XN −ηMN∂0XN = 0, (3.23)
which leads to (cid:0) (cid:1)
g∂1X −∂0X˜ = f1(τ), (3.24)
g−1∂1X˜ −∂0X = f2(τ), (3.25)
where f1(τ) and f2(τ) are arbitrary regular functions solely depending on τ. It turns out that after
imposing the self-duality condition f = 0, the action (3.21) recedes back to the Polyakov action [5].
i
This self-duality condition corresponds to the strong constraint in the framework of double field theory
[15]. Butwewillmomentarily keepitunfixedandshowthat, foropenstrings,theself-duality condition
is a derived consequence under O(D,D) covariant boundary condition but not a premise. The EOM
(3.23) can be put into decoupled form
2 2
(∂1 −∂0 )X = ∂0f2(τ), (3.26)
(∂12−∂02)X˜ = ∂0f1(τ). (3.27)
The boundary conditions of open strings are determined by the EOM
1
δXM hMN∂1XN − ηMN∂0XN = 0, (3.28)
2
(cid:18) (cid:19)(cid:12)∂Σ
(cid:12)
which can be expanded as (cid:12)
(cid:12)
1 1
δX g∂1X − ∂0X˜ +δX˜ g−1∂1X˜ − ∂0X = 0, (3.29)
2 2
(cid:18) (cid:19) (cid:18) (cid:19)(cid:12)∂Σ
(cid:12)
It looks like we have four options for the boundary conditions. However(cid:12), O(D,D) covariance excludes
(cid:12)
two of them and we are only left with two equally good boundary conditions
δX|∂Σ = ∂0X|∂Σ = 0,
1
g−1∂1X˜ − ∂0X = 0⇒ ∂1X˜|∂Σ = 0. (3.30)
2
(cid:12)∂Σ
(cid:12)
or
(cid:12)
(cid:12)
δX˜ = ∂0X˜ = 0,
∂Σ ∂Σ
1 (cid:12) (cid:12)
g∂1X − ∂0X˜(cid:12)(cid:12) = 0 ⇒(cid:12)(cid:12)∂1X|∂Σ = 0. (3.31)
2
(cid:12)∂Σ
(cid:12)
(cid:12)
(cid:12) 7
Apparently, thesetwosetsofboundaryconditionsagreewithT-dualityofopenstrings: ifoneboundary
conditionisNeumann, itsT-dualboundaryconditionisDirichlet. Sinceweeventually wanttocompare
with the results in [1], hereafter we will select the second boundary condition (3.31). After applying
(3.24)ontheboundary,onegetsf1(τ) = 0. Ontheotherhand,wecanmakeashiftX → X− dτf2(τ)
´
which does not affect the boundary conditions, EOM and the action. We are therefore free to set
f2(τ) = 0. The decoupled EOM now can be casted into
2 2
(∂1 −∂0 )X = 0,
∂1X|∂Σ = 0, (3.32)
and
(∂12−∂02)X˜ = 0,
∂0X˜ = 0. (3.33)
∂Σ
(cid:12)
From now on, we work with Euclidean signature (cid:12)(cid:12)by setting τ → −iτ (∂0 → i∂0), and use complex
coordinates: ∂0 = ∂+∂¯ and ∂1 = i ∂ −∂¯ . It is easy to figure out the propagators
(cid:0) (cid:1)
Xi(z,z¯)Xj z′,z¯′ = −α′ gijlog z−z′ +gijlog z−z¯′ . (3.34)
(cid:10) (cid:0) (cid:1)(cid:11) (cid:0) (cid:12) (cid:12) (cid:12) (cid:12)(cid:1)
(cid:12) (cid:12) (cid:12) (cid:12)
X˜ (z,z¯)X˜ z′,z¯′ = −α′ g log z−z′ −g log z−z¯′ . (3.35)
i j ij ij
From these two propDagators, X an(cid:0)d X˜ (cid:1)aEre both c(cid:0)ommut(cid:12)(cid:12)ative. (cid:12)(cid:12)Our next s(cid:12)(cid:12)tep is(cid:12)(cid:12)t(cid:1)o calculate the mixed
propagators X˜ (z,z¯)Xj(z′,z¯′) and Xi(z,z¯)X˜ (z′,z¯′) . From eqn. (3.24) and (3.25) with f = 0,
i j i
we have D E D E
gij∂1Xj = i∂0X˜i, gij∂1X˜j = i∂0Xi. (3.36)
Therefore, the propagators satisfy the following equations
g ∂−∂¯ Xi(z,z¯)X˜ z′,z¯′ = ∂ +∂¯ X˜ (z,z¯)X˜ z′,z¯′ , (3.37)
ℓi j ℓ j
gℓi(cid:0)∂−∂¯(cid:1)DX˜ (z,z¯)Xj(cid:0)z′,z¯′(cid:1)E = (cid:0)∂ +∂¯(cid:1)DXℓ(z,z¯)Xj(cid:0)z′,z¯′(cid:1)E , (3.38)
i
D E D E
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
The solutions of these two equations are
′ ′ ′
α z−z z−z¯
Xi(z,z¯)X˜ z′,z¯′ = − gikg log −log , (3.39)
j 2 kj z¯−z¯′ z¯−z′
(cid:18) (cid:19)
D (cid:0) (cid:1)E α′ z−z′ z−z¯′
X˜ (z,z¯)Xj z′,z¯′ = − g gkj log +log . (3.40)
i 2 ik z¯−z¯′ z¯−z′
(cid:18) (cid:19)
D E
(cid:0) (cid:1)
8
It is easy to check that these propagators also satisfy the boundary conditions. Considering the
′ ′
propagators on the boundary z = τ and z = τ , we can get the commutators
[Xi(τ),Xj(τ′)] = [X˜i(τ),X˜j(τ′)] = 0,
[X˜ (τ),Xj(τ)] = i2πα′δ j. (3.41)
i i
Therefore, we find that X or X˜ alone is commutative, but they do not commute with each other. This
striking feature comes from the requirements that the double sigma model is O(D,D) invariant and
can reduce back to the Polyakov action. However, this non-commutativity is hidden from the ordinary
picture. SincetheactionisO(D,D)invariant, wenowgotoageneralphaseframebyapurecoordinate
transformation
1 −Bij
Ω = , (3.42)
0 1
(cid:18) (cid:19)
where Bij is an antisymmetric tensor. The generalized metric h is then rotated to
MN
g −gB−1
HMN = ΩThMNΩ = B−1g g−1−B−1gB−1 , (3.43)
(cid:18) (cid:19)
accompanied by the coordinate transformation
Xi′ = Xi+BijX˜ ,
j
X˜′ = X˜ . (3.44)
j j
It is straightforward to calculate the propagators in the new frame
Xi′(z,z¯)Xj′ z′,z¯′
= (cid:10)Xi(z,z¯)+B(cid:0)ikX˜ ((cid:1)z(cid:11),z¯),Xj z′,z¯′ +BjℓX˜ z′,z¯′
k l
= DXi(z,z¯),Xj z′,z¯′ − Xi((cid:0)z,z¯),(cid:1)X˜ z′,z¯′ (cid:0) Bℓj(cid:1)E
ℓ
+(cid:10) Bik X˜ (z,z¯(cid:0)),Xj(cid:1)(cid:11)z′,z¯D′ −Bik X˜(cid:0)(z,z¯)(cid:1),EX˜ z′,z¯′ Blj
k k ℓ
= −α′ Dgij −Bikg B(cid:0)ℓj log(cid:1)Ez−z′ +D gij +Bikg (cid:0)Bℓj (cid:1)lEog z−z¯′ +Bij log z−z¯′ .
kℓ kℓ z¯−z′
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:16) (cid:17) (cid:12) (cid:12) (cid:16) (cid:17) (cid:12) (cid:12) (3.45)
(cid:12) (cid:12) (cid:12) (cid:12)
We see that the non-commutativity becomes visible in the last term. It is of importance to note that
this propagator natively uses the open string metric and parameters. That is why it looks different
from the propagator in [1] where the propagator was explicitly expressed with the closed string metric
and parameters. Therefore, in order to make comparison, we need to rotate the parameters back to
the closed string parameters. Rotating H back to the closed string parameters gˆ and Bˆ is achieved
MN
by the transformation
9
gˆ−1 −gˆ−1Bˆ gˆ−Bˆgˆ−1Bˆ Bˆgˆ−1 g −gB−1
η η = = . (3.46)
Bˆgˆ−1 gˆ−Bˆgˆ−1Bˆ −gˆ−1Bˆ gˆ−1 B−1g g−1−B−1gB−1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
It gives us the following relations
g = gˆ−Bˆgˆ−1Bˆ ,
ij
ij
(cid:16) (cid:17) ij
1 1
Bij = − Bˆ ,
gˆ+Bˆ gˆ−Bˆ
(cid:18) (cid:19)
gˆij = g−1−B−1gB−1 ij
Bˆij = (cid:0)B−1−g−1Bg−1(cid:1)ij (3.47)
(cid:0) (cid:1)
whicharepreciselythetransformationsbetweenclosedandopenparameters. Therefore,thepropagator
(3.45) can be rewritten with the closed string (hat) parameters as
Xi′(z,z¯)Xj′ z′,z¯′ = −α′ gˆijlog z−z′ −gˆijlog z−z¯′ (3.48)
(cid:20)
(cid:10) (cid:0) (cid:1)(cid:11) (cid:12) (cid:12) ij (cid:12) (cid:12) ij ′
+ 1 gˆ(cid:12) 1 (cid:12) log z−(cid:12) z¯′ 2+(cid:12) − 1 Bˆ 1 log z−z¯ ,
gˆ+Bˆ gˆ−Bˆ gˆ+Bˆ gˆ−Bˆ z¯−z′ #
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
which is completely identical to the Seiberg-Witten result in [1]. It is of help to give the other three
propagators with the closed string (hat) parameters
X˜′(z,z¯)X˜′ z′,z¯′ = −α′ gˆ−Bˆgˆ−1Bˆ log z−z′ − gˆ−Bˆgˆ−1Bˆ log z−z¯′ ,(3.49)
i j
ij ij
(cid:20) (cid:21)
D (cid:0) (cid:1)E α′ (cid:16) z−z′(cid:17) (cid:12)z−z¯′ (cid:12) (cid:16) (cid:17) (cid:12) (cid:12)
Xi′(z,z¯)X˜′ z′,z¯′ = − δi log −log(cid:12) (cid:12) (cid:12) (cid:12)
j 2 j z¯−z¯′ z¯−z′
(cid:18) (cid:19)
D E
(cid:0) (cid:1) i
1 1
−α′ Bˆ log z−z′ −log z−z¯′ , (3.50)
gˆ+Bˆ gˆ−Bˆ
(cid:18) (cid:19)j
′ ′ (cid:0) (cid:12) ′ (cid:12) (cid:12) (cid:12)(cid:1)
X˜′(z,z¯)Xj′ z′,z¯′ = −α δj log z−z +log z−(cid:12)z¯ (cid:12) (cid:12) (cid:12)
i 2 i z¯−z¯′ z¯−z′
(cid:18) (cid:19)
D E
(cid:0) (cid:1) j
1 1
+α′ Bˆ log z−z′ −log z−z¯′ . (3.51)
gˆ+Bˆ gˆ−Bˆ
(cid:18) (cid:19)i
(cid:0) (cid:12) (cid:12) (cid:12) (cid:12)(cid:1)
(cid:12) (cid:12) (cid:12) (cid:12)
We can easily see the commutators
ij
1 1
[Xi′(τ),Xj′(τ)] = iπα′ − Bˆ
gˆ+Bˆ gˆ−Bˆ
(cid:18) (cid:19)
[X˜′(τ),X˜′(τ)] = 0
i j
[X˜′(τ),Xj′(τ)] = i2πα′δ j.
i i
10