T-Duality and Conformal Invariance at Two Loops

21 Pages·0.15 MB·English

by  S. Parsons

Liverpool Preprint LTH470
hep-th/9912105
Revised Version
T-Duality and Conformal Invariance at Two Loops
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2 S. Parsons

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a Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK
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v Abstract
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1 We show that the conformal invariance conditions for a general σ-model with
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9 torsion are invariant under T-duality through two loops.
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- I. INTRODUCTION
p
e
h
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v Dualityinvarianceisatremendously powerfulconceptinstringtheory. Oneoftheearliest
i
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formsofdualitytoberecognisedwasthatnowknownasT-duality[1]. This actstotransform
r
a
the background fields of a σ-model so as to map one conformally-invariant background (or
string vacuum) into another conformally-invariant background, at least at lowest order. In
fact the σ-model and its dual should be equivalent, again at least at lowest order [2] [3].
The duality can be understood as a consequence of an isometry of the theory; upon gauging
the isometry, by performing the path integral over the gauge field and the path integral
over a lagrange multiplier in different orders, one obtains two equivalent descriptions of the
theory with backgrounds related by the duality. This duality is straightforwardly checked
at lowest order in α; conformal invariance requires the vanishing of “B”-functions, one for
′
each background field (the metric g, the antisymmetric tensor b and the dilaton φ), which
1
are related to the β-functions for the corresponding background fields in a way which we will
describe later. One can check that the duality transformations on backgrounds for which
the B-functions vanish lead to dual backgrounds which also have vanishing B-functions. In
fact, this property is equivalent to requiring that the B-functions are form-invariant under
duality, and hence satisfy certain consistency conditions which were derived in Ref. [4] [5]
(see also Ref. [6]). However, it is far less clear whether, and if so how, the T-duality is
maintained at higher orders in α. In Ref. [7] for a restricted background, and in Ref. [8]
′
for the general case, it has been found that the two-loop string effective action can be made
duality invariant by a redefinition of the background fields (see also Ref. [9]). Now the B-
functions are related to the string effective action, and so this is grounds for hoping that
the B-functions will also be invariant. However, this is not a fait accompli, as was explained
in Refs. [4] [10]. Certainly the invariance of the one-loop effective action guarantees the
invariance of the one-loop B-functions. On the other hand, the relation between the action
and the B-functions is more complicated at higher orders. Moreover, once one has decided
that the fields need to be redefined at higher orders to maintain duality invariance, then
the B-functions will also be modified in a non-trivial way, because field redefinitions lead
to changes in the β-functions. For these reasons it seems desirable to investigate explicitly
whether the field redefinitions which make the action duality-invariant also lead to duality-
invariant B-functions. This verification was carried out in Ref. [4] for the restricted case of
Ref. [7]. Here we shall carry out the analysis for a different, complementary case which we
believe displays most of the features of the full general situation.
II. DUALITY AT FIRST ORDER
The two-dimensional non-linear σ-model is defined by the action
2
1
S = d2σ √γg (X)∂ Xµ∂ XνγAB +ib (X)∂ Xµ∂ XνǫAB +√γ R(2)φ(X) ,
µν A B µν A B
4πα
′ Z h i
(2.1)
whereg isthemetric, b istheantisymmetrictensorfieldoftenreferredtoastorsion,andφ
µν µν
is the dilaton. The indices µ,ν run over 1,...D+1. γ is the metric on the two-dimensional
AB
worldsheet,withA,B =1,2. γ =detγ ,ǫ isthetwo-dimensionalalternatingsymboland
AB AB
R(2) the worldsheet Ricci scalar. Note that b is only defined up to a gauge transformation
µν
b b + ζ . Conformal invariance of the σ-model requires the vanishing of the Weyl
µν µν [µ ν]
7→ ∇
anomaly coefficients Bg , Bb and Bφ (we will refer to these as the B-functions), which are
defined as follows [11]:
Bg = βg +2α φ+ S , (2.2)
µν µν ′∇µ∇ν ∇(µ ν)
1
Bb = βb +αHρ φ+ Hρ S , (2.3)
µν µν ′ µν∇ρ 2 µν ρ
1
Bφ = βφ +α ρφ φ+ ρφS . (2.4)
′ ρ ρ
∇ ∇ 2∇
Here βg , βb and βφ are the renormalisation group β-functions for the σ-model, and H is
µνρ
the field strength tensor for b , defined by H = 3 b = b + cyclic. The vector
µν µνρ [µ νρ] µ νρ
∇ ∇
Sµ arises in the process of defining the trace of the energy-momentum tensor as a finite
composite operator, and can be computed perturbatively. It will be sufficient to assume
Sµ = (S0,0). At one loop we have
1
βg(1) = R H H ρσ, (2.5)
µν µν − 4 µρσ ν
1
βb(1) = Hρ , (2.6)
µν −2∇ρ µν
1 1
βφ(1) = 2φ H2, (2.7)
−2 − 24
3
where H2 = H Hµνρ. (Here and henceforth we set α = 1.)
µνρ ′
The conformal invariance conditions are equivalent to the equations of motion of a string
effective action
Γ = dD+1X√gL(λ ) (2.8)
M
Z
where λ (g ,b ,φ). More explicitly, we have to leading order
M µν µν
≡
1
Γ(1) = dD+1X√ge 2φ R(g)+4( φ)2 H Hµνρ . (2.9)
− µνρ
∇ − 12
Z (cid:20) (cid:21)
From this action it can be shown that
∂Γ(1)
= 2√ge 2φK(0) B(1) (2.10)
∂λ − MN N
M
where
gρgσ 0 g
µ ν − µν
 
K(0) 0 gρgσ 0 (2.11)
MN ≡  − µ ν 
 
 
 0 0 8 
 
 
 
and we have a basis for B such that B = (Bg ,Bb ,B˜φ) and
M M µν µν
1
B˜φ = Bφ gµνBg . (2.12)
− 4 µν
We now consider the dual σ-model. This involves introducing an abelian isometry in the
target space background of the model such that one can perform duality transformations.
Background fields will now be locally independent of the coordinate θ X0 and locally
≡
dependent on Xi ,i = 1,...,D. The σ-model action now reads as
1
S = d2σ √γγAB(g (Xk)∂ θ∂ θ +2g (Xk)∂ θ∂ Xi +g (Xk)∂ Xi∂ Xj)
4πα 00 A B 0i A B ij A B
Z ′ h
+√γR(2)φ(Xk))+iǫAB(2b (Xk)∂ θ∂ Xi +b (Xk)∂ Xi∂ Xj) (2.13)
oi A B ij A B
i
The classical T-duality transformations act on the background fields g ,b to give dual
µν µν
{ }
background fields g˜ ,˜b given by [1]:
µν µν
{ }
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1
g˜ = ,
00 g
00
b g
g˜ = 0i, ˜b = 0i,
0i g 0i g
00 00
g g b b
g˜ = g 0i 0j − 0i 0j, (2.14)
ij ij
− g
00
g b b g
˜b = b 0i 0j − 0i 0j.
ij ij
− g
00
The origin of the the two distinct models lies in the order in which one performs simple parts
of the path integral. We shall deal with the transformation on the dilaton later.
We now parametrize the metric and torsion tensors in terms of reduced fields which
appear in the Kaluza-Klein reduction to D-dimensions [12] [4] [8].
g g a av
00 0j j
gµν =   =  , (2.15)
g g av g¯ +av v
 i0 ij   i ij i j 
   
   
b b 0 w
00 0j j
bµν =   =  . (2.16)
b b w b
 i0 ij   − i ij 
   
   
This choice simplifies the form of the classical transformations to
1
a , v w ,
i i
7→ a ↔
b ˜b = b +w v w v . (2.17)
ij ij ij i j j i
7→ −
Note that g¯ is unchanged under duality. It has been shown that a further simplification
ij
in the context of conformal invariance conditions can be employed, since the transformation
properties of the one loop B-functions are manifest when mapped to tangent space [4]. That
is, we construct a vielbein, e µ, such that
a
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δ = e µe νg . (2.18)
ab a b µν
In fact we can choose a block diagonal form
e 0 e i 1 0
e µ =  ˆ0 ˆ0  =  √a . (2.19)
a
e 0 e i v e¯ i
 α α   − α α 
   
   
with v e¯ iv . We can now define the tangent space anomaly coefficients as:
α ≡ α i
Bg = e µe νBg , Bb = e µe νBb , (2.20)
ab a b µν ab a b µν
Bg˜ = e µe νBg˜ , B˜b = e µe νB˜b . (2.21)
ab a b µν ab a b µν
These coefficients transform as
Bg˜ = Bg ,
ˆ0ˆ0 − ˆ0ˆ0
Bg˜ = Bb , B˜b = Bg , (2.22)
ˆ0α ˆ0α ˆ0α ˆ0α
Bg˜ = Bg , B˜b = Bb .
αβ αβ αβ αβ
With the exception of Bb , we find that we can express each tangent space B-function in
αβ
terms of a B-function for one reduced field, up to factors of g .
00
1
Bg = e 0e 0Bg = Ba
ˆ0ˆ0 ˆ0 ˆ0 00 a
1
= βa +ak¯ φ, (2.23)
k
a ∇
Bg = e 0e¯ iaBv = e¯ i√aBv
ˆ0α ˆ0 α i α i
1
= e¯ i√a βv F k¯ φ+ ¯ S0, (2.24)
α i − i ∇k 2∇i
(cid:20) (cid:21)
1
Bb = e 0e¯ iBw = e¯ i Bw
ˆ0α ˆ0 α i α √a i
1
= e¯ i βw G k¯ φ , (2.25)
α √a i − i ∇k
h i
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Bg = e¯ ie¯ jBg¯
αβ α β ij
= e¯ ie¯ j βg¯ +2¯ ¯ φ , (2.26)
α β ij ∇i∇j
h i
Bb = e¯ ie¯ jBb⋆
αβ α β ij
1
= e¯ ie¯ j βb⋆ +Hˆk ¯ φ G S0 . (2.27)
α β ij ij∇k − 2 ij
(cid:20) (cid:21)
We define
Bb⋆ = Bb v Bw +v Bw. (2.28)
ij ij − i j j i
The one loop B-functions for the reduced tensors are
a 1
Ba(1) = ¯ ai +aai¯ Φ+ (a2F Fkm G Gkm), (2.29)
i i km km
−2∇ ∇ 4 −
1 1 1 1
Bv(1) = F k(¯ Φ a )+ ¯kF + Hˆ Gkm + ¯ S0, (2.30)
i − i ∇k − 2 k 2∇ ik 4a ikm 2∇i
1 1 a
Bw(1) = G k(¯ Φ+ a )+ ¯kG + Hˆ Fkm, (2.31)
i − i ∇k 2 k 2∇ ik 4 ikm
1 1 1 1
Bg¯(1) = R¯ a a (aF F k + G G k) Hˆ Hˆ km +2¯ ¯ Φ, (2.32)
ij ij − 4 i j − 2 ik j a ik j − 4 ikm j ∇i∇j
1 1
Bb⋆(1) = ¯kHˆ +Hˆ ¯kΦ G S0, (2.33)
ij −2∇ kij kij∇ − 2 ij
where a = ∂ lna, F = 2¯ v , G = 2¯ w = H , and Hˆ = H + 3v G . All
i i ij [i j] ij [i j] 0ij ijk ijk [i jk]
∇ ∇ −
barred tensors and covariant derivatives only have dependence on g¯ , i.e. that part of metric
ij
that is invariant under the classical T-duality transformations. Under the mapping (2.17)
we have a a , F G and Hˆ invariant. Φ is the reduced dilaton defined by
i i ij ij ijk
7→ − ↔
1
Φ = φ lna. (2.34)
− 4
7
Φ is another invariant under the duality transformation and this is seen to be case if one
combines the shift in φ needed to keep the one loop action invariant [8] with the transforma-
tion on a. The corresponding B-function can be calculated, though it need not be worked
out in full detail. (2.12) simplifies to
1
B˜φ = BΦ Bg . (2.35)
− 4 αα
This is manifestly invariant and indeed this fact is closely related to the invariance of the
reduced one loop string effective action which can be similarly expressed [8]
1
Γ(1) = ddX√g¯e 2Φ R¯(g¯)+4¯iΦ¯ Φ aia
R − ∇ ∇i − 4 i
Z (cid:20)
1 1 1
aF Fij G Gij Hˆ Hˆijk . (2.36)
ij ij ijk
− 4 − 4a − 12
(cid:21)
From a reduced effective action one would expect the equations of motion for the reduced
tensors to reproduce the appropriate conformal invariance conditions. In fact, we have the
following (see (A4)—(A7)):
∂Γ ∂Γ
e µe ν R = 2a R, (2.37)
ˆ0 ˆ0 ∂gµν(cid:12) − ∂a
(cid:12)bµν
(cid:12)
(cid:12)
(cid:12)
∂Γ ∂Γ
e µe ν R = e¯ ie¯ j R, (2.38)
α β ∂gµν(cid:12) α β ∂g¯ij
(cid:12)bµν
(cid:12)
(cid:12)
(cid:12)
∂Γ ∂Γ
e µe ν R = e¯ ie 0 R, (2.39)
ˆ0 α ∂gµν(cid:12) − α ˆ0 ∂vi
(cid:12)bµν
(cid:12)
(cid:12)
(cid:12)
∂Γ ∂Γ
e µe ν R = e¯ ie 0a R, (2.40)
ˆ0 α ∂bµν(cid:12) α ˆ0 ∂wi
(cid:12)gµν
(cid:12)
(cid:12)
(cid:12)
∂Γ ∂Γ
e µe ν R = e¯ ie¯ j R. (2.41)
α β ∂bµν(cid:12) α β ∂bij
(cid:12)gµν
(cid:12)
The validity of this set of equations at lea(cid:12)ding order is clear upon inspection of (2.29)—(2.33)
(cid:12)
and (2.36), together with (2.2)—(2.7) and (2.9).From now on we drop the R subscript, since
all quantities will be assumed to be reduced.
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III. DUALITY AT SECOND ORDER
We shall now proceed to illustrate the transformation properties of the next to leading
order conformal invariance conditions. We would like to show that the two-loop conformal
invariance conditions behave under duality as in (2.22), but we shall see that we have to
redefine the fields in order to achieve this. There are two ways in which we might proceed.
The more direct approach is to calculate the B-functions explicitly for the fields we are
considering. These expressions are very complicated, so we abstain from this. The indirect
method is to consider the corresponding result to (2.10) at second order
∂Γ(2)
= 2√ge 2φ K(0) B(2) +K(1) B(1) , (3.1)
∂λ − MN N MN N
M
h i
where Γ(2) is given in Ref. [13]. K(1) can be read off the following expressions [13],
MN
1 ∂Γ(2)
= 8B˜φ(2) +2Bb(1)Bb(1)ρσ +2Bg(1)Bg(1)ρσ 2(B˜φ(1))2, (3.2)
√ge 2φ ∂φ ρσ ρσ −
−
1 1 ∂Γ(2)
= Bg(2) Bg(1)Bg(1)ρ +R ρσ Bg(1) +B˜φ(1)Bg(1)
2√ge 2φ ∂gµν µν − µρ ν µ ν ρσ µν
−
3 1
Bb(1)Bb(1)ρ + H Hτ Bg(1)ρσ + H ρσBb(1)
− µρ ν 4 µρτ νσ 2∇(µ ν) ρσ
+g 4B˜φ(2) Bb(1)Bb(1)ρσ Bg(1)Bg(1)ρσ +(B˜φ(1))2 , (3.3)
µν − − ρσ − ρσ
h i
1 1 ∂Γ(2) 1 1
= Bb(2) +R Bb(1)ρσ H HτρσBb(1) + H H τ Bb(1)ρσ
2√ge 2φ ∂bµν − µν ρµνσ − 4 τµν ρσ 4 µτρ ν σ
−
1
2Bg(1)Bb(1)ρ σHρ Bg(1) B˜φ(1)Bb(1). (3.4)
− [µρ ν] − 2∇ µν ρσ − µν
| |
With these relations, we could show the properties under duality of B(2) if we knew the
M
properties of the second order action and those of K(1) . After all we are well informed to
MN
the behaviour of B(1). However we know already that Γ(2) is not the appropriate action.
In fact in Ref. [8] it was proved that a shift in the reduced fields is required to obtain an
invariant second order action. The shift in the one loop reduced effective action is then
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1 a 1
δΓ(1) = ddX√g¯e 2Φ ai δa δaFijF +FijδF
− i ij ij
−2 ∇ − 2 2a
Z (cid:20) (cid:18) (cid:19)
1 1 1
+ δaGijG GijδG HˆijkδHˆ . (3.5)
ij ij ijk
2a 2a − − 6
(cid:18) (cid:19) (cid:21)
The authors of Ref. [8] find the required leading order corrections to the reduced fields to be
1 1
δa = aa ai + a2FijF + GijG , (3.6)
i ij ij
8 8
1 1
δv = F ka Hˆ Gkm, (3.7)
i −4 i k − 8a ikm
1 1
δw = G ka + aHˆ Fkm, (3.8)
i −4 i k 8 ikm
3
δHˆ = ¯ G mF +3F δw +3G δv . (3.9)
ijk −2∇[i j k]m [ij k] [ij k]
(cid:16) (cid:17)
We now have at our disposal an invariant action which we shall write as
Γ(2) = Γ(2) +δΓ(1). (3.10)
′
However the consequence of (3.6)—(3.9) in (3.1) is not just that we have to replace Γ(2)
with Γ(2) but also that the two-loop β-functions change, leading consequently to a modified
′
K-matrix. The new β-functions are in general given by:
β = β δβ (3.11)
M′ M − M
where
δ d
δβ (λ) = δλ. β (λ) µ δλ . (3.12)
M M M
δλ − dµ
The precise details of how we apply (3.12) will be given in the Appendix, and moreover we
will leave for later the definition of the redefined K-matrix. We now seem to have all the
required apparatus to achieve our task. In the expressions that follow, after all functional
10