International Journal of Geometric Methods in Modern Physics Vol. 10, No. 8 (2013) 1360003 (13 pages) c World Scientific Publishing Company DOI: 10.1142/S0219887813600037 DUALITY IN STRING THEORY YOLANDA LOZANO Department of Physics, University of Oviedo Av. Calvo Sotelo 18, 33007 Oviedo, Spain [email protected]Received 9 November 2012 Accepted 20 January 2013 Published 26 April 2013 Duality symmetries have played a key role in the discovery that the five consistent superstring theories in 10 dimensions emerge as different corners of the moduli space of a single unifying theory, known as M-theory. Focusing on the target space, or T, duality symmetry, we show how it can be formulated in spacetimes with abelian and non-abelian isometries. Finally, we discuss some recent work that realizes non-abelian T-duality as a consistent truncation to seven-dimensional supergravity, thus generalizing to the non-abelian case the realization of abelian T-duality at the supergravity level. Keywords : String theory; dualities; T-duality. 1. Introduction Few words have been used with more different meanings than the word “duality”. Even within the restricted framework of string theories, duality originally meant a symmetry between the s and the t-channels in strong interactions. Somewhat related ideas, also termed “duality”, appeared in the context of conformal field the- ories as simple consequences of locality and associativity of the operator product expansions. In statistical mechanics duality can be understood as a way to show the equivalence between two apparently different theories. On a lattice system described by a Hamiltonian H (g i ) with coupling constants g i the duality transformation pro- duces a new Hamiltonian H ∗ (g ∗ i ) with coupling constants g ∗ i on the dual lattice. In this way one can often relate the strong coupling regime of H (g) with the weak coupling regime of H ∗ (g ∗ ). In string theory duality symmetries have been key in the discovery that what had been once regarded as five distinct superstring theories in 10 dimensions were in fact special points in the moduli space of consistent vacua of a single underlying theory, named M-theory. In this setting two types of duality symmetries play a fundamental role. T-duality is a genuine stringy symmetry which relates physical properties corresponding to big spacetime radius with quantities corresponding to small radius. It maps the two type II theories into each other as well as the two 1360003-1 Int. J. Geom. Methods Mod. Phys. 2013.10. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 09/22/13. For personal use only.
Transcript
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Received 9 November 2012Accepted 20 January 2013Published 26 April 2013
Duality symmetries have played a key role in the discovery that the five consistentsuperstring theories in 10 dimensions emerge as different corners of the moduli spaceof a single unifying theory, known as M-theory. Focusing on the target space, or T,duality symmetry, we show how it can be formulated in spacetimes with abelian andnon-abelian isometries. Finally, we discuss some recent work that realizes non-abelianT-duality as a consistent truncation to seven-dimensional supergravity, thus generalizingto the non-abelian case the realization of abelian T-duality at the supergravity level.
Keywords: String theory; dualities; T-duality.
1. Introduction
Few words have been used with more different meanings than the word “duality”.Even within the restricted framework of string theories, duality originally meanta symmetry between the s and the t-channels in strong interactions. Somewhatrelated ideas, also termed “duality”, appeared in the context of conformal field the-ories as simple consequences of locality and associativity of the operator productexpansions. In statistical mechanics duality can be understood as a way to show theequivalence between two apparently different theories. On a lattice system describedby a Hamiltonian H(gi) with coupling constants gi the duality transformation pro-duces a new Hamiltonian H∗(g∗i ) with coupling constants g∗i on the dual lattice.In this way one can often relate the strong coupling regime of H(g) with the weakcoupling regime of H∗(g∗).
In string theory duality symmetries have been key in the discovery that whathad been once regarded as five distinct superstring theories in 10 dimensions werein fact special points in the moduli space of consistent vacua of a single underlyingtheory, named M-theory. In this setting two types of duality symmetries play afundamental role. T-duality is a genuine stringy symmetry which relates physicalproperties corresponding to big spacetime radius with quantities corresponding tosmall radius. It maps the two type II theories into each other as well as the two
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Y. Lozano
heterotic ones. S-duality is a symmetry relating the strong coupling regime with theweak coupling one, a bold generalization of the original conjecture by Montonenand Olive in field theory. In string theory it is a symmetry of type IIB and relatesthe type I and heterotic SO(32) superstrings.
This lecture will be devoted to T-duality. After reviewing its formulation inspacetimes with abelian and non-abelian isometries we will present preliminarywork that explores the realization of non-abelian T-duality at the supergravitylevel, in an attempt to generalize the results in [1] for abelian T-duality to thenon-abelian setting.
T-duality was first described in the context of toroidal compactifications [2–4].For the simplest case of a single compactified dimension of radius R, the entirephysics of the interacting theory is left unchanged under the replacement R →α′/R provided one also transforms the dilaton field φ → φ− log(R/
√α′) [5]. This
simple case was generalized to non-flat conformal backgrounds in [6, 7]. In Buscher’sconstruction one starts with a manifold M with metric gij , antisymmetric tensorbij and dilaton field φ(xi), with an abelian continuous isometry. The final outcomeis that for any continuous isometry of the background which is a symmetry ofthe corresponding nonlinear σ-model action, one obtains the equivalence of twoapparently very different nonlinear σ-models. The transformation is referred to inthe literature as abelian T-duality due to the abelian character of the isometryof the original σ-model. A particularly useful derivation of the dual σ-model is interms of the gauging of the isometry symmetry [8], as we show in Sec. 2. Of morerecent history is the notion of non-abelian duality [9–11], which has no analoguein statistical mechanics. The basic idea of [9], inspired in the treatment of abelianduality presented in [8], is to consider a conformal field theory with a non-abeliansymmetry group G, as we show in Sec. 3.
We refer the interested reader to the review articles [12, 13] for more detailsabout the material contained in Secs. 2–4 and a more complete set of references.Section 5 contains recent results derived in [14].
2. Abelian T-Duality
The formulation of abelian T-duality by Rocek and Verlinde [8] starts with a σ-model defined on a d-dimensional manifold M :
S =1
4πα′
∫d2ξ[
√hhµνgij∂µx
i∂νxj + iεµνbij∂µx
i∂νxj + α′√hR(2)φ(x)], (1)
where gij is the target space metric, bij the torsion and φ the dilaton field, coupledto the two-dimensional scalar curvature in the world-sheet R(2). hµν is the world-sheet metric and α′ the inverse of the string tension. We assume that the σ-modelhas an abelian isometry represented by a translation in a coordinate θ in the targetspace, θ → θ+ε. The coordinates θ, xα, α = 1, . . . , d−1, adapted to the isometry,the metric, torsion and dilaton fields are then θ-independent.
1360003-2
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Duality in String Theory
It is possible to gauge the isometry by introducing some gauge fields Aµ trans-forming as δAµ = −∂µε. With a Lagrange multiplier term the gauge field strengthis required to vanish, forcing the constraint that the gauge field is pure gauge. Aftergauge fixing the original model is then recovered.
Gauging the isometry in (1) and adding the Lagrange multiplier term leads to:
Sd+1 =1
4πα′
∫d2ξ[
√hhµν(g00(∂µθ +Aµ)(∂νθ +Aν) + 2g0α(∂µθ +Aµ)∂νx
α
+ gαβ∂µxα∂νx
β) + iεµν(2b0α(∂µθ +Aµ)∂νxα + bαβ∂µx
α∂νxβ)
+ 2iεµν θ∂µAν + α′√hR(2)φ(x)]. (2)
The dual theory is obtained by integrating the A fields:
Aµ = − 1g00
(g0α∂µx
α + iενµ√h
(b0α∂νxα + ∂ν θ)
), (3)
and fixing θ = 0:
S =1
4πα′
∫d2ξ[
√hhµν(g00∂µθ∂ν θ + 2g0α∂µθ∂νx
α + gαβ∂µxα∂νx
β)
+ iεµν(2b0α∂µθ∂νxα + bαβ∂µx
α∂νxβ) + α′√hR(2)φ(x)], (4)
where
g00 =1g00
, g0α =b0α
g00, b0α =
g0α
g00,
gαβ = gαβ − g0αg0β − b0αb0β
g00, bαβ = bαβ − g0αb0β − g0βb0α
g00.
(5)
These are the celebrated Buscher’s formulae. They show that duality relatesvery different geometries. Moreover, as we will see, it may also lead to differenttopologies. The integration on Aµ produces a factor in the measure det g00, whichregularized by requiring conformal invariance of the dual theory yields the shift ofthe dilaton:
φ = φ− 12
log g00. (6)
For a genus g world-sheet Σg and compact isometry orbits we may have largegauge transformations. We can then consider multivalued gauge functions:∮
γ
dε = 2πn(γ), n(γ) ∈ Z, (7)
where γ is a non-trivial homology cycle in Σg. Since we are dealing with abelianisometries, it suffices to consider only the toroidal case g = 1. The variation ofSd+1 is
δSd+1 =12π
∫(∂θ∂ε− ∂ε∂θ) =
i
4π
∫T
dθ ∧ dε
=i
4π
(∮a
dθ
∮b
dε−∮
a
dε
∮b
dθ
), (8)
1360003-3
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Y. Lozano
where a and b are the two generators of the homology group of the torus T. Sinceε is multivalued by 2πZ, we learn from (8) that θ is multivalued by 4πZ:∮
γ
dθ = 4πm(γ), m(γ) ∈ Z. (9)
For a non-compact isometry δSd+1 = 0 and dθ may in general have real periods.The original theory is recovered integrating the Lagrange multiplier, which appearsin the action in the form of a closed 1-form. In non-trivial world-sheets these 1-formshave exact and harmonic components. The θ-dependence in (2) is
Sθ = − 12π
∫(dθ0 + θh) ∧A. (10)
Integrating by parts in the exact part and using Riemann’s bilinear identity weobtain
Sθ =12π
∫θ0 ∧ dA− 1
2π
(∮a
θh
∮b
A−∮
a
A
∮b
θh
). (11)
Integration on θ0 yields the constraint dA = 0 and integration on the harmoniccomponents leads to ∮
a
A =∮
b
A = 0. (12)
Both constraints imply that A must be an exact 1-form. Fixing the gauge theoriginal theory is recovered. Locally the dual manifold is equivalent to (M/S1)×S1
(for compact isometries), where the quotient means that the gauge is fixed bydividing the orbits of the isometry group. Generically we expect topology changeas a consequence of duality. However, the more delicate issue is whether the dualmanifold M is indeed a product or a twisted product (non-trivial bundle). It is alsouseful to notice that in the previous arguments the structure of π1(M) played norole. This rises some questions concerning the way the operators in both theoriesare mapped under duality. The nature of the product relating the gauged originalmanifold and the Lagrange multipliers space turns out to be dictated by the gaugefixing procedure, in particular by Gribov problems. We can use an example to laborthis point. This is the SU(2) principal chiral model, which represents a σ-model inS3. The dual with respect to a fixed-point free abelian isometry is locally S2 × S1.One knows that this also holds globally when performing the gauge fixing. Thisreveals that the dual manifold is S2 × S1 and not a squeezed S3. The reader isreferred to [15] for more details.
3. Non-Abelian T-Duality
The same procedure a la Rocek and Verlinde was generalized in [9] to constructthe dual with respect to a given non-abelian isometry group G. The gauge fieldstake values in the Lie algebra associated to the isometry group and they transform
1360003-4
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Duality in String Theory
under gauge transformations xm → gmn x
n,m, n = 1, . . . , N , where g ∈ G, as A →g(A+ ∂)g−1. The isometry is gauged by introducing covariant derivativesa:
∂xm → Dxm = ∂xm +Aα(Tα)mn x
n, (13)
where Tα is a N -dimensional representation for the α generator of the Lie algebraof G. The flatness of the gauge fields is imposed by the term:∫
Tr(χF ), (14)
with F = ∂A− ∂A+ [A, A]. The χ-fields take values in the Lie algebra associatedto G and transform in the adjoint representation to preserve gauge invariance.Integration on χ fixes F = 0 in semisimple groups, then A is pure gauge (in sphericalworld-sheets) and after gauge fixing we recover the original model. As before thedual model is obtained integrating on A and then fixing the gauge. We can writethe gauged σ-model action as:
Sgauge =12π
∫d2z
[QmnDx
mDxn +QmµDxm∂xµ +Qµn∂x
µDxn +Qµν∂xµ∂xν
+ Tr(χF ) +12R(2)φ
], (15)
where Q = g + b, Latin indices are associated to coordinates adapted to the non-abelian isometry and Greek indices to inert coordinates. We can write (15) as:
Sgauge = S[x] +12π
∫d2z
[AαfαβA
β + hαAα + hαA
α +12R(2)φ
], (16)
with
hα = (Qmn∂xm +Qµn∂x
µ)(Tα)nq x
q − ∂χαTRηαα,
hα = (Qnµ∂xµ +Qnm∂x
m)(Tα)nq x
q + ∂χαTRηαα,
fαβ = Qmn(Tβ)mr (Tα)n
pxrxp + Cγ
βαχγTRηγγ ,
(17)
where [Tα, Tβ] = CγαβTγ and Tr(TαTβ) = TRηαβ .
Integrating A, A:
S = S[x] +12π
∫d2z
[hα(fαβ)−1hβ +
12R(2)φ
], (18)
where φ is given by: φ = φ− 12 log(det f), after regularizing the factor det f coming
from the measure as in the previous section. In all the examples considered the dualmodel with this dilaton satisfies the conformal invariance conditions to first orderin α′, but a general proof analogous to that in the abelian case is lacking.
The construction above seems to be a straightforward extension of abelian dual-ity. However this is not so. Non-abelian duality is quite different from abelian
aNote that this way of gauging a continuous global isometry is only valid for certain σ-modelsand isometry groups [16].
1360003-5
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Y. Lozano
duality, as it is clearly manifested in the context of statistical mechanics. In thiscontext duality transformations are applied to models defined on a lattice L withphysical variables taking values on some abelian group G. The duality transforma-tion takes us from the triplet (L,G, S[g]), where S[g] is the action depending onsome coupling constants labeled collectively by g to a model (L∗, G∗, S∗[g∗]) onthe dual lattice L∗ with variables taking values on the dual group G∗ and withsome well-defined action S∗[g∗]. For abelian groups, G∗ is the representation ring,itself a group, and when we apply the duality transformation once again we obtainthe original model. As soon as the group is non-abelian the previous constructionbreaks down because the representation ring of G is not a group. In particular thenon-abelian duality transformation cannot be performed again to obtain the modelwe started with. In the context of string theory the major problems in stating non-abelian duality as an exact symmetry come when trying to extend it to non-trivialworld-sheets and when performing the operator mapping (a detailed explanationon this can be found in [15]).
4. The Canonical Approach
The procedure to implement T-duality explained in the previous sections looksunnecessarily complicated. In this section we are going to show that when thebackground admits an abelian isometry, it is possible to formulate it in terms of asimple canonical transformation [17], laying T-duality as a (privileged) subgroup ofthe whole group of (non-anomalous, that is implementable in quantum field theory[18]) canonical transformations on the phase space of the theory. Moreover, we willshow that all standard results in the abelian case (and more) are easily recoveredin this approach. The extension to the non-abelian case is explored in Sec. 4.2.
4.1. The abelian case
We start with a bosonic sigma model written in arbitrary coordinates on a manifoldM with Lagrangian
L =12(gab + bab)(φ)∂+φ
a∂−φb, (19)
where x± = (τ ± σ)/2, a, b = 1, . . . , d = dimM . The corresponding Hamiltonian is
H =12(gab(pa − bacφ
′c)(pb − bbdφ′d) + gabφ
′aφ′b), (20)
where φ′a ≡ dφa/dσ. We assume moreover that there is a Killing vector field ka,Lkgab = 0 and ikH = −dv for some 1-form v, where (ikH)ab ≡ kcHcab and H =db locally. This guarantees the existence of a particular system of coordinates,“adapted coordinates”, which we denote by xi ≡ (θ, xα), such that k = ∂/∂θ. Wedenote the Jacobian matrix by ei
a ≡ ∂xi/∂φa.This defines a point transformation in the original Lagrangian (19) which
acts on the Hamiltonian as a canonical transformation with generating function
1360003-6
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Duality in String Theory
Φ = xi(φ)pi, and yields:
pa = eiapi,
xi = xi(φ).(21)
Once in adapted coordinates we can write the sigma model Lagrangian as
In finding the dual with a canonical transformation we can use the Routh func-tion with respect to θ, i.e. we only apply the Legendre transformation to (θ, θ). Thecanonical momentum is given by
pθ = Gθ + (J+ + J−) (24)
and the Hamiltonian
H = pθθ − L
=12G−1p2
θ −G−1(J+ + J−)pθ +12Gθ′2
+12G−1(J+ + J−)2 + θ′(J+ − J−) − V. (25)
The generator of the canonical transformation we choose is
F =12
∫D,∂D=S1
dθ ∧ dθ =12
∮S1
(θ′θ − θθ′)dσ, (26)
that is
pθ =δF
δθ= −θ′,
pθ = −δFδθ
= −θ′.(27)
This generating functional does not receive any quantum corrections (as explainedin [18]) since it is linear in θ and θ. If θ was not an adapted coordinate to acontinuous isometry, the canonical transformation would generically lead to a non-local form of the dual Hamiltonian. Since the Lagrangian and Hamiltonian in ourcase only depend on the time- and space-derivatives of θ, there are no problemswith non-locality. The transformation (27) in (25) gives:
H =12G−1θ′2 +G−1(J+ + J−)θ′
+12Gp2
θ− (J+ − J−)pθ +
12G−1(J+ + J−)2 − V. (28)
1360003-7
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Y. Lozano
Since
˙θ =δH
δpθ
= Gpθ − (J+ − J−), (29)
we can perform the inverse Legendre transform:
L =12G−1( ˙
θ2 − θ′2) +G−1J+( ˙θ − θ′)
−G−1J−( ˙θ + θ′) + V − 2G−1J+J−. (30)
From this expression we can read the dual metric and torsion and check that theyare given by Buscher’s formulae. The way to get the shift of the dilaton within thecanonical approach is however not completely clear (see [19] for preliminary workin this direction).
The dual manifold M is automatically expressed in coordinates adapted to thedual Killing vector k = ∂/∂θ. We can now perform another point transformation,with the same Jacobian as (21) to express the dual manifold in coordinates whichare as close as possible to the original ones. The transformations we perform arethen: First a point transformation φa → θ, xα, to go to adapted coordinates in theoriginal manifold. Then a canonical transformation θ, xα → θ, xα, which is thetrue duality transformation. And finally another point transformation θ, xα →φa, with the same Jacobian as the first point transformation, to express the dualmanifold in general coordinates.
From the generating functional (26) we can learn about the multivaluedness andperiods of the dual variables [15]. Since θ is periodic and in the path integral thecanonical transformation is implemented by [18]:
ψk[θ(σ)] = N(k)∫
Dθ(σ)eiF [θ,θ(σ)]φk[θ(σ)], (31)
where N(k) is a normalization factor, φk(θ+ a) = φk(θ) implies for θ: θ(σ + 2π)−θ(σ) = 4π/a, which means that θ must live in the dual lattice of θ. Note that (31)suffices to construct the dual Hamiltonian. It is a simple exercise to check thatacting with (28) on the left-hand side of (31) and pushing the dual Hamiltonianthrough the integral, we obtain the original Hamiltonian acting on φk[θ(σ)]:
Hψk[θ(σ)] = N(k)∫
Dθ(σ)eiF [θ,θ(σ)]Hφk[θ(σ)]. (32)
This makes the duality transformation very simple conceptually, and it alsoimplies how it can be applied to arbitrary genus Riemann surfaces, because thestate φk[θ(σ)] could be the state obtained by integrating the original theory on anarbitrary Riemann surface with boundary.
4.2. The non-abelian case
As we have mentioned the conventional gauging approach to non-abelian T-dualityhas two important drawbacks. The first one is that the transformation is in general
1360003-8
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Duality in String Theory
non-invertible (i.e. it is not possible to recover the original theory by repeatingthe gauging procedure starting from the dual) and the second that it is not validfor non-spherical world-sheets. A canonical transformation description would bein this sense very useful, since it is invertible and the generalization to arbitrarygenus Riemann surfaces is straightforward, as we have seen. Such a description is infact known for those sigma models in which the non-abelian isometry acts withoutisotropy, i.e. without fixed points. The most general sigma model of this kind is
S[g, x] =∫dσ+dσ−[Eab(x)(∂+gg
−1)a(∂−gg−1)b + FRaα(x)(∂+gg
−1)a∂−xα
+FLαa(x)∂+x
α(∂−gg−1)a + Fαβ(x)∂+xα∂−xβ ], (33)
where g ∈ G, a Lie group (which we take to be compact), and ∂±gg−1 =(∂±gg−1)aTa with Ta the generators of the corresponding Lie algebra.b This modelis invariant under right transformations g → gh, with h ∈ G. Let us parametrizethe Lie group using the Maurer–Cartan forms Ωa
k, such that
(∂±gg−1)a = Ωak(θ)∂±θk. (34)
The following canonical transformation from θi,Πi to χa, Πa:Πi = −(Ωa
i χ′a + fabcχ
aΩbjΩ
ciθ
′j),
Πa = −Ωai θ
′i,(35)
yields the non-abelian dual of (33) with respect to its isometry g → gh:
S =∫dσ+dσ−[(E + adχ)−1
ab (∂+χa + FL
αa(x)∂+xα)(∂−χb − FR
bβ(x)∂−xβ)
+Fαβ∂+xα∂−xβ ]. (36)
This was first realized in [20] for the case of SU(2) principal chiral models (whereEab = δab, FR
aα = FLαa = Fαβ = 0), generalized in [21, 22] to arbitrary group, and
shown to apply also to this more general case in [23].Equation (35) is generated by:
F [χ, θ] =∮dσ Tr(χ∂σgg
−1), (37)
which is linear in the dual variables but nonlinear in the original ones. This meansthat in general it will receive quantum corrections when implemented at the levelof the Hilbert spaces [18], the reason being that we cannot prove a relation like
|χa〉 =∫
Dθi(σ)eiF [χa,θi(σ)]|θi(σ)〉, (38)
using the eigenfunctions of the respective Hamiltonians. However, it was shownin [20, 21] that such a relation can in fact be proven using the eigenfunctions of the
bTa are normalized such that Tr(TaTb) = δab.
1360003-9
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Y. Lozano
respective conserved currents in the initial and dual theories. Of course for this tobe true we need to have a symmetry in the dual theory, which is not the case forarbitrary backgrounds.
5. Non-Abelian T-Duality as a Consistent Truncationto Seven-Dimensional Supergravity
Our understanding of abelian T-duality was considerably advanced by showinghow to implement the transformation at the level of type II supergravities [1].This allowed to prove that any solution to type IIA (respectively, IIB) supergravitywith non-trivial RR fluxes was mapped under T-duality to a solution of type IIB(respectively, IIA).
Partial understanding of non-abelian T-duality at the supergravity level hasbeen achieved recently in [14]. Taking a general class of SO(4) symmetric back-grounds non-abelian T-duality was examined with respect to an SU(2) subgroup. Aconsistent reduction ansatz to the same underlying seven-dimensional theory wasunearthed, meaning that any solution to these lower-dimensional actions upliftssimultaneously to a solution of both type IIA and type IIB supergravities. Themain difficulty in setting general transformation rules in the non-abelian case lieson the non-trivial transformation of the background fields, which depends moreoveron the symmetry group that is used to generate the T-dual.
As stated, we are interested in spacetimes of the form
ds2 = ds2(M7) + e2Ads2(S3), (39)
where M7 is a seven-dimensional Minkowskian spacetime and the warp factor Adepends only on the coordinates on M7. The above metric has an SO(4) group ofisometries, together with the isometries of the M7 manifold in which we will notbe interested in our general discussion. The NS-sector fields are comprised also bya 2-form b with field strength H = db with non-components along S3, as well as adilaton φ which may depend on the coordinates of M7. Consequently, these fieldsare also invariant under the SO(4) isometry group. Irrespective of the chirality ofthe theory, one can write the SO(4) as SU(2)L × SU(2)R and perform an SU(2)transformation with respect to one of these factors. The end result is a spacetimewith fields in the NS-sector given by
ds2 = ds2(M7) + e−2Adr2 +r2e2A
r2 + e4Ads2(S2),
b = b+ b, b =r3
r2 + e4AVol(S2),
e−2φ = e−2φe2A(r2 + e4A).
(40)
Observe that the SU(2) isometry left untouched by the transformation is capturedin the symmetries of the resulting S2.
1360003-10
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Duality in String Theory
To understand the realization of non-abelian T-duality in type II supergravity,we need to complement our original spacetime ansatz (39), with knowledge of theRR fields. To do this we begin by considering type IIB supergravity, and we incor-porate into our type IIB ansatz the following RR fluxes that respect the symmetryof the round S3 appearing in the metric
F5 = G2 ∧ Vol(S3) − e−3A 7 G2,
F3 = G3 −mVol(S3),
F1 = G1.
(41)
Note that the self-duality of the 5-form has already been imposed. We take theforms Gi, i = 1, 2, 3 to live on M7. Thanks to the type IIB Bianchi identities,the parameter m is a constant and as it turns out, it will be mapped to the massparameter of the massive IIA supergravity.
Through the existence of known examples, we know that our SU(2) transforma-tion takes solutions of the equations of motion in type IIB to solutions in massiveIIA [24], strongly suggesting that by examining the equations of motion one canunearth some deeper structure. Indeed, this is the case and the underlying structurethat emerges is a unifying gravity description in seven dimensions via parallel con-sistent truncations on the original spacetime and on its non-abelian T-dual. In thisformulation non-abelian T-duality is then invertible, unlike the case of the standardσ-model approach.
The reduction to seven dimensions implies that the forms we used in our ansatzcan be expressed in terms of some potentials as G1 = dC0, G2 = dC1 − mb,G3 = dC2 − C0H . Hence the field content arising from the RR sector is a scalar,C0, a 1-form, C1 and a 2-form, C2. These supplement the metric, the 2-form b-fieldand the dilaton φ from the NS-sector. All these arise from the seven-dimensionaleffective action which is given in string frame by
L = e3A−2Φ
(R+ 6(∂A)2 + 4(∂Φ)2 − 12∂A · ∂Φ − 1
12H2
)
− 12
(m2e−3A − 3eA−2Φ + e3AG2
1 +e−3A
2G2
2 +e3A
6G2
3
)
+G2 ∧ C2 ∧H. (42)
The same action (42) can be obtained by reducing massive type IIA on theT-dual background. In this case, since from earlier work [24] it is known how thefluxes transform, we can simply generate the appropriate ansatz for the dual RRfluxes using the type IIB flux ansatz as a seed. Following [24] one constructs thetype IIB flux bispinor from the ansatz (41)
P =eφ
2
4∑n=0
/F 2n+1
(2n+ 1)!, (43)
1360003-11
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Y. Lozano
where we have employed the usual notation /Fp ≡ Fi1···ipΓi1···ip , and reads off theT-dual bispinor from the transformation P = PΩ−1 where Ω is the Lorentz trans-formation matrix acting on the spinors, given by
Ω = Γ11e2AΓ789 + x · Γ√
r2 + e4A. (44)
This procedure gives the massive IIA fluxes:
F0 = m,
F2 =mr3
r2 + e4AVol(S2) + rdr ∧G1 −G2,
F4 =r2e4A
r2 + e4AG1 ∧ dr ∧ Vol(S2) − r3
r2 + e4AG2 ∧ Vol(S2)
+ rdr ∧G3 + e3A 7 G3.
(45)
One can now use these fluxes in tandem with the T-dual spacetime and plugthem both into the massive IIA action. After convenient regularization of the rintegral the action (42) is reproduced. The reader is referred to [14] for more details.
Acknowledgments
I would like to thank E. Alvarez, L. Alvarez-Gaume, J. L. F. Barbon, G. Itsios,E. O. Colgain and K. Sfetsos for collaboration on some of the results presented inthis review.
References
[1] E. Bergshoeff, C. M. Hull and T. Ortin, Duality in the type II superstring effectiveaction, Nucl. Phys. B 451 (1995) 547.
[2] L. Brink, M. B. Green and J. H. Schwarz, N = 4 Yang–Mills and N = 8 supergravityas limits of string theories, Nucl. Phys. B 198 (1982) 474.
[3] K. Kikkawa and M. Yamasaki, Casimir effects in superstring theories, Phys. Lett. B149 (1984) 357.
[4] N. Sakai and I. Senda, Vacuum energies of string compactified on torus, Progr. Theor.Phys. 75 (1986) 692.
[5] E. Alvarez and M. A. R. Osorio, Duality is an exact symmetry of string perturbationtheory, Phys. Rev. D 40 (1989) 1150.
[6] T. H. Buscher, A symmetry of the string background field equations, Phys. Lett. B194 (1987) 51.
[7] T. H. Buscher, A symmetry of the string background field equations, Phys. Lett. B201 (1988) 466.
[8] M. Rocek and E. Verlinde, Duality, quotients, and currents, Nucl. Phys. B 373 (1992)630.
[9] X. De la Ossa and F. Quevedo, Duality symmetries from non-abelian isometries instring theory, Nucl. Phys. B 403 (1993) 377.
[10] A. Giveon and M. Rocek, On non-abelian duality, Nucl. Phys. B 421 (1994) 173.[11] E. Alvarez, L. Alvarez-Gaume and Y. Lozano, On non-abelian duality, Nucl. Phys.
B 424 (1994) 155.
1360003-12
Int.
J. G
eom
. Met
hods
Mod
. Phy
s. 2
013.
10. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by M
ON
ASH
UN
IVE
RSI
TY
on
09/2
2/13
. For
per
sona
l use
onl
y.
2nd Reading
April 24, 2013 14:56 WSPC/S0219-8878 IJGMMP-J043 1360003
Duality in String Theory
[12] A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys.Rep. 244 (1994) 77.
[13] E. Alvarez, L. Alvarez-Gaume and Y. Lozano, An introduction to T-duality in stringtheory, Nucl. Phys. Proc. Suppl. 41 (1995) 1.
[14] G. Itsios, Y. Lozano, E. O. Colgain and K. Sfetsos, Non-abelian T-duality and con-sistent truncations in type-II supergravity, J. High Energy Phys. 1208 (2012) 132.
[15] E. Alvarez, L. Alvarez-Gaume, J. L. F. Barbon and Y. Lozano, Some global aspectsof duality in string theory, Nucl. Phys. B 415 (1994) 71.
[16] C. Hull and B. Spence, The gauged nonlinear sigma model with Wess–Zumino term,Phys. Lett. B 232 (1989) 204.
[17] E. Alvarez, L. Alvarez-Gaume and Y. Lozano, A canonical approach to duality trans-formations, Phys. Lett. B 336 (1994) 183.
[18] G. I. Ghandour, Effective generating functions for quantum canonical transforma-tions, Phys. Rev. D 35 (1987) 1289.
[19] Y. Lozano, Duality and canonical transformations, Mod. Phys. Lett. A 11 (1996)2893.
[20] T. Curtright and C. Zachos, Currents, charges, and canonical structure of pseudodualchiral models, Phys. Rev. D 49 (1994) 5408.
[21] Y. Lozano, Non-abelian duality and canonical transformations, Phys. Lett. B 355(1995) 165.
[22] O. Alvarez and C.-H. Liu, Target space duality between simple compact Lie groupsand Lie algebras under the Hamiltonian formalism: 1. Remnants of duality at theclassical level, Commun. Math. Phys. 179 (1996) 185.
[23] K. Sfetsos, Non-abelian duality, parafermions and supersymmetry, Phys. Rev. D 54(1996) 1682.
[24] K. Sfetsos and D. C. Thompson, On non-abelian T-dual geometries with Ramondfluxes, Nucl. Phys. B 846 (2011) 21.
