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JHEP08(2017)116

Published for SISSA by Springer

Received: June 9, 2017

Accepted: August 1, 2017

Published: August 25, 2017

Hidden isometry of “T-duality without isometry”

Peter Bouwknegt,

Mark Bugden,

Ctirad Klimˇc´ık

and Kyle Wright

Mathematical Sciences Institute, Australian National University,

Canberra, ACT, 2601 Australia

Institut de Math´ematiques de Luminy, Aix Marseille Universit´e, CNRS,

Centrale Marseille I2M, UMR 7373, Marseille, 13453 France

Department of Theoretical Physics, Research School of Physics and Engineering, and

Mathematical Sciences Institute, Australian National University,

Canberra, ACT, 2601 Australia

E-mail:

peter.bouwknegt@anu.edu.au, mark.bugden@anu.edu.au,

ctirad.klimcik@univ-amu.fr, wright.kyle.j@gmail.com

Abstract: We study the T-dualisability criteria of Chatzistavrakidis, Deser and Jonke [3]

who recently used Lie algebroid gauge theories to obtain sigma models exhibiting a “T-

duality without isometr y ”. We point out that those T-d ual i sabi l i ty criteria are not written

invariantly in [

3] and depend on the choice of the algebroid framing. We then show that

there always exists an isometri c framing for which the Lie algebroid gauging boils down to

standard Yang-Mills gauging. The “T-duality without isometry” of [

3] is therefore nothing

but traditional isometric non-Abelian T-duality in disguise.

Keywords: Gauge Symmetry, Sigma Models, String Duality, Discr e t e Symmetries

ArXiv ePrint:

1705.09254

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP08(2017)116

JHEP08(2017)116

Contents

1 Introduction

2 Preliminaries on the non-Abelian T-duality 3

3 CDJ gauge theory 4

4 Inclusion of non-exact 3-form background H 6

5 Lie algebroid gauged sigma models 8

6 Examples 12

7 Conclusion and outlook 16

1 Introduction

T-dualisability is a rare property of non-linear sigma-models and it is not k nown what

necessary conditions must be imposed on a targe t space m et r i c G, and closed 3-form ﬁeld

H, such that the corresponding sigma-model has a T-dual with (

H). On the other

hand, several suﬃcient conditions are known, giv i ng various T-dualities like the Abelian

one [

9, 16] or non-Abelian one [1, 5–7], both in turn included as special cases of Poisson-

Lie T-duality [

10, 11]. Chatzistavrakidis, Deser and Jonke (CDJ in what follows) recently

proposed a new set of suﬃcient conditions which, they claimed, would give rise to new

examples of T-dual pair s [

3]. Their dualisability conditions appear much less restr i c ti ve

than those previously describ ed in T-duality research. It is the purpose of the present work

to show that, in reality, they are not less restr i c t i ve as they give rise to the same dual i ty

pattern as that of traditional non-Abelian T-dual i ty.

The proposal of CDJ for dualising a given sigma model on a target M , is an extension of

the Roˇcek-Verlinde approach [5, 15], which amounts to the introduction of an intermediate

gauge theory yi el di n g the T-dual pair of sigma models upon eliminating di ﬀe re nt sets of

ﬁelds. It was traditionally thought that the Roˇcek-Verlinde intermediate gauge theory

can be constructed only if the background of the sigma-model is isometric with respect

to the action of the Lie alge br a g of the gauge group. However, CDJ have argued that

more general gaugings are possible if one uses the re ce ntly introduc e d L i e alge br oi d gauge

theory [

12, 14, 17]. The construction of the Li e alge br oi d ge ne r al i sati on of the Roˇcek-

Verlinde intermediate gauge theory requires the existence of a Lie algebroid bundl e Q, over

the target M, as well as a ﬁxed connection ∇

on Q compatible wi t h the sigma mod el

– 1 –

JHEP08(2017)116

background. As CDJ show, the compatibility of ∇

, G and H can be expressed in a

particularly simple way for exact 3-form backgrounds H = dB where it reads:

ρ(e

)

G = ω

∨ ι

ρ(e

)

G, L

ρ(e

)

B = ω

∧ ι

ρ(e

)

B . (1.1)

Here e

form local frames of the Lie algebroid, the Lie derivatives are taken with respect

to the anchored frames ρ(e

), the symbols ∨ and ∧ stand respectively for the symmetrised

and anti-symmetrised direct products of 1-forms on M, and the 1-forms ω

are deﬁned by

the relations

∇

:= ω

⊗ e

. (1.2)

Since the choice of the connection ∇

seems l ar ge l y arbitrar y, it may appear from (

1.1) that

a vast set of non-isometric backgrounds could be gauged, thus producing a new and rich T-

duality pattern. However, as we shall argue in this paper, this is not the case. The simplest

way to understand what is happening is to r e al i se that th e compatibility conditions (

1.1),

as given by CDJ in ref. [

3] are not written invariantly; upon a local changes of frames

′

= P

, P

∈ C

∞

(M), they change to

ρ(e

′

)

G = ω

′b

∨ ι

ρ(e

′

)

G, L

ρ(e

′

)

B = ω

′b

∧ ι

ρ(e

′

)

B , (1.3)

where

∇

′

:= ω

′b

⊗ e

′

. (1.4)

The components of the connection form ω

transform non-homogeneously upon a change

of the framing, and we may naturally question whether there exists a dist i ngui sh ed frame

ˆe

for which they all vanish. This question can be answered in the aﬃrmative, and this

fact follows from the Lie algebroid gauge invariance of the Roˇcek-Verlinde intermediate

gauge theory. It is therefore always possible to write down an equivalent version of the

CDJ compatibility conditions (

1.1) in the standard isometric form

ρ(ˆe

)

G = 0 , L

ρ(ˆe

)

B = 0 . (1.5)

Moreover, the gauge invariance of the intermediate gauge theory also require s that the

structure functions

deﬁned by the Lie algebroid b r ackets

[ˆe

, ˆe

] ≡

ˆe

, (1.6)

be constants, and we thus rec over the standard intermediate Yang-Mills gauge theor y

leading to traditi onal non-Abelian T-duality [

5, 7].

The p l an of our paper is as follows: in section

2 we expose some useful preliminary

background on traditional non- Abelian T-duality. In section

3 we review the “T-duality

without isometry” proposal of CDJ and detail the ﬁeld redeﬁnit i ons which reproduce stan-

dard non-Abelian T-duality. In section

4 we work out the case of non-exact 3-form back-

ground H. In section

5 we p rovide a geometric interpretation of the ﬁeld redeﬁnitions from

the invariant perspective of Lie algebroid gauge theory. In section

6, we il l us tr at e a few ex-

amples where, by sim pl e ﬁeld redeﬁnitions, the traditional isometric Roˇcek-Verlinde gauge

theory may look like a non-trivial Lie algebroid gauge theory. In particular, we unmask the

“non-isometric T-duality” example of CDJ presented in [

3]. Finally, we end with a short

discussion.

– 2 –

JHEP08(2017)116

2 Preliminaries on the non-Abelian T-duality

To set up some technical and notational background, as well as remind the reader of the

gauging approach to T-duality, we review traditional non-Abelian T-duali ty obtained by

the Roˇcek-Verl i nde procedure [

5–7]. We ﬁrst rest r i ct our attention to backgrounds for

which H = dB is an exact 3-form , postponing the study of cohomologically non-trivi al

backgrounds to section

Let a Lie group G act from the right on the target manifold M , let T

be a basis of

the Lie algebra g ≡ Lie(G), and v

the set of vector ﬁelds on M correspondin g to the

inﬁnitesimal right actions of the elements T

. The Lie derivatives L

then satisfy

= [v

, v

] = C

, (2.1)

where C

are the structu re constants of g in the basis T

Denoting the (Lorentzian) cylindri c al world-sheet by Σ and introducing coordinates

on M, we write the sigma model action with the background metric G and the 3-form

ﬁeld H = dB as

S(X

) =

∧



∗ dX

+ B



. (2.2)

Here d denotes the de Rham diﬀere ntial, ∗ (∗

= 1) the Hodge star on the world-sheet Σ,

and the X

are viewed as functions on Σ describing a string moving in M.

If the Lie derivatives of the metri c and the B ﬁeld vanish

G = 0, L

B = 0 , (2.3)

then the sigma mode l (

2.2) c an be gauged in the standard Yang-Mills way. This means

that one introduces a world-sheet one-form A valued in the Lie algebra g ≡ Lie(G), a

world-sheet scalar η val ue d in the dual g

∗

, and the gauged action

S(X

, A, η) =

∧



∗ DX

+ B



hη, F (A)i . (2.4)

Here h· , ·i is the canonical pairing between g

∗

and g, F (A) is the standard Yang-Mills ﬁe l d

strength

F (A) := dA + A ∧ A ≡



∧ A



, (2.5)

and DX

are the covariant derivatives

:= dX

− v

. (2.6)

If the isometry conditions (

2.3) hold, the action (2.4) i s gauge invariant with respect to the

following local inﬁn i te si mal gauge transformations:

= v

, δ

A = dǫ + [A, ǫ] ≡



dǫ

+ C



, δ

η = − ad

∗

η ≡ −C

∗a

(2.7)

Here ǫ is a function on the world-sheet valued in g, and ad

∗

denotes the co-adjoint action

of g on g

∗

– 3 –

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