Comments on non-isometric T-dualitythphys.irb.hr/dualities2017/files/Jun08Bugden.pdfComments on...
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Comments on non-isometric T-duality
Mark Bugden
Mathematical Sciences InstituteAustralian National University
Based on [1705.09254]with P. Bouwknegt, C. Klimcık, and K. Wright
June 2017
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Outline
1 Isometric T-duality
2 Non-isometric T-duality
3 Equivalence
4 Examples
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Setting up notation
Consider a non-linear sigma model X : Σ→ M described by thefollowing action:
S =
∫Σgij dX
i ∧ ?dX j +
∫ΣBij dX
i ∧ dX j
In this talk we will ignore the dilaton, and assume that both g andB are globally defined fields on M.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Gauging isometries
Suppose now that there are vector fields generating the followingglobal symmetry:
δεXi = v ia ε
a
for εa constant. The sigma model action is invariant under thistransformation if
Lvag = 0 LvaB = 0
If this is the case, we can gauge the model by promoting the globalsymmetry to a local one.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
The gauged action
Introducing gauge fields Aa and Lagrange multipliers ηa, thegauged action is
SG =
∫Σgij DX
i ∧ ?DX j +
∫ΣBij DX
i ∧ DX j +
∫ΣηaF
a
where
F = dA + A ∧ A is the standard Yang-Mills field strength
DX i = dX i − v iaAa are the gauge covariant derivatives.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Gauge invariance
The gauged action is invariant with respect to the following (local)gauge transformations:
δεXi = v ia ε
a
δεAa = dεa + C a
bcAbεc
δεηa = −C cabε
bηc
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Isometric T-duality Non-isometric T-duality Equivalence Examples
T-duality
S [X ]
SG [X ,A, η]
gaug
eiso
met
ries
inte
grat
eη
and
fixga
uge
S [η]integrate
A
andfix
gauge
Varying the Lagrange multipliers forces the field strength F tovanish. If we then fix the gauge A = 0 we recover the originalmodel.
On the other hand, we can eliminate the non-dynamical gaugefields A, obtaining the dual sigma model.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
T-duality
S [X ]
SG [X ,A, η]
gaug
eiso
met
ries
inte
grat
eη
and
fixga
uge
S [η]integrate
A
andfix
gauge
Varying the Lagrange multipliers forces the field strength F tovanish. If we then fix the gauge A = 0 we recover the originalmodel.
On the other hand, we can eliminate the non-dynamical gaugefields A, obtaining the dual sigma model.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
T-duality
S [X ]
SG [X ,A, η]
gaug
eiso
met
ries
inte
grat
eη
and
fixga
uge
S [η]integrate
A
andfix
gauge
Varying the Lagrange multipliers forces the field strength F tovanish. If we then fix the gauge A = 0 we recover the originalmodel.
On the other hand, we can eliminate the non-dynamical gaugefields A, obtaining the dual sigma model.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Can we do it without isometries?
The existence of global symmetries is a very stringent requirement.A generic metric will not have any Killing vectors.
Question
Is it possible to follow the same procedure when the vector fieldsare not Killing vectors?
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Gauging without isometry
Kotov and Strobl1 introduced a method of gauging a sigma modelwithout requiring the model to possess isometries.
Their method uses Lie algebroids, and generalises the standardgauging in two notable ways:
The structure constants of the Lie algebra are promoted tostructure functions:
[va, vb] = C cab(X ) vc
The gauge invariance of the gauged action doesn’t require theoriginal vector fields to be isometries:
Lvag 6= 0 LvaB 6= 0
1[1403.8119]9
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Non-isometric T-duality
Chatzistavrakidis, Deser, and Jonke2 applied this non-isometricgauging to the Buscher procedure we just reviewed.
They promote the structure constants to functions, and introducea matrix-valued one-form ωb
a satisfying
Lvag = ωba ∨ ιvbg
LvaB = ωba ∧ ιvbB
2[1509.01829] and [1604.03739]10
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Isometric T-duality Non-isometric T-duality Equivalence Examples
The gauged action
The gauged action is almost the same:
SωG =
∫Σgij DX
i ∧ ?DX j +
∫ΣBij DX
i ∧ DX j +
∫ΣηaF
aω
where the curvature is now given by
F aω = dAa +
1
2C abc(X )Ab ∧ Ac − ωa
biAb ∧ DX i
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Modified gauge invariance
The modified gauge transformations are now
δεXi = v ia ε
a
δεAa = dεa + C a
bcAbεc + ωa
biεbDX i
δεηa = −C cabε
bηc + v iaωcbiε
bηc
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Isometric T-duality Non-isometric T-duality Equivalence Examples
T-duality
S [X ]
SωG [X ,A, η]
exot
ical
lyga
uge
inte
grat
eη
and
fixga
uge
Sω[η]
integrateA
andfix
gauge
As with isometric T-duality, we can integrate out the fields in twodifferent ways, obtaining the original model or a dual model
In principle, we could use this to construct T-duals of spaces whichhave no isometries.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
T-duality
S [X ]
SωG [X ,A, η]
exot
ical
lyga
uge
inte
grat
eη
and
fixga
uge
Sω[η]
integrateA
andfix
gauge
As with isometric T-duality, we can integrate out the fields in twodifferent ways, obtaining the original model or a dual model
In principle, we could use this to construct T-duals of spaces whichhave no isometries.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
T-duality
S [X ]
SωG [X ,A, η]
exot
ical
lyga
uge
inte
grat
eη
and
fixga
uge
Sω[η]
integrateA
andfix
gauge
As with isometric T-duality, we can integrate out the fields in twodifferent ways, obtaining the original model or a dual model
In principle, we could use this to construct T-duals of spaces whichhave no isometries.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
T-duality
S [X ]
SωG [X ,A, η]
exot
ical
lyga
uge
inte
grat
eη
and
fixga
uge
Sω[η]
integrateA
andfix
gauge
As with isometric T-duality, we can integrate out the fields in twodifferent ways, obtaining the original model or a dual model
In principle, we could use this to construct T-duals of spaces whichhave no isometries.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
T-duality
S [X ]
SωG [X ,A, η]
exot
ical
lyga
uge
inte
grat
eη
and
fixga
uge
Sω[η]
integrateA
andfix
gauge
As with isometric T-duality, we can integrate out the fields in twodifferent ways, obtaining the original model or a dual model
In principle, we could use this to construct T-duals of spaces whichhave no isometries.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
The problem?
This proposal is equivalent to non-abelian T-duality.3
That is, if we can find a set of vector fields and ωba which give a
non-isometric T-dual, then there exists a set of Killing vectors forthe model. The T-dual with respect to these Killing vectors is thesame as the non-isometric T-dual.
3[1705:09254] P. Bouwknegt, M.B., C. Klimcık, K. Wright14
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Isometric T-duality Non-isometric T-duality Equivalence Examples
A necessary condition for gauge invariance
Gauge invariance of the action requires the structure functions tobe constant, as well as the vanishing of the following variation:
δε(ηaFaω) = ηa(dωa
b + ωac ∧ ωc
b)εb +O(A) +O(A2).
We therefore require that ωba is flat:
Rba = dωb
a + ωbc ∧ ωc
a = 0,
and this tells us that ωba is of the form K−1dK for some Kb
a (X ).
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Isometric T-duality Non-isometric T-duality Equivalence Examples
A field redefinition
Using this K , we can perform the following field redefinitions:
Aa = K abA
b
ηa = ηb(K−1)ba
va = v ib(K−1)ba
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Isometric T-duality Non-isometric T-duality Equivalence Examples
The non-abelian action!
The gauged action can now be rewritten in terms of the new fields(X i , Aa, ηa).
SωG [X , A, η] =
∫Σgij DX
i∧ ?DX
j+
∫ΣBij DX
i∧ DX
j+
∫ΣηaF
a
= SG [X , A, η]
where
F a = dAa +1
2C abc A
b ∧ Ac
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Isometric T-duality Non-isometric T-duality Equivalence Examples
The gauge transformations become the usual non-abelian gaugetransformations, and a short computation reveals
Lvag = 0 LvaB = 0
Conclusion
This proposal is equivalent, via a field redefinition, to the standardnon-abelian T-duality
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Isometric T-duality Non-isometric T-duality Equivalence Examples
First example
Consider the 3D Heisenberg Nilmanifold, or twisted torus. It has ametric given by
ds2 = dx2 + (dy − x dz)2 + dz2
The non-abelian T-dual of this space is given by
ds2 = dY 2 +1
1 + Y 2
(dX 2 + dZ 2
)B =
Y
1 + Y 2dX ∧ dZ
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Isometric T-duality Non-isometric T-duality Equivalence Examples
First example
Consider the 3D Heisenberg Nilmanifold, or twisted torus. It has ametric given by
ds2 = dx2 + (dy − x dz)2 + dz2
The non-abelian T-dual of this space is given by
ds2 = dY 2 +1
1 + Y 2
(dX 2 + dZ 2
)B =
Y
1 + Y 2dX ∧ dZ
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Isometric T-duality Non-isometric T-duality Equivalence Examples
First example
We can gain a better understanding of the geometry by writing themanifold as a group:
Heis :=
1 x y
0 1 z0 0 1
: x , y , z ∈ R
(left-invariant) MC forms = (dx , dy − xdz , dz)(right-invariant) vector fields = (∂x + z∂y , ∂y , ∂z)
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Isometric T-duality Non-isometric T-duality Equivalence Examples
First example
We could instead try to gauge this space non-isometrically usingthe left-invariant vector fields: {∂x , ∂y , x∂y + ∂z}.
These are not all isometries:
Lv1g = −dy ⊗ dx − dz ⊗ dy + 2xdz ⊗ dz
Lv2g = 0
Lv3g = dx ⊗ dy + dy ⊗ dx − xdx ⊗ dz − xdz ⊗ dx
and they don’t commute:
[v1, v3] = v2,
however...
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Isometric T-duality Non-isometric T-duality Equivalence Examples
First example4
If we take ω23 = dx and ω2
1 = −dz , with other componentsvanishing, the non-isometric gauging constraints are satisfied andwe can calculate the non-isometric T-dual model.
ds2 = dY 2 +1
1 + Y 2
(dX 2 + dZ 2
)B =
Y
1 + Y 2dX ∧ dZ
Unsurprisingly, it is also the T-fold.
4Gauged non-isometrically in [1509:01829]A. Chatzistavrakidis, A. Deser, L. Jonke
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Isometric T-duality Non-isometric T-duality Equivalence Examples
First example4
If we take ω23 = dx and ω2
1 = −dz , with other componentsvanishing, the non-isometric gauging constraints are satisfied andwe can calculate the non-isometric T-dual model.
ds2 = dY 2 +1
1 + Y 2
(dX 2 + dZ 2
)B =
Y
1 + Y 2dX ∧ dZ
Unsurprisingly, it is also the T-fold.
4Gauged non-isometrically in [1509:01829]A. Chatzistavrakidis, A. Deser, L. Jonke
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Second example
Consider S3 with the round metric and B = 0.
This metric has an SO(4) group of isometries, and we can find thenon-abelian T-dual with respect to an SU(2) subgroup of this.
The non-abelian T-dual is well-known. The metric is the ‘cigar’metric, and there is also a non-zero B-field.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Second example
Consider S3 with the round metric and B = 0.
This metric has an SO(4) group of isometries, and we can find thenon-abelian T-dual with respect to an SU(2) subgroup of this.
The non-abelian T-dual is well-known. The metric is the ‘cigar’metric, and there is also a non-zero B-field.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Second example
Consider S3 with the round metric and B = 0.
This metric has an SO(4) group of isometries, and we can find thenon-abelian T-dual with respect to an SU(2) subgroup of this.
The non-abelian T-dual is well-known. The metric is the ‘cigar’metric, and there is also a non-zero B-field.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Second example
We can write the round metric as
g = λ1 ⊗ λ1 + λ2 ⊗ λ2 + λ3 ⊗ λ3
where the λi are the left-invariant Maurer-Cartan forms.
The right-invariant vector fields are isometries of this metric, solet’s try gauging with respect to the left-invariant vector fields5.
5These also happen to be isometries of the metric, but let’s try to gaugethem non-isometrically
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Second example
The Lie derivatives of the metric with respect to the left-invariantvector fields, La are
LLag = −∑b
Cbacλ
c ∨ λb
= −Cbacλ
c ∨ ιLbg
We can do non-isometric T-duality by taking ωba = −Cb
acλc .
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Second example
The remaining gauging constraints are satisfied, and we cancalculate the non-isometric T-dual. It is the ’cigar’ metric, asexpected.
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Isometric T-duality Non-isometric T-duality Equivalence Examples
Comments
The equivalence of non-isometric and non-abelian T-dualityremains valid for non-exact H
Geometric interpretation of ωba as a connection on a Lie
algebroid
There are proposals for alternate gauging. Unknown how toincorporate into T-duality
non-flat ωba
include a term φbaiεb ? DX i into δεA
a
Thanks!
27
![Page 38: Comments on non-isometric T-dualitythphys.irb.hr/dualities2017/files/Jun08Bugden.pdfComments on non-isometric T-duality Mark Bugden Mathematical Sciences Institute Australian National](https://reader033.fdocuments.in/reader033/viewer/2022060814/6092712033a5e7351d64fc77/html5/thumbnails/38.jpg)
Isometric T-duality Non-isometric T-duality Equivalence Examples
Comments
The equivalence of non-isometric and non-abelian T-dualityremains valid for non-exact H
Geometric interpretation of ωba as a connection on a Lie
algebroid
There are proposals for alternate gauging. Unknown how toincorporate into T-duality
non-flat ωba
include a term φbaiεb ? DX i into δεA
a
Thanks!
27