The structure of non-geometric frames
in string theory
Erik Plauschinn
University of Padova
PRIN Meeting Pisa — 17.05.2013
this talk is based on ...
This talk is based on work with R. Blumenhagen, A. Deser, F. Rennecke and
C. Schmid ::
◾
A bi-invariant Einstein-Hilbert action for the non-geometric string
arXiv:1210.1591
◾
Non-geometric strings, symplectic gravity and differential geometry of
Lie algebroids
arXiv:1211.0030
The intriguing structure of non-geometric frames in string theory
arXiv:1304.2784
◾
this talk is based on ...
This talk is based on work with R. Blumenhagen, A. Deser, F. Rennecke and
C. Schmid ::
◾
A bi-invariant Einstein-Hilbert action for the non-geometric string
arXiv:1210.1591
◾
Non-geometric strings, symplectic gravity and differential geometry of
Lie algebroids
arXiv:1211.0030
The intriguing structure of non-geometric frames in string theory
arXiv:1304.2784
◾
Related work is by D. Andriot, O. Hohm, M. Larfors, D. Lüst, P. Patalong ::
◾
A ten-dimensional action for non-geometric fluxes
arXiv:1106.4015
◾
A geometric action for non-geometric fluxes
arXiv:1202.3060
◾
Non-geometric fluxes in supergravity and double field theory
arXiv:1204.1979
outline
1. introduction
2. field redefinitions
3. actions
4. discussion
outline
1. introduction
2. field redefinitions
3. actions
4. discussion
introduction :: motivation
String theory
introduction :: motivation
String theory
Conformal field theories
◾
◾
◾
◾
free boson CFTs
orbifold models
Gepner models
WZW models
introduction :: motivation
String theory
Conformal field theories
◾
◾
◾
◾
Geometric spaces
◾
◾
◾
◾
Rn , T n , S n
group manifolds
Calabi-Yau manifolds
...
free boson CFTs
orbifold models
Gepner models
WZW models
introduction :: motivation
String theory
Conformal field theories
◾
◾
◾
◾
Geometric spaces
◾
◾
◾
◾
Rn , T n , S n
group manifolds
Calabi-Yau manifolds
...
free boson CFTs
orbifold models
Gepner models
WZW models
Non-geometric spaces
◾
◾
◾
◾
asymmetric orbifolds
T-folds
non-commutative spaces
non-associative spaces
introduction :: motivation
String theory
Conformal field theories
◾
◾
◾
◾
free boson CFTs
orbifold models
Gepner models
WZW models
Geometric spaces
◾
◾
◾
◾
Rn , T n , S n
group manifolds
Calabi-Yau manifolds
...
Non-geometric spaces
..., T-duality, ...
◾
◾
◾
◾
asymmetric orbifolds
T-folds
non-commutative spaces
non-associative spaces
introduction :: motivation
String theory
Conformal field theories
◾
◾
◾
◾
free boson CFTs
orbifold models
Gepner models
WZW models
Geometric spaces
◾
◾
◾
◾
Rn , T n , S n
group manifolds
Calabi-Yau manifolds
...
Non-geometric spaces
..., T-duality, ...
◾
◾
◾
◾
asymmetric orbifolds
T-folds
non-commutative spaces
non-associative spaces
introduction :: example
Example :: apply successive T-duality transformations to a three-torus with H-flux
Habc
Tc
! fab
c
Tb
! Qa
bc
Ta
! Rabc
introduction :: example
Habc
Tc
! fab
c
Tb
! Qa
bc
Ta
! Rabc
introduction :: h-flux background
Habc
Tc
! fab
c
Tb
! Qa
bc
Ta
Consider string theory compactified on a three-torus with H-flux:
◾
The geometry is characterized by
ds2 = dx2 + dy 2 + dz 2 ,
Byz = N x .
◾
The H-flux is determined by
1
(2
)3
Z
H=N.
! Rabc
introduction :: f-flux background
Habc
Tc
! fab
c
Tb
! Qa
Ta
bc
! Rabc
After a T-duality in the z-direction, one arrives at a twisted torus:
◾
The geometry is characterized by
ds2 = dx2 + dy 2 + (dz + N x dy)2 ,
B = 0.
◾
The geometric flux follows from
ex = dx ,
z
xy
=
[ex , ey ] =
ey = dy ,
ez = dz + N x dy ,
N /2 ,
N ez .
Scherk, Schwarz - 1979
Kachru, Schulz, Tripathy, Trivedi - 2002
introduction :: f-flux background
Habc
Tc
! fab
c
Tb
! Qa
Ta
bc
! Rabc
After a T-duality in the z-direction, one arrives at a twisted torus:
◾
The geometry is characterized by
ds2 = dx2 + dy 2 + (dz + N x dy)2 ,
B = 0.
◾
The geometric flux follows from
ex = dx ,
z
xy
=
[ex , ey ] =
ey = dy ,
ez = dz + N x dy ,
N /2 ,
N ez .
Scherk, Schwarz - 1979
Kachru, Schulz, Tripathy, Trivedi - 2002
introduction :: q-flux background
Habc
Tc
! fab
c
Tb
! Qa
bc
Ta
! Rabc
After a second T-duality in the y-direction, one arrives at a T-fold:
The geometry is characterized by
1
2
2
dy
+
dz
,
ds = dx +
2
2
1+N x
Nx
Byz =
.
2
2
1+N x
◾
The non-geometric Q-flux reads
Qx yz = N .
◾
The metric and B-field are well-defined locally, but not globally.
◾
Transition functions between local charts involve T-duality transformations,
hence the name T-fold.
◾
2
2
Hellermann, McGreevy, Williams - 2002
Dabholkar, Hull - 2002
Hull - 2004
introduction :: q-flux background
Habc
Tc
! fab
c
Tb
! Qa
bc
Ta
! Rabc
After a second T-duality in the y-direction, one arrives at a T-fold:
◾
◾
The geometry is characterized by
1
2
2
dy
+
dz
,
ds = dx +
2
2
1+N x
Nx
Byz =
.
2
2
1+N x
The non-geometric Q-flux reads
Qx yz = N .
(G, B)(x = 1)
T-duality
2
2
(G, B)(x = 0)
◾
The metric and B-field are well-defined locally, but not globally.
◾
Transition functions between local charts involve T-duality transformations,
hence the name T-fold.
x
Hellermann, McGreevy, Williams - 2002
Dabholkar, Hull - 2002
Hull - 2004
introduction :: q-flux background
Habc
Tc
! fab
c
Tb
! Qa
bc
Ta
! Rabc
After a second T-duality in the y-direction, one arrives at a T-fold:
The geometry is characterized by
1
2
2
dy
+
dz
,
ds = dx +
2
2
1+N x
Nx
Byz =
.
2
2
1+N x
◾
The non-geometric Q-flux reads
Qx yz = N .
◾
The metric and B-field are well-defined locally, but not globally.
◾
Transition functions between local charts involve T-duality transformations,
hence the name T-fold.
◾
2
2
Hellermann, McGreevy, Williams - 2002
Dabholkar, Hull - 2002
Hull - 2004
introduction :: r-flux background
Habc
Tc
! fab
c
Tb
! Qa
bc
Ta
! Rabc
After formally applying a third T-duality, one obtains an R-flux background:
◾
The metric and B-field are not even locally well-defined.
◾
The non-geometric R-flux is formally written as Rxyz = N .
◾
This background gives rise to a non-associative structure.
Bouwknegt, Hannabuss, Mathai - 2004
Shelton, Taylor, Wecht - 2005
Ellwood, Hashimoto - 2006
Blumenhagen, EP & Lüst - 2010
introduction :: summary, goals, ...
Summary ::
Goals ::
◾
Non-geometric spaces form an important class of string
theory backgrounds.
◾
Some explicit examples are T-duals of a torus with H-flux.
◾
Construct an action for a non-geometric background.
◾
Study its symmetries.
Strategy ::
Andriot, [Hohm,] Larfors, Lüst, Patalong - 2011 & 2012
Blumenhagen, Deser, EP, Rennecke, [Schmid] - 2012 & 2013
introduction :: summary, goals, ...
Strategy ::
introduction :: ... strategy
Strategy ::
non-geometric
background
action in a nongeometric frame
introduction :: ... strategy
Strategy ::
non-geometric
background
T-duality and O(d,d)
transformations
generalized
geometry
action in a nongeometric frame
Lie algebroids &
differential geometry
field
redefinitions
introduction :: ... strategy
Strategy ::
non-geometric
background
T-duality and O(d,d)
transformations
generalized
geometry
action in a nongeometric frame
Lie algebroids &
differential geometry
field
redefinitions
outline
1. introduction
2. field redefinitions
3. actions
4. discussion
field redefinitions :: generalized geometry I
Consider a generalized tangent bundle for a d-dimensional manifold M with sections
X + ⇠ 2 TM
T ⇤M .
A natural bi-linear form, or scalar product, on this bundle is given by
✓ ◆t✓
◆✓ ◆
X
Y
0 1
hX + ⇠, Y + ⇣i =
= ⇠a Y a + ⇣a X a .
1 0
⇠
⇣
Transformations M leaving this form invariant constitute the group O(d,d), determined by
✓
◆
0 1
t
⌘=
.
M ⌘M = ⌘,
1 0
Hitchin - 2002
Gualtieri - 2004
Graña, Minasian, Petrini, Waldram - 2008
field redefinitions :: generalized geometry II
The metric G and a two-form B of a manifold can be combined into a generalized metric
✓
◆
1
1
G BG B BG
H=
.
G 1B
G 1
Under O(d,d) transformations M, the metric H changes as follows
✓
◆
a b
t
b
M=
2 O(d, d) .
H = M HM,
c d
field redefinitions :: O(d,d) transformations
Three particular examples for O(d,d) transformations are the following ::
◾
diffeomorphism
Mdi↵ =
✓
A
0
0
(At )
1
◆
,
Mtdi↵ H(G, B) Mdi↵ = H At G A, At B A ,
◾
gauge transformation
(B-transform)
Mgauge =
✓
◆
1
d⇤
0
,
1
Mtgauge H(G, B) Mgauge = H(G, B + d⇤) ,
◾
-transform
M =
✓
1
0
1
◆
.
field redefinitions :: O(d,d) transformations
Three particular examples for O(d,d) transformations are the following ::
◾
diffeomorphism
Mdi↵ =
✓
A
0
0
(At )
1
◆
,
Mtdi↵ H(G, B) Mdi↵ = H At G A, At B A ,
◾
gauge transformation
(B-transform)
Mgauge =
✓
◆
1
d⇤
0
,
1
Mtgauge H(G, B) Mgauge = H(G, B + d⇤) ,
◾
-transform
M =
✓
1
0
1
◆
.
Remark :: -transforms are not part of the geometric group, but lead to new
non-geometric frames.
field redefinitions :: main idea
O(d,d) transformations induce field redefinitions of the metric G and B-field via
H(G, B)
generalized metric in
variables G and B
Mt HM
b
H(G,
B)
O(d,d)-transformed
generalized metric
Ĝ(G, B), B̂(G, B)
H(Ĝ, B̂)
generalized metric in
new variables
field redefinitions :: main idea
O(d,d) transformations induce field redefinitions of the metric G and B-field via
H(G, B)
Mt HM
generalized metric in
variables G and B
b
H(G,
B)
Ĝ(G, B), B̂(G, B)
O(d,d)-transformed
generalized metric
More concretely, this reads
Mt H(G, B) M = H(Ĝ, B̂)
H(Ĝ, B̂)
generalized metric in
new variables
field redefinitions :: main idea
O(d,d) transformations induce field redefinitions of the metric G and B-field via
H(G, B)
Mt HM
generalized metric in
variables G and B
More concretely, this reads
✓
◆t ✓
a b
G BG 1 B
G 1B
c d
Ĝ(G, B), B̂(G, B)
b
H(G,
B)
generalized metric in
new variables
O(d,d)-transformed
generalized metric
BG
G 1
1
◆✓
a
c
b
d
◆
=
H(Ĝ, B̂)
✓
Ĝ
1
B̂ Ĝ B̂
Ĝ 1 B̂
B̂ Ĝ
Ĝ 1
1
◆
,
field redefinitions :: main idea
O(d,d) transformations induce field redefinitions of the metric G and B-field via
Mt HM
H(G, B)
Ĝ(G, B), B̂(G, B)
b
H(G,
B)
generalized metric in
variables G and B
generalized metric in
new variables
O(d,d)-transformed
generalized metric
More concretely, this reads
✓
◆t ✓
a b
G BG 1 B
G 1B
c d
BG
G 1
1
◆✓
a
c
b
d
◆
=
H(Ĝ, B̂)
✓
Ĝ
1
B̂ Ĝ B̂
Ĝ 1 B̂
B̂ Ĝ
Ĝ 1
leading to
b =
G
b =
B
1
G(
⇥
1
1 t
t
) ,
⇤
G (
1 t
) ,
= d + (G
B) b ,
= c + (G
B) a .
1
◆
,
field redefinitions :: example
As an example, consider the O(d,d) transformation
M=
✓
G
0
BG
(G
1
B
BG
0
1
B)
1
◆
,
which implies
= G(G + B)
1
,
= (G + B) G
1
(G
B) ,
b =
G
b =
B
(1 + B G
1
) G (1
G
1
B) ,
(1 + B G
1
) B (1
G
1
B) .
Andriot, Hohm, Larfors, Lüst, Patalong - 2012
Blumenhagen, Deser, EP, Rennecke, Schmid - 2013
For the Q-flux background discussed earlier, this reads
1
2
2
dy
+
dz
,
ds = dx +
2
2
1+N x
2
Byz =
2
Nx
.
2
2
1+N x
c2 = dx2 + dy 2 + dz 2 ,
ds
byz = N x .
B
field redefinitions :: summary
Summary ::
◾
T-duality transformations are part of the group O(d,d).
◾
Every O(d,d) transformation gives rise to a field
redefinition of the metric and B-field.
◾
The example of the Q-flux background has been
discussed.
outline
1. introduction
2. field redefinitions
3. actions
4. discussion
actions :: lie algebroids
Lie algebroids provide a framework for the description of non-geometric backgrounds.
Hull - 2004
Halmagyi - 2008 & 2009
Berman, Perry - 2010
Blumenhagen, Deser, EP, Rennecke - 2012
actions :: construction
...
Blumenhagen, Deser, EP, Rennecke - 2012
Blumenhagen, Deser, EP, Rennecke, Schmid - 2013
actions :: result
The (NS-NS) action in a non-geometric frame reads
✓
◆
Z
q
1
1
n
2
abc
a
b
S=
d
x
Ĝ
|
|
e
R
⇥
⇥
+
4
D
D
.
abc
a
2
2
12
The following quantities appear ::
◾
⟷ O(d,d) transformation
= d + (G
B) b ,
◾
Ricci scalar for the metric Ĝ
◾
analogue of the B-field
◾
analogue of the H-flux
R̂(Ĝ) = R(G) ,
⇥ 1
bab =
B(
◾
partial derivative Da
Da = (
1 t
b [a bbc] ,
⇥abc = 3 r
1
)
)a m @m .
⇤
ab
,
actions :: symmetries
The above action in a non-geometric frame is invariant under diffeomorphisms.
The action is furthermore invariant under modified gauge transformations ::
1
2
◾
Consider the object
b=
◾
Introduce a differential via
dea =
Fab
◾
c
bab ea ^ eb ,
{ea } 2 (T ⇤ M ).
Fbc a eb ^ ec ,
⇥
t c
= ( ) m Da ( 1 )b m
1
2
The flux ⇥ can be written as
⇥ = db ,
with gauge transformations
b ! b + da .
Db (
1
)a
m
⇤
.
actions :: summary
Summary ::
◾
An action corresponding to a field redefinition can be
constructed (using Lie algebroids).
◾
Its symmetries are diffeomorphisms and modified gauge
transformations.
outline
1. introduction
2. field redefinitions
3. actions
4. discussion
discussion :: recap
1
2
2
dy
+
dz
ds = dx +
1 + N 2 x2
2
non-geometric
background
Byz =
2
Nx
1 + N 2 x2
field redefinition
"good-looking"
background
c2 = dx2 + dy 2 + dz 2
ds
byz = N x
B
Lie algebroids
non-geometric action
& symmetries
...
discussion :: example I
Consider again the Q-flux background ::
MQ =
✓
1
0
0
1
0
1
= @0
0
◆
0
1
1+N 2 x2
Nx
1+N 2 x2
0
1
+N x A
.
1+N 2 x2
1
1+N 2 x2
The non-trivial structure constants of the differential read
y
Fxy = Fxz
z
N 2x
,
=
2
2
1+N x
z
Fxy = Fzx =
The proper gauge field in this non-geometric frame is
b=
y
N x dy ^ dz .
N
.
2
2
1+N x
discussion :: example II
To obtain a globally defined gauge field, one has to employ gauge transformations ::
da = N dy ^ dz
b=
N (x + 1)dy ^ dz
b=
x
N xdy ^ dz
discussion :: example II
To obtain a globally defined gauge field, one has to employ gauge transformations ::
da = N dy ^ dz
b=
N (x + 1)dy ^ dz
b=
x
But :: da = N dy ^ dz has no solution!
N xdy ^ dz
discussion :: a general argument
H1 = T t H2 T
H1
H2
discussion :: a general argument
H1 = T t H2 T
H1
H2
O(d,d) transformation M
Mt H1 M = H10
H20 = Mt H2 M
t
H10 = T 0 H2 T 0
discussion :: a general argument
H1 = T t H2 T
H1
Ggeom
H2
O(d,d) transformation M
T0=M
Mt H1 M = H10
1
H20 = Mt H2 M
t
H10 = T 0 H2 T 0
G0geom
TM
discussion :: interpretation
Two possible interpretations of this example are ::
1. Non-geometric frames (field redefinitions) are not suitable to obtain globally
well-defined non-geometric backgrounds.
2. Different local charts of non-geometric backgrounds are described by
different actions.
summary & conclusions
Summary ::
Conclusion ::
◾
Non-geometric spaces form an important class of string
theory backgrounds.
◾
Non-geometric backgrounds give rise to field
redefinitions.
◾
Actions in non-geometric frames can be constructed
(using Lie algebroids).
◾
These actions and field redefinitions may not be suitable
to obtain globally well-defined backgrounds.
◾
Non-geometric backgrounds deserve further studies.