Transcript The effective action of Double Field Theory
Flux formulation of Double Field Theory
Quantum Gravity in the Southern Cone VI Maresias, September 2013 Carmen Núñez IAFE-CONICET-UBA
Outline
• Introduction to Double Field Theory • Applications • Flux formulation • Double geometry • Open questions and problems • Work with G. Aldazabal, W. Baron, D. Geissbhuler, D. Marqués, V. Penas
T-duality
• Closed string theory on a torus T d exhibits O(d,d) symmetry • • Strings experience geometry in a rather different way to point particles.
T-duality establishes equivalence of theories formulated on very different backgrounds
G kk
1
G kk
,
G ki
B ki G kk
,
G ij
G ij
G ki G kj
G kk B ki B kj
,
B ki
G ki G kk
,
B ij
B ij
G ki B kj
G kk B ki G kj
DOUBLE FIELD THEORY
• DFT is constructed from the idea to incorporate the properties of T-duality into a field theory • • • • • Conserved momentum and winding quantum numbers have associated coordinates in T d
p a
x a
,
w
~
a
...,
d
Double all coordinates
p i
x i
,
a w i
x a x
,
i
,
i
1 ,
1 2 , ,...,
D
Every object in a duality invariant theory must belong to some representation of the duality group. In particular,
x i
have to be supplemented with
x i X M
x
~
i i
,
M
i i
, Raise and lower indices with the O(D,D) metric
MN
Introduce doubled fields (
x i
,
M i
) 1 ,..., 2
D
and write
X M
fundamental rep. O(D,D)
S DFT
0 1
1 0
MN
d D xd D
L (
x
, ) with manifest global O(D,D) symmetry
Field content
• Focus on bosonic universal gravity sector
G ij , B ij ,
S sugra
d D x G e
2
R
4
i
i
1 12
H ijk H ijk
• Fields are encoded in a
2D × 2D GENERALIZED METRIC
H
MN
G ij B ik G kj G ij
G ik B kj
B ik G kl B lj
O
(
D
,
D
)
,
H
MP
PQ
H
QN
MN
O(D,D) INVARIANT GENERALIZED DILATON
d
G e
2
The generalized metric spacetime action
Hull and Zwiebach (2009)
S DFT
d D xd D
O. Hohm, C. Hull and B. Zwiebach (2010)
e
2
d R
( H ,
d
) •
R
( H ,
d
) 4 H
MN
M
N d
M
N
H
MN
4 H
MN
M d
N d
4
M
H
MN
N d
1 8 H
MN
M
H
KL
N
H
KL
1 2 H
MN
M
H
KL
K
H
NL
O(D,D) symmetry is manifest 0
S sugra
d D x G e
2
R
4
i
i
1 12
H ijk H ijk
DFT also has a gauge invariance generated by a pair of parameters
M
(
i
,
i
) • Gauge invariance and closure of the gauge algebra lead to a set of differential constraints that restrict the theory. In particular, the constraints can be solved enforcing a stronger condition named strong constraint
Strong constraint
• A ll fields , gauge parameters and products of them satisfy
M
M
[
A
(
x
,
x
)
B
(
x
,
x
)] 0 • • • • It implies there is some dual frame (
x
' , ' ) where fields are not doubled Strongly constrainted DFT displays the O(D,D) symmetry but it is not physically doubled Gauge invariance and closure of gauge transformations weaker condition Certain backgrounds allow relaxations of the strong constraint, producing a truly doubled theory: – – – – Massive type IIA O. Hohm, S. Kwak (2011) Suggested by Scheck-Schwarz compactifications of DFT G. Aldazabal, W. Baron, D. Marqués, C.N. (2011) D. Geissbhuler (2011) Sufficient but not necessary for gauge invariance and closure of gauge algebra M. Graña, D. Marqués (2012) Explicit double solutions found in D. Geissbhuler, D. Marqués, C.N., V. Penas (2013)
Applications of DFT
• DFT has been a powerful tool to explore string theoretical features beyond supergravity and Riemanian geometry • Some recent developments include: – Geometric interpretation of non-geometric gaugings in flux compactifications of string theory G. Aldazabal, W. Baron, D. Marqués, C.N. (2011) D. Geissbhuler (2011) – Identification of new geometric structures – – – – D. Andriot, R. Blumenhagen, O. Hohm, M. Larfors, D. Lust, P. Patalong (2011, 2012) Description of exotic brane orbits F. Hassler, D., Lust (2013) J. de Boer, M. Shigemori (2010, 2012), T. Kikuchi, T. Okada, Y. Sakatani (2012) Non-commutative/non-associative structures in closed string theory R. Blumenhagen, E. Plauschinn, D. Andriot, C. Condeescu, New perspectives on C. Floriakis, M. Larfors, D. Lust , P. Patalong (2010-2012) ‘ corrections, O. Hohm, W. Siegel, B. Zwiebach (2012,2013) New possibilities for upliftings, moduli fixing and dS vacua, Roest et al. (2012)
Application I: Missing gaugings in geometric compactifications
(see Aldazabal’s talk)
10D string sugra
T-duality in D-dim
Double Field Theory
???
SS reduction on twisted T 6,6
SS reduction on twisted T
6
4D gauged sugra geometric fluxes
ab c
& H
abc
T-duality in d-dim
4D gauged sugra
all dual (geometric & non- geometric) gaugings
Moduli fixing & dS vacua
•
Application II: New geometric structures Non geometry, Generalized Geometry
Diffeomorphisms of GR
G ij
G ij
L
L
G ij
,
B ij
B ij
L
B ij
, and gauge transformations of 2-form
B ij
are combined in generalized diffeomorphisms
B ij
i j
j
i
and Lie derivatives
L
ˆ
A M
P
P A M
(
M
P
P
M
)
A P
;
M
(
i
,
i
)
New term needed so that
L
ˆ
MN
0
• Gauge transformations H
MN
L
ˆ
H
MN
,
e
2
d
M
(
M e
2
d
)
The action of the generalized metric formulation is gauge invariant because
R(
H ,
d
) is a generalized scalar under the strong constraint
R
M
M R
DFT vs Generalized Geometry
• • The double geometry underlying DFT differs from ordinary geometry. DFT is a small departure from Generalized Geometry (Hitchin, 2003; Gualtieri, 2004) • Given a manifold M, GG puts together vectors V i V + TM T*M and one-forms . Structures on this larger space i as The Courant bracket V and [
V
1 generalizes the Lie bracket 1 ,
V
2 2 ] [
V
1 ,
V
2 ] are not treated symmetrically
L V
1 2
L V
2 1 1 2
d
(
i V
1 2
i V
2 1 ) • DFT puts TM and T*M on similar footing by doubling the underlying manifold. Gauge parameters
M
( ~
i
,
i
) and then C-bracket [ 1 For non-doubled , M 2 ]
M C
N
[ 1
N
2
M
] 1
P
[ 1
M
2 ]
P
2 the C-bracket reduces to the Courant bracket
Geometry, connections and curvature
•
S DFT
d D xd D e
2
d R
(
H
,
d
)
It can be shown that the action and EOM of DFT can be obtained from traces and projections of a generalized Riemann tensor
R MNPQ
• The construction goes beyond Riemannian geometry because it is based on generalized rather than standard Lie derivatives • The notions of connections, torsion and curvature have to be generalized • E.g. the vanishing torsion and compatibility conditions do not completely determine the connections and curvatures, but only fix some of their projections I. Jeon, K. Lee, J. Park (2011), O. Hohm, B. Zwiebach (2012) • Strong constraint was assumed in these constructions. Can it be relaxed?
•
Flux formulation of DFT
W. Siegel (1993) D. Geissbhuler, D. Marqués, C.N., V. Penas (2013) Basic fields are generalized vielbeins E A M and dilaton H
MN
E A M S AB E B N
;
MN
E A M
AB E B N
• • • E A M can be parametrized in terms of vielbein of D-dimensional metric
G ij
e a i s ab e b j
,
s ab
D-dimensional Minkowski metric
E A M
e a i
0
e a k B ki
,
e a i S AB
s ab
0 0
s ab
Arrange the fields in dynamical fluxes:
F ABC
3
E
[
A M
M E B N E C
]
N
E C M L E A E B M F A
E B M
M E B N E AN
2
E A M
M d
e
2
d L E A e
2
d
Field dependent and non-constant fluxes, that give rise to gaugings or constant fluxes upon compactification (e.g. F abc =H abc )
The action
S DFT
d D xd D e
2
d R
(
E
,
d
)
R
S AB
2
E A M
M F B
F A F B
F ABC F DEF
1 4
S AD
BE
CF
1 12
S AD S BE S CF
2
E A M
M F A
F A F A
1 6
F ABC F ABC
The action takes the form of the electric sector of the scalar potential of N=4 D=4 gauged supergravity Vanishes under strong constraint Generalized metric DFT action modulo one strong constraint violating term This action generalizes the generalized metric formulation, including all terms that vanish under the strong constraint
d
M
M d
Generalized diffeomorphisms
1 2
M
M
,
E A M
P
P E A M
M
P
P
M
E A P
• The closure constraints (generalized Lie derivatives generate closed transformations) take the form:
Z AB
3
Z ABCD
D
[
A F BCD
]
F
[
AB E F CD
]
E
4
D
M C
F CAB E
[
A
N
2
D
[
E B A
]
F B M N
]
F C
2
C F CAB AB D C d
3 4
E
[
AB
E CD
] 0 , 0
D B
E B M
M
and they asure that F ABC , F A transform as scalars and S is gauge invariant • Imposing these conditions only requires a relaxed version of strong constraint the theory admits truly double fields • Constraints can be interpreted as Bianchi identities for generalized Riemann tensor
Geometric formulation of DFT
• Define covariant derivative on tensors
M V A K
M V A K
MN K V A N
MA B V B K
• Determine the connections imposing set of conditions: – Compatibility with generalized frame:
M E A N
0
MK N
MK N
E A K E B N
MA B
– – – – – – Compatibility with O(D,D) invariant metric
M
NP
0
M
AB
0 Compatibility with generalized metric
M
H
NK
0
M S AB
0 Covariance under generalized diffeomorphisms:
MNP
MPN C AB
M V A M
Vanishing generalized torsion: Standard torsion non covariant
P
(
P
AB C
MAB
)
V
M
MBA
QP M
Q V P
Compatibility with generalized dilaton
PM P
2
M d
B BA
F A
Only determine some projections of the connections
ABC
3 [
ABC
]
F ABC
Generalized curvature
• The standard Riemann tensor in planar indices is not a scalar under generalized diffeomorphisms • It can be modified adding new terms, leading to
R MNKL
R MNKL
R KLMN
QMN
Q KL
QMN
Q KL
• • Projections with and similarly EOM
P
MN
1 2 (
MN
H
MN
)
R MN
P
KL R MKNL
Bianchi identities give
R
P
MK P
NL R MNKL R
[
ABCD
] 4 3
D
[
A F BCD
]
F
[
AB E F CD
]
E
4 3
Z ABCD
Scherk-Schwarz solutions
• All the constraints can be solved restricting the fields and gauge parameters as
E A M
(
X
) ˆ
A I
(
x
)
U I M
(
Y
) ,
d
d
ˆ (
x
) (
Y
) where
X
( , ;
x
,
y
) ;
Y
M
(
X
) (
y
,
y
) and
A
(
x
) ˆ
A I f
(
x
)
U I M
(
Y
)
IJK
3
U
[
I M
M U J N U K
]
N f I
U J M
M U JN U IN
const
2
U I M
M
quadratic constraints of N=4 gauged sugra
f H
[
IJ f H KL
] 0 • • For these configurations all the consistency constraints are satisfied.
The dynamical fluxes become:
F ABC
ˆ
ABC
f IJK
ˆ
A I
ˆ
B J
ˆ
C K
• This ansatz contains the usual decompactified strong contrained case (U=1, =0,
x i
,
i=1,…, D
). It is a particular limit in which all the compact dimensions are decompactified.
Conclusions
• Presented formulation of DFT in terms of dynamical and field dependent fluxes.
• The gauge consistency constraints take the form of quadratic constraints for the fluxes, that admit solutions that violate the SC allows to go beyond supergravity • Computed connections and curvatures on the double space under assumption that covariance is achieved upon generalized quadratic constraints, rather than SC, which can be interpreted as BI.
• Interestingly, this procedure gives rise to all the SC-violating terms in the action, which are gauge invariant and appear systematically • This completes the original formulation of DFT, incorporating the missing terms that allow to make contact with half-maximal gauged sugra, containing all duality orbits of non-geometric fluxes (F ABC F ABC ).
Open questions
• • Some elements of the O(D,D) geometry have been understood, but it is important to better understand the geometry underlying DFT • Can this construction be extended beyond tori? Calabi-Yau?
• ’ corrections. Inner product and C-bracket are corrected deformation of Courant bracket and other structures in GG Beyond T-duality? U-duality?
• • Relation between DFT and string theory. Is this a consistent truncation of string theory? No massive states, but fully consistent Worldsheet theory?