Flux formulation of Double Field Theory

Quantum Gravity in the Southern Cone VI Maresias, September 2013

Carmen NúñezIAFE-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 Td 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

• Is there a more appropriate geometrical language with which to understand string theory ?

kk

kjkikjkiijij

kk

kiki

kk

kjkikjkiijij

kk

kiki

kkkk

GGBBG

BBGGB

GBBGG

GGGBG

GG

,

,,,1

• 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 Td

• Double all coordinates• Every object in a duality invariant theory must belong to some representation

of the duality group. In particular, xi have to be supplemented with

XM fundamental rep. O(D,D)

• Raise and lower indices with the O(D,D) metric

• Introduce doubled fields and write

with manifest global O(D,D) symmetry

DMii

Mi

iM

xx

X 2,...,1,~,~

DOUBLE FIELD THEORY

Dixwxp iii

i ,...,1,~,

MNMN

0110

)~,( ii xx )~,(~ xxxxddS DD

DFT L

daxwxp aaa

a ...,,2,1,~,

ix~

Field content

• Focus on bosonic universal gravity sector Gij, Bij,

• Fields are encoded in a

2D × 2D GENERALIZED METRIC

, O(D,D) INVARIANT GENERALIZED DILATON

ijk

ijkiiD

sugra HHReGxdS12142

2 eGd

),( DDOBGBGGB

BGG

ljkl

ikijkj

ik

kjikij

MN

H MNQNPQ

MP HH

The generalized metric spacetime action Hull and Zwiebach (2009)

O. Hohm, C. Hull and B. Zwiebach (2010)

),(~ 2 dexxddS dDDDFT HR

NLKKL

MMN

KLNKL

MMN

NMN

MNMMNMN

NMNMMN ddddd

HHHHHH

HHHHH

21

81

444),(R

0~

ijk

ijkiiD

sugra HHReGxdS12142

• DFT also has a gauge invariance generated by a pair of parameters

• 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

O(D,D) symmetry is manifest

)~( ,i

iM

Strong constraint

• All fields, gauge parameters and products of them satisfy

• It implies there is some dual frame 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)

0)]~,()~,([ xxBxxAMM

)'~,'( xx

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, C. Floriakis, M. Larfors, D. Lust , P. Patalong (2010-2012) – New perspectives on ‘ corrections, O. Hohm, W. Siegel, B. Zwiebach (2012,2013)– New possibilities for upliftings, moduli fixing and dS vacua, Roest et al. (2012)

10D string sugra

SS reduction

on twisted T6

4D gauged sugra T-duality 4D gauged sugra geometric fluxes in d-dim all dual (geometric & ab

c & Habc non- geometric) gaugings Moduli fixing & dS vacua

T-duality

in D-dimDouble Field Theory

SS reductionon twisted T6,6???

Application I: Missing gaugings in geometric compactifications (see Aldazabal’s talk)

Application II: New geometric structuresNon geometry, Generalized Geometry

• Diffeomorphisms of GR

and gauge transformations of 2-form are combined in generalized diffeomorphisms and Lie derivatives

; New term needed so that

• Gauge transformations

The action of the generalized metric formulation is gauge invariant because R(H,d) is a generalized scalar under the strong constraint

L

BLBBGLGG ijijijijijij

,,

ijjiijij BB ~~

)~( ,i

iM PM

PPMMP

PM AAA )(ˆ L

)(,ˆ 22 dMM

dMNMN ee HH L

RR MM

0ˆ MNL

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 Vi and one-forms i as

V + TM T*M . Structures on this larger space The Courant bracket generalizes the Lie bracket

V and are not treated symmetrically

• DFT puts TM and T*M on similar footing by doubling the underlying manifold. Gauge parameters and then C-bracket

For non-doubled M the C-bracket reduces to the Courant bracket

)(21],[],[ 1212212211 2121

VVVV iidVVVV LL

)~( ,i

iM

PMPM

NNM

C ]21[]21[21 21],[

Geometry, connections and curvature

• The action was tendentiously written as It can be shown that the action and EOM of DFT can be obtained from

traces and projections of a generalized Riemann tensor RMNPQ

• 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?

),(~ 2 dexxddS dDDDFT HR

• Basic fields are generalized vielbeins EAM and dilaton

• EAM can be parametrized in terms of vielbein of D-dimensional metric

D-dimensional Minkowski metric

• Arrange the fields in dynamical fluxes:

• Field dependent and non-constant fluxes, that give rise to gaugings or constant fluxes upon compactification (e.g. Fabc=Habc)

Flux formulation of DFT W. Siegel (1993) D. Geissbhuler, D. Marqués, C.N., V. Penas (2013)

NB

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ABMA

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MA

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612

121

412

R

),(~ 2 dEexxddS dDDDFT R

Vanishes under strong constraint

Generalized metric DFT action modulo one strong constraint violating term

The action

The action takes the form of the electric sector of the scalar potential of N=4 D=4 gauged supergravity

This action generalizes the generalized metric formulation, including all terms that vanish under the strong constraint

PA

MPP

MMA

PP

MAM

MMM EEEdd ,

21

Generalized diffeomorphisms

• The closure constraints (generalized Lie derivatives generate closed transformations) take the form:

and they asure that FABC, FA 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

M

MBB

CABC

NBN

AMM

CABC

BACABC

AB

CDE

ABEECDE

ABBCDAABCD

ED

dDEE

FFFDFDZ

FFFDZ

,02

2

043

43

][

][

][][][

Geometric formulation of DFT

• Define covariant derivative on tensors

• Determine the connections imposing set of conditions:

– Compatibility with generalized frame:

– Compatibility with O(D,D) invariant metric

– Compatibility with generalized metric

– Covariance under generalized diffeomorphisms: – Covariance under double Lorentz transformations: Lorentz scalar– Vanishing generalized torsion: Standard torsion non covariant

– Compatibility with generalized dilaton

Only determine some projections of the connections

KB

BMA

NA

KMN

KAM

KAM VVVV

BMA

NBK

ANMK

NMK

NAM EEE 0

MBAMABMPNMNP

ABM

NPM

00

00 ABMNKM SHC

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AB M

AMV

ABCABCABC

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M

F

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][3

)(

ABAB

MP

PM Fd 2

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

• Projections with give and similarly EOM

• Bianchi identities

KLQ

QMNKLQ

QMNKLMNMNKLMNKL RR R

)(21

MNMNMNP H MNKLNLMK PP RR

ABCDECDE

ABBCDAABCD ZFFFD34

34

][][][ R

MKNLKL

MN P RR

Scherk-Schwarz solutions

• All the constraints can be solved restricting the fields and gauge parameters as

where and

quadratic constraints of N=4 gauged sugra

• For these configurations all the consistency constraints are satisfied.• The dynamical fluxes become:

• This ansatz contains the usual decompactified strong contrained case (U=1, =0, xi, i=1,…, D). It is a particular limit in which all the compact dimensions are decompactified.

)()(ˆ,)()(ˆ)( YxddYUxXE MI

IA

MA

)()(ˆ)()( YUxxX MI

IA

AM

),~(;),;~,~( yyYyxyxX

0

2

3

][

][

KLH

IJH

MM

IINJN

MM

JI

NKN

JMM

IIJK

ff

UUUUf

constUUUf

KC

JB

IAIJKABCABC fFF ˆˆˆˆ

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 (FABCFABC).

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?

THANK YOU