double field theory and the geometry of duality › research › hep › seminars › string... ·...
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Double Field Theory and the Geometry of
DualityCMH & Barton Zwiebach
arXiv:0904.4664, 0908.1792
CMH & BZ & Olaf HohmarXiv:1003.5027, 1006.4664
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Closed String Field Theory
• Captures exotic and complicated structure of interacting string
• Non-polynomial, algebraic structure not Lie algebra, cocycles.
• On torus: winding modes, T-duality.
• Seek subsector capturing exotic structure & duality, simple enough to analyse explicitly
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Strings on d-Torus
• Supergravity limit: symmetry O(d,d)
• String: Perturbative T-duality O(d,d;Z)
• Kaluza-Klein theory: includes momentum modes on torus
• Include string winding or brane wrapping modes? Duality symmetry?
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Strings on Td
X = XL(! + ") + XR(! ! "), X = XL ! XR
X Xconjugate to momentum, to winding no.
dX = !dX !aX = "ab!bX
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Strings on Td
X = XL(! + ") + XR(! ! "), X = XL ! XR
X Xconjugate to momentum, to winding no.
dX = !dX !aX = "ab!bX
Need “auxiliary” for interacting theoryi) Vertex operators ii) String field Kugo & Zwiebach
X
eikL·XL , e
ikR·XR
![x, x, a, a]
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Strings on Td
X = XL(! + ") + XR(! ! "), X = XL ! XR
X Xconjugate to momentum, to winding no.
dX = !dX !aX = "ab!bX
Need “auxiliary” for interacting theoryi) Vertex operators ii) String field Kugo & Zwiebach
X
eikL·XL , e
ikR·XR
![x, x, a, a]
Doubled Torus 2d coordinatesTransform linearly underDoubled sigma model CMH 0406102
O(d, d; Z) X !!
xi
xi
"
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String Field Theoryon Minkowski Space
String field ![X(!), c(!)]
Expand to get infinite set of fields
Xi(!)! xi, oscillators
gij(x), bij(x), !(x), . . . , Cijk...l(x), . . .
Integrating out massive fields gives field theory for
gij(x), bij(x), !(x)
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String Field Theoryon a torus
String field ![X(!), c(!)]
Expand to get infinite set of double fields
Seek double field theory for
Xi(!)! xi, xi, oscillators
gij(x, x), bij(x, x), !(x, x), . . . , Cijk...l(x, x), . . .
gij(x, x), bij(x, x), !(x, x)
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Double Field Theory
• Construct from SFT, DFT of “massless” fields
• Double field theory on doubled torus
• Novel symmetry, reduces to diffeos + B-field trans. in any half-dimensional subtorus
• Backgrounds depending on seen by particles, on seen by winding modes. Backgrounds with both: unfamiliar.
Earlier versions: Siegel, Tseytlin
{xa}{xa}
gij(x, x), bij(x, x), !(x, x)
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• Restriction to “massless” fields NOT a low-energy limit
• Lowest terms in level expansion
• T-duality a manifest symmetry
• General solution of SFT: double fields
• DFT needed for non-geometric backgrounds
• Real dependence on full doubled geometry, dual dimensions not auxiliary or gauge artifact. Double geom. physical and dynamical
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Strings on a Torus• Coordinates
• Momentum
• Winding
• Fourier transform
• Doubled Torus
• String Field Theory gives infinite set of fields
Rn!1,1 ! T d
xa ! xa + 2!
pi = (kµ, pa)
wa (pa, wa) ! Z2d
(kµ, pa, wa)! (yµ, xa, xa)
xa ! xa + 2!Rn!1,1 ! T 2d
n + d = D = 26 or 10
!(yµ, xa, xa)
xi = (yµ, xa)
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T-Duality
• Interchanges momentum and winding
• Equivalence of string theories on dual backgrounds with very different geometries
• String field theory symmetry, provided fields depend on both Kugo, Zwiebach
• For fields not Buscher
• Aim: generalise to fields
!(y) !(y, x, x)
!(y, x, x)
x, x
Dabholkar & CMHGeneralised T-duality
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Free field equn, M mass in D dimensions
M2 ! "(k2 + p2 + w2) =2!! (N + N " 2)
L0 ! L0 = N ! N ! pawa = 0
hij ! {hµ! , hµa, hab}
Constraint
N = N = 1 M2 = 0 pawa = 0Massless states
hij(yµ, xa, xa), bij(yµ, xa, xa), d (yµ, xa, xa)
Constrained fields
!! = 0
!(y, x, x)
! ! !
!xa
!
!xa
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Constrained fields
!! = 0
!(y, x, x)
! ! !
!xa
!
!xa
Momentum space !(kµ, pa, wa) ! = pawa
Momentum space: Dimension n+2dCone: dimension n+2d-1pawa = 0
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Constrained fields
!! = 0
!(y, x, x)
! ! !
!xa
!
!xa
Momentum space !(kµ, pa, wa) ! = pawa
Momentum space: Dimension n+2dCone: dimension n+2d-1pawa = 0
DFT: fields on this cone, with discrete p,wProblem: naive product of fields on cone do not lie on cone. Vertices need projectors
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Constrained fields
!! = 0
!(y, x, x)
! ! !
!xa
!
!xa
Momentum space !(kµ, pa, wa) ! = pawa
Momentum space: Dimension n+2dCone: dimension n+2d-1pawa = 0
DFT: fields on this cone, with discrete p,wProblem: naive product of fields on cone do not lie on cone. Vertices need projectors
Restricted fields: Fields that depend on d of 2d torus momenta, e.g. orSimple subsector, no projectors needed, easier
!(kµ, pa) !(kµ, wa)
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Torus Backgrounds
Gij =!
!µ! 00 Gab
", Bij =
!0 00 Bab
"
xi = {yµ, xa} = {0, xa}xi = {yµ, xa}
Eij ! Gij + Bij
Left and Right Derivatives
Di =!
!xi! Eik
!
!xk, Di =
!
!xi+ Eki
!
!xk
!! = 1
! =12(D2 ! D2) = !2
!
!xi
!
!xi
! =12(D2 + D2) D2 = GijDiDj
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Quadratic Action
S(2) =!
[dxdx]" 1
2eij!eij +
14(Djeij)2 +
14(Dieij)2
!2 d DiDjeij ! 4 d ! d#
!eij = Dj"i + Di"j ,
!d = !14D · "! 1
4D · "
Invariant under
using constraint !! = !! = 0
eij ! eji , Di ! Di , Di ! Di , d! dDiscrete Symmetry
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Comparison with Conventional Actions
Di = !i ! !i , Di = !i + !i
Take Bij = 0 !i ! Gik!
!xk
! = !2 + !2 ! = !2 !i!i
eij = hij + bij
Usual quadratic action!
dx L[h, b, d; ! ]
L[h, b, d; ! ] =14
hij!2hij +
12(!jhij)2 ! 2 d !i!j hij
!4 d !2 d +14
bij!2bij +
12(!jbij)2
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Action + dual action + strange mixing terms
Double Field Theory Action
!hij = "i#j + "j#i + "i#j + "j #i ,
!bij = !("i#j ! "j#i)! ("i#j ! "j #i) ,
!d = ! " · # + " · # .
Diffeos and B-field transformations mixed. Cubic action found for full DFT
S(2) =!
[dxdx]"
L[h, b, d; ! ] + L[!h,!b, d; ! ]
+ (!khik)(!jbij) + (!khik)(!jbij)! 4 d !i!jbij
#
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T-Duality Transformations of Background
g =!
a bc d
"! O(d, d; Z)
X !!
xi
xi
"
E! = (aE + b)(cE + d)"1
X ! =!
x!
x!
"= gX =
!a bc d
" !xx
"
T-duality
transforms as a vector
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T-Duality is a Symmetry of the Action
Fields eij(x, x), d(x, x)
Background Eij
E! = (aE + b)(cE + d)"1
X ! =!
x!
x!
"= gX =
!a bc d
" !xx
"
M ! dt " E ct
M ! dt + Etct
Action invariant if:
eij(X) = Mik Mj
l e!kl(X
!)
d(X) = d!(X !)
With general momentum and winding dependence!
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Projectors and Cocycles
Naive product of constrained fields does not satisfy constraint
String product has explicit projection
Double field theory requires projections, novel formsLeads to a symmetry that is not a Lie algebra, but is a homotopy lie algebra
SFT has non-local cocycles in vertices, DFT should tooCocycles and projectors not needed in cubic action
L!0 !1 = 0, L!0 !2 = 0 but
!A = 0,!B = 0 !(AB) != 0but
L!0 (!1!2) != 0
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General fields
!(x)Fields on Spacetime M
!(x, x)
Restricted Fields on N, T-dual to M !(x!)
Subsector with fields and parameters all restricted to M or N
• Constraint satisfied on all fields and products of fields• No projectors or cocycles• T-duality covariant: independent of choice of N• Can find full non-linear form of gauge transformations• Full gauge algebra, full non-linear action
M,N null wrt O(D,D) metric ds2 = 2dxidxi
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Background Independence?
Action S(E, e, d)
!Eij = !"ij
!eij = "ij !12!"i
kekj ! "kjeik
"+ O(e2)
!!S = 0is background independent:
Constant shift
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Background Independence?
Action S(E, e, d)
!Eij = !"ij
!eij = "ij !12!"i
kekj ! "kjeik
"+ O(e2)
!!S = 0
Background independent field: !!Eij = 0
Eij ! Eij + eij +12ei
kekj +O(e3)
is background independent:
Constant shift
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Background Independence?
Action S(E, e, d)
!Eij = !"ij
!eij = "ij !12!"i
kekj ! "kjeik
"+ O(e2)
!!S = 0
Background independent field: !!Eij = 0
Eij ! Eij + eij +12ei
kekj +O(e3)
E ! E +!1" 1
2e"!1
e
is background independent:
Constant shift
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• Write action and transformations in terms of
• and derivatives
• Find manifestly background independent forms that agree with action and transformations to lowest order
• Unique BI terms that agree with lowest order results. Complete non-linear structure!
Eij , dDi = !i ! Eik!k
Di = !i + Eki!k
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S =!
dxdx e!2d"! 1
4gikgjlDpEkl DpEij
+14gkl
#DjEikDiEjl + DjEki DiElj
$
+#Did DjEij + Did DjEji
$+ 4DidDid
%
BI Action and Transformations
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S =!
dxdx e!2d"! 1
4gikgjlDpEkl DpEij
+14gkl
#DjEikDiEjl + DjEki DiElj
$
+#Did DjEij + Did DjEji
$+ 4DidDid
%
!Eij = "M#MEij
+Di"j ! Dj "i +Di"kEkj + Dj"
kEik
!!d = !12"M#M + #M"M d
BI Action and Transformations
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• Remarkable action, reduces to familiar ones
• uses stringy combination
• checked gauge invariant
• Invariant under O(D,D) if dimensions all non-compact. Broken to discrete subgroup by boundary conditions.
Eij = gij + bij
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2-derivative action
S = S(0)(!, !) + S(1)(!, !) + S(2)(!, !)
Write in terms of usual fieldsS(0)
Eij = gij + bij e!2d =!"ge!2!
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2-derivative action
S = S(0)(!, !) + S(1)(!, !) + S(2)(!, !)
Write in terms of usual fieldsS(0)
Eij = gij + bij e!2d =!"ge!2!
!dx!"ge!2!
"R + 4(!")2 " 1
12H2
#Gives usual action (+ surface term)
S(0) = S(E , d, !)
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2-derivative action
S = S(0)(!, !) + S(1)(!, !) + S(2)(!, !)
Write in terms of usual fieldsS(0)
Eij = gij + bij e!2d =!"ge!2!
!dx!"ge!2!
"R + 4(!")2 " 1
12H2
#Gives usual action (+ surface term)
S(0) = S(E , d, !) S(2) = S(E!1, d, !)T-dual!
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2-derivative action
S = S(0)(!, !) + S(1)(!, !) + S(2)(!, !)
Write in terms of usual fieldsS(0)
Eij = gij + bij e!2d =!"ge!2!
!dx!"ge!2!
"R + 4(!")2 " 1
12H2
#Gives usual action (+ surface term)
S(0) = S(E , d, !) S(2) = S(E!1, d, !)T-dual! strange mixed termsS(1)
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E !(X !) = (aE(X) + b)(cE(X) + d)"1
Generalised T-duality transformations:
d!(X !) = d(X)
h in O(d,d) acts on toroidal coordinates only
X !M !!
x!i
x!i
"= hXM =
#a bc d
$ #xi
xi
$
Buscher if fields independent of toroidal coordinatesGeneralisation to case without isometries
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!M !!
!i
!i
"
!MN =!
0 II 0
"
O(D,D) Covariant Notation
(!, !)! !M
M = 1, ..., 2D
XM !!
xi
xi
"
Parameters
Gauge Algebra
C-Bracket:
[!!1 , !!2 ] = ![!1,!2]C
[!1,!2]C ! [!1,!2]"12
!MN!PQ !P[1 "N !Q
2]
Lie bracket + metric term
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Parameters restricted to NDecompose into vector + 1-form on NC-bracket reduces to Courant bracket on N
!M (X)
Same covariant form of gauge algebra found in similar context by Siegel
Symmetry is Reducible
!M = !MN"N#Parameters of the formdo not act
cf 2-form gauge fieldParameters of the formdo not act
!B = d"! = d"
Gauge algebra determined up to such transformations
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J(!1,!2,!3) ! [ [!1,!2] ,!3 ] + cyclic "= 0
Jacobi Identities not satisfied!
for both C-bracket and Courant-bracket
How can bracket be realised as a symmetry algebra?
[ [!!1 , !!2 ] , !!3 ] + cyclic = !J(!1,!2,!3)
![Page 40: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/40.jpg)
J(!1,!2,!3) ! [ [!1,!2] ,!3 ] + cyclic "= 0
Jacobi Identities not satisfied!
for both C-bracket and Courant-bracket
How can bracket be realised as a symmetry algebra?
Resolution:
J(!1,!2,!3)M = !MN"N#
!J(!1,!2,!3) does not act on fields
[ [!!1 , !!2 ] , !!3 ] + cyclic = !J(!1,!2,!3)
![Page 41: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/41.jpg)
Generalised Metric Formulation
HMN =!
gij !gikbkj
bikgkj gij ! bikgklblj
".
2 Metrics on double space HMN , !MN
![Page 42: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/42.jpg)
Generalised Metric Formulation
HMN =!
gij !gikbkj
bikgkj gij ! bikgklblj
".
HMN ! !MPHPQ!QN
HMPHPN = !MN
2 Metrics on double space HMN , !MN
Constrained metric
![Page 43: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/43.jpg)
Generalised Metric Formulation
HMN =!
gij !gikbkj
bikgkj gij ! bikgklblj
".
hPMhQ
NH!PQ(X !) = HMN (X)
X ! = hX h ! O(D,D)
HMN ! !MPHPQ!QN
HMPHPN = !MN
2 Metrics on double space HMN , !MN
Constrained metric
Covariant Transformation
![Page 44: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/44.jpg)
L =18HMN!MHKL !NHKL !
12HMN!NHKL !LHMK
! 2 !Md !NHMN + 4HMN !Md !Nd
S =!
dxdx e!2d L
O(D,D) covariant action
![Page 45: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/45.jpg)
L =18HMN!MHKL !NHKL !
12HMN!NHKL !LHMK
! 2 !Md !NHMN + 4HMN !Md !Nd
S =!
dxdx e!2d L
!!HMN = "P #PHMN
+ (#M"P ! #P "M )HPN + (#N"P ! #P "N )HMP
Gauge Transformation
O(D,D) covariant action
![Page 46: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/46.jpg)
L =18HMN!MHKL !NHKL !
12HMN!NHKL !LHMK
! 2 !Md !NHMN + 4HMN !Md !Nd
S =!
dxdx e!2d L
!!HMN = "P #PHMN
+ (#M"P ! #P "M )HPN + (#N"P ! #P "N )HMP
!!HMN = !L!HMN
Gauge Transformation
Rewrite as “Generalised Lie Derivative”
O(D,D) covariant action
![Page 47: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/47.jpg)
Generalised Lie Derivative
!L!AMN ! !P "P AM
N
+("M!P""P !M )APN + ("N!P " "P !N )AM
P
A M1...N1...
![Page 48: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/48.jpg)
Generalised Lie Derivative
!L!AMN ! !P "P AM
N
+("M!P""P !M )APN + ("N!P " "P !N )AM
P
!L!AMN = L!AM
N ! !PQ!MR "Q#R APN
+ !PQ!NR "R#Q AMP
A M1...N1...
![Page 49: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/49.jpg)
Generalised Lie Derivative
!L!AMN ! !P "P AM
N
+("M!P""P !M )APN + ("N!P " "P !N )AM
P
!L!AMN = L!AM
N ! !PQ!MR "Q#R APN
+ !PQ!NR "R#Q AMP
A M1...N1...
! "L!1 , "L!2
#= ! "L[!1,!2]C
Algebra given by C-bracket
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D-Bracket!A, B
"D! #LAB
!A, B
"M
D=
!A, B
"M
C+
12!M
#BNAN
$
![Page 51: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/51.jpg)
D-Bracket!A, B
"D! #LAB
!A, B
"M
D=
!A, B
"M
C+
12!M
#BNAN
$
Not skew, but satisfies Jacobi-like identity!A,
!B, C
"D
"D
=!!
A, B"D
", C
"D
+!B,
!A, C
"D
"D
![Page 52: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/52.jpg)
D-Bracket!A, B
"D! #LAB
On restricting to null subspace NC-bracket ➞ Courant bracketD-bracket ➞ Dorfman bracketGen Lie Derivative ➞ GLD of Grana, Minasian, Petrini and Waldram
!A, B
"M
D=
!A, B
"M
C+
12!M
#BNAN
$
Not skew, but satisfies Jacobi-like identity!A,
!B, C
"D
"D
=!!
A, B"D
", C
"D
+!B,
!A, C
"D
"D
![Page 53: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/53.jpg)
Generalized scalar curvature
R ! 4HMN!M!Nd" !M!NHMN
" 4HMN!Md !Nd + 4!MHMN !Nd
+18HMN!MHKL!NHKL "
12HMN!MHKL !KHNL
![Page 54: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/54.jpg)
Generalized scalar curvature
R ! 4HMN!M!Nd" !M!NHMN
" 4HMN!Md !Nd + 4!MHMN !Nd
+18HMN!MHKL!NHKL "
12HMN!MHKL !KHNL
S =!
dx dx e!2dR
![Page 55: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/55.jpg)
Generalized scalar curvature
R ! 4HMN!M!Nd" !M!NHMN
" 4HMN!Md !Nd + 4!MHMN !Nd
+18HMN!MHKL!NHKL "
12HMN!MHKL !KHNL
S =!
dx dx e!2dR
!!R = !L!R = "M#MR!! e!2d = "M (#Me!2d)
Gauge Symmetry
![Page 56: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/56.jpg)
Generalized scalar curvature
R ! 4HMN!M!Nd" !M!NHMN
" 4HMN!Md !Nd + 4!MHMN !Nd
+18HMN!MHKL!NHKL "
12HMN!MHKL !KHNL
S =!
dx dx e!2dR
!!R = !L!R = "M#MR!! e!2d = "M (#Me!2d)
Gauge Symmetry
Field equations give gen. Ricci tensor
![Page 57: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/57.jpg)
M-Theory
• Strings: Doubled torus e.g. T7→ T14
• M-Theory: M-torus e.g. T7→ T56
• U-duality E7 acts on T56
• 7 torus coordinates + 49 coords conjugate brane wrapping modes
![Page 58: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/58.jpg)
Extended Geometry
• Extended geometry: metric and 3-form gauge field
• Generalised metric
• Rewrite of 11-d sugra action in terms of this
CMH; Pacheco & Waldram
Hillman; Berman & Perry
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Double Field Theory• Captures some of the magic of string theory
• Constructed cubic action, quartic could have new stringy features
• T-duality symmetry, cocycles, homotopy Lie, constraints
• For fields restricted to null subspace, have full non-linear action and gauge transformations.
• Background independent, duality covariant
• Courant bracket gauge algebra
![Page 60: Double Field Theory and the Geometry of Duality › research › hep › seminars › string... · Double Field Theory • Construct from SFT, DFT of “massless” fields • Double](https://reader035.vdocuments.mx/reader035/viewer/2022070806/5f0442c37e708231d40d19eb/html5/thumbnails/60.jpg)
• Stringy issues in simpler setting than SFT
• Geometry? Meaning of curvature?
• Use for non-geometric backgrounds?
• Generalised Geometry doubles Tangent space, DFT doubles coordinates.
• Full theory (no restriction)? Does it close on a geometric action with just these fields?
• General spaces, not tori?
• Doubled geometry physical and dynamical