Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If...
Transcript of Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If...
![Page 1: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/1.jpg)
1
From
Double to Extended Field Theory, Stringy Geometries and Gauged Supergravities
G. Aldazabal, CAB-IB,Bariloche
Frontiers in String Phenomenology,Ringberg, 2014
![Page 2: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/2.jpg)
2
G.A, Andriot, Baron, Bedoya, Berkeley, Berman, Betz, Blair, Blumenhagen, Dall Agata, Dibitetto, Cederwall, Coimbra, Copland, Geissbuller, Fernandez-Melgarejo, Fuchs, Graña, Hassler, Hohm, Hull, Jeon, Kleinschmidt, Kwak, Larfors, Lee, Lust, Malek, Marques, Musaev, Nibbelink, Nuñez, Park, Patalong, Penas, Perry, Renecke, Roest, Rosabal, Rudolph, Samtleben,Thompson, Waldram, West, Zweibach, …
Hitchin, Gualtieri, Petrini, Strickland-Constable, Waldram, ....(GG)
Duff, Siegel, Tseytlin, (1990-1993)
Hull, Zwiebach (2009)
Hohm, Hull, Zwiebach (2010)
O.Hohm talk
![Page 3: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/3.jpg)
3
Motivations:
● Low energy effective field theories for strings, 10(11) dimensional “sugras” miss “stringy dualities”. D(E)FT, could capture duality information
● Like Riemann Geometry describes Gravity, D(E)FT could provide a “Geometry” for strings
● New configurations, non derivable from effective 10 dimensional sugra theories could be reached from D(E)FT. Relevant for Phenomenology, susybreaking, moduli stabilization..
● Metric and 2-form field (RR fields) geometrically unified (Berman's talk)
![Page 4: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/4.jpg)
4
Plan:
● T(U)-duality as a symmetry of a field theory: D(E)FT
● DFT(EFT) a “Geometry” for strings?
● Generalized Scherk-Schwarz reductions. Fluxes and gaugings
● Link with (bosonic) sector of Supergravities
● Comments, problems and outlook
● D(E)FT, “dynamical fluxes” formulation
![Page 5: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/5.jpg)
5
Extended Field Theory(U-duality) explicit in a field theory
Double Field Theory(T-duality) explicit in a field theory
![Page 6: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/6.jpg)
6
Double(Extended) Field Theory
● Coordinate space
● Fields
● Symmetries
● Action
![Page 7: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/7.jpg)
7
DFTdual coordinates
coordinates
space-time
internal, fundamental representation of
space time duals, fictitious
d
![Page 8: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/8.jpg)
8
D1+D3+D5 charge=6+20+6
NS charge=6
KK monopole=6
6 KK momentum
KK +winding=6+6=12
12D fund. of
56D fund. of
T-duality
U-duality
56D fund. ofEFT
string charge
DFT
![Page 9: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/9.jpg)
9
fields
restrict to
Generalized metric Invariant dilaton
Massless bosonic modes of string theory ( )
DFT
EFT
![Page 10: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/10.jpg)
10
is fixed by requiring inside the group
Symmetries
i.e
departure form Lie derivative
leads to
Usual diffeos and gauge transformations in sugra
if
![Page 11: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/11.jpg)
11
In general
Berman, Cederwall, Kleinschmidt,Thompson (2013)
![Page 12: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/12.jpg)
12
Consistency
Two successive gauge transformations parameterized by and , acting on a given field , must reproduce a new gauge transformation parameterized by some given acting on the same vector
define
with
requires
a solution is
DFT is a constrained theory
![Page 13: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/13.jpg)
13
“section condition” or “strong constraint”
Fields depend on half of the coordinates
rotation to only dependence
not really doubled !
Are there other solutions to consistency constraints?
![Page 14: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/14.jpg)
14
DFT action
● up to two derivatives ● up to cubic terms in the metric● invariant● Invariant under generalized diffeos if strong contrained● If
Hull-Hohm-Zwiebach
![Page 15: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/15.jpg)
15
DFT action with dynamical fluxesA frame formulation:
generalized frame
can be parametrized as
with and
Geissbuhler, (2011) Marques, Nuñez, Penas, G.A, Marques, Nuñez (2014)
Hohm, Kwak (2010)
![Page 16: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/16.jpg)
16
Generalized (dynamical) fluxes
then
with generalized Weitzsenbock connection
transforms as a vector
in particular
fluxes
![Page 17: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/17.jpg)
17
DFT action
![Page 18: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/18.jpg)
18
if closure is satisfied
scalar !
under generalized diffeos
then the action is invariant
Is a generalized Ricci scalar?
![Page 19: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/19.jpg)
19
A Double Geometry?Can we “generalize” Riemann geometry to a “Double Geometry”?
diffeo generalized diffeo?
covariant derivative generalized covariant derivative?
connections
Ricci tensor
scalar curvature generalized scalar curvature?
generalized ….?
![Page 20: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/20.jpg)
20
Double Geometry
generalized covariant derivative
metric compatibility, torsionless...
only part of (and )is determined
Generalized Ricci tensor,
covariant if consistency constraints are satisfied, partially determined
![Page 21: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/21.jpg)
21
![Page 22: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/22.jpg)
22
![Page 23: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/23.jpg)
23
Action from generalized Ricci scalar:
![Page 24: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/24.jpg)
24
Scherk-Schwarz dimensional reductions
split coordinates internal
s-time
ansatz
twist encode dependence on internal coordinates
G. A, Baron, Marques, Nuñez, (2011)Geissbuller
![Page 25: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/25.jpg)
25
Dynamical fluxes split
with
choose
( )
strong constraint
![Page 26: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/26.jpg)
26
Gauged DFT
quadratic constraints
strong constraint
M. Grana, D. Marques, (2012)
![Page 27: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/27.jpg)
27
From gauged DFT to gauged supergravity
parametrization
consider
![Page 28: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/28.jpg)
28
space -time fields
gauge transformations
![Page 29: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/29.jpg)
29
and
with
Electric bosonic sector of gauged supergravity
![Page 30: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/30.jpg)
30
gaugings electric
Global symmetry
magnetic
sugra
![Page 31: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/31.jpg)
31
quadratic constraints
![Page 32: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/32.jpg)
32
● There is a subset of new solutions not annihilated by the strong constraint.
quadratic constraints
strong constraint
● Although many non-geometric backgrounds are related to geometric ones through T-duality, there are genuinely non-geometric orbits of fluxes
closure
constant fluxes
![Page 33: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/33.jpg)
33
fluxes=sugra gaugings
Only appear in
SS compactifications of Sugras
here we have also
non-geometric fluxes
geometric
(Andriot' s talk)
![Page 34: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/34.jpg)
34
more explicitely
![Page 35: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/35.jpg)
35
Supergravity 10-dim
SS
Gauged supergravity d-dim (geometrical fluxes)
Gauged supergravity d-dim (all fluxes)
Generalized SS
DFT (10+10)
6-dim 12-dim + 4
![Page 36: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/36.jpg)
36
EFT● Coordinate space
d=4 space-time
internal
● to start with: restrict to internal sector
expectator
Berman, Dall Agata, Cederwall, Coimbra, Hohm, Kleinschmidt, Musaev, Samtleben,Thompson, Waldram, West,Strickland-Constable, Waldram, ....
GA, Graña,Marques, Rosabal ´13, ´14
![Page 37: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/37.jpg)
37
● Fields
Integration measure
![Page 38: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/38.jpg)
38
must preserve invariants symplectic metric
projector to adjoint
● Symmetries
![Page 39: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/39.jpg)
39
Extended (dynamical) fluxes
Namely
![Page 40: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/40.jpg)
40
closure of the algerba
Leibnitz
Are there other solutions to consistency constraints?
“section condition” or “strong constraint”A solution:
fields depend on a 6d slice of the 56d space
Consistency constraints
![Page 41: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/41.jpg)
41
J.I. +
intertwining tensor.
● SS
● constant fluxes
● quadratic constraints
Consistency reads
![Page 42: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/42.jpg)
42
An Extended Geometry?Can we “generalize” Riemann geometry to a “Extended Geometry”?
diffeo extended diffeo?
covariant derivative extended covariant derivative?
connections
Ricci tensor
scalar curvature extended scalar curvature?
extended ….?
![Page 43: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/43.jpg)
43
generalized covariant derivative
torsion free
only part of (and ) is determined
Coimbra, Strickland-Constable,Waldram(2012)
metric compatibility
![Page 44: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/44.jpg)
44
Ricci scalar
quadratic constraints
![Page 45: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/45.jpg)
45
chose and
Internal action
Scalar potential for gauged supergravityde Wit, H.Samtleben, M.Trigiante (2007)
![Page 46: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/46.jpg)
46
Extended Field Theory?
Is it possible to build up a ?EFT
Mixed terms terms require extra structure!
Tensor hierarchy
● start in with gaugings
● then try to uplift to
Gauged Sugra
![Page 47: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/47.jpg)
47
algebra
as in DFT
quadratic constraints
![Page 48: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/48.jpg)
48
Tensor hierarchy
closes if
solved, due to quadratic constraints ( ), by
Intertwining tensor
( )
![Page 49: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/49.jpg)
49
Next step in the hierarchy ( )
intertwining tensor
![Page 50: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/50.jpg)
50
Extended tangent space
Cosistent with: F. Riccioni, D. Steele and P. West, The E(11) origin of all maximal supergravities: The Hierarchy of field-strengths, JHEP 0909, 095 (2009).
![Page 51: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/51.jpg)
51
Generalized diffeomorphisms in gauged maximal supergravity
vectors
Gen. diff
Intertwinig tensor.
projector onto irreps.
![Page 52: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/52.jpg)
52
vectors
frame parametrization
gauge trasformations
dynamical fluxes
Extended tangent space
![Page 53: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/53.jpg)
53Democratic formulation of maximal (bosonic) gauged Sugra in
BI
![Page 54: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/54.jpg)
54
An Extended space-time geometry?
uplift?
Obstructions to uplifting at different levels depending on
Extended gauged sugra
See Berman et al., Hohm et al. ,
![Page 55: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/55.jpg)
55
Intertwinig tensor. Intertwining operartor
Obstruction in
i.e.
hierarchy
obstruction in
![Page 56: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/56.jpg)
56
Summary and Outlook ● Low energy effective field theories for strings miss “stringy dualities”,
D(E)FT could capture duality information
● Like Riemann Geometry describes Gravity, DFT(EFT) could provide a “Geometry” for strings
● Twisted (SS) compactifications lead to full gauged supergravities
DFT
EFT (under construction?, O. Hohm talk)
● All (electric) “gaugings” are obtained from DFT(EFT). New configurations, not derivable from effective 10 dimensional sugra theories can be reached from D(E)FT
Non-Geometry geometrized
(electric bosonic sector of ) half-maximal gauge supergravity
Scalar potential of maximal gauged supergravityDemocratic maximal gauged supergravity
![Page 57: Double to Extended Field Theory, fileInvariant under generalized diffeos if strong contrained If Hull-Hohm-Zwiebach. 15 DFT action with dynamical fluxes A frame formulation: generalized](https://reader030.fdocuments.in/reader030/viewer/2022041204/5d529efa88c99378528bbbda/html5/thumbnails/57.jpg)
57
● Can the strong constraint be generically relaxed ? i.e.: SS (see Betz,B.L,R )
● Can we really include windings?
Strong constraint: no windings
SS: only zero mode
● Is there a double(extended) geometry?: patchings involving all coordinates? (see Hohm,L.Z,.; Berman et al., Hull,..)
truly double manifold with a globally defined basis?
● A consistent truncation of string (field?) theory?
● Truly stringy states? .
● Massive states? Quantum corrections?