Recently, various techniques of solution generating transformations in SUGRA are developed.

Usually, Solution of SUGRA Yang-Baxter deformation or Non-Abelian T-duality

Solution of massive IIA SUGRA Solution of SUGRA NATD sometimes

Sometimes, solutions of Generalized SUGRA are produced.

Generalized Supergravity Equations of motion (GSE) Killing vector

If a target space is a solution of Generalized SUGRA, superstring theory has the kappa-invariance and the rigid scale-invariance. However, the Weyl invariance seems to be broken. Weyl invariance may not be broken! talk by String theory may be consistently defined!

Solution of GSE Solution of SUGRA sometimes YB-deformation/NATD

I will explain that massive IIA SUGRA Generalized SUGRA massive IIA SUGRA

/ are solution generating transformations of SUGRA

If time allows, SUGRA deformed SUGRAs

1. Review of DFT 2. How to derive GSE from DFT? How to derive massive IIA SUGRA? 67 pages 3. Exceptional Field Theory (EFT) (U-duality) more deformed SUGRAs can be derived

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momenta windings conjugate winding coordinates (dual coordinates) Suggested in

d-dim. Generalized coordinates: 2d-dim. d-dim. On the doubled space, there is a natural metric, known as the generalized metric:

String theory / winding momenta O(d,d) T-duality symmetry:

O(d,d) matrix is defined by O(d,d) metric

We raise/lower the O(d,d) indices M, N, by using or

In the usual spacetime, diffeomorphisms are generated by the Lie derivative: The diffeomorphism-invariant gravitational theory is the Einstein gravity:

In the doubled space, diffeomorphisms are generated by the generalized Lie derivative: generalized Derivations Hamiltonian formulation of string: Gauge symmetry of CSFT:

Double Field Theory is the generalized diffeomorphism-invariant theory. Fundamental fields: DFT dilaton T-duality invariant.

Lagrangian of DFT (NS-NS sector): This combination is invariant under the generalized diffeomorphism.

In fact, the generalized diffeo.-invariance of the action requires a condition, called the section condition. This is also required for the closure of the generalized Lie derivative.

fields or diffeomorphism parameter This is trivially satisfied if all fields are independent of the dual coordinates

Another solution is Section condition always removes the dependence on a half of the doubled coordinates. In fact, depending on the choice of the coordinates we can reproduce the SUGRA or GSE.

canonical section Parameterization

Generalized diffeomorphism unifies the usual diffeo. and B-field gauge transformation.

Gravitational theory on the doubled space Fundamental fields (NS-NS sector): 2d dim. generalized diffeo. d dim. diffeo. gauge sym. of

differential geometry on the doubled space

Lagrangian of DFT (NS-NS sector): Einstein-Hilbert action generalized Ricci scalar curvature

Generalized connection torsion free projection not covariant not-yet-fully covariant covariant curvatures

Action E.O.M. generalized Ricci flatness manifestly covariant under generalized diffeomorphisms. E.O.M.

based on creation annihilation

matter Energy-momentum tensor O(d,d) matrix

Introduce double vielbein two vielbeins Local flat metric double Lorentz symmetry O(d) x O(d)

Local flat metric double Lorentz symmetry O(d) x O(d) Ramond-Ramond fields (bi-spinor): In this formulation, there are 2 vielbeins: In order to reproduce the usual SUGRA, we should make an identification:

Under a general O(d,d) rotation The two vielbeins transform differently: In order to keep the identification we should combine O(d,d) rotation and a Lorentz transf.

Under a general O(d,d) rotation only barred index is rotated by the Lorentz transf. The same idea has been applied in the context of the Non-Abelian T-duality. Field strength: (covariant under generalized diffeo.)

In type II SUGRA, we have gravitino: (IIA/IIB) dilatino: (IIA/IIB) Covariant combinations Type II DFT action.

Neglecting the R-R fields, GSE take the form: Constraint : non-dynamical vector I should be a Killing vector of a GSE solution.

1. Non-unimodular YB deformations talk by 2. Non-Abelian T-duality for tracefull structure constants

Original Background (Type V Bianchi universe): We consider 3 Killing vectors, tracefull!

This background does not satisfy the SUGRA e.o.m.

They considered a general ansatz for dilaton, but no solution.

Solution of Generalized Supergravity!

More GSE solutions were obtained from Non-Abelian T-dualities of other Bianchi universes. General formula: The Killing vector is given by SUGRA Generalized SUGRA

Without the R-R fields, GSE have the form:

Due to the Killing property, we can always take an adapted coordinates where is constant. Killing equations become

Now, we can derive GSE by introducing a linear dual-coordinate dependence into the DFT dilaton: Section condition is not violated:

According to the ansatz, derivative of dilaton becomes canonical section

By substituting the ansatz into e.o.m. of DFT, we can reproduce the GSE!

We make an ansatz

Type IIA/IIB GSE:

We have also determined the T-duality rule in the presence of the Killing vector:

canonical section DFT e.o.m. SUGRA e.o.m. GSE with Choice of the section is different.

DFT action is invariant under a constant O(d,d) rotation Unlike the Abelian T-duality of string theory, this is a symmetry even without isometries. formal T-duality

Arbitrary solution of GSE, Perform a formal T-duality along the z-direction Buscher rule Solution of SUGRA A solution of GSE can be mapped to a solution of SUGRA

Conversely, for an arbitrary linear-dilaton solution a formal T-duality GSE solution GSE solution is just a SUGRA solution described in a non-canoical section.

Type II DFT massive type IIA SUGRA annihilation operator

D8-brane solution in DFT D7-brane solution formal T-duality

massive IIA SUGRA Generalized SUGRA massive IIA SUGRA formal T-duality formal T-duality

SUGRA e.g. deformed SUGRAs

We introduced the doubled space. T-dual momentum (P) string winding (F1) canonical conjugate T-dual

DFT There are more branes, which are connected by U-dualities.

It depends on the dimension of the compactification torus Example : type IIB on 10 branes are related by U-duality cannot wrap on

It depends on the dimension of the compactification torus Example : type IIB on

5 3 1 1 0

x

Einstein-frame metric

EFT F1/D1 D3 NS5/D5 M2 M5 KKM Type IIB section M-theory section

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In all of the domain-wall brane backgrounds were constructed as solutions of EFT. They have a linear dual-coordinate dependence. solution of a certain deformed SUGRA. much like D8-brane solution

Redefinition of We can (in principle) obtain the action of a deformed SUGRA.

Similarly, we can obtain deformed SUGRAs for all of the domain-wall branes.