㻰㼛㼡㼎㼘㼑㻛㻱㼤㼏㼑㼜㼠㼕㼛㼚㼍㼘㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥㼍㼚㼐㻌㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㻿㼡㼜㼑㼞㼓㼞㼍㼢㼕㼠㼥㼅㼡㼔㼛㻌㻿㼍㼗㼍㼠㼍㼚㼕㻔㻷㼥㼛㼠㼛㻌㻼㼞㼑㼒㻚㻌㼁㼚㼕㼢㻚㻌㼛㼒㻌㻹㼑㼐㼕㼏㼕㼚㼑㻕㻿㼠㼞㼕㼚㼓㻦㻌㼀㻙㼐㼡㼍㼘㼕㼠㼥㻘㻌㻵㼚㼠㼑㼓㼞㼍㼎㼕㼘㼕㼠㼥㻌㼍㼚㼐㻌㻳㼑㼛㼙㼑㼠㼞㼥㻠㻙㻤㻌㻹㼍㼞㼏㼔㻌㻞㻜㻝㻥㻿㼛㼘㼡㼠㼕㼛㼚㻌㼓㼑㼚㼑㼞㼍㼠㼕㼚㼓㻌㼠㼞㼍㼚㼟㼒㼛㼞㼙㼍㼠㼕㼛㼚㼟
Recently, various techniques of solution generatingtransformations in SUGRA are developed.

‫؞‬㼅㼅㼍㼚㼓㻙㻮㼍㼤㼠㼑㼞㻌㼐㼑㼒㼛㼞㼙㼍㼠㼕㼛㼚㼇㻷㼘㼕㼙㼏㼕㼗㻘㻌㻰㼑㼘㼐㼡㼏㻘㻌㻹㼍㼓㼞㼛㻘㻌㼂㼕㼏㼑㼐㼛㻘㻌㻷㼍㼣㼍㼓㼡㼏㼔㼕㻘㻌㻹㼍㼠㼟㼡㼙㼛㼠㼛㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻭㼞㼡㼠㼥㼡㼚㼛㼢㻘㻌㻮㼛㼞㼟㼍㼠㼛㻘㻌㻲㼞㼛㼘㼛㼢㻘㻌㻴㼛㼍㼞㼑㻘㻌㻾㼛㼕㼎㼍㼚㻘㻌㼀㼟㼑㼥㼠㼘㼕㼚㻘㻌㻷㼥㼛㼚㼛㻘㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㻻㼞㼘㼍㼚㼐㼛㻘㻌㻾㼑㼒㼒㼑㼞㼠㻘㻌㼃㼡㼘㼒㼒㻘㻌㼢㼍㼚㻌㼀㼛㼚㼓㼑㼞㼑㼚㻘㻌㻻㼟㼠㼑㼚㻘㻌㼀㼔㼛㼙㼜㼟㼛㼚㻘㻌㻭㼞㼍㼡㼖㼛㻘㻌㻮㼍㼗㼔㼙㼍㼠㼛㼢㻘㻌㽞㻌㻯㼛㼘㼓㽬㼕㼚㻘㻌㻿㼔㼑㼕㼗㼔㻙㻶㼍㼎㼎㼍㼞㼕㻘 䈈㼉‫؞‬㻺㼛㼚㻙㻭㼎㼑㼘㼕㼍㼚㻌㼀㻙㼐㼡㼍㼘㼕㼠㼥㼇㼐㼑㻌㼘㼍㻌㻻㼟㼟㼍㻘㻌㻽㼡㼑㼢㼑㼐㼛㻘㻌㻳㼕㼢㼑㼛㼚㻘㻌㻾㼛㼏㼑㼗㻘㻌㻭㼘㼢㼍㼞㼑㼦㻘㻌㻌㻭㼘㼢㼍㼞㼑㼦㻙㻳㼍㼡㼙㼑㻘㻌㻸㼛㼦㼍㼚㼛㻘㻌㻷㼘㼕㼙㼏㼕㼗㻘㻌㻿㼑㼢㼑㼞㼍㻘㻌㻿㼒㼑㼠㼟㼛㼟㻘㻌㼀㼔㼛㼙㼜㼟㼛㼚㻘㻌㽞㻌㻯㼛㼘㼓㽬㼕㼚㻘㻌㻴㼛㼍㼞㼑㻘㼀㼟㼑㼥㼠㼘㼕㼚㻘㻌㻮㼛㼞㼟㼍㼠㼛㻘㻌㼃㼡㼘㼒㼒㻘㻌䈈䈈㼉㻿㼛㼘㼡㼠㼕㼛㼚㻌㼓㼑㼚㼑㼞㼍㼠㼕㼚㼓㻌㼠㼞㼍㼚㼟㼒㼛㼞㼙㼍㼠㼕㼛㼚㼟
Usually,Solution of SUGRA

Yang-Baxter deformationor Non-Abelian T-duality㻿㼛㼘㼡㼠㼕㼛㼚㻌㼓㼑㼚㼑㼞㼍㼠㼕㼚㼓㻌㼠㼞㼍㼚㼟㼒㼛㼞㼙㼍㼠㼕㼛㼚㼟
Solution of massive IIA SUGRA
Solution of SUGRA

NATDsometimes㼇㻿㼒㼑㼠㼟㼛㼟㻘㻌㼀㼔㼛㼙㼜㼟㼛㼚㻘㻌㻝㻜㻝㻞㻚㻝㻟㻞㻜㻧㻸㼛㼦㼍㼚㼛㻘㻌㽞㻌㻯㼛㼘㼓㽬㼕㼚㻘㻌㻿㼒㼑㼠㼟㼛㼟㻘㻌㼀㼔㼛㼙㼜㼟㼛㼚㻘㻌㻝㻝㻜㻠㻚㻡㻝㻥㻢㻧㻌䈈㼉㻿㼛㼘㼡㼠㼕㼛㼚㻌㼓㼑㼚㼑㼞㼍㼠㼕㼚㼓㻌㼠㼞㼍㼚㼟㼒㼛㼞㼙㼍㼠㼕㼛㼚㼟
Sometimes, solutions of “Generalized SUGRA”
are produced.
㼅㼍㼚㼓㻙㻮㼍㼤㼠㼑㼞㻌㼐㼑㼒㼛㼞㼙㼍㼠㼕㼛㼚㼇㻭㼞㼡㼠㼥㼡㼚㼛㼢㻘㻌㻮㼛㼞㼟㼍㼠㼛㻘㻌㻲㼞㼛㼘㼛㼢㻘㻌㻝㻡㻜㻣㻚㻜㻠㻞㻟㻥㻧㻭㼞㼡㼠㼥㼡㼚㼛㼢㻘㻌㻲㼞㼛㼘㼛㼢㻘㻌㻴㼛㼍㼞㼑㻘㻌㻾㼛㼕㼎㼍㼚㻘㻌㼀㼟㼑㼥㼠㼘㼕㼚㻘㻌㻝㻡㻝㻝㻚㻜㻡㻣㻥㻡㻧㻷㼥㼛㼚㼛㻙㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻢㻜㻡㻚㻜㻞㻡㻝㻥㻧㻻㼞㼘㼍㼚㼐㼛㻘㻌㻾㼑㼒㼒㼑㼞㼠㻘㻌㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻢㻜㻣㻚㻜㻜㻣㻥㻡㻧㻌㻲㼑㼞㼚㼍㼚㼐㼑㼦㻙㻹㼑㼘㼓㼍㼞㼑㼖㼛㻘㻌㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻣㻝㻜㻚㻜㻢㻤㻠㻥㻧㻌䈈㼉㻺㼛㼚㻙㻭㼎㼑㼘㼕㼍㼚㻌㼀㻙㼐㼡㼍㼘㼕㼠㼥㼇㻳㼍㼟㼜㼑㼞㼕㼚㼕㻘㻌㻾㼕㼏㼏㼕㻘㻌㼂㼑㼚㼑㼦㼕㼍㼚㼛㻘㻌㼔㼑㼜㻙㼠㼔㻛㻥㻟㻜㻤㻝㻝㻞㻧㻲㼑㼞㼚㼍㼚㼐㼑㼦㻙㻹㼑㼘㼓㼍㼞㼑㼖㼛㻘㻌㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻣㻝㻜㻚㻜㻢㻤㻠㻥㻧㻴㼛㼚㼓㻘㻌㻷㼕㼙㻘㻌㽞㻌㻯㼛㼘㼓㽬㼕㼚㻘 㻝㻤㻜㻝㻚㻜㻥㻡㻢㻣㻧㻌㻌䈈㼉㻔㼀㼥㼜㼑㻌㻵㻵㻮㻕㻌㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㻿㼡㼜㼑㼞㼓㼞㼍㼢㼕㼠㼥㼇㻭㼞㼡㼠㼥㼡㼚㼛㼢㻙㻲㼞㼛㼘㼛㼢㻙㻴㼛㼍㼞㼑㻙㻾㼛㼕㼎㼍㼚㻙㼀㼟㼑㼥㼠㼘㼕㼚 䇻㻝㻡㻧㻔㻺㻿㻙㻺㻿㻌㼟㼑㼏㼠㼛㼞㻕 㻴㼡㼘㼘㻙㼀㼛㼣㼚㼟㼑㼚㼐㻌䇻㻤㻢㻌㼉
Generalized Supergravity
Equations of motion (GSE)

Killing vector㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㻿㼡㼜㼑㼞㼓㼞㼍㼢㼕㼠㼥
If a target space is a solution of Generalized SUGRA,superstring theory has the kappa-invariance㼇㼀㼟㼑㼥㼠㼘㼕㼚㻘㻌㼃㼡㼘㼒㼒㻌㻝㻢㻜㻡㻚㻜㻠㻤㻤㻠㼉and the rigid scale-invariance.
㼇㻴㼡㼘㼘㻘㻌㼀㼛㼣㼚㼟㼑㼚㼐㻌䇻㻤㻢㻧㻭㼞㼡㼠㼥㼡㼚㼛㼢㻘㻌㻲㼞㼛㼘㼛㼢㻘㻌㻴㼛㼍㼞㼑㻘㻌㻾㼛㼕㼎㼍㼚㻘㻌㼀㼟㼑㼥㼠㼘㼕㼚 䇻㻝㻡㼉

However, the Weyl invariance seems to be broken.
㼇㻲㼑㼞㼚㼍㼚㼐㼑㼦㻙㻹㼑㼘㼓㼍㼞㼑㼖㼛㻘㻌㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍 㻝㻤㻝㻝㻚㻝㻜㻢㻜㻜㼉

Weyl invariance may not be broken!
talk by 㻷㼑㼚㼠㼍㼞㼛㼔 㼅㼛㼟㼔㼕㼐㼍String theory may be consistently defined!

㻿㼛㼘㼡㼠㼕㼛㼚㻌㼓㼑㼚㼑㼞㼍㼠㼕㼚㼓㻌㼠㼞㼍㼚㼟㼒㼛㼞㼙㼍㼠㼕㼛㼚㼟
Solution of GSE
Solution of SUGRA

sometimes

YB-deformation/NATD㻵㼚㻌㼠㼔㼕㼟㻌㼠㼍㼘㼗I will explain that㻰㼛㼡㼎㼘㼑㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥

massive IIASUGRAGeneralizedSUGRA

massive IIASUGRA㼅㻮㻙㼐㼑㼒㼛㼞㼙㼍㼠㼕㼛㼚/㻺㻭㼀㻰 aresolution generating transformations of 㻰㻲㼀㻰㼛㼡㼎㼘㼑㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥

SUGRAIf time allows,㻱㼤㼏㼑㼜㼠㼕㼛㼚㼍㼘㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥㻰㼛㼡㼎㼘㼑㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥

SUGRAdeformed SUGRAs㻼㼘㼍㼚
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㻝㻚㻌㻾㼑㼢㼕㼑㼣㻌㼛㼒㻌㻰㻲㼀㼇㻿㼕㼑㼓㼑㼘㻌䇻㻥㻟㻧㻝㻜㻌㼥㼑㼍㼞㼟㻌㼍㼓㼛㻍㻴㼡㼘㼘㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻘㻌㻜㻥㻜㻠㻚㻠㻢㻢㻠㻧㻴㼛㼔㼙㻘㻌㻴㼡㼘㼘㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻘㻌㻝㻜㻜㻢㻚㻠㻤㻞㻟㻧㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻚㻙㻴㻚㻌㻼㼍㼞㼗㻘㻌㻝㻜㻝㻝㻚㻝㻟㻞㻠㻧㻌䈈㼉㻿㼑㼢㼑㼞㼍㼘㻌㼒㼛㼞㼙㼡㼘㼍㼠㼕㼛㼚㼟㻌㼛㼒㻌㻰㻲㼀‫؞‬㻳㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㼙㼑㼠㼞㼕㼏㻌㼒㼛㼞㼙㼡㼘㼍㼠㼕㼛㼚㻦㻌㻌㼠㼛㼞㼛㼕㼐㼍㼘㻌㻰㻲㼀㼇㻴㼛㼔㼙㻘㻌㻴㼡㼘㼘㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻧㻌㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻚㻙㻴㻚㻌㻼㼍㼞㼗㻧㻌䈈㼉‫؞‬㻲㼘㼡㼤㻌㼒㼛㼞㼙㼡㼘㼍㼠㼕㼛㼚㼇㻳㼑㼕㼟㼟㼎㼡㼔㼘㼑㼞㻘㻌㻹㼍㼞㼝㼡㼑㼟㻘㻌㻺㼡㼚㼑㼦㻘㻌㻼㼑㼚㼍㼟㻘㻌㻝㻟㻜㻠㻚㻝㻠㻣㻞㼉talk by 㻲㼍㼘㼗㻌㻴㼍㼟㼟㼘㼑㼞‫؞‬㻰㻲㼀㻌㼛㼚㻌㼓㼞㼛㼡㼜㻌㼙㼍㼚㼕㼒㼛㼘㼐㼇㻮㼘㼡㼙㼑㼚㼔㼍㼓㼑㼚㻘㻌㻴㼍㼟㼟㼘㼑㼞㻘㻌㻸㼡㼟㼠㻘㻌㻝㻠㻝㻜㻚㻢㻟㻣㻠㼉‫؞‬㻰㻲㼀㻌㼕㼚㻌㼟㼡㼜㼑㼞㼙㼍㼚㼕㼒㼛㼘㼐 㼒㼛㼞㼙㼡㼘㼍㼠㼕㼛㼚㼇㻯㼍㼞㼛㼣㻙㼃㼍㼠㼍㼙㼡㼞㼍㻘㻌㻵㼗㼑㼐㼍㻘㻌㻷㼍㼚㼑㼗㼛㻘㻌㼃㼍㼠㼍㼙㼡㼞㼍㻘㻌㻝㻤㻝㻞㻚㻜㻟㻠㻢㻠㼉…

talk by 㻺㼛㼞㼕㼍㼗㼕 㻵㼗㼑㼐㼍㼀㻙㼐㼡㼍㼘㼕㼠㼥

momenta

windings

conjugate

winding coordinates(dual coordinates)
Suggested in 㼇㻰㼡㼒㼒㻌䇻㻥㻜㻧㻌㼀㼟㼑㼥㼠㼘㼕㼚㻌䇺㻥㻝㻧㻷㼡㼓㼛㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔 䇻㻥㻞㻧㻌㻿㼕㼑㼓㼑㼘㻌䇻㻥㻟㻧㻌䈈㼉㻰㼛㼡㼎㼘㼑㼐㻌㼟㼜㼍㼏㼑d-dim.
Generalized coordinates:

2d-dim.

d-dim.

On the doubled space, there is a natural metric,
known as the generalized metric:㻰㼛㼡㼎㼘㼑㼐㻌㼟㼜㼍㼏㼑ࢀࢊString theory / ࢀࢊwinding
momentaO(d,d) T-duality symmetry:㻻㻔㼐㻘㼐㻕㻌㼙㼑㼠㼞㼕㼏O(d,d) matrix is defined byO(d,d) metric㻿㼠㼍㼚㼐㼍㼞㼐㻌㼏㼛㼚㼢㼑㼚㼠㼕㼛㼚We raise/lower the O(d,d) indices M, N, … by usingor㻰㼕㼒㼒㼑㼛㼙㼛㼞㼜㼔㼕㼟㼙In the usual spacetime,diffeomorphisms are generated by the Lie derivative:

The diffeomorphism-invariant gravitational theoryis the Einstein gravity:㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㼐㼕㼒㼒㼑㼛㼙㼛㼞㼜㼔㼕㼟㼙In the doubled space,diffeomorphisms are generated by
the generalized Lie derivative:“generalized”

Derivations
Hamiltonian formulation of string: 㼇㻿㼕㼑㼓㼑㼘㻌㼔㼑㼜㻙㼠㼔㻛㻥㻟㻜㻡㻜㻣㻟㼉
Gauge symmetry of CSFT: 㼇㻴㼡㼘㼘㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻌㻜㻥㻜㻠㻚㻠㻢㻢㻠㼉㻰㼛㼡㼎㼘㼑㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥Double Field Theory is
the generalized diffeomorphism-invariant theory.
Fundamental fields:

DFT dilaton

T-duality invariant.

㻰㻲㼀㻌㼍㼏㼠㼕㼛㼚
Lagrangian of DFT (NS-NS sector):㼇㻴㼛㼔㼙㻘㻌㻴㼡㼘㼘㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻘㻌㻝㻜㻜㻢㻚㻠㻤㻞㻟㼉This combination is invariant
under the generalized diffeomorphism.

㻯㼛㼚㼟㼕㼟㼠㼑㼚㼏㼥㻌㼏㼛㼚㼐㼕㼠㼕㼛㼚
In fact, the generalized diffeo.-invariance of the action
requires a condition, called the section condition.

This is also required forthe closure of the generalized Lie derivative.

㻿㼑㼏㼠㼕㼛㼚㻌㼏㼛㼚㼐㼕㼠㼕㼛㼚fields or diffeomorphism parameter
This is trivially satisfied if all fields
are independent of the dual coordinates㻿㼑㼏㼠㼕㼛㼚㻌㼏㼛㼚㼐㼕㼠㼕㼛㼚

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

㻯㼛㼚㼢㼑㼚㼠㼕㼛㼚㼍㼘㻌㻿㼁㻳㻾㻭Parameterization

canonicalsection㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㼐㼕㼒㼒㼑㼛㼙㼛㼞㼜㼔㼕㼟㼙

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

㻿㼔㼛㼞㼠㻌㻿㼡㼙㼙㼍㼞㼥Gravitational theory
on the doubled space
Fundamental fields (NS-NS sector):

2d dim. generalized diffeo.

d dim. diffeo.
gauge sym. of㻿㼕㼐㼑㻌㼞㼑㼙㼍㼞㼗differential geometryon the doubled space㼇㻿㼕㼑㼓㼑㼘㻌䇻㻥㻟㻧㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻙㻴㻚㻌㻼㼍㼞㼗 䇻㻝㻜㻧㻴㼛㼔㼙㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻌䇻㻝㻞㼉㻰㻲㼀㻌㼍㼏㼠㼕㼛㼚
Lagrangian of DFT (NS-NS sector):“Einstein-Hilbert” action

generalized Ricci scalar curvature㼇㻿㼕㼑㼓㼑㼘㻌䇻㻥㻟㻧㻌㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻙㻴㻚㻌㻼㼍㼞㼗 䇻㻝㻜㻧㻴㼛㼔㼙㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻌䇻㻝㻞㼉㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㻾㼕㼏㼏㼕㻌㼠㼑㼚㼟㼛㼞
Generalized connection㼇㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻙㻴㻚㻌㻼㼍㼞㼗 䇻㻝㻜㻧㻴㼛㼔㼙㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻌䇻㻝㻞㼉“torsion” free

not covariant
not-yet-fully covariant
projection

covariant curvatures㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㻾㼕㼏㼏㼕㻌㼠㼑㼚㼟㼛㼞
ActionE.O.M.

generalized
Ricci flatness

manifestly covariant undergeneralized diffeomorphisms.
E.O.M.

㻾㻙㻾㻌㼒㼕㼑㼘㼐㼟㻛㼒㼑㼞㼙㼕㼛㼚㼟㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞㻌㻝㻛㻞㼇㻴㼛㼔㼙㻘㻌㻷㼣㼍㼗㻘㻌㼆㼣㼕㼑㼎㼍㼏㼔㻘㻌䇻㻝㻝㼉
based on 㼇㻲㼡㼗㼡㼙㼍㻘㻌㻻㼛㼠㼍㻘㻌㼀㼍㼚㼍㼗㼍㻌䇻㻥㻥㼉㻻㻔㼐㻘㼐㻕㻌㼟㼜㼕㼚㼛㼞

creation annihilation㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞㻌㻝㻛㻞
matter㻱㻚㻻㻚㻹㻚㻌Energy-momentum tensor㻻㻔㼐㻘㼐㻕㻌㼠㼞㼍㼚㼟㼒㼛㼞㼙㼍㼠㼕㼛㼚O(d,d) matrix㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞㻌㻞㻛㻞
Introduce double vielbein㼇㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻚㻙㻴㻚㻌㻼㼍㼞㼗 䇻㻝㻞㼉㼎㼍㼟㼑㼐㻌㼛㼚㻌㼇㻴㼍㼟㼟㼍㼚㻌䇺㻜㻝㻌㼉two vielbeins

Local flat metricdouble Lorentz symmetry O(d) x O(d)㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞㻌㻞㻛㻞
Local flat metricdouble 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:㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞㻌㻞㻛㻞
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.

㼇㻴㼍㼟㼟㼍㼚㻌䇻㻜㻝㻧㻌㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻚㻙㻴㻚㻌㻼㼍㼞㼗㻌䇻㻝㻞 㼉㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞㻌㻞㻛㻞
Under a general O(d,d) rotation
only barred index is rotated by the Lorentz transf.

The same idea has been appliedin the context of the Non-Abelian T-duality.
㼇㻿㼒㼑㼠㼟㼛㼟㻘㻌㼀㼔㼛㼙㼜㼟㼛㼚㻘㻌㻝㻜㻝㻞㻚㻝㻟㻞㻜㼉

Field strength:(covariant undergeneralized diffeo.)㼇㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻚㻙㻴㻚㻌㻼㼍㼞㼗㻌䇻㻝㻞 㼉㻲㼑㼞㼙㼕㼛㼚㼟In type II SUGRA, we havegravitino:(IIA/IIB)

dilatino:(IIA/IIB)

Covariant combinations
Type II DFT action. 㼇㻵㻚㻌㻶㼑㼛㼚㻘㻌㻷㻚㻌㻸㼑㼑㻘㻌㻶㻚㻙㻴㻚㻌㻼㼍㼞㼗㻌㻘㻌㼅㻚㻌㻿㼡㼔㻌䇻㻝㻞 㼉㼏㻚㼒㻚㻌㼇㻯㼛㼕㼙㼎㼞㼍㻘㻌㻿㼠㼞㼕㼏㼗㼘㼍㼚㼐㻙㻯㼛㼚㼟㼠㼍㼎㼘㼑㻘㻌㼃㼍㼘㼐㼞㼍㼙 㻓㻝㻝㼉㻞㻚㻌㻳㻿㻱㻌㼒㼞㼛㼙㻌㻰㻲㼀Neglecting the R-R fields, GSE take the form:

non-dynamical vector
Constraint :I should be a Killing vector of a GSE solution.

㻱㼤㼍㼙㼜㼘㼑㻌㼛㼒㻌㻳㻿㻱㻌㼟㼛㼘㼡㼠㼕㼛㼚㻿㼛㼘㼡㼠㼕㼛㼚㼟㻌㼛㼒㻌㻳㻿㻱
1. Non-unimodular YB deformations㼇㻮㼛㼞㼟㼍㼠㼛㻘㻌㼃㼡㼘㼒㼒㻘㻌㻝㻢㻜㻤㻚㻜㻟㻡㻣㻜㼉

talk by 㻶㼡㼚㼕㼏㼔㼕㻌㻿㼍㼗㼍㼙㼛㼠㼛

2. Non-Abelian T-duality
for tracefull structure constants㻱㼤㼍㼙㼜㼘㼑㻌㼒㼞㼛㼙㻌㻺㻭㼀㻰㼇㻳㼍㼟㼜㼑㼞㼕㼚㼕㻘㻌㻾㼕㼏㼏㼕㻘㻌㼂㼑㼚㼑㼦㼕㼍㼚㼛㻌㼔㼑㼜㻙㼠㼔㻛㻥㻟㻜㻤㻝㻝㻞㼉

Original Background (Type V Bianchi universe):

We consider 3 Killing vectors,tracefull!

㻰㼡㼍㼘㻌㼓㼑㼛㼙㼑㼠㼞㼥㼇㻳㼍㼟㼜㼑㼞㼕㼚㼕㻘㻌㻾㼕㼏㼏㼕㻘㻌㼂㼑㼚㼑㼦㼕㼍㼚㼛㻌㼔㼑㼜㻙㼠㼔㻛㻥㻟㻜㻤㻝㻝㻞㼉㼇㼐㼑㻌㼘㼍㻌㻻㼟㼟㼍㻘㻌㻽㼡㼑㼢㼑㼐㼛㻌䇻㻥㻞㼉This background does not satisfy the SUGRA e.o.m.

䇾㻭㻌㻼㼞㼛㼎㼘㼑㼙㻌㼣㼕㼠㼔㻌㻺㼛㼚㻙㻭㼎㼑㼘㼕㼍㼚㻌㻰㼡㼍㼘㼕㼠㼥㻫䇿㼇㻳㼍㼟㼜㼑㼞㼕㼚㼕㻘㻌㻾㼕㼏㼏㼕㻘㻌㼂㼑㼚㼑㼦㼕㼍㼚㼛㻌㼔㼑㼜㻙㼠㼔㻛㻥㻟㻜㻤㻝㻝㻞㼉

They considereda general ansatz for dilaton,but no solution.

㻳㻿㻱㻌㼟㼛㼘㼡㼠㼕㼛㼚㼇㻲㼑㼞㼚㼍㼚㼐㼑㼦㻙㻹㼑㼘㼓㼍㼞㼑㼖㼛㻘㻌㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻣㻝㻜㻚㻜㻢㻤㻠㻥㼉

Solution of Generalized Supergravity!

㻳㻿㻱㻌㼟㼛㼘㼡㼠㼕㼛㼚㼟More GSE solutions were obtained fromNon-Abelian T-dualities of other Bianchi universes.
㼇㻴㼛㼚㼓㻘㻌㻷㼕㼙㻘㻌㽞㻌㻯㼛㼘㼓㽬㼕㼚㻘㻌㻝㻤㻜㻝㻚㻜㻥㻡㻢㻣㼉

General formula:
The Killing vector is given bySUGRA

Generalized SUGRA㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㻿㼁㻳㻾㻭㻌㼒㼞㼛㼙㻌㻰㻲㼀㼇㼅㻿㻘㻌㼁㼑㼔㼍㼞㼍㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻢㻝㻝㻚㻜㻡㻤㻡㻢㼉㼇㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻣㻜㻟㻚㻜㻥㻞㻝㻟㼉

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

Due to the Killing property,

we can always take an adapted coordinates
whereis constant.

Killing equations become㻳㻿㻱㻌㼒㼞㼛㼙 㻰㻲㼀㼇㼅㻿㻘㻌㼁㼑㼔㼍㼞㼍㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻢㻝㻝㻚㻜㻡㻤㻡㻢㻧㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻣㻜㻟㻚㻜㻥㻞㻝㻟㼉

Now, we can derive GSE by introducing
a linear dual-coordinate dependence
into the DFT dilaton:

Section condition is not violated:㻳㻿㻱㻌㼒㼞㼛㼙㻌㻰㻲㼀
According to the ansatz,

derivative of dilaton becomescanonical section

By substituting the ansatz㼇㼅㻿㻘㻌㼁㼑㼔㼍㼞㼍㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻢㻝㻝㻚㻜㻡㻤㻡㻢㼉

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

㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞㼇㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻣㻜㻟㻚㻜㻥㻞㻝㻟㼉We make an ansatz㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞Type IIA/IIB GSE:

㼇㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻣㻜㻟㻚㻜㻥㻞㻝㻟㼉㻾㼍㼙㼛㼚㼐㻙㻾㼍㼙㼛㼚㼐 㼟㼑㼏㼠㼛㼞㼇㻭㼞㼡㼠㼥㼡㼚㼛㼢㻙㻲㼞㼛㼘㼛㼢㻙㻴㼛㼍㼞㼑㻙㻾㼛㼕㼎㼍㼚㻙㼀㼟㼑㼥㼠㼘㼕㼚 䇻㻝㻡㻧㻿㼍㼗㼍㼙㼛㼠㼛㻘㻌㼅㻿㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻣㻜㻟㻚㻜㻥㻞㻝㻟㼉We have also determined the T-duality rulein the presence of the Killing vector:㻿㼡㼙㼙㼍㼞㼥

canonicalsection

SUGRA e.o.m.

DFT e.o.m.

GSE withChoice of the section is different.

㻯㼛㼙㼙㼑㼚㼠

DFT action is invariant undera constant O(d,d) rotation

Unlike the Abelian T-duality of string theory,
this is a symmetry even without isometries.
“formal T-duality”㻳㻿㻱㻌㼟㼛㼘㼡㼠㼕㼛㼚
Arbitrary solution of GSE,

Perform a formal T-duality
along the z-direction

Buscher rule
Solution of SUGRA
A solution of GSE㼇㻴㼛㼍㼞㼑㻘㻌㼀㼟㼑㼥㼠㼘㼕㼚㻘㻌㻝㻡㻜㻤㻚㻜㻝㻝㻡㻜㻧can be mapped to㻭㼞㼡㼠㼥㼡㼚㼛㼢㻙㻲㼞㼛㼘㼛㼢㻙㻴㼛㼍㼞㼑㻙㻾㼛㼕㼎㼍㼚㻙㼀㼟㼑㼥㼠㼘㼕㼚 䇻㻝㻡㻧a solution of SUGRA 㼅㻿㻘㻌㼁㼑㼔㼍㼞㼍㻘㻌㼅㼛㼟㼔㼕㼐㼍㻘㻌㻝㻢㻝㻝㻚㻜㻡㻤㻡㻢㼉㻳㻿㻱㻌㼟㼛㼘㼡㼠㼕㼛㼚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.

㻹㼍㼟㼟㼕㼢㼑㻌㻵㻵㻭㻌㻿㼁㻳㻾㻭Type II DFT

massive type IIA SUGRA㼇㻴㼛㼔㼙㻘㻌㻷㼣㼍㼗㻘㻌㻝㻝㻜㻤㻚㻠㻥㻟㻣㼉㻻㻔㼐㻘㼐㻕㻌
㼟㼜㼕㼚㼛㼞㻾㻙㻾㻌㻜㻙㼒㼛㼞㼙annihilation
operator㻰㻤㻙㼎㼞㼍㼚㼑㻌㼟㼛㼘㼡㼠㼕㼛㼚㼇㻮㼑㼞㼓㼟㼔㼛㼑㼒㼒㻘㻌㼐㼑㻌㻾㼛㼛㻘㻌㻳㼞㼑㼑㼚㻘㻌㻼㼍㼜㼍㼐㼛㼜㼛㼡㼘㼛㼟㻘㼀㼛㼣㼚㼟㼑㼚㼐㻘㻌䇺㻥㻢㼉

D8-brane solution in DFT

D7-brane solution

formal T-duality㻿㼡㼙㼙㼍㼞㼥㻰㼛㼡㼎㼘㼑㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥

massive IIASUGRAGeneralizedSUGRA

massive IIA
SUGRAformal T-duality

formalT-duality㻱㼤㼏㼑㼜㼠㼕㼛㼚㼍㼘㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥㻰㼛㼡㼎㼘㼑㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥

SUGRAe.g.
deformed SUGRAs㻟㻚㻌㻱㼤㼏㼑㼜㼠㼕㼛㼚㼍㼘㻌㻲㼕㼑㼘㼐㻌㼀㼔㼑㼛㼞㼥㻔㻱㻲㼀㻕㼁㻙㼐㼡㼍㼘 㼢㼑㼞㼟㼕㼛㼚㻌㼛㼒㻌㻰㻲㼀㼇㼃㼑㼟㼠㻌䇻㻜㻜㻧㻌㻮㼑㼞㼙㼍㼚㻘㻌㻼㼑㼞㼞㼥㻌䇻㻝㻝㻧㻮㼑㼞㼙㼍㼚㻘㻌㻳㼛㼐㼍㼦㼓㼍㼞㻘㻌㻼㼑㼞㼞㼥㻘㻌㼃㼑㼟㼠㻌䇻㻝㻞㻧㻴㼛㼔㼙㻘㻌㻿㼍㼙㼠㼘㼑㼎㼑㼚㻌䇻㻝㻟㻧㻌㻚㻚㻚㼉㻵㼚㻌㻰㻲㼀We introduced the doubled space.

momentum (P)T-dualcanonical
conjugate

T-dualstring winding (F1)DFT…

㻵㼚㻌㼠㼥㼜㼑㻌㻵㻵㻌㼠㼔㼑㼛㼞㼥

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

㻴㼛㼣㻌㼙㼍㼚㼥㻌㼎㼞㼍㼚㼑㼟㻫It depends on the dimensionof the compactification torus
Example : type IIB on ࢀ૜㻼㻔㻝㻕㻘㻌㻼㻔㻞㻕㻘㻌㻼㻔㻟㻕㻲㻝㻔㻝㻕㻘㻌㻲㻝㻔㻞㻕㻘㻌㻲㻝㻔㻟㻕㻰㻝㻔㻝㻕㻘㻌㻰㻝㻔㻞㻕㻘㻌㻰㻝㻔㻟㻕

10 branes are relatedby U-duality㻰㻟㻔㻝㻞㻟㻕㻰㻡㻛㻺㻿㻡cannot wrap on ࢀ૜㻴㼛㼣㻌㼙㼍㼚㼥㻌㼎㼞㼍㼚㼑㼟㻫It depends on the dimensionof the compactification torus
Example : type IIB on ࢀ૜㻼㻔㻝㻕㻘㻌㻼㻔㻞㻕㻘㻌㻼㻔㻟㻕㻲㻝㻔㻝㻕㻘㻌㻲㻝㻔㻞㻕㻘㻌㻲㻝㻔㻟㻕㻰㻝㻔㻝㻕㻘㻌㻰㻝㻔㻞㻕㻘㻌㻰㻝㻔㻟㻕㻰㻟㻔㻝㻞㻟㻕㻱㼤㼠㼑㼚㼐㼑㼐㻌㼟㼜㼍㼏㼑㻦㻿㻸㻔㻞㻕㻌㼐㼛㼡㼎㼘㼑㼠㼀㼥㼜㼑㻌㻵㻵㻮㻌㼠㼔㼑㼛㼞㼥 㻛 ࢀ㻼㻲㻝㻛㻰㻝㻰㻟࢔ି૚㻰㻡㻛㻺㻿㻡‫ۺ܁‬ሺ૞ሻ‫۽܁‬ሺ૞ǡ ૞ሻ䈈㼇㻻㼎㼑㼞㼟㻘㻌㻼㼕㼛㼘㼕㼚㼑 䇻㻥㻥㼉㻌㻌䇾㼜㼍㼞㼠㼕㼏㼘㼑㻌㼙㼡㼘㼠㼕㼜㼘㼑㼠䇿㻌㼛㼒㻌۳࢔۳૟૟ૠ㼀㼥㼜㼑㻌㻵㻵㻮㻌㼠㼔㼑㼛㼞㼥 㻛 ࢀ 㼛㼞㻌ࢀ㻔㻰㻣㻘㻌㻺㻿㻣㻘㻌㻖㻖㻣㻕㻼 㻲㻝㻛㻰㻝 㻰㻟㻡૛૛ 㻛㻌5૛૜㻰㻡㻛㻺㻿㻡㻷㻷㻹3૝૜㼠㼞㼕㼜㼘㼑㼠㻌㼛㼒㻌㻣㻙㼎㼞㼍㼚㼑㼟1૟૝ 㻛㻌1૟૜ሺ૚ǡ૟ሻ0૝㻱㼤㼛㼠㼕㼏㻌㼎㼞㼍㼚㼑㼟㼇㻱㼘㼕㼠㼦㼡㼞㻘㻌㻳㼕㼢㼑㼛㼚㻘㻌㻷㼡㼠㼍㼟㼛㼢㻘㻌㻾㼍㼎㼕㼚㼛㼢㼕㼏㼕 䇺㻥㻣㻧㻌㻌㻮㼘㼍㼡㻘㻌㻻䇻㻸㼛㼡㼓㼔㼘㼕㼚㻌䇻㻥㻣㻧㻌㻌㻻㼎㼑㼞㼟㻘㻌㻼㼕㼛㼘㼕㼚㼑 䇻㻥㻥㻧㻌㻱㼥㼞㼍㼟㻘㻌㻸㼛㼦㼍㼚㼛 䇻㻜㻜㻧㻌㻌㻸㼛㼦㼍㼚㼛㻙㼀㼑㼘㼘㼑㼏㼔㼑㼍㻘㻌㻻㼞㼠㼕㼚 䇻㻜㻜㼉㻱㼤㼏㼑㼜㼠㼕㼛㼚㼍㼘㻌㼟㼜㼍㼏㼑㼁㻙㼐㼡㼍㼘㼕㼠㼥㻌㼓㼞㼛㼡㼜x㼐㻙㼐㼕㼙㻚㼑㼤㼠㼑㼞㼚㼍㼘ࢀ࢔ି૚㻱㼤㼏㼑㼜㼠㼕㼛㼚㼍㼘㻌㼟㼜㼍㼏㼑㻳㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㼙㼑㼠㼞㼕㼏

Einstein-frame metric㻱㻲㼀㻌㼍㼏㼠㼕㼛㼚㼇㻴㼛㼔㼙㻘㻌㻿㼍㼙㼠㼘㼑㼎㼑㼚 䇻㻝㻟㻧㻌䈈㻚㼉㼀㼥㼜㼑㻌㻵㻵㻮㻌㻿㼁㻳㻾㻭㻌㼍㼏㼠㼕㼛㼚㻿㼑㼏㼠㼕㼛㼚㻌㼏㼛㼚㼐㼕㼠㼕㼛㼚㻌㼕㼚㻌㻱㻲㼀۳࢔ EFT
F1/D1

D3

NS5/D5

Type IIB section㼇㻮㼘㼍㼕㼞㻘㻌㻹㼍㼘㼑㼗㻘㻌㻶㻚㻙㻴㻚㻌㻼㼍㼞㼗 䇻㻝㻟㻧㻴㼛㼔㼙㻘㻌㻿㼍㼙㼠㼘㼑㼎㼑㼚㻌䇺㻝㻟㼉
M2

M5

KKMM-theory section㻱㻲㼀㻌㼍㼏㼠㼕㼛㼚㼇㻴㼛㼔㼙㻘㻌㻿㼍㼙㼠㼘㼑㼎㼑㼚 䇻㻝㻟㻧㻌䈈㻚㼉㻝㻝㻰㻌㻿㼁㻳㻾㻭㻌㼍㼏㼠㼕㼛㼚㼀㼥㼜㼑㻌㻵㻵㻌㼠㼔㼑㼛㼞㼥㻘㻌㼙㼍㼚㼥㻌㼎㼞㼍㼚㼑㼟㻌㼍㼞㼑㻌㼏㼛㼚㼚㼑㼏㼠㼑㼐㻌㼎㼥㻌㼀㻙㻛㻿㻙㼐㼡㼍㼘㼕㼠㼕㼑㼟㻵㻵㻭㻵㻵㻮㻿㻙㼐㼡㼍㼘㻵㻵㻭㻵㻵㻮㻵㻵㻭㻵㻵㻮䈈㼀㼥㼜㼑㻌㻵㻵㻮㻌㼎㼞㼍㼚㼑㼟㼀㼥㼜㼑㻌㻵㻵㻮 / ࢀૠ㼇㻷㼕㼙㼡㼞㼍㻘㻌㻌㻲㼑㼞㼚㼍㼚㼐㼑㼦㻙㻹㼑㼘㼓㼍㼞㼑㼖㼛㻘㻌㼅㻿 䇺㻝㻤㼉㼐㼑㼒㼑㼏㼠㻌㼎㼞㼍㼚㼑㼟㻘㻌㻌㼐㼛㼙㼍㼕㼚㻙㼣㼍㼘㼘㻌㼎㼞㼍㼚㼑㼟㻘㻌㻌㼟㼜㼍㼏㼑㻙㼒㼕㼘㼘㼕㼚㼓㻌㼎㼞㼍㼚㼑㼟㻰㼛㼙㼍㼕㼚㻙㼣㼍㼘㼘㼟In 㼇㻷㼕㼙㼡㼞㼍㻘㻌㻌㻲㼑㼞㼚㼍㼚㼐㼑㼦㻙㻹㼑㼘㼓㼍㼞㼑㼖㼛㻘㻌㼅㻿 䇺㻝㻤㼉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㻱㼤㼍㼙㼜㼘㼑ሺ૜ǡ૜ሻ㻞૞㻙㼎㼞㼍㼚㼑㻌㼟㼛㼘㼡㼠㼕㼛㼚㻌㻦㼐㼡㼍㼘㻌㼏㼛㼛㼞㼐㼕㼚㼍㼠㼑㼛㼒㻌㻰㻟㻙㼎㼞㼍㼚㼑
Redefinition ofWe can (in principle) obtain
the action of a deformed SUGRA.

㻰㼛㼙㼍㼕㼚㻙㼣㼍㼘㼘㼟
Similarly, we can obtain deformed SUGRAs
for all of the domain-wall branes.

㻿㼡㼙㼙㼍㼞㼥㼅㻮㻙㼐㼑㼒㼛㼞㼙㼍㼠㼕㼛㼚㻛㻺㻭㼀㻰㻌㼏㼍㼚㻌㼓㼑㼚㼑㼞㼍㼠㼑㻌㼟㼛㼘㼡㼠㼕㼛㼚㼟㻌㼛㼒㻌㻳㻿㻱 㼛㼞㻌㼙㼍㼟㼟㼕㼢㼑㻌㻵㻵㻭㻌㻿㼁㻳㻾㻭㻚㻰㼑㼒㼛㼞㼙㼑㼐㻌㻿㼁㻳㻾㻭㼟㻌㼏㼍㼚㻌㼎㼑㻌㼐㼑㼞㼕㼢㼑㼐㻌㼒㼞㼛㼙㻌㻰㻲㼀㼎㼥㻌㼠㼍㼗㼕㼚㼓㻌㼚㼛㼚㻙㼏㼍㼚㼛㼚㼕㼏㼍㼘㻌㼟㼑㼏㼠㼕㼛㼚㼟㻚㻮㼥㻌㼏㼛㼚㼟㼕㼐㼑㼞㼕㼚㼓㻌㻱㻲㼀㻘㼣㼑㻌㼏㼍㼚㻌㼏㼛㼚㼟㼕㼐㼑㼞㻌㼙㼛㼞㼑㻌㼚㼛㼚㻙㼏㼍㼚㼛㼚㼕㼏㼍㼘㻌㼟㼑㼏㼠㼕㼛㼚㼟㻘㼍㼚㼐㻌㼢㼍㼞㼕㼛㼡㼟㻌㼐㼑㼒㼛㼞㼙㼑㼐㻌㻿㼁㻳㻾㻭㼟 㼏㼍㼚㻌㼎㼑㻌㼐㼑㼞㼕㼢㼑㼐㻚