㻭㻌㼙㼛㼐㼕㼒㼕㼑㼐㻌㻼㼑㼚㼞㼛㼟㼑㻌㼘㼕㼙㼕㼠㻌㼍㼚㼐㻌㼓㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㼜㼜㻙㼣㼍㼢㼑㻌㼟㼛㼘㼡㼠㼕㼛㼚㼟㻿㼡㼓㼡㼞㼡 㻻㼗㼡㼙㼡㼞㼍
Department of physics, Kyoto U.

㼣㼛㼞㼗㻌㼕㼚㻌㼜㼞㼛㼓㼞㼑㼟㼟㻌㼣㼕㼠㼔㻌㻶㼡㼚㻙㼕㼏㼔㼕 㻿㼍㼗㼍㼙㼛㼠㼛㻛㻷㼑㼚㼠㼍㼞㼛㼔 㼅㼛㼟㼔㼕㼐㼍String: T-duality, Integrability and Geometry 3/7 2019 @ Tohoku U.

1

Introduction
The generalized supergravity [Arutyunov-Frolov-Hoare-Roiban-Tseytlin,1511.05795]
Classical consistency Kappa symmetry of GS action[Tseytlin-Wulf, 1605.04884]

Quantum consistency Weyl anomaly cancellation[Sakamoto-Sakatani-Yoshida, 1703.09213][Jose J. Fernandez-Melgarejo-Sakamoto-Sakatani-Yoshida, 1811.10600]

The gSUGRA may give a new consistent b.g. of the superstring2

Introduction
A main interest

How to analyze the superstrings on such deformed backgrounds
It is difficult to analyze the superstring on a curved background.

We assume one can obtain a non-trivial, but solvable backgroundby the Penrose limit
cf. Penrose limit of AdS5㽢S5 Ѝ maximally supersymmetric pp-wave

3Motivation
Undeformed AdS5㽢㽢S5

TsT transformed background

EOM w/ periodic b.c.

EOM w/ twisted periodic b.c.

[Frolov, 0503201]
Assumption ᾉ A similar relation holds for YB deformed AdS5ᶣS5.
Penrose limit
Such a relation still holds for deformed pp-wave b.g.

We may analyze the superstring on deformed b.g.

4

Plan of this talk

0. Introduction
1. Review of the standard Penrose limit (Poincare AdS)
2. A (modified) Penrose limit of deformed b.g.
¾ A gravity dual of a NCYM
¾ A non-unimodular e.g.

3. Summary and discussion

5

Review of the standard Penrose limit
AdS5㽢S5 metric䠯䡐䡁䡌䠍 Expanding the background around a null geodesics on(And rescale other coordinates like䠯䡐䡁䡌䠎plane)

limit
6

Review of a standard Penrose limit
By takinglimit,The maximally supersymmetric pp-wave sol.

7Plan of this talk

0. Introduction
1. Review of the standard Penrose limit (Poincare AdS)
2. A (modified) Penrose limit of deformed b.g.
¾ A gravity dual of a NCYM
¾ A non-unimodular e.g.

3. Summary and discussion8An example of the Penrose limit of the deformed b.g.
Ex. A gravity dual of a non-commutative gauge theory[Matsumoto, Yoshida, 1404.3657]
Abelian r-matrix

A solution to the standard type IIB SUGRA䠃 dilaton F3, F5 flux

[Hashimoto, Itzhaki / Maldacena, Russo , 1999]
9A modified Penrose limit䕔 㼀㼔㼑㻌㼟㼠㼍㼚㼐㼍㼞㼐㻌㻼㼑㼚㼞㼛㼟㼑㻌㼘㼕㼙㼕㼠㼀㼔㼑㻌㼟㼠㼍㼚㼐㼍㼞㼐㻌㼜㼜㻙㼣㼍㼢㼑 㼟㼛㼘㻚

Fixing the deformation parameterNo deformation effect

AdS radius䕔 㼀㼔㼑㻌㼙㼛㼐㼕㼒㼕㼑㼐㻌㻼㼑㼚㼞㼛㼟㼑㻌㼘㼕㼙㼕㼠Rescaling the deformation parameter㼀㼔㼑㻌㼓㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㼜㼜㻙㼣㼍㼢㼑(next slide)by the radius

10An example of the modified Penrose limit
Results of the modified Penrose limit withhere䠃 dilaton F3, F5 flux
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The dual metric and the beta field
Let me introduce

andsuch that

Killing vectors commuting each other

Beta trans. of the pp-wave

Comments¾ One can also check it including the dilaton and RR-flux.

¾ The deformed b.g. can be understood as TsT trans. pp-wave.
12A more complicated example䠄
䠄1/2䠅
Ex .

Non-unimodular r-matrix[Fernandez-Melgarejo, Sakamoto,Sakatani, Yoshida, 1710.06849]
A solution to the generalized supergravity

䠃dilaton, F1, F3, F5 flux

Rescale the deformation parameter asand takelimit.
13

A more complicated example䠄
䠄2/2䠅
Results of the modified Penrose limit [S. O. , J. Sakamoto, K. Yoshida, 19xx.xxxxx]

here䠃dilaton, F5 flux
Comments¾ The generalized pp-wave with

эA solution to GSE¾ cannot be understood as the beta trans. pp-wave
14Summary
Concerning YB deformed backgrounds,䕔 㼀㼔㼑㻌㼟㼠㼍㼚㼐㼍㼞㼐㻌㻼㼑㼚㼞㼛㼟㼑㻌㼘㼕㼙㼕㼠㼀㼔㼑㻌㼟㼠㼍㼚㼐㼍㼞㼐㻌㼜㼜㻙㼣㼍㼢㼑 㼟㼛㼘㻚䕔 㼀㼔㼑㻌㼙㼛㼐㼕㼒㼕㼑㼐㻌㻼㼑㼚㼞㼛㼟㼑㻌㼘㼕㼙㼕㼠㼀㼔㼑㻌㼓㼑㼚㼑㼞㼍㼘㼕㼦㼑㼐㻌㼜㼜㻙㼣㼍㼢㼑Rescaling the deformation parameterby the radius :¾ A gravity dual of a NCYM э TsT trans. of pp-wave¾ A non-unimodular e.g. э The generalized pp-wave with15

Discussion -Future directionTsT transformed background

EOM w/ periodic b.c.

Undeformed pp-wave

EOM w/ twisted periodic b.c.

䕔 Light-cone quantization of superstrings on the generalized pp-wave
HIMR case in [S. O. , J. Sakamoto, K. Yoshida, 19xx.xxxxx]
light-cone hamiltonian, vertex operator, 2-pt correlation fnc …
䕔 Is it possible to analyze more general cases?
(including the case of the generalized SUGRA background)16

Back up17

Introduction
AdS/CFT
Type II superstring on AdS5㽢S5

4d N=4 SU(N) SYM (Large N)

The integrability has played an important role to analyze this conjecture.

A next issue : Integrable deformation of AdS5㽢S5 superstring

Yang-Baxter deformations(as a 2d non-linear sigma model)

Deformed AdS5㽢S5 background[Sakamoto-san’s talk]18