O ( d , d ) preserve classical integrability Yuta Sekiguchi - - PowerPoint PPT Presentation

• transformations

preserve classical integrability

O(d, d)

Yuta Sekiguchi

• 1. Motivation
• 2. Classical integrability of WZW models
• 3. Doubled formalism and

transf.

• 4. Application
• 5. Outlook

O(d, d)

The plan of my talk

• 1. Motivation

(Quick!)

1.2 Classical integrability of string theory

• AdSd/CFTd-1: attractive examples of Gauge/Gravity duality

• Intriguing: integrable structures

• f integrability (=YRISW)

• AdSd/CFTd-1: attractive examples of Gauge/Gravity duality

• Significant: integrable deformations

1.2 Classical integrability of string theory

• Intriguing: integrable structures

1.3 A bit about Yang-Baxter deformation

• The YB deformed σ-model action (w/ WZ-term: )

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• A linear operator R : g → g

• The classical r-matrix is a solution of classical YB equation (CYBE)

1.3 Yang-Baxter (YB) deformation

• 1. Put classical r-matrix into the YB deformed sigma model action:
• const. deformation parameter↑
• 2. Rewrite the action to read off the deformed background data

• −

• em

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• cl. r-matrix inserted.

1.3 Yang-Baxter (YB) deformation

• invariant formalism

• deformation (see Kentaroh’s talk)

1.4 Upshot

• 1. Classical integrability of string theory
• 2. Duality invariant approach to string theory

}

• Systematically obtained
• deformed Lax pairs

O(d, d)

• map)

O(d, d)

• Global

O(d, d; ℝ)

WZW

• 2. Classical integrability

(=the existence of Lax pairs)

• f WZW models

2.1 Basics of WZW model (on w/ -flux)

S3 H

2.2 Lax pair and monodromy matrix

• = 1 +

L → ˆ L = h−1L h + h−1dh

2.3 Concrete gauged Lax pairs for w/ -flux

S3 H

}

2.3 Concrete gauged Lax pairs for w/ -flux

S3 H

}

• O(d, d)
• 3. Doubled formalism

and

• transformations

O(d, d)

• Xi ˜

X i

3.1 Setting of doubled formalism

Td

˜ Td

}

T2d

3.2 Action of doubled formalism

No

3.2 Action of doubled formalism

No

• ne condition imposed (self-duality constraint) :

• GijdXj + Bij ? dXj

• isometry currents.

}

}

• f physical sigma model
• f doubled sigma model

3.3 Constraint on doubled formalism

3.4 (-duality) map

O(d, d)

• deformed Lax pairs satisfy zero-curvature condition?

3.5 Flatness of

• deformed Lax pairs

O(d, d)

• deformed Lax pairs satisfy zero-curvature condition?

3.5 Flatness of

• deformed Lax pairs

O(d, d)

• deformed Lax pairs satisfy zero-curvature condition?

3.5 Flatness of

• deformed Lax pairs

O(d, d)

• f the undeformed model

• deformed Lax pairs satisfy zero-curvature condition?

3.5 Flatness of

• deformed Lax pairs

O(d, d)

• f the undeformed model

• 2. The EoMs for of the undeformed model

• 1. The EoM for of the undeformed model

3.5 Flatness of

• deformed Lax pairs

O(d, d)

3.5 Flatness of

• deformed Lax pairs

O(d, d)

• f the deformed model.

✔

• JJ
• 4. Application

4.1 (current-current) deformation

J ¯ J

4.1 (current-current) deformation

J ¯ J

}

4.1 (current-current) deformation

J ¯ J

}

4.1 (current-current) deformation

J ¯ J

}

• 5. Conclusions and Outlook

5 Conclusions and Outlook

• transformation using the doubled formalism.

Thank you!