Sine-Square Deformation (SSD) and its Relevance to String Theory - PowerPoint PPT Presentation

Sine-Square Deformation (SSD) and its Relevance to String Theory Tsukasa Tada Riken Nishina Center Based on work with N. Ishibashi ! and [arXiv:1404.6343]

Conformal Field Theory in 2 dim. Let us consider a simple (almost trivial) modification to the Hamiltonian

Global Conformal Transformation ! on the Riemann surface Introduce Casimir Operator

Now the modification

����������������������� No way to realize

What does ! suggest? Gap or “Mass” “Continuous Spectrum” c.f. “Level” structure of excited states in CFT

To motivate further, let me introduce an interesting work by A. Gendiar, R. Krcmar and T. Nishino Prog. Theor. Phys. 122 (2009) 953; ibid. 123 (2010) 393. !

Gendiar, Krcmar, Nishino (2009) They Started With 1d systems w/ nearest neighbor coupling and Open Boundary Condition

Open Boundary Condition J

Remedy Edge Effect J

Sine Square Deformation J

Sine Square Closed Deformation Same Ground State A. Gendiar, R. Krcmar and T. Nishino ! Prog. Theor. Phys. 122 (2009) 953; ibid. 123 (2010) 393. !

The mechanism behind this deformation was clarified by H. Katsura and his collaborators. H. Katsura, J. Phys. A:Math.Theor. 44 (2011) 252001 ! I. Maruyama, H. Katsura and T. Hikihara, ! Phys.Rev.B84(2011)165132 !

Closed Hamlitonian h N, 1 � = 0

Coupling ... 1 - N N 1 2 ... Sites

Coupling ... 1 - N N 1 2 ... Sites

Katsura (2011), Maruyama, Katsura, Hikihara (2011) Provided | vac � annihilates ’s vacuum H c Either ’s vacuum is unique H SSD or is bounded below H SSD | vac � H SSD ’s vacuum is also

2D Cft On A Cylinder = 2 � − � c L 0 + ¯ � � H c L 0 � 6 � = 2 � L ± 1 + ¯ � � L � 1 � H c ’s vacuum sl(2,c) invariance

= 2 � = 2 � − � c L ± 1 + ¯ L 0 + ¯ � � � � H c L � 1 L 0 � 6 � � H SSD | 0 � = E 0 2 | 0 � H. Katsura, J. Phys. A: Math. Theor. 45 (2012) 115003.

����������������������� = 2 � = 2 � − � c L ± 1 + ¯ L 0 + ¯ � � � � H c L � 1 L 0 � 6 � � H SSD | 0 � = E 0 2 | 0 � H. Katsura, J. Phys. A: Math. Theor. 45 (2012) 115003.

Implication For String Theory? Non-Trivial Modification (Deformation) Affects Boundary Condition D-Brane World Sheet Dynamics Of Open/Closed Duality

Implication For String Theory? Non-Trivial Modification (Deformation) Affects Boundary Condition Modification Of World Sheet Metric D-Brane World Sheet Dynamics Of Open/Closed Duality Worth Further Exploration

Let Me Elaborate Boundary condition — set by hand Compartmentalize characteristic physics Useful to concentrate each idiosyncrasy Often non-perturbative effects involve different boundary conditions D-brane, open closed duality Understanding Non-perturbative dynamics in terms of the world sheet gravity

Lagrangean Z ` 1 L ↵ = dx { ( @ t ' ) F ( x ) ( @ t ' ) − ( @ x ' ) G ( x ) ( @ x ' ) } 2 0 g ` X and G ( x ) = 1 � ↵ cos 2 ⇡ x X r | k | e 2 ⇡ ikx/ ` F ( x ) = N ` , k ∈ Z Z g ` � n ˙ ˙ X � − n − k Nr | k | X = 2 n,k − 2 ⇡ 2 g n 2 � n � − n − ↵ n n o 2 ( n ( n + 1) � n � − n − 1 + n ( n − 1) � n � − n +1 ) . `

Z g ` X � n ˙ ˙ � − n − k Nr | k | X L α = 2 n,k − 2 ⇡ 2 g n 2 � n � − n − ↵ n n o 2 ( n ( n + 1) � n � − n − 1 + n ( n − 1) � n � − n +1 ) . ` Now conjugate momenta are Nr | k | ˙ X ⇡ n = g ` � − n − k k → ⇡ n ˙ X n √ Provided X = H α � n − L α r = 1 − 1 − ↵ 2 1 N = , √ ↵ 1 − ↵ 2 n 1 ⇡ n ⇡ − n − ↵ 2 ⇡ n ⇡ − n +1 − ↵ 1 h X = 2 ⇡ n ⇡ − n − 1 h 2 g ` ↵ + (2 ⇡ g ) 2 n 2 � n � − n − ↵ 2 (2 ⇡ g ) 2 n ( n + 1) � n � − n − 1 − 1 − ↵ 2 (2 ⇡ g ) 2 n ( n − 1) � n � − n +1 i

n X 1 ⇡ n ⇡ − n − ↵ 2 ⇡ n ⇡ − n +1 − ↵ h H α = 2 ⇡ n ⇡ − n − 1 h 2 g ` ↵ + (2 ⇡ g ) 2 n 2 � n � − n − ↵ 2 (2 ⇡ g ) 2 n ( n + 1) � n � − n − 1 − 1 − ↵ 2 (2 ⇡ g ) 2 n ( n − 1) � n � − n +1 i = 2 ⇡ L 0 � ↵ ⇣ �⌘ L 0 + ¯ L 1 + ¯ L 1 + L − 1 + ¯ � L − 1 2 ` ⌘ � 1 ⌘ n = 2 ⇡ ⌘ n X X L 1 + ¯ L 1 + L � 1 + ¯ = ⇡ n ⇡ − ( n +1) + ⇡ n ⇡ − ( n − 1) � � L � 1 2 g ` ` n ∈ Z n ∈ Z o (2 ⇡ g ) 2 n ( n + 1) � n � − ( n +1) + (2 ⇡ g ) 2 n ( n � 1) � n � − ( n − 1) +

Z ` 1 L ↵ = dx { ( @ t ' ) F ( x ) ( @ t ' ) − ( @ x ' ) G ( x ) ( @ x ' ) } 2 0 g ` X X r | k | e 2 ⇡ ikx/ ` F ( x ) = N = N δ ( x ) k ∈ Z = 2 ⇡ L 0 � ↵ ⇣ �⌘ L 0 + ¯ L 1 + ¯ L 1 + L − 1 + ¯ � H α L − 1 2 ` and G ( x ) = 1 � ↵ cos 2 ⇡ x 1 ` , p , N ⌘ 1 � ↵ 2 . = 2 sin 2 � x α = 1 p world sheet metric � 1 � ↵ 2 r ⌘ 1 � , N ↵ L 0 � L 1 + L � 1 + ¯ L 1 + ¯ ✓ ◆ ) = ⇡ L � 1 � ⇡ c L 0 + ¯ 2 12 ` `

Worldsheet Metric � � L SSD = 1 ( � t � ) N � ( x ) ( � t � ) − ( � x � ) 2 sin 2 � x � � � ( � x � ) dx 2 0 N → ∞ L 0 � L 1 + L � 1 + ¯ L 1 + ¯ ✓ ◆ ) = ⇡ L � 1 � ⇡ c L 0 + ¯ 2 12 ` `

Non-Trivial Divergence Confirmed Difficult To Tackle Directly Explore States Other Than

Other Than ❖ “Excited” states - work in progress A candidate for the implied “continuous” states : continuous parameter ❖ Exotic states

Other Than ❖ Exotic states by H. Katsura The lowest energy state ! for H SSD

Other Than ❖ Exotic states by H. Katsura The lowest energy state ! for H SSD

Other Than ❖ Exotic states by H. Katsura The lowest energy state ! for H SSD So as the previously mentioned candidate states

Other Than ❖ Exotic states e L − 1 | h i , The lowest energy state ! for H SSD

Other Than ❖ Exotic states The lowest energy state ! for H SSD e L − 1 | h i , Need More Work To Understand Note The Whole Structure

Summary Sine Square Deformation String Theory Duality Divergence In Worldsheet Condensation of Dynamics world sheet metric

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