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

Transcription

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

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

3 Global Conformal Transformation! on the Riemann surface Introduce Casimir Operator

4 Now the modification

5

6 No way to realize

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

8 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.!

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

10 J Open Boundary Condition

11 J Remedy Edge Effect

12 J Sine Square Deformation

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

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

15 Closed Hamlitonian h N,1 =0

16

17 Coupling N N Sites

18 Coupling N N Sites

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

20

21 2D Cft On A Cylinder H c = 2 L 0 + L 0 c 6 = 2 L ±1 + L 1 H c s vacuum sl(2,c)invariance

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

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

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

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

26 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

27 Lagrangean L = 1 2 Z Z ` 0 dx {(@ t ') F (x)(@ t ') g` X (@ x ') G(x)(@ x ')} Z X 2 2 g ` n F (x) =N X r k e 2 ikx/` k2z Z G(x) =1 cos 2 x `, { = g` 2 2 n 2 n n X n,k n e the following notation n knr k n o 2 (n (n + 1) n n 1 + n (n 1) n n+1 )

28 X L 2 2 g ` n X = g` 2 2 n 2 n n X n,k { n e the following notation X Now conjugate momenta are X H = n X 1 n knr k n o 2 (n (n + 1) n n 1 + n (n 1) n n+1 ) X X n = g` Nr k X = 1 h 2g` n X n n X h n n n k L n n 8 < : k 2 n n+1 o Provided r = 1 p 1 2 o, N =! 1 p 1 2 nd (31). In the following we understand t 2 n n 1 + (2 g) 2 n 2 n n 2 (2 g)2 n (n + 1) n n 1 i 2 (2 g)2 n (n 1) n n+1

29 n = 2 ` H n = 1 h h n n 2g` 2 n n+1 o 2 n n 1 + (2 g) 2 n 2 n n 2 (2 g)2 n (n + 1) n n 1 o i 2 (2 g)2 n (n 1) n n+1 o 2 L 1 + L 1 + L 1 + L 1 X = 2 ` L 0 + L 0 15), (21). L 1 + L 1 + L 1 + L 1 X n = 1 2g` n2z X n2z n n (n+1) + n (n 1) + (2 g) 2 n (n + 1) n (n+1) + (2 g) 2 n (n 1) n (n 1) o

30 L = 1 2 H Z ` 0 15), (21). dx {(@ t ') F (x)(@ t ') (@ x ') G(x)(@ x ')} g` X X F (x) =N r k e 2 ikx/` = N (x) = 2 ` k2z L 0 + L 0 2 L 1 + L 1 + L 1 + L 1 G(x) =1 =2sin 2 cos 2 x `, x =1 N 1 p 1 2. world sheet metr r 1 p 1 2 = ` L 0 + L 0 L 1 + L 1 + L 1 + L 1 2 c 12`

31 Worldsheet Metric L SSD = dx ( t ) N (x) ( t ) ( x ) 2 sin 2 x ( x ) N = ` L 0 + L 0 L 1 + L 1 + L 1 + L 1 2 c 12`

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

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

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

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

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

37 Other Than Exotic states e L 1 hi The lowest energy state! for H SSD

38 Other Than Exotic states The lowest energy state! for H SSD e L 1 hi Note Need More Work To Understand The Whole Structure

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

40 Thank You For Your Attention