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)
Sine-Square Deformation (SSD) and its Relevance to String Theory
and [arXiv:1404.6343]
Tsukasa Tada
Based on work with N. Ishibashi !
Riken Nishina Center
Let us consider a simple (almost trivial) modification to the Hamiltonian
in 2 dim.
Global Conformal Transformation !
Introduce
Casimir Operator
Now the modification
No way to realize
!
suggest?
What does
“Continuous Spectrum”
c.f. “Level” structure of excited states in CFT
Gap or “Mass”
1d systems w/ nearest neighbor coupling
They Started With
and
Gendiar, Krcmar, Nishino (2009)
Open Boundary Condition
J
Open Boundary Condition
J
Remedy Edge Effect
J
Sine Square Deformation
Sine Square Deformation Closed
Same Ground State
Phys.Rev.B84(2011)165132!
Closed Hamlitonian
hN,1 = 0
Coupling Sites
1 2
N N
1
...
Coupling Sites
1 2
N N
1
...
Katsura (2011), Maruyama, Katsura, Hikihara (2011)
Provided annihilates ’s vacuum Either ’s vacuum is unique is bounded below
Hc
HSSD
|vac
HSSD
|vac
HSSD
is also ’s vacuum
= 2
L0
6
= 2
L1
Hc’s vacuum
sl(2,c) invariance
2D Cft On A Cylinder
= 2
L0
6
= 2
L1
HSSD|0 = E0 2 |0
= 2
L0
6
= 2
L1
HSSD|0 = E0 2 |0
Implication For String Theory? Non-Trivial Modification (Deformation) World Sheet Dynamics Of D-Brane Open/Closed Duality Affects Boundary Condition
Implication For String Theory? Non-Trivial Modification (Deformation) World Sheet Dynamics Of D-Brane Open/Closed Duality Affects Boundary Condition Worth Further Exploration Modification Of World Sheet Metric
Let Me Elaborate
Understanding Non-perturbative dynamics in terms of the world sheet gravity 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
L↵ = 1 2 Z ` dx {(@t') F(x) (@t') − (@x') G(x) (@x')} g` X
F(x) = N X
k∈Z
r|k|e2⇡ikx/`
and G(x) = 1 ↵ cos 2⇡x ` ,
Z = g` 2 X
n,k
˙ n ˙ −n−kNr|k| n
X −2⇡2g ` n n2n−n − ↵ 2 (n (n + 1) n−n−1 + n (n − 1) n−n+1)
Lagrangean
Z = g` 2 X
n,k
˙ n ˙ −n−kNr|k| n
X −2⇡2g ` n n2n−n − ↵ 2 (n (n + 1) n−n−1 + n (n − 1) n−n+1)
Lα
⇡n = g` X
k
Nr|k| ˙ −n−k Now conjugate momenta are
Hα = X
n
⇡n ˙ n − Lα X 1
→ r = 1 − √ 1 − ↵2 ↵ , N = 1 √ 1 − ↵2
Provided
X n = 1 2g` h ⇡n⇡−n − ↵ 2 ⇡n⇡−n+1 − ↵ 2 ⇡n⇡−n−1 ↵
h + (2⇡g)2 n2n−n − ↵ 2 (2⇡g)2 n (n + 1) n−n−1 −
1 − ↵
2 (2⇡g)2 n (n − 1) n−n+1 i
X n = 1 2g` h ⇡n⇡−n − ↵ 2 ⇡n⇡−n+1 − ↵ 2 ⇡n⇡−n−1 ↵
h + (2⇡g)2 n2n−n − ↵ 2 (2⇡g)2 n (n + 1) n−n−1 −
1 − ↵
2 (2⇡g)2 n (n − 1) n−n+1 i
Hα
= 2⇡ ` ⇣ L0 + ¯ L0 ↵ 2
L1 + L−1 + ¯ L−1 ⌘
⌘ = 1 2g` X
n∈Z
n ⇡n⇡−(n+1) + ⇡n⇡−(n−1) ⌘ X
n∈Z
n + (2⇡g)2n (n + 1) n−(n+1) + (2⇡g)2n (n 1) n−(n−1)
`
L1 + L1 + ¯ L1
Hα
= 2⇡ ` ⇣ L0 + ¯ L0 ↵ 2
L1 + L−1 + ¯ L−1 ⌘
L↵ = 1 2 Z ` dx {(@t') F(x) (@t') − (@x') G(x) (@x')} g` X
F(x) = N X
k∈Z
r|k|e2⇡ikx/`
and G(x) = 1 ↵ cos 2⇡x ` ,
r ⌘ 1 p 1 ↵2 ↵ , N
, N ⌘ 1 p 1 ↵2 . world sheet metric
) = ⇡ ` ✓ L0 + ¯ L0 L1 + L1 + ¯ L1 + ¯ L1 2 ◆ ⇡c 12`
α = 1
= Nδ(x)
= 2 sin2 x
) = ⇡ ` ✓ L0 + ¯ L0 L1 + L1 + ¯ L1 + ¯ L1 2 ◆ ⇡c 12` LSSD = 1 2 dx
(x)
Worldsheet Metric
Other Than
❖ “Excited” states - work in progress ❖ Exotic states
A candidate for the implied “continuous” states
: continuous parameter
❖ Exotic states
Other Than
by H. Katsura The lowest energy state! for HSSD
❖ Exotic states
Other Than
by H. Katsura The lowest energy state! for HSSD
❖ Exotic states
Other Than
by H. Katsura The lowest energy state! for HSSD
So as the previously mentioned candidate states
eL−1|hi,
Other Than
❖ Exotic states
The lowest energy state! for HSSD
Note
eL−1|hi,
Other Than
❖ Exotic states
The lowest energy state! for HSSD
Need More Work To Understand The Whole Structure
Sine Square Deformation String Theory Duality Divergence In Worldsheet Dynamics
Condensation of world sheet metric