Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Sine square deformation and its implications to string theory†
T. Tada∗1
It was recently found that certain 1d quantum systems with an open boundary condition can share thesame vacuum state with a similar system having aclosed boundary condition, if the coupling constants ofthe system with an open boundary are modulated ina certain way called sine square deformation1–3) . Sinesquare deformation works similarly in two-dimensional
conformal field theories, which describe string theory4). We have investigated sine square deformation in
the context of string theory, focusing in particular onopen/closed duality.
Sine square deformation is the modulation of thecoupling of open boundary systems so thatn π(1)Ji,i+1 ≡ J sin2Nkeeping the boundary coupling J0,1 = JN,N +1 = 0 at
the both ends (Fig. 1) for the following 1d quantumsystem:(2)H=−Jn,n+1 (σn · σn+1 ) .


xfx (x) = 1 − α cos , ft (x) = N
r|k| e2πikx/l , (6)lk∈Zandr≡

while the original Hamiltonian isH0 = πc2π L0 + L̄0 −.
l6l(4)

Then, the vacuum |0� for H0 is also the vacuum ofHSSD with half the energy.
To interpret the sine square deformation in termsof the dynamics of the world sheet, we need to findthe corresponding Lagrangean. We found that the Lagrangean corresponding to HSSD is obtained by taking
α to 1 in the following expression:
1dx {(∂t ϕ) ft (∂t ϕ) − (∂x ϕ) fx (∂x ϕ)} , (5)Lα =2where†∗1√11 − α2, N≡√.
α1 − α2(7)One can readily see that the g00 component of theworld sheet metric in the Lagrangean, ft , diverges
severely as we apply sine square deformation. Thisis in some sense expected because at the SSD point
there occurs an event as singular as the change of theboundary condition.
L1 −L−1
toOne can apply sl(2, C) transformation ea 2(the holonomic part of) H0 to obtaincosh aL0 − sinh a

L1 + L−1.
2(8)The right-hand side of the above would have corresponded to HSSD if cosh a = sinh a, which is a directcontradiction with the identity cosh2 a − sinh2 a = 1.
One, therefore, needs to take a → ∞ and suitably
rescale. Hence, HSSD is not connected with H0 throughthe ordinary sl(2,c) transformation, but through a certain limiting procedure.
We also found that HSSD has the following different
vacua other than |0�Fig. 1. Sine square deformation of the coupling for a 1dquantum system.

In the case of conformal field theory, the Hamiltonian HSSD that is sine square deformed isπcπL1 + L−1 + L̄1 + L̄−1L0 + L̄0 −−, (3)l212l

1−eL−1 |h�,(9)where |h� is the state corresponding to the primary
fields of CFT. However, the norm of (9) is divergent.
One also needs a certain limiting process to properly
define (9).
In summary, we have investigated sine square deformation of string theory to shed light on the relation
between open and closed strings. Recent studies ofstring dualities suggest that one needs to go beyond
the realm where open and closed are inseparable to understand the true dynamics of string theory. We hopethat we can further uncover the nature of this realm
through insights offered by sine square deformation.
References
1) A. Gendiar, R. Krcmar, T. Nishino: Prog. Theor. Phys.
122 953 (2009) ; ibid. 123 393 (2010).
2) H. Katsura: J. Phys. A: Math. Theor. 44 252001(2011).
3) I. Maruyama, H. Katsura, T. Hikihara: Phys. Rev. B
84 165132(2011) .
4) H. Katsura: J. Phys. A: Math. Theor. 45 115003(2012).

Condensed from the article in Proceedings of the 12th Asia
Pacific Physics Conference (APPC12)
RIKEN Nishina Center- 132 -