RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-6. Particle Physics

SSD and the infinite circumference limit of CFT†
T. Tada∗1Among all the Virasoro generators of Conformal
Field Theory (CFT), three of them, L0 , L1 and L−1 ,form a subalgebra that is isomorphic to sl(2, R) andcorresponds to the global conformal transformation.
The Casimir operator of the subalgebra can be expressed as
C2 = L20 − L2+ − L2− ,(1)L1 + L−1
L1 − L−1, L− =.
22i(2)In analogy with the 2+1 dimensional Lorentz transformation, the space spanned by L0 , L+ and L− isapparently divided into three distinctive regions. The
first region is the “time-like” region that contains L0and small perturbations around it. Any vector within
this region can be transformed to L0 upto some numerical multiplication, by the global conformal transformatiom or the sl(2, R). This is actually the region
one would have in mind, when one demanded the invariance of the vacuum on the basis of the physicalequivalence for the states connected by the global conformal transformation. The second is the “space-like”
region, which contains the linear combination of L+and L− . The region between these two is the last one,and could be called the “light-cone” region. This region is represented by either L0 − L+ or L0 − L− .
If one further invokes the analogy with the Lorentz
geometry, the “time-like” region corresponds to the“massive” representation. Since one observes the spectrum of L0 in this region, the “mass” in this case shouldbe the inverse of the circumference, or the finite scale ofCFT1,2) . Then, it is natural to induce that the “lightcone” region corresponds to the “massless” representation and the infinite circumference. In this letter, wewill argue that if one takes the generator in the “lightcone” region, say L0 − L+ (plus anti-holomorphic partL̄0 − L̄+ , to be exact), as a Hamiltonian, one can obtaina CFT with the infinite circumference.
Should we adopt a generator that corresponds toL0 − L+ as a Hamiltonian, we can define the following
conserved charges:
 t=const.
2κ
11Lκ ≡dz(− (z − 1)2 )e z−1 T (z), (3)2πi2where T (z) = Tzz (z) is the energy momentum tensor
of the original CFT. Note that for κ = 0 t=const.
11dz(− (z − 1)2 )T (z)
L0 =
2πi2
∗1L1 + L−1.
2(4)One can further calculate the commutation relations
among the charges defined above using the operator
product expansion of the energy momentum tensor
T (z)T (w) ∼c/2
2T (w)∂w T (w)+· · · .(5)++(z − w)4 (z − w)2z−w

The result reads

whereL+ == L0 −[Lκ , Lκ ] = (κ − κ )Lκ+κ +c 3κ δ(κ + κ ).
12(6)We have thus obtained the continuous Virasoro algebrawith the central charge c, establishing that we have thetheory that exhibits the continuous spectrum. This isconsistent with the argument presented at the beginning.
This also nicely explains the feature observed in thephenomena called sine-square deformation (SSD) atleast for the case that involves CFT. It was found
3–6)
that a certain class of quantum systems, systems
with closed and open boundary conditions, have identical vacua provided that the coupling constants of theopen-boundary system are modulated in a specific way.
In particular, SSD works for two-dimensional conformal field theories and it’s implications for string theory were discussed by the present author7,8) . SSD for
CFT adopts exactly (4) as the (holomorphic part of) Hamiltonian. At that time, it had been somewhat
enigmatic that these two systems with different boundary conditions share the same vacuum state, but thiscan be explained through the discovery of the continuous spectrum for the SSD system. Because the continuous spectrum suggest that SSD system has an infinitely large space, the distinction between the open
and closed condition at the ends that located at infinitely away, is no longer physically relevant.
References
1) A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov: Nucl. Phys. B 241, 333 (1984).
2) J. L. Cardy: J. Phys. A 17, L385 (1984).
3) A. Gendiar, R. Krcmar and T. Nishino: Prog. Theor.
Phys. 122 (2009) 953; ibid. 123 (2010) 393.
4) H. Katsura: J. Phys. A 45, 115003 (2012)5) H. Katsura: J. Phys. A: Math. Theor. 44 (2011) 252001.
6) I. Maruyama, H. Katsura and T. Hikihara, Phys. Rev.
B 84 (2011) 165132.
7) T. Tada: Proceedings of the 12th Asia Pacific Physics
Conference JPS Conf. Proc. 1, 013003 (2014) .
8) T. Tada: Mod. Phys. Lett. A 30, 1550092 (2015).

RIKEN Nishina Center- 162 -完全版2014_本文.indd 162

15/10/16 17:50