II-6. Particle Physics

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Study of Lorentzian sine-square deformed CFT†
X. Liu∗1,∗2 and T. Tada∗2,∗3
The concept of sine-square deformation (SSD) wasﬁrst introduced into two-dimensional conformal ﬁeldtheory (2D CFT) in Ref. 1). In previous studies, wefound that the introduction of SSD in 2D CFT deﬁnes a new time translation generated by the SSD
Hamiltonian.1) The SSD CFT processes the Virasoroalgebra with a continuum index.2) Further, inspiredby the discovery in SSD CFT, we generalized thisstudy to CFT using more general modular Hamiltonians, and found three diﬀerent types of Virasoro algebra in the Euclidean CFT.3) In this study, we extend
our analysis to Lorentzian CFT. To consider CFT in
the Minkowski spacetime, the universal covering spaceof the Minkowski spacetime must be introduced. Generally, a spacetime is mapped to a Penrose diamondon the cylinder, which is the universal covering of thePenrose diamond. The time translation is deﬁned on
the universal covering using the Luscher-Mack Hamiltonian P̂0 +2K̂0 .4) Consider the worldline for a particle ina Penrose diamond; if we perform a special conformaltransformation, the worldline may cross the boundary of the Penrose diamond and move on to the next
patch. This problem can be solved if the time translation for the CFT is deﬁned to be conﬁned in a single Penrose diamond, implying that the CFT becomes
eﬀectively non-compact under such time translations.
This can be achieved by applying SSD in Lorentzian
CFT; the non-compactness of CFT can be observed
from the continuum index of the Virasoro Algebra.5)
We performed similar analysis as that in the Euclideansignature.2,3) Next, we investigate the three diﬀerent
cases with Hamiltonians having plus, minus, and zero
values in the quadratic Casimir element. We selectthree examples from the three cases:i0L∗3− π2i−Fig. 1. The universal covering space of the Penrose diamond (shaded region) is charted by u+ and u− . Whilethe conformal symmetry is represented using the entire
covering space, the time flow by P0 (arrowed line) isconfined within a single Penrose diamond.

must bear discrete indices to remove the ambiguities and preserve the local conformal symmetry.
There is UV divergence in the central extensionalterm in the Virasoro Algebra.
• The Naive type: The Virasoro generators can havecontinuum indices. This corresponds to the SSD
case, where the theory is eﬀectively non-compact.
Thus, the time translation is conﬁned inside a single Penrose diagram. See Fig. 1.
The conformal invariance requires the introductionof the universal covering space; moreover, occasionally,
the Lorentzian CFT is considered to be unphysical owing to the existence of a closed time-like curve. In thisstudy, our analysis ensured that there is no closed timelike curve if the Naive Hamiltonian is selected insteadof the Luscher-Mack Hamiltonian.5)
References

1) H. Katsura, J. Phys. A 45, 115003 (2012).
2) N. Ishibashi, T. Tada, J. Phys. A 48, 315402 (2015).
3) T. Tada, J. Phys. A 53, 255401 (2020).
4) M. Luscher, G. Mack, Commun. Math. Phys. 41, 203(1975).
5) X. Liu, T. Tada, Prog. Theor. Exp. Phys. 2020, 061B01(2020).

• Luscher Mack type: The Virasoro generators inthis type must bear discrete indices; otherwise,
ambiguities are observed in the generators.
• Rindler type: The Virasoro generators in this type∗2i0R− π2

We deﬁne diﬀerent mode decompositions of theenergy-stress tensor with 0th modes corresponding toeach Hamiltonian in the three cases. By applying thecommutation relation of the energy-stress tensor, we
can obtain three types of Virasoro Algebra:∗1u+π2π2• The Luscher Mack Hamiltonian P̂0 +2K̂0 ,• The Rindler Hamiltonian M̂01 ,• The Naive Hamiltonian P̂0 .

†i+u−Condensed from the article in Prog. Theor. Exp. Phys.
2020, 061B01 (2020)
Department of Physics, University of Tokyo
RIKEN Nishina Center
RIKEN iTHEMS- 68 -1