II-6. Particle Physics

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Time development of conformal field theories associated with L1 andL−1 operators†
T. Tada∗1,∗2
An unconventional time development of the twodimensional conformal ﬁeld theory (CFT) induced by
the L1 and L−1 operators was studied by employing
the formalism previously developed in a study of sinesquare deformation (SSD).1) Consequently, we foundthat the retention of the Virasoro algebra naturally
leads to the presence of a cut-oﬀ near the ﬁxed points(shown as gray blobs in Fig. 1). The introduction of ascale by the cut-oﬀ may appear at odds with the conformal symmetry; however, it is essential to derive theformula for entanglement entropy.2,3)
The SSD of two-dimensional CFT4) has been explained with a formalism developed in Refs. 5, 6), in
which unconventional time developments, other than
the radial time development, were utilized to study
CFTs. One such time development suited with thestudy of SSD was named “dipolar quantization.” Further generalization was investigated by Wen, Ryuand Ludwig,7) where the entanglement Hamiltonianand other interesting deformations of two-dimensional
CFT were studied.
In this study, we visit the following particular timedevelopment, which was not studied in Ref. 6):

are not accessible from each other by the sl(2, R) action. Typical operators for each group are: L0 represents c(2) > 0, L0 − 12 (L1 + L−1 ) represents c(2) = 0,
and L1 + L−1 represents c(2) < 0. Thus, we could
investigate the L1 + L−1 operator by applying the formalism developed in Refs. 5, 6) and demonstrate thatthe aforementioned three cases, including L1 + L−1 ,can be studied in a uniﬁed manner.
zLLLc(1)L1 + L−1 + L̄1 + L̄−1 ,Fig. 1. Flow of time t can be considered to begin at a section of space with length L (solid line). The remaining
space, including infinity, is depicted as a dashed line
and denoted by Lc . The setup corresponds to the entanglement entropy for the segment with length L.

instead of ordinary L0 + L̄0 . We found that this timedevelopment corresponds to the entanglement Hamiltonian as discussed in Ref. 7).
The signiﬁcance of the operator (1) can be understood in the following way. The L0 , L1 , and L−1 operators constitute sl(2, R) algebra. A linear combinationof these operators,
(0)

(1)

(−1)

L−1 ,

(2)

x′(0) L0 + x′(1) L1 + x′(−1) L−1 ,(3)x

L0 + x

L1 + xcReferences

1) A. Gendiar, R. Krcmar, T. Nishino, Prog. Theor. Phys.
122, 953 (2009), Erratum: [Prog. Theor. Phys. 123,393 (2010)].
2) C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424,443 (1994).
3) H. Casini, M. Huerta, R. C. Myers, J. High Energy
Phys. 1105, 036 (2011).
4) H. Katsura, J. Phys. A 45, 115003 (2012).
5) N. Ishibashi, T. Tada, J. Phys. A 48, 315402 (2015).
6) N. Ishibashi, T. Tada, Int. J. Mod. Phys. A 31, 1650170(2016).
7) X. Wen, S. Ryu, A. W. W. Ludwig, Phys. Rev. B 93,235119 (2016).

can be mapped toby the adjoint action of sl(2, R); however, the following
quadratic form of the coeﬃcients, which is known as
the quadratic Casimir element, remains the same:()2c(2) ≡ x(0) − 4x(1) x(−1) .

(4)

Using this c(2) , the general linear combinations (2)can be categorized into three distinctive groups that†∗1∗2

Condensed from the article in J. Phys. A Math Theor. 53,255401 (2020)
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