Time development of conformal field theories associated with $L_{1}$ and $L_{-1}$ operators

In this study, we examined consequences of unconventional time development of two-dimensional conformal field theory induced by the $L_{1}$ and $L_{-1}$ operators, employing the formalism previously developed in a study of sine-square deformation. We discovered that the retainment of the Virasoro algebra requires the presence of a cut-off near the fixed points. The introduction of a scale by the cut-off makes it possible to recapture the formula for entanglement entropy in a natural and straightforward manner.


Introduction
In [1,2], a formalism was conceived in which unconventional time developments other than the radial time development can be used to study conformal field theories (CFTs). In particular, the sine-square deformation (SSD) [3] of twodimensional (2D) CFT [4] has been explained with this formalism as a particular time development called "dipolar quantization". There are earlier studies [5,6] on SSD, studies in the context of string theory [7,8], and more recent studies on this topic [9,10]. An elegant generalization has also been developed by Wen, Ryu and Ludwig [11], which involves the entanglement Hamiltonian and other interesting deformations of 2D CFT. The most recent studies on the subject include Refs. [12][13][14][15][16][17].
In the present study, we examine the case of a particular time development which was left for further study in Ref. [2]. The time development in question can be achieved using as the time-development operator, instead of L 0 +L 0 . The holomorphic part of Eq. (1), L 1 + L −1 has also been investigated in a different context [18] in which the Hamiltonian was retained as L 0 +L 0 . In this study, we change the Hamiltonian itself to (1), and examine its consequences. In fact, this case turned out to correspond to the entanglement Hamiltonian as discussed in Ref. [11], and further explored in Ref. [14,15], thus providing further motivation for the present research.
Let us elucidate the significance of the operator (1) in the context of radial quantization [19] and dipolar quantization [1,2]. As is well-known, the L 0 , L 1 and L −1 operators constitute sl(2, R) algebra. The combination of these operators, can be mapped to by 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: Using c (2) , the general linear combinations (2) can be classified into three distinctive classes that are not accessible from each other by the sl(2, R) action. Each class can be represented by a typical operator up to the overall rescaling: L 0 represents case c (2) > 0 and L 0 − 1 2 (L 1 + L −1 ) represents case c (2) = 0, which correspond to radial quantization and dipolar quantization, respectively. The final case, c (2) < 0, can be represented by L 1 + L −1 , which signifies the importance of the operator in question. Below, we investigate the L 1 + L −1 operator by applying the formalism developed in [1,2] and demonstrate that the three cases mentioned above, including L 1 + L −1 , can be studied in a unified manner.

Time-development vector field
First, let us recapitulate the analysis in [1,2]. We introduce a set of differential operators, l κ , over the complex plane with label κ, in the following form: where g(z) and f κ (z) are both holomorphic functions on the complex plane. Among l κ , we select the l 0 operator and require so that each f κ is an eigenfunction of l 0 with the eigenvalue −κ. The solution to Eq. (6) is which yields an simple expression for l 0 as follows: It can be seen that which enables the calculation of the commutation relations among l κ 's, It should be noted that the algebra generated by l κ can be represented over the linear space spanned by f κ (z)'s, since Because κ is the index for the basis that spans the representation space, the representation theory may incur restrictions on κ. For example, if g(z) is set to be z, up to possible κ depending multipliers, which are omitted here. It is natural for f κ = z κ to be single-valued on the complex plane thus, each κ should take an integer value 1 . Another choice of g(z) = z − z 2 +1 2 [1,2] leads to In this case, κ can take any real number without inflicting multiple values on f κ (z). The procedure described above can be extended to the anti-holomorphic variablez by introducing,l The most natural choice forg(z) is the complex conjugate of g(z) 2 , It then follows thatf l κ satisfies the following commutation relation: which is isomorphic to Eq. (10). We can define the time-translation operator, using Eq. (8) and through the following expression: which yields the time coordinate, t. The space coordinate, s, can be obtained by the following orthogonal operator: The above two equations can be summarized in the following explicit matrix form: Then, the relation between the coordinates t, s and z,z can be expressed succinctly [1,2] as follows: We now investigate the effect of selecting Eq. (1) as the time-development operator of the system using the aforementioned formalism. The choice of g(z) that corresponds to Eq. (1) should be The time translation on z is then expressed as z i −i Figure 1: Flow of time t generated by (z 2 + 1) ∂ ∂z + (z 2 + 1) ∂ ∂z . t = −π/2 (or π/2) on the solid line between i and −i, and t = 0 on the remainder of the imaginary axis. The value of t is unavoidably periodic. A line with constant t is shown in gray.
The relation between the above vector field (24) (or the choice of g(z)) and the operator (1) can be easily deduced by noting the expression of the Virasoro generators as follows: The selection (23) yields [2] f κ = exp(κ and, finally, Possible constant multiplications are omitted for the sake of brevity. A notable idiosyncrasy of the solution (26) is the appearance of the multivalued function κ arctan z, which may yield ambiguous multiplicative factors without a proper specification of the principal value. The reason for these multiple values is clear from Fig. 1. With the proper selection of the principal value for arctan, t = − π 2 at the thick line between z = i and z = −i as illustrated in Fig.  1. Along the time flow, however, t becomes π 2 after encircling either z = i or z = −i. As t develops further, it returns to the same point on the z coordinate with different values for t. To be more precise, the time and space coordinates can be given by Eq. (22) as follows: If we convert the argument of the logarithm in Eq. (29)to polar coordinates it is then simple to discern the time and space coordinates in terms of R and θ as follows:

Conserved charges and the Virasoro algebra
The analysis above produces a set of (conformal) Killing vectors, in the complex notation. We can now define the conserved charges by integrating the Noether current, which is the product of the energy momentum tensor and the Killing vector: Here, C denotes a contour on which t is constant and s takes all possible values. An example of such a contour is depicted as a gray line in Fig. 1. The definition of the anti-holomorphic charge,L κ , should be trivial. It should be noted that L 0 andL 0 are significant charges among others because for the selection of g(z), as in Eq. (23). The operator product expansion of the energy momentum tensor is governed by conformal symmetry on the z-plane and takes the following form: where c CFT is the central charge of CFT in question. Then, the commutation relations among the conserved charges L κ 's lead to the following integration: Performing the contour integral around z ′ in Eq. (36) yields where the last term of the righthand side is nothing but If we denote the integral part of the central extension in the first term of the righthand side of Eq. (37) as the commutation relations read For the charges, L κ , to satisfy the Virasoro algebra in Eq. (39), CI[κ|κ ′ ] must vanish unless κ + κ ′ = 0. Otherwise, a certain part of the Jacobi identity is breached (see, for example, [20]). In the following, we evaluate CI[κ|κ ′ ] explicitly to verify whether CI[κ|κ ′ ] can be zero for κ + κ ′ = 0. Evaluating the value of CI[κ|κ ′ ] involves function g. Function g is explicitly given by Eq. (23) for the case at hand; however, it is useful to consider a more general case, where we assume Solid lines with arrows represent the flow of t. The gray line connecting z ± is a possible contour, C, where t is constant and s changes; in particular, s → ±∞ near z ± , respectively.
to keep the quadratic Casimir element c (2) negative (see Eq. (4)). We also limit a to be positive for the sake of notational simplicity. The flow generated by g(z) in Eq. (40) is illustrated in Fig. 2.
The terms inside the braces in the definition of CI (38) can be easily demonstrated to amount to thus yielding A useful formula can be obtained by differentiating Eq. (7): which further simplifies the integral in Eq. (43) as for κ + κ ′ = 0. If we denote two roots of g(z) = 0 as z ± , whose imaginary parts are positive and negative, respectively, they constitute the boundary of C. Thus, It should be noted that expression (26) is generalized as follows: where possible multiplicative constants are neglected.
It is now apparent that f κ+κ ′ (z + ) and f κ+κ ′ (z − ) are divergent; as a result, the evaluation of Eq. (46) is non-trivial. Therefore, we introduce the cut-off, ε, near the fixed points z = z ± , bearing in mind the application to entanglement entropy (Fig. 3). Figure 3: Cut-off region near z + (colored in gray). The radius of the cut-off is ε in the z plane.
The structure near z ± in terms of t and s can be determined from the following generalization of Eq. (29): Introducing L as the length of the separation between z + and z − in the imaginary direction, and setting z as the cut-off boundary z = z ± + εe iθ in the above equation, we obtain Thus, in terms of t and s, the cut-off boundaries are located at respectively; Fig. 4 provides a depiction. Equation (46) can then be evaluated using the cut-off, and the expression of CI[κ|κ ′ ] for κ + κ ′ = 0 can be obtained as follows: Finally, we uncover that there should be an integer n so that for Eq. (52) to vanish. We could require that κ, which was originally introduced as the label for the (differential) operator, takes the following values: However, we rather homogeneously rescale the parameters a, b, c in g(z), which governs the time development, and demand πaL ln (L/ε) = 1 so that κ can be either an integer or half-integer. Noting that L is invariant under the rescaling (55), it is evident that selecting ξ as satisfies Eq. (56). The rescaling (55) also affects the range of t and s as − π < s < π, − π 2 ln L ε < t < π 2 ln L ε . (58) Thus, there is now a torus with the moduli parameter, on which the path integral should be performed, as depicted in Fig. 5. The introduction of cut-off ε and the corresponding rescaling of g(z) eliminate the undesirable contribution to CI[κ|κ ′ ] for κ = −κ ′ . However, the case for κ = −κ ′ yields by invoking Eqs. (7) and (43). Here we denote the contour with the cut-off as C ε . The above expression can be further evaluated as Employing the divergent rescaling (55) and neglecting the finite iπ/2 term, Thus, we arrive at the following Virasoro algebra: where κ is either an integer or half-integer. In addition, a, b, and c are the original values before rescaling, as introduced in Eq. (40). However, L κ is defined by Eq. (33) with the rescaled ξg(z) and f κ (z) which are also defined by the rescaled ξg(z) in Eq. (6).
Thus, we have established the Virasoro algebra with the exception of the divergent term in the central charge. However, this central extension term that is proportional to κδ κ,−κ ′ can be absorbed into the constant shift in L 0 as follows: A similar procedure is performed when one considers the CFT on a cylinder by mapping from a complex plane to a cylinder. In this case, however, the shift is cCFT 24 . The difference is that the entire complex plane or the Riemann sphere is mapped into a cylinder, whereas in our case, the degrees of freedom in the two disks with radius ε are discarded. We simply take advantage of the coincidence of the values of f κ and interpret the space as a torus. However, once the shift in Eq. (64) is performed, the resulting set of Virasoro generators L κ yields the same central extension term as the one on the torus. The procedure performed up to now is recapitulated below. We began with the time foliation of the Riemann sphere governed by g(z) = az 2 + bz + c, where parameters a, b and c satisfy b 2 − 4ac ≤ 0. The selection of the parameters is exemplified by the case a = c = 1 and b = 0, which corresponds to the L 1 + L −1 operator. The flow of time exhibits two fixed points separated by L as depicted in Fig. 6. Along this time development, we can define a set of conserved charges as Eq. (33). Requiring these conserved charges to form a Virasoro algebra, we are led to introduce the cut-off around the two fixed points (Fig. 3) and the periodic (or anti-periodic) boundary condition by means of rescaling (55). Now, we have the CFT on the torus with central charge c and modular parameter τ , as defined in Eq. (59). On the torus, the time development is invoked by L 0 +L 0 , which is shifted from the original definition (33) by Eq. (64). The procedure described above is connected with entanglement entropy as explained in the next section.

Entanglement Entropy
Consider a function defined on the line between the two fixed points, which we refer to as L (Fig. 6): where subscript i denotes the index for the basis that spans the Hilbert space of such functions. Thus, φ L i (x) can also be considered a state vector of the Hilbert space that corresponds to space L, The time-dependent state is constructed by applying the time development as where H is the generator of the time development, namely the Hamiltonian, and L ′ is the region developed from the original segment L during time period t. If the time-flow generated by H has a fixed point as t increases to infinity as illustrated in Fig. 7 (a), any state can flow into the lowest energy state (i.e., the vacuum): The situation can be succinctly summarized in the following path integral: where L c is the space compliment to L (Fig. 7 (b)). We attach either L c or superscript c to the functions, states, coordinates that are associated with L c (see Fig. 7 (d)). In addition, S is the appropriate conformal symmetric action. It should be noted that t should take only positive values, hence the subscript ≥ 0 is attached. The Hermitian conjugation of Eq. (69) with a different state |φ L j on the L can be written as where t takes negative values (Fig. 7 (c)). One can the glue Eqs. (69) and (70) ((c) and (d) of Fig.7 ) to obtain the following expression: Another way to explicate Eq. (71) is to consider a cut with length L on each side of which φ L j | and |φ L i reside respectively. The path integration is performed on the entire Riemann sphere with the exception of cut L (Fig.8(a)). One can convince oneself of Eq. (71) by integrating over cut L with the condition φ L j (x) = φ L i (x), since the integration (or the trace) yields the partition function 0|0 , which is simply the path integral over the entire sphere.
Note that the left-hand side of Eq. (71) takes the form of a density matrix that corresponds to the vacuum. It is implicit that this density matrix depends only on the sector related to L, since the sector originating from the complement space L c is already integrated in the right-hand side of Eq. (71). Therefore, this matrix is the reduced density matrix of the vacuum: where Z is the partition function of the entire system: The reduced density matrix ρ is normalized as If the reduced density matrix can be written as the exponentiation of a Hermitian operator, the operator is called the modular Hamiltonian in the context of axiomatic quantum field theory [21]. In the context of statistical physics, this operator is also called the entanglement Hamiltonian [22][23][24][25][26][27][28]. For the case at hand, the modular Hamiltonian in question is simply with b 2 − 4ac ≤ 0, which we have been examining in this study. This is because we require the time flow that begins on one side of L, where the state |φ L i is located, and returns to the other side of L where φ L j | is assigned, after covering the entire sphere. See Fig. 8 (b) and compare it with Fig. 6.
With these setups, it is almost trivial to derive the entanglement hamiltonian for section L. Following the treatment in [29], it is useful to introduce the following generalization of the partition function: which includes the expression trρ n . It is a well-known trick that the derivative of trρ n yields the entropy for the system governed by the density matrix ρ: Because the reduced density matrix is considered, we obtain the following expression for the entanglement entropy: The partition function Z can be calculated through the path integral over the sphere; however, the existence of the cut-off alters the integral to that on the torus with the moduli parameter τ (59), as demonstrated in the previous section. By introducing the partition function can be expressed in the following familiar form: where the subscript explicitly denotes the moduli parameter. As L n represents the Virasoro charge on the torus, there should be a corresponding Virasoro algebra L n on the sphere, whose energy-momentum tensor differs as due to the Schwarzian derivative. Therefore, the partition function can also be expressed as follows: To obtain Z(n) or Z τ (n), one can simply replace q with q n in the above expression. Noting that the n dependence only enters in the following combination, ln q n = n ln q , lnq n = n lnq, one can replace the n derivative in Eq. (79) with the derivative by ln q and lnq: One can further exploit the modular invariance of the partition function on the torus by the following modular transformation: The expression for the entanglement entropy becomes and Then, as argued in [29], the contribution from term trq L 0 to the partition function is exponentially suppressed provided that L 0 is a positive definite operator. Because the relevant contribution originates only from the part q (− cCFT 24 )q(−c CFT 24 ) , we arrive at which is in accordance with known results.

Discussion
Thus, we have established the relation between our formalism and entanglement entropy. In our treatment, the cut-off that is required in the expression of entanglement entropy is naturally and geometrically introduced. The boundary condition at the cut-off is also determined from the consistency of the Virasoro algebra. The entanglement entropy and the cut-off boundary condition have also been discussed in the literature [30][31][32].
Another interesting association with our formalism can be seen by considering the infinitesimal limit of section L: This limit can be achieved by, for example, taking b → 1 while retaining a = c = − 1 2 , which in the limit yields and the following time development operator: The above time development operator is simply the SSD Hamiltonian [1,2,4].
To further investigate this limit, it is rather convenient to retain the original notion of κ in Eq. (54) instead of rescaling by ξ. It is then apparent from Eq. (54) that κ takes continuous values in the L → 0 limit. The factor between κ and integer πaL ln(L/ε) also appears as the (inverse) factor in the expression of the central charge (62), yielding the delta function in the L → 0 limit. Therefore, simply by taking the L → 0 limit, we obtain the continuous Virasoro algebra, which was found in [1,2]: [L κ , L κ ′ ] = (κ − κ ′ )L κ+κ ′ + c CFT 12 κ 3 δ(κ + κ ′ ).
Here, we also take the step of shifting L 0 as Eq. (64). In this limit, the new structure of the continuous Virasoro algebra emerges. The details of this limiting procedure will be addressed in future publications. It would be interesting to explore the implication of the SSD Hamiltonian and continuous Virasoro structure by taking the L → 0 limit in the study of entanglement entropy. Figure 9: The hatched disks are removed from the sphere, and their boundaries (dotted circles) are identified with each other, changing the topology from a sphere to a torus. Note, in particular, that the shape of each boundary of the disks is congruent with one of the time flows (arrowed lines).
The formalism developed in this study has a wider and intriguing application. In the course of the above analysis, the existence of the cut-off, which itself is introduced to preserve the consistency of the Virasoro algebra, leads us to the Virasoro algebra on a torus. Therefore, the same analysis with an arbitrary large "cut-off " should yield the Virasoro algebra on a torus with an arbitrary moduli parameter. This also opens up the possibility of constructing the Virasoro algebra on a general two-dimensional surface with the higher genus than the sphere and torus by contriving an appropriate time-flow and applying the gluing procedure, as illustrated in Fig. 9. It would be also interesting to apply the present formalism to the calculation of entanglement entropy for multiple sections [33][34][35]. These possibilities will be pursued in future research.
In summary, we have shown that the time-flow associated with the class of L 1 + L −1 operator leads to the Virasoro algebra on a torus. This fact was utilized to re-derive the entanglement entropy in a rather straightforward manner without resorting to a contrived transformation.