## Abstract

Motivated by sine-square deformation (SSD) for quantum critical systems in 1+1 dimensions, we discuss a Möbius quantization approach to the 2D conformal field theory, which bridges the conventional radial quantization and the dipolar quantization recently proposed by Ishibashi and Tada [N. Ishibashi and T. Tada, J. Phys. A: Math. Theor. **48**, 315402 (2015)], [N. Ishibashi and T. Tada, arXiv:1602.01190 [hep-th] [Search INSPIRE]. We then find that the continuous Virasoro algebra of the dipolar quantization can be interpreted as a continuum limit of the Virasoro algebra for scaled generators in the SSD limit of the Möbius quantization approach.

## Introduction

Recently, intriguing properties of sine-square deformation (SSD) for 1+1-dimensional quantum many-body systems have been revealed by a series of intensive theoretical research [1–13]. The SSD was originally introduced as a spatial deformation of interaction couplings with the sine-square function for 1+1D quantum lattice models [1–3]. Several numerical studies have clarified that critical ground states of SSD systems with an open boundary are identical to those of uniform systems with a periodic boundary within numerical accuracy. Later, this correspondence between the ground states of the uniform and SSD systems was proved for some exactly solved models that can be reduced to free fermionic models [4, 5]. In Ref. [6], an interesting connection of the SSD to supersymmetric quantum mechanics was pointed out for a nonrelativistic free fermion system in continuous space. Moreover, the correspondence of the open and periodic boundary systems under the SSD has been successfully applied to numerical estimation of bulk quantities via finite-size systems [7–9]. These scalabilities of SSD research suggest that there exists rich physics behind the SSD.

The SSD of the 2D conformal field theory (CFT) [14] was first investigated by Katsura, where the operator $$L_0- \frac{L_1+L_{-1}}{2}+ \bar{L}_0- \frac{\bar{L}_1+\bar{L}_{-1}}{2}$$ is regarded as a Hamiltonian of the SSD system [10]. Here, $$L_n$$ denotes the Virasoro generator of CFT. Then, an essential point is that the CFT vacuum is annihilated by the deformation terms $$\frac{L_1+L_{-1}}{2}$$ and $$\frac{\bar{L}_1+\bar{L}_{-1}}{2}$$, because of SL(2,R) invariance, implying that the ground state of the SSD system is identical to that of the uniform system. Also, the relevance of the SSD to string theory was discussed on the basis of the free boson in Ref. [11]. Very recently, Ishibashi and Tada have proposed a more direct approach to the SSD of CFT, which is called “dipolar quantization”; this novel quantization scheme, unlike conventional radial quantization, provides a continuous Virasoro algebra for the SSD of CFT [12, 13]. However, the connection between the dipolar quantization and the radial quantization, or equivalently between the SSD system and the uniform system, is not yet clear at the present stage of research. How can we interpolate between the dipolar quantization for the SSD system and the usual CFT based on the radial quantization?

In order to address the above problem, we introduce a parameterized Hamiltonian that bridges the uniform system and the SSD system:^{1}

^{2}Note that $$\theta=0$$ corresponds to the uniform system and $$\theta\to \infty$$ to the SSD system except for the overall normalization. An essential observation of this Hamiltonian is that its ground state is always identical to that of the uniform system of $$\theta=0$$, since Eq. (2) can be regarded as a “Lorentz transformation” reflecting the SL(2,R) symmetry in CFT [11], and $$\frac{L_1+L_{-1}}{2}$$ annihilates the CFT vacuum. In this paper, we derive the classical Virasoro (Witt) algebra based on the Hamiltonian (2), where the Möbius transformation coordinate plays a crucial role. We then discuss quantization of the Virasoro algebra for (1), which we call “Möbius quantization”, interpolating between the radial quantization and the dipolar quantization. In particular, we find that the continuous Virasoro algebra of the dipolar quantization can be interpreted as a continuum limit of the Virasoro algebra obtained in the SSD limit of the Möbius quantization. We also mention that the Möbius quantization approach gives a consistent result with the Möbius-type conformal mapping of CFT, except at the SSD point.

This paper is organized as follows. In the next section, we briefly explain the classical Virasoro algebra and the role of the Möbius coordinate. In Sect. 3, we explain the Möbius quantization approach to the central extension of the classical Virasoro algebra. In particular, we derive the Virasoro algebra for Eq. (1) and discuss its continuum limit corresponding to the SSD point. We also describe the vacuum state and the primary field in the Möbius quantization. In Sect. 4, we comment on the conformal mapping associated with the Möbius quantization. In Sect. 5, we discuss the dipolar/SSD limit of the Möbius quantization in detail. In Appendix A, the relation between the Virasoro generators of the Möbius quantization and the radial quantization is provided as a series expansion form, where expansion coefficients are exactly represented by Gauss’s hypergeometric function.

## Classical Virasoro algebra

In this section, let us discuss the classical Virasoro (Witt) algebra for Eq. (2). For the usual classical Virasoro algebra, the generator is defined as

By analogy with Refs. [12, 13], we define a new differential operator for Eq. (2) as

Note that $$\theta=0$$ describes the uniform system and $$\theta \to \infty$$ corresponds to the SSD point. This definition of $${\mathfrak l}_0$$ contains a divergent factor if $$\theta \to \infty$$. Thus, we also define a scaled generator

It should be noted that the scaled generator $$\tilde{\mathfrak l}_0$$ was dealt with in the previous papers [12, 13], where the dipolar quantization at the SSD point ($$\theta \to \infty$$) was directly discussed.

According to Refs. [12, 13], we consider the eigenfunction of $${\mathfrak l}_0$$,

On the basis of $$f_n(z)$$, we next define the differential operator with an index $$n$$:

Then, a straightforward calculation yields

If $$A_n = (\tanh(\theta))^{- n }$$ is adopted, $$A_n A_{n'} = A_{n+n'}$$ is satisfied. We can then verify $$f_n(z) f_{n'}(z) = f_{n+n'}(z)$$ with

An interesting point on the eigenfunction $$f_n(z)$$ is that it contains an identical form to the Möbius transformation of SL(2,R) for the complex variable $$z$$. If $$\theta =0 $$, $$f_n(z) = -z^{n}$$, which reproduces the conventional classical Virasoro algebra of the radial quantization approach. In the SSD limit ($$\theta\to \infty$$), on the other hand, $$\kappa = n/N_\theta$$ becomes continuous, and

*Möbius coordinate*

We briefly analyze the relation of $${\mathfrak l}_0$$ with the Mönius coordinate, which plays an essential role in quantizing $$ {\mathfrak l}_n $$. As in the usual CFT, we may regard the time development operator and the spatial translation operator as $${\mathfrak l}_0+\bar{\mathfrak l}_0$$ and $${\mathfrak l}_0-\bar{\mathfrak l}_0$$, respectively. On this basis, we can specify the complex coordinate $$\zeta=\tau+i s$$ with

Then, the real part of this equation is reduced to

In Fig. 1, we show a typical time-flow diagram with constant-$$\tau$$ contours for $$\theta=1$$. At $$\tau=-\infty$$, an infinitely small circle is located at $$X=\tanh(\theta)$$, indicating the source of the time flow. As $$\tau$$ increases to $$0$$, the center position moves toward $$X= -\infty$$; at the same time, the radius $$R$$ also becomes larger. At $$\tau=0$$, the contour (20) reduces to the vertical line of

We next consider the spatial range of $$s$$ for a fixed $$\tau$$. Taking the imaginary part of Eq. (19), we obtain

^{3}Together with (25) for the $$\tau=0$$ case, we conclude that the spatial coordinate $$s$$ always moves in

## Möbius quantization

In the usual CFT, the stress tensor $$T(z)$$ (and $$\bar{T}(\bar{z})$$) is formally expanded by the Laurent series around $$z=0$$ as

*Conserved charge and generator*

On the basis of the space-time structure of $$\zeta$$ in the previous section, we consider a conserved charge of the stress tensor for the constant-$$\tau$$ contour:

The integrand of Eq. (30) has no singularity for $$n=0,\pm 1$$, reflecting the global conformal symmetry of SL(2,R). Then, setting up the contour surrounding $$z=0$$ and $$1/\tanh(\theta)$$, we have

Moreover, these operators satisfy the sl(2) subalgebra:

For calculation of $$n >1 $$, the algebraic series expansion with respect to $$z \tanh(\theta)$$ gives

For $$n=2$$, for instance, we explicitly have

For the analytic continuation of $$n \le -2$$, we expand the integrand of (30) around $$z=\infty$$, because $$z/\tanh(\theta)$$ is beyond the convergence radius of the algebraic series. Using $$w=1/z$$, we have

Here, we make some comments on the above results. First, we can easily see that $${\cal L}_{\pm 2} \to L_{\pm 2}$$ in the $$\theta \to 0$$ limit, implying that $${\cal L}_{n}$$ recovers the conventional Virasoro generator of the radial quantization. In the SSD limit ($$\theta \to \infty$$), on the other hand, we can see that all of $${\cal L}_{\pm 1}$$ and $${\cal L}_{\pm2}$$ reduce to $${\rm const} \times {\cal L}_0$$. This suggests that $${\cal L}_n$$ of a finite $$n$$ collapses to $${\cal L}_0$$ in the $$\theta \to \infty$$ limit, reflecting $$\kappa=n/N_\theta \to 0$$. In other words, a continuous spectrum may be well defined for $$n,\, N_\theta \to \infty$$ with fixing $$n/N_\theta=\kappa$$, as will be seen in the next subsection.

Next, we discuss the Hermitian conjugate of the generator $${\cal L}_n$$. We can see that Eqs. (35) and (37) have a symmetric form with respect to exchanging $$n\leftrightarrow -n$$. If $$L_n$$ is equipped with the standard Hermitian conjugate of $$L_{-n}=L_n^\dagger$$, this symmetric relation of (35) and (37) ensures the Hermiticity of the generator,

*Virasoro algebra*

We next construct the commutator for the generator $${\cal L}_n$$. The operator product expansion (OPE) of the stress tensor is given by

Thus, the Lorentz transformation (2) basically provides the same physics as CFT based on the radial quantization. Moreover, this fact indicates that the Möbius quantization can be obtained through the conformal map of the Möbius transformation. We will comment on this point in the next section. Nevertheless, we would like to emphasize that the Möbius quantization clarifies an origin of the continuous Virasoro algebra obtained by the dipolar quantization, as follows.

An important viewpoint of Eq. (41) is that the SSD limit, i.e., $$N_\theta(=\cosh(2\theta))\to \infty$$, should correspond to the dipolar quantization by Ishibashi and Tada. Recalling Eqs. (7) and (10), we define the scaled spectrum and generator as

In the $$N_\theta\to \infty$$ limit, then, $$\kappa$$ becomes the continuous index and Eq. (41) is reduced to

*Vacuum*

In the usual CFT based on radial quantization, the vacuum is defined as

For the case of Möbius quantization, we may define the vacuum as

Here, we should recall that Eqs. (31), (32), (33), and (35) imply that $${\cal L}_n$$ for $$n\ge -1$$ can only be expanded by $$L_n$$ with $$n\ge -1$$. Thus, we trivially have

*Primary field*

We next discuss the primary field. In general, for a conformal transformation $$z\to w(z)$$, a primary field of a scaling dimension $$h$$ is defined by

In particular, the primary field specifies the highest weight state as

For the Möbius quantization, we have the commutator

If we define the primary state for the Möbius quantization as

From Eq. (53), moreover, we straightforwardly read

For $$n\le -1$$, on the other hand, $${\cal L}_n$$ generates a new state $${\cal L}_n|h\rangle_{\theta}$$, as in the usual CFT. Using the commutator (41) for $${\cal L}_n$$, we can thus construct the representation of the Virasoro algebra for the primary state (55).

The relation between $$|h\rangle_{\theta}$$ and $$|h\rangle$$ can be directly seen through the conventional Virasoro generator by the radial quantization; since $$L_{-1}$$ is the translation operator in the $$z$$ plane, we have

Then, Eq. (55) is rewritten as

This result is of course consistent with Eq. (56). Moreover, the norm of $$|h\rangle_{\theta}$$ can be evaluated as

If $$\alpha=\tanh(\theta) <1$$,

Thus, the series of Eq. (62) is convergent for $$\tanh(\theta) <1 $$, implying that $$|h\rangle_{\theta}$$ is normalizable. In the SSD limit, however, the series is divergent [11]. This suggests that it may be difficult to construct the primary field just at the SSD point within the present approach where the series of (35) and (37) is also divergent. In this sense, the primary field for the dipolar quantization is a nontrivial problem.

## The conformal mapping approach

In the previous section, we have quantized CFT on the Möbius coordinate, which yields a Virasoro algebra of the same form as that of the radial quantization. Of course, this is a natural consequence of the SL(2,R) symmetry in CFT, which suggests a more direct approach to obtain the Virasoro algebra of $${\cal L}$$. Let us consider the Möbius-type conformal map of SL(2,R):

We then define the Virasoro generator in the mapped coordinate ($$w$$ plane) as

Now, we rewrite $${\cal L}_n$$ in terms of the $$z$$ plane. Recall that, for the conformal map (65), the stress tensor transforms as

## Discussion

We have discussed the 2D CFT in terms of the Möbius quantization, which bridges the radial quantization and the dipolar quantization. An essential feature of the Möbius coordinate is that the source and sink of the time flow are, respectively, located at $$z=\tanh(\theta)$$ and $$1/\tanh(\theta)$$, and they move toward $$z=1$$ as the deformation parameter $$\theta$$ increases. As long as $$\theta$$ is finite, then, we have constructed the Virasoro algebra for the generator $${\cal L}_n$$ defined by Eq. (30), consistent with the result of the conformal mapping of SL(2,R). In the SSD ($$\theta\to\infty$$) limit, the locations of the source and sink finally fuse together at $$z=1$$, which corresponds to the dipolar quantization, where the continuous Virasoro algebra emerges for the scaled generator $$\tilde{\cal L}_n$$ defined by Eq. (42).

An important implication of the Möbius quantization approach is that the continuous Virasoro algebra for $$\theta\to\infty$$ can be viewed as a continuum limit of the conventional Virasoro algebra. As in the previous works, it is natural to deal with the scaled Hamiltonian $$\tilde{H}=\tilde{\cal L}_0 + {\bar{ \tilde{\cal L}}}_0 $$ at the SSD point, so that $${\mathfrak l}_0$$ and $$\bar{\mathfrak l}_0$$ in Eqs. (17) and (18) should be replaced by $$\tilde{\mathfrak l}_0$$ and $$\bar{\tilde{\mathfrak l}}_0$$. Then, the $$\tilde{\zeta}$$ coordinate involves the scale factor $$N_\theta$$:

We can also discuss the role of the scale factor $$N_\theta$$ by analogy with the finite-size scaling of CFT. For the radial quantization, the finite-size-scaling analysis for a cylinder is obtained through the conformal mapping $$z'= \frac{\ell}{2\pi} \ln z$$, where $$\ell$$ is a free parameter representing the strip width of the cylinder [15, 16]. For the scaled Möbius coordinate (69), we can rewrite $$\tilde{\zeta}$$ as

As was seen above, some properties of the spectrum of the scaled generator $$\tilde{\cal L}$$ can be understood by analogy with finite-size scaling. Also, the continuous Virasoro algebra (43) for the scaled generator $$\tilde{\cal L}$$ is basically consistent with the dipolar quantization. However, we should mention that there remain some difficulties in the SSD limit. $$\ell_\theta$$ in Eq. (71) is not a free parameter but the scale factor embedded in the Lorentz transformation, and thus $$\tilde{\zeta}$$ has the limit $$\tilde{\zeta}= \frac{2}{z-1}$$ for $$\theta \to \infty$$. This function corresponds to the mapping of $$u=e^{\tilde{\zeta}}=\exp(\frac{2}{z-1})$$ containing the essential singularity at $$z=1$$, which causes peculiar behaviors in the dipolar quantization [11–13]. Accordingly, the primary field (55) is unnormalizable at the SSD point, and the Hermitian conjugate (39) is inconsistent with that defined in Ref. [13]. A significant point of the SSD limit is that the source and sink of the time flow fuse at $$z=1$$, which never occurs for the case of the usual CFT, where $$\tau=-\infty$$ is assumed to be isolated in the complex plane. Further investigations are clearly required to understand the SSD and the continuous Virasoro algebra. We believe that the relation (42) provides a solid footing from which to address the SSD limit. In addition, it should be noted that a connection to similar quantization approaches based on the global conformal symmetry [17], such as N-S quantization [18], is also an interesting problem.

## Acknowledgements

The author would like to thank H. Katsura and T. Tada for useful discussions and comments. This work was supported in part by Grants-in-Aid No. 26400387 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

## Funding

Open Access funding: SCOAP^{3}.

#### Expansion coefficients of $${\mathcal{L}}_{n}$$

The Virasoro generator $${\cal L}_n$$ of the Möbius quantization can be represented by a series of the conventional Virasoro generator $$L_l$$ of the radial quantization. We assume $$n>1$$ for Eq. (30). Using $$t=\tanh(\theta)(<1)$$ for simplicity, we have

The integration path is the contour surrounding $$y=0$$ and 1 in the complex $$y$$ plane. The pole in the contour is located only at $$y=0$$ if $$n\ge -1$$. Using Goursat’s formula, then, we have

#### Useful commutators

For the Virasoro generators $$L_0$$ and $$L_{\pm 1}$$, we have the following commutators:

## References

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