Infinite circumference limit of conformal field theory

We argue that an infinite circumference limit can be obtained in 2-dimensional conformal field theory by adopting $L_0-(L_1+L_{-1})/2$ as a Hamiltonian instead of $L_0$. The theory obtained has a circumference of infinite length and hence exhibits a continuous and heavily degenerated spectrum as well as the continuous Virasoro algebra. The choice of this Hamiltonian was inspired partly by the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The enigmatic behavior of sine-square deformed systems such as the sharing of their vacuum states with the closed boundary systems can be understood by the appearance of an infinite circumference.

We argue that an infinite circumference limit can be obtained in 2-dimensional conformal field theory by adopting L0 − (L1 + L−1)/2 as a Hamiltonian instead of L0. The theory obtained has a circumference of infinite length and hence exhibits a continuous and heavily degenerated spectrum as well as the continuous Virasoro algebra. The choice of this Hamiltonian was inspired partly by the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The enigmatic behavior of sine-square deformed systems such as the sharing of their vacuum states with the closed boundary systems can be understood by the appearance of an infinite circumference.
Among all Virasoro generators L n that play essential roles in the analysis of a conformal field theory (CFT), three of them, L 0 , L 1 and L −1 , form a subalgebra that is isomorphic to sl(2, R) and corresponds to the global conformal transformation. The Casimir operator of this subalgebra can be expressed as if we introduce the notation as follows; In analogy to the (2+1)-dimensional Lorentz transformation, the space spanned by L 0 , L + , and L − is apparently divided into three distinctive regions. The first region is "time like" and contains L 0 and small perturbations around it. Any vector within this region can be transformed to L 0 up to some numerical factor, by the global conformal transformation, SL(2, R). This is actually the region that we would have in mind when we demand the invariance of the vacuum on the basis of the physical equivalence of the states connected by the global conformal transformation. The second region is "space like" and contains the linear combination of L + and L − . The region between these two is the last one and is called the "light like" region. This region is represented by ei- If we further evoke the analogy with the Lorentz geometry, the "time like" region corresponds to a "massive" representation. Because we observe the spectrum of L 0 in this region, the "mass" in this case should be the inverse of the circumference, or the finite scale of a CFT [1,2]. Thus, it is natural to induce that the "light like" region corresponds to the "massless" representation, having an infinite circumference. In this study, we argue that if we consider the generator in the "light like" region, say L 0 − L + (plus the anti-holomorphic partL 0 −L + , to be exact), as a Hamiltonian, we can obtain a CFT with an infinite circumference. One might say that the following analysis is the "light-cone quantization" of a CFT in terms of the global conformal SL(2, R). Let us start with the generator that corresponds to L 0 − L + , given by in the complex coordinate. We denote the above generator as l 0 . The following analysis may exhibit a partial similarity with those in [3,4]. Let us introduce the eigenfunctions of l 0 as follows: We can readily solve Eq. (4) as where A κ denotes a κ dependent constant. Using f κ (z), we can introduce a set of differential operators as follows; Note that the above definition includes the case for l 0 (3), as its κ = 0 case, provided that A 0 = 1. Therefore, we take A 0 to be unity in the equations given below. The commutation relation among the l κ generators is easily calculated to be Noting that from the formal solution of f κ (z) (5), we arrive at the Witt algebra or classical Virasoro algebra; if we impose the condition that A κ A κ = A κ+κ , which is satisfied by We have introduced the eigenfunctions of l 0 , f κ (z) as a method of defining l κ , but let us now calculate the action of l κ on f κ (z). It is easy to derive that (11) The action of l κ on f κ (z) shifts the eigenvalue of f κ (z) in the amount of κ yielding f κ+κ (z), as well as multiplies it by −κ . Therefore, the Witt algebra (9) can be represented over the (Hilbert) space spanned by f κ (z)'s.
The argument presented above can be duplicated for another set of differential operators starting from wherez denotes the complex conjugate of z. Thus we have constructed two independent sets of the classical Virasoro algebra. Now we would like to consider the time-development of the system driven by the generator l 0 +l 0 . If we parametrize the time by t and space by s, we obtain the following set of equations: It solves in a consice form; If we use the Cartesian coordinates z = x + iy, where it follows readily that Therefore, the trajectory of the constant t is a circle with a radius 1 |t| whose center is located at z = 1 + 1 t . The significance of this contour is that it always passes through the point z = 1. Note that as the contour starts from z = 1 and encircles back to z = 1, the parameter s ranges from −∞ to ∞.
If we rewrite the time translational vector field in the Cartesian coordinates, we obtain This yields a dipole field located at z = 1 in accordance with the previously mentioned fact that the constant time contour always passes through z = 1. While the conventional treatment [1] leads to the so called radial quantization [5], after the configuration of the time-translational vector field, the present situation may be called as the "dipolar quantization." Once we understand the constant time contour, we can define the conserved charges as follows; where T (z) = T zz (z) is the energy momentum tensor of the original CFT. Note that for κ = 0 (20) We can further calculate the commutation relations among the charges defined above using the operator product expansion of the energy momentum tensor The result reads We have now obtained the continuous Virasoro algebra with central charge c [11] [12] . When deriving the central charge in Eq. (22) , it is probably the easiest to consider the t = 0 contour, which is a line that connects z = 1 with the infinity: Then, the term in question becomes Now let us exploit the algebraic structure of Eq. (22) to obtain information about its spectrum. If we denote an eigenstate of L 0 with an eigenvalue α with an additional index σ denoting possible degeneracy as operating L κ on |α, σ yields based on commutation relation (22). Therefore, starting from the vacuum or any other energy eigenstate, we can construct an eigenstate for L 0 with an arbitrary eigenvalue because κ can assume any real value. On the other hand, the integrand in Eq. (20) can be expressed in terms of the integral of L κ : If we assume the regularity of the above operator when it acts on the vacuum state |0 while considering the t → −∞ limit, we must demand that as L n |0 = 0 is usually required for n ≥ −1 in the analysis of ordinary 2d CFTs. Eq. (29) suggests that the spectrum of the system is bounded below by |0 , at least for the states that can be constructed by the multiplication of L κ . Thus, we have established that the theory with the Hamiltonian given by exhibits a continuous spectrum. This is consistent with the argument at the beginning of the present correspondence, as well as the previously noted observation that the variable s which is conjugate to the momentum, takes values from −∞ to ∞. This observation offers an elucidation of the phenomenon known as the sine-square deformation (SSD) [6], at least for the case involving a CFT [7]. It was found in [6] that a certain class of quantum systems exhibited identical vacua regardless of whether closed-or open-boundary conditions were imposed, if the coupling constants of the open-boundary system were modulated as Here g is the original coupling constant and L is the size of the system. Note that the coupling constant becomes zero at the both ends x = 0 and x = L as it should be for the open-boundary condition, while it assumes its original strength g just at the middle of the system x = L 2 . If SSD is applied for 2-dimensional conformal field theories [7], it is easy to see that the resulting Hamiltonian [7] becomes just SSD for a 2d CFT produces exactly the same Hamiltonian discussed in the present study. The implications of this fact for string theory were previously discussed by one of the present authors in [8,9]. It had been somewhat enigmatic that these two systems with different boundary conditions share the same vacuum state because intuitively, the lowest energy state should be the most affected by the global structure of the system. Now, we know that SSD systems possess continuum spectra implying that the system has an infinitely large space. The distinction between the open-and closed-conditions at the ends that located infinitely far away becomes irrelevant. It would be interesting to see if this explanation is applicable for other SSD systems besides CFT considered here. As a matter of fact, it was shown in [10] that the first excitation of several SSD fermionic systems on finite lattice correspond to the inverse-square of the system size, hence hinting that this is indeed the case. Another point worth mentioning in regard of the relation between the present analysis and SSD is the significance of the point z = 1. There found [8] new solutions for SSD, by one of the present authors: e L−1 φ(0)|0 , where φ is a primary field, and by H. Katsura: n>1 L −n |0 . These solutions can be recast as respectively, thus signify the the point z = 1. This resonates the peculiar role played by the point z = 1 in the dipolar quantization explained earlier.
The structure of the Hilbert space of the proposed theory could be very complex. It is apparent that for any given eigenvalue, there are many ways to multiply L κ ; hence, the eigenstates are likely to be degenerated in general. Nonetheless, a set of vectors expressed as offers an orthonormal set of vectors because amounts to the equation given below owing to the commutation relation (22): if we assume the Hermite conjugate takes the form The justification of the above assumption will be subject to further study. A straightforward generalization of the present analysis would also be worth considering: where n is an integer larger than unity. It is easy to see that these systems involve disconnected spaces. The detailed analysis for these generalizations is left for future study.
In summary, we argued that by adapting L 0 − (L 1 + L −1 )/2 as the holomorphic part of the Hamiltonian in stead of L 0 , we can derive a CFT with the continuous Virasoro algebra. The new theory can be considered to be the infinite circumference limit of the original one. Another viewpoint might be that the continuous spectrum can be attributed to the non-trivial world-sheet metric incorporated by the present procedure. Should we adopt this viewpoint, it would be natural to expect that the present result can be extended to other quantum statistical systems that are susceptible to SSD. While the present analysis uses the conformal symmetry, it would be interesting to see if the other systems also show continuous spectra [10]. Of course, CFT's are ubiquitous in the study of string theory, and hence there are many subjects wherein this new system can be put in good use.