Nevanlinna Analytical Continuation of the Central Charge in 2D Conformal Field Theory

Abstract

We present an analytic continuation of the central charge c in two-dimensional conformal field theory (2D CFT), modeled as a Nevanlinna function—an analytic map from the upper half-plane to itself. Motivated by the structure of vacuum energies arising from the quantization of spin-j conformal fields on the circle, we derive a discrete spectrum of central charges $c\left(j\right)=1+6j(j+1)$ and extend it continuously via $c\left(z\right)=1+6z$. The Möbius-inverted form $f\left(z\right)=1-6/z$ satisfies the conditions of a Nevanlinna function, providing a physically consistent analytic structure that captures both the unitarity of minimal models ($c<1$) and the continuous spectrum for $c\ge 1$. This unified framework highlights the connection between spectral theory, analyticity, and conformal symmetry in quantum field theory.

1. Introduction

Two-dimensional conformal field theory (2D CFT) has emerged as a cornerstone of modern theoretical physics, offering exact solutions and powerful symmetry principles that apply across string theory, statistical mechanics, and quantum field theory. A key feature of 2D CFT is the central charge c, which governs anomalies in the stress-energy tensor and characterizes different conformal universality classes. The classification of unitary minimal models with discrete central charges less than one was a major breakthrough, initiated by Belavin, Polyakov, and Zamolodchikov [1] and completed by Friedan, Qiu, and Shenker [2]. These models exhibit rich algebraic structures governed by the Virasoro algebra and contain only a finite number of primary fields. The central charge also manifests physically in the Casimir energy of fields on the circle [3] and appears in universal finite-size corrections [4,5]. The c-theorem [6] further links it to renormalization group flows, reinforcing its interpretation as a measure of degrees of freedom. In this paper, we revisit the computation of vacuum energy for conformal tensor fields of arbitrary spin j and show how it gives rise to a discrete spectrum of central charges $c\left(j\right)=1+6j(j+1)$. We then promote $j(j+1)$ to a complex variable z, enabling analytic continuation of the central charge to $c\left(z\right)=1+6z$. A Möbius inversion $z\mapsto -1/z$ leads to a Nevanlinna function $f\left(z\right)=1-6/z$, which satisfies the analyticity and positivity properties in the upper half-plane. These conditions are characteristic of physically significant Green functions and causal response functions [7,8]. The resulting analytical framework not only recovers the central charges of unitary minimal models $c=1-6/m(m+1)$ for $m>2$ but also connects the discrete and continuous parts of the conformal spectrum through a single spectral function. Beyond its formal structure, the proposed analytic continuation $f\left(z\right)$ has potential applications in the study of renormalization group (RG) flows in two-dimensional quantum field theories. By promoting the central charge to a Nevanlinna function $f\left(z\right)$, the construction naturally encodes causality and analytic monotonicity—that is, the imaginary part of the function remains positive and behaves smoothly without erratic oscillations. These features reflect the behavior prescribed by the c-theorem, suggesting that this framework could be used to interpolate between discrete conformal fixed points and explore continuous RG trajectories while preserving unitarity, causality, and an entropy-like flow of degrees of freedom. Therefore, the Nevanlinna property ensures that $f\left(z\right)$ behaves like a physically meaningful RG flow function: analytic, causal, and unitary—all in one.

2. Quantization of Conformal Tensor Fields on the Circle and the Casimir Energy

To explore the emergence of central charge in 2D CFT, we consider a conformal field $x\left(\theta \right)$ of spin j, defined on the unit circle in the complex plane. Under a rotation $\theta \mapsto \theta +\varphi$, the field transforms covariantly as

$$x^{\prime }\left(\theta \right)=e^{-ij\varphi }x\left(\theta \right),$$

(1)

reflecting its spin-j character. This transformation law ensures that the field acquires a phase proportional to its spin under angular displacements. The field admits a Fourier expansion of the form

$$x\left(\theta \right)=\sum _{n=-\infty }^{\infty }{x}_{n-j}{e}^{i(n-j)\theta },$$

(2)

where the mode index $n-j$ accounts for the spin shift. Each mode behaves like a harmonic oscillator with a frequency proportional to $n-j$. This sum includes both positive and negative modes, as is standard in quantum field theory on the circle. The presence of both annihilation and creation operators (associated with negative and positive n, respectively) is essential for the consistency of quantum theory.

To compute the vacuum energy (Casimir energy), we consider the sum over non negative-frequency modes

$$E_0\left(j\right)={\displaystyle \frac{1}{2}}\sum _{n=0}^{\infty }(n-j)={\displaystyle \frac{1}{2}}\sum _{n=1}^{\infty }n-{\displaystyle \frac{1}{2}}\sum _{n=1}^{j}n,$$

(3)

The expression in Equation (3) separates into spin-independent and spin-dependent contributions. The spin independent part resembles the vacuum energy contribution of an infinite set of bosonic harmonic oscillators, each with zero-point energy. This divergent sum can be regularized using an exponential cut-off, as shown in [3]:

$$\sum _{n=1}^{\infty }n⟶\sum _{n=1}^{\infty }n{e}^{-\epsilon n}=-{\displaystyle \frac{\partial }{\partial \epsilon }}\sum _{n=1}^{\infty }{e}^{-\epsilon n}=-{\displaystyle \frac{\partial }{\partial \epsilon }}\left({\displaystyle \frac{1}{1-e^{-\epsilon }}}\right).$$

(4)

As $\epsilon \to 0$, the leading divergence behaves like $1/{\epsilon }^2$ and must be subtracted through renormalization. The finite part that remains is

$$\sum _{n=1}^{\infty }n=-{\displaystyle \frac{1}{12}}.$$

(5)

This value corresponds to the physically observed Casimir energy in a one-dimensional bosonic system, such as a vibrating string. It is important to note that this regularized value is universal and arises independently of the specific regularization technique used, whether zeta function regularization or exponential cutoff [3]. The second term, which carries the spin dependence, is a finite sum and evaluates to the j-th triangular number:

$$T_j=\sum _{n=1}^{j}n={\displaystyle \frac{j(j+1)}{2}}.$$

(6)

Thus, the total vacuum energy becomes

$$E_0\left(j\right)={\displaystyle \frac{1}{2}}·\left(-{\displaystyle \frac{1}{12}}\right)-{\displaystyle \frac{1}{2}}·{\displaystyle \frac{j(j+1)}{2}}=-{\displaystyle \frac{1}{24}}-{\displaystyle \frac{j(j+1)}{4}}.$$

(7)

In 2D CFT, the vacuum energy on a circle of unit radius is related to the central charge c via

$$E_0=-{\displaystyle \frac{c}{24}}.$$

(8)

Equating this with the expression for $E_0\left(j\right)$ yields the central charge as a function of spin:

$$c\left(j\right)=1+6j(j+1).$$

(9)

This formula defines a discrete spectrum of central charges indexed by the spin j. The central charge grows quadratically with j, starting from $c=1$ for $j=0$, corresponding to a scalar field. For $j=1$, one obtains $c=13$, and for $j=2$, $c=37$. The discreteness of this spectrum reflects the quantization of spin and the representation theory of the Virasoro algebra. Moreover, the positivity of $c\left(j\right)$ for all $j\ge 0$ ensures unitarity in the corresponding conformal field theory representations.

3. Nevanlinna Analyticity and Unitarity

The central charges previously derived can be analytically continued by promoting the total spin factor $j(j+1)$ to a complex variable z and defining a function of the form

$$c\left(z\right)=1+6z.$$

(10)

This extension interpolates continuously between the discrete central charges associated with the conformal fields of spin j. A particularly interesting aspect of this analytic continuation is its connection to the theory of Nevanlinna functions, which are holomorphic functions on the upper half-plane.

$${\mathbb{C}}_{+}=\{z\in \mathbb{C}\mid \mathrm{Im}\left(z\right)>0\}$$

(11)

and have non-negative imaginary part. These functions arise naturally in contexts where causality and positivity constraints are imposed [7,8]. Consider the Hermitian conjugate of $c\left(z\right)$, defined as

$$f\left(z\right)=c(-1/z)=1-{\displaystyle \frac{6}{z}},$$

(12)

Such an inversion $z\mapsto -1/z$ should be interpreted within the framework of radial causality in 2D conformal field theory, as discussed in Ref. [3]. In this context, the transformation symbolically exchanges the roles of the ‘initial’ and ‘final’ configurations under conformal evolution, with the origin $z=0$ representing the initial state.

The function $f\left(z\right)$ is analytic in ${\mathbb{C}}_{+}$, and its imaginary part is given by

$$\mathrm{Im}\left[f\left(z\right)\right]={\displaystyle \frac{6\mathrm{Im}\left(z\right)}{{\left|z\right|}^2}}>0\mathrm{for}\mathrm{all}z\in {\mathbb{C}}_{+}.$$

(13)

Thus, $f\left(z\right)$ satisfies the defining property of a Nevanlinna function. By the Nevanlinna representation theorem, any such function admits the integral form

$$f\left(z\right)=a+bz+{\int }_{-\infty }^{\infty }\left({\displaystyle \frac{1}{t-z}}-{\displaystyle \frac{t}{1+t^2}}\right)d\mu \left(t\right),$$

(14)

where $a\in \mathbb{R}$, $b\ge 0$, and $\mu$ is a finite positive Borel measure satisfying

$${\int }_{-\infty }^{\infty }{\displaystyle \frac{1}{1+t^2}}d\mu \left(t\right)<\infty .$$

(15)

For the specific case of $f\left(z\right)=1-{\displaystyle \frac{6}{z}}$, this reduces to

$$f\left(z\right)=1+{\int }_{-\infty }^{\infty }\left({\displaystyle \frac{1}{t-z}}-{\displaystyle \frac{t}{1+t^2}}\right)d\mu \left(t\right),\mathrm{with}\mu \left(t\right)=6\delta \left(t\right).$$

(16)

At large $\left|z\right|$, we have the limiting behavior

$$\underset{\left|z\right|\to \infty }{lim}f\left(z\right)=1,$$

(17)

which corresponds to the central charge $c=1$ of the free boson CFT compactified on a unit circle. At first glance, this result may appear to contradict the derivation in Section 2, where the free boson corresponds to $j=0$ and hence $z=0$. However, this discrepancy is resolved by observing that $f\left(z\right)=c(-1/z)$ performs a spectral inversion. In this dual picture, the free boson limit of $c\left(z\right)$ at $z=0$ corresponds to the large-$\left|z\right|$ limit of $f\left(z\right)$. Thus, the function $c\left(z\right)$ describes the direct spectrum of spin states, while $f\left(z\right)$ is associated with another part of the spectrum.

The function $f\left(z\right)$ reproduces the central charges of unitary minimal models when $z=m(m+1)$ for integers $m\ge 3$, yielding:

$$c=f\left(m(m+1)\right)=1-{\displaystyle \frac{6}{m(m+1)}}.$$

(18)

This gives the familiar discrete sequence

$$c={\displaystyle \frac{1}{2}},{\displaystyle \frac{7}{10}},{\displaystyle \frac{4}{5}},\dots ,$$

(19)

corresponding to the Ising model ($m=3$), tricritical Ising model ($m=4$), and three-state Potts model ($m=5$), respectively. These values accumulate at $c=1$, which serves as a threshold separating the discrete unitary minimal models ($c<1$) from the continuum of conformal field theories with $c\ge 1$.

4. Conclusions

We have shown that the central charge in two-dimensional conformal field theory, traditionally viewed as a discrete label for unitary representations of the Virasoro algebra, can be systematically derived from vacuum fluctuations via the Casimir effect of spin-j tensor fields on the circle. The resulting expression $c\left(j\right)=1+6j(j+1)$, while initially discrete, admits a natural analytic continuation to a complex parameter z, revealing a more general structure encoded in the function $c\left(z\right)=1+6z$. Applying the inversion $z\mapsto -1/z$, we arrive at a Nevanlinna function $f\left(z\right)=1-6/z$, which satisfies the analyticity and positivity conditions in the upper half-plane and provides a physically meaningful extension of the central charge into the complex domain. This framework not only recovers the unitary minimal models for $z=m(m+1)$ with $m>2$ but also connects the discrete and continuous spectra of conformal field theories under a common analytic umbrella.