Notes on Liouville Theory at c<=1

TL;DR

The paper investigates continuing Liouville conformal field theory to $c\le 1$, examining a timelike boson interpretation for 2D de Sitter geometries and applying the conformal bootstrap to obtain three-point functions. It shows a unique analytic factor identical to the minimal-model case, but a non-analytic factor is necessary to produce a diagonal two-point function, which exists only for central charges of topological minimal models with $h>(c-1)/24$, while the full spectrum remains continuous and typically non-degenerate. By constructing shift relations and their solutions, the work connects spacelike Liouville amplitudes to a non-rational $c\le 1$ sector (the $C_M$ theory) and discusses their implications for minimal strings, showing that sensible diagonal metrics arise only in special topological cases. The analysis reveals a nuanced landscape: a timelike interpretation is problematic in general, but the $c=13-6(q^{-1}+q)$ topological sector yields well-defined amplitudes and a diagonal inner product, suggesting a constrained but meaningful link between Liouville continuation, non-rational CFTs, and minimal-string theories. Overall, the results delineate the precise conditions under which non-rational Liouville continuations yield consistent CFTs and illuminate the role of dual potentials in 2D gravity contexts.

Abstract

The continuation of the Liouville conformal field theory to c<=1 is considered. The viability of an interpretation involving a timelike boson which is the conformal factor for two-dimensional asymptotically de Sitter geometries is examined. The conformal bootstrap leads to a three-point function with a unique analytic factor which is the same as that which appears along with the fusion coefficients in the minimal models. A corresponding non-analytic factor produces a well-defined metric on fields only when the central charge is restricted to those of the topological minimal models, and when the conformal dimensions satisfy h>(c-1)/24. However, the theories considered here have a continuous spectrum which excludes the degenerate representations appearing in the minimal models. The c=1 theory has been investigated previously using similar techniques, and is identical to a non-rational CFT which arises as a limit of unitary minimal models. When coupled to unitary matter fields, the non-unitary theories with c<=-2 produce string amplitudes which are similar to those of the minimal string.

Figures (4)

• Figure 1: The figure at left above shows the complex $a$ plane in spacelike Liouville quantum mechanics, with the spectrum of normalized states shown in blue. There is a continuum of states for $p\in\mathbb{R}_{+}\,$, where $2a=Q+ip\,$. The figure at right is a plot of the potential $\exp(2b\hat{\phi})+Q^2/2\,$ for the unit mass Schroedinger equation corresponding to (\ref{['SLZM']}) with energy $2h\,$, where the substitution $2b\hat{\phi}=2b\phi+\ln(2\pi\mu)\,$ has been made. Also shown is a plot of the solution ${\rm K}(ip/b,\exp(b\hat{\phi})\sqrt{2/b^2})\,$ for $p=10\,b\,$, where $h=(Q^2+p^2)/4\,$. The choice $b=0.3$ has been made in both figures.
• Figure 2: The figure at left above shows the complex $\alpha$ plane in timelike Liouville quantum mechanics, with the spectrum of normalized states shown in blue. For each $\nu\in(0,1]\,$, which corresponds to a particular self-adjoint extension of the Hamiltonian, there is a continuum of states (\ref{['aboveshelf']}) with $\omega\in\mathbb{R}_{+}\,$, where $2\alpha=-\Lambda+i\omega\,$. In addition, there is a discrete set of states (\ref{['belowshelf']}) with $i\omega=2\beta(n+\nu)$ for $n\in\mathbb{Z}_{\geq 0}\,$. The figure at right is a plot of the potential $-\exp(2\beta\hat{\varphi})+\Lambda^2/2\,$ for the unit mass Schroedinger equation corresponding to (\ref{['TLZM']}) with energy $-2h\,$, where the substitution $2\beta\hat{\varphi}=2\beta\varphi+\ln(2\pi\rho)\,$ has been made. Also shown is a plot of the discrete solution ${\rm J}(2(n+\nu),\exp(\beta\hat{\varphi})\sqrt{2/\beta^2})\,$ for $\nu=0.6\,$ and $n=3\,$, for which $h=-(\Lambda^2+\omega^2)/4>-\Lambda^2/4\,$. Not shown is an example of the continuum of solutions (\ref{['aboveshelf']}) corresponding to $h<-\Lambda^2/4\,$ for this value of $\nu\,$. The choice $\beta=0.3$ has been made in both figures.
• Figure 3: The figure above shows the complex $b^2$ plane. The black curve is the branch solution of the equation $c=1+6(b+1/b)^2$ for which $b\in(0,1]$ for $c\geq 25$ and $\beta=ib\in(0,1]$ for $c\leq 1$. The blue half-line ($b^2\in\mathbb{R}_{+}$) is the domain on which $C_{ L}$ is the unique analytic solution to (\ref{['LV3ptshift']}). The red half-line ($b^2\in\mathbb{R}_{-}$) is the domain on which $C_{ M}$ is the unique analytic solution to (\ref{['MM3ptshift']}). Thus (\ref{['eq3ptshift']}) requires that $C_{ L}$ and $C_{ M}$ are non-analytic on the red and blue half-lines, respectively.
• Figure 4: The figure at left is the coefficient of the dual potential $\pi\tilde{\mu}\,e^{2\phi/b}$ as a function of $b^{-2}$ for the choice $\pi\mu=1$ in Liouville theory. Here the real part is shown in blue and the imaginary part is shown in red. It may be seen that the dual potential is not bounded from below for all $c\geq 25\,$, despite the assumption of a single vertex operator per conformal dimension utilized in the conformal bootstrap. The figure at right is the corresponding coefficient of the dual potential $\pi\tilde{\rho}\,e^{-2\varphi/\beta}$ as a function of $\beta^{-2}$ for the choice $\pi\rho=1$ in the continuation of Liouville theory to $c\leq 1$. In this case the dual potential is complex and vanishes at the central charges $c=13-6(q^{-1}+q)$ for $q\in\mathbb{Z}_+$ of the topological minimal models.