Minimal string theories and their limits

This text is an introduction to recent work by Collier, Eberhardt, Mühlmann and Rodriguez:

• arXiv:2309.10846 – The Virasoro Minimal String

• arXiv:2409.18759 – The complex Liouville string: the worldsheet

• arXiv:2410.07345 – The complex Liouville string: the matrix integral

I am grateful to the authors for helpful discussions and correspondence, and to SciPost for invitations to review two of these articles. (As always with SciPost, the reviews are online.) I am also grateful to the string theory group at IPhT Saclay for inviting me to discuss this subject in their journal club.

 

Minimal string theories

In the worldsheet approach, a string theory may be constructed from a two-dimensional conformal field theory with the central charge c = 26, where we consider primary fields of conformal dimensions $\Delta=\bar{\Delta}=1$. These conditions on the central charge and conformal dimensions, called respectively criticality and marginality, are necessary for the string theory to be independent from the parametrization of the worldsheet. Alternatively, if we take the physical spacetime to be the worldsheet itself rather than the target space, we obtain a model of two-dimensional quantum gravity. In this interpretation, criticality allows gravity to remain topological at the quantum level.

Which conformal field theories give rise to string theories that are simple enough to be tractable, but complicated enough to be interesting? A simple recipe is to take a product of two theories: a theory called the matter CFT, which can in principle be arbitrary, and Liouville theory, which allows arbitrary complex values of c and Δ, allowing us to fulfill criticality and to build a marginal field from any diagonal field of the matter CFT. The resulting string theory is tractable provided the matter CFT is, given that Liouville theory is exactly solved.

This recipe was first implemented by taking the matter CFT to be a diagonal minimal model, giving rise to minimal Liouville gravity. Other solvable diagonal CFTs give rise to other string theories which I will still call minimal:

• Diagonal minimal model (MM) → minimal Liouville gravity.

• Generalized minimal model (GMM) → no name.

• Liouville theory with c ≤ 1 (sometimes called timelike or imaginary Liouville theory) → Virasoro minimal string.

• Liouville theory with c ∈ ℂ ∖ ( − ∞, 1)∪(25, ∞) → complex Liouville string. (The original articles assume c ∈ 13 + iℝ, I am giving the largest domain that can be reached by analytic continuation in c.)

These diagonal CFTs are related by various limits, which suggests that the resulting minimal string theories should be related too. Let me quickly review these CFTs and their limits. (For more details see my review article arXiv:1406.4290.)

Solvable diagonal CFTs and their limits

We use standard notations for the central charge and conformal dimension:
$$\begin{aligned} c = 13+6b^2+6b^{-2} \ \  , \ \  \Delta = \frac{c-1}{24}-P^2\ \   , \ \  P_{(r, s)} = \frac{1}{2}\left(br +b^{-1}s\right) \end{aligned}$$
Let V_P be a primary field of momentum P, and for r, s ∈ ℕ^* let V_(r, s)^d be a degenerate field of momentum P_(r, s), i.e. a primary field whose OPE with any other primary field yields finitely many (namely rs) primary fields. In particular, the identity field V_(1, 1)^d has trivial OPEs V_(1, 1)^dV ∼ V.
$$\begin{aligned} \renewcommand{\arraystretch}{1.3} \begin{array}{|c||c|c|c|c|} \hline \text{CFT} & \text{MM} & \text{GMM} & \text{Liouville}_{c\leq 1} & \text{Liouville} \\ \hline\hline b & i\sqrt{\frac{q}{p}} & \mathbb{C}^* & i\mathbb{R} & \mathbb{C}\backslash i\mathbb{R} \\ \hline \text{Spectrum} & \{P_{(r, s)}\}_{\substack{1\leq r \leq q-1\\ 1\leq s \leq p-1}} & \{P_{(r, s)}\}_{r,s\in\mathbb{N}^*} & i\mathbb{R} & i\mathbb{R} \\ \hline \end{array}\end{aligned}$$
GMMs depend analytically on b, in particular we get b ∈ iℝ as a limit from b ∈ ℂ, and b ∈ ℂ by analytic continuation from b ∈ iℝ. This is not true for Liouville theory: the theory with b ∈ iℝ is a genuinely different CFT. In Liouville theory, degenerate fields can be recovered from generic fields for c generic, but not for c ≤ 1:
$$\begin{aligned} \lim_{P\to P_{(r, s)}} V_P \underset{\text{Liouville}}{=} V^d_{(r, s)} \quad , \quad \lim_{P\to P_{(r, s)}} V_P \underset{\text{Liouville}_{c\leq 1}}{=} V_{P_{(r, s)}}\end{aligned}$$
At the level of correlation functions, it follows that
$$\begin{aligned} \lim_{P\to P_{(r, s)}} \text{Liouville} = \text{GMM}\end{aligned}$$
Moreover,
$$\begin{aligned} b\in i\mathbb{R}\backslash i\mathbb{Q} \quad \implies \quad \text{Closure}\left(\{P_{(r, s)}\}_{r,s\in\mathbb{N}^*}\right) = i\mathbb{R}\end{aligned}$$
leading to the limit
$$\begin{aligned} \lim_{P_{(r, s)}\to P} \text{GMM}_{c\leq 1} = \text{Liouville}_{c\leq 1}\end{aligned}$$
We can also take limits in the central charge:
$$\begin{aligned} \lim_{b \to i\sqrt{\frac{q}{p}}} \text{GMM} = \text{MM} \ , \ \lim_{i\sqrt{\frac{q}{p}} \to b} \text{MM} = \text{GMM}_{c\leq 1}\ , \ \lim_{\substack{i\sqrt{\frac{q}{p}} \to b\\ P_{(r, s)}\to P}} \text{MM} = \text{Liouville}_{c\leq 1}\end{aligned}$$
Some of these limits hold at the level of certain correlation functions, but not all of them. Ignoring this subtlety, let me summarize all these limits:

Correlation numbers

The observables of minimal string theories are correlation numbers. They are defined as the integrals of correlation functions over the moduli. The moduli include the positions z_i of the fields: integrating over z_i is compatible with conformal symmetry thanks to marginality. Calling ℳ_g, n the moduli space of genus g Riemann surfaces with n punctures, correlation numbers read
$$\begin{aligned} V^\text{CFT}_{g,n}(P_1,\dots, P_n) = \int_{\mathcal{M}_{g,n}} \left<\prod_{i=1}^n V^{\text{CFT}_b}_{P_i}(z_i)V^{\text{Liouville}_{ib}}_{iP_i}(z_i)V^{\text{ghost}}(z_i) \right>_g \label{vgn}\end{aligned}$$
Here CFT ∈{MM, GMM, Liouville, Liouville_c ≤ 1}. Criticality and marginality are guaranteed by the identities
$$\begin{aligned} c(b)+c(ib)=26 \quad ,\quad \Delta(b,P)+\Delta(ib,iP)=1\end{aligned}$$
For example, ℳ_0, 4 is parametrized by the cross-ratio, and ℳ_1, 1 by the modulus. The integrand factorizes into correlation functions of the matter CFT, Liouville theory, and ghost CFT. Matter CFT correlation functions $\left<\prod_{i=1}^n V_{P_i}(z_i)\right>_g^\text{CFT}$ obey the various limits that we have discussed: we therefore expect that V_g, n obeys the same limits.

Let us start with the case (g, n)=(0, 3) of sphere 3-point functions. There are no moduli to integrate over, and CFT 3-point functions are proportional to structure constants $\left<V_{P_1}V_{P_2}V_{P_3}\right>\propto C_{P_1,P_2,P_3}$. From crossing symmetry of $\left< V^d_{(r, s)}V_{P_1}V_{P_2}V_{P_3}\right>$ with r, s ∈ ℕ^*, follow shift equations of the type
$$\begin{aligned} \frac{C_{P_1+b,P_2,P_3}}{C_{P_1,P_2,P_3}} = \text{known} \quad ,\quad \frac{C_{P_1+b^{-1},P_2,P_3}}{C_{P_1,P_2,P_3}} = \text{known}\end{aligned}$$
After coupling the Liouville and matter CFTs, it follows that 3-point correlation numbers are doubly periodic:
$$\begin{aligned} V_{0,3}(P_1+b,P_2,P_3)= V_{0,3}(P_1+b^{-1},P_2,P_3)= V_{0,3}(P_1,P_2,P_3)\end{aligned}$$
In the case of MM and GMM, this implies that V_0, 3 is a P_i-independent constant. Taking limits, this must also hold in the case of Liouville_c ≤ 1, as can be checked from the known expressions of 3-point structure constants. In the case of Liouville theory, V_0, 3 is a non-trivial, doubly periodic function, which was computed in arXiv:2409.18759, resulting in several equivalent expressions:
$$\begin{aligned} V_{0,3}^\text{Liouville} &= \frac{ib\eta(b^2)^3\prod_{j=1}^3 \vartheta_1(2bP_j|b^2)}{2\prod_{\pm,\pm}\vartheta_3(bP_1\pm bP_2\pm bP_3|b^2)} = \frac{b}{2}\sum_{k=0}^\infty \sum_{\sigma_1,\sigma_2,\sigma_3=\pm}\frac{\sigma_1\sigma_2\sigma_3}{1+e^{2\pi ib(\sum_{i=1}^3\sigma_iP_i+b(k+\frac12))}} \\ &= 2b\sum_{m=1}^\infty \frac{\prod_{i=1}^3 \sin(2\pi mbP_i)}{\sin(\pi m(b^2+1))}\end{aligned}$$
The first expression uses Jacobi theta functions ϑ₃(z|τ)=∑_{n ∈ ℤ}e^{iπ(τn2 + 2zn)} and $\vartheta_1(z|\tau)=e^{i\pi(\frac{\tau}{4}+z-\frac12)}\vartheta_3(z+\frac{\tau}{2}+\frac12|\tau)$. The properties of these functions ensure that V_0, 3 is invariant under P_i → P_i + b and also b → b⁻¹.

Sphere with 4 punctures

V_0, 4 is hard to compute from its definition , because there is no explicit formula for the integrand, and the integral is very nontrivial.

In the case of Liouville theory with c ≤ 1, the calculation of arXiv:2309.10846 is neither straightforward nor conceptually transparent, but the result is remarkably simple:
$$\begin{aligned} V_{0,4}^{\text{Liouville}_{c\leq 1}} = \tfrac14(b^2+b^{-2}) - \sum_{j=1}^4 P_j^2\end{aligned}$$
This is broadly compatible with the partial results that are known for V_0, 4^MM.

In the case of Liouville theory, there is a conceptually straightforward but technically complicated calculation, based on a direct study of the analytic properties of the integrand and integral arXiv:2409.18759:

• The integrand has poles as a function of P_i, leading to discontinuities of the integral. The jump at a discontinuity can be explicitly computed, because the integral localizes in z (the cross-ratio).

• When P_i → P_(r, s) becomes degenerate, the integrand can be written as a total derivative, and the integral can be performed explicitly. This phenomenon is well-known in the case CFT=MM, it gives rise to the so-called ground ring operators in the string theory. A ground ring operator is a descendant of a degenerate primary field at level rs − 1, so it does not survive in the limit P_(r, s) → P, which requires r, s → ∞.

These are the main constraints, and they do not fully determine V_0, 4^Liouville. However, there is a unique solution that grows polynomially as bP_j → +∞: the other solutions grow exponentially:
$$\begin{gathered} V_{0,4}^\text{Liouville} = 2b^2\left(\tfrac14(b^2+b^{-2}) - \sum_{j=1}^4 P_j^2\right) \sum_{m=1}^\infty \frac{\prod_{j=1}^4 \sin(2\pi mbP_j)}{\sin(\pi m(b^2+1))^2} \\ -\sum_{m,n=1}^\infty \frac{\prod_{j=1}^2 \sin(2\pi mbP_j)\prod_{j=3}^4\sin(2\pi nbP_j)}{\pi^2\sin(\pi m(b^2+1))\sin(\pi n(b^2+1))}\left[\frac{1}{(m+n)^2}-\frac{\delta_{m\neq n}}{(m-n)^2}\right] + 2\text{ perms.}\end{gathered}$$
Among many nontrivial properties, let us focus on the value for P₄ = P_(r, s) (using invariance under b → ib, P → iP):
$$\begin{aligned} \left. V_{0,4}^\text{Liouville}\right|_{P_4=P_{(r, s)}}= V_{0,3}^\text{Liouville}(P_1+P_{(r, s)}-P_{(1,1)},P_2,P_3) \left[\frac{rsP_{(r,-s)}}{4}-\sum_{j=1}^3\sum_{r'\overset{2}{=}1-r}^{r-1}\sum_{s'\overset{2}{=}1-s}^{s-1} \sqrt{\left(P_j+P_{(r',s')}\right)^2}\right]\end{aligned}$$
Here the factor V_0, 3 depends very weakly on r, s due to its invariance under shifts P_j → P_j + b, P_j → P_j + b⁻¹. And it is permutation-symmetric. We see that the dependence on r, s has become quadratic: much simpler than the dependence on P₄, in general agreement with the limits of CFTs. Following the limits, we should indeed be able to deduce the quadratic V_0, 4^{Liouville_c ≤ 1} from the complicated V_0, 4^Liouville. We also see that V_0, 4^Liouville knows the fusion rules of degenerate fields,
$$\begin{aligned} V_{(r, s)} V_P \sim \sum_{r'\overset{2}{=}1-r}^{r-1}\sum_{s'\overset{2}{=}1-s}^{s-1} V_{P+P_{(r',s')}}\end{aligned}$$
This is consistent with $\lim V_P \underset{\text{Liouville}}{=} V^d_{(r,s)}$ and therefore limLiouville = GMM. See also recent results by Artemev on minimal Liouville gravity arXiv:2506.09222.

Conclusion

More work would be needed to fully understand the limits of minimal string theories, and compute all correlation numbers in the MM and GMM cases. And this is just one aspect of the very rich results of the articles in question. Other aspects are explored in other articles by the same authors. In particular, they have a matrix model construction of the correlation numbers. According to arXiv:2410.07345, the relevant spectral curves are
$$\begin{aligned} & \text{Liouville}: \quad x(z) = -2\cos(\pi b^{-1}\sqrt{z}) \quad ,\quad y(z) = 2\cos(\pi b\sqrt{z}) \\ & \text{Liouville}_{c\leq 1}: \quad x(z) = -z^2 \quad ,\quad y(z) = \tfrac{1}{z}\sin(2\pi bz)\sin(2\pi b^{-1}z) \\ & \text{MM}: \quad x(z) = -2T_p(z) \quad , \quad y(z)=2T_q(z)\end{aligned}$$
where T_p, T_q are Chebyshev polynomials, which can also be recovered from the Liouville case.