After assuming conformal symmetry, it is natural to make assumptions on the theory’s spectrum, i.e. its space of states. For any complex value of the central charge \(c\), the spectrum of Liouville theory is

\[\mathcal{S} = \int_{\frac{c-1}{24}}^{\frac{c-1}{24}+\infty} d\Delta\ \mathcal{V}_\Delta \otimes \bar{\mathcal{V}}_\Delta\ ,\]

where \(\mathcal{V}_\Delta\) is a Verma module (with conformal dimension \(\Delta\)) of the left-moving Virasoro algebra, and \(\bar{\mathcal{V}}_\Delta\) is the same Verma module of the right-moving Virasoro algebra. Some features of this spectrum are:

- It is continuous, with no discrete terms.
- Each representation appears with multiplicity one.
- It is diagonal, i.e. a given left-moving representation is always paired with the same right-moving representation.
- For \(c>1\), the spectrum is unitary, i.e. each Verma module has a positive definite bilinear form such that the Virasoro generator \(L_0\) is self-adjoint.

**Higher multiplicities:**Allowing each representation to have a finite multiplicity leads to theories that may be called \(N\times\) Liouville theory with \(N\geq 2\). Then each representation is characterized not only by its conformal dimension, but also by a multiplicity index that can take \(N\) values.**Discrete terms:**Correlation functions of Liouville theory depend analytically on the fields’ conformal dimensions. So these dimensions can be analytically continued outside the spectrum without much happening. More precisely, the operator product of two fields of dimensions \(\Delta_1\) and \(\Delta_2\) is a linear combinations of fields with dimensions \(\Delta \in (\frac{c-1}{24},\frac{c-1}{24}+\infty)\), provided \(\Delta_1\) and \(\Delta_2\) are close enough to this same interval. However, discrete terms have to be included if \(\Delta_1\) and \(\Delta_2\) go too far, in particular if \(c>1\) and \(\Delta_1=\Delta_2<\frac{c-1}{32}\). (See Exercise 3.5 of my review article for more details.)**Non-analytic structure constants:**For some values of the central charge, the assumption that structure constants are analytic in the conformal dimensions is known to be necessary for Liouville theory to be unique. Namely, for \(c=1-6\frac{(p-q)^2}{pq}\) for \(p,q\) positive integers (i.e. a central charge where a minimal model exists), there exists an alternative consistent theory with the same spectrum as Liouville theory, but non-analytic structure constants. A theory of this type was first found by Runkel and Watts at \(c=1\). The generalization to other rational central charges was proposed by McElgin and confirmed by myself and Santachiara.

#### Degenerate fields and Teschner’s trick

#### The basic idea of the conformal bootstrap approach is to solve crossing symmetry equations. The basic problem of the approach is that these equations involve sums over a basis of the spectrum, and such a basis is infinite is most cases, including in the case of Liouville theory. To circumvent this problem, an idea is to use crossing symmetry equations for four-point functions that involve one degenerate field. Since the fusion of a degenerate field with any other field produces only finitely many primary fields, the corresponding crossing symmetry equations involve finitely many terms. In the case of the simplest nontrivial degenerate fields, these equations actually involve only two terms, i.e. the four-point functions are combinations of only two conformal blocks. The equations can be written explicitly, and solved analytically.

It is far from obvious that degenerate fields can be used in Liouville theory, since such fields are absent from the spectrum. Nevertheless, the assumption that such fields exist leads to the complete determination of the three-point structure constants, and the results agree with the DOZZ formula for these structure constants. The original derivation of the DOZZ formula relied on the Lagrangian formulation of Liouville theory, and therefore said nothing about uniqueness. In contrast, Teschner’s degenerate crossing symmetry equations imply a statement of uniqueness, under the assumption that degenerate fields exist.

Having an analytic formula for structure constants makes it possible to study the appearance of discrete terms, when the conformal dimensions are analytically continued. Moreover, the degenerate crossing symmetry equations are very suggestive that non-analytic structure constants can exist for certain discrete values of the central charge. Restricting to \(c\leq 1\) and writing \(c = 1-6(\beta - \frac{1}{\beta})^2\) with \(\beta \in \mathbb{R}\), the degenerate crossing symmetry equations indeed dictate how the three-point structure constant behaves when momentums are shifted by \(\beta\) or \(\frac{1}{\beta}\). (The momentum \(P\) associated to the conformal dimension \(\Delta\) is defined by \(\Delta = \frac{c-1}{24}+P^2\).) Assuming that the structure constant is a continuous function of dimensions, this determines it uniquely if \(\beta^2\) is an irrational number. But uniqueness fails if \(\beta^2 = \frac{p}{q}\) is rational, and this correspond to central charges where Liouville theory is known not to be unique. Of course, more work is needed in order to make more precise statements, including explaining why uniqueness fails if \(\beta^2\) is a positive rational number, but not if it is a negative rational number.

#### Analytic crossing symmetry equations

#### Since the assumption that degenerate fields exist is strong, the corresponding statement of uniqueness is weak. What can we deduce from crossing symmetry of four-point functions that do not involve degenerate fields? In diagonal conformal field theories, crossing symmetry can be reduced to an algebraic relation between the three-point structure constant (\(C\)) and the fusing matrix (\(F\)),

\[C_{12s} C_{s34} F_{\Delta_s,\Delta_t}\begin{bmatrix} \Delta_2 & \Delta_3 \\ \Delta_1 & \Delta_4 \end{bmatrix} = C_{23t}C_{t41} F^{-1}_{\Delta_t,\Delta_s}\begin{bmatrix} \Delta_2 & \Delta_3 \\ \Delta_1 & \Delta_4 \end{bmatrix}\]

(See for example my review article for the derivation.) This relation easily implies that the three-point structure constant is unique. However, it is difficult to determine under which assumptions that result holds. The derivation of our relation relies on the linear independence of a family of conformal blocks: this independence is intuitively obvious in the limit \(z\to 0\), where the blocks behave as powers of \(z\), although proving it rigorously in a well-defined space of functions may not be that easy.

We could well accept the above relation in the case where all involved representations belong to the Liouville spectrum, and try to analytically continue it in conformal dimensions and/or the central charge, in order to find out when uniqueness breaks down. However, this would require working with the fusing matrix \(F\). While explicitly known in principle, the fusing matrix is a complicated object, and not easy to work with.

Therefore, the argument that Liouville theory is unique based on crossing symmetry equations with no degenerate fields may be more a formal manipulation than a genuine proof, and is hard to use for testing the limits of uniqueness.

#### Numerics with the spectral function

#### In their recent article, Collier, Kravchuk, Lin and Yin build on ideas, technique and code from the ongoing conformal bootstrap renaissance, i.e. the successful numerical implementation of the conformal bootstrap methods to conformal field theories in arbitrary dimensions (not just two dimensions) that started about ten years ago. Their first implementation of the numerical bootstrap relies on the assumption of unitarity. This assumption implies that squares of three-point structure constants are positive, and crossing symmetry leads to inequalities on correlation functions. These inequalities are in general used for deriving bounds on conformal dimensions and/or structure constants. In their article, Collier, Kravchuk, Lin and Yin find it easier to derive bounds on the spectral function \(f(\Delta_*)\), which they define as the contribution of Virasoro primary fields with dimensions at most \(\Delta_*\), to a four-point function with cross-ratio \(z=\frac12\). Known numerical bootstrap techniques usually deal with discrete spectrums, and the use of the spectral function is an adaptation to the case of continuous spectrums.

So they derive bounds on the spectral function, assuming not only unitarity, but also that the spectrum is diagonal, i.e. that all primary fields have zero spin. The spectral function of Liouville theory sits right in the middle of their bounds, although the allowed region is a bit large, and the authors admit that their bounds “have not quite converged convincingly”. They attribute this lack of convergence to the computational complexity of their method.

As a result, this method produces only weak evidence that Liouville theory is unique. The method can also be used to study the analytic continuation of Liouville theory, and more specifically the four-point function of a scalar field whose dimension obeys \(\Delta \geq \frac{c-1}{32}\) rather than the Liouville theory bound \(\Delta \geq \frac{c-1}{24}\). (The lower bound \(\Delta = \frac{c-1}{32}\) corresponds to the appearance of the discrete terms.)

#### Second numerical approach: the linear method

#### The linear method, used in Section 3.1, consists in writing crossing symmetry as a linear equation for the squared structure constant \(C^2_\Delta\),

\[\forall z \in \mathbb{C}\ , \quad \int d\Delta\ C^2_\Delta \mathcal{F}_\Delta(z) = 0 \ ,\]

where \(\mathcal{F}_\Delta(z)\) is a conformal block. Dealing with a continuous spectrum, and therefore a continuous set of conformal blocks, is not easy. The trick is to consider the conformal blocks not as a \(\Delta\)-parametrized family of functions of \(z\), but as a \(z\)-parametrized family of functions of \(\Delta\). After Taylor expanding near \(z=\frac12\), this family becomes discrete, and therefore easier to deal with. Schematically, the crossing symmetry equation becomes

\[\forall m \in \mathcal{M}\ , \quad \int d\Delta\ C^2_\Delta \mathcal{F}_m(\Delta) = 0 \ ,\]

with \(\mathcal{M}\) some discrete set of parameters. There is also a normalization condition of the type \(\int d\Delta\ C^2_\Delta \mathcal{F}_0(\Delta)=1\), which eliminates the trivial solution \(C^2_\Delta = 0\).

Given a finite subset \(\mathcal{M}_N\) of \(\mathcal{M}\), the idea is to build an approximate solution \((C_N)^2_\Delta\) as the linear combination of \(\mathcal{F}_0\) and \((\mathcal{F}_m)_{m\in\mathcal{M}_N}\) that satisfies the crossing symmetry equation for \(m\in\mathcal{M}_N\). Choosing the subsets \(\mathcal{M}_N\) such that \(\lim_{N\to \infty} \mathcal{M}_N = \mathcal{M}\), the question is then whether \(\lim_{N\to \infty} (C_N)^2_\Delta\) exists and agrees with the Liouville theory structure constant. (For this to work, it is actually necessary to rescale the conformal blocks, structure constant and integration measure \(d\Delta\) by \(\Delta\)-dependent normalization factors.) The result is that \((C_N)^2_\Delta\) indeed converges relatively quickly towards the Liouville theory structure constant. The mathematical interpretation of this convergence is that \((\mathcal{F}_m)_{m\in\mathcal{M}}\) form a basis of a certain space of functions of \(\Delta\), and their linear independence is the reason why Liouville theory is unique.

What I do not understand is why the authors formulate this argument in terms of the spectral function. My impression is that the argument should work with the structure constant \(C^2_\Delta\) itself, and that using the spectral function is an unnecessary complication.

Now what are the consequences for Liouville theory? Keeping the assumption that the theory is diagonal, but dropping the assumption of unitarity, this argument provides good evidence that Liouville theory is unique. However, the argument is not rigorous, in particular when dealing with functional spaces. This argument may therefore be seen as a numerical counterpart of the argument with analytic crossing symmetry equations that I reviewed above.

#### Side results

#### In Section 3.2, Collier, Kravchuk, Lin and Yin analytically derive the spectral density of Liouville theory, as the only solution of modular invariance of the torus partition function. To do this, they need assume not only unitarity of the theory, but also consistency on the torus. The only useful piece of information that follows from such strong assumptions is a confirmation that the lower bound on the spectrum must be \(\frac{c-1}{24}\). This meagre result only reinforces my skepticism about using modular invariance in non-rational CFT.

In Section 3.3, the problem of higher multiplicities is analyzed, and it is argued that allowing finite multiplicities while insisting that the theory remains diagonal only allows \(N\times\) Liouville theory, which the authors call Liouville theory with superselection sectors. The argument assumes that the theory is consistent on arbitrary Riemann surfaces, not only on the sphere.

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