27 problems

Liouville spectrum conjecture. The spectrum of Liouville theory is

For a surface of genus $0$ with $n$ punctures, let the Liouville action be the suitably regularized critical value of the action functional of Liouville's equation, and let the acc…

Let $Q=b+b^{-1}$, and let $\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_{23}$ be complex Liouville momenta. The fusion coefficients are denoted by…

Let $mathbb S$ be the contour of intermediate Liouville momenta, and let ${{\mathcal F}}^s_{\alpha_{21}}$ and ${{\mathcal F}}^t_{\alpha_{32}}$ denote the s- and t-channel conformal…

Faddeev's conjecture. The constant part of the inversion factor on the right-hand side, namely $\exp\left(-\frac{\mathsf i\pi}{12}(\lambda^2+\lambda^{-2})\right)$, is related to th…

David–Distler–Kawai conjecture. The Jacobian arising from the use of this translation-invariant measure is given by the exponential of the Liouville action with modified coefficien…

Let $C$ be the three-punctured sphere and let $\mathcal{F}^{\mathrm{GS}}(\alpha_0,\alpha_1,\alpha_\infty)$ denote the Goncharov–Shen block associated with the configuration shown i…

The paper studies an extension of the free-field formalism to smooth spectral coverings, using the Maulik–Okounkov R-matrix to incorporate wall-crossing effects in spectral-network…

Let $C$ be a Riemann surface of genus $g$ with $n$ punctures, let $V_{\alpha_k}(z_k)$ be Liouville vertex operators, and set $h=3g-3+n$. Write $Q=b+b^{-1}$ and…

Let $\rho(x,y)$ be the conformal factor of a two-dimensional metric … with $\triangle=\partial_x^2+\partial_y^2$, satisfying the Liouville equation … Liouville theory is a classica…

Liouville completeness conjecture. Liouville conformal blocks form a complete basis with respect to the inner product above. This is the second mathematical gap in the rigorous def…

Let $c\geq 25$ and consider the inner product on conformal blocks … where $Z_{bc}$ is the ghost partition function and $Z_{26-c}$ is the timelike Liouville correlation function wit…

Let $c$ be the central charge, and let $Z_4(P_s,P_t,P_u)$ denote the four-point functions defined over the space $\{\Re \beta^2>0\}$, where $c$ is related to $\beta^2$ as in the mo…

Let $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ be real external Liouville momenta, let $z\in(0,1)$ be real, and let $C_{\gamma,\mu}^{\mathrm{DOZZ}}$ denote the DOZZ structure constant.…

Let $Q$ be the Liouville background-charge parameter, let the spectrum be … and let $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ be the external Liouville momenta. Write…

Let $\gamma$ and $Q$ be the Liouville parameters, with $\beta\in(\frac{\gamma}{2},Q)$ and $\mu,\mu_1,\mu_2>0$. For $i=1,2$, define $\sigma_i$ by … Let $R(\beta,\sigma_1,\sigma_2)$…

For the self-dual Liouville model at $b=1$, consider the Mathieu equation … By Floquet theory, let $\nu(\theta,P^2)$ be its Floquet index, so that there are quasi-periodic solution…

Liouville path-integral conjecture. The path integral of the quantised Liouville action carries the expected algebraic structure of a representation of the Virasoro algebra.

Let $cSigma^{z_1,z_2,z_3}(x)d^2x$ be the scaling-limit field of an observable defined on a random planar map conformally embedded in the sphere, with embedding points $z_1,z_2,z_3$…

Let $mathcal H$ be the physical Hilbert space obtained from the reflection-positive Liouville measure, and let $L_n$ and $bar L_n$ be the generators of the two Virasoro algebra rep…

Let $\rho_{mu,gamma}$ be the random measure arising as a scaling limit, and let $mathcal M=e^{gamma c}M_{hat g,gamma}$ under $hat{mathbb P}_{mu,gamma}$. For the Ising model, take…

Let $C$ be a punctured Riemann surface with refined pants decomposition $\widehat\sigma$, let $S$ be a set of $d$ degenerate insertions, and let $\varpi$ denote the associated disc…

Let $P_1,P_2,P_3,P_4$ be external Liouville momenta, let $p$ and $q$ determine the rational central charge, and define $p_i$ by … For the family of three-point functions parametriz…

Let $\mathcal{V}_\alpha$ denote a Liouville representation labelled by momentum $\alpha$, let $Q$ be the Liouville background-charge parameter, and let $b$ be the Liouville couplin…

The partition function of certain $\mathcal{N}=2$ gauge theories is related, after an appropriate translation of parameters, to correlation functions in two-dimensional Liouville o…