35 problems

Combinatorial-map polynomiality conjecture. Solutions in this basis should have polynomial structure constants.

Let $D_{(r,s)}$ be the structure constants in the conformal-block decomposition of a torus one-point function, with contractible loop weight $n$, channel loop weight $w$, and exter…

Let $V$ be a non-diagonal field in the critical loop model, and let $\langle V\rangle$ denote its torus one-point function. Vanishing conjecture for non-diagonal fields. The one-po…

Consider a torus correlation function depending on the loop weight $w(P)$ associated with a diagonal channel field $V_P$. Holomorphicity conjecture. The correlation function is hol…

Let $V_P$ be a diagonal channel field and let $D_P$ be its torus structure constant. Let $\hat D_P$ denote the corresponding reference structure constant. Diagonal-field reference-…

Let $M$ be a one-point map on the torus, and let $Z_M$ denote its corresponding one-point function. Let $M'$ be a four-point map on the sphere, and let $Z^*_{M'}$ denote the associ…

A conformal field theory (CFT) on $\mathbf{R}^d$ has a state space $\mathcal{H}$ carrying an $\operatorname{\mathfrak{so}}(2,d)$-action. Its operator product expansion (OPE) is spe…

Three-dimensional derivation conjecture. The Liouville CFT partition function of $\Sigma$ can be derived from quantities associated with $M$. This conjecture proposes a three-dimen…

Exact lattice-bootstrap agreement conjecture. The non-constant ratios should be numerical artefacts, and the lattice and bootstrap results should exactly coincide for all $r,s$.

Universal amplitude-ratio conjecture. The ratios of amplitudes should have the observed independence from $L$, $K$, and $\rho$ for all amplitudes in all 4-point correlation functio…

Pole-free polynomiality conjecture. The normalized 4-point structure constant should remain polynomial when it has no pole, namely when $s\notin\mathbb{Z}$, and even when there is…

Holomorphic-term polynomiality conjecture. The holomorphic term of the normalized 4-point structure constant should also be polynomial.

Amplitude-ratio conjecture. The equality

Polynomial-degree conjecture. The quantity $d^{(x)}_{(r,s)}$ is a polynomial in $n$ whose coefficients are $\beta$-independent numbers, and

Universal polynomial-factor conjecture. All structure constants should be of this form.

Let $M$ be a combinatorial map and $W$ a choice of loop weights, and write $Z_{M,W}$ for the associated loop-model correlation function. Assume the critical limit exists. The corre…

Let $\mathcal{S}_{r_0}$ denote the relevant channel spectrum indexed by the smallest first Kac index $r_0$, and let $\mathcal{M}_{0,4}(r_i\mid\sigma_s,\sigma_t,\sigma_u)$ be the se…

Consider four-point conformal-bootstrap spectra in the $s$-, $t$-, and $u$-channels. Let $V_\Delta$ be a diagonal field with generic conformal dimension, meaning that…

Let an $n$-point function of diagonal and non-diagonal fields lie on a Riemann surface of genus $g$. In each channel, consider conformal-bootstrap spectra containing all allowed no…

Consider an $n$-point function of diagonal and non-diagonal fields on a Riemann surface of genus $g$. Let $\mathcal{M}^c_{g,n}(r_i)$ denote the set of weakly connected maps with th…

Let $\mathcal{M}_{0,4}(r_1,r_2,r_3,r_4)$ be the set of maps with the specified vertex data, and order signatures componentwise: $sigma ot greater than or equal tosigma'$ when…

In two-dimensional statistical physics, correlation functions of the $O(N)$ and Potts models can be expressed as sums over configurations of non-intersecting loops. For a large cla…

Contour-deformation conjecture. Physicists conjecture that, for a general bootstrap formula, the integration contour should be the spectrum line $Q+iP$ together with small circles…

Let $Q$, $\beta$, $\mu$, $\mu_\partial$, $P$, $q$, $G$, $U_{\rm FZZ}$, $\eta$, and $H$ be the quantities in the annulus Liouville CFT bootstrap formula. For $\mu\;\geqslant\; 0$,…

Equality conjecture. For every $g$,