25 problems

Twining-character modular transformation conjecture. The matrix entries satisfy

Let an RCFT have characters whose leading exponents are $-\frac{c}{24}+h_A$, and let the associated contour integrals have leading exponents $\frac12(p+\Delta_A)$. Suppose these ex…

A rational vertex operator algebra (RVOA) has a graded character, denoted by $\operatorname{ch}_V$, as in the source's equation (1.1). A congruence subgroup of $SL(2,\mathbb{Z})$ i…

Let $\Gamma(1)=SL(2,\mathbb{Z})$ be the modular group, and let the kernel be the normal subgroup of $\Gamma(1)$ whose elements are represented by the identity matrix in the modular…

Completeness conjecture. Conversely, every CFT built from conformal blocks of the rational vertex algebra ${\mathfrak V}$ defined on world sheets of types (3) and (4) can be obtain…

Modular-data conjecture. Consistent genus-one couplings should be governed by the modular data of the chiral categories $\mathcal{C}$ and $\widetilde{\mathcal{C}}$, for example thr…

Let the two-dimensional conformal field theory under discussion be the theory realizing the symmetry associated with the unitary Haagerup fusion category $H_3$. Let a Haagerup rati…

Let $W_k(\mathfrak{g},f)$ be a unitary minimal $W$-algebra, with corresponding conformal net denoted by $\mathcal{A}_{W_k(\mathfrak{g},f)}$. The algebra is strongly rational when i…

Let $W$ be a unitary minimal $W$-algebra, and let its corresponding conformal net be denoted by $d49c_W$. The algebra $W$ is strongly rational when it has the strong rationality pr…

An RCFT has vanishing gravitational anomaly when its left- and right-moving gravitational anomaly cancels. A boundary condition has finite $g$-function when its boundary entropy pa…

Let $\mathcal{T}$ and $\mathcal{T}'$ be RCFTs with positive chiral algebras. RCFT Witt-to-interface conjecture. If $\mathcal{T}$ and $\mathcal{T}'$ belong to the same Witt class, t…

Let $V$ be a positive, strong-rational vertex operator algebra with central charge $c=1$. A rank-$1$ lattice vertex operator algebra is the lattice construction associated with a p…

Let ${_V}\mathcal{H}_W$ and ${_{V'}}\mathcal{H}'_{W'}$ be RCFTs, and let $\mathsf{W}(\mathcal{H}),c(V),c(W))$ and $(\mathsf{W}(\mathcal{H}'),c(V'),c(W'))$ be their respective invar…

Regularity conjecture. The fixed-point subalgebra $\mathcal{V}^G$ is completely rational.

Let $M$ be a real $2n$-dimensional manifold, let $B$ be a $2$-form on $M$, and let $G$ be a Riemannian metric that can be Ricci-flat and Kähler for a suitable complex structure $I$…

Let $\mathbf{c}$ be the central charge of the gravity theory, and let $\mathbf{c}_{\rm crit}$ be the critical central charge determined by the given boundary chiral algebra. The Po…

A rational conformal field theory (RCFT) is a rational conformal field theory arising as the full theory on a surface between two opposite-chirality boundary theories of a bulk Che…

Contour-integral and fusion-rule conjecture. Contour integrals should describe the characters of certain families of RCFTs, and integrality of the resulting fusion rules may provid…

Let $\ell$ denote the parameter measuring the excess in the modular differential equation, and let quasi-characters be the integer-coefficient, not necessarily positive, solutions…

Let an extended topological field theory be equipped with the algebraic data entering the full center construction and with the relevant defects and morphisms of all codimensions.…

Let $V$ be an affine Lie algebra and fix a positive integral level. Consider the category generated by the integrable highest weight modules at that level, and, more generally, mod…

A modular tensor category is a semisimple ribbon category with finitely many inequivalent simple objects and a braiding whose matrix of traces of the double-braiding morphisms on s…

Let $Y_{ab}^c$ be the scalars indexed by primary fields $a,b,c$ of a rational conformal field theory, and let $[s_{ab}]$ denote the $S$-matrix of the RCFT. Borisov–Halpern–Schweige…

Let $Y_{ab}^c$ be the scalars indexed by primary fields $a,b,c$ of a rational conformal field theory, as introduced by Pradisi, Sagnotti and Stanev. Pradisi–Sagnotti–Stanev's conje…

A unitary rational conformal field theory (RCFT) has a modular functor, and a Chern–Simons–Witten (CSW) theory is labeled by a pair $(G,\lambda)$ with $G$ a compact group and…