74 problems

Let $\mathbf{Vir}$ be the Virasoro algebra, and call a $\mathbf{Vir}$-module a Harish-Chandra module when it is a weight module with finite-dimensional weight spaces. Kac's conject…

Norm formula conjecture. The norm has the expansion

Virasoro conjecture. The partition function $Z^M(t)$ is annihilated by $L_n$ for every $n\geq -1$:

Let $V$ be a space whose Gromov–Witten theory has large phase space potential $\Phi(\mathbf{t})$, and let the Virasoro algebra act on the corresponding generating functions. Write…

Let $V$ be a smooth projective manifold, let $Z(T;\lambda)$ be its total Gromov–Witten generating function, and write … The genus-$g$ $L_n$ constraint is the equation…

Let $Z(\mathbf{t})$ be the exponential of the $\mathrm{KdV}_r$ potential function. Let $W_r^+$ denote the relevant algebra of differential operators, containing the Virasoro genera…

The Virasoro characters are represented by $q$-series whose coefficients have a constant sign. Stony Brook's conjecture. There exist constant-sign $q$-series expressions for the Vi…

Conjecture 1.2. There are no irreducible mixed modules over the Virasoro algebra.

Nonexistence conjecture. There are no simple mixed $\mathcal{V}$-modules.

Let $\langle\!\langle\mathcal L_n\rangle\!\rangle_g$ denote the genus-$g$ one-point Gromov–Witten correlator with the Virasoro vector field $\mathcal L_n$, and let $\rho_{g,n}$ be…

Topological string theories on suitable target spaces lead to gravitational quantum cohomology, whose generating function is expected to be governed by an integrable hierarchy. Vir…

Let $M$ be a Grassmannian manifold, and let the Virasoro operators be the operators described in equations (47), (48), together with $L_0$ from equation (29), using the quantum-coh…

Let $|\chi_{r,s}\rangle$ be the Virasoro singular vector at level $n=rs$, normalized so that the coefficient of $L_{-1}^n|A_{r+1,s+1}\rangle$ is $1$. Let $\beta$ be the bosonizatio…

Let $L$ be the level of an eigenvalue, let $ell$ be real, and let $A_{+}(s)$ and $A_{-}(s)$ be the entire functions associated with the eigenvalues of the operators…

Let $r$ be such that the half-integer rank $r-1/2$ and the integer rank $r-1$ are defined. An irregular vector of rank $r-1/2$ is a Virasoro irregular vector associated with an irr…

An irregular vector of rank $5/2$ is a vector in a completed Virasoro Verma module associated with an irregular singularity of rank $5/2$; an irregular vector of rank $2$ is define…

Let $b$ tend to infinity, let $\tilde h,\tilde h_2,\tilde h_3,\tilde h_4$ be fixed parameters, and write $x$ for the conformal-block expansion variable. In the one-sided special ca…

Rank-$5/2$ Whittaker-state conjecture. The state $|\widetilde\Psi\rangle$ exists and is uniquely determined by $\{\alpha_n\}_{n\in\mathbb N}$ and $c$. Moreover,

Rank-$5/2$ irregular-block conjecture. The irregular state $|I^{(5/2)}\rangle$ exists, and the defining equations admit a solution at every order in $\varepsilon$, with

Gaiotto–Teschner conjecture. The differential-operator constraints arising from the Feigin–Fuchs realization of the positive Virasoro algebra uniquely fix a chain of embeddings

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\Ua(\g)$ be its enveloping algebra. ACC conjecture. The algebra $\Ua(\g)$ satisfies the ascending chain condition on primitive…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\Ua(\g)$ be its enveloping algebra. Complete-primality conjecture. Every primitive ideal of $\Ua(\g)$ is completely prime. The…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$. Let $\operatorname{P.Prim}\operatorname{S}(\g)$ denote the Poisson primitive spectrum of the symmetric algebra and let…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote the associated Poisson primitive ideals, and l…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be one-point local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote their associated Poisson primitive i…