49 problems

Let $\mathcal M(\Sigma,t)$ be the moduli space associated with a surface $\Sigma$ and parameter $t$. In the higher-genus setting, model $\mathcal M(\Sigma,t)$ by a toric manifold,…

Quantum $SO_3$ spin-Chern-Simons conjecture. The quantum $SO_3$ spin-Chern-Simons theory is, in fact, a TQFT and at the $1$ (mod $2$)-valued $SO_3$ level $2m-1$, it is isomorphic t…

Let $\phi$ be a pseudo-Anosov mapping class of a surface of arbitrary genus, and let $k$ denote the level of the associated TQFT representation. Infinite-order TQFT representation…

The p-modular TQFT structure conjecture.

Let $p$ be a fixed prime with $p\geq 3$. Let ${\cal V}^I_p$ denote the integral TQFT, and let $\dov {\cal V}^{FN}_p$ denote the corresponding Fibonacci-type TQFT; the tensor produc…

Let $M$ be an oriented connected closed three-manifold, possibly equipped with a banded trivalent colored graph, and let $p$ be a prime. Define … For a connected three-manifold $M$…

Let ${\cal V}$ be a non-semisimple TQFT whose mapping class group representations have precisely the Torelli groups as their kernels. Let ${\cal P}^+$ be the class of polynomials d…

Let ${\cal V}^{DF}$ be Donaldson's TQFT associated with moduli spaces of connections on a non-trivial $SO(3)$ bundle. For $x_1,x_2,\ldots$, define … Let ${\cal V}^{(D)}$ be the pol…

A trialgebra is an algebraic structure with three compatible algebraic operations, as envisioned here for constructing four-dimensional topological quantum field theories. Trialgeb…

Gluing conjecture. The continuum amplitudes satisfy

The objects are closed 1-dimensional manifolds equipped with chosen bounding 2-manifolds, the 1-morphisms are 2-dimensional cobordisms equipped with bounding 3-manifolds, and the 2…

Let $\mathcal{C}$ be a unimodular finite braided tensor category, and let $\operatorname{Ver}^{\zeta^{1/2}}_{p^{(n)}}$ denote a mixed higher Verlinde category. Regard these categor…

Let $\sigma_2(q/p)$ denote the genus-two signature at a rational number $q/p$. For coprime odd integers $0<q<p$, consider the transformed fraction $q/(2q+p)$. Modular transformatio…

Let $k$ be as in the paper, let $x_1$ and $y$ specify the twist, and let $\alpha_{\mathrm{DW}}$ and $\alpha_{\mathrm{GW}}$ be the Dijkgraaf–Witten and Gu–Wen classes in…

Chain-level $\mathfrak{gl}_N$ functoriality conjecture. The chain-level $\mathfrak{gl}_N$ link invariant arises as the truncation of an $\mathbb{E}_3$-monoidal functor

Moore–Tachikawa conjecture. There exists a two-dimensional TQFT

Completeness conjecture for the $S^n$ functional. The $S^n$ functional $Z$ is complete positive and complete finite. The conjecture occurs in the paper's discussion of extending th…

Let $G$ be a connected complex semisimple affine algebraic group with Lie algebra $\mathfrak{g}$. Consider a principal $\mathfrak{sl}_2$-triple $(e,h,f)\in\mathfrak{g}^{\times 3}$,…

Let $G$ be a connected semisimple affine algebraic group with Lie algebra $\mathfrak{g}$. Consider a principal $\mathfrak{sl}_2$-triple $(e,h,f)\in\mathfrak{g}^{\times 3}$, and set…

Let $\Sigma$ be a surface of genus $h$ with $n$ boundary circles, whose boundary circles are triangulated into $k_1,\ldots,k_n$ intervals. Pachner-complex conjecture. There exists…

Let $\mathsf{c}_\Delta^k$ be the BV cycle defined by the displayed construction referred to as (cDelta^n). BV-cycle nontriviality conjecture. The cycle $\mathsf{c}_\Delta^k$ is non…

Crane–Frenkel conjecture. There exists a categorification of the quantum group $U_q(\mathfrak{sl}_2)$ at roots of unity giving rise to a 4D TQFT.

Let $\Sigma^{n-1}\mathsf{Vect}^{fd}$ and $\Sigma^{n-1}\mathsf{sVect}^{fd}$ be the stable suspensions used as codomains for bosonic and fermionic fully extended TQFTs, respectively.…

Let $Y$ be an $n$-finite space, meaning that $\pi_0Y$ and all homotopy groups $\pi_q(Y,y)$ for $0\leq q\leq n$ and $y\in Y$ are finite. An $n$-dimensional bosonic or fermionic $Y$-…

Let $\mathscr{V}$ be a non-semisimple modular tensor category. Consider the non-compact filled 321-dimensional cobordism category $Cob^{nc,filled}_{321}$, its quotient…