167 problems

Kashaev–Luo–Vartanov partition-function conjecture. The partition function of the Kashaev–Luo–Vartanov model is conjectured to satisfy

In a $(2+1)$D topological quantum field theory, the entanglement spectrum is the spectrum obtained from the reduced density matrix of a spatial subsystem, while a two-dimensional r…

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…

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 $\Gamma$ be a connected ribbon graph. Denote its bigraded $n$-color homology by $CH_n^{i,j}(\Gamma;\mathbb{C})$, where $i$ is the homological grading and $j$ is the quantum gra…

Let $X$ be a hyper-Kähler manifold and consider the Hilbert spaces of states arising from Rozansky–Witten theory on a manifold with boundary. Rozansky–Witten Hilbert-space conjectu…

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…

Let $p(d)$ denote the universal sesquilinear pairing obtained by gluing $d$-manifolds with a common boundary, with positivity meaning that $\langle x,x\rangle\neq 0$ whenever…

The p-modular TQFT structure conjecture.

Let $\2C$ satisfy the assumptions of Theorem, and let $\2Z_1(\2C)$ be its quantum double. Write $RT(M,\2Z_1(\2C))$ for the surgery invariant and $Tr(M,\2C)$ for the triangulation i…

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 $M$…

An $A_{\infty}$-category of lagrangian submanifolds in a symplectic manifold is the category whose objects are such lagrangian submanifolds and whose morphisms and compositions hav…

Rozansky–Witten conjecture. The theory has an associated TQFT with Hilbert spaces given by certain cohomology groups of $X$.

Let … where $q$ is an odd $p$-th root of unity and the standard generators satisfy $E^p=F^p=0$ and $K^{2p}=1$. Let ${\cal V}^{FN}$ be the Frohman–Nicas TQFT,…

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…

Let ${\cal V}^{PSU(n)}_k$ be the Frohman–Nicas $PSU(n)$ TQFT, and let ${\cal Q}^{[+]}$ be the ring of finite direct sums of irreducible homological TQFTs with non-negative multipli…

Let ${\cal Q}_5={\cal W}_5/{\cal U}_5$ and ${\cal U}_5\subset{}\cal W_5$ be the homological quotient and sub-TQFT over $\mathbb Z/5$, and let $Q$ and $U$ be the linear polynomials…

Let $I_{k,n,\varphi}^{FN}$ denote the higher $PSU(n)$ knot invariant of Frohman and Nicas, and let $\Delta^{(\gamma)}$ denote a higher irreducible Alexander Character. Positivity c…

Let the Hennings TQFT be constructed from quantum-$\mathfrak{sl}_2$, and let a homological TQFT be one of the theories arising from the homological constructions discussed above. H…

Let $K$ be a knot and let ${\cal O}[K,\lambda]$ be a proper solution, meaning a solution depending only on the triple $\Sigma_K,K,{x_0}$ and reducing to $\exp\gamma^{\Sigma_K}$ whe…

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

At perturbative order $2n$, let $c_v$ and $c_2(R)$ be defined by … and by $c_2(R){\bf 1}=\sum_a R(T^a)R(T^a)$, respectively, where $R$ is a representation of $SU(N)$. Common Casimi…