55 problems

All-order higher-spin deformation conjecture. This generalized family includes deformations by arbitrary functions of higher-spin currents at all orders.

Let $s$ be the trigonometric deformation parameter and let $tau$ denote RG time. A flow is called linear when $log(s)$ is proportional to $tau$. Ricci flow conjecture. For Hermitia…

All-orders effective-action conjecture. The lambda-deformed action $S_{k,\lambda}(g)$ is the effective action for the non-Abelian Thirring model, valid to all orders in $\lambda$ a…

Let $\Pi=P_X\oplus P_Z$ be the positive plane determining a supersymmetric non-linear K3 sigma model, and let $G_\Pi$ be the corresponding group of automorphisms. For $g\in G_\Pi$,…

Let the $S^2$ sigma model be considered in the large-radius limit, and let its zero-mode Hamiltonian be the $U$-invariant Laplace-type operator on $G/G_{\mathbb R}$. In the radial…

Zero-mode distribution conjecture. The pushforward measure is

Let $P(u)$ be either $P^{(N)}(u)$ or $P^{(-N)}(u)$, and let $F$ and $\epsilon$ be the operators appearing in the fuzzy sigma-model action. The saturation condition is … The fuzzy s…

A Lie algebroid on a manifold is given, and one considers a proposed construction that produces a Lie algebroid on its loop space. A Dirac structure is the relevant Lie algebroid i…

Let $O(N)$ denote the two-dimensional nonlinear sigma model with $N$ components, where $N$ is a finite integer satisfying $3\leq N$. Let $\eta$ be its anomalous-dimension exponent.…

The spanning-forest model is mapped perturbatively to an $OSP(1|2)$ sigma-model and, formally, to the $N$-vector model at $N=-1$. The usual correspondence fails nonperturbatively b…

Let a mixing dynamical system have Liouville operator $\hat L$, let a regularization produce a regularized Liouvillian operator with discrete eigenvalues, and let the Ruelle resona…

Let $\hat U_t$ be the classical evolution operator and let $\hat L=\frac{d}{dt}\hat U_t$ be the Liouville operator. Consider the suitably averaged correlation functions of quantum…

Let $N\geq 3$, let $\Phi(x)\in\sqrt{\beta}\,\mathbf{S}^{N-1}\subset\mathbb{R}^{N}$ for $x\in\mathbb{Z}^{2}$, and consider an infinite-volume limit formally corresponding to … Here…

Let $X$ be the target of a 3-dimensional $\mathcal{N}=1$ supersymmetric sigma model, and let $X_+$ denote $X$ with a disjoint basepoint. Sigma-model boundary-condition conjecture.…

A compact Ricci-flat manifold is a compact manifold equipped with a Ricci-flat Riemannian metric, where the metric is considered up to multiplication by a positive constant. A targ…

Let a sigma model have metric and $B$-field depending on finitely many couplings, and suppose the model is classically integrable. Call it RG-stable when renormalization preserves…

Let $\tilde G_{\rm reg}$ and $\tilde b$ be the trans-series for the regular quantity $G_{\rm reg}$ and the physical coupling relation, respectively, and let $S^+$ denote lateral Bo…

Let $F(v)$, $u(v)$, $w_1(v)$, and $\sigma(v)$ be the physical quantities, and let $\tilde F$, $\tilde u$, $\tilde w_1$, and $\tilde\sigma$ denote their corresponding normalized tra…

T-duality conjecture. With respect to the $\mathbb{T}$-action on $Z$ and the $\widehat{\mathbb{T}}$-action on $\widehat Z$, respectively,

For $N,P>2$, the coset sigma models $SU(N)/SO(N)$ and $O(2P)/O(P) \times O(P)$ satisfy … so they have a ${\mathbb Z}_2$ topological charge and a discrete theta angle with values…

The $O(3)$ sigma model has topological charge … where … The Euclidean action may include the theta term ${\rm i}\vartheta {\cal Q}$. Haldane's conjecture. When $\vartheta=\pi$, the…

The $O(3)$ sigma model, the $SU(N)/SO(N)$ sigma models, and the $O(2P)/O(P) \times O(P)$ models have a ${\mathbb Z}_2$ instanton number, so their theta angle takes the values…

Let $\mathsf{M}(\lambda)$ be the quantum monodromy matrix, let $P_3$ be the classical zero-mode variable, and let $V_\pm,V_0$ be the classical chiral fields defined from the limiti…

Some non-linear sigma models are classically integrable. Integrability-survival conjecture. The classical integrability of many such models should survive quantisation. This is a b…

Let the bosonic $\mathds{CP}^{n-1}$ model be a sigma model whose integrability is obstructed by known anomalies, and consider models with vanishing chiral anomalies. Bykov's anomal…