68 problems

Let the axioms ${\bf A0}$ and ${\bf A1}$ hold approximately on an intermediate length scale, with sufficiently small violations. Coarse-graining conjecture. Coarse-graining the lat…

The asymptotic-safety program seeks a non-trivial ultraviolet fixed point of the renormalization-group flow that can provide a quantum-field-theoretic ultraviolet completion of gra…

The two enhanced tensor field theory models considered here have tensor rank $d$, background group dimension $D$, and kinetic and vertex parameters $a$ and $b$; model $+$ is studie…

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…

The Hitchin spin chains and Whitham theories are considered as renormalization-group equations, with Hitchin times identified with the spatial RG scale $t\sim \log(r)$. The fast To…

Let $\alpha$ denote the physical fine structure constant, and consider loopwise summation of quantum electrodynamics with an eigenvalue for $\alpha$, uniformly across all fermion s…

Momentum-space critical-region conjecture. As in real-space renormalization, one cannot in general trust that the critical region is free of problems for momentum-space transformat…

Zero-denominator cancellation conjecture. All numerators corresponding to zero denominators are zero.

In the elliptic case, let $C=\mathbb{T}_\tau$ be the spectral parameter space, with complex structure modulus $\tau$, and let $\mathfrak{t}_{\rm RG}$ denote RG time. Elliptic-modul…

Consider rational integrable order defects in the 4d Chern–Simons construction, with chiral defect positions $z_i^+$ and antichiral defect positions $z_j^-$, and let $c_G$ be the g…

In the 4d Chern–Simons construction, let $z_i^+$ and $z_j^-$ denote the positions of chiral and antichiral order defects on the spectral parameter space, and let…

RG-trajectory conjecture. Any quantum integrable model with constant coupling strengths should retain integrability when its coupling strengths are made time dependent, provided th…

Let $K_{\mathrm{UV}}$ denote the ultraviolet value of the scaling quantity $K$ in the renormalization-group trajectory. Ultraviolet fixed-point conjecture. The ultraviolet value is…

Let $\zeta_{\mathrm{R}}\in[0.62,0.67]$ be the regularity index of the zero potential $V_Z(x)$, measured at finite scale $L$, and let $\kappa$ be the formal algebraic generator. Und…

Let $R_\Lambda:\mathcal{K}\to\mathcal{K}$ be the renormalization-group map, let $\beta(k)$ be its beta function, and let $k^*$ be an RG fixed point satisfying $\beta(k^*)=0$. Let…

Let a generalized fusion categorical symmetry emerge in the infrared, and let its simple objects have quantum dimensions $d_i$. Weak integrality conjecture. The squares of the dime…

For an integrable Hamiltonian with time-dependent interaction strengths, let the interaction strengths evolve along temporal trajectories coinciding with the renormalization-group…

Let $\operatorname{\bf NG}_{s}$ be the probability measure whose moment generating function is specified by … Under the assumptions of Theorem, let $w_N$ denote the critical-window…

Let $omega$ be the differential governing the RG flow, with zeroes at $gamma_-$ and $gamma_+$, and let $mu$ be the RG scale. The parameters $k_1$ and $k_2$ are RG-invariant WZW lev…

Ring-homomorphism conjecture. A ring homomorphism generally corresponds to integrable or partially solvable systems.

Consider the stochastic porous medium equation in one dimension with large-$h$ behavior $V(h)\sim |h|^s$ for $s>0$, and the renormalization-group equations governing its flow. Let…

Let $\Phi^{(N)}$ be the sequence of flow kernels produced by stochastic RG iteration, and let $\Delta\Phi^{(N)}=\Phi^{(N+1)}-\Phi^{(N)}$. Near the attractor, the Cauchy differences…

Let $\Phi^\infty$ be the limiting flow kernel and fixed point of the stochastic RG operator $\mathfrak{R}$, so that … The limiting kernel represents the vanishing-regularization li…

The fixed-point RG attractor is described by a flow whose states and leading eigenmode are analyzed through Cauchy differences of successive RG iterates. Classical stability conjec…

Let the O(3) nonlinear sigma-model have a topological charge density $\mathcal{L}_{\rm top}$, and let the theta angle fluctuate dynamically under renormalization-group evolution. I…