論文

In this talk, we give the lattice regularized formulation of the mixed 't Hooft anomaly between the $\mathbb{Z}_N$ $1$-form symmetry and the $θ$ periodicity for $4$d pure Yang-Mills theory, which was originally discussed by Gaiotto $\textit{et al.}$ in the continuum description. For this purpose, we define the topological charge of the lattice $SU(N)$ gauge theory coupled with the background $\mathbb{Z}_N$ $2$-form gauge fields $B_p$ by generalizing Lüscher's construction of the $SU(N)$ topological charge. We show that this lattice topological charge enjoys the fractional $1/N$ shift completely characterized by the background gauge field $B_p$, and this rigorously proves the mixed 't Hooft anomaly with the finite lattice spacings. As a consequence, the Yang-Mills vacua at $θ$ and $θ+2π$ are distinct as the symmetry-protected topological states when the confinement is assumed.

In $U(1)$ lattice gauge theory with compact $U(1)$ variables, we construct the symmetry operator, i.e.\ the topological defect, for the axial $U(1)$ noninvertible symmetry. This requires a lattice formulation of chiral gauge theory with an anomalous matter content and we employ the lattice formulation on the basis of the Ginsparg--Wilson relation. The invariance of the symmetry operator under the gauge transformation of the gauge field on the defect is realized, imitating the prescription by Karasik in continuum theory, by integrating the lattice Chern--Simons term on the defect over \emph{smooth\/} lattice gauge transformations. The projection operator for allowed magnetic fluxes on the defect then emerges with lattice regularization. The resulting symmetry operator is manifestly invariant under lattice gauge transformations. In an appendix, we give another way of constructing the symmetry operator on the basis of a 3D $\mathbb{Z}_N$ topological quantum field theory, the level-$N$ BF theory on the lattice.

Background and objective Musculoskeletal factors, such as myofascial pain, are often overlooked in chronic headaches. This study aimed to evaluate the effectiveness of a structured self-care program, myofascial pain management (MPM), for patients with chronic headaches. Methods This single-arm observational study involved 37 patients with chronic headaches who were referred from a neurology clinic. The patients were categorized into two groups: suspected myofascial pain (n=19) and medication-refractory (n=18). All patients underwent the MPM program, which included instructions for self-massage and stretching techniques. Treatment effectiveness was evaluated using the pain item in the Integrated Palliative Care Outcome Scale (IPOS), with a score of ≤1 indicating improvement. Results The overall improvement rate in the patients was 76%, with significant improvements observed in the myofascial (84%; p=0.00017) and refractory (67%; p=0.00038) groups. Of the 28 patients who demonstrated improvements, 25 (89%) achieved this outcome without changing their prescribed headache medications. Conclusions The MPM program was effective for patients with chronic headache, including those with medication-refractory headache. Evaluation of the masticatory and pericranial muscles, along with the introduction of a structured self-care program, could prove to be a valuable nonpharmacological strategy for headache management.