Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator

Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS index theorem is too abstract and general (allowing non-trivial metric and so on) and also the connection between the APS boundary condition and the physical boundary condition on the surface of topological material is unclear. For this reason, in contrast to the Atiyah-Singer index theorem, derivation of the APS index theorem in physics language is still missing. In this talk, we attempt to reformulate the APS index in a "physicist-friendly" way, similar to the Fujikawa method on closed manifolds, for our familiar domain-wall fermion Dirac operator in a flat Euclidean space. We find that the APS index is naturally embedded in the determinant of domain-wall fermions, representing the so-called anomaly descent equations.Comment: 8 pages, Proceedings of the 35th annual International Symposium on Lattice Field Theor

Atiyah-Patodi-Singer index on a lattice

We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the $\eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion, we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $\eta$ invariant comes entirely from the massless edge-localized modes.Comment: 14 pages, appendices added, details of key equations added, typos corrected, to appear in PTE

Tomographic Image Reconstruction Based on Minimization of Symmetrized Kullback-Leibler Divergence

Iterative reconstruction (IR) algorithms based on the principle of optimization are known for producing better reconstructed images in computed tomography. In this paper, we present an IR algorithm based on minimizing a symmetrized Kullback-Leibler divergence (SKLD) that is called Jeffreys’ J-divergence. The SKLD with iterative steps is guaranteed to decrease in convergence monotonically using a continuous dynamical method for consistent inverse problems. Specifically, we construct an autonomous differential equation for which the proposed iterative formula gives a first-order numerical discretization and demonstrate the stability of a desired solution using Lyapunov’s theorem. We describe a hybrid Euler method combined with additive and multiplicative calculus for constructing an effective and robust discretization method, thereby enabling us to obtain an approximate solution to the differential equation.We performed experiments and found that the IR algorithm derived from the hybrid discretization achieved high performance

Activation of Satellite Glial Cells in Rat Trigeminal Ganglion after Upper Molar Extraction

The neurons in the trigeminal ganglion (TG) are surrounded by satellite glial cells (SGCs), which passively support the function of the neurons, but little is known about the interactions between SGCs and TG neurons after peripheral nerve injury. To examine the effect of nerve injury on SGCs, we investigated the activation of SGCs after neuronal damage due to the extraction of the upper molars in rats. Three, 7, and 10 days after extraction, animals were fixed and the TG was removed. Cryosections of the ganglia were immunostained with antibodies against glial fibrillary acidic protein (GFAP), a marker of activated SGCs, and ATF3, a marker of damaged neurons. After tooth extraction, the number of ATF3-immunoreactive (IR) neurons enclosed by GFAP-IR SGCs had increased in a time-dependent manner in the maxillary nerve region of the TG. Although ATF3-IR neurons were not detected in the mandibular nerve region, the number of GFAP-IR SGCs increased in both the maxillary and mandibular nerve regions. Our results suggest that peripheral nerve injury affects the activation of TG neurons and the SGCs around the injured neurons. Moreover, our data suggest the existence of a neuronal interaction between maxillary and mandibular neurons via SGC activation

欠損した投影の推定を伴う断層逆問題

Image reconstruction in computed tomography can be treated as an inverse problem, namely, obtaining pixel values of a tomographic image from measured projections. However, a seriously degraded image with artifacts is produced when a certain part of the projections is inaccurate or missing. A novel method for simultaneously obtaining a reconstructed image and an estimated projection by solving an initial-value problem of differential equations is proposed. A system of differential equations is constructed on the basis of optimizing a cost function of unknown variables for an image and a projection. Three systems described by nonlinear differential equations are constructed, and the stability of a set of equilibria corresponding to an optimized solution for each system is proved by using the Lyapunov stability theorem. To validate the theoretical result given by the proposed method, metal artifact reduction was numerically performed