No abstract

We study the mean dimensions of the spaces of Brody curves. In particular we give the formula of the mean dimension of the space of Brody curves in the Riemann sphere. A key notion is a non-degeneracy of Brody curves introduced by Yosida (1934). We develop a deformation theory of non-degenerate Brody curves and apply it to the calculation of the mean dimension. Moreover we show that there are sufficiently many non-degenerate Brody curves. ∂ 2 ∂y 2 . |df |(z) is classically called a spherical derivative. It evaluates the dilatation of the map f with respect to the Euclidean metric on C and the Fubini-Study metric on CP N . (See the equation (6) in Section 4.2.) A holomorphic map f : C → CP N is called a Brody curve ([3]) if it satisfies |df |(z) ≤ 1 for all z ∈ C. Let M(CP N ) be the space of Brody curves in CP N . It is endowed with the compact-open topology (the topology of uniform convergence on compact subsets): A sequence of Brody curves {f n } ⊂ M(CP N ) converges to f ∈ M(CP N ) if and only if for any compact subset K ⊂ C we have sup z∈K d(f n (z), f (z)) → 0 as n → ∞. ( d(·, ·) is the distance on CP N with respect to the Fubini-Study metric.) M(CP N ) is an infinite dimensional compact metrizable space, and it admits the following continuous C-action.

We study a moduli space of ASD connections over S 3 × R. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the mean dimension by using a "Runge-approximation" for ASD connections, and we prove its lower bound by constructing an infinite dimensional deformation theory of periodic ASD connections.

We reformulate the mod-two Atiyah-Patodi-Singer (APS) index in a physicist-friendly way using the domain-wall fermion. Our new formulation is given on a closed manifold, which is extended from the original manifold with boundary, where we instead give a fermion mass term changing its sign at the location of the original boundary. This new setup does not need the APS boundary condition, which is non-local. A mathematical proof of equivalence between the two different formulations is given by two different evaluations of the same index of a Dirac operator on a higher dimensional manifold. The domain-wall fermion allows us to separate the edge and bulk mode contributions in a natural but not in a gauge invariant way, which offers a straightforward description of the global anomaly inflow.

The Atiyah-Patodi-Singer index theorem describes the bulk-edge correspondence of symmetry protected topological insulators. The mathematical setup for this theorem is, however, not directly related to the physical fermion system, as it imposes on the fermion fields a non-local and unnatural boundary condition known as the "APS boundary condition" by hand. In 2017, we showed that the same integer as the APS index can be obtained from the η invariant of the domain-wall Dirac operator. Recently we gave a mathematical proof that the equivalence is not a coincidence but generally true. In this contribution to the proceedings of LATTICE 2019, we try to explain the whole story in a physicist-friendly way.