Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis
in topological quantum field theory. We develop a direct relationship between Fedosov’s deformation
quantization of a symplectic manifold X and the Batalin-Vilkovisky (BV) quantization of a one-dimensional
sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously
using Costello’s homotopic theory of effective renormalization. We show that Fedosov’s Abelian connections
on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration
produces a natural trace map on the deformation quantized algebra. This formulation allows us to
exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem
via semi-classical analysis, i.e., one-loop Feynman diagram computations.
In this note, we study submanifold geometry of the Atiyah--Hitchin manifold, a double cover of the 2-monopole moduli space, which plays an important role in various settings such as the supersymmetric background of string theory. When the manifold is naturally identified as the total space of a line bundle over S^2, the zero section is a distinguished minimal 2-sphere of considerable interest. In particular, there has been a conjecture about the uniqueness of this minimal 2-sphere among all closed minimal 2-surfaces. We show that this minimal 2-sphere satisfies the ``strong stability condition" proposed in our earlier work, and confirm the global uniqueness as a corollary.
There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other
is the Bondi mass loss formula, which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect function of the Vaidya spacetime.