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Title:
Perturbative renormalization and supersymmetric localization
Speaker:
Abstract:
Index theorems are "proved" in physics via the SUSY localization in QFT, generating a great beauty and simplicity conceptually together with a substantial difficulty rigorously. In this lecture, I will explain an analogue realization of such QFT idea within Costello's perturbative renormalization theory. In lecture I, I will describe a Chern-Simons type theory for the one dimensional sigma model with target a symplectic manifold. We establish an equivalence between the BV quantization of our model and Fedosov's deformation quantization. In lecture II, I will describe the trace map on quantum observables. We will show that a rescaling symmetry of Fedosov's deformation quantization leads to an exact computation of the trace via the semi-classical approximation. The homotopy of the rescaling plays the role of SUSY localization, giving rise to a simple proof of algebraic index theorem.
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