Heat kernel

Rough Outline of Heat Kernel Proof of Atiyah-Singer Index Theorem 1. Proof Strategy Theorem 1 (Mckean-Singer Formula):$$\text{ind}(D^+) = \text{Str}(e^{-tD^2}) = \int_{x \in M} \text{Str}(K(x,y)).$$ From this, we know Fredholm operator deformation invariance. At the same time, we need Chern-Weil theory. Main Challenge: Proof:Our proof road: Mckean-Singer formula $\rightarrow$ local-index thm $\rightarrow$ A-S index thm $\rightarrow$ Riemann-Roch-Hirzebunch theorem. $\Box$ 1.2. Preliminary Work S…

  1. framework of atiyah singer index theory 1.1. A genus form $ {(M,g)} $ campact,complete,Riemann manifold without boundary,dim $ { M=2m ,m \in N^* } $. $ {\bigtriangledown^g} $ is the Levi-civita connection on $ {TM} $,$ {R=R_g\in \Omega^2(End(TM))} $. $ {\widehat A} $ genus form: $ \widehat A(M,g)=det^{\frac{1}{2}}(\frac{\frac{i}{4\pi}R_g}{sinh(\frac{i}{4\pi}R_g)})\in \Omega(M). $ by chern-weil theory,we know: $ { 1. \widehat A } $ is closed. $ { 2. \widehat A_{g1}-\widehat A_{g2} } $ i…