Heat kernel

1. rough outline of heat kernel proof of Atiyah-singer index theorem 1.1. proof strategy Theorem 1 (Mckean-Singer formula.) $latex \displaystyle ind(D^+)=Str(e^{-tD^2})=\int\limits_{x \in M} Str(K(x,y)). &fg=000000$ from this we know Fredholm operator deformation invariance,in the same time we need chern-weil theory. Main challenge: 1. in the expansion on heat kernel ,we need to proof when $latex { t \rightarrow 0}&fg=000000$, the limit exist and find a way to calculate it. 2. indentify …

  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…