The A-hat form and relative Chern character

For the real-coefficient cohomological interpretation below, assume full AC as in Characteristic forms represent topological characteristic classes over the reals. The displayed form definitions themselves use only the supplied bundles and connections.

Let $M$ be an oriented Riemannian manifold of dimension $2m$ and let ${\mathrm{\Omega }}^{LC}$ be the skew-symmetric matrix of curvature two-forms of the Levi–Civita connection, with curvature convention $R(X,Y)=[{\mathrm{\nabla }}_X,{\mathrm{\nabla }}_Y]-{\mathrm{\nabla }}_{[X,Y]}$. Define $\widehat{A}(TM,{\mathrm{\nabla }}^{LC})=\det {}^{1\mathrm{/}2}\text{\hspace{0.17em}⁣}\left(\frac{i{\mathrm{\Omega }}^{LC}\mathrm{/}4\pi }{\sinh (i{\mathrm{\Omega }}^{LC}\mathrm{/}4\pi )}\right)\mathrm{.}$ The determinant and square root are formal power series in commuting even-degree differential forms; the square-root branch has constant term $1$, and terms above degree $2m$ are discarded. This fixes the A-hat normalization using the curvature convention of Chern, Pontryagin, and Euler characteristic forms.

For $m\ge 1$ and a graded Hermitian Clifford module $V$ with compatible connection, use the local multiplicity space and relative curvature from Spinor and geometric Clifford bundles and connections are well defined ↗. Define the relative Chern character form by

$$\mathrm{ch}(V\mathrm{/}S)={\mathrm{str}}_W\exp \text{\hspace{0.17em}⁣}(-\frac{F^{V\mathrm{/}S}}{2\pi i})\mathrm{.}$$

Here $\mathrm{str}$ is the ordinary trace if $W$ is ungraded and the even trace minus odd trace if it is graded. The cited well-definedness lemma proves this local expression patches to a global even form. For $V=S\widehat{\otimes }E$ with the product connection, it is $\mathrm{tr}\exp (-F^E\mathrm{/}(2\pi i))$. The Chern–Weil supplier Characteristic forms represent topological characteristic classes over the reals identifies the resulting cohomology classes after passage to real coefficients; this definition asserts no integral torsion identity.

For $m=0$, let ${\epsilon }_x\in \{1,-1\}$ be the supplied orientation sign at $x$ (Oriented smooth manifolds and oriented charts). The reference zero-dimensional Clifford module $S_{0,x}$ is the complex line with grading operator ${\mathrm{\Gamma }}_{S_{0,x}}={\epsilon }_{x}I$: it is even at a positive point and odd at a negative point. The zero-dimensional Clifford algebra is $\mathbb{C}$, acting by scalars. Set $W_x={\mathrm{Hom}}_{\mathbb{C}}(S_{0,x},V_x)$ with the grading for which evaluation $S_{0,x}\widehat{\otimes }{W}_x\to V_x$ is even. Identifying a map with its value at $1$ gives ${\mathrm{\Gamma }}_{W_x}={\epsilon }_x{\mathrm{\Gamma }}_{V_x}$. All curvature two-forms vanish, so ${\widehat{A}}_0=1$, $F^{V\mathrm{/}S}=0$, and $\mathrm{ch}(V\mathrm{/}S{)}_0(x)={\mathrm{str}}_{W_x}I={\epsilon }_x(\dim V_x^{+}-\dim V_x^{-})\mathrm{.}$ This formula is independent of any basis of $V_x$ and leaves its arbitrary given grading unrestricted. For $V=S_0\widehat{\otimes }E$ with $E$ in even degree, evaluation identifies $W=E$ in even degree, so the relative character is the ordinary rank, consistently with the product formula above. This reference-line convention makes no assertion about a Spin$(0)$ lift or its isomorphism classes.