Dirac heat curvature

Operator and coefficient

Let $D$ be the untwisted spin Dirac operator on a smooth Riemannian spatial manifold, away from any boundary, with spinor rank $r_s$. With positive connection Laplacian, the Lichnerowicz identity and local expansion give

$$D^2=\nabla^*\nabla+\frac R4 I,\qquad K_{D^2}(t;x,x)\sim\frac1{(4\pi t)^{d/2}} \left[I-\frac{tR(x)}{12}I+O(t^2)\right].$$

Indeed a Laplace-type operator $L=\nabla^*\nabla+V$ has first local correction $RI/6-V$. Substitution gives $-RI/12$. Tracing multiplies the scalar terms by $r_s$. See Vassilevich, equations (3.27) and (4.27). Twisting connections or additional potentials change the endomorphism; this formula is not a substitute for calculating the specified V2 operator.

Here $t$ is a heat parameter with length-squared units when $D$ has inverse-length units. It is not the shared physical time coordinate.

Independent sphere benchmark

For the round two-sphere of radius $a$, the untwisted spin Dirac eigenvalues are $\pm j/a$, each of multiplicity $2j$, $j=1,2,\ldots$. Thus

$$\operatorname{Tr}e^{-tD^2}=4\sum_{j=1}^\infty j e^{-tj^2/a^2} =\frac{2a^2}{t}-\frac13+O(t/a^2).$$

Euler–Maclaurin gives the constant term from $-B_2 f'(0)/2!=-1/12$ for $f(s)=s e^{-(t/a^2)s^2}$, before multiplying by four. The local coefficient agrees: integrating rank two over area $4\pi a^2$ with $R=2/a^2$ gives exactly the constant $-1/3$.

Curvature is not an Euler characteristic

Rescaling a metric by $a^2$ changes its scalar curvature by $a^{-2}$, whereas the Euler characteristic is unchanged. On a closed surface,

$$\int R\,d\mu=4\pi\chi.$$

This is an integrated identity. It does not make $R(x)=\chi$. For the round sphere, their numerical equality at radius one is a choice of units and metric. In particular, the round three-sphere has $\chi(S^3)=0$ but $R=6/a^2$, so its local Dirac heat correction is nonzero.

Applications to finite mode bands or mixing vertices require the associated operator, physical scale, and normalized matrix element. Numerical use at a particular nonzero $t$ additionally requires control of the asymptotic remainder.

Verification and related

interaction_operator_checks.py compares the sphere heat sum with its constant term at several radii and heat parameters, and bounds the omitted positive tail by an integral.

• Interaction operator tools — summarizes this curvature tool.
• Round-sphere Dirac counts — supplies the independently specified spectral benchmark.
• Simplex heat trace — concerns a different operator, with one count eigenvalue per index.