Dirac operator ($/D$)

The Dirac operator is the first-order elliptic operator obtained by composing the spin connection with Clifford multiplication. On an even-dimensional compact spin manifold it exchanges positive and negative chirality spinors.

For the oriented round unit two-sphere, positive and negative spinor bundles have degrees $-1,+1$. Its Dirac operator has $D^2=C+1/4$ on their homogeneous sections, where $C$ is the angular-momentum Casimir operator; the eigenvalues of $D$ are the nonzero integers. Therefore the chiral phase is a unitary between the two L2 chiral spaces, commuting with $SU(2)$. For a smooth function $f$, $[D,f]=c(df)$ is bounded. The sign-operator resolvent integral expresses its phase commutator as a norm-convergent integral of compact resolvent products bounded by a constant times $(1+t^2{)}^{-1}$. Thus it intertwines multiplication by every continuous function modulo compacts. Its full graded phase is self-adjoint; the chiral phase is the component relevant to nonzero compressed indices.

Let $P$ project onto a degree-one equivariant line bundle on the complex-oriented round two-sphere, and let $V$ be its positive-to-negative chiral Dirac operator phase. The compressed map has parametrix $P{V}^{\ast }P$ modulo compacts. Its domain bundle has degree zero and its target degree two. Their homogeneous line-bundle representations on the two-sphere are respectively $E_0\oplus E_2\oplus E_4\oplus \cdots$ and $E_2\oplus E_4\oplus \cdots$, so the equivariant Fredholm index is the trivial representation and the ordinary index is one. Reversing chirality negates the index. Compressing the full self-adjoint phase instead gives ordinary index zero.

For a Twisted Dirac operator on a closed even-dimensional spin manifold, the theorem identifies its analytic Fredholm index with the integral of the top-degree part of the A-hat form times the Chern character. This is one geometric specialization of the general theorem for elliptic differential operators, which expresses the analytic index in terms of the topological class of the principal symbol. The heat proof pairs all positive eigenvalues by the odd Dirac operator, leaving only the index in the supertrace; Getzler rescaling computes its local limit from curvature.

Getzler rescaling combines the change $x\mapsto \sqrt{t}x$ in geodesic normal coordinates with the Clifford algebra filtration, assigning degree one to a derivative and to a Clifford generator and degree minus one to a coordinate. The associated graded Clifford algebra is the exterior algebra. This is useful because the spinor supertrace annihilates every Clifford monomial except the top-degree one. Applied to the Lichnerowicz formula for a twisted Dirac operator, it retains a quadratic curvature model, whose heat kernel is computed by a Gaussian ansatz; all terms of smaller rescaling degree disappear from the local index limit. Thus the cancellations in a supersymmetric heat trace can be computed before solving the complete curved heat equation.

On a spin manifold, tensor the spinor bundle $S$ with a Hermitian vector bundle $W$ carrying a unitary connection. Composing the induced tensor-product connection on a vector bundle with Clifford multiplication defines the twisted Dirac operator. In even dimension it maps $S^{+}\otimes W$ to $S^{-}\otimes W$ and conversely. On a closed manifold its chiral restriction $D_W^{+}:H^1(S^{+}\otimes W)\to L^2(S^{-}\otimes W)$ is a Fredholm operator.

At the centre of geodesic normal coordinates, expand the square of the Twisted Dirac operator. The diagonal terms give the rough Laplacian; the off-diagonal terms are ${\sum }_{i<j}c(e^i)c(e^j)[{\nabla }_i,{\nabla }_j]$. The tensor-product vector-bundle curvature is the sum of spin curvature and $F^W$. Substituting spin curvature $\frac{1}{4}{\sum }_{a,b}⟨R(e_i,e_j)e_a,e_b⟩c(e^a)c(e^b)$ and using the Clifford algebra relations yields the scalar term $\mathrm{Scal}/4$: the four-generator contribution vanishes by the algebraic Bianchi identity and the contracted terms are the scalar curvature. This proves the displayed identity at every point.

Let a compact Lie group $G$ act on a closed manifold and on the vector bundles of a $G$-equivariant elliptic differential operator $D$. Its finite-dimensional kernel and cokernel define the virtual representationAn equivariant index theorem computes this character of a representation from local principal symbol and curvature data on the fixed-point set of $g$. At the identity element it recovers the ordinary Atiyah-Singer index theorem; for isolated fixed points the fixed-set integral becomes a finite sum.

For example, the holomorphic fixed-point formula for the trivial line bundle on a compact complex manifold giveswhen the fixed points are nondegenerate. On the Riemann sphere, $f(z)=az$ with $|a|=1$ and $a\ne 1$ has fixed points zero and infinity, with tangent multipliers $a$ and $a^{-1}$. Their contributions areThe global answer is also one: only the constant holomorphic functions contribute. Local fixed-point terms can thus be nonintegral complex numbers even though their sum is the character of an actual virtual representation.^[ref]