the Dirac operator
/ dih-RACK /
Physicists wanted a wave operator that was the square root of the Laplacian — a first-order operator D whose square is the second-order Laplacian, D^2 = -Delta (up to lower-order curvature terms). You cannot take such a square root with ordinary numbers, because of the cross terms; Dirac's trick was to use anticommuting matrices (a Clifford algebra) so that the cross terms cancel. The geometric Dirac operator is the manifold version of that idea: a first-order elliptic operator acting on spinor fields whose square is, beautifully, the Laplacian plus a curvature term. It is the single most important operator in modern geometry — it is the operator in the Atiyah-Singer index theorem from which almost every classical case is derived.
Here is how it is built. On a Riemannian spin manifold (M, g) you have a spinor bundle S, a vector bundle whose fibers carry a representation of the Clifford algebra: Clifford multiplication lets a tangent vector v act on a spinor, with the defining relation v . w + w . v = -2 g(v, w). Using the Levi-Civita connection lifted to S, the Dirac operator is D = sum_i e_i . nabla_{e_i}, where {e_i} is a local orthonormal frame and the dot is Clifford multiplication — you covariantly differentiate the spinor in each direction and then Clifford-multiply by that direction and sum. The miracle is the Lichnerowicz formula D^2 = nabla* nabla + S/4, where nabla* nabla is the connection Laplacian and S is the scalar curvature: squaring the first-order Dirac operator returns a Laplacian plus exactly one quarter of the scalar curvature, no other curvature terms. D is elliptic and, on a closed manifold, self-adjoint (in the ungraded case) with discrete spectrum.
Why it is central: the index of the Dirac operator (dimension of its kernel minus dimension of its cokernel, after the natural Z/2 grading) is computed by the A-hat genus, a characteristic number, and this is the cleanest instance of Atiyah-Singer. The Lichnerowicz formula alone gives a famous obstruction — if S > 0 everywhere then D has no kernel, so the A-hat genus must vanish, which forbids many spin manifolds from admitting positive scalar curvature metrics. The honest prerequisites: the construction needs a spin structure (or at least a spin-c structure), which not every manifold admits — the obstruction is the second Stiefel-Whitney class w_2 — so 'the Dirac operator' presupposes that topological condition. There are also several Dirac-type operators (spin, spin-c, the de Rham operator d + d* viewed as a Dirac operator on forms, twisted Dirac operators on S tensor E); be explicit which one you mean, because their indices compute different characteristic numbers.
On flat R^4 with constant spinors, the Dirac operator in coordinates is D = sum_{k=1}^4 gamma^k partial_k with the gamma^k the 4x4 Dirac gamma-matrices satisfying gamma^j gamma^k + gamma^k gamma^j = -2 delta^{jk}; squaring gives D^2 = -(partial_1^2 + ... + partial_4^2) = -Delta since the cross terms cancel by anticommutation, the flat case of the Lichnerowicz formula with S = 0.
Anticommuting gamma-matrices make the first-order Dirac operator square to the Laplacian — the algebraic core of the construction.
The Dirac operator needs a spin structure, which exists iff w_2(M) = 0; without one you fall back to spin-c (always available on oriented 4-manifolds) whose Dirac operator is the one used in Seiberg-Witten theory, so 'the' Dirac operator is ambiguous until you fix the structure.