8 · spin geometry and atiyah-singer index theory

clifford algebras, pin and spin groups, and periodicity

The Clifford algebra $\mathrm{Cl}(V,q)$ of a real or complex vector space $V$ equipped with a quadratic form $q$ is the associative algebra generated by $V$ subject to the relation $$ v\cdot w+w\cdot v=2q(v,w)\,1. $$ When $V=\mathbb{R}^{p,q}$ one writes $\mathrm{Cl}_{p,q}$; the complexification yields $\mathrm{Cl}_n(\mathbb{C})$. These algebras are classified by an 8-fold periodicity in the real case and a 2-fold periodicity in the complex case: $$ \mathrm{Cl}_{n+8}\simeq\mathrm{Cl}_n\otimes\mathrm{Cl}_8,\qquad \mathrm{Cl}_{n+2}(\mathbb{C})\simeq\mathrm{Cl}_n(\mathbb{C})\otimes\mathrm{Cl}_2(\mathbb{C}). $$ The groups $\mathrm{Pin}(p,q)$ and $\mathrm{Spin}(p,q)$ are the multiplicative subgroups of the units in $\mathrm{Cl}_{p,q}$ generated by vectors of unit length (and the even subgroup, respectively). They furnish double covers of the orthogonal groups $O(p,q)$ and $\mathrm{SO}(p,q)$. The Atiyah-Bott-Shapiro construction realizes the $K$-theory of a point in terms of Clifford modules, establishing a canonical isomorphism between the representation rings of the Clifford algebras and the Bott-periodic $K$-groups $KO^{-n}(\mathrm{pt})$ and $K^{-n}(\mathrm{pt})$.

definition (clifford algebra). let $(V,q)$ be a finite-dimensional real (resp. complex) quadratic space. $\mathrm{Cl}(V,q)$ is the quotient of the tensor algebra $T(V)$ by the two-sided ideal generated by $v\otimes v-q(v)1$ for $v\in V$.

definition (pin and spin). for a unit vector $v\in V$ with $q(v)=\pm 1$, the reflection map $w\mapsto w-2\frac{q(v,w)}{q(v)}v$ is realized by conjugation by $v$ in $\mathrm{Cl}(V,q)$. $\mathrm{Pin}(V,q)$ is the group generated by such unit vectors inside $\mathrm{Cl}(V,q)^\times$; $\mathrm{Spin}(V,q)=\mathrm{Pin}(V,q)\cap\mathrm{Cl}^{\mathrm{even}}(V,q)$ is the even subgroup. the twisted adjoint representation induces double covers $\mathrm{Pin}\to O$ and $\mathrm{Spin}\to\mathrm{SO}$ (with discrete kernel $\{\pm 1\}$ in the connected cases of interest).

theorem (clifford periodicity). there are graded algebra isomorphisms $$ \mathrm{Cl}_{n+8}(\mathbb{R})\simeq\mathrm{Cl}_n(\mathbb{R})\otimes\mathbb{R}(16),\qquad \mathrm{Cl}_{n+2}(\mathbb{C})\simeq\mathrm{Cl}_n(\mathbb{C})\otimes\mathrm{Cl}_2(\mathbb{C}), $$ where $\mathbb{R}(16)=M_{16}(\mathbb{R})$. consequently the stable isomorphism classes of Clifford modules depend only on $n\bmod 8$ (real) or $n\bmod 2$ (complex).

proof. explicit presentations give $\mathrm{Cl}_0=\mathbb{R}$, $\mathrm{Cl}_1=\mathbb{C}$, $\mathrm{Cl}_2=\mathbb{H}$, and recursive identities $\mathrm{Cl}_{n+2}\simeq\mathrm{Cl}_n\otimes\mathrm{Cl}_2$ up to grading conventions for the real signatures of definite quadratic forms (Cartan's periodic table). tensoring eight times multiplies the matrix size by $16$ and returns an isomorphic type shifted by $8$. the complex case follows from $\mathrm{Cl}_1(\mathbb{C})\simeq\mathbb{C}\oplus\mathbb{C}$ and $\mathrm{Cl}_2(\mathbb{C})\simeq M_2(\mathbb{C})$, yielding period $2$. $\square$

theorem (atiyah-bott-shapiro, statement). the abelian group of virtual $\mathrm{Cl}_n$-modules (irreducible Clifford modules modulo those extendable to $\mathrm{Cl}_{n+1}$) is isomorphic to $KO^{-n}(\mathrm{pt})$ (real case) or $K^{-n}(\mathrm{pt})$ (complex case). this realizes Bott periodicity as Clifford periodicity.

proof (outline). a Clifford module $W$ for $\mathrm{Cl}_n$ determines a virtual representation of $\operatorname{Spin}(n)$ and hence a class in the representation ring. the difference between modules that do and do not extend across one more Clifford generator matches the kernel of restriction maps that model the connecting maps in the Bott sequence. identifying the resulting graded ring with the known values of $KO^{-n}(\mathrm{pt})$ (period $8$) and $K^{-n}(\mathrm{pt})$ (period $2$) completes the isomorphism. $\square$

remark (link to chapter 7). complex and real $K$-theory of a point thereby acquire an algebraic model before one geometricizes the construction to manifolds via symbols of elliptic operators.

spin structures, clifford bundles, and the spin connection

An oriented Riemannian (or Lorentzian) manifold $M$ admits a spin structure if and only if its second Stiefel-Whitney class vanishes: $w_2(M)=0$. A spin structure is a principal $\mathrm{Spin}(n)$-bundle $P_{\mathrm{Spin}}$ that double-covers the oriented orthonormal frame bundle. The associated spinor bundles are obtained by taking representations of $\mathrm{Spin}(n)$ on the irreducible Clifford modules. Clifford multiplication endows the spinor bundle $S$ with a bundle map $$ TM\otimes S\to S. $$ The Levi-Civita connection on $TM$ lifts uniquely to a spin connection $\nabla^S$ on $S$. When torsion is present the lift still exists but acquires additional contorsion terms, recovering the Einstein-Cartan setting of earlier chapters.

definition (spin structure). for an oriented Riemannian $n$-manifold $(M,g)$, a spin structure is a principal $\operatorname{Spin}(n)$-bundle $P_{\operatorname{Spin}}\to M$ together with a twofold covering $P_{\operatorname{Spin}}\to P_{\mathrm{SO}}$ of the oriented orthonormal frame bundle intertwining the structure-group map $\operatorname{Spin}(n)\to\mathrm{SO}(n)$.

definition (clifford and spinor bundles). the Clifford bundle is $\mathrm{Cl}(TM)=\bigcup_p\mathrm{Cl}(T_p M,g_p)$. a spinor bundle $S\to M$ is an associated bundle of $P_{\operatorname{Spin}}$ for an irreducible left $\mathrm{Cl}_n$-module, so that Clifford multiplication $T^*M\otimes S\to S$ is well-defined.

theorem (existence of spin structures). an oriented manifold admits a spin structure if and only if $w_2(TM)=0$. when one exists, the set of spin structures is a torsor over $H^1(M;\mathbb{Z}/2)$.

proof. as in chapters 3 and 6, the short exact sequence $1\to\mathbb{Z}/2\to\operatorname{Spin}\to\mathrm{SO}\to 1$ produces a long exact sequence in nonabelian cohomology; the primary obstruction to lifting $P_{\mathrm{SO}}$ is $w_2(TM)$, and residual freedom is $H^1(M;\mathbb{Z}/2)$. $\square$

theorem (unique metric spin connection). given a spin structure on $(M,g)$, there is a unique connection $\nabla^S$ on $S$ compatible with Clifford multiplication and projecting to the Levi-Civita connection on $TM$.

proof. the Levi-Civita connection form $\omega_{\mathrm{LC}}$ on $P_{\mathrm{SO}}$ takes values in $\mathfrak{so}(n)$. the Lie algebra isomorphism $\lambda_*:\mathfrak{spin}(n)\xrightarrow{\simeq}\mathfrak{so}(n)$ pulls $\omega_{\mathrm{LC}}$ back to a unique $\mathfrak{spin}(n)$-valued form on $P_{\operatorname{Spin}}$ covering $\omega_{\mathrm{LC}}$. the associated covariant derivative on $S$ preserves the Clifford action because $\operatorname{Spin}$ acts by algebra automorphisms. uniqueness follows because any two such connections differ by a section of $T^*M\otimes\mathfrak{spin}(n)$ annihilated by the soldering condition, hence zero. $\square$

the dirac operator and vanishing theorems

The Dirac operator is the first-order elliptic operator $$ D=\sum_i e_i\cdot\nabla_{e_i}^S:C^\infty(S)\to C^\infty(S) $$ obtained by composing the spin connection with Clifford multiplication. Its square satisfies the Schrodinger-Lichnerowicz formula $$ D^2=\nabla^*\nabla+\frac14 R, $$ where $R$ is the scalar curvature. Consequently, on a compact spin manifold with strictly positive scalar curvature there are no harmonic spinors: $\ker D=0$. This vanishing theorem imposes severe topological restrictions on the existence of positive-scalar-curvature metrics. In the asymptotically flat setting the same identity underlies the Positive Mass Theorem: the ADM mass of a spin manifold satisfying the dominant-energy condition is non-negative and vanishes only for flat space.

definition (dirac operator). in a local orthonormal frame $\{e_i\}$, $Ds=\sum_i e_i\cdot\nabla_{e_i}^S s$. the definition is frame-independent. on even-dimensional manifolds chirality $\gamma$ splits $S=S^+\oplus S^-$ and $D$ off-diagonalizes into $D^\pm:C^\infty(S^\pm)\to C^\infty(S^\mp)$.

definition (principal symbol). the principal symbol of $D$ at $(x,\xi)\in T^*M$ is Clifford multiplication by $\xi^\sharp$: $\sigma(D)(x,\xi)=i\,\xi\cdot$. it is invertible for $\xi\neq 0$, so $D$ is elliptic.

theorem (schrodinger-lichnerowicz). for the Levi-Civita spin connection on a Riemannian spin manifold, $$ D^2=\nabla^*\nabla+\frac14\mathrm{Scal}, $$ where $\nabla^*\nabla$ is the connection Laplacian (Bochner Laplacian) on spinors and $\mathrm{Scal}$ is the scalar curvature acting by multiplication.

proof. expand $$ D^2s =\sum_{i,j}e_i\cdot\nabla_{e_i}(e_j\cdot\nabla_{e_j}s). $$ at a point $p$ choose a normal frame so that $\nabla e_i|_p=0$. then $$ D^2s =\sum_{i,j}e_i\cdot e_j\cdot\nabla_{e_i}\nabla_{e_j}s =-\sum_i\nabla_{e_i}\nabla_{e_i}s +\sum_{i\lt j}e_i\cdot e_j\cdot\bigl(\nabla_{e_i}\nabla_{e_j}-\nabla_{e_j}\nabla_{e_i}\bigr)s. $$ the first sum is $\nabla^*\nabla s$ (up to the usual sign conventions for the connection Laplacian). the commutator $[\nabla_{e_i},\nabla_{e_j}]$ acts on spinors by the spin curvature, whose Clifford contraction $\sum_{i\lt j}e_i\cdot e_j\cdot R(e_i,e_j)$ reduces by the first Bianchi identity and Clifford relations to $\frac14\mathrm{Scal}$. $\square$

theorem (lichnerowicz vanishing). if $M$ is a closed Riemannian spin manifold with $\mathrm{Scal}>0$ everywhere, then $\ker D=\{0\}$. in particular the $\widehat{A}$-genus vanishes whenever a positive scalar curvature metric exists (via Atiyah-Singer below).

proof. if $Ds=0$ then $0=\|Ds\|_{L^2}^2=\langle D^2s,s\rangle=\|\nabla s\|_{L^2}^2 +\frac14\int_M\mathrm{Scal}\,|s|^2$. positive scalar curvature forces $\nabla s=0$ and $s=0$. $\square$

theorem (positive mass, schematic spinor proof). let $(M^3,g)$ be a complete asymptotically flat Riemannian $3$-manifold with nonnegative scalar curvature, admitting a spin structure. then the ADM mass is nonnegative and vanishes if and only if $(M,g)$ is isometric to Euclidean $\mathbb{R}^3$.

proof (strategy of witten). solve a Dirac equation with asymptotic constant-spinor boundary data at infinity. the Lichnerowicz identity integrated by parts against that spinor produces a boundary term at infinity proportional to the ADM mass plus a bulk integral of $|\nabla\psi|^2+\frac14\mathrm{Scal}|\psi|^2\ge 0$. nonnegativity of the mass follows; vanishing forces $\nabla\psi=0$ and $\mathrm{Scal}=0$, hence flatness by the asymptotic conditions. $\square$

the atiyah-singer index theorem

For an elliptic pseudodifferential operator $P$ acting between sections of vector bundles over a closed manifold, the analytical index $$ \mathrm{index}\,P=\dim\ker P-\dim\operatorname{coker} P $$ is a well-defined integer. The Atiyah-Singer theorem asserts that this integer equals a topological index constructed from the $K$-theory class of the principal symbol of $P$ via the Chern character and the Todd class of the tangent bundle: $$ \mathrm{index}\,P =\bigl\langle\operatorname{ch}([\sigma(P)])\,\mathrm{Td}(TM\otimes\mathbb{C}),[M]\bigr\rangle. $$ One modern proof proceeds by heat-kernel methods: the supertrace of the heat operators $e^{-tD^*D}$ and $e^{-tDD^*}$ admits an asymptotic expansion whose constant term is both the analytical index and a local density built from curvature forms; the local density integrates to the topological index.

definition (analytical index). if $P:C^\infty(E)\to C^\infty(F)$ is elliptic on a closed manifold, then $\ker P$ and $\operatorname{coker} P$ are finite-dimensional and $\mathrm{index}\,P=\dim\ker P-\dim\operatorname{coker} P$ depends only on the homotopy class of the principal symbol in $K$-theory.

theorem (elliptic regularity and fredholm property). an elliptic differential (or pseudodifferential) operator of order $m$ on a closed manifold extends to a Fredholm operator $H^s(E)\to H^{s-m}(F)$ between Sobolev spaces. smooth elliptic regularity implies that kernels consist of smooth sections and that the analytical index is independent of $s$.

proof. existence of a parametrix $Q$ with $QP-I$ and $PQ-I$ smoothing follows from the symbolic calculus (invert the principal symbol on the cosphere bundle and quantize). smoothing operators are compact on Sobolev spaces, so $P$ is Fredholm. if $Pu=f$ with $f$ smooth then $u=Qf-(QP-I)u$ is smooth because $Qf$ is smoothing applied to a distribution plus a smooth remainder. $\square$

theorem (atiyah-singer for the twisted dirac operator). let $M$ be a closed even- dimensional Riemannian spin manifold and $E\to M$ a Hermitian vector bundle with connection. for the chiral Dirac operator $D_E^+:C^\infty(S^+\otimes E)\to C^\infty(S^-\otimes E)$, $$ \mathrm{index}\,D_E^+ =\int_M\operatorname{ch}(E)\wedge\widehat{A}(TM), $$ where $\widehat{A}$ is the A-hat genus of $TM$, a multiplicative sequence in Pontryagin classes. both sides are integers.

proof (heat-kernel strategy). the McKean-Singer formula asserts that for every $t>0$, $$ \mathrm{index}\,D_E^+ =\operatorname{Tr}\bigl(e^{-tD^-D^+}\bigr) -\operatorname{Tr}\bigl(e^{-tD^+D^-}\bigr). $$ nonzero eigenvalues of $D^-D^+$ and $D^+D^-$ match, so only harmonic modes survive; the difference of heat traces equals $\dim\ker D^+-\dim\ker D^-$. as $t\to 0^+$, the local heat-kernel expansion of the supertrace density admits an asymptotic series in powers of $t$ whose coefficients are universal polynomials in the jets of the metric and connection. Getzler's rescaling of the Clifford variables shows that the constant term in $t$ is exactly the Chern-Weil form of $\operatorname{ch}(E)\widehat{A}(TM)$. integrating over $M$ yields the topological index. $\square$

theorem (atiyah-singer, general elliptic operator). for an elliptic pseudodifferential operator $P$ on a closed oriented manifold, $$ \mathrm{index}\,P =\bigl\langle\operatorname{ch}([\sigma(P)])\,\mathrm{Td}(TM\otimes\mathbb{C}),[T^*M]\bigr\rangle $$ after Thom isomorphism / pushforward from $T^*M$ to a point (equivalently the cohomological formula on $M$ recalled above for Dirac-type operators).

proof (outline of $K$-theoretic path). the principal symbol defines a class $[\sigma(P)]\in K^0(T^*M)$. the topological index is the composition of the Thom isomorphism in $K$-theory for the tangent bundle with the pushforward $K^0(TM)\to K^0(\mathrm{pt})\simeq\mathbb{Z}$. homotopy invariance of the analytical index and agreement on generators (Gysin maps, products, Bott elements) identify analytical and topological indices. the cohomological formula is the Chern character image of that $K$-theory equality. $\square$

The theorem extends in several directions: families of elliptic operators, yielding a class in the $K$-theory of the parameter space; the $G$-index for compact group actions, equivariant with respect to the action; and the $\mathrm{Cl}_k$-index theorem for operators commuting with a Clifford-algebra action, which refines the ordinary index to an element of $KO^{-k}(\mathrm{pt})$.

remark (families and equivariance). a continuous family $P_b$ parametrized by a compact space $B$ defines $\mathrm{index}(P)\in K^0(B)$. for a compact group $G$ acting on $M$ and on the bundles, $\mathrm{index}_G(P)$ takes values in the representation ring $R(G)$. Clifford-linear Dirac operators produce the refined analytical index in KO-theory matching Atiyah-Bott-Shapiro.

applications and further geometric structures

The index theorem implies classical integrality results (for example, the integrality of the $\widehat{A}$-genus of a spin manifold). It solves the vector-field problem on spheres by computing the maximal number of linearly independent vector fields. Equivariant versions constrain possible group actions on manifolds. Combined with the Lichnerowicz vanishing theorem it classifies, to a large extent, those simply connected manifolds that admit metrics of positive scalar curvature.

On Kahler manifolds Clifford multiplication interacts with the complex structure: pure spinors define almost-complex structures, and the existence of a parallel pure spinor reduces the holonomy group to a subgroup of $\mathrm{SU}(n)$. Twistor theory reinterprets these constructions by replacing the spinor bundle with a projective pure-spinor bundle whose holomorphic geometry encodes anti-self-dual conformal structures.

theorem (integrality of $\widehat{A}$). if $M$ is a closed spin manifold then $\langle\widehat{A}(TM),[M]\rangle\in\mathbb{Z}$. if $\mathrm{Scal}>0$ then this integer vanishes.

proof. Atiyah-Singer for the untwisted Dirac operator yields $\mathrm{index}\,D^+=\langle\widehat{A}(TM),[M]\rangle$. the left-hand side is an integer. Lichnerowicz vanishing forces $\mathrm{index}\,D^+=0$ when $\mathrm{Scal}>0$. $\square$

theorem (adams on vector fields of spheres, via index ideas). the maximal number of pointwise linearly independent vector fields on $S^{n-1}$ is $\rho(n)-1$, where $\rho(n)$ is the Radon-Hurwitz number determined by writing $n=(2a+1)2^{c+4d}$ and setting $\rho(n)=2^c+8d$.

proof (pointer). Clifford module structures on $\mathbb{R}^n$ produce $\rho(n)-1$ orthonormal vector fields on $S^{n-1}$ by $x\mapsto e_i\cdot x$. Adams' theorem shows this number is sharp, historically by K-theory operations and Adams operations on $KO(S^n)$; the Clifford / ABS model of $KO^{-n}(\mathrm{pt})$ organizes the same count. $\square$

remark (pure spinors and holonomy). a pure spinor $\psi$ satisfies $\xi\cdot\psi=0$ for a maximally isotropic subspace of $T\otimes\mathbb{C}$. if $\nabla\psi=0$ for the spin connection, the holonomy reduces inside the stabilizer of $\psi$, which for Calabi-Yau and special holonomy geometries is contained in $\mathrm{SU}(n)$, $G_2$, or $\operatorname{Spin}(7)$ as appropriate. twistor space packages the projective pure spinors into a complex manifold whose holomorphic data reconstruct anti-self-dual metrics (chapter 9).

Spin geometry and the Atiyah-Singer theorem therefore close the circle that began with local differential forms and characteristic classes: they convert the topological data of a spin manifold into precise analytical statements about the spectra of natural differential operators, with direct consequences for the classical energy conditions, the existence of positive-mass metrics, and the global consistency of fermionic fields.

remark (to yang-mills and twistors). chapter 9 returns to connections as dynamical gauge fields, instantons as absolute minima, and twistor methods that linearize anti-self-duality.

exercises

exercise 1 (symbol of $D$). verify that $\sigma(D)(x,\xi)=i\,\xi\cdot$ is invertible for $\xi\neq 0$ and compute $\sigma(D^2)(x,\xi)=\|\xi\|^2\operatorname{id}$.

exercise 2 (mckean-singer cancellation). if $D\begin{pmatrix}0&D^-\\ D^+&0\end{pmatrix}$ is self-adjoint on $S^+\oplus S^-$ and $\lambda\neq 0$ is an eigenvalue of $D^2$, show that the $\lambda$-eigenspaces of $D^-D^+$ and $D^+D^-$ are isomorphic via $D^\pm$. conclude that they cancel in the supertrace of $e^{-tD^2}$.

exercise 3 (sphere index). using $\widehat{A}(TS^{2m})=1$ in degree $0$ and vanishing of higher $\widehat{A}$ classes on spheres (or direct knowledge of $\ker D$ on $S^2$), compute $\mathrm{index}\,D^+$ on $S^2$ and reconcile with $\dim\ker D^+-\dim\ker D^-$.

exercise 4 (clifford relation in dimensions 2 and 3). construct Pauli matrices $\sigma_i$ satisfying $\{\sigma_i,\sigma_j\}=2\delta_{ij}I$ and note they furnish a representation of $\operatorname{Cl}(\mathbb{R}^3)$. explain how $\operatorname{Spin}(3)\simeq\operatorname{SU}(2)$ sits inside the even Clifford algebra.

exercise 5 (index additivity under direct sum). if $D_i:H_i\to H_i'$ are Fredholm, $i=1,2$, show that $D_1\oplus D_2$ is Fredholm with $\operatorname{index}(D_1\oplus D_2)=\operatorname{index}D_1+\operatorname{index}D_2$.