Spin Structures and Dirac Operators

An ordinary spinor bundle is not produced merely by choosing gamma matrices in every tangent space. First the relevant orthonormal-frame bundle must lift through the double cover from a Spin group. For an oriented Riemannian manifold, such a lift exists exactly when $w_2(TM)=0$; if it exists, its inequivalent choices form a torsor for $H^1(M;\mathbb Z_2)$. A chosen lift and complex spin module then form the spinor bundle. The Levi–Civita connection lifts to that bundle, and Clifford contraction defines the Dirac operator.

The same construction has sharply different analytic consequences in the two signatures relevant here. The Riemannian Dirac operator is elliptic and, with the standard factor of $i$, formally self-adjoint. On an oriented and time-oriented Lorentzian spacetime it has null characteristic covectors and belongs to causal, hyperbolic analysis instead. This page constructs the global data, fixes the convention translation, and follows one circle sector into its QFT interpretation. Curved-space dynamics, currents, quantization, index formulas, and anomalies remain with their later destinations.

Required background. Levi–Civita Connections, Geodesics, and Riemann Curvature supplies the metric connection and curvature convention; Vector, Principal, and Associated Bundles supplies frame bundles, transition functions, and associated bundles; and Clifford Algebras and Pin and Spin Groups supplies the double cover and its spin representations.

Spin-geometric setting and Clifford convention

Begin with an oriented Riemannian $n$-manifold $(M,g)$, where $g$ is positive definite. The site’s Clifford convention is

$$\gamma(v)\gamma(w)+\gamma(w)\gamma(v) =2g(v,w)\mathbf1.$$

Thus the matrices $\gamma^a$ in an oriented orthonormal frame may be chosen Hermitian and obey $\{\gamma^a,\gamma^b\}=2\delta^{ab}\mathbf1$. Many spin- geometry sources instead use a skew-Hermitian action

$$\mathbf c(v):=i\gamma(v), \qquad \mathbf c(v)\mathbf c(w)+\mathbf c(w)\mathbf c(v) =-2g(v,w)\mathbf1.$$

That factor of $i$ will matter when the operator is squared. In the Lorentzian crosswalk, the site uses signature $(+---)$ in four dimensions, $\{\gamma^a,\gamma^b\}=2\eta^{ab}\mathbf1$, and the proper-orthochronous cover $\operatorname{Spin}^+(1,3)\to SO^+(1,3)$.

The curvature convention inherited from the Levi–Civita page is

$$[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma,$$

with a round sphere having positive scalar curvature.

A spin structure lifts the oriented frame bundle

Let $P_{SO}(M,g)$ be the principal right $SO(n)$-bundle of oriented orthonormal frames, and let

$$\lambda:\operatorname{Spin}(n)\longrightarrow SO(n)$$

be the double cover. A spin structure is a principal right $\operatorname{Spin}(n)$-bundle $P_{\mathrm{Spin}}\to M$ together with a twofold bundle map

$$\Lambda:P_{\mathrm{Spin}}\longrightarrow P_{SO}(M,g)$$

covering the identity on $M$ and satisfying

$$\Lambda(\widetilde u\mathbin{\cdot}s) = \Lambda(\widetilde u)\mathbin{\cdot}\lambda(s).$$

Two such structures are equivalent only when a principal-bundle isomorphism between them commutes with their maps to $P_{SO}$. This map is the global object that local gamma matrices do not supply. Lawson and Michelsohn 1989, Chapter II develops this principal-bundle definition and the associated spinor construction.

In pseudo-Riemannian signature the frame component must be stated because $SO(p,q)$ can be disconnected. For Lorentzian QFT, orientation and time orientation select the principal $SO^+(1,n-1)$-bundle; an ordinary Lorentzian spin structure lifts that chosen bundle through $\operatorname{Spin}^+(1,n-1)\to SO^+(1,n-1)$. Without orientation one is instead led to a Pin problem. Time orientation is a separate requirement of the physical Lorentzian reduction, not another name for $w_2=0$.

Triple-overlap signs give the obstruction

Choose a good cover $\{U_i\}$ and oriented orthonormal frames on its sets. With the bundle convention used throughout this volume, their transition functions satisfy

$$g_{ij}g_{jk}=g_{ik}.$$

Every $g_{ij}:U_i\cap U_j\to SO(n)$ lifts locally to a map $\widetilde g_{ij}$ with $\lambda(\widetilde g_{ij})=g_{ij}$. The lifted maps need not obey the cocycle law. On a triple overlap their failure is

$$\varepsilon_{ijk} := \widetilde g_{ij}\widetilde g_{jk} \widetilde g_{ik}^{-1} \in\ker\lambda=\{\pm1\}.$$

The signs $\varepsilon_{ijk}$ form a Čech $2$-cocycle. Replacing any local lift by its negative multiplies this cocycle by a coboundary, so the cohomology class is independent of the arbitrary lifts. That class is

$$[\varepsilon]=w_2(TM)\in H^2(M;\mathbb Z_2).$$

It vanishes exactly when signs can be changed so that $\widetilde g_{ij}\widetilde g_{jk}=\widetilde g_{ik}$ everywhere. The correct existence statement is therefore

$$M\text{ admits an ordinary spin structure} \quad\Longleftrightarrow\quad w_1(TM)=0\ \text{and}\ w_2(TM)=0.$$

After an orientation has already been chosen, only the $w_2$ condition remains. Nakahara 2003, §§ 11.6.1–11.6.3 and Wernli 2019, §§ 2.3.1–2.3.2 construct this cocycle and prove the obstruction criterion.

Existence is not uniqueness. If $\widetilde g_{ij}$ and $\widetilde g'_{ij}$ are two valid lift systems, then

$$\alpha_{ij} = \widetilde g'_{ij}\widetilde g_{ij}^{-1} \in\{\pm1\}$$

is a Čech $1$-cocycle. Changing local spin frames changes $\alpha$ by a coboundary. Consequently, when at least one spin structure exists, the set of its isomorphism classes carries a free and transitive action of

$$H^1(M;\mathbb Z_2).$$

It is a torsor, not canonically the group itself: choosing one spin structure as an origin creates a bijection, but geometry need not provide a preferred origin.

Two examples separate commonly conflated properties. Every oriented surface is spin, so $S^2$ is spin; because $H^1(S^2;\mathbb Z_2)=0$, its spin structure is unique. Yet $S^2$ is not parallelizable: every tangent vector field on $S^2$ must vanish somewhere, whereas a global frame would contain a nowhere-zero field. Frankel 2012, § 16.2, p. 423 gives this Poincaré–Hopf argument explicitly. Thus a spin lift is weaker than a global frame. Conversely, $\mathbb{CP}^2$ is oriented and simply connected but not spin. Neither orientability nor simple connectivity alone settles the question. Simple connectivity can remove nonuniqueness after existence; it cannot force existence. Dai 2015, § 2.2, pp. 15–16, PDF gives the obstruction and classification theorem together with the spin and projective-space examples and the circle and torus classifications used below.

The lift forms the spinor bundle

Choose a complex Clifford module $\Delta_n$ and let $\rho_\Delta$ be the induced representation of $\operatorname{Spin}(n)$. The associated complex spinor bundle is

$$\Sigma M = P_{\mathrm{Spin}} \times_{\rho_\Delta}\Delta_n,$$

with the volume’s associated-bundle convention

$$(\widetilde u,z) \sim (\widetilde u\mathbin{\cdot}s, \rho_\Delta(s^{-1})z).$$

Local spinor representatives therefore glue by $\psi_i=\rho_\Delta(\widetilde g_{ij})\psi_j$. Clifford equivariance,

$$\rho_\Delta(s)\gamma(v)\rho_\Delta(s)^{-1} = \gamma(\lambda(s)v),$$

turns the fiberwise matrices into a global bundle map

$$\gamma:T^*M\otimes\Sigma M\longrightarrow\Sigma M,$$

where the metric identifies vectors and covectors. In a local orthonormal coframe $e^a$, this map is represented by constant matrices $\gamma^a$; in coordinates, $\gamma^\mu=e_a{}^\mu\gamma^a$. The matrices are local representatives of the global map, not a substitute for its gluing data. Dai 2015, § 2.4, pp. 19–20, PDF constructs the associated spinor bundle, its Clifford action, and the even-dimensional splitting used below.

This construction is complex. A real or Majorana condition requires dimension- and signature-dependent conjugation data and does not follow from the existence of $\Sigma M$ alone. Those additional choices are treated in Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities.

The Levi–Civita connection lifts to spinors

Let $e_a$ be a local oriented orthonormal frame and define

$$\nabla^{\mathrm{LC}}_\mu e_b = \omega_\mu{}^a{}_b e_a, \qquad \omega_{\mu ab} =g(e_a,\nabla^{\mathrm{LC}}_\mu e_b) =-\omega_{\mu ba}.$$

The Lie algebras of the two covering groups are isomorphic, so the Levi–Civita principal connection lifts uniquely once the spin structure is chosen. In the site’s gamma convention, the induced connection is

$$\boxed{ \nabla^\Sigma_\mu = \partial_\mu +\frac18\omega_{\mu ab}[\gamma^a,\gamma^b] = \partial_\mu +\frac14\omega_{\mu ab}\gamma^a\gamma^b. }$$

The two coefficients are the same statement: antisymmetry of $\omega_{\mu ab}$ removes the Clifford anticommutator. A common source instead writes $\widetilde\omega_{ab}=g(\nabla e_a,e_b)=-\omega_{ab}$; that transposed frame-index convention cancels the minus sign introduced by $\mathbf c=i\gamma$. The explicit definition above therefore matters when translating the local formula. The invariant characterization is compatibility with Clifford multiplication,

$$\nabla_X^\Sigma\!\bigl(\gamma(v)\psi\bigr) = \gamma(\nabla_X^{\mathrm{LC}}v)\psi +\gamma(v)\nabla_X^\Sigma\psi.$$

With the inherited curvature sign, the round-trip check is

$$[\nabla^\Sigma_\mu,\nabla^\Sigma_\nu]\psi = \frac14R_{ab\mu\nu} \gamma^a\gamma^b\psi,$$

where $R_{ab\mu\nu}=g(e_a,[\nabla_\mu,\nabla_\nu]e_b)$. This relation fixes the sign of the lifted curvature relative to the tangent-bundle convention. Wernli 2019, §§ 2.3.3–2.3.5 develops the associated spinor bundle, Clifford compatibility, lifted connection, and contraction.

Clifford contraction defines the Dirac operator

Literal contraction with the site’s plus-sign Clifford action gives

$$\mathcal D_0 = \gamma\circ\nabla^\Sigma = \sum_{a=1}^n\gamma^a\nabla^\Sigma_{e_a}.$$

Most Riemannian analysis instead uses the skew-Hermitian action $\mathbf c=i\gamma$. Its standard formally self-adjoint Dirac operator is

$$\boxed{ \mathcal D_R = \sum_{a=1}^n\mathbf c(e_a)\nabla^\Sigma_{e_a} =i\mathcal D_0. }$$

In local coordinates,

$$\mathcal D_R = i\gamma^a e_a{}^\mu \left( \partial_\mu +\frac14\omega_{\mu bc}\gamma^b\gamma^c \right).$$

Choosing $\mathbf c=-i\gamma$ would reverse the whole operator and leave its square unchanged. What is not allowed is to choose one sign in the definition and the other in a later formula.

Use the Fourier-symbol convention $\partial_\mu\mapsto-i\xi_\mu$. Then

$$\sigma_1(\mathcal D_R)(x,\xi) = \gamma(\xi^\sharp), \qquad \sigma_1(\mathcal D_R)(x,\xi)^2 = |\xi|_g^2\mathbf1.$$

For every nonzero Riemannian covector the symbol is invertible, so $\mathcal D_R$ is elliptic. This is a local statement and needs no compactness hypothesis. On compactly supported smooth spinors, with no boundary contribution, the Hermitian spinor metric and compatible connection make $\mathcal D_R$ formally self-adjoint. On a closed manifold there is no boundary contribution. A boundary or an incomplete metric requires an operator domain and, where appropriate, boundary conditions; formal symmetry by itself is not self-adjointness. Dai 2015, § 2.6, pp. 23–26, PDF gives the operator, symbol, Hermitian structure, formal-adjoint calculation, and boundary Green formula under the minus-Clifford convention translated above.

Squaring the operator detects scalar curvature

Define the nonnegative connection Laplacian by

$$(\nabla^\Sigma)^*\nabla^\Sigma = -\sum_a \left( \nabla^\Sigma_{e_a}\nabla^\Sigma_{e_a} -\nabla^\Sigma_{\nabla^{\mathrm{LC}}_{e_a}e_a} \right).$$

The Schrödinger–Lichnerowicz identity is

$$\boxed{ \mathcal D_R^2 = (\nabla^\Sigma)^*\nabla^\Sigma +\frac14\operatorname{Scal}_g. }$$

Equivalently, the raw plus-Clifford contraction obeys

$$\mathcal D_0^2 = -(\nabla^\Sigma)^*\nabla^\Sigma -\frac14\operatorname{Scal}_g.$$

On flat Euclidean space, where the spin connection and scalar curvature vanish, this reduces to

$$\mathcal D_R^2 =-\sum_{a=1}^n\partial_a^2.$$

The formula itself is local. Closedness matters only when it is integrated. For example, if $M$ is closed and $\operatorname{Scal}_g>0$ everywhere, then a harmonic spinor would satisfy

$$0 =\|\mathcal D_R\psi\|_{L^2}^2 =\|\nabla^\Sigma\psi\|_{L^2}^2 +\frac14 \int_M\operatorname{Scal}_g|\psi|^2\,\mathrm{vol}_g,$$

so $\psi=0$. This is an obstruction to harmonic spinors, not a converse existence theorem. Dai 2015, §§ 2.5–2.8, pp. 22–31, PDF derives the lifted connection, symbol, boundary term, formal self-adjointness, and Lichnerowicz formula under its explicitly stated minus-Clifford convention.

Chirality and the twisting handoff

In oriented even Riemannian dimension $n=2m$, define

$$\Gamma_E =i^m\gamma^1\gamma^2\cdots\gamma^{2m}.$$

It is a parallel Hermitian involution, anticommutes with Clifford multiplication, and splits the spinor bundle:

$$\Sigma M=\Sigma^+M\oplus\Sigma^-M, \qquad \mathcal D_R: \Gamma(\Sigma^\pm M)\longrightarrow\Gamma(\Sigma^\mp M).$$

Dai 2015, § 2.4, p. 20, PDF gives the complex volume element, its involution and anticommutation properties, and the resulting half-spinor decomposition.

Changing the overall sign of $\Gamma_E$ merely exchanges the labels. In odd dimension the complex volume element acts within an irreducible module and does not furnish an analogous half-spin decomposition. On a closed even-dimensional manifold the completed chiral operator is elliptic and Fredholm. Its index formula and zero-mode applications belong to the later index page. Wernli 2019, §§ 3.2–3.3, pp. 85–87 supplies the elliptic/Fredholm step, formal chiral adjoint, and the even-dimensional operator splitting.

With the additional input of a Hermitian bundle $W\to M$ and a unitary connection $\nabla^W$, one may instead use $\Sigma M\otimes W$, its product connection, and the twisted operator

$$\begin{aligned} \nabla^{\Sigma\otimes W} &=\nabla^\Sigma\otimes\mathbf1 +\mathbf1\otimes\nabla^W, \\ \mathcal D_{R,W} &=\sum_a\mathbf c(e_a) \nabla^{\Sigma\otimes W}_{e_a}. \end{aligned}$$

This twist is not part of the spin structure. Its curvature contributes when the operator is squared; the detailed twisted formula and its index- theory uses belong to the later index treatment. Dai 2015, § 2.6, p. 26, and § 2.8, p. 32, PDF defines the twisted operator and identifies the additional curvature contraction in its square.

One local formula can have different global spectra

Let $S^1_r$ have circumference $2\pi r$, angle $\theta\sim\theta+2\pi$, unit frame $e=r^{-1}\partial_\theta$, and trivial local spin connection. Because

$$H^1(S^1;\mathbb Z_2)\cong\mathbb Z_2,$$

there are two spin structures. They may be represented by the boundary conditions

$$\psi(\theta+2\pi) =e^{2\pi i\alpha}\psi(\theta), \qquad \alpha=0\ \text{or}\ \frac12.$$

Take $\gamma(e)=1$. The same local differential expression applies in both sectors:

$$\mathcal D_R =\frac{i}{r}\frac{\mathrm d}{\mathrm d\theta}.$$

For modes $\psi_n(\theta)=e^{i(n+\alpha)\theta}$,

$$\mathcal D_R\psi_n =-\frac{n+\alpha}{r}\psi_n,$$

and hence

$$\operatorname{spec}\mathcal D_R = \begin{cases} r^{-1}\mathbb Z, &\alpha=0, \\[2pt] r^{-1}(\mathbb Z+\tfrac12), &\alpha=\tfrac12. \end{cases}$$

Only the periodic sector has a zero mode. Both satisfy $\mathcal D_R^2=-r^{-2}\mathrm d^2/\mathrm d\theta^2$, as the Lichnerowicz identity predicts because a one-dimensional metric has zero scalar curvature. The example isolates the global content: identical local gamma matrices, metric, and connection coefficients can define different operator domains and spectra.

On the Lorentzian cylinder $\mathbb R_t\times S^1_r$, the same choice gives integer or half-integer spatial momenta. Turning those modes into a quantized fermion field requires a state, anticommutation relations, and dynamics, none of which is supplied by the spin structure alone. Sanders 2010, pp. 1–2 states the additional locally covariant QFT problem on globally hyperbolic spin spacetimes and uses the same $(+---)$ Clifford convention as this page.

Lorentzian QFT bridge

Now let $(M,g)$ be an oriented, time-oriented Lorentzian spin manifold. In four dimensions use $\eta_{ab}=\operatorname{diag}(1,-1,-1,-1)$ and $\operatorname{Spin}^+(1,3)\to SO^+(1,3)$. For a local tetrad,

$$\gamma^\mu=e_a{}^\mu\gamma^a, \qquad \{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbf1.$$

The bounded local crosswalk for the geometric Lorentzian operator is

$$\boxed{ \mathcal D_L =i\gamma^\mu \left( \partial_\mu +\frac18\omega_{\mu ab}[\gamma^a,\gamma^b] \right) }$$

In a flat global tetrad, $\omega_{\mu ab}=0$, and the operator becomes $i\gamma^\mu\partial_\mu$. A change of local orthonormal frame and its Spin lift changes the component fields, gamma matrices, and connection together, leaving the resulting section $\mathcal D_L\psi$ covariant. Mass terms, internal gauge twists, and their physical interpretation are additional lower-order data handled downstream.

With the declared Fourier convention, the principal symbol is

$$\sigma_1(\mathcal D_L)(x,\xi) =\gamma(\xi^\sharp), \qquad \sigma_1(\mathcal D_L)(x,\xi)^2 =g^{-1}(\xi,\xi)\mathbf1.$$

A nonzero null covector therefore makes the symbol singular. The Lorentzian operator is not elliptic; its connection term is lower order and does not change that conclusion. With the additional hypothesis of global hyperbolicity, the classical Dirac operator is Green-hyperbolic; Bär 2015, § 3.5, p. 15 proves this by relating its square to a normally hyperbolic operator. Its adjoint, conserved current, causal propagation, admissible backgrounds, tetrad examples, and quantization are developed on Spinors, Tetrads, and Spin Connections.

Common pitfalls

Equating orientability with spin. Orientation removes the $w_1$ obstruction and produces $P_{SO}$; the independent class $w_2(TM)$ may still obstruct its Spin lift. The simply connected manifold $\mathbb{CP}^2$ is the standard warning.

Inferring a global lift from local gamma matrices. Every small contractible chart admits local frames and matrices. The obstruction lives on triple overlaps, where the signs of the local lifts must fit together.

Equating spin with parallelizable. A global frame trivializes much more data than a double-cover lift. The spin but nonparallelizable sphere $S^2$ separates the notions.

Treating the set of spin structures as a canonically based group. It is an $H^1(M;\mathbb Z_2)$-torsor. Subtracting two choices is meaningful; calling one of them zero requires an additional choice.

Mixing the two Riemannian Clifford signs. The site uses $\gamma(v)^2=g(v,v)$, whereas the standard analytic action is $\mathbf c(v)=i\gamma(v)$. Forgetting the factor of $i$ reverses the sign of the squared operator and the Lichnerowicz crosswalk.

Calling a formally symmetric expression self-adjoint. Self-adjointness is a claim about an operator and its domain. Boundaries and incomplete ends cannot be ignored.

Transferring Riemannian ellipticity to spacetime. Positive-definite covectors have nonzero norm, but Lorentzian geometry has nonzero null covectors. They are precisely where the Dirac symbol fails to be invertible.

Assuming ordinary Spin is the only fermionic global structure. Pin, Spin$^c$, and combined gauge-spacetime lifts answer different questions. Their possible existence does not retroactively make an ordinary non-spin manifold spin.

Exercises

Retrieval

List the geometric data in order from a metric manifold to its Dirac operator. Which step is a genuine global choice?

Solution

The chain is

$$(M,g) \longrightarrow P_{SO} \longrightarrow P_{\mathrm{Spin}} \longrightarrow\Sigma M \longrightarrow\nabla^\Sigma \longrightarrow\mathcal D_0.$$

After orientation, $P_{SO}$ is fixed by the metric. The lift $P_{\mathrm{Spin}}\to P_{SO}$ is the global existence-and-choice step. A complex spin module forms $\Sigma M$, Levi–Civita lifts to $\nabla^\Sigma$, and literal Clifford contraction forms $\mathcal D_0=\gamma\circ\nabla^\Sigma$. The standard Riemannian analytic convention then sets $\mathcal D_R=i\mathcal D_0$.

Hypothesis and counterexample

Why do neither orientability nor simple connectivity guarantee a spin structure? What can simple connectivity guarantee after existence?

Solution

Orientability is $w_1(TM)=0$, while a Spin lift also requires $w_2(TM)=0$. The manifold $\mathbb{CP}^2$ is oriented and simply connected but has $w_2\ne0$, so it is not spin. If a connected manifold is simply connected, then $H^1(M;\mathbb Z_2)=0$; once a spin structure exists, the torsor therefore has only one element. This proves uniqueness, not existence.

Derivation

Given local lifts $\widetilde g_{ij}$, explain why $\varepsilon_{ijk}=\widetilde g_{ij}\widetilde g_{jk} \widetilde g_{ik}^{-1}$ is a sign and how its class tests existence.

Solution

Applying $\lambda$ gives

$$\lambda(\varepsilon_{ijk}) =g_{ij}g_{jk}g_{ik}^{-1} =\mathbf1,$$

so $\varepsilon_{ijk}\in\ker\lambda=\{\pm1\}$. Associativity and the ordinary cocycle law make these signs a Čech $2$-cocycle. Changing lift signs changes it by a coboundary. Its class is $w_2(TM)$, and the class vanishes exactly when the lifts can be adjusted to satisfy their own cocycle law.

QFT transfer

On $\mathbb R_t\times S^1_r$, what momenta follow from the two spin structures, and why does this not yet quantize a Dirac field?

Solution

Periodic spinors have spatial momenta $n/r$, while antiperiodic spinors have $(n+\tfrac12)/r$, with $n\in\mathbb Z$. The first sector admits a spatial zero mode and the second does not. These are global domain and spectrum statements for the spatial Dirac differential operator. A quantum field still needs dynamics, canonical anticommutation relations, a state or representation, and the relevant Lorentzian analytic construction.

Synthesis and continuations

Spin geometry is a sequence of compatible lifts. Orientation produces the relevant orthonormal-frame bundle; $w_2(TM)=0$ permits a Spin lift; a chosen lift and module form the spinor bundle; Levi–Civita lifts to a Clifford- compatible connection; and contraction produces the Dirac operator. The $H^1(M;\mathbb Z_2)$ torsor records the remaining global choices, which can alter spectra without altering any local formula. Riemannian positivity makes the operator elliptic and yields the Lichnerowicz identity, while Lorentzian null covectors force a different analytic theory.

Continue according to the next question:

• Spinors, Tetrads, and Spin Connections develops the curved-spacetime Dirac field, local Lorentz covariance, adjoints, currents, and admissible Lorentzian backgrounds;
• after characteristic-class and spectral preparation, Fredholm and Dirac Index Theorems and Zero-Mode Counting computes the chiral index rather than merely preparing its operator;
• Generalized Killing Spinors and Global Spin–R Bundles studies the additional global structures required on rigid supersymmetric backgrounds.

References

• Christian Bär, “Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes,” Communications in Mathematical Physics 333 (2015), 1585–1615, doi:10.1007/s00220-014-2097-4, arXiv:1310.0738, § 3.5, p. 15. This supplies the globally-hyperbolic causal handoff for classical Dirac and twisted Dirac operators.
• Xianzhe Dai, Lectures on Dirac Operators and Index Theory — Open PDF, lecture notes, 2015, §§ 2.2 and 2.4–2.8, pp. 15–32. This provides an accessible specialist account of the obstruction and classification theorem, associated spinor bundle, connection, symbol, formal adjoint and boundary term, twisting, and the Lichnerowicz formula.
• Theodore Frankel, The Geometry of Physics: An Introduction, third edition, Cambridge University Press, 2012, § 16.2, p. 423, and § 19.5, pp. 515, 518, and 521. The former supplies the nonparallelizability check for $S^2$; the latter supplies an independent four-dimensional check of the lifted connection and local $1/4$ coefficient.
• H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989, Chapter II. This chapter develops spin structures, spinor bundles, lifted connections, Dirac operators, and the Lichnerowicz formula.
• Mikio Nakahara, Geometry, Topology and Physics, second edition, Taylor & Francis, 2003, § 11.6, pp. 449 and 451, and § 12.6, pp. 468–469 and 471. These sections provide a physics-oriented independent check of the transition-function obstruction, Euclidean chirality, ellipticity, and the twisted Dirac operator.
• Ko Sanders, “The Locally Covariant Dirac Field,” Reviews in Mathematical Physics 22, no. 4 (2010), 381–430, doi:10.1142/S0129055X10003990, arXiv:0911.1304, pp. 1–2. This supplies the QFT-application boundary beyond the geometric operator.
• Konstantin Wernli, Lecture Notes on Spin Geometry, arXiv:1911.09766, 2019, §§ 2.3 and 3.2–3.3, especially pp. 60–72 and 85–87. These notes supply the Čech obstruction, the $H^1(M;\mathbb Z_2)$ classification, the compatible connection, and the Weitzenböck–Lichnerowicz formula.