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The zero-mode anticommutator $\{d_0^I,d_0^J{\}}_{+}={\delta }^{IJ}$ becomes the Euclidean Clifford algebra after ${\gamma }^I=\sqrt{2}{d}_0^I$. It acts within the oscillator-vacuum space. Eight transverse Gamma matrices give a minimal complex representation of dimension sixteen, split into two eight-dimensional chirality sectors.

In flat conformal gauge, take $\mathcal{L}=\partial X\cdot \partial X+i{\alpha }^{\prime }\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi$ and rigid transformations $\delta X=iA\overline{ϵ}\psi$, $\delta \psi =B{\gamma }^{\nu }{\partial }_{\nu }Xϵ$, with $A={\alpha }^{\prime }B$. The Majorana Grassmann bilinear interchange and Clifford algebra combine the variations intoThe identity is off shell. The integrated action is invariant when its boundary flux vanishes, for example on a closed worldsheet or with compatible supersymmetric endpoint conditions. Constant spinors and ordinary derivatives here refer to flat gauge; arbitrary curved worldsheets require covariant spinor derivatives and the locally supersymmetric completion.

The spin group $\mathrm{Spin}(n)$ is the connected double cover of $SO(n)$ constructed inside the units of a Clifford algebra.

A spinor is an element of a representation space for a Spin group; algebraically, spinor spaces arise from irreducible representations of a Clifford algebra or its even part. The Spin group is a double cover of a rotation group, so its action can distinguish rotations that ordinary vectors cannot.

For rotations in three-dimensional Euclidean space, a spinor can be a column in ${\mathbb{C}}^2$, acted on by $\mathrm{SU}(2)$. A rotation through angle $\theta$ about a unit direction $n$ acts as $\exp (-i\theta n\cdot \sigma /2)$, where $\sigma$ are the Pauli matrices. At $\theta =2\pi$ this is $-I$, and at $4\pi$ it is $I$. The sign change explains the distinction between spinors and spatial vectors and underlies the representation of spin one-half. A spinor is not a literal square root of a vector; the relationship is through the Clifford algebra action.

Lift the Levi-Civita connection to the spinor bundle using the spin representation of the Lorentz algebra. Under a local Lorentz transformation, its matrix ${\Omega }_{\mu }=\frac{1}{4}{\omega }_{\mu AB}{\gamma }^{AB}$ obeys ${\Omega }_{\mu }^{\prime }=S{\Omega }_{\mu }{S}^{-1}-({\partial }_{\mu }S)S^{-1}$, so ${\nabla }_{\mu }^{\prime }{\psi }^{\prime }=S{\nabla }_{\mu }\psi$. The Clifford algebra identity $[{\gamma }^C,{\gamma }^{AB}]=2({\eta }^{CA}{\gamma }^B-{\eta }^{CB}{\gamma }^A)$ fixes the coefficient. The tetrad postulate then gives ${\partial }_{\mu }{\gamma }^{\nu }+{\Gamma }_{\mu \rho }^{\nu }{\gamma }^{\rho }+[{\Omega }_{\mu },{\gamma }^{\nu }]=0$, establishing compatibility with curved Clifford multiplication.

An irreducible complex Clifford algebra module in even spacetime dimension $2k$ has dimension $2^k$ and splits under the connected Spin group into two Weyl spinor modules of dimension $2^{k-1}$. In odd dimension $2k+1$, an irreducible spinor still has dimension $2^k$ but has no Weyl splitting. Reality conditions depend on signature and are additional to these complex counts.

Define ${\sigma }^{\mu \nu }=\frac{i}{2}[{\gamma }^{\mu },{\gamma }^{\nu }]$ using the gamma matrix Clifford algebra. Moving ${\gamma }^{\rho }$ through the two factors with $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }$ proves the displayed operator commutator. For real antisymmetric ${\omega }_{\mu \nu }$ and $S=1-\frac{i}{4}{\sigma }^{\mu \nu }{\omega }_{\mu \nu }$, it gives $S^{-1}{\gamma }^{\rho }S={\gamma }^{\rho }+{\omega }^{\rho }{}_{\nu }{\gamma }^{\nu }+O({\omega }^2)$. This is the infinitesimal Lorentz covariance identity required by the Dirac equation.

A spinor representation is a representation of a Spin group, or infinitesimally of a Special orthogonal Lie algebra, constructed from a Clifford algebra. In even dimensions it can split into two chiral spinor representations.