Given Spin and Spin-c structures and their spinor bundles

Let V→X be an oriented Euclidean real vector bundle of rank n, with oriented frame bundle P_SO(V). Use Spin(n)=Spin_N=1(ℝⁿ,−‖·‖²), the norm-one even subgroup of the negative-square Clifford algebra, and its standard twisted-Clifford homomorphism λ_n:Spin(n)→SO(n). A Spin structure on V is a principal Spin(n)-bundle P with a Spin(n)-equivariant two-fold map $\tilde{\lambda }:P\to P_{SO}(V)$ covering the identity on X and satisfying $\tilde{\lambda }(pg)=\tilde{\lambda }(p){\lambda }_n(g)$. Put ${\mathrm{Spin}}^c(n)=(\mathrm{Spin}(n)\times U(1))\mathrm{/}⟨(-1,-1)⟩$ and define ${\lambda }_n^c:{\mathrm{Spin}}^c(n)\to SO(n)$ by ${\lambda }_n^c([s,z])={\lambda }_n(s)$. A Spin${}^c$ structure is a principal Spin${}^c(n)$-bundle $P_c$ with a ${\mathrm{Spin}}^c(n)$-equivariant map ${\tilde{\lambda }}_c:P_c\to P_{SO}(V)$ over the identity satisfying ${\tilde{\lambda }}_c(pg)={\tilde{\lambda }}_c(p){\lambda }_n^c(g)$. A spinor bundle is the associated bundle for the specified real Clifford representation; a Spin${}^c$ spinor bundle is $P_c{\times }_{{\tau }^c}S$, where the specified complex representation $\tau$ of $\mathrm{Spin}(n)$ and integer $U(1)$ weight $k$ satisfy $\tau (-1)(-1{)}^k=I_S$, and ${\tau }^c([s,z])=z^k\tau (s)$. This condition is exactly that $(-1,-1)$ act trivially, so the representation descends to the quotient. For the complex Clifford spinor representation, $\tau (-1)=-I_S$ and the standard weight is $k=1$. These are given structures; this definition asserts no existence criterion.