Spin structure and parallel spinors from Calabi–Yau data

Identify $U(1)$ with the multiplicative unit circle. The based homeomorphism $ϵ:\mathbb{R}\mathrm{/}\mathbb{Z}\to U(1)$ from [F13] and [F14] show ${\pi }_1(U(1),1)\cong \mathbb{Z}$ with generator $t\mapsto e^{2\pi it}$: composing based loops with $ϵ$ and its inverse gives inverse maps on loop classes and preserves concatenation. Every $z\in U(1)$ is joined to $1$ by a path $s\mapsto e^{2\pi ist}$ for a representative $z=e^{2\pi it}$, so $U(1)$ is path-connected.

Write $\pi ([s,z])=\lambda (s)$ and $\delta ([s,z])=z^2$, and put $\mathrm{\Pi }=(\pi ,\delta ):{\mathrm{Spin}}^c(2n)\to SO(2n)\times U(1)$. The projection $\pi$ is onto with kernel $\{[1,z]:z\in U(1)\}\cong U(1)$, so it is not a double cover. The pair map $\mathrm{\Pi }$ is onto and has kernel $\{[1,1],[1,-1]\}$; local covering charts follow from [F11] and local square-root charts for $z\mapsto z^2$, so $\mathrm{\Pi }$ is the two-fold cover.

Put $E=T_X^{1,0}$. By [F5], choose a homotopy equivalence $g:X\to Y$ to a path-connected CW complex and a homotopy inverse $f:Y\to X$. Set $E_Y=f^{\ast }E$. It is numerable; because $f\circ g$ is homotopic to the identity, [F6] gives a bundle isomorphism $g^{\ast }{E}_Y\cong E$. Apply [F4] to $E_Y$ over the actual CW base $Y$. Pulling that equality back along $g$, [F7] gives Chern-class naturality, [F8] gives Stiefel–Whitney naturality, and reduction modulo two commutes with pullback; the bundle isomorphism then yields $w_2(E_{\mathbb{R}})={\rho }_2{c}_1(E)$ on $X$. This uses the CW theorem only on $Y$, not directly on the smooth manifold $X$.

The determinant map $d:U(n)\to U(1)$ is globally trivial as a bundle: the homeomorphism over $U(1)$ is $SU(n)\times U(1)⟶U(n),(B,z)⟼B\mathrm{diag}(z,1,\dots ,1),$ with inverse $A\mapsto (A\mathrm{diag}((\det A{)}^{-1},1,\dots ,1),\det A)$. Under this homeomorphism $d$ is the product projection and has the homotopy-lifting property for every space: for any initial map $x\mapsto (B_x,z_x)$ and homotopy $H:X\times [0,1]\to U(1)$ starting at $z_x$, the lift is $(x,t)\mapsto (B_x,H(x,t))$. Hence $d$ is a Serre fibration and the exact sequence [F10] applies. For $n\ge 2$, [F9] gives $SU(n)$ path-connected with ${\pi }_1(SU(n))=0$, and step 1.1 gives ${\pi }_1(U(1))\cong \mathbb{Z}$. Exactness therefore makes $d_{\ast }:{\pi }_1(U(n),I)\to {\pi }_1(U(1),1)$ an isomorphism. For $n=1$, $d$ is the identity. In every dimension $U(n)$ is path-connected by this product description and the path-connectedness of $SU(n)$ and $U(1)$. The loop $A(t)=\mathrm{diag}(e^{2\pi it},1,\dots ,1)$ therefore generates ${\pi }_1(U(n),I)$, because its image under $d_{\ast }$ is the generator from step 1.1.

The matrix group $U(n)$ is locally path-connected. Use the operator norm induced by the Euclidean norm on ${\mathbb{C}}^n$. For $\mathrm{\parallel }A-I\mathrm{\parallel }<1$, the convergent principal series $L=\log A={\sum }_{k\ge 1}(-1{)}^{k+1}(A-I{)}^k\mathrm{/}k$ satisfies $\exp L=A$ and $L^{\ast }=-L$: since $A$ is unitary, $\mathrm{\parallel }{A}^{-1}-I\mathrm{\parallel }=\mathrm{\parallel }A-I\mathrm{\parallel }$, and the absolutely convergent power-series identities $\log (e^z)=z$ and $\log (z^{-1})=-\log z$ near $1$ give these relations. For sufficiently small $\delta >0$, the open set $V_{\delta }=\{A\in U(n):\mathrm{\parallel }A-I\mathrm{\parallel }<1,\mathrm{\parallel }\log A\mathrm{\parallel }<\delta \}$ is path-connected by $s\mapsto \exp (s\log A)$: these exponentials are unitary since $L$ is skew-Hermitian; choose $\delta <\log 2$, so $\mathrm{\parallel }\exp (s\log A)-I\mathrm{\parallel }\le e^{\delta }-1<1$ and $\log (\exp (s\log A))=s\log A$. As $\log A\to 0$ when $A\to I$, these path-connected open sets form a neighborhood basis at $I$; left translations give local path-connectedness everywhere. Define $\mathrm{\Phi }(A)=(A_{\mathbb{R}},\det A)$. The loop $A(t)$ of step 2.1 has image $\mathrm{\Phi }(A(t))$ with the closed lift under $\mathrm{\Pi }$ given by $t\mapsto [\cos (\pi t)+\sin (\pi t)e_1{e}_2,e^{\pi it}]$: the spin factor covers the real-plane rotation through $2\pi t$, the second coordinate squares to $e^{2\pi it}$, and the endpoint $[-1,-1]$ equals $[1,1]$. Since step 2.1 shows that $A(t)$ generates ${\pi }_1(U(n),I)$, every loop class lies in the covering subgroup for $\mathrm{\Pi }$. The based lifting criterion [F15] therefore gives a unique based lift $\tau :U(n)\to {\mathrm{Spin}}^c(2n)$ with $\mathrm{\Pi }\circ \tau =\mathrm{\Phi }$. It is a homomorphism because the lifts of $(A,B)\mapsto \tau (A)\tau (B)$ and $(A,B)\mapsto \tau (AB)$ differ by a discrete kernel element, agree at $(I,I)$, and $U(n)\times U(n)$ is connected by step 2.1. The local smooth spin section in [F16] and the local square-root charts make $\mathrm{\Pi }$ a smooth covering; since $\mathrm{\Phi }$ is smooth, its lift $\tau$ is smooth in these local charts. In particular, $\delta \circ \tau (A)=\det A$.

Let $P_U\to X$ be the unitary frame bundle of $(T_X^{1,0},h)$ and $P_{SO}\to X$ the oriented orthonormal frame bundle of $(T{X}_{\mathbb{R}},h_{\mathbb{R}})$, where the orientation is the complex orientation in the Statement. The map $A\mapsto A_{\mathbb{R}}$ makes $P_U$ a reduction of $P_{SO}$. Form the induced principal bundle $P_c=P_U{\times }_{\tau }{\mathrm{Spin}}^c(2n)$, with equivalence $(uA,s)\sim (u,\tau (A)s)$. Its map to the oriented frame bundle is ${\lambda }_c([u,s])=u\cdot \pi (s)$; this is well-defined and Spin${}^c$-equivariant because $\pi \circ \tau (A)=A_{\mathbb{R}}$. Its fiber is $U(1)=\ker \pi$, so this structure map is not a double cover. Define the determinant line by $L_{\det }:=P_c{\times }_{\delta }\mathbb{C}$ with its Hermitian metric induced by the unitary character $\delta$, and let $P_{L_{\det }}$ be its unit-circle bundle. Since $\delta \circ \tau ={\det }_{\mathbb{C}}$, the associated-bundle construction identifies $L_{\det }\cong {\det }_{\mathbb{C}}(T_X^{1,0})=K_X^{-1}$ with the determinant metric induced by $h$. The pair map is explicitly ${\widehat{\lambda }}_c(p)=({\lambda }_c(p),[p,1])\in P_{SO}{\times }_X{P}_{L_{\det }}$; on each fiber its group map is $\mathrm{\Pi }$, hence it is a two-fold cover. This is the canonical Spin${}^c$ structure under the definition in [F1].

The nowhere-zero section ${\mathrm{\Omega }}^{-1}$ of $L_{\det }\cong K_X^{-1}$ has positive smooth norm for the determinant metric induced by $h$. Its pointwise unit normalization $s={\mathrm{\Omega }}^{-1}\mathrm{/}\mathrm{\parallel }{\mathrm{\Omega }}^{-1}\mathrm{\parallel }$ is a smooth unit section of the determinant line, even when $\mathrm{\parallel }\mathrm{\Omega }\mathrm{\parallel }$ is not constant. Use the pair cover from step 4.1 to define $P_{\mathrm{Spin}}:={\widehat{\lambda }}_c^{-1}(P_{SO}{\times }_{X}s(X)),$ where ${\widehat{\lambda }}_c:P_c\to P_{SO}{\times }_X{P}_{L_{\det }}$ is the pair map. Since $\ker \delta \cong \mathrm{Spin}(2n)$ by $s_0\mapsto [s_0,1]$, this is a principal Spin$(2n)$ bundle. Its map to $P_{SO}$ is the two-fold cover $\mathrm{Spin}(2n)\to SO(2n)$. Thus the topological Spin structure exists for every complex manifold and Hermitian metric in the Statement. Construct the connection in the Statement as follows. The smooth local section in [F16], together with the covering charts [F11], gives smooth local inverse branches for $\lambda$; hence $d\lambda :\mathfrak{s}\mathfrak{p}\mathfrak{i}\mathfrak{n}(2n)\to \mathfrak{s}\mathfrak{o}(2n)$ is an isomorphism. In the negative-square Clifford convention, differentiating $s(t)=\exp (t{e}_a{e}_b\mathrm{/}2)$ gives $[e_a{e}_b\mathrm{/}2,e_a]=e_b$ and $[e_a{e}_b\mathrm{/}2,e_b]=-e_a$, so these bivectors have the stated elementary-rotation normalization. The same local smooth Spin branch and local square-root branches make $\mathrm{\Pi }$ a smooth covering; since $\mathrm{\Phi }(A)=(A_{\mathbb{R}},\det A)$ is smooth, its lift $\tau$ from step 3.1 is smooth. For any smooth Spin structure $\widehat{\lambda }:Q\to P_{SO}$, its equivariance implies that ${\omega }_Q=(d\lambda {)}^{-1}{\widehat{\lambda }}^{\ast }{\omega }_{LC}$ reproduces every vertical generator and obeys $R_g^{\ast }{\omega }_Q={\mathrm{Ad}}_{g^{-1}}{\omega }_Q$; thus it is a principal connection. The local spinor formula in the Statement transforms under $q\mapsto qg$ and $f\mapsto \sigma (g{)}^{-1}f$ by the product rule, so it defines a global covariant derivative. Its horizontal paths project to Levi-Civita horizontal paths, and conversely the unique covering lift of a Levi-Civita horizontal path with specified initial spin frame is horizontal. In the additional compact Ricci-flat Kähler case, the Levi-Civita connection preserves $J,h$ and hence $P_U$. Extend this unitary connection to $P_c$ via $\tau$: in a local unitary frame its connection form is $d\tau ({\theta }_U)$, where ${\theta }_U$ is the unitary connection form. Its pair projection is the Levi-Civita connection together with the induced determinant connection. By [F2], $\mathrm{\Omega }$ and the unit determinant section $s$ are parallel, so this connection is tangent to $P_{\mathrm{Spin}}$. Its restriction projects to ${\omega }_{LC}$ and therefore equals the uniquely defined ${\omega }_Q$ above. More explicitly, let $P_{SU}\subset P_U$ consist of frames whose determinant is $s$. It is preserved by parallel transport, and $[u,g]\mapsto [u,g]$ identifies $P_{SU}{\times }_{\tau {\mathrm{\mid }}_{SU(n)}}\mathrm{Spin}(2n)$ with this very $P_{\mathrm{Spin}}$, because any unitary frame can be adjusted to determinant $s$ and the remaining Spin-c coordinate then has determinant one. For a general Hermitian metric the induced Spin connection still exists by the pullback formula, but preservation of this unitary connection or of $s$ is not asserted.

Model the negative-square complex Clifford module on $S={\mathrm{\Lambda }}^{\ast }{\mathbb{C}}^n$ with creation ${ϵ}_j$ and contraction ${\iota }_j$ for each unitary coordinate: put $c(e_{2j-1})={ϵ}_j-{\iota }_j$ and $c(e_{2j})=i({ϵ}_j+{\iota }_j)$. The identities ${\iota }_j{ϵ}_j+{ϵ}_j{\iota }_j=1$ and the cross-coordinate anticommutation give $c(e_a)c(e_b)+c(e_b)c(e_a)=-2{\delta }_{ab}$, so this is the Clifford spin module. For diagonal $A=\mathrm{diag}(e^{i{\theta }_1},\dots ,e^{i{\theta }_n})$, the canonical lift from step 3.1 is represented by $[{\prod }_j(\cos ({\theta }_j\mathrm{/}2)+\sin ({\theta }_j\mathrm{/}2)e_{2j-1}{e}_{2j}),e^{i{\sum }_j{\theta }_j\mathrm{/}2}]$. Since $e_{2j-1}{e}_{2j}=i(2N_j-1)$ with $N_j={ϵ}_j{\iota }_j$, the Spin factor acts on a wedge basis vector indexed by $I$ by $\exp (i{\sum }_j{\theta }_j(1_{j\in I}-1\mathrm{/}2))$; the central weight-one factor makes the total character ${\prod }_{j\in I}{e}^{i{\theta }_j}$. For a diagonal unitary frame change, the induced right-frame action on anti-holomorphic covectors has weights $z_j$, so the calculation is the exterior action on ${\mathrm{\Lambda }}^{0,\ast }{T}_X^{\ast }$. To justify equality for every $A\in U(n)$, let $R(A)={\mathrm{\Lambda }}^{\ast }A$ on this exterior model and let $B(A)$ denote the action of $\tau (A)$. Creation by $v\in {\mathbb{C}}^n$ and its Hermitian-adjoint contraction obey $R(A)ϵ(v)R(A{)}^{-1}=ϵ(Av)$ and $R(A)ϵ(v{)}^{\ast }R(A{)}^{-1}=ϵ(Av{)}^{\ast }$, since $A$ is unitary. These identities and the displayed Clifford formulas give $R(A)c(v)R(A{)}^{-1}=c(A_{\mathbb{R}}v)$ for real $v$; Clifford conjugation gives the same identity for $B(A)$. Therefore $R(A{)}^{-1}B(A)$ commutes with all creation and contraction operators. Their common contraction kernel is the vacuum line $\mathbb{C}1$; commuting with creation then forces this operator to be a scalar $a(A)I$ on every wedge basis vector. Since $R$ and $B$ are representations, these scalars form a character: $a(AB)=a(A)a(B)$, hence $a(UA{U}^{-1})=a(A)$. The diagonal calculation gives $a=1$ on diagonal matrices, and unitary diagonalization now gives $a=1$ everywhere. Thus $B=R$. The unitary representation on anti-holomorphic covectors is ${\bar{A}}^{-\mathsf{T}}=A$, so this proves the asserted associated bundle identification with ${\mathrm{\Lambda }}^{0,\ast }{T}_X^{\ast }$, intertwining Clifford multiplication. The determinant section identifies the reduced Spin spinor bundle with this same complex vector bundle; this identification is parallel when the additional compact Ricci-flat Kähler hypotheses of step 5.1 hold. For $n=1$, $SU(1)=\{1\}$ and $S={\mathrm{\Lambda }}^{0,0}\oplus {\mathrm{\Lambda }}^{0,1}$, so both degrees $0$ and $n$ are exactly the invariant subspace. For $n\ge 2$, restrict to the determinant-one diagonal torus $T=\{\mathrm{diag}(z_1,\dots ,z_n):{\prod }_j{z}_j=1\}$. The wedge vector indexed by $I$ has character ${\chi }_I(z)={\prod }_{j\in I}{z}_j$, which is trivial on $T$ exactly when its indicator vector $(1_{j\in I}{)}_j$ is an integer multiple of $(1,\dots ,1)$; since its entries are $0$ or $1$, this occurs exactly for $I=\mathrm{\varnothing }$ or $I=\{1,\dots ,n\}$. Thus the $T$-fixed subspace is these two lines, and both are fixed by all of $SU(n)$ because their characters are $1$ and $\det A=1$. The exterior-module identification also intertwines connections in the Kähler case: both are induced by the same unitary connection ${\theta }_U$ and the same exterior representation, while step 5.1 identifies its determinant-reduced connection with ${\omega }_Q$. Fix $u_0\in P_{SU}$ over a base point and the compatible spin frame $[u_0,1]$. For any based piecewise-smooth loop $\gamma$, let $u(t)$ be its Levi-Civita parallel unitary frame with $u(0)=u_0$. By [F2], $u(t)$ stays in $P_{SU}$ and $u(1)=u_0{A}_{\gamma }$ for $A_{\gamma }\in {\mathrm{Hol}}_{u_0}(g)\subseteq SU(n)$. The path $[u(t),1]$ is horizontal for the lifted connection of step 5.1, and its endpoint is $[u_0{A}_{\gamma },1]=[u_0,\tau (A_{\gamma })]$. Uniqueness of the horizontal covering lift therefore proves, for all based loops and not only contractible ones, ${\mathrm{Hol}}_{[u_0,1]}({\omega }_Q)=\tau ({\mathrm{Hol}}_{u_0}(g))$. In particular no additional central minus sign occurs for this chosen Spin structure. The associated spinor transport is the representation $\sigma$ of this lifted transport, so both invariant vectors are fixed by full spinor holonomy. Parallel transport of either vector from the base point is path independent: two paths differ by a based loop whose holonomy fixes it. Local parallel transport in coordinate neighborhoods makes the resulting section smooth and covariantly constant. Connectedness thus gives the two global parallel spinors, without a claim for any other Spin structure.

Apply [F3] to the numerable bundle $E_Y$ on $Y$. The homotopy equivalence and naturality [F7] transfer the equality $c_1(E_Y)=c_1({\det }_{\mathbb{C}}{E}_Y)$ through $g$ to $c_1(E)=c_1({\det }_{\mathbb{C}}E)$. Since ${\det }_{\mathbb{C}}E=K_X^{-1}$ and $K_X$ is holomorphically trivial, [F12] gives $c_1(E)=0$. Step 1.3 now yields $w_2(T{X}_{\mathbb{R}})={\rho }_2{c}_1(T_X)=0$. This is consistent with the explicit Spin lift in step 5.1 and is not used as an unproved existence criterion.