A global SU(2) determinant-line phase and its mod-two inflow cancellation

Use natural units $c=\mathrm{\hslash }=1$. Assume full Axiom of Choice for the analytic product-kernel and mod-two-index calculation on every index-one background below, as well as for the concrete round-sphere benchmark. The finite-dimensional inverse-determinant sign calculation itself needs no additional choice principle. Let $M=S^4$ have its boundary orientation, a specified smooth Riemannian metric, and the standard spin structure induced from the Euclidean $5$-ball. Here a spin structure means a principal right $\mathrm{Spin}(4)$-bundle $Q\to M$ with a smooth equivariant double-cover map to the principal $\mathrm{S}\mathrm{O}(4)$-bundle $F_{\mathrm{S}\mathrm{O}}^{+}(M)$ of positively oriented orthonormal frames, covering $\mathrm{Spin}(4)\to \mathrm{S}\mathrm{O}(4)$. Let ${\mathrm{\Delta }}^{\pm }$ be the complex half-spin representations with representation maps ${\rho }_{\pm }$, and let $S^{\pm }=Q{\times }_{{\rho }_{\pm }}{\mathrm{\Delta }}^{\pm }$ be their associated bundles; the Levi-Civita connection lifts to $Q$. Let $E\to M$ be a smooth rank-two Hermitian bundle with a specified $SU(2)$ reduction of its unitary frame bundle. Let $A$ be a smooth unitary connection compatible with that reduction, meaning that in each $SU(2)$ frame its local connection matrix is traceless anti-Hermitian, so is valued in $\mathfrak{s}\mathfrak{u}(2)$. Treat $SU(2)$ as a nondynamical background global symmetry and consider the chiral Dirac family for one massless left-handed Weyl doublet in the fundamental representation. Use the Clifford convention $c(v)c(w)+c(w)c(v)=-2g(v,w)$ for $v,w\in T_{x}M$, and the tensor-product spin and gauge connection in $D_A^{+}={\sum }_{i}c(e_i){\mathrm{\nabla }}_{e_i}:H^1(M;S^{+}\otimes E)\to L^2(M;S^{-}\otimes E)$. Here $(e_i)$ is any positively oriented local orthonormal frame and the expression is frame-independent. The $H^1$ norm is $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^2$ and $L^2$ uses the bundle metric and Riemannian volume; these are complex Hilbert spaces, and the smooth first-order expression defines a bounded map between them. For complex Banach spaces $X,Y$, a bounded complex-linear map $T:X\to Y$ is Fredholm when its range is closed and both $\ker T$ and $\mathrm{coker}T:=Y\mathrm{/}\mathrm{ran}T$ are finite dimensional; its index is ${\mathrm{ind}}_{\mathbb{C}}T:={\dim }_{\mathbb{C}}\ker T-{\dim }_{\mathbb{C}}\mathrm{coker}T$. Put $n_{+}={\dim }_{\mathbb{C}}\ker D_A^{+}$ and $n_{-}={\dim }_{\mathbb{C}}\mathrm{coker}{D}_A^{+}$. Assume this bounded chiral realization is Fredholm with $n_{+}-n_{-}={\mathrm{ind}}_{\mathbb{C}}{D}_A^{+}=1$; this is the index-one-sector hypothesis. The chosen boundary line in this theorem is the inverse determinant line ${\mathcal{L}}_A=({\det }_{\mathbb{C}}\ker D_A^{+}{)}^{\ast }\otimes {\det }_{\mathbb{C}}\mathrm{coker}{D}_A^{+}$, equipped with the supplied Quillen metric and Bismut--Freed unitary connection. This explicitly selects the complex chiral determinant line; no identification with the fermionic Pfaffian line of a pseudoreal path-integral presentation is asserted. For $N$ independent identical doublets use ${\mathcal{L}}_A^{\otimes N}$. For a proved concrete instance, assume full AC and take the unit round metric, $E=S^{+}$ and the positive-spin Levi-Civita connection: its complex chiral kernel has dimension $1$, its cokernel dimension $0$, and $⟨c_2(E),[S^4]⟩=-1$ in the convention $c(E)=\det (I-F_A\mathrm{/}(2\pi i))$. The conditional conclusion below still applies to every stipulated index-one background; this concrete instance supplies that index by proof rather than as a physical postulate.

Let ${\mathcal{R}}^{TM}$ be the curvature $2$-form of the Levi-Civita connection, viewed as a skew-symmetric matrix of real $2$-forms, and fix the Chern--Weil convention

$p_1(TM)=-\frac{1}{8{\pi }^2}{\mathrm{tr}}_{\mathbb{R}}({\mathcal{R}}^{TM}\wedge {\mathcal{R}}^{TM}),\widehat{A}(TM)={\det }_{\mathbb{C}}^{1\mathrm{/}2}\text{\hspace{0.17em}⁣}\left(\frac{i{\mathcal{R}}^{TM}\mathrm{/}(4\pi )}{\sinh (i{\mathcal{R}}^{TM}\mathrm{/}(4\pi ))}\right),$

where the determinant and square root are formal power series with constant term $1$, so ${\widehat{A}}_4=-p_1(TM)\mathrm{/}24$. With $F_A$ anti-Hermitian, set $\mathrm{ch}(E)={\mathrm{tr}}_{\mathbf{2}}\exp (-F_A\mathrm{/}(2\pi i))$ and, for $k\ge 0$, ${\mathrm{ch}}_k(E)=\frac{1}{k!}{\mathrm{tr}}_{\mathbf{2}}((-F_A\mathrm{/}(2\pi i){)}^k)$. The adopted degree-six index-density polynomial is $I_6=[\widehat{A}(TM)\mathrm{ch}(E){]}_6={\mathrm{ch}}_3(E)-p_1(TM){\mathrm{ch}}_1(E)\mathrm{/}24$. This is the formal Chern--Weil polynomial used in this theorem; no descent relation between it and an infinitesimal BRST variation is assumed here.

Let $z=-I\in SU(2)$ act on the fermion fibers and form the corresponding order-two loop ${\gamma }_z$ of the background/gauge-field groupoid. Since $z$ acts as $-1$ in the fundamental representation, its action on this fermion is the same twist as $(-1{)}^F$; use the fermion-parity mapping-torus convention represented by $Y_z=S^4\times S^1$ with product metric $g_{S^4}+d{t}^2$, product orientation $\mathrm{or}(S^4)\wedge dt$, and gauge connection pulled back from $S^4$, and periodic total spinors around $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$, meaning $\mathrm{\Psi }(t+1)=\mathrm{\Psi }(t)$. Equivalently, the central $z$ gauge clutch combined with an antiperiodic spin clutch gives this periodic total fermion boundary condition. A periodic spin clutch combined with the same $z$ gauge clutch instead gives antiperiodic total fermions; the two conventions must not be conflated in an eta-holonomy formula.

For this test take the candidate bulk line to be the dual of the boundary line with its specified unitary connection. Adopt the five-dimensional mod-two-index response $Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(Y_z,A)=(-1{)}^{N{\mathrm{ind}}_2{D}_{Y_z}}$, where $S=S^{+}\oplus S^{-}$, $D_4={\sum }_{i}c(e_i){\mathrm{\nabla }}_{e_i}$ is the full Dirac operator on $H^1(S^4;S\otimes E)$, $\mathrm{\Gamma }$ acts by $+1$ on $S^{+}$ and $-1$ on $S^{-}$, and $D_{Y_z}=D_4+i\mathrm{\Gamma }{\mathrm{\partial }}_t$ on periodic $H^1$ sections over $Y_z$. The pseudoreal $\mathrm{Spin}(5)$ and $SU(2)$ structures give a specified parallel antiunitary involution $J$ with $J^2=1$ preserving this kernel; define ${\mathrm{ind}}_2{D}_{Y_z}={\dim }_{\mathbb{R}}((\ker D_{Y_z}{)}^J)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$, and stipulate that this closed response is the bulk-line holonomy ${\mathrm{Hol}}_{{\gamma }_z}({\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}})$ in the mapping-torus orientation and endpoint-identification convention above. For $N\ge 1$ copies the adopted polynomial $I_6$ vanishes and the boundary holonomy on this loop is $(-1{)}^N$; the bulk phase is also $(-1{)}^N$, so the combined boundary/bulk holonomy is $+1$. For the finite normalization check take $K$ to be a single-vertex ordered simplicial complex and the ordinary field groupoid $\mathcal{F}(K)=B{C}_2$, with one object and automorphism group $C_2=\{1,z\}$. The boundary line alone has no invariant scalar gauging weight when $N$ is odd, while its tensor product with the assigned bulk line, which is its dual has the normalized finite groupoid sum $1\mathrm{/}2$. Even $N$ makes the boundary character trivial for this tested loop; this statement does not exclude other global anomalies or prove vanishing of the infinitesimal local anomaly outside the stated polynomial calculation.

An explicit round-S4 spin twist has charge one and chiral index one

Periodic real product kernel and central action on the inverse determinant line

The Axiom of Choice

Fredholm operator cokernel and index

Wang, Wen, and Witten, §2.2

Dai and Freed, §5

Freed, Lecture 11, §§11.1–11.2