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

Assume full AC. A complex Banach space is a complex normed vector space complete in its norm. 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 ${\mathrm{ind}}_{\mathbb{C}}T:={\dim }_{\mathbb{C}}\ker T-{\dim }_{\mathbb{C}}\mathrm{coker}T$. Here $L^2$ uses the bundle metric and Riemannian volume, and $H^1$ is the completion of smooth sections in the norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^2$, equivalent to the finite-chart Sobolev norm. These are complex Hilbert, hence Banach, spaces. A smooth first-order differential operator on this compact base is bounded from $H^1$ to $L^2$. Fredholm and index below refer to this bounded Sobolev realization; the full selfadjoint Dirac operators are separately regarded as unbounded operators on $L^2$ with domain $H^1$.

For the concrete case, set $M=S^4$ with the unit round metric and spin structure induced from the Euclidean ball, take $E=S^{+}$ with its positive-spin Levi-Civita connection, and use the Clifford action and full Dirac operator $D_4=D_{A_{+}}={\sum }_{i}c(e_i){\mathrm{\nabla }}_{e_i}$ on the complete twisted spinor bundle specified in the round-sphere spin-twist result. Let $z=-I_2\in SU(2)$ be the central gauge automorphism. Write $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ and $t$ for the coordinate on its universal cover. Call the circle spin structure periodic (nonbounding) when lifted spinors satisfy $\psi (t+1)=+\psi (t)$, and antiperiodic (bounding) when they satisfy $\psi (t+1)=-\psi (t)$. If the gauge bundle is glued at the ends by a scalar ${ϵ}_g\in \{+1,-1\}$ on $E$ and the spin bundle has multiplier ${ϵ}_s\in \{+1,-1\}$, the total spinor section satisfies $\mathrm{\Psi }(t+1)={ϵ}_s{ϵ}_g\mathrm{\Psi }(t)$. In particular, a trivial gauge gluing with periodic spin structure gives periodic total spinors; gluing gauge fibers by $z$ with an antiperiodic spin clutch also gives periodic total spinors, while the same $z$ clutch with periodic spin clutch gives antiperiodic total spinors.

Let $Y=S^4\times S^1$ have product metric and pulled-back gauge connection. On the pulled-back complex bundle $S\otimes E$, put $c_5=i\mathrm{\Gamma }$ and $D_5=D_4+i\mathrm{\Gamma }{\mathrm{\partial }}_t$, with domain the periodic Sobolev space $H^1(Y;S\otimes E)$, so total spinors obey $\mathrm{\Psi }(t+1)=\mathrm{\Psi }(t)$. There is a parallel antiunitary involution $J$ on this bundle with $J^2=1$ and $J{D}_5=-D_{5}J$; antiunitary means conjugate-linear and isometric. Thus $J$ preserves $\ker D_5$, and $i{D}_5$ restricts to a real skew-adjoint operator on the fixed real Hilbert space $(L^2(Y;S\otimes E){)}^J:=\{\mathrm{\Psi }\in L^2(Y;S\otimes E):J\mathrm{\Psi }=\mathrm{\Psi }\}$. The kernel of this restriction has

$${\dim }_{\mathbb{R}}(\ker D_5{)}^J={\dim }_{\mathbb{C}}\ker D_4=1,{\mathrm{ind}}_2(i{D}_5):={\dim }_{\mathbb{R}}(\ker D_5{)}^J<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2=1.$$

Independently, for the general version let $M$ be a nonempty compact closed oriented Riemannian four-manifold. Let $F_{SO}^{+}(M)$ be its principal $SO(4)$-bundle of positively oriented orthonormal frames, and specify a spin lift $Q\to F_{SO}^{+}(M)$, meaning a principal $\mathrm{Spin}(4)$-bundle with an equivariant double covering for $\mathrm{Spin}(4)\to SO(4)$. Use $\mathrm{Spin}(4)=Sp(1{)}_{+}\times Sp(1{)}_{-}$, where $Sp(1)$ is the unit-quaternion group identified with $SU(2)$ and $(a,b)$ acts on $\mathbb{H}\cong {\mathbb{R}}^4$ by $q\mapsto aq{b}^{-1}$. Let ${\mathrm{\Delta }}^{+}$ and ${\mathrm{\Delta }}^{-}$ be the two defining two-dimensional complex half-spin representations, labeled by the Clifford chirality operator $\mathrm{\Gamma }=-c(e_1)c(e_2)c(e_3)c(e_4)$ so that it acts by $+1$ on ${\mathrm{\Delta }}^{+}$ and $-1$ on ${\mathrm{\Delta }}^{-}$. Define $S^{\pm }:=Q{\times }_{{\mathrm{\Delta }}^{\pm }}{\mathbb{C}}^2$. The Clifford action is skew-Hermitian and satisfies $c(v)c(w)+c(w)c(v)=-2g(v,w)$. An $SU(2)$ structure on a Hermitian rank-two bundle is a specified reduction of its unitary frame bundle to $SU(2)$. Let $E\to M$ be such a smooth bundle and let $A$ be a supplied unitary connection compatible with this reduction, meaning that its local connection matrices in $SU(2)$ frames lie in $\mathfrak{s}\mathfrak{u}(2)$. Let $\mathrm{\nabla }$ be the tensor-product connection induced by the lifted Levi-Civita connection and $A$. Define the full Dirac operator and its positive chiral restriction by

$D_A={\sum }_{i=1}^{4}c(e_i){\mathrm{\nabla }}_{e_i}:H^1(M;S\otimes E)⟶L^2(M;S\otimes E),S:=S^{+}\oplus S^{-};D_A^{+}:=D_A{\mathrm{\mid }}_{H^1(M;S^{+}\otimes E)}:H^1(M;S^{+}\otimes E)⟶L^2(M;S^{-}\otimes E),$

where the displayed local expression is independent of the chosen oriented orthonormal frame. Define $\mathrm{coker}{D}_A^{+}:=L^2(M;S^{-}\otimes E)\mathrm{/}\mathrm{ran}{D}_A^{+}$. Assume $D_A^{+}$ is Fredholm; put $n_{+}={\dim }_{\mathbb{C}}\ker D_A^{+}$, $n_{-}={\dim }_{\mathbb{C}}\mathrm{coker}{D}_A^{+}$, and ${\mathrm{ind}}_{\mathbb{C}}{D}_A^{+}:=n_{+}-n_{-}$. This hypothesis assigns no index value to an arbitrary bundle.

For either fixed four-dimensional background, set $H_{+}=H^1(M;S^{+}\otimes E)$ and $H_{-}=L^2(M;S^{-}\otimes E)$, and write $\mathfrak{F}=\mathrm{Fred}(H_{+},H_{-})$ for the norm-topological space of Fredholm operators from $H_{+}$ to $H_{-}$. For a finite-dimensional complex vector space $V$, define ${\det }_{\mathbb{C}}V={\bigwedge }^{{\dim }_{\mathbb{C}}V}V$ and ${\det }_{\mathbb{C}}0=\mathbb{C}$. Use the inverse determinant-line bundle ${\mathrm{Det}}^{-1}\to \mathfrak{F}$ whose fiber at $T\in \mathfrak{F}$ is

${\mathcal{L}}_T=({\det }_{\mathbb{C}}\ker T{)}^{\ast }\otimes {\det }_{\mathbb{C}}\mathrm{coker}T,\mathrm{coker}T:=H_{-}\mathrm{/}\mathrm{ran}T\mathrm{.}$

Suppose a connection ${\mathrm{\nabla }}^{\mathrm{Det}}$ on this line bundle is supplied. The central gauge automorphism $z$ fixes $A$ and induces multiplication by $-1$ on both chiral section spaces, hence on $\ker D_A^{+}$ and $\mathrm{coker}{D}_A^{+}$. Its induced fiber map $z_{\ast }:{\mathcal{L}}_{D_A^{+}}\to {\mathcal{L}}_{D_A^{+}}$ is the dual top-exterior-power action on $\ker D_A^{+}$ tensored with the top-exterior-power action on $\mathrm{coker}{D}_A^{+}$; thus it acts by $(-1{)}^{n_{+}+n_{-}}$. The constant connection path $A_t=A$ gives the constant operator path $\gamma (t)=D_A^{+}$ in $\mathfrak{F}$, so parallel transport ${\mathrm{PT}}_{\gamma }$ for any supplied line-bundle connection is the identity. For the loop closed by the endpoint gauge identification, the automorphism $z_{\ast }^{-1}\circ {\mathrm{PT}}_{\gamma }$ is therefore $(-1{)}^{n_{+}+n_{-}}=(-1{)}^{{\mathrm{ind}}_{\mathbb{C}}{D}_A^{+}}$. On $M\times S^1$, the product spin and compatible $SU(2)$ structures give a parallel antiunitary involution $J$ of the same type for $D_5=D_A+i\mathrm{\Gamma }{\mathrm{\partial }}_t$ with periodic domain $H^1$. The real kernel of its restriction to the $J$-fixed space has dimension $n_{+}+n_{-}$, so its parity is ${\mathrm{ind}}_{\mathbb{C}}{D}_A^{+}<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$. This calculation does not assign an index value to an arbitrary bundle.

In particular, for the explicit round background, the fiber ${\mathcal{L}}_{D_A^{+}}$ has action $-1$, since its chiral kernel dimension is $1$ and cokernel dimension is $0$. The $N$-fold tensor power has action $(-1{)}^N$. For the fixed product background, antiperiodic total spinor boundary conditions give no product zero mode because a zero mode must be constant in $t$. This distinguishes the tested fermion-parity mapping-torus convention from an automatic eta-holonomy formula. No general spectral-flow or Atiyah–Singer theorem is asserted.

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

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Fredholm operator cokernel and index