Determinant-line holonomy for a global anomaly test

Assume Countable Choice: every countable family of nonempty sets admits a choice function. Fix an even integer $d\ge 2$ and a compact smooth oriented Riemannian $d$-manifold $M$ without boundary. Use the negative-square real Clifford algebra ${\mathrm{C}\mathrm{l}}_{0,d}$, generated by ${\mathbb{R}}^d$ with $vw+wv=-2⟨v,w⟩$. The group $\mathrm{Spin}(d)$ is its subgroup generated by even products of unit vectors; its standard double cover acts by $v\mapsto sv{s}^{-1}$ on ${\mathbb{R}}^d$ and has kernel $\{1,-1\}$. These are the Spin and covering conventions of the given-Spin-structure supplier. A smooth principal right $G$-bundle has a free, fiberwise transitive smooth right action and equivariant local trivializations by diffeomorphisms to $U\times G$; $G$ is a finite-dimensional real Lie group. A connection is a supplied smooth right-invariant horizontal complement $H$ to $\ker d\pi$ in the tangent bundle of its total space. Let $F_{\mathrm{S}\mathrm{O}}^{+}(M)$ be its principal right $\mathrm{S}\mathrm{O}(d)$-bundle of positively oriented orthonormal frames. A specified spin structure is a principal right $\mathrm{Spin}(d)$-bundle $Q\to M$ with a smooth equivariant double-cover map $\lambda :Q\to F_{\mathrm{S}\mathrm{O}}^{+}(M)$ covering the standard double covering $\rho :\mathrm{Spin}(d)\to \mathrm{S}\mathrm{O}(d)$. The Levi-Civita connection lifts to a connection on $Q$. Fix an irreducible complex Clifford representation on a Hermitian space $\mathrm{\Delta }$ of dimension $2^{d\mathrm{/}2}$, with skew-Hermitian vector action $c_{\mathrm{\Delta }}$. Put $\mathrm{\Gamma }=i^{d\mathrm{/}2}{c}_{\mathrm{\Delta }}(e_1)\cdots c_{\mathrm{\Delta }}(e_d)$ in a positively oriented orthonormal basis and ${\mathrm{\Delta }}^{\pm }=\ker (\mathrm{\Gamma }\mp I)$. The restrictions of the even Clifford action to these eigenspaces are the half-spin representations ${\rho }_{\pm }$ of $\mathrm{Spin}(d)$. Define $S^{\pm }=Q{\times }_{{\rho }_{\pm }}{\mathrm{\Delta }}^{\pm }$. Let $P\to M$ be a smooth principal right $G$-bundle with connection $A$, let $V_R$ be a finite-dimensional Hermitian complex vector space, and let $R:G\to U(V_R)$ be a smooth group homomorphism. Set $E_R=P{\times }_R{V}_R=(P\times V_R)\mathrm{/}G$ for the right action $(p,v)g=(pg,R(g{)}^{-1}v)$, so $[pg,v]=[p,R(g)v]$. The associated-bundle and covariant-derivative supplier supplies its smooth complex bundle charts (using the underlying real representation for the real associated-bundle definition). In a smooth local section $s$ of $P$, write $A_s=s^{\ast }\omega$, where $\omega$ is the principal connection form of $H$, and define the induced connection by ${\mathrm{\nabla }}^A\psi =d\psi +(dR{)}_e(A_s)\psi$. The supplier proves that this formula patches with the stated quotient convention under Countable Choice. Since $R$ is unitary, the fiber metric $⟨[p,v],[p,w]⟩=⟨v,w{⟩}_{V_R}$ is well-defined and this connection preserves it. Let $c:TM\to {\mathrm{End}}_{\mathbb{C}}(S^{+}\oplus S^{-})$ be the skew-Hermitian Clifford action, with $c(v)c(w)+c(w)c(v)=-2g(v,w)$ for $v,w\in T_{x}M$, and let $\mathrm{\nabla }$ be the tensor-product connection induced by the lifted Levi-Civita connection and $A$. The chiral Dirac operator is the extension to the stated Sobolev spaces of the first-order expression in any positively oriented local orthonormal frame $(e_i)$; this expression is independent of that frame.

$D_{A,R}^{+}:H^1(M;S^{+}\otimes E_R)⟶L^2(M;S^{-}\otimes E_R),D_{A,R}^{+}\psi ={\sum }_{i=1}^{d}c(e_i){\mathrm{\nabla }}_{e_i}\psi (\psi \in C^{\mathrm{\infty }}(M;S^{+}\otimes E_R)),$

where $L^2$ is the completion for the bundle metric and Riemannian volume, and $H^1$ is the completion of smooth sections in the norm $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{H^1}^2=\mathrm{\parallel }\psi {\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\nabla }\psi {\mathrm{\parallel }}_2^2$. These are complex Hilbert spaces. The smooth first-order expression is bounded from $H^1$ to $L^2$. Assume it is Fredholm in the bounded Banach-space sense: its range is closed and its kernel and quotient cokernel $L^2(M;S^{-}\otimes E_R)\mathrm{/}\mathrm{ran}{D}_{A,R}^{+}$ are finite dimensional. Its index is their complex dimension difference. Its inverse determinant line is

${\mathcal{L}}_A={({\det }_{\mathbb{C}}\ker D_{A,R}^{+})}^{\ast }\otimes {\det }_{\mathbb{C}}\mathrm{coker}{D}_{A,R}^{+},$

Here ${\det }_{\mathbb{C}}W={\bigwedge }_{\mathbb{C}}^{{\dim }_{\mathbb{C}}W}W$, including $\det 0=\mathbb{C}$. Our phrase inverse determinant line means the dual of ${\mathrm{\Delta }}_A=\det \ker D^{+}\otimes (\det \mathrm{coker}{D}^{+}{)}^{\ast }$; the displayed $\mathcal{L}$ is the line called $\mathrm{D}\mathrm{e}\mathrm{t}$, not ${\mathrm{D}\mathrm{e}\mathrm{t}}^{-1}$, in Dai–Freed §2. This explicit formula fixes the convention throughout.

The named regularized geometry means the following supplied admissible analytic data, rather than an arbitrary Hermitian line relabeled by a name. Test smooth dependence on each finite-dimensional smooth parameter manifold $T$ of connections on the fixed $M,P,R$. Write ${\mathcal{H}}^{\pm }=T\times L^2(M;S^{\pm }\otimes E_R)$ with common first-Sobolev domains. Supply unitary parameter connections ${\tilde{\mathrm{\nabla }}}^{\pm }$ on these Hilbert bundles, preserving smooth sections and the Sobolev domains; for the fixed product identification one may take the ordinary derivative in $T$. Fix a positive inverse-length scale $\mu$ and put $\mathcal{D}=D^{+}\mathrm{/}\mu$, with Hilbert adjoint ${\mathcal{D}}^{\ast }$. These choices and the local-counterterm convention are fixed before phases are compared.

Require $Q^{+}={\mathcal{D}}^{\ast }\mathcal{D}$ and $Q^{-}=\mathcal{D}{\mathcal{D}}^{\ast }$ to have the supplied nonnegative self-adjoint realizations with discrete spectrum, finite multiplicities and smooth eigensections, and the same positive spectrum with multiplicity. A spectral cut is a number $a>0$ not in either spectrum on an open parameter patch $U_a\subseteq T$. The orthogonal projections onto eigenvalues $\lambda <a$, including zero modes, must form smooth finite-rank bundles $V_a^{\pm }\subset {\mathcal{H}}^{\pm }$; their complements are denoted ${\mathcal{H}}_{>a}^{\pm }$. The finite cut line is $L_a=(\det V_a^{+}{)}^{\ast }\otimes \det V_a^{-}$. For $a<b$, the operator $\mathcal{D}$ is invertible between the eigenbundles $V_{(a,b)}^{+}$ and $V_{(a,b)}^{-}$. Glue $L_a$ to $L_b$ by tensoring with its top exterior power $\det {\mathcal{D}}_{(a,b)}$. Finite-dimensional cancellation of these invertible modes identifies each fiber with the displayed kernel/cokernel line, even where kernel dimensions jump. Tensor factors are kept in increasing cut order; parity is $\mathrm{ind}{D}^{+}<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$ if graded lines are used.

For each cut, define the scalar function ${\zeta }_a(z)={\sum }_{\lambda >a}{\lambda }^{-z}$ using the positive $Q^{+}$ eigenvalues with multiplicity and real $\log \lambda$. Admissibility requires convergence on an initial right half-plane, a specified meromorphic continuation to a connected domain reaching $z=0$, regularity there, and smooth parameter dependence of ${\zeta }_a^{\mathrm{\prime }}(0)$. The Quillen squared norm is $\mathrm{\parallel }\mathrm{\ell }{\mathrm{\parallel }}_Q^2=\exp (-{\zeta }_a^{\mathrm{\prime }}(0))\mathrm{\parallel }{\mathrm{\ell }}_a{\mathrm{\parallel }}_{L^2}^2$. Finite bands satisfy ${\zeta }_a-{\zeta }_b={\sum }_{a<\lambda <b}{\lambda }^{-z}$ and $\mathrm{\parallel }{\mathrm{\ell }}_b{\mathrm{\parallel }}_{L^2}^2=({\prod }_{a<\lambda <b}\lambda )\mathrm{\parallel }{\mathrm{\ell }}_a{\mathrm{\parallel }}_{L^2}^2$, which give cut independence of this norm. No existence of that continuation for an arbitrary Fredholm family is inferred.

Project ${\tilde{\mathrm{\nabla }}}^{\pm }$ to the cut subbundles and their complements. They induce a connection ${\mathrm{\nabla }}^a$ on $L_a$. On the invertible complement let $\tilde{\mathrm{\nabla }}{\mathcal{D}}_{>a}={\tilde{\mathrm{\nabla }}}_{>a}^{-}{\mathcal{D}}_{>a}-{\mathcal{D}}_{>a}{\tilde{\mathrm{\nabla }}}_{>a}^{+}$. For $t>0$ define the complex one-form $h_a(t)={\mathrm{Tr}}_{{\mathcal{H}}_{>a}^{+}}({\mathcal{D}}_{>a}^{-1}(\tilde{\mathrm{\nabla }}{\mathcal{D}}_{>a})e^{-t{Q}_{>a}^{+}})$; the trace is the absolutely convergent sum of diagonal matrix coefficients in an orthonormal eigenbasis, independent of choices within each eigenspace. Require this convergence and the smooth coefficient expansion $h_a(t)={\sum }_{j=-d\mathrm{/}2}^{-1}{b}_j{t}^j+b_0+b_{\log }\log t+o(1)$ as dimensionless $t\downarrow 0$. The finite-part convention is $\mathrm{FP}{h}_a=b_0+{\gamma }_1{b}_{\log }$, where ${\gamma }_1={\int }_0^{\mathrm{\infty }}{e}^{-r}\log rdr={\mathrm{\Gamma }}^{\mathrm{\prime }}(1)$; the equality names the usual gamma-function derivative and fixes the log-term subtraction. The Bismut–Freed connection in this convention is the local formula ${\mathrm{\nabla }}^{\mathrm{B}\mathrm{F}}={\mathrm{\nabla }}^a+\mathrm{FP}{h}_a$, as in Dai–Freed (3.6)–(3.7).

Admissibility further requires that these forms are smooth, intertwine the specified cut transitions, and are compatible with the Quillen norm: $d\mathrm{\parallel }s{\mathrm{\parallel }}_Q^2=⟨{\mathrm{\nabla }}^{\mathrm{B}\mathrm{F}}s,s{⟩}_Q+⟨s,{\mathrm{\nabla }}^{\mathrm{B}\mathrm{F}}s{⟩}_Q$ for every local section $s$. Thus in a unit frame the connection form is purely imaginary. Gauge identifications must intertwine all cut data, preserve this norm and preserve the connection after the corresponding parameter pullback. These are explicit mathematical conditions on the supplied regularization. Cut gluing reduces on a finite band to the determinant derivative $(\det \mathcal{D}{)}^{-1}\mathrm{\nabla }(\det \mathcal{D})=\mathrm{Tr}({\mathcal{D}}^{-1}\tilde{\mathrm{\nabla }}\mathcal{D})$ with the induced finite-dimensional connections; analytic existence, smooth heat expansions and naturality for all geometric families are not asserted here. In particular a Hermitian line alone does not supply this connection. Parallel transport means the solution of ${\mathrm{\nabla }}_{{\dot{A}}_t}^{\mathrm{B}\mathrm{F}}s(t)=0$ with a specified initial value; in a unit frame with connection $i\beta$, it is multiplication by $\exp (-i\int \beta )$.

When the supplied real or pseudoreal fermion structure instead yields a skew-Fredholm quadratic form, include its actual Pfaffian data. Concretely, on a common dense Sobolev domain $W^1$ of a complex Hilbert space $W^0$, supply a bounded linear Fredholm map $B:W^1\to (W^0{)}^{\ast }$ such that $b(w,z)=(Bw)(z)$ is complex bilinear and skew on $W^1\times W^1$, with $\mathrm{coker}B\cong (\ker B{)}^{\ast }$ induced by the pairing. Supply finite spectral subspaces $W_a$ of $B^{\ast }B$ containing the kernel, with orthogonal cut bands on which $b$ is nondegenerate and no cross terms between bands. For a finite $2m$-dimensional space $W$ the Pfaffian is $\mathrm{pf}(b)=b^{\wedge m}\mathrm{/}m!\in \det W^{\ast }$, interpreting $b\in {\bigwedge }^2{W}^{\ast }$; it is zero in odd dimension. Glue $P_a=\det W_a^{\ast }$ by the nonzero Pfaffians of the even-dimensional invertible cut bands. The resulting graded line $P$ has parity $\dim \ker B<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$ and the specified natural square identification $P^{\otimes 2}\cong L_B$, where $L_B$ is the displayed determinant convention applied to $B$. Its squared Quillen norm has high-mode factor $\exp (-{\zeta }_{B,a}^{\mathrm{\prime }}(0)\mathrm{/}2)$, with ${\zeta }_{B,a}$ defined from $B^{\ast }B$ exactly as above. Its unitary connection is the one whose tensor square, under that identification, is the supplied Bismut–Freed connection of $L_B$; locally its connection form is half that determinant form in square-compatible frames. Smooth gluing, the square identification, spectral continuations, metric compatibility and gauge naturality are required Pfaffian admissibility data, not an arbitrary choice of a square root or an existence assertion for every real/pseudoreal theory.

Let $\mathcal{A}(P)$ be the affine space of smooth connections on $P$, and let $\mathcal{G}(P)$ be the group of smooth bundle automorphisms covering the identity on $M$. A gauge arrow $u:A\to A^u$ induces a unitary map $u_{\ast }:{\mathcal{L}}_A\to {\mathcal{L}}_{A^u}$ by its action on kernel and cokernel. A closed loop in background/gauge-field space consists of a smooth path $A_t$ in $\mathcal{A}(P)$, $0\le t\le 1$, and an endpoint gauge arrow $u:A_0\to A_1$. Let ${\mathrm{P}\mathrm{T}}_{A_t}:{\mathcal{L}}_{A_0}\to {\mathcal{L}}_{A_1}$ be parallel transport and $u_{\ast }:{\mathcal{L}}_{A_0}\to {\mathcal{L}}_{A_1}$ the endpoint gauge map. The closed-loop automorphism is $u_{\ast }^{-1}\circ {\mathrm{P}\mathrm{T}}_{A_t}$ on ${\mathcal{L}}_{A_0}$. Its scalar is the holonomy ${\mathrm{Hol}}_{\gamma }(\mathcal{L})\in U(1)$. Fix the regularization and local-counterterm convention before comparing this loop phase with the infinitesimal local anomaly; the holonomy itself is additional global loop data.

Countable Choice

Given spin structures and Clifford covering convention

Smooth associated bundles and induced covariant derivative

Bounded Fredholm operator, cokernel and index

Dai and Freed, §§1–2 and §5

Freed, Lecture 11, §§11.1–11.2