Relative boundary theory and invertible anomaly inflow

Assume the Countable Choice convention of the global-holonomy definition. For $n\ge 1$ use ${\mathrm{C}\mathrm{l}}_{0,n}$ with $vw+wv=-2⟨v,w⟩$, and define $\mathrm{Spin}(n)$ as the subgroup of even products of unit vectors. Its standard cover ${\rho }_n(s)(v)=sv{s}^{-1}$ is onto $SO(n)$ with kernel $\{1,-1\}$; in particular $\mathrm{Spin}(1)=\{1,-1\}$. Principal bundles and connections have the smooth right-action and horizontal-complement meaning stated in the global-holonomy definition. Fix an integer $d\ge 1$. Let $Y$ be a compact oriented smooth Riemannian $(d+1)$-manifold with boundary $M=\mathrm{\partial }Y$, oriented so that the outward unit normal followed by a positively oriented frame of $M$ gives the orientation of $Y$. Let $F_{\mathrm{S}\mathrm{O}}^{+}(Y)$ be the principal bundle of positively oriented orthonormal frames and choose a principal right $\mathrm{Spin}(d+1)$-bundle $Q_Y$ with a smooth equivariant double-cover map ${\lambda }_Y:Q_Y\to F_{\mathrm{S}\mathrm{O}}^{+}(Y)$ over $\mathrm{Spin}(d+1)\to \mathrm{S}\mathrm{O}(d+1)$. This is the spin structure on $Y$; its connection is the lift of the Levi-Civita connection. On $M$, let $F_{\mathrm{S}\mathrm{O}}^{+}(TM)$ denote the positively oriented orthonormal frame bundle for the induced boundary metric. The frames $({\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}},e_1,\dots ,e_d)$ in $F_{\mathrm{S}\mathrm{O}}^{+}(TY){\mathrm{\mid }}_M$ form a reduction identified with $F_{\mathrm{S}\mathrm{O}}^{+}(TM)$, where ${\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the outward unit normal. The preimage of this reduction under ${\lambda }_Y$, with structure group $\mathrm{ȷ}(\mathrm{Spin}(d))={\rho }_{d+1}^{-1}(\mathrm{diag}(1,SO(d)))$, is a principal $\mathrm{Spin}(d)$-bundle $Q_M\to F_{\mathrm{S}\mathrm{O}}^{+}(TM)$ double-covering that frame bundle; this is the induced boundary spin structure. The embedding $\mathrm{ȷ}$ is induced by ${\mathbb{R}}^d\to {\mathbb{R}}^{d+1}$, $(x_1,\dots ,x_d)\mapsto (0,x_1,\dots ,x_d)$, on Clifford algebras and their even unit-vector products; it intertwines ${\rho }_d$ with the block embedding $h\mapsto \mathrm{diag}(1,h)$. The preimage equality is verified below. Thus the subgroup fixes the first vector through its orthogonal action, rather than by the spinor representation. A background $b$ consists of these metrics and spin structures, together with any specified principal-bundle connections and associated fields on $Y$; restriction gives the boundary background $b{\mathrm{\mid }}_M$.

A $d$-dimensional boundary theory is relative to an invertible bulk theory when its partition value is a vector $Z_{\mathrm{\partial }}(M,b{\mathrm{\mid }}_M)$ in a specified one-dimensional Hermitian line ${\mathcal{L}}_{\mathrm{\partial }}(M,b{\mathrm{\mid }}_M)$, and the bulk assigns a one-dimensional Hermitian line ${\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(M,b{\mathrm{\mid }}_M)$ together with a specified unitary isomorphism ${\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(M,b{\mathrm{\mid }}_M)\cong {\mathcal{L}}_{\mathrm{\partial }}(M,b{\mathrm{\mid }}_M{)}^{\ast }$ and a vector $Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(Y,b)$ in that bulk line. The canonical evaluation

$\mathrm{e}\mathrm{v}:{\mathcal{L}}_{\mathrm{\partial }}(M,b{\mathrm{\mid }}_M)\otimes {\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(M,b{\mathrm{\mid }}_M)⟶\mathbb{C}$

then defines the combined scalar amplitude. The term invertible bulk theory requires the following supplied functorial data, not only a collection of lines and vectors.

The selected model specifies an admissible geometric bordism category ${\mathsf{B}}_{d+1}$. Its objects $X=(M,b_M)$ are closed compact oriented smooth spin $d$-manifolds with admitted metrics, principal-bundle connections and associated background fields, including the empty object $\mathrm{\varnothing }$. A nonidentity bordism $W:X_0\to X_1$ is represented by a compact oriented smooth spin $(d+1)$-manifold with boundary identified with ${\bar{M}}_0\bigsqcup M_1$, where the bar reverses orientation with the induced spin convention, and with compatible background fields restricting to $b_0,b_1$ under these identifications. Representatives carry specified smooth product collars of their boundary components; all metric and field data are of product form on these collars. Bordisms are identified by orientation-, spin- and background-preserving diffeomorphisms respecting the boundary identifications and collar germs. Admissibility is closed under disjoint union and gluing of matching outgoing and incoming backgrounds. Composition $W_2\circ W_1$ glues these matching collars and fields; the model specifies the collar/germ convention so that iterated gluing is associative on morphisms. Identity arrows are formal zero-thickness product cylinders, with $W\circ 1_{X_0}=W=1_{X_1}\circ W$; positive-thickness metric cylinders remain actual bordisms and are not identified with these identities merely because they are cylinders. Background-preserving spin and gauge isomorphisms are also admitted arrows, compose by their actual isomorphism law, and act on boundary identifications when composed with bordisms. Disjoint union supplies the symmetric monoidal product, with unit $\mathrm{\varnothing }$ and the usual component-permutation isomorphisms. These admissibility, collar and isomorphism conventions are required model data satisfying the stated category and tensor laws, not an existence assertion for arbitrary proposed fields. The particular filling $Y$ used above must be an admitted morphism $\mathrm{\varnothing }\to (M,b{\mathrm{\mid }}_M)$ with its specified collar-compatible background. Neither equivalence of bordisms nor admissibility forgets their metric or connection data.

Let ${\mathsf{L}\mathsf{i}\mathsf{n}\mathsf{e}}_{\mathbb{C}}^{\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{m}}$ have one-dimensional complex Hermitian spaces as objects and complex-linear isomorphisms as morphisms, with ordinary composition, Hermitian tensor product, unit $\mathbb{C}$, and the tensor flip as symmetry. An invertible bulk assignment is a strong symmetric monoidal functor $Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}:{\mathsf{B}}_{d+1}\to {\mathsf{L}\mathsf{i}\mathsf{n}\mathsf{e}}_{\mathbb{C}}^{\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{m}}$. Explicitly, it assigns the line ${\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(X)$ to $X$, an isomorphism

$Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W):{\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(X_0)⟶{\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(X_1)$

to $W:X_0\to X_1$, and specified unitary tensor and unit identifications $J_{X,X^{\mathrm{\prime }}}:{\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(X)\otimes {\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(X^{\mathrm{\prime }})\to {\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(X\bigsqcup X^{\mathrm{\prime }})$ and $J_0:\mathbb{C}\to {\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(\mathrm{\varnothing })$. Require

$Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(1_X)=1_{{\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(X)},Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W_2\circ W_1)=Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W_2)\circ Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W_1),$

and $Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W\bigsqcup W^{\mathrm{\prime }})\circ J_{X_0,X_0^{\mathrm{\prime }}}=J_{X_1,X_1^{\mathrm{\prime }}}\circ (Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W)\otimes Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W^{\mathrm{\prime }}))$. The $J$ maps satisfy the associativity and left/right unit coherence equations for a strong monoidal functor, and the symmetry equation $Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}({\mathrm{s}\mathrm{w}\mathrm{a}\mathrm{p}}_{X,X^{\mathrm{\prime }}})\circ J_{X,X^{\mathrm{\prime }}}=J_{X^{\mathrm{\prime }},X}\circ \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{p}$. For orientation reversal supply unitary identifications ${\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(\bar{X})\cong {\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(X{)}^{\ast }$, compatible with tensor products and evaluation. If $W^{\vee }:{\bar{X}}_1\to {\bar{X}}_0$ denotes an admitted bordism with incoming/outgoing roles reversed, require that its map is the algebraic dual $Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W{)}^{\ast }$ under those identifications; this is part of the supplied duality data. For a filling $Y:\mathrm{\varnothing }\to X$ the vector already used above is precisely $Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(Y)(J_0(1))$, and for a closed bulk bordism $W:\mathrm{\varnothing }\to \mathrm{\varnothing }$ its scalar response is the scalar of $J_0^{-1}{Z}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(W)J_0$. Every assigned bordism map is invertible, but need not be unitary; the metric-dependent response is not thereby assumed topological. The boundary/bulk duality pairing used for the relative amplitude must respect these declared gluing and tensor identifications.

For every finite-dimensional smooth parameter manifold $T$ and smooth family of compatible boundary backgrounds over $T$, assume the boundary and bulk lines form Hermitian line bundles ${\mathcal{L}}_{\mathrm{\partial }}$ and ${\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ over $T$ with specified unitary connections ${\mathrm{\nabla }}^{\mathrm{\partial }}$ and ${\mathrm{\nabla }}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$. Here a unitary connection is a complex-linear covariant derivative satisfying $d⟨s,t⟩=⟨\mathrm{\nabla }s,t⟩+⟨s,\mathrm{\nabla }t⟩$ for smooth local sections. Parallel transport along the smooth background path $\gamma :[0,1]\to T$ evaluates at $1$ the unique section along the path satisfying ${\mathrm{\nabla }}_{\dot{\gamma }}s=0$ with its prescribed initial value at $0$. This transport definition uses the supplied line bundles and connections in any boundary dimension $d\ge 1$; it does not assume a chiral Dirac construction of either line. The unitary line-duality isomorphism above is part of the background data and is equivariant under endpoint gauge maps, but it is not assumed to preserve these connections. A loop $\gamma$ consists of a smooth path of such backgrounds and an endpoint gauge identification $u$; let ${\mathrm{P}\mathrm{T}}_{\gamma }^{\mathrm{\partial }}$ and ${\mathrm{P}\mathrm{T}}_{\gamma }^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ denote parallel transport for these connections, and let $u_{\ast }^{\mathrm{\partial }}$ and $u_{\ast }^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ be the induced endpoint maps on the two lines, required to be complex-linear unitary isomorphisms for their specified Hermitian metrics. Denote the scalars of $(u_{\ast }^{\mathrm{\partial }}{)}^{-1}\circ {\mathrm{P}\mathrm{T}}_{\gamma }^{\mathrm{\partial }}$ and $(u_{\ast }^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}{)}^{-1}\circ {\mathrm{P}\mathrm{T}}_{\gamma }^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ by ${\mathrm{Hol}}_{\gamma }({\mathcal{L}}_{\mathrm{\partial }})$ and ${\mathrm{Hol}}_{\gamma }({\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}})$, respectively; they lie in $U(1)$ because both parallel transports and endpoint gauge maps are unitary. Its mapping torus $Y_{\gamma }$ has underlying manifold $M\times S^1$, with the principal-bundle and associated fields glued at the circle seam using $u$ and any specified spin clutch, and product orientation $\mathrm{or}(M)\wedge dt$. Whenever the closed bulk response on this mapping torus is used to compute the bulk-line holonomy, the convention is

$Z_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(Y_{\gamma },b_{\gamma })={\mathrm{Hol}}_{\gamma }({\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}),$

not its inverse. This equality is part of the adopted bulk/line data. The boundary/bulk pair cancels that anomaly along $\gamma$ exactly when

${\mathrm{Hol}}_{\gamma }({\mathcal{L}}_{\mathrm{\partial }}){\mathrm{Hol}}_{\gamma }({\mathcal{L}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}})=1.$

The diagram records this condition: the vertical maps are the line evaluation at the two endpoints, and the top map is parallel transport on both lines.

Parallel transport for a supplied connection, independent of the spacetime dimension

Existence and uniqueness on the compact background path

Invertibility and real linearity of the underlying transport

Metric-compatible transport preserves the underlying real Hermitian metric

Exact associativity and unit coherence equations for the bulk tensor identifications

Symmetry-preservation equation, with both braidings the symmetric tensor swaps

Spin structures and negative-square Clifford convention

Surjectivity and two-element kernel of the spin cover

Freed, Lecture 11, §§11.1–11.2

Dai and Freed, §5