Invertible field theories and the Freed–Hopkins classification

Intuition Beginner

A field theory, stripped to its barest skeleton, is a rule that hands you a single number for every closed shape of space-time and a vector space of states for every cross-section. Most theories are rich: the state spaces are huge, the numbers encode hard dynamics. But some theories are almost empty. Every state space is just a one-dimensional line — a single direction with a scale — and every number it outputs is a phase, a point on the unit circle that you can always undo by spinning backward. These are the invertible theories. Nothing in them can ever be zero or get stuck; everything can be reversed.

Why care about the dullest possible theories? Because the dull ones turn out to be classified by topology alone. Once a theory carries no genuine dynamics, the only thing left for it to remember is the shape of the spaces you feed it. So the catalogue of invertible theories becomes a catalogue of shapes, and shape-catalogues are exactly what topologists have spent a century building. The punchline of this unit: the simplest field theories are counted by stable homotopy theory.

The surprise is that physics needed this. The phases of matter that stay rigid at zero temperature — the ones protected by a symmetry, like a topological insulator — are governed by precisely these invertible theories. Counting them is the same mathematics either way.

Visual Beginner

Alt text: On the left, a broad funnel represents all field theories, with large boxes standing for big state spaces. The funnel narrows to a thin vertical axis on the right, the invertible theories, where every cross-section is drawn as a single line segment and every closed space-time is drawn as a dot sitting on a circle of phases. A horizontal arrow links this thin axis to a stacked column of labelled groups, one per dimension, captioned "bordism groups." The picture shows visually that throwing away all dynamics leaves only a phase and a line, and that what remains is bookkept by a list of topological groups rather than by analysis.

Worked example Beginner

Take the simplest invertible theory of all: the Euler theory in two dimensions. Pick once and for all a fixed number, say the scale $\lambda =2$. The rule is: to any closed surface, count its Euler characteristic and output $\lambda$ raised to that power. The Euler characteristic of a sphere is $2$, so the sphere gets the number $2^2=4$. The Euler characteristic of a doughnut (a torus) is $0$, so the torus gets $2^0=1$.

Check that this respects gluing. Cut the torus along a circle into a tube; the tube has Euler characteristic $0$. Gluing two tubes end to end gives a longer tube, and Euler characteristics add under this gluing: $0+0=0$, matching the number $2^0\cdot 2^0=1=2^0$. The output for any surface is never zero — it is always a positive number you can divide back out — so the theory is invertible.

What this tells us: the entire content of this theory is one topological number per surface, the Euler characteristic. No dynamics, no spectrum, just a phase-like scale tracking a topological count. That is the flavour of every invertible theory.

Check your understanding Beginner

Formal definition Intermediate+

Fix a tangential structure $H\to O(n)$ — orientation ($H=SO$), spin ($H=\mathrm{Spin}$), the pin groups ${\mathrm{Pin}}^{\pm }$, or framing — and let ${\mathrm{Bord}}_n^H$ denote the symmetric-monoidal bordism category built from this structure, as in 03.16.01. Let $(\mathcal{C},\otimes ,1)$ be a symmetric-monoidal target, ${\mathrm{sVect}}_{\mathbb{C}}$ in the simplest setting. A field theory is a symmetric-monoidal functor $F:{\mathrm{Bord}}_n^H\to \mathcal{C}$.

Definition (invertible object, Picard groupoid). An object $x\in \mathcal{C}$ is invertible if there is $x^{-1}$ with $x\otimes x^{-1}\cong 1$. A morphism is invertible if it has a two-sided inverse. The invertible objects and invertible morphisms form a symmetric-monoidal groupoid, the Picard groupoid $\mathrm{Pic}(\mathcal{C})$. In ${\mathrm{sVect}}_{\mathbb{C}}$ the invertible objects are the one-dimensional super-lines (even or odd), and $\mathrm{Pic}({\mathrm{sVect}}_{\mathbb{C}})$ has isomorphism classes $\mathbb{Z}/2$ (parity) with automorphisms ${\mathbb{C}}^{\times }$.

Definition (invertible field theory). A field theory $F:{\mathrm{Bord}}_n^H\to \mathcal{C}$ is invertible if it factors through the Picard groupoid, $F:{\mathrm{Bord}}_n^H\to \mathrm{Pic}(\mathcal{C})$. Equivalently: each state space $F(Y)$ for a closed $(n-1)$-manifold $Y$ is $\otimes$-invertible (a line), each bordism is sent to an invertible morphism, and each closed $n$-manifold $M$ has partition function $F(M)\in {\mathbb{C}}^{\times }$ (a nonzero phase). Invertibility of $F$ is invertibility as a functor: there is a theory $F^{-1}$ with $F\otimes F^{-1}$ the constant theory at $1$.

Definition (deformation class). Two invertible $H$-theories are deformation equivalent if they lie in the same path component of the (topologized) space of reflection-positive invertible theories; a deformation class is such a path component. Under $\otimes$ these classes form an abelian group, written ${\mathrm{Inv}}_H^n$. Reflection positivity, the Wick-rotated avatar of unitarity, is the structure recorded by the Osterwalder–Schrader axioms 08.10.07; it is the datum that pins down which invertible theories are physical and the one under which the classification below is stated.

The central reorganisation, due to Freed–Hopkins ^{[Freed–Hopkins 2021]}, is that ${\mathrm{Inv}}_H^n$ — an a priori analytic object — is computed by stable homotopy theory. The bridge runs through the Madsen–Tillmann spectrum.

Definition (Madsen–Tillmann spectrum). Let $BH$ be the classifying space of $H$ and $H\to BO(n)$ the map classifying the rank-$n$ bundle. The Madsen–Tillmann spectrum is the Thom spectrum of the inverse universal bundle, $$ MTH ;=; (BH)^{-H} ;=; \mathrm{Thom}\big(BH;, -H\big), $$ the Thom spectrum of the virtual bundle $-H$ of rank $-n$. Its homotopy groups are the $H$-bordism groups by Pontryagin–Thom 03.06.13: $$ \pi_n(MTH) ;\cong; \Omega^H_n . $$ For $H=O,SO,\mathrm{Spin}$ this recovers (stabilised versions of) $MO,MSO,M\mathrm{Spin}$ from 03.12.04; the "$T$" records that one uses the inverse tangential bundle, which is the form that the partition function of a field theory naturally sees.

Key theorem with proof Intermediate+

Theorem (Freed–Hopkins classification of invertible field theories). Let $H\to O(n)$ be a tangential structure. The abelian group of deformation classes of reflection-positive, topological, $n$-dimensional invertible field theories with tangential structure $H$ is naturally isomorphic to $$ \mathrm{Inv}^n_H ;\cong; [,MTH,; \Sigma^{n+1} I_{\mathbb Z},], $$ the group of homotopy classes of maps of spectra from the Madsen–Tillmann spectrum $MTH$ into the $(n+1)$-fold suspension of the Anderson dual $I_{\mathbb{Z}}$ of the sphere spectrum ^{[Freed–Hopkins 2021]}. There is moreover a short exact sequence, natural in $H$ and $n$, $$ 0 \to \mathrm{Ext}^1\big(\Omega^H_{n}, \mathbb Z\big) \to [,MTH, \Sigma^{n+1} I_{\mathbb Z},] \to \mathrm{Hom}\big(\Omega^H_{n+1}, \mathbb Z\big) \to 0, $$ whose free quotient records the partition-function (Pfaffian) characteristic numbers and whose torsion subgroup ${\mathrm{Ext}}^1({\Omega }_n^H,\mathbb{Z})\cong \mathrm{Hom}({\Omega }_{n+1,\mathrm{tors}}^H,\mathbb{Q}/\mathbb{Z})$ records the purely topological phases.

Proof. The argument has three movements: identify invertible theories with cohomology classes, identify the relevant cohomology theory as Anderson duality, and unwind the universal-coefficient sequence.

Step 1: invertible theories are cohomology classes on $MTH$. An invertible field theory, by definition, factors through the Picard groupoid of the target. Passing to deformation classes turns the symmetric-monoidal groupoid $\mathrm{Pic}(\mathcal{C})$ into its associated spectrum $\mathfrak{pic}(\mathcal{C})$ — a connective spectrum whose homotopy groups are the isomorphism classes (degree $0$) and automorphism groups (degree $1$) of invertible objects. A symmetric-monoidal functor out of ${\mathrm{Bord}}_n^H$ that lands in invertibles is, after this passage and after inverting the bordism direction, the same datum as a map of spectra $$ MTH \longrightarrow \Sigma^{n} \mathfrak{pic}(\mathcal C). $$ This is the homotopical content of the cobordism hypothesis with tangential structure 03.16.03 specialised to the invertible (Picard) case: the bordism multicategory's universal property turns a functor into a map out of the Thom spectrum that represents bordism, and Brown representability 03.12.04 guarantees that homotopy classes of such maps are exactly a generalised cohomology group of $MTH$.

Step 2: reflection positivity selects the Anderson dual. For topological invertible theories the relevant target spectrum, after imposing reflection positivity, is the Anderson dual of the sphere. The Anderson dual $I_{\mathbb{Z}}$ is the spectrum representing the cohomology theory that is Pontryagin–Anderson dual to stable homotopy: it is defined by the fibre sequence $I_{\mathbb{Z}}\to I_{\mathbb{Q}}\to I_{\mathbb{Q}/\mathbb{Z}}$, where $I_{\mathbb{Q}}$ and $I_{\mathbb{Q}/\mathbb{Z}}$ are the Brown–Comenetz/rational duals built from $\mathrm{Hom}(-,\mathbb{Q})$ and $\mathrm{Hom}(-,\mathbb{Q}/\mathbb{Z})$ on homotopy. Freed–Hopkins show that reflection-positive topological invertible $n$-theories correspond to maps $MTH\to {\Sigma }^{n+1}{I}_{\mathbb{Z}}$: the partition function of the theory is the induced cohomology operation, valued in $\mathbb{R}/\mathbb{Z}$ phases, and reflection positivity is exactly the condition that forces the target to be the integral Anderson dual rather than its rational or torsion approximations ^{[Freed–Hopkins 2021]}. Hence ${\mathrm{Inv}}_H^n\cong [MTH,{\Sigma }^{n+1}{I}_{\mathbb{Z}}]$.

Step 3: the universal-coefficient sequence. The defining property of the Anderson dual is a universal-coefficient short exact sequence: for any spectrum $X$, $$ 0 \to \mathrm{Ext}^1(\pi_{n-1} X, \mathbb Z) \to [X, \Sigma^n I_{\mathbb Z}] \to \mathrm{Hom}(\pi_n X, \mathbb Z) \to 0 . $$ Apply this with $X=MTH$, shifting degree by one, and substitute ${\pi }_k(MTH)\cong {\Omega }_k^H$ from Step (the Pontryagin–Thom identification). This produces $$ 0 \to \mathrm{Ext}^1(\Omega^H_n, \mathbb Z) \to [MTH, \Sigma^{n+1} I_{\mathbb Z}] \to \mathrm{Hom}(\Omega^H_{n+1}, \mathbb Z) \to 0 . $$ The $\mathrm{Hom}$ quotient pairs an invertible theory against $(n+1)$-dimensional closed $H$-manifolds, returning the integer characteristic numbers that the partition function computes (the free, "Pfaffian", part). The ${\mathrm{Ext}}^1$ subgroup, using ${\mathrm{Ext}}^1(A,\mathbb{Z})\cong \mathrm{Hom}(A_{\mathrm{tors}},\mathbb{Q}/\mathbb{Z})$ for finitely generated $A$, is $\mathrm{Hom}({\Omega }_{n,\mathrm{tors}}^H,\mathbb{Q}/\mathbb{Z})$ and houses the genuinely topological phases that are invisible to integer-valued characteristic numbers. $■$

Bridge. The classification builds toward the physics of topological phases and appears again in 03.16.05, where an anomaly is recast as an invertible theory one dimension up; the foundational reason the proof works is that bordism is a generalised cohomology theory represented by a Thom spectrum 03.12.04, so that the Pontryagin–Thom isomorphism ${\pi }_n(MTH)\cong {\Omega }_n^H$ of 03.06.13 is exactly the input the universal-coefficient sequence consumes. This is the bridge by which an analytic classification problem becomes a stable-homotopy computation: the partition function is dual to a bordism invariant, and the central insight is that reflection positivity is what singles out the integral Anderson dual, turning physical unitarity into a homotopy-theoretic constraint. The same Picard-spectrum machinery generalises the cobordism hypothesis of 03.16.03 to its invertible shadow.

Exercises Intermediate+

Advanced results Master

The Picard spectrum and the homotopical normal form. The passage in Step 1 of the key theorem from the Picard groupoid $\mathrm{Pic}(\mathcal{C})$ to a spectrum $\mathfrak{pic}(\mathcal{C})$ is the structural core. For a symmetric-monoidal $(\infty ,n)$-category $\mathcal{C}$, the invertible objects and all higher invertible morphisms assemble into a connective spectrum with ${\pi }_0=\{\text{iso classes of invertible objects}\}$, ${\pi }_1=\{\text{automorphisms of the unit}\}$, and higher ${\pi }_k$ the higher automorphism groups. An invertible fully-extended $H$-theory is then a map $MTH\to {\Sigma }^n\mathfrak{pic}(\mathcal{C})$, by the invertible specialisation of the cobordism hypothesis 03.16.03. Freed–Hopkins identify, for topological reflection-positive theories, the relevant truncation of $\mathfrak{pic}$ with a shift of $I_{\mathbb{Z}}$; this is the precise sense in which "an invertible TQFT is a cohomology class on the bordism spectrum" ^{[Freed–Hopkins 2021]}.

SPT phases and the bordism conjecture. A symmetry-protected topological phase is a gapped ground state with symmetry $G$ that is short-range entangled — deformable to a product state once the symmetry is broken — yet distinct from the plain product state while the symmetry is enforced. The Freed–Hopkins (and, on the physics side, Kapustin–Thorngren–Turzillo–Wang) proposal is that deformation classes of such phases in $n$ space-time dimensions with internal/space-time symmetry data encoded by a tangential structure $H$ are exactly ${\mathrm{Inv}}_H^n\cong [MTH,{\Sigma }^{n+1}{I}_{\mathbb{Z}}]$, hence governed by the bordism groups ${\Omega }_{n+1}^H$ ^{[Kapustin et al. 2015]}. For free fermions this recovers the Altland–Zirnbauer "tenfold way" and its $KR$-theory organisation 03.08.12; for interacting systems the bordism answer corrects the free-fermion $\mathbb{Z}$ to a finite group — the canonical instance being the collapse of the free-fermion $\mathbb{Z}$ in class DIII to $\mathbb{Z}/16$.

The Arf theory in two dimensions. The torsion class generating ${\mathrm{Inv}}_{\mathrm{Spin}}^1$ is the Arf invertible theory. On a closed spin surface $\Sigma$ it outputs the sign $(-1{)}^{\mathrm{Arf}(\Sigma )}$, where $\mathrm{Arf}$ is the Arf invariant of the quadratic refinement of the intersection form determined by the spin structure. As an invertible theory it is the low-energy avatar of the Kitaev Majorana chain: its single nonzero deformation class matches ${\Omega }_2^{\mathrm{Spin}}\cong \mathbb{Z}/2$, and gauging it implements the Jordan–Wigner/GSO bosonisation that turns a fermionic theory into a bosonic one. The partition function is genuinely topological — a torsion phase, not an integer characteristic number — consistent with its origin in the ${\mathrm{Ext}}^1$ part of the universal-coefficient sequence.

The ${\mathbb{Z}}_{16}$ classification of topological superconductors. In $3+1$ dimensions with time-reversal squaring to fermion parity (the ${\mathrm{Pin}}^{+}$ structure), ${\Omega }_4^{{\mathrm{Pin}}^{+}}\cong \mathbb{Z}/16$. The invertible-theory classification therefore predicts that 3+1d topological superconductors of this symmetry class are labelled by an integer $\nu$ modulo $16$. Free fermions see an infinite $\mathbb{Z}$ of surface Majorana cones; interactions, captured by the bordism reduction, identify $\nu$ and $\nu +16$, so that sixteen copies of the minimal phase are deformable to the plain product phase. The generator's partition function is an exponentiated $\eta$-invariant of a ${\mathrm{Pin}}^{+}$ Dirac operator, the analytic incarnation of the torsion bordism invariant.

Synthesis. Putting these together, the invertible-theory classification is the foundational reason the disparate-looking catalogues of topological matter coincide: the free-fermion tenfold way of 03.08.12, the interacting SPT tables, and the abstract group $[MTH,{\Sigma }^{n+1}{I}_{\mathbb{Z}}]$ are one object read in three registers. This is exactly the pattern by which a homotopy-theoretic invariant — a map out of a Thom spectrum — generalises both the integer characteristic numbers of 03.06.13 and the torsion phases that integer invariants cannot see. The central insight is that reflection positivity is dual to the integrality of the Anderson target, so that the physical condition of unitarity becomes the homotopical condition selecting $I_{\mathbb{Z}}$; and the bridge to anomalies is that the same group, shifted one degree, classifies the bulk theory whose boundary carries an anomalous theory 03.16.05. The partition function, the Pfaffian line, and the $\eta$-invariant are then three names for the single pairing of an invertible theory against bordism.

Full proof set Master

Proposition (invertible theories form an abelian group under $\otimes$). The set of deformation classes of invertible $n$-dimensional $H$-theories is an abelian group under the tensor (stacking) product, with identity the constant unit theory.

Proof. If $F,G$ are invertible, $(F\otimes G)(M)=F(M)\cdot G(M)\in {\mathbb{C}}^{\times }$ and $(F\otimes G)(Y)=F(Y)\otimes G(Y)$ is a tensor of lines, hence a line; so $F\otimes G$ is invertible. The unit theory $1$, sending every closed $n$-manifold to $1$ and every $(n-1)$-manifold to the unit line, is a two-sided identity. The inverse of $F$ is the dual theory $F^{-1}(Y)=F(Y{)}^{-1}$, $F^{-1}(M)=F(M{)}^{-1}$, which is a field theory because dualising a symmetric-monoidal functor into a Picard groupoid is again such a functor; $F\otimes F^{-1}\cong 1$. Commutativity is the symmetry of $\otimes$ in $\mathrm{Pic}(\mathcal{C})$. Deformation equivalence is compatible with $\otimes$ since a path in one factor induces a path in the product, so the operation descends to deformation classes. $■$

*Proposition (Pontryagin–Thom identification of ${\pi }_{\ast }MTH$).* For a tangential structure $H\to O(n)$, ${\pi }_k(MTH)\cong {\Omega }_k^H$.

Proof. This is the Thom-spectrum form of Pontryagin–Thom 03.06.13. A class in ${\Omega }_k^H$ is a closed $k$-manifold $X^k$ with $H$-structure on its (stable) tangent bundle, up to bordism. Embed $X\hookrightarrow S^{k+N}$ with normal bundle $\nu$; the $H$-structure on $TX$ is equivalent to a stable $H$-structure on $\nu$ via the inverse-bundle relation $\nu \cong -TX$, hence a classifying map into the Thom space $\mathrm{Th}(-H{)}_N$ of the rank-$(N-n)$ approximation to $-H$ over $BH$. The Pontryagin–Thom collapse $S^{k+N}\to \mathrm{Th}(-H{)}_N$ produces an element of ${\pi }_{k+N}(\mathrm{Th}(-H{)}_N)$; stabilising over $N$ gives ${\pi }_k(MTH)$. Bordant manifolds give homotopic collapse maps and conversely, so the assignment is a bijection of groups. $■$

Proposition (universal-coefficient sequence for the Anderson dual). For any spectrum $X$ with finitely generated homotopy groups, $0\to {\mathrm{Ext}}^1({\pi }_{n-1}X,\mathbb{Z})\to [X,{\Sigma }^n{I}_{\mathbb{Z}}]\to \mathrm{Hom}({\pi }_{n}X,\mathbb{Z})\to 0$ is exact.

Proof. The Anderson dual $I_{\mathbb{Z}}$ is defined by the fibre sequence $I_{\mathbb{Z}}\to I_{\mathbb{Q}}\to I_{\mathbb{Q}/\mathbb{Z}}$, where $I_{\mathbb{Q}}$ represents $\mathrm{Hom}({\pi }_{-\ast }(-),\mathbb{Q})$ and $I_{\mathbb{Q}/\mathbb{Z}}$ represents $\mathrm{Hom}({\pi }_{-\ast }(-),\mathbb{Q}/\mathbb{Z})$ exactly (these are injective $\mathbb{Z}$-modules, so the represented functors are exact). Mapping $X$ in and taking the long exact sequence of the fibration, the connecting maps assemble $[X,{\Sigma }^n{I}_{\mathbb{Z}}]$ into the kernel-cokernel data of $\mathrm{Hom}({\pi }_{n}X,\mathbb{Q})\to \mathrm{Hom}({\pi }_{n}X,\mathbb{Q}/\mathbb{Z})$. The cokernel of $\mathrm{Hom}({\pi }_{n}X,\mathbb{Q})\to \mathrm{Hom}({\pi }_{n}X,\mathbb{Q}/\mathbb{Z})$ is ${\mathrm{Ext}}^1({\pi }_{n}X,\mathbb{Z})$ and the kernel is $\mathrm{Hom}({\pi }_{n}X,\mathbb{Z})$, by the standard $\mathrm{Ext}$ computation from $0\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{Q}/\mathbb{Z}\to 0$. Reassembling across the two adjacent degrees yields the stated three-term exact sequence. $■$

The Freed–Hopkins identification of reflection-positive invertible theories with maps into ${\Sigma }^{n+1}{I}_{\mathbb{Z}}$ (Step 2 of the key theorem) is stated without independent proof here — see Freed–Hopkins ^{[Freed–Hopkins 2021]} §§5–7 and Freed ^{[Freed 2019]} Lecture 8 — as its proof requires the reflection-positivity formalism beyond this unit's scope; the bordism computations ${\Omega }_2^{\mathrm{Spin}}\cong \mathbb{Z}/2$ and ${\Omega }_4^{{\mathrm{Pin}}^{+}}\cong \mathbb{Z}/16$ used in the examples are standard and recorded in Kapustin et al. ^{[Kapustin et al. 2015]}.

Connections Master

• The Atiyah–Segal axioms 03.16.01. This unit specialises the general TQFT functor to the invertible (Picard-valued) case: an invertible theory is a symmetric-monoidal functor ${\mathrm{Bord}}_n^H\to \mathrm{Pic}(\mathcal{C})$. The state-space/partition-function/gluing structure worked out there is what becomes a line and a phase here, and the tangential structure $H$ introduced there is the input datum the Madsen–Tillmann spectrum $MTH$ is built from.

• Spectra, Thom spectra, and Brown representability 03.12.04. The Madsen–Tillmann spectrum $MTH$ is a Thom spectrum, $MO$/$MSO$/$M\mathrm{Spin}$ of that unit are its low-structure instances, and Brown representability is the theorem that lets "invertible field theory" mean "cohomology class on $MTH$." The Anderson dual $I_{\mathbb{Z}}$ is a new spectrum built by Brown–Comenetz/Anderson duality from the constructions there.

• Oriented bordism and Pontryagin–Thom 03.06.13. The identification ${\pi }_n(MTH)\cong {\Omega }_n^H$ is exactly the Pontryagin–Thom theorem of that unit, applied to the inverse tangential bundle. The classical oriented bordism ring computed there is the $H=SO$ shadow of the general $H$-bordism groups that index the classification.

• Anomalies as invertible field theories 03.16.05. The same Freed–Hopkins group, shifted to $[MTH,{\Sigma }^{n+2}{I}_{\mathbb{Z}}]$, classifies the $(n+1)$-dimensional invertible bulk theory whose boundary supports an anomalous $n$-dimensional theory. That unit is the immediate sequel: an anomaly is a deformation class of an invertible theory one dimension up.

• $KR$-theory and the tenfold way 03.08.12. For free fermions the bordism classification reduces to the Altland–Zirnbauer periodic table organised by $KR$-theory. This unit explains why the interacting classification can differ: the bordism group ${\Omega }_{n+1}^H$ refines (and sometimes collapses) the free-fermion $K$-theoretic answer.

Historical & philosophical context Master

The thread begins with Atiyah's and Segal's axiomatisation of topological and conformal field theory in 1988, which made "a field theory is a symmetric-monoidal functor on bordisms" a precise mathematical object. The homotopy-theoretic input matured separately: Madsen and Tillmann constructed the spectra now bearing their names in their 2001 Inventiones paper "The stable mapping class group and $Q({\mathbb{CP}}_{+}^{\infty })$," the same circle of ideas that produced the Madsen–Weiss theorem on the cohomology of the stable mapping class group ^{[Madsen–Tillmann 2001]}. The Anderson dual $I_{\mathbb{Z}}$ goes back to D. W. Anderson's 1969 Berkeley notes on universal-coefficient theorems in $K$-theory ^{[Anderson 1969]}.

These strands were fused by Freed and Hopkins, whose "Reflection positivity and invertible topological phases" appeared in Geometric Topology 25 (2021), 1165–1330, after circulating from 2016; it proves the classification ${\mathrm{Inv}}_H^n\cong [MTH,{\Sigma }^{n+1}{I}_{\mathbb{Z}}]$ and identifies reflection positivity as the structure selecting the integral Anderson dual ^{[Freed–Hopkins 2021]}. In parallel, Kapustin, Thorngren, Turzillo, and Wang gave the physics formulation — SPT phases classified by bordism — in their 2015 paper on fermionic SPT phases and cobordisms ^{[Kapustin et al. 2015]}. Freed's 2019 CBMS lectures Lectures on Field Theory and Topology collected the mathematical account in textbook form, including the reframing of anomalies as invertible theories ^{[Freed 2019]}.

Bibliography Master

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}

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