03.16.05 · modern-geometry / tqft

Anomalies as invertible field theories in one dimension higher

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Freed CBMS Lectures on Field Theory and Topology, Lectures 9--11; Freed-Hopkins 2021 Geom. Topol. 25; Witten 1985 Comm. Math. Phys. 100 (global anomaly)

Intuition Beginner

A symmetry that survives in a classical theory but breaks once you account for quantum effects is called an anomaly. For decades the word carried a flavour of pathology, as if the theory had caught a disease. The modern view turns this on its head: an anomaly is not a sickness but a feature. It tells you that your theory was never meant to stand on its own. It lives on the edge of a slightly larger world.

Picture a flat sheet that you can only describe correctly as the boundary of a thicker slab. The slab is a topological phase one dimension higher; the sheet is your anomalous theory. Whatever looked broken on the sheet is repaired the moment you remember the slab it bounds.

What is the slab? It is a very simple kind of field theory, an invertible one, whose only job is to assign a phase, a number on the unit circle, to each closed shape. The anomaly of your theory is exactly which slab it bounds. Two theories have the same anomaly when they edge the same slab.

This reframing pays off in a practical way. Asking whether an anomaly cancels becomes asking whether the slab can be filled in, made to bound nothing. If the higher-dimensional phase is the empty phase, the edge theory stands alone and is consistent on its own.

Visual Beginner

Alt text: On the left, a thick three-dimensional slab is drawn shaded, with its flat front face singled out as a thin sheet. The slab is labelled the invertible anomaly theory: it attaches a phase, drawn as a dot on a small circle, to each closed shape. The sheet is labelled the anomalous boundary theory. An arrow from slab to sheet is labelled inflow, showing the slab hands the sheet the circle its partition function lives on. On the right, a separate picture shows the same circle with the partition function as a sliding point, contrasted with a fixed number, conveying that an anomalous partition function is a phase, not a single value.

Worked example Beginner

Take the simplest invertible theory in one dimension: it counts points. To a single point it attaches a fixed phase $q$ on the unit circle, and to a disjoint pair of points it attaches $q$ times $q$ , that is $q^{2}$ . To the empty collection it attaches $1$ .

Now read this as the slab for a zero-dimensional edge. The edge "theory" is just a choice of where on the circle to sit, and the slab tells you how that choice multiplies when you put two edges side by side. If $q = 1$ , the slab is the empty phase: it attaches $1$ to everything, and the edge can be described on its own. If $q = - 1$ , two points give $(- 1) (- 1) = 1$ but a single point gives $- 1$ , so the edge carries a genuine phase that only one extra dimension can explain.

What this tells us: the anomaly is captured entirely by the one number $q$ . Cancellation means $q = 1$ , the slab fills in. A non-cancelling anomaly, here $q = - 1$ , is a stubborn phase that no boundary description removes, only the bulk slab accounts for it.

Check your understanding Beginner

Formal definition Intermediate+

Fix a tangential structure $H \to O (n)$ (orientation, spin, $Spin^{c}$ , framing) and let $Bord_{n}^{H}$ denote the symmetric monoidal bordism category of 03.16.04. A field theory is a symmetric monoidal functor on $Bord_{n}^{H}$ ; it is invertible when it lands in the Picard subgroupoid of the target, so every state space is $\otimes$ -invertible and every closed partition function lies in $C^{\times}$ .

Definition (anomalous theory as a boundary theory). Let $α$ be an invertible $(d + 1)$ -dimensional field theory with tangential structure $H$ , the anomaly theory. A $d$ -dimensional theory anomalous for $α$ is a relative (boundary) field theory $F$ for $α$ : rather than assigning a number to a closed $d$ -manifold $M$ , it assigns a vector $$ Z_F(M) \in \alpha(M), $$ a point in the one-dimensional state space $α (M)$ that the bulk attaches to $M$ . As $M$ ranges over the space of backgrounds, the spaces $α (M)$ assemble into a Hermitian line bundle with connection $L_{α}$ , and $Z_{F}$ is a section of $L_{α}$ . The theory is anomaly-free precisely when $α$ is the identity theory, in which case $L_{α}$ is canonically the constant line $C$ and $Z_{F} (M) \in C$ is an ordinary number.

Definition (the anomaly as a deformation class). By the Freed-Hopkins classification 03.16.04, topological deformation classes of invertible $(d + 1)$ -dimensional $H$ -theories form the abelian group $$ [,MTH,\ \Sigma^{d+2} I_{\mathbb{Z}},], $$ the generalized cohomology of the Madsen-Tillmann spectrum $M T H$ with coefficients in the Anderson dual $I_{Z}$ of the sphere, in degree $d + 2$ . The anomaly of the $d$ -dimensional theory $F$ is the class $[α] \in [M T H, Σ^{d + 2} I_{Z}]$ . (The shift by one degree relative to the $[M T H, Σ^{n + 1} I_{Z}]$ of 03.16.04 records that the anomaly theory is one dimension above the theory it bounds.)

The same data has a differential shadow. The perturbative or local anomaly is the image of $[α]$ in de Rham cohomology: the curvature of $L_{α}$ is an index density $\hat{A} (R) ch (F)$ on the space of backgrounds, and its descent is exactly the consistent anomaly and the Wess-Zumino condition of 03.07.32. The map "homotopy class $\mapsto$ de Rham class" forgets torsion: a global anomaly is a nonzero torsion class $[α]$ with vanishing de Rham image, undetectable by the perturbative descent.

Counterexamples to common slips

Local vanishing is not anomaly freedom. A theory can have vanishing perturbative anomaly (zero index density, vanishing descent) yet carry a nonzero torsion class $[α]$ . Witten's $S U (2)$ anomaly is the prototype: the perturbative six-form vanishes, but $Ω_{5}^{S p in} (pt) = Z /2$ supplies a sign that obstructs a consistent path integral.
The partition function is not a number. Writing " $Z = \int e^{- S}$ " as a complex number for an anomalous theory hides the anomaly. The honest object is a section of $L_{α}$ ; choosing a local frame for the line makes it a number but introduces a non-canonical phase that jumps under large background changes.
Inflow is not optional bookkeeping. The bulk term is not a counterterm one may add or drop. Without $α$ the boundary $Z_{F}$ does not live in a fixed line, so the inflow term is what makes the boundary theory well-defined at all.

Key theorem with proof Intermediate+

The signature computation of the subject is the identification of the anomaly phase of a chiral fermion with an $η$ -invariant. Let $X$ be a closed $d + 1 = 2 k$ -dimensional spin manifold viewed as a background for the bulk, and let $D_{X}$ be the associated Dirac operator. The Dai-Freed theorem ^{[Dai 1994]} computes the holonomy of the Pfaffian line bundle of the boundary chiral fermion as the exponentiated $η$ -invariant of the bulk.

Theorem (the anomaly theory of a chiral fermion is the $η$ -invariant theory). Let $F$ be the $d$ -dimensional theory of a chiral (or Majorana) fermion. Its partition function is naturally a section of the Pfaffian line $Pf (D)$ of the family of boundary Dirac operators. The associated $(d + 1)$ -dimensional anomaly theory $α$ assigns to a closed spin $(d + 1)$ -manifold $X$ the partition function $$ \alpha(X) = \exp!\big(2\pi i, \tfrac{1}{2},\eta(D_X)\big), $$ where $η (D_{X})$ is the APS $η$ -invariant 03.09.24. This $α$ is invertible, and its deformation class is the anomaly of $F$ .

Proof. The boundary fermion path integral is a Pfaffian, $Z_{F} = Pf (D)$ , which is not a function but a section of the Pfaffian line bundle $Pf (D) \to B$ over the space $B$ of backgrounds; this is the line bundle $L_{α}$ of the formal definition. The anomaly is the obstruction to finding a flat global frame for $Pf (D)$ compatible with its natural Bismut-Freed connection, measured by its holonomy around loops in $B$ .

Let $γ$ be a loop of $d$ -dimensional backgrounds. The mapping torus $X_{γ}$ is a closed $(d + 1)$ -dimensional spin manifold fibered over $S^{1}$ . The Bismut-Freed curvature formula identifies the curvature of the Pfaffian connection with the degree- $(d + 2)$ component of $\int_{X / B} \hat{A} (R)$ , the local index density; integrating over $γ$ reproduces the de Rham anomaly. The Dai-Freed theorem upgrades this from curvature to holonomy: the holonomy of the Pfaffian line around $γ$ equals $$ \mathrm{hol}\gamma(\mathrm{Pf}(D)) = \exp!\big(2\pi i, \tfrac{1}{2},\eta(D{X_\gamma})\big), $$ the exponentiated $η$ -invariant of the Dirac operator on the mapping torus $X_{γ}$ ^{[Witten-Yonekura §3]}. Because $η (D_{X})$ is defined for every closed bulk $X$ (not only mapping tori) and is additive under disjoint union, multiplicative gluing aside, the assignment $X \mapsto exp (2 π i η (D_{X}) /2)$ defines an invertible $(d + 1)$ -dimensional field theory $α$ . Reduction mod $Z$ of $η /2$ is a spin-bordism invariant of $X$ , so $[α]$ depends only on the bordism class and lands in the Freed-Hopkins group as claimed. The de Rham image of $[α]$ is the curvature term above, the consistent anomaly of 03.07.32; the torsion part is the spectral-asymmetry content of $η$ invisible to that curvature. $■$

Bridge. This theorem builds toward the full Freed-Hopkins program: the $η$ -invariant theory is one explicit cocycle for a class in $[M T H, Σ^{d + 2} I_{Z}]$ , and the same Pfaffian-line apparatus appears again in the reflection-positivity (unitarity) refinement that pins down which classes are physical. The bridge is the equality of two computations of one phase: the curvature side is exactly the descent and index density of 03.07.32, while the holonomy side is the spectral asymmetry of 03.09.24; putting these together, the de Rham anomaly and the global anomaly are the local and torsion parts of a single deformation class. This is the foundational reason the homotopy-theoretic anomaly refines, and never contradicts, the differential-form anomaly: the de Rham face generalises to the $I_{Z}$ -valued class by adding the information that $η$ carries beyond its derivative.

Exercises Intermediate+

Exercise (hard, short-answer). Explain, in the invertible-theory language, what anomaly cancellation means for a collection of chiral fermions, and why summing $\eta$-invariants is the right operation.

Hint

Stacking theories multiplies their partition functions; what does that do to the bulk classes?

Answer

Rubric. Stacking decoupled fermions tensors their boundary theories and multiplies their partition functions, so the bulk anomaly theories tensor and their deformation classes add in $[M T H, Σ^{d + 2} I_{Z}]$ . Since $α_{i} (X) = exp (2 π i η_{i} /2)$ , the total phase is $exp (2 π i \sum_{i} η_{i} /2)$ , so the classes add via the sum of $η$ -invariants. Cancellation is $\sum_{i} [α_{i}] = 0$ , i.e. the total anomaly theory bounds and the combined boundary theory stands alone. Full credit links tensoring of theories to addition of classes and to the additivity of $η$ .

Exercise (hard, short-answer). The 3d parity anomaly of a single two-component Dirac fermion forces a half-integer Chern-Simons level. Frame this as a boundary-of-invertible-theory statement and say which invertible 4d theory plays the role of $\alpha$.

Hint

The bulk theory is built from the same $η$ -invariant; a half-level Chern-Simons term on the boundary is the inflow of a 4d $θ$ -type invertible theory.

Answer

Rubric. The 3d massless fermion partition function is a section of a Pfaffian line, not a number; making it a number requires a half-integer Chern-Simons counterterm whose fractional part is fixed, not free. In the invertible-theory language the 3d fermion is the boundary of a 4d invertible theory $α$ given by $exp (2 π i η /2)$ of the 4d Dirac operator (equivalently a $θ = π$ / level- $\frac{1}{2}$ bulk). The "parity anomaly" is that no boundary-only choice makes $α$ the identity while preserving parity; the bulk must absorb the half-unit. Full credit identifies $α$ as the $η$ /half-level 4d invertible theory and states that the fractional Chern-Simons level is inflow.

Advanced results Master

The Anderson-dual target and the universal coefficient sequence. The coefficient spectrum $I_{Z}$ for invertible theories is the Anderson dual of the sphere, characterized by a short exact sequence $$ 0 \to \mathrm{Ext}^1!\big(\pi_{n-1} MTH, \mathbb{Z}\big) \to [,MTH, \Sigma^{n+1} I_{\mathbb{Z}},] \to \mathrm{Hom}!\big(\pi_n MTH, \mathbb{Z}\big) \to 0 . $$ For anomalies one replaces $n + 1$ by $d + 2$ . The free quotient $Hom (π_{d + 1} M T H, Z)$ is the perturbative anomaly, paired against bordism classes by integrating the index density; the $Ext^{1}$ subgroup is precisely the torsion, the global anomalies. The $η$ -invariant theory realizes both pieces at once: its curvature computes the free pairing and its mod- $Z$ reduction computes the torsion pairing. This is the structural source of the slogan that $η$ "sees" anomalies the descent cannot.

Global anomalies and the mod-2 index. Witten's $S U (2)$ anomaly is the first global anomaly identified ^{[Witten 1982]}. A 4d theory with an odd number of $S U (2)$ doublet Weyl fermions has a partition function that changes sign under a large gauge transformation in $π_{4} (S U (2)) = Z /2$ . In the bordism language this sign is the mod-2 index of the 5d Dirac operator, valued in $Ω_{5}^{S p in} (B S U (2)) \supset Z /2$ , the $Ext$ -torsion of the relevant $I_{Z}$ -cohomology. The anomaly theory $α$ is the nonzero element; it has zero curvature (vanishing perturbative anomaly) and is detected only by its holonomy $exp (2 π i η /2) = \pm 1$ . Witten's later treatment of global gravitational anomalies ^{[Witten 1985]} is the same mechanism with $H = Spin$ and no gauge bundle.

Reflection positivity selects the physical classes. Not every class in $[M T H, Σ^{d + 2} I_{Z}]$ arises from a unitary theory. Freed-Hopkins impose reflection positivity, the Euclidean form of unitarity (the Osterwalder-Schrader positivity pointed to in 08.10.07), which cuts the deformation classes down to the physical subgroup and is what makes the classification a theorem about actual quantum field theories rather than formal functors ^{[Freed-Hopkins 2021]}. The $η$ -invariant theory is reflection-positive, which is why it is the canonical representative.

Synthesis. The invertible-theory reframing is dual to the descent picture of 03.07.32 in a precise sense: the central insight is that one deformation class in $[M T H, Σ^{d + 2} I_{Z}]$ carries both the de Rham anomaly (its free, curvature part) and the global anomaly (its torsion, $η$ part), so the two registers are images of one object under the universal coefficient sequence. This is exactly the bordism-theoretic completion of the index-theorem anomaly: the foundational reason a perturbatively anomaly-free theory can still be inconsistent is that the map to de Rham cohomology forgets the $Ext$ -torsion that $η$ retains. The construction generalises the Pfaffian-line and determinant-line geometry of 03.09.24 from a single number to a field theory, and it builds toward the reflection-positive classification of 03.16.04 by exhibiting $exp (2 π i η /2)$ as a physical cocycle. Putting these together, anomaly cancellation, anomaly inflow, the parity anomaly, and Witten's $S U (2)$ anomaly become four facets of the single question of whether a bulk invertible theory bounds, and the answer appears again in the SPT/SET classification of gapped phases, where the same $M T H$ -cohomology groups label topological phases of matter.

Full proof set Master

Proposition (invertibility of the $η$ -theory). The assignment $α (X) = exp (2 π i η (D_{X}) /2)$ , with state spaces the Pfaffian lines on codimension-one data, is an invertible field theory: every closed value lies in $C^{\times}$ and every state space is $\otimes$ -invertible.

Proof. For closed $X$ the value $exp (2 π i η /2)$ is a unit-modulus complex number, hence in $C^{\times}$ . The state space assigned to a closed $d$ -manifold $M$ is the Pfaffian line $Pf (D_{M})$ , a one-dimensional complex line; a one-dimensional line is $\otimes$ -invertible with inverse its dual $Pf (D_{M})^{*}$ . Gluing compatibility follows from the APS gluing law for $η$ under cutting along $M$ : the bulk $η$ splits into boundary contributions plus an integer determined by the index on the closed-up pieces, and the integer drops out of $exp (2 π i η /2)$ up to the controlled Maslov/APS correction absorbed into the Pfaffian-line identification. Thus $α$ is monoidal, invertible, and lands in the Picard subgroupoid. $■$

Proposition (the de Rham image is the consistent anomaly). The curvature of the Pfaffian line bundle $L_{α} = Pf (D) \to B$ equals the degree- $(d + 2)$ component of the index density $\int_{X / B} \hat{A} (R) ch (F)$ , and its descent reproduces the Wess-Zumino consistent anomaly of 03.07.32.

Proof. The Bismut-Freed curvature theorem for the determinant/Pfaffian line bundle of a family of Dirac operators states that the curvature of the Bismut-Freed connection is the appropriate component of the families index density $\int_{X / B} \hat{A} (R) ch (F)$ ^{[Dai 1994]}. Restricting to the relevant degree gives a closed form on $B$ whose cohomology class is the de Rham anomaly. Transgressing this closed form along the gauge/diffeomorphism directions of $B$ yields the Chern-Simons-type secondary form, and the descent equations $s ω^{0} = - d ω^{1}$ of 03.07.32 express the variation of the local effective action; the integrated $ω^{1}$ is the consistent anomaly satisfying the Wess-Zumino condition. Hence the curvature (free) part of $[α]$ is exactly the perturbative anomaly. $■$

Proposition (global anomaly as a mod-2 index). For a 4d theory with an odd number of $S U (2)$ -doublet Weyl fermions, the anomaly class $[α]$ is the nonzero element of $Z /2 \subset [M T H, Σ^{6} I_{Z}]$ , with zero de Rham image and holonomy $- 1$ .

Proof. The relevant bulk is 5d spin geometry with an $S U (2)$ bundle; the deformation group receives a $Z /2$ from $Ω_{5}^{S p in} (B S U (2))$ , the torsion captured by the $Ext^{1}$ term of the universal coefficient sequence above. The bulk partition function $exp (2 π i η /2)$ reduces, for this $Z /2$ -bordism, to $(- 1)^{ind_{2} D}$ , the mod-2 index of the 5d Dirac operator. A single doublet contributes the nonzero class; an odd number sums to the nonzero class, an even number to zero. The free quotient $Hom (π_{5} M T H, Z)$ vanishes on this class, so the de Rham image is zero, matching Witten's observation that the six-form perturbative anomaly of $S U (2)$ vanishes while the partition function still changes sign ^{[Witten 1982]}. $■$

The reflection-positivity reduction and the full Dai-Freed holonomy theorem are stated above without reproof; see Dai-Freed ^{[Dai 1994]} for the holonomy formula and Freed-Hopkins ^{[Freed-Hopkins 2021]} for the reflection-positive classification.

Connections Master

Invertible field theories and the Freed-Hopkins classification 03.16.04. This unit is the anomaly specialization of that one: the anomaly theory $α$ is an invertible $(d + 1)$ -theory, and the anomaly is its deformation class in $[M T H, Σ^{d + 2} I_{Z}]$ , the same $M T H$ /Anderson-dual machinery shifted up one degree. The reflection-positivity datum that selects physical invertible theories there is what makes the $η$ -theory here the canonical anomaly representative.
Anomalies via descent equations and the index theorem 03.07.32. That unit develops the de Rham face: consistent anomaly, Wess-Zumino condition, inflow $2 n + 2 \to 2 n$ via Chern-Simons descent and the index density. This unit shows that face is the free, curvature part of a single homotopy class; the global, torsion anomalies that $η$ detects are exactly what the descent cannot see, so the two are the local and torsion halves of one object.
Eta invariant and the Atiyah-Patodi-Singer index theorem 03.09.24. The APS $η$ -invariant is the value of the bulk invertible theory: $α (X) = exp (2 π i η (D_{X}) /2)$ . The spectral asymmetry, holonomy dependence, and Pfaffian-line geometry developed there are precisely the inputs that compute the global anomaly phase here.
Chiral ABJ anomaly and the triangle diagram 12.18.05. This is the perturbative physics face of the same phenomenon: the triangle-diagram axial anomaly is the lowest-order avatar of the curvature/index density, the de Rham image of the deformation class. The invertible-theory framing is the nonperturbative completion that also accounts for the torsion the triangle diagram misses.
Wightman/Osterwalder-Schrader axioms 08.10.07. Reflection positivity, the Euclidean form of unitarity from the OS framework, is the datum Freed-Hopkins impose to cut the formal deformation classes down to those realized by genuine quantum theories; it is why the anomaly classification is a statement about physics and not only about functors.

Historical & philosophical context Master

The reinterpretation of anomalies as invertible field theories is due to Daniel Freed, in lectures and in the 2014 paper "Anomalies and invertible field theories" ^{[Freed 2014]}, building on the determinant-line geometry he and Bismut developed in the 1980s and on the Dai-Freed holonomy theorem ^{[Dai 1994]}. The global-anomaly phenomenon that this framework subsumes was found earlier by Edward Witten: the $S U (2)$ anomaly in 1982 ^{[Witten 1982]} and the bordism/ $η$ -invariant formulation of global gravitational anomalies in 1985 ^{[Witten 1985]}. The classification of the resulting invertible theories by $[M T H, Σ^{n + 1} I_{Z}]$ was proved by Freed and Hopkins ^{[Freed-Hopkins 2021]}, using the Anderson dual of the sphere as the coefficient spectrum and reflection positivity as the unitarity constraint.

The modern synthesis by Witten and Yonekura ^{[Witten-Yonekura §3]} makes the inflow precise via the APS theorem on manifolds with boundary, identifying the boundary phase of an anomalous theory with the bulk $η$ -invariant. The de Rham descent treatment of Stora and Zumino, recorded in 03.07.32, remains the perturbative shadow of this homotopy-theoretic statement; the index theorem of Atiyah-Singer and its APS extension supply the bridge between the two.

Bibliography Master

@article{Freed2014Anomalies,
  author  = {Freed, Daniel S.},
  title   = {Anomalies and invertible field theories},
  journal = {Proceedings of Symposia in Pure Mathematics},
  volume  = {88},
  pages   = {25--46},
  year    = {2014},
  note    = {arXiv:1404.7224}
}

@article{FreedHopkins2021,
  author  = {Freed, Daniel S. and Hopkins, Michael J.},
  title   = {Reflection positivity and invertible topological phases},
  journal = {Geometry \& Topology},
  volume  = {25},
  pages   = {1165--1330},
  year    = {2021}
}

@article{Witten1982SU2,
  author  = {Witten, Edward},
  title   = {An {SU(2)} anomaly},
  journal = {Physics Letters B},
  volume  = {117},
  pages   = {324--328},
  year    = {1982}
}

@article{Witten1985Global,
  author  = {Witten, Edward},
  title   = {Global gravitational anomalies},
  journal = {Communications in Mathematical Physics},
  volume  = {100},
  pages   = {197--229},
  year    = {1985}
}

@article{DaiFreed1994,
  author  = {Dai, Xianzhe and Freed, Daniel S.},
  title   = {{$\eta$}-invariants and determinant lines},
  journal = {Journal of Mathematical Physics},
  volume  = {35},
  pages   = {5155--5194},
  year    = {1994}
}

@article{WittenYonekura2019,
  author  = {Witten, Edward and Yonekura, Kazuya},
  title   = {Anomaly inflow and the eta-invariant},
  journal = {arXiv preprint},
  year    = {2019},
  note    = {arXiv:1909.08775}
}

@book{FreedCBMS2019,
  author    = {Freed, Daniel S.},
  title     = {Lectures on Field Theory and Topology},
  series    = {CBMS Regional Conference Series in Mathematics},
  number    = {133},
  publisher = {American Mathematical Society},
  year      = {2019}
}

Prerequisites

03.16.04 pending
03.07.32 pending
03.09.24 pending

Tier anchors

beginner: Freed CBMS Lectures on Field Theory and Topology, Lecture 11 (informal anomaly-as-edge picture); 3Blue1Brown-paced phase / holonomy account
intermediate: Freed 2014 'Anomalies and invertible field theories' (arXiv:1404.7224) §§1--3; Witten-Yonekura 2019 (arXiv:1909.08775) §§2--4
master: Freed CBMS Lectures on Field Theory and Topology, Lectures 9--11; Freed-Hopkins 2021 Geom. Topol. 25; Witten 1985 Comm. Math. Phys. 100 (global anomaly)

References

free-downloads/Freed-CBMS_Field_Theory_and_Topology.pdf · Lectures 9--11 (long-range theories, invertible theories, anomalies)
Freed, D. S. — Anomalies and invertible field theories · arXiv:1404.7224; Proc. Symp. Pure Math. 88 (2014), 25--46 — anomaly = deformation class of an invertible theory
Freed, D. S. & Hopkins, M. J. — Reflection positivity and invertible topological phases · Geom. Topol. 25 (2021), 1165--1330 — the classification group [MTH, Σ^{n+1} I_Z]
Witten, E. — An SU(2) anomaly · Phys. Lett. B 117 (1982), 324--328 — the global anomaly as a mod-2 index
Witten, E. — Global gravitational anomalies · Comm. Math. Phys. 100 (1985), 197--229 — eta-invariant / bordism formulation of global anomalies
Witten, E. & Yonekura, K. — Anomaly inflow and the eta-invariant · arXiv:1909.08775 — Dai-Freed theorem, APS bulk and the boundary phase
Dai, X. & Freed, D. S. — eta-invariants and determinant lines · J. Math. Phys. 35 (1994), 5155--5194 — the Pfaffian line and the APS holonomy

Estimated time

beginner: 16m
intermediate: 42m
master: 82m