Eta-Invariants and Determinant Lines

Daniel S. Freed; Xianzhe Dai

REVIEW

Eta-Invariants and Determinant Lines

Xianzhe Dai, Daniel S. Freed

High Energy Physics - Theory · Submitted 2 May 1994 · 8 references

Pith's verdict: A solid, well-executed paper that recasts the exponentiated eta-invariant as a unit-norm element of the boundary determinant line and gets a cleaner holonomy proof out of it; the delicate §4 symmetry step holds up.

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arXiv IDhep-th/9405012 (version 4)

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CitationDaniel S. Freed, Xianzhe Dai. "Eta-Invariants and Determinant Lines." arXiv:hep-th/9405012 (1994). https://pith.science/paper/KP27HDL4

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The thing to know: this is the paper that puts the exponentiated xi-invariant where it belongs, as a unit-norm element of the inverse determinant line of the boundary, and then cashes that in for a genuinely simpler proof of the Bismut-Freed holonomy (global anomaly) formula. The geometric move is the contribution, and it is correct as far as I can tell. Proposition 2.15 is the load-bearing identification, and it is done cleanly: the boundary-condition dependence (1.5)-(1.6) is exactly what makes the patching construction of the line work, and the Quillen metric comes out with unit norm for free. That is the right way to see the invariant, and it is why people cite this paper rather than the earlier boundary-condition computations it supersedes in spirit. The gluing law (Theorem 2.20) is the other real result, and it is more general than Bunke's because it handles nonzero index by working with graded determinant lines. The graded-line bookkeeping in §2 is fussy but the signs are right, and the cylinder computation (Prop 4.7) is done directly rather than assumed, so there is no circularity between §2 and §4. The §3 variation formula is proven with the analytic details pushed to Appendix A, and the boundary-term cancellation in Prop 3.14 is standard APS-type local cancellation, consistent with the known (3.9). The stress-test's worry about the §4 symmetry argument (eq. 4.21) does not land: IQ = -QI follows from the evenness of theta under x -> -x, which is exactly what f_L(-x) = f_R(x) in (4.19) buys you, and Q e^{-tD^2} is trace class for t > 0, so the cyclic trace chain is fine. Soft spots, in proportion: the novelty is benchmarked partly against 1993 preprints (Bunke, Lesch-Wojciechowski, Muller, Wojciechowski) that a referee at the time could not check, so the 'new and simpler' claim rests on unpublished comparanda; and the headline is a re-proof of a known theorem, so significance is bounded even if the proof is nicer. The §6 conjecture is explicitly labeled and not load-bearing. This is for people working in geometric index theory and anomalies; it deserves a serious referee and a place in the literature.

Referee report

Significance

If correct, this is a substantial and clarifying contribution. The geometric reformulation of τ_X as a section of the inverse determinant line is the right home for the invariant, and it converts the awkward boundary-condition dependence of Theorem 1.4 into a statement about a line bundle with a natural metric and connection. The gluing law is genuinely more general than the usual cut-a-closed-manifold-into-two-pieces statement, because the graded determinant line formalism handles nonzero index on the cutting hypersurface; the authors are explicit that this generality is what §5 needs when gluing cylinders. The proof of the holonomy formula avoids long-time heat kernel asymptotics and non-pseudodifferential operators, which is a real simplification over the earlier route, and the argument that the adiabatic limit defines parallel transport for a connection, which is then identified with the natural one via the variation formula, is conceptually clean. The paper also ships a falsifiable conjecture (6.8) with the degree-one case verified, and it is candid that the general case is expected to follow from independent work of Melrose and Piazza. The analytic foundations are supplied in

Major objections (1)

[§2 and §5] The family extension of the main results is asserted rather than proved, and it is load-bearing for the headline claim. In §2, immediately after (2.18), the text states that the properties of L_Y and τ_X 'work as stated for families, and the proofs are designed to work with the patching construction of §1.' §5 then uses the family version twice: to define τ_γ as a map L_{γ(0)} → L_{γ(1)} for a path of spin maps Y_γ → [0,1], and to conclude the composition law τ_{γ2∘γ1} = τ_{γ2}∘τ_{γ1} 'from the gluing law (Theorem 2.20).' Theorem 2.20 itself is stated and proved (Proposition 4.5) for a single manifold, and Proposition 4.5's reduction to (4.6) fixes a single pair ⟨a',T'⟩. Since the construction of ∇' in §5 and the identification ∇' = ∇ in Proposition 5.7 both rest on this extension, the paper should either state the family version of Theorem 2.20 as a separate theorem with its own proof, or state explicitly and prominently that it is being used as a hypothesis. As written, a reader cannot tell from §4 alone whether the symmetry argument of (4.21) survives the patching over U_{a,i}.

Minor objections (10)

[§1, eq. (1.7) and following sentence] The hermitian metric on L_∂X is written as (τ1,τ2) = τ1(a,T)τ2(a,T). For this to be a hermitian metric it must be sesquilinear, i.e. τ1(a,T) times the complex conjugate of τ2(a,T); as printed the expression is bilinear and (τ,τ) = τ² is not positive. This is presumably a typesetting slip, but it occurs in the definition of the metric that Proposition 2.15 later identifies with the Quillen metric, so it should be corrected.
[§2, proof of Proposition 2.15] The proof says 'Then (1.7) and (2.21) imply that the isomorphism (2.16) is an isometry.' Equation (2.21) is the gluing law, which is not what is needed here; the relevant input is the Quillen metric normalization following (2.12), namely |θ(a,b)ω(a)|²_{K(b)} = (∏_{a<λ<b} λ)|ω(a)|²_{K(a)}. The cross-reference should be corrected.
[§2, after Theorem 2.20] The sentence 'We prove Theorem 2.20 in §3' is inconsistent with §4, which opens 'In this section we prove Theorem 2.20.' The reference should read §4.
[Theorem 1.9] The theorem is stated for a spin map π : X → Y, whereas §1 and the right-hand side of (1.10) use Z for the base. The notation should be harmonized, since the same symbol Y is used throughout §2 and §4 for a closed even-dimensional spin manifold, which is a different object.
[§2, eq. (2.19)] The identity ι̃* = (-1)^{index D_Y} for the spin involution is stated without justification, and (2.25) depends on it to fix the sign of τ_{S¹_bounding × Y}. A one-line indication of where the sign comes from, or a pointer to the determinant-line computation, would make the sign check in (2.24)–(2.25) self-contained.
[§2, after (2.18)] The relations τ_{-X} = τ_X^{-1} and τ_{X1⊔X2} = τ_{X1} ⊗ τ_{X2} are asserted immediately after the corresponding line isometries (2.17) and (2.18). Since these are used implicitly in §5 when τ_{Y_γ}(ε) is regarded as a map L_{γ(0)} → L_{γ(1)}, a sentence spelling out the compatibility with the isometries would help.
[§5, proof of Lemma 5.4] The existence of the adiabatic limit is the crux of the section, and the argument is compressed to 'any term in the integrand involving dt approaches zero as ε → 0.' Since (5.5) only controls the limiting connection and curvature, one more sentence explaining why the surviving terms in [∫_p Â(Ω_p)]^{(1)} are exactly those carrying dt would make the step verifiable by a reader rather than plausible.
[Appendix A, Lemma A.9(ii)] Hypothesis (Hc), used in Proposition A.17 and hence in Lemma 3.10 and Theorem 3.3, requires the coefficients a_j(P(z)) to be smooth in z. Lemma A.9(ii) states the expansion for a fixed operator and describes a_j, b_j as integrals of local densities computable from the symbol, but does not say that these densities depend smoothly on the parameter. A sentence closing that loop would make the analytic foundation of the variation formula fully explicit.
[References] Reference [Q] gives the journal as 'Funk. Anal. iprilozen', which appears to be a garbled transliteration of Funktsional'nyi Analiz i Ego Prilozheniya. The reference should be corrected.
[§6, Conjecture 6.8] The conjecture should restate explicitly the hypotheses used in the paragraph above it, namely that ker D_∂X/Z has constant rank and that T is a smooth family of isometries between the kernel bundles. Theorem 6.9 is proved only for the degree-one component, and the text is appropriately clear about this, but the conjecture's statement would read better if its hypotheses were self-contained.

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The paper

The paper claims

The exponentiated eta-invariant of a manifold with boundary is a unit-norm element of the inverse determinant line of its boundary; that placement yields a gluing law, a variation formula, and a shorter proof of the global anomaly formula.

The reading

The eta-invariant of Atiyah–Patodi–Singer measures the spectral asymmetry of a Dirac operator, and on a closed odd-dimensional spin manifold its exponentiated form $e^{2i}$ is the piece that moves smoothly with the metric. Dai and Freed ask what happens when the manifold has a boundary and the Dirac operator is made self-adjoint by a global boundary condition encoded in a unitary $T$ between the kernel pieces of the boundary operator. Their answer is that the dependence on $T$ is exactly the patching rule for the inverse determinant line of the boundary, so the exponentiated invariant is best read not as a complex number but as a unit-norm vector in that line. Once stated that way, two structural properties fall out: a gluing law for cutting along a closed hypersurface, and a formula for the covariant derivative with respect to the natural connection on the determinant line bundle. The payoff is a shorter proof of the holonomy formula, or global anomaly formula, which computes the parallel transport of that connection as an adiabatic limit of exponentiated eta-invariants.

What carries the argument

The load-bearing device is the reinterpretation of a boundary condition as a trivialization of a graded line. A boundary condition is an isometry $T: K^+_{X}(a)K^-_{X}(a)$; changing $T$ multiplies $X$ by $(T_1^{-1}T_2)$, which is exactly how a trivialization of $^{-1}X$ changes, so the whole collection of numbers $X(a,T)$ is one element of that line. Graded determinant lines — where swapping two factors costs a sign and $(-V) (V)^{-1}$ — are what make the gluing signs come out; the authors say plainly that this is the best way to avoid a 'cauchemar de signes'. The gluing proof patches spinor fields as Bunke did, then builds an explicit unitary $U$ and a difference operator $Q$ supported on two cylinders, and kills the variation $d/du$ by an involution $I$ on those cylinders with $I^2=-1$, $ID=-DI$, $IQ=-QI$, forcing $(Qe^{-tD^2})$ to equal its own negative. The variation formula is proved by conjugating a varying boundary condition to a fixed one and extracting a finite part of a heat-kernel trace, with the boundary contribution shown to vanish.

Extended reading notes

Core claim

Proposition 2.15 identifies the hermitian line $L_X$ built from boundary conditions in (1.7) with the inverse determinant line $^{-1}X$ of the boundary Dirac operator, and exhibits $X = e^{2i_X(a,T)}$ as a unit-norm element of it; the boundary-condition change law $X(a,T_2) = (T_1^{-1}T_2)X(a,T_1)$ is precisely the transition function of that line. Theorem 2.20 then states a gluing law: cutting $X$ along a closed hypersurface $Y$ and contracting the two copies of $L_Y$ with the supertrace recovers $X$, with a sign $(-1)^{D_Y}$ that only makes sense inside the graded determinant lines of Knudsen–Mumford and Deligne. Theorem 1.9 states that the covariant derivative of the section $_{X/Z}$ with respect to the Bismut–Freed connection is $2i_{X/Z}(_{X/Z})^{(1)}$ times $_{X/Z}$. Section 5 turns these two results into the holonomy theorem: gluing shows the adiabatic limit of $ $ on cylinders defines a connection, and the variation formula alone then identifies it with the Bismut–Freed connection. Section 6 conjectures the corresponding correction term in the odd-dimensional families index theorem, and proves its degree-one case.

Load-bearing premise

The gluing law, and through it the holonomy proof, rests on the involution $I$ on the two cylinders satisfying $IQ=-QI$, which is what forces the variation of the xi-invariant to vanish; if that relation or the trace manipulation behind it fails, the gluing law and the short holonomy proof fail with it.

Editorial extensions

If this is right

The dependence of the exponentiated eta-invariant on the choice of global boundary condition is fully accounted for by the determinant line, so no separate bookkeeping is needed when boundary conditions vary in a family.
The holonomy of the Bismut–Freed connection on the inverse determinant line bundle equals the adiabatic limit of exponentiated eta-invariants, with the sign $(-1)^{D_Y}$ fixed by the grading (Corollary 5.11).
The proof avoids long-time heat kernel asymptotics and non-pseudodifferential operators, replacing them with gluing plus the variation formula.
The degree-one component of the odd Chern character of the index bundle of a family of Dirac operators on odd-dimensional manifolds with boundary is represented by $_{X/Z}(_{X/Z}) - - _T$ (Theorem 6.9), with the full odd-degree statement left as Conjecture 6.8.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the exponentiated eta-invariant really is a local action taking values in a line, the same packaging should work for other invariants on odd-dimensional manifolds with boundary; the paper itself draws the analogy with classical Chern–Simons theory while noting that, unlike Chern–Simons, this invariant is not multiplicative under coverings.
The gluing-plus-variation template looks portable: any invariant that obeys a cutting law with graded signs and has a computable covariant derivative is pinned down by those two facts alone, which is a general strategy for identifying a connection rather than a fact about Dirac operators.
Conjecture 6.8, if proved in all degrees, would supply the missing boundary-kernel correction in the odd-dimensional families index theorem; the degree-one case already follows because the correction term collapses to $-(T^{-1}^a T)$.
The residual ambiguity in the answer is the phase $(T_1^{-1}T_2)$ attached to a change of trivialization, so a reader wanting a parameter-free statement should expect to quotient by that choice — which is exactly what working inside the line accomplishes.

Original abstract

We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). A postscript file with figures was submitted separately in uuencoded tar-compressed format.

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arXiv listing abstract v4. Results and the main claim are underlined; show every role.

We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). A postscript file with figures was submitted separately in uuencoded tar-compressed format.

background contribution method result interpretation main claim

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s0	We study eta-invariants on odd dimensional manifolds with boundary.	contribution 0.63	0.84	0.04	asserted 0.89
s1	The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary.	interpretation 0.79	0.36	0.28	asserted 0.96
s2	We prove a gluing law and a variation formula for this invariant.	contribution 0.55	0.96	0.34	asserted 1.00
s3	This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula.	result 0.68	0.90	0.48	asserted 1.00
s4	This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included).	method 0.70	0.19	0.01	asserted 0.99
s5	A postscript file with figures was submitted separately in uuencoded tar-compressed format.	method 0.53	0.54	0.01	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v4

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References

Works cited (8)

[1]

[APS] M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69; II, Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432; III, Math. Proc. Cambridge Philos. Soc. 79 (1976), 71–99. [Bis] J. M. Bismut, The Atiyah-Singer Index Theorem for families of Dirac opera tors: two he...

View on Pith 1975
[8]

[W] E. Witten, Global gravitational anomalies , Commun. Math. Phys. 100 (1985), 197–229. 52 [Wo1] K. P. Wojciechowski, The additivity of the η-invariant. The case of an invertible tangential operator, preprint. [Wo2] K. P. Wojciechowski, The additivity of the η-invariant. The case of a singular tangential operator, preprint. Department of Mathematics, Uni...

View on Pith Pith review arXiv 1985
[37]

[RS] D. B. Ray, I. M. Singer, R-torsion and the laplacian on Riemannian manifolds , Adv. Math. 7 (1971), 145–210. [Se] R. T. Seeley, Topics in pseudo-diﬀerential operators , CIME, Edizioni Cremonese, Roma 1969, pp. 169–305. [S] G. Segal, The deﬁnition of conformal ﬁeld theory , preprint. [Si] I. M. Singer, The η-invariant and the index , Mathematical Aspe...

View on Pith 1971
[1965]

[Q] D. Quillen, Determinants of Cauchy-Riemann operators over a Riemann su rface, Funk. Anal. iprilozen 19 (1985),

View on Pith 1985
[1982]

[B1] U. Bunke, A glueing-formula for the η-invariant, 1993 preprint. [B2] U. Bunke, The η-invariant as a lagrangian of a topological quantum ﬁeld the ory, 1993 preprint. [BGV] N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators. , Springer-Verlag, Berlin,

View on Pith 1993
[1987]

[F2] D. S. Freed, Classical Chern-Simons Theory, Part 1 , Adv. Math. (to appear). [F3] D. S. Freed, Characteristic numbers and generalized path integrals , to appear in 70th birthday volume in honor of Raoul Bott. [F4] D. S. Freed, A gluing law for the index of Dirac operators , Global Analysis in Modern Mathematics (K. K. Uhlenbeck, ed.), Publish or Peri...

View on Pith 1983
[1992]

[C1] J. Cheeger, Spectral geometry of singular Riemannian spaces , J. Diﬀ. Geom. 18 (1983), 575–657. [C2] J. Cheeger, η-invariants, the adiabatic approximation and conical sing ularities, J. Diﬀ. Geom. 26 (1987), 175–221. [D1] X. Dai, Eta invariants on manifolds with boundary , 1994 preprint. [D2] X. Dai, Adiabatic limits, the non-multiplicativity of sign...

View on Pith 1983
[1994]

[MP] R. B. Melrose, P. Piazza, Families of Dirac operators, boundaries and the b-calculus, 1993 preprint. [M¨ u] W. M¨ uller, Eta invariants and manifolds with boundary , 1993 preprint. [P] R. S. Palais, ed., Seminar on the Atiyah-Singer index theorem , Ann. of Math. Studies 57, Princeton University Press, Princeton,

View on Pith 1993

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The paper derives the nonperturbative anomaly inflow formula Z = |Pf D+| exp(-iπη/2) from the Euclidean path integral, justifying a result previously expected from the Dai-Freed theorem.
Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
hep-th 2018-12 conditional novelty 6.0 of 10

A generalized Thom-Madsen-Tillmann and Adams spectral sequence machinery is used to compute bordism groups of higher classifying spaces, giving tables of bordism invariants that classify higher-symmetry-protected topo...
A Non-Perturbative Definition of the Standard Models
hep-th 2018-09 conditional novelty 6.0 of 10

Cobordism classification of Spin(10) and SU(5) anomalies shows the mirror sector of the corresponding chiral fermion theories can be gapped, giving the Standard Model a non-perturbative lattice definition.
Dai-Freed anomalies in particle physics
hep-th 2018-07 conditional novelty 6.0 of 10

Bordism-based Dai-Freed anomaly checks leave the Standard Model and its unified extensions anomaly-free, but impose a mod-9 constraint on baryon triality that only three generations satisfy.
Why are fractional charges of orientifolds compatible with Dirac quantization?
hep-th 2018-05 accept novelty 6.0 of 10

The apparent conflict between fractional orientifold charges and Dirac quantization is resolved because the worldvolume fermion anomaly of the dual probe brane supplies a compensating phase, computed via eta invariant...
On time-reversal anomaly of 2+1d topological phases
hep-th 2016-10 conditional novelty 6.0 of 10

A crosscap-geometry momentum calculation determines the Z16 time-reversal anomaly of 2+1d topological phases, reproducing known values for semion-fermion, T-Pfaffian, and gapped free-fermion theories.
The Standard Model on a finite bosonic lattice with continuum Weyl fermions
hep-th 2026-10 accept novelty 5.0 of 10

A finite-bosonic-lattice regulator for the Standard Model is constructed with continuum Weyl fermions and a gauge-invariant chiral fermion determinant including Yukawa interactions.
Minimal Majoron Dark Matter from a Discrete $Z_N$ Gauge Symmetry
hep-ph 2026-07 conditional novelty 5.0 of 10

Discrete Z_N-protected majoron dark matter excludes Z_5, leaves Z_7/Z_11/Z_13 viable, and predicts a 1–10 MeV Z_7 majoron testable by COSI through 511 keV and γγ lines.
New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and $\mathbb{CP}^{\mathrm{N}-1}$ Sigma Model
hep-th 2018-12 conditional novelty 5.0 of 10

The authors propose a correspondence between 5d cobordism invariants for 4d SU(N) Yang-Mills at theta = pi and 3d cobordism invariants for 2d CP^{N-1} sigma models at theta = pi, derived via dimensional reduction on a...
Dai-Freed theorem and topological phases of matter
hep-th 2016-07 conditional novelty 5.0 of 10

A massive fermion's ground-state wave function, equal to det T in the large-mass limit, yields a physics derivation of the Dai-Freed eta-invariant theorems.
Ghosts that Connect
physics.hist-ph 2026-07 conditional novelty 4.0 of 10

The ghost field is a principal connection on the field-space bundle A→A/G, and BRST is the vertical exterior derivative along gauge orbits.
On the cobordism classification of symmetry protected topological phases
hep-th 2018-03 conditional novelty 4.0 of 10

Unitary invertible topological field theories with symmetry group H are in one-to-one correspondence with cobordism invariants Hom(Omega^H_d, U(1)), proved from Atiyah's axioms.
Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond
hep-th 2022-05 unverdicted novelty 2.0 of 10

This review summarizes transformative examples of generalized symmetries in QFT and their applications to anomalies and dynamics.
Spectral Networks: Bridging higher-rank Teichm\"uller theory and BPS states
math-ph 2024-11 unverdicted

A comprehensive introduction to spectral networks that develops higher-rank Teichmüller theory in parallel with class S gauge theory and BPS spectra.

How the literature uses it

80 arXiv papers cite this in 149 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 97 sentences without a confident label

Builds on it 1 paper

A particularly interesting example would involve $η$-invariants on odd-dimensional manifolds with corners, which should be related to chiral anomalies in even dimensions and extend Dai-Freed theories [ref] .

Extended quantum field theory, index theory and the parity anomaly · The extended quantum field theory $𝒜_parity^ζ$ · 0.88

Uses its method or tool 13 papers

Then, the path integral ([ref]) is changed by a gauge-diffeomorphism transformation $π$ on the boundary as [equation] In the third line, we use the Dai-Freed theorem [ref] .

Anomaly Inflow of Rarita-Schwinger Field in 3 Dimensions · Spin Manifold · 0.89
To identify which anomaly class is realized by the type II worldsheet theory, we relate these bordism invariants to reduced eta invariants of three-dimensional Dirac operators [ref] .

What makes spacetime spin in string theory? · Introduction · 0.97
This non-perturbative nature of the anomalies requires the use of the techniques developed in [ref] , see also [ref] .

(Quadratically) Refined Discrete Anomaly Cancellation · Anomalies for chiral fields · 0.97

Compares with it 1 paper

Then we have[There is a few sign differences between the formulas in this paper and those in [ref] .

Dai-Freed theorem and topological phases of matter · Description of the theorems · 0.91

Cites it as support 1 paper

As a nonextended field theory of 2- and 3-manifolds this follows from the theorems in [ref] .

The Sum Over Topological Sectors and $\theta$ in the 2+1-Dimensional $\mathbb{C}\mathbb{P}^1$ $\sigma$-Model · $Spin^c$ manifolds · 0.98

Cites it as background 29 papers

What we call the Dai-Freed theorem [ref] is actually a set of theorems regarding Dirac operators on manifolds with boundary.

Dai-Freed theorem and topological phases of matter · Introduction and summary · 0.85
This ambiguity is called the Dai-Freed anomaly [ref] .

Anomaly Inflow of Rarita-Schwinger Field in 3 Dimensions · Global Anomalies · 0.88
More concretely, the anomaly inflow is generalized by the $η$ invariant of a Dirac operator on the $d+1$-dimensional manifold [ref] , where a nontrivial boundary condition known as the APS boundary condition [ref] is imposed.

Mod-two APS index and domain-wall fermion · Introduction · 0.83

Discussion

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[W] E. Witten, Global gravitational anomalies , Commun. Math. Phys. 100 (1985), 197–229. 52 [Wo1] K. P. Wojciechowski, The additivity of the η-invariant. The case of an invertible tangential operator, preprint. [Wo2] K. P. Wojciechowski, The additivity of the η-invariant. The case of a singular tangential operator, preprint. Department of Mathematics, Uni...

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[B1] U. Bunke, A glueing-formula for the η-invariant, 1993 preprint. [B2] U. Bunke, The η-invariant as a lagrangian of a topological quantum ﬁeld the ory, 1993 preprint. [BGV] N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators. , Springer-Verlag, Berlin,

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[F2] D. S. Freed, Classical Chern-Simons Theory, Part 1 , Adv. Math. (to appear). [F3] D. S. Freed, Characteristic numbers and generalized path integrals , to appear in 70th birthday volume in honor of Raoul Bott. [F4] D. S. Freed, A gluing law for the index of Dirac operators , Global Analysis in Modern Mathematics (K. K. Uhlenbeck, ed.), Publish or Peri...

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[C1] J. Cheeger, Spectral geometry of singular Riemannian spaces , J. Diﬀ. Geom. 18 (1983), 575–657. [C2] J. Cheeger, η-invariants, the adiabatic approximation and conical sing ularities, J. Diﬀ. Geom. 26 (1987), 175–221. [D1] X. Dai, Eta invariants on manifolds with boundary , 1994 preprint. [D2] X. Dai, Adiabatic limits, the non-multiplicativity of sign...

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[MP] R. B. Melrose, P. Piazza, Families of Dirac operators, boundaries and the b-calculus, 1993 preprint. [M¨ u] W. M¨ uller, Eta invariants and manifolds with boundary , 1993 preprint. [P] R. S. Palais, ed., Seminar on the Atiyah-Singer index theorem , Ann. of Math. Studies 57, Princeton University Press, Princeton,

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