Anomalies in String Theory with D-Branes

Daniel S. Freed; Edward Witten

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Anomalies in String Theory with D-Branes

Daniel S. Freed, Edward Witten

High Energy Physics - Theory · Submitted 26 Jul 1999 · 6 references

Pith's verdict: A genuine perturbative derivation of the Spin-c condition on D-branes, with one acknowledged geometric gap that matters less than it looks because the relevant connection is flat.

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arXiv IDhep-th/9907189 (version 2)

Pith ID52EFWUXY, Pith's permanent record of this paper and its review

CitationDaniel S. Freed, Edward Witten. "Anomalies in String Theory with D-Branes." arXiv:hep-th/9907189 (1999). https://pith.science/paper/52EFWUXY

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Editor's note

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The thing to know: this paper computes the open-string global anomaly directly on the worldsheet and gets w2(nu) as the sign, which forces the A-field to be a Spin-c connection rather than a U(1) one. That condition was already known from K-theory in Witten's earlier work, and the authors say so plainly in section 1 and again in section 6. What is new is the derivation, and it is a real derivation, not a restatement. The reader's take is right on this and the circularity worry is correctly low: [W3] and [W4] are used as a consistency check on a condition obtained independently here, not as premises. The abstract's framing at [abstract:s2] is a bit grander than the body, which is really about the anomaly itself, but that is a small thing. What the paper does well: Proposition 5.8 gets three genuinely different proofs (Agranovich-Dynin boundary symbol, KO excision, characteristic classes), and having all three is not padding, because each checks a different link. The flat index theorem argument in 5.9 is the right tool and is used correctly. The CP^2 and CP^2-bundle-over-S^1 examples are concrete and checkable, and the D0-brane charge integrality payoff at (1.14) is the kind of thing that makes the abstract claim at [abstract:s2] more than rhetoric. Where it is soft: section 6 says outright that the geometric index theory for local boundary conditions is only outlined, and the metric and connection on the physical pfaﬃan bundle are defined by fiat from the double. The stress-test is right that this is the weak link, but I think it overstates the damage. The ratio in 5.11 is flat by 5.9, and its class is computed topologically by excision, so the holonomy is a topological invariant of the ratio; a mismatch in the doubling map would have to produce a flat discrepancy with nontrivial holonomy, and nothing in the argument suggests one. Footnote 3 is the other acknowledged limit: (1.12) is sufficient, not necessary, with the converse deferred to factorization and unitarity. That is honest and minor. This is for anyone working on D-branes, anomalies, or the worldsheet origin of K-theory classifications; the first two sections are readable without the index theory. It deserves referees, and I would cite it.

Referee report

Significance

This is the perturbative derivation of the Spin^c/K-theory condition on D-branes that had previously been known only from the nonperturbative analysis of [MM, W4, W3], and it is the paper that made the w2(ν) anomaly a standard tool. The argument is unusually well supported for a result of this kind: Proposition 5.8 is given three independent proofs (the Agranovich-Dynin boundary-symbol computation, KO-theoretic excision, and a characteristic-class computation), and the topological Theorem 5.5 does not depend on the geometric subtleties discussed below. The paper also supplies concrete, checkable consequences: CP^2 is allowed but must carry half-integral flux, the non-Spin^c CP^2 bundle over S^1 is excluded in R^10, and the D0-brane charge formula (1.14) becomes integral via Atiyah-Singer once A is read as a Spin^c connection. The attribution of the K-theory interpretation to prior work is handled carefully, and the abstract's claim [abstract:s2] is presented as a consistency statement rather than a new derivation.

Major objections (2)

[§6, Dirac Operators with Local Boundary Conditions] The geometric index theory for Dirac operators with local boundary conditions is explicitly left incomplete: the section states 'We only indicate a rough outline of the arguments; we have not carried through the details,' and the fallback is that 'we adapted the practical point of view of defining the metric and connection directly from the Dirac operator on the double.' Read literally, Theorem 5.6 is then a theorem about the double-induced connection, and the identification of that connection with the physical Quillen connection of the boundary-value problem rests on two unproved ingredients: that the gluing map U of Figures 1–2 is an isometry of the relevant spinor bundles intertwining the Dirac operators up to the stated cutoﬀs, and that the determinant line of the cylinder factor C3 with the antiperiodic condition χ(1) = −χ(−1) is canonically trivial as a bundle with metric and connection. A concrete test is to verify the second point directly and to check that U commutes with the Dirac operators to the order needed for the determinant lines to agree. If either fails, the holonomy computed in Theorem 5.6 belongs to a bundle other than the physical one. The topological Theorem 5.5 is unaﬀected, so the Spin^c condition itself is not at risk; what is at risk is the stronger physical claim that A cannot be a conventional U(1) connection. I would ask the authors to surface this caveat in §5 rather than only in §6, and to state the C3 triviality as an explicit lemma with its proof or a precise reference.
[Footnotes 2 and 3] The exclusion of branes with w2(ν) ≠ 0 is established only when the anomaly is detectable by a map from the torus ∂Σ × C; footnote 2 defers detection by higher-genus surfaces to factorization and unitarity arguments that the paper does not carry out. Similarly, footnote 3 concedes that (1.12) is only a suﬀicient condition and that the full requirement is expected to follow from factorization. Since the physical conclusion is that such branes are 'ruled out,' the argument as written establishes the exclusion only for the torus-detectable case. The authors acknowledge this, so my request is not for a new computation but for precision: either make the factorization step explicit, or state the exclusion conditionally on the torus-detectability of the class, so that a reader can see exactly what has been proved.

Minor objections (7)

[Introduction and §1] The cross-references to the closed-string results are inconsistent. The introduction cites 'Corollary 4.7' for closed-string cancellation and 'Theorem 4.8' for the R^10 statement, but in §4 these are Corollary 4.6 and Theorem 4.7; §1 cites 'Theorem 4.6' for the same cancellation statement. Please reconcile the numbering.
[Eq. (5.11)] The symbol L_{α,β,θ} is used both for the full determinant line defined earlier in §5 and for the ratio (Pfaﬀ D_{α,β,θ}) ⊗ (Pfaﬀ D'_{α,β,θ})^{-1} introduced in (5.11). One of the two should be renamed.
[§1 and §5] The letter L is heavily overloaded: it denotes the 'line bundle' on Q on which A is a connection, the line bundles L^±_α over the space of bosons, and the line bundle L_B over loop space. Distinct notation would make the Spin^c discussion in §1 and the determinant-line discussion in §5 easier to follow.
[References] [DZ] (Dai–Zhang) appears in the reference list but I could not locate a citation to it in the text; either cite it or remove it.
[References] The [CY] entry has a doubled comma ('Yin,,') and the arXiv number given, hep-th/9803931, does not appear to correspond to the Cheung–Yin paper; please verify the bibliographic data.
[Third proof of Proposition 5.8] The odd-rank case is handled by 'replace F' with F' plus a trivial line bundle and apply the previous,' which changes the pfaﬃan by a factor. A sentence explaining why the reduction is legitimate in the pfaﬃan line would help the reader.
[Eq. (1.13)] Please state explicitly what the pairing (Σ, w2(Q)) denotes and why the formula is well defined for a flat B-field, since this is the example that makes the twisted case concrete.

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The paper

The paper claims

A hidden minus sign forces D-branes to be Spin-c

The reading

Freed and Witten ask whether the Type II string worldsheet path integral is globally well defined when the worldsheet has a boundary ending on a D-brane. For closed worldsheets the answer is yes, given that spacetime is spin and ten-dimensional; once a boundary is present, the fermion path integral — the pfaffian of the worldsheet Dirac operator — is not a number but a section of a line bundle, and the question becomes whether that bundle admits a flat section of unit norm. The paper computes the obstruction: the bundle's natural connection is flat, yet transporting the pfaffian around a loop of worldsheets multiplies it by $(-1)^{(C, w_2(Q))}$, the pairing of the boundary torus with the second Stiefel-Whitney class of the brane. The consequence is that the brane's apparent $U(1)$ gauge field cannot be an ordinary gauge field, since it must supply exactly the same sign; instead it is a Spin$^c$ connection, and the brane worldvolume must be Spin$^c$. That restriction is precisely the one already required for D-brane charge to be a class in K-theory, so a perturbative worldsheet argument lands on the same condition as the nonperturbative classification.

What carries the argument

The load-bearing device is a doubling trick. The Dirac operator on a surface with boundary and the local boundary conditions (5.3) — which mix right- and left-handed spinors and flip the sign on the normal-bundle part — is identified with a chiral Dirac operator on the closed double surface, with the sign encoded in a real line bundle $2$ supported in a collar. That identification turns an index problem with boundary into a closed-manifold problem, where the determinant-line-bundle machinery applies: the curvature formula, the holonomy as an adiabatic limit of the exponentiated $2$-invariant, and the Atiyah-Patodi-Singer flat index theorem. The difference between the two boundary conditions is then computed by the Agranovič-Dynin formula as the index of a family of Dirac operators on the boundary circle, which reduces to the standard statement that the pfaffian line bundle of a family of spin circles coupled to a real bundle $F'$ is the pullback of $w_2(F')$.

Extended reading notes

Core claim

The central result, Theorem 5.6, is that the canonical connection on the pfaffian line bundle of the open-string Dirac operator is flat, and that its holonomy around a family of worldsheets fibered over a circle is $(-1)^{(C,w_2(Q))}$, where $Q$ is the D-brane worldvolume and the pairing is over $W = (1.2)$'s $W = (1.2)$'s boundary torus $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s boundary torus $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s boundary torus $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s boundary torus $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s $W = (1.2)$'s boundary torus $W = (1.2)$'s boundary torus, that is, the integral of $w_2(Q)$ over $W = (1.2)$'s boundary torus. Equivalently (Theorem 5.5), the line bundle is the complexification of a real line bundle whose class in $H^1(Z;Z/2)$ is the pushforward of $w_2(2)$, and the topological class of the complex line bundle is the pushforward of $W_3(2)$.

Load-bearing premise

The anomaly is computed for a Dirac operator on the doubled surface, and the metric and connection on the physical boundary-value problem are defined by declaring them to come from that double; the paper states in section 6 that the geometric index theory for local boundary conditions is only outlined and the details are not carried through.

Editorial extensions

If this is right

A D-brane can wrap $Q$ only if $W_3(2)=0$, so $Q$ must be Spin$^c$; the paper's own example $Q=\mathbb{CP}^2$ is allowed but requires half-integral flux, while the non-Spin$^c$ five-manifold built as a $\mathbb{CP}^2$ bundle over $S^1$ with conjugation monodromy is excluded even though it embeds in $\mathbb{R}^{10}$.
With a topologically nontrivial $B$-field the condition becomes $\zeta|_Q = W_3(\nu)$, where $\zeta\in H^3(Y;\mathbb{Z})$ is the characteristic class of $B$, so the allowed branes are exactly those whose normal bundles have Thom classes in the appropriate twisted K-group.
The D0-brane charge of a brane wrapping $Q_0$, naively $\int_{Q_0}\hat A(Q_0)\exp(c_1(L))$, is an integer automatically, because with $L$ read as a Spin$^c$ line bundle the integral is the index of a Dirac operator; for $\mathbb{CP}^2$ with trivial $L$ the naive value $\pm 1/8$ is avoided without postulating new conservation laws.
For closed worldsheets the anomaly cancels whenever $Y$ is spin of dimension ten, and for $Y=\mathbb{R}^{10}$ it cancels separately for left- and right-movers, so the whole obstruction is localized at the boundary of the worldsheet.
The product of the Levi-Civita holonomy in the spin representation with the $A$-field holonomy, $\mathrm{Tr}\,P\exp(\oint\omega)\cdot\exp(i\oint A)$, is globally well defined even though neither factor is, which is exactly the statement that the brane carries a Spin$^c$ structure built from the metric and $A$.

Reading between the lines

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

The argument is a clean template for other boundary conditions: any brane whose worldvolume fermions see a sign ambiguity in a real bundle should be re-read as a twisted (Spin$^c$-like) object, and the same doubling computation should produce the relevant characteristic class without any input from the closed-string sector.
Because the anomaly is a flat $±1$ holonomy rather than a curvature, the condition is invisible to any local differential-geometric check of the low-energy effective action; one should expect the same to be true for the analogous conditions in M-theory and for orientifold and Type I settings the paper explicitly sets aside.
The paper's heuristic section suggests that the result could also be reached by imagining a 9-$ar 9$ pair and quantizing 9-$p$ strings; that route depends on assuming brane-antibrane pairs can be created, so the anomaly computation is the conservative version of the same statement, and the two should agree wherever both apply.
The demand that the pfaffian bundle be flat with holonomy in $±1$ is a stronger statement than the topological condition $W_3(\nu)=0$; a brane could pass the K-theory test and still fail the geometric one, so the paper's footnote that the topological class may vanish while the holonomy does not is worth pursuing as a separate constraint.

Original abstract

We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.05 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We analyze global anomalies for elementary Type II strings in the presence of D-branes.	contribution 1.00	0.92	0.06	asserted 0.91
s1	Global anomaly cancellation gives a restriction on the D-brane topology.	result 0.99	0.70	0.37	asserted 1.00
s2	This restriction makes possible the interpretation of D-brane charge as an element of K-theory.	interpretation 0.99	0.62	0.56	hedged possibility 0.59

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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Figures

**Figure 1.** Gluing spinor fields -1 0 1 0.5 1 f L f R

Figures reproduced from arXiv:hep-th/9907189 by the authors.

References

Works cited (6)

[1]

[A] J. F. Adams, Vector-ﬁelds on spheres , Bull. Amer. Math. Soc 68 (1962), 39–41. [AB] M. F. Atiyah, R. Bott, The index problem for manifolds with boundary , Diﬀerential Analysis, Bombay Colloq., 1964, Oxford Univ. Press, London , 1964, pp. 175–186. [APS1] M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc...

View on Pith 1962
[3]

[BH] A. Borel, F. Hirzebruch, Characteristic classes and homogeneous spaces. II , Amer. J. Math. 81 (1959), 315–382. 38 [CY] Y.-K. E. Cheung and Z. Yin,, Anomalies, Branes, and Currents , Nucl. Phys. B517 (1998), 185-196, hep-th/9803931. [DF] X. Dai, D. S. Freed, η-invariants and determinant lines , J. Math. Phys. 35 (1994), 5155–5194. [DZ] X. Dai, W. Zha...

View on Pith arXiv 1959
[513]

[GHM] M. Green, J. A. Harvey, and G. Moore, I-Brane Inﬂow And Anomalous Couplings On D-Branes, Class. Quant. Grav. 14 (1997), 47-52, hep-th/9605033. [H] P. Horava, Type IIA D-Branes, K Theory, And Matrix Theory , Adv. Theor. Math. Phys. 2 (1999), 1373–1404, hep-th/9812135. [MM] G. Moore and R. Minasian, K Theory And Ramond-Ramond Charge , JHEP 9711:002, 1...

View on Pith Pith review arXiv 1997
[609]

[P] P. Piazza, Determinant bundles, manifolds with boundary and surgery , Commun. Math. Phys. 178 (1996), 597–626; II. Spectral sections and surgery rules for anomalies , Com- mun. Math. Phys. 193 (1998), 105–124. [S] A. Sen, Tachyon Condensation On The Brane Antibrane System , JHEP 9808:012, 1998, hep-th/9805170. [W1] E. Witten, Global gravitational anom...

View on Pith Pith review arXiv 1996
[1987]

[F2] D. S. Freed, Determinants, torsion, and strings , Commun. Math. Phys 107 (1986), 483–513. [F3] D. S. Freed, Two index theorems in odd dimensions , Commun. Anal. Geom. 6 (1998), 317–329, dg-ga/9601005. [FM] D. S. Freed, J. W. Morgan, Appendix to [F2], Commun. Math. Phys 107 (1986), 510–

View on Pith Pith review arXiv 1986
[1993]

[BF] J. M. Bismut, D. S. Freed, The analysis of elliptic families I: Metrics and connection s on determinant bundles , Commun. Math. Phys. 106 (1986), 159–176; II: Dirac operators, eta invariants, and the holonomy theorem of Witten , Commun. Math. Phys. 107 (1986), 103–163. [BW] B. Booss-Bavnbek, K. P. Wojciechowski, Elliptic Boundary Problems for Dirac O...

View on Pith 1986

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Anomaly inflow on Sugimoto model branes supports the conclusion that the 10d gauge group is Sp(16)/Z2, and the same inflow matching rules out most alternative chiral spectra.
Comments on classical AdS flux vacua with scale separation
hep-th 2022-02 conditional novelty 6.0 of 10

The refined Strong AdS Distance Conjecture fails for the known scale-separated AdS3 flux vacua because the required discrete higher-form symmetry is absent, and the putative dual CFT2 has no integer operator dimensions.
Small Cosmological Constants in String Theory
hep-th 2021-07 conditional novelty 6.0 of 10

Explicit type IIB orientifolds with quantized fluxes admit supersymmetric AdS4 vacua in which all closed-string moduli are stabilized and the cosmological constant is below 10^-123 in Planck units.
The Standard Model Quiver in de Sitter String Compactifications
hep-th 2021-06 conditional novelty 6.0 of 10

An explicit type IIB Calabi-Yau compactification is constructed in which D3-branes at a dP5 singularity give a three-family left-right symmetric model, with closed-string moduli stabilised in a de Sitter minimum.
Conifold Vacua with Small Flux Superpotential
hep-th 2020-09 accept novelty 6.0 of 10

Flux vacua of type IIB string theory are constructed in which the flux superpotential is as small as 7.4e-9 and a complex structure modulus is stabilized exponentially near a conifold point, yielding a warped throat w...
Chiral Global Embedding of Fibre Inflation Models
hep-th 2017-09 conditional novelty 6.0 of 10

A Calabi-Yau orientifold with four Kähler moduli, a suitable D7-brane and flux choice, and one Fayet-Iliopoulos term reduces to a chiral fibre-inflation model whose Kähler cone bounds tightly limit the inflaton range.
Global Orientifolded Quivers with Inflation
hep-th 2017-06 conditional novelty 6.0 of 10

An explicit type IIB Calabi-Yau orientifold is built in which fractional D3 branes give a chiral quiver sector, a T-brane hidden sector stabilises all closed-string moduli in de Sitter, and a del Pezzo divisor drives ...
Extended Riemannian Geometry I: Local Double Field Theory
hep-th 2016-11 accept novelty 6.0 of 10

A graded-manifold framework called extended Riemannian geometry reproduces double field theory's symmetries and yields a coordinate-invariant form of its strong section condition.
Global String Embeddings for the Nilpotent Goldstino
hep-th 2015-12 conditional novelty 6.0 of 10

Two explicit Calabi-Yau compactifications are built that embed the nilpotent Goldstino (an anti-D3-brane on an O3-plane at a warped conifold tip) globally, one via an F-theory construction and one by retrofitting an e...
A Flux-Scaling Scenario for High-Scale Moduli Stabilization in String Theory
hep-th 2015-03 conditional novelty 6.0 of 10

A flux-scaling construction stabilizes all closed-string moduli at tree level in non-supersymmetric, tachyon-free AdS vacua, but the same scaling that makes an axion light also pushes the heavy moduli above the Kaluza...
Chow groups, Deligne cohomology and massless matter in F-theory
hep-th 2014-02 conditional novelty 6.0 of 10

A Chow-group and Deligne-cohomology construction extracts a line bundle on each F-theory matter curve whose cohomology groups are conjectured to count the exact charged massless spectrum.
Non-associative Deformations of Geometry in Double Field Theory
hep-th 2013-12 conditional novelty 6.0 of 10

In the flux formulation of double field theory, both candidate non-associative tri-product deformations reduce to total derivatives or vanish identically on-shell, so the effective action shows no associativity violation.
Explicit de Sitter Flux Vacua for Global String Models with Chiral Matter
hep-th 2013-11 conditional novelty 6.0 of 10

An explicit fluxed type IIB orientifold with chiral matter at a dP6 singularity has every closed string modulus stabilised, giving a de Sitter vacuum with TeV-scale soft terms.
Zp charged branes in flux compactifications
hep-th 2012-11 accept novelty 6.0 of 10

Background fluxes in 4d string compactifications induce Z_p discrete gauge symmetries, realized as broken U(1) subgroups, under which wrapped branes carry charges conserved only modulo p.
Building an explicit de Sitter
hep-th 2012-08 conditional novelty 6.0 of 10

An explicit type IIB compactification on CP^4_{11169}[18] is constructed in which all geometric moduli are stabilized and the Kähler sector lands in a metastable de Sitter vacuum at volume about 52.
D-Branes at del Pezzo Singularities: Global Embedding and Moduli Stabilisation
hep-th 2012-06 conditional novelty 6.0 of 10

The authors construct compact Calabi-Yau orientifolds with two del Pezzo singularities hosting a trification-like visible sector, and show all Kaehler moduli can be stabilised, giving LARGE Volume de Sitter vacua with...
G_4 flux, chiral matter and singularity resolution in F-theory compactifications
hep-th 2011-09 conditional novelty 6.0 of 10

A U(1)-restricted F-theory compactification with SU(5) gauge group yields chirality-inducing G4 fluxes whose chiral index equals the U(1) charge times the flux integral over the matter curve, realized in an explicit t...
Massive Abelian Gauge Symmetries and Fluxes in F-theory
hep-th 2011-07 conditional novelty 6.0 of 10

Massive Abelian gauge symmetries in F-theory arise from expanding the M-theory three-form on non-harmonic two-forms tied to non-Kahler resolutions, and fluxes along them reproduce known Type IIB tadpoles and chirality.
Non-geometric Fluxes, Asymmetric Strings and Nonassociative Geometry
hep-th 2011-06 conditional novelty 6.0 of 10

At linear order in a constant three-form flux, closed-string tachyon amplitudes can be repackaged into a nonassociative tri-product, and the R-flux background is conjectured to flow to a left-right asymmetric conforma...
Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS4/CFT3
hep-th 2011-05 conditional novelty 6.0 of 10

Reducing M-theory on a circle yields Chern-Simons quiver gauge theories dual to AdS4 x Y^{p,q}(CP2) for the full range of q and for every torsion G4 flux, with ranks and levels that differ from the earlier equal-rank ...
Twisted differential String and Fivebrane structures
math.AT 2009-10 conditional novelty 6.0 of 10

The Green-Schwarz mechanism and its magnetic dual are shown to be cocycles in twisted differential nonabelian cohomology, whose local L-infinity data reproduces the twisted Bianchi identities.
Global F-theory GUTs
hep-th 2009-08 conditional novelty 6.0 of 10

A compact Calabi-Yau fourfold fibered over a del Pezzo transition of P^4[4] is shown to host a global F-theory SU(5) GUT with three chiral generations and the required Yukawa couplings, using a split S[U(4)xU(1)_X] sp...
D-brane Charges in Gravitational Duals of 2+1 Dimensional Gauge Theories and Duality Cascades
hep-th 2009-06 accept novelty 6.0 of 10

Careful tracking of Page versus Maxwell versus brane charges implies the infrared B-field through the CP1 cycle of the ABJM/ABJ dual is b0 = -N4/k + 1/2, not -N4/k as previously claimed.
F-theory Compactifications for Supersymmetric GUTs
hep-th 2009-04 accept novelty 6.0 of 10

An explicit almost-Fano three-fold base is constructed on which elliptically fibered Calabi-Yau four-folds realize SU(5) GUTs with a massless hypercharge boson, and three chiral generations plus flavor-hierarchy geome...
D-instantons and twistors
hep-th 2008-12 conditional novelty 6.0 of 10

All D-instanton corrections to the hypermultiplet moduli space metric in Type II Calabi-Yau compactifications can be packaged as a sum of dilogarithms controlled by a single holomorphic function, in the leading instan...
GUTs in Type IIB Orientifold Compactifications
hep-th 2008-11 accept novelty 6.0 of 10

Globally consistent SU(5) GUT models are built on intersecting D7-branes in Type IIB orientifolds of Calabi-Yau threefolds obtained from del Pezzo transitions.
Branes and Quantization
hep-th 2008-09 conditional novelty 6.0 of 10

Quantization of a symplectic manifold (M, omega) is identified with the space of (Bcc, B') strings in the A-model of a complexification of M, with Bcc a space-filling coisotropic brane and B' a Lagrangian brane on M.
D-brane Deconstructions in IIB Orientifolds
hep-th 2008-05 accept novelty 6.0 of 10

The correct D3 charge, moduli count, and flux lattice dimension for singular D7-branes in IIB orientifolds are given by the topology of an auxiliary surface that resolves the double intersection with the O7-plane.
New Aspects of Heterotic--F Theory Duality
hep-th 2008-05 conditional novelty 6.0 of 10

The direct image of the anti-symmetric square of a Heterotic vector bundle is a line bundle on a 'covering matter curve', and the divisors defining that line bundle correspond to codimension-3 singularities in the dua...
Moduli Stabilisation versus Chirality for MSSM like Type IIB Orientifolds
hep-th 2007-11 conditional novelty 6.0 of 10

A chiral D-brane sector cannot be added after moduli are stabilised, because it induces a D-term potential and charged instanton zero modes; only combined F- and D-term stabilisation works, demonstrated on one swiss-c...
A Farey Tail for Attractor Black Holes
hep-th 2006-08 conditional novelty 6.0 of 10

The elliptic genus counting BPS states of 4d attractor black holes admits a Rademacher (Farey tail) expansion whose terms map to semiclassical supergravity saddles dressed by wrapped M2-branes.
Strong CP Problem in Type IIA Toroidal Orientifold String Theory
hep-th 2026-08 reject novelty 5.0 of 10

A Type IIA string construction is claimed to cancel the QCD theta angle via four-form fluxes, producing an axion with mass and coupling in a detectable range.
Moduli Stabilisation for ADD and the Dark Dimension Scenario
hep-th 2026-06 unverdicted novelty 5.0 of 10

A perturbative stabilization mechanism in the Large Volume Scenario for K3-fibered Calabi-Yau threefolds generates exponentially large 2D base volumes while keeping the 4D fibre small, enabling ADD or Dark Dimension l...
Dark Matter and Strong CP Problem in Type IIA String Theory
hep-th 2026-03 unverdicted novelty 5.0 of 10

Type IIA orientifold model with D6-branes stabilizes moduli via STU and KL, predicts axion-neutralino dark matter, and uses four-form flux to address strong CP problem.
($0,6$) AdS$_3$/CFT$_2$ and surface defects
hep-th 2025-04 conditional novelty 5.0 of 10

A brane box and quiver are proposed as the 2d dual of N=(0,6) AdS3 vacua, with a central charge formula and Seiberg-like dualities, though key claims remain conjectural.
The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories
hep-th 2021-07 accept novelty 5.0 of 10

The Mordell-Weil group of the Seiberg-Witten elliptic fibration determines the global form of the flavour symmetry group for rank-one 4d N=2 theories and their 5d parents on a circle.

See all 75 Pith citations

How later papers used it

204 later papers cite it

Evidence: substantial. 200 of the citing papers share no author with it, and 18 of those carry a confident label from Pith's citation classifier.

Pith read 31 of the 305 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 2 built on it and 1 cited it as support.

Builds on it 2 of 6 papers with this label

Said that the spin^c condition on the bundle ν is the appropriate generalization of this paper's Freed-Witten anomaly cancellation condition to this situation.

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory (2007) · 0.99 · paraphrased: the licence does not allow quoting
Used KK-theory's noncommutative version of K-orientation, which generalizes this paper's Freed-Witten anomaly cancellation condition, to select consistent sets of D-branes from its category.

D-branes and bivariant K-theory (2008) · 0.88 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 7 papers with this label

Said that if the D7 stack wraps a non-spin manifold, this gauge field is already sufficient to make the path integral measure of open strings attached to the D7 stack well-defined, citing this paper.

On Flux Quantization in F-Theory (2010) · 0.96 · paraphrased: the licence does not allow quoting

How the literature uses it

205 arXiv papers cite this in 306 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 · 244 sentences without a confident label

Builds on it 6 papers

This is the orbifold generalization of the Freed-Witten condition [ref] formulated in terms of obstruction classes in Bredon cohomology.

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory · Consistency conditions · 0.99
KK-theory also provides us with a noncommutative version of K-orientation, which generalizes the Freed-Witten anomaly cancellation condition [ref] and enables us to select the consistent sets of D-branes from our category.

D-branes and bivariant K-theory · Introduction · 0.88
This extends the observation in [ref] , where it was shown that, for smooth target spaces, the existence of a spin structure implies the absence of global gauge anomalies.

What makes spacetime spin in string theory? · Introduction · 0.99

Uses its method or tool 1 paper

Although it may not be evident reading this work, we used Freed-Witten anomaly as a “guide” (see [ref] , [ref] ).

Topics on topology and superstring theory · Introduction · 0.91

Compares with it 6 papers

This may be thought of as mod 3 analog in M-theory of the mod 2 expression in type II string theory, namely the Freed-Witten anomaly cancelation formula for D-branes [ref] $(Sq^3 + H_3∪)F=0$, since the class $Θ$ measures the anomaly of the M-branes [ref] .

An approach to anomalies in M-theory via KSpin · The Degree eight `dual' field $Θ$ · 0.94
Then a K-orientation for $f$ is given by [equation] In this way our notion of K-orientation coincides with the Freed-Witten anomaly cancellation condition [ref] .

D-branes and bivariant K-theory · K-orientation and Gysin homomorphisms · 0.81
This can be compared with the Freed-Witten anomaly-free condition [ref] for untwisted open strings, which involves the $w_2$ of the normal bundle of the world-volume.

Non-orientable surfaces and electric-magnetic duality · Dimensional reduction from an orientable $4$-manifold · 0.98

Cites it as support 4 papers

If the D7 stack wraps a non-spin manifold, this gauge field is already sufficient to make the path integral measure of open strings attached to the D7 stack well-defined [ref] .

On Flux Quantization in F-Theory · Sp(N) singularities · 0.96
Before we proceed let us briefly note that the Freed-Witten anomaly [ref] actually rules out cohomologically non-trivial 4-form flux $ι^* G_4 ≠0$ on free M5-instantons (with $G_4 = d C_3$).

Fluxed M5-instantons in F-theory · The partition function of vertical M5-brane instantons · 0.96
First of all, anomalies prevent D-branes from wrapping submanifolds whose normal bundles are not spin$^c$ [ref] , justifying our assumption.

D-branes and $K$-homology · D-brane Charges · 0.99

Cites it as background 38 papers

These problems have already been analyzed in the literature, starting from the seminal paper of D.Freed and E.Witten [ref] .

Classifying A-field and B-field configurations in the presence of D-branes · Introduction · 0.98
In this paper we will address the the quantization rule for the M-theory $G_4$ flux [ref] in its relation to the Freed-Witten quantization condition for gauge fluxes on 7-brane configurations [ref] .

On Flux Quantization in F-Theory II: Unitary and Symplectic Gauge Groups · Introduction · 0.87
We recall that, for a single D-brane, the world-sheet path-integral measure contains the terms [ref] : [equation] where $Σ$ is the world-sheet, $ϕ$ is the trajectory of the world-sheet in the space-time, $D_ϕ$ is the pfaffian of the Dirac operator on the world-sheet, coupled to the space-time via $ϕ$, and $A$ and $B$ are the $A$-field and the $B$-field.

Freed-Witten anomaly and D-brane gauge theories · Single D-brane · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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