Evidence for F-Theory

Cumrun Vafa

REVIEW

Evidence for F-Theory

Cumrun Vafa

High Energy Physics - Theory · Submitted 5 Feb 1996 · 39 references

Pith's verdict: The founding F-theory paper, and the core construction holds up; the section 4 spectrum claims are conditional on an adiabatic step the author flags but does not establish.

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arXiv IDhep-th/9602022 (version 1)

Pith IDE6HAJTT5, Pith's permanent record of this paper and its review

CitationCumrun Vafa. "Evidence for F-Theory." arXiv:hep-th/9602022 (1996). https://pith.science/paper/E6HAJTT5

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Editor's note

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This is the founding paper of F-theory, and the core of it holds up. The elliptic-fibration dictionary for type IIB, the 24 seven-branes on S2, and the F-theory/M-theory circle relation are all here, and the checkable arithmetic is right as far as it goes. Worth your time if you work on IIB vacua or seven-branes; it is short and still the standard reference for the geometry. What is genuinely new: identifying the axion-dilaton with the complex structure of an elliptic fiber, which turns SL(2,Z) into a geometric symmetry and turns seven-branes into degeneration loci. The compact D-manifold with 24 seven-branes is the first compact example of the D-manifold program, and the moduli count checks out: 9+13-3-1 = 18 complex moduli plus one real Kaehler parameter for S2, matching heterotic on T2, including the anomaly coefficient that forces exactly 24 branes. The E8 deformation argument, where deforming gives two A4's and four A2's so you predict SU(5) and SU(3) rather than E8, is a real piece of reasoning that could have failed and did not. Soft spots, in proportion. The twelve-dimensional theory is postulated from the N=2 string analogy, not derived; the author says so. The choice of tau over rho as the dynamical modulus is a conjecture, also flagged. The (2,2) worldvolume discussion in section 3 is programmatic, with no amplitudes and no quantization. None of that is hidden. The one place I would push: the abstract's 'equivalent' [abstract:s3] is stronger than the text, which only claims the F-theory/M-theory relation 'to the extent that we can trust the adiabatic argument.' The stress-test note is right that the fiberwise duality is not obviously valid at degeneration loci, and the local (p,q) patch does not establish that the small-S1 and small-fiber limits commute. But the paper is titled 'Evidence,' and the section 2 checks, the heterotic spectrum match, the anomaly coefficient, and the E8 deformation, do not depend on the adiabatic step. Section 4's T = k-1 and V = h11-k-1 formulas do, and should be read as conditional. The Schoen example's T=9, V=8 is independently cross-checked by Sen [36], which helps. Send it out. This is a landmark, and referees should engage with the adiabatic step rather than the framing.

Referee report

Significance

If the proposal holds, it is a structural result of the first rank: the SL(2,Z) symmetry of type IIB becomes the modular group of a physical torus, the axion-dilaton becomes its complex structure, seven-branes become degeneration loci of an elliptic fibration, and a large new class of six-dimensional N=1 theories appears. The evidence offered is unusually concrete for a paper of this kind and is largely parameter-free: the degree-24 discriminant, the 18 complex moduli of an elliptically fibered K3, and the coefficient of p1(R) requiring exactly 24 seven-branes are independent algebraic and anomaly facts that could have come out otherwise, and they agree with heterotic string theory on T2. The Schoen example's T = 9, V = 8 is independently cross-checked against Sen [36], and the Spin(7) discussion yields a falsifiable structural prediction, namely four-dimensional vacua with no supersymmetry that become supersymmetric on a small circle. The paper is also honest about its status, repeatedly using 'conjecture' and 'evidence' rather than claiming derivation.

Major objections (3)

[Abstract, sentence s3, versus §4] The abstract states flatly that 'compactification of M-theory on a manifold K which admits elliptic fibration is equivalent to compactification of F-theory on K x S1', whereas §4 states the same relation only 'at least if ... the adiabatic argument can be trusted'. The unconditional wording of the abstract is stronger than what the body establishes, and since this sentence is the paper's central structural claim it should carry the same qualification as §4, for instance 'is on the same moduli space, to the extent that the adiabatic argument applies'. This is a calibration issue rather than a physics error, but it is the one place where the manuscript's own hedging is dropped.
[§4, first paragraph, and §4.1] The relation between F-theory on K x S1 and M-theory on K is obtained by applying the adiabatic argument of [19] fiberwise over a base that carries 24 degenerate fibers, and the spectrum formulas T = k - 1, V = h^{1,1} - k - 1, H = h^{2,1} + 1 in §4.1 inherit this. The local patch near each degeneration locus, where one passes to the perturbative (p,q) string description as in §2, is a statement about a neighborhood and does not by itself establish that the global moduli space factorizes, nor that the small-S1 limit used for the M-theory circle and the small-fiber limit used for the F-theory limit commute. The author flags the assumption but does not test it. A short paragraph in §4.1 stating explicitly which of the two limits is taken first, and whether the Schoen example's T = 9, V = 8 is stable under the other order, would materially strengthen the section; the independent cross-check against Sen [36] mitigates the concern but does not remove it.
[§3, paragraph beginning 'Even though the BRST invariance rules out non-zero momentum states'] The identification of the physical modulus with the complex structure tau rather than the Kaehler parameter rho of the internal Gamma^{2,2} lattice is the step that turns the twelve-dimensional bookkeeping into a geometric torus, and it is asserted rather than derived: 'We conjecture that the zero mode corresponding to one of the two moduli, say tau is physical'. The supporting analogy with N=2 string amplitudes depending only on tau and not on rho is cited to [25], but no argument is given for why the same selection should hold here, where the internal space has signature (1,1) and the null reduction is along a different direction. Since the axion-dilaton identification is load-bearing for everything in §4, a sentence or two explaining what distinguishes tau from rho in the BRST cohomology would be worth adding; if no such argument is currently available, the conjecture should be labelled as such at the point of use in §4 as well as here.

Minor objections (7)

[Eq. (2.1)] The kinetic term is typeset as 'partial tau partial-bar tau tau_2^2' with the denominator garbled; it should presumably read partial tau partial-bar tau / tau_2^2. The equation as printed cannot be read unambiguously and should be corrected.
[§2, paragraph beginning 'Let us see how many gauge fields'] The statement that 'we can get at most 18 of the D-branes to be the 7-branes of a given theory' is the key step in reducing the naive U(1)^24 to U(1)^18, but it is asserted without derivation. A short justification, or a pointer to where the argument is given, would make the gauge-field counting checkable rather than plausible.
[§2, paragraph beginning 'Let us see how many gauge fields'] The two additional U(1) factors from the overall complex shift of all seven-branes are called 'graviphotons'. In eight dimensions the N=1 gravity multiplet contains no vector, so the term is nonstandard here; presumably these are the Kaluza-Klein vectors of the would-be T2, matching the two torus U(1)'s of heterotic string theory on T2. Clarifying the terminology would help the comparison with the heterotic spectrum that follows.
[§2, paragraph beginning 'Note that we have managed to find a compact manifold'] The 'no-go theorem' for D-branes in compact manifolds is paraphrased in one sentence and not attributed. Since the resolution of this obstruction via the non-abelian structure of (p,q) charges is one of the paper's selling points, the statement of the obstruction and a reference would be appropriate.
[§4.1, Schoen example] The value k = 10 for the elliptic Calabi-Yau with h^{1,1} = 19, h^{2,1} = 19 is quoted with the parenthetical 'consistent with preserving one elliptic curve fibered' but without a derivation. A brief indication of how the Kaehler parameters are split between the base and the two fibers would let the reader verify T = 9 and V = 8 directly.
[References [22], [29], [31], [39]] Several substantive points rest on work listed as unpublished or in preparation: the (10,2) origin of type IIB attributed to Banks and Seiberg via [29], the turning on of string condensates and membranes via [39], and the heterotic/type II duality of [22]. Where these are cited in support of a claim rather than as priority acknowledgement, the argument should either be sketched in the text or the dependence made explicit.
[§4.3] The resolution of the apparent mismatch between a non-supersymmetric four-dimensional theory and an N=1 three-dimensional theory is attributed to [9] and is convincing, but the sentence 'it is quite amusing that at least the uncompactified dimensions (if not the signature) in each of these three cases agrees with the dimension of worldvolume of the corresponding theory' is cryptic as written and could be rephrased or dropped.

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The paper

The paper claims

Type IIB string theory is reformulated as a twelve-dimensional F-theory whose elliptic fibrations reproduce the type IIB coupling constant and generate new six-dimensional vacua.

The reading

Type IIB string theory has a strange symmetry, SL(2,Z), that acts on its two scalar fields exactly the way the modular group acts on the shape of a torus. But ten-dimensional type IIB contains no torus, so the symmetry has no geometric home. Vafa's move is to suppose the torus is real but hidden: two extra dimensions of signature (1,1), for twelve in total. The complex structure of that torus is the type IIB coupling, so strong and weak coupling become two points in the same geometry. A seven-brane is then simply a place where the torus degenerates, and the axion-dilaton logarithm near a brane is the standard monodromy of a degenerate elliptic fiber. Wrapping the extra dimensions 24 times around the fundamental domain of SL(2,Z) gives a compact space: a sphere carrying 24 seven-branes. The counting works out. An elliptically fibered K3 has 18 complex moduli plus one real Kaehler class, which is exactly the spectrum of the heterotic string on a two-torus, so Vafa conjectures the two are dual, with the size of the sphere playing the heterotic coupling. The general rule that follows is that F-theory on K times a circle matches M-theory on K, which lets one read off supersymmetry and massless spectra for compactifications to six, five and four dimensions on Calabi-Yau threefolds, G2 holonomy and Spin(7) holonomy manifolds. The four-dimensional case has no supersymmetry, which Vafa connects to Witten's proposal for explaining why the cosmological constant is small.

Extended reading notes

Core claim

The load-bearing assertion is that type IIB string theory admits a twelve-dimensional reformulation, F-theory, and that compactification of M-theory on an elliptically fibered manifold K is equivalent to compactification of F-theory on K x S1 (abstract:s3). If correct, the SL(2,Z) symmetry of type IIB is the modular group of a physical torus of signature (1,1), the axion-dilaton is its complex structure, seven-branes are degeneration loci of an elliptic fibration, and the 24 seven-branes on S2 with 18 complex moduli plus one real Kaehler parameter are exactly heterotic string theory on T2. The paper's own framing is calibrated: it offers evidence for a reformulation, not a proof.

Load-bearing premise

The adiabatic argument, invoked in section 2 ('to the extent that we can trust the adiabatic argument') and again in section 4 to move the M-theory/type IIB duality fiberwise over a base. This is what licenses the claim that F-theory on K x S1 sits on the same moduli space as M-theory on K, and therefore what licenses every statement in section 4 about supersymmetry and spectra of the six-, five- and four-dimensional compactifications. The argument is cited to Vafa and Witten [19] and is assumed rather than proved; if it fails for a fibration with degenerating fibers, the spectrum claims in section 4 lose their derivation and the four-dimensional Spin(7) discussion loses its link to M-theory.

Original abstract

We construct compact examples of D-manifolds for type IIB strings. The construction has a natural interpretation in terms of compactification of a 12 dimensional `F-theory'. We provide evidence for a more natural reformulation of type IIB theory in terms of F-theory. Compactification of M-theory on a manifold $K$ which admits elliptic fibration is equivalent to compactification of F-theory on $K\times S^1$. A large class of $N=1$ theories in 6 dimensions are obtained by compactification of F-theory on Calabi-Yau threefolds. A class of phenomenologically promising compactifications of F-theory is on $Spin(7)$ holonomy manifolds down to 4 dimensions. This may provide a concrete realization of Witten's proposal for solving the cosmological constant problem in four dimensions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,528 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.29 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We construct compact examples of D-manifolds for type IIB strings. The construction has a natural interpretation in terms of compactification of a 12 dimensional `F-theory'. We provide evidence for a more natural reformulation of type IIB theory in terms of F-theory. Compactification of M-theory on a manifold $K$ which admits elliptic fibration is equivalent to compactification of F-theory on $K\times S^1$. A large class of $N=1$ theories in 6 dimensions are obtained by compactification of F-theory on Calabi-Yau threefolds. A class of phenomenologically promising compactifications of F-theory is on $Spin(7)$ holonomy manifolds down to 4 dimensions. This may provide a concrete realization of Witten's proposal for solving the cosmological constant problem in four dimensions.

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 construct compact examples of D-manifolds for type IIB strings.	contribution 0.99	0.95	0.06	asserted 1.00
s1	The construction has a natural interpretation in terms of compactification of a 12 dimensional `F-theory'.	interpretation 1.00	0.74	0.20	asserted 0.94
s2	We provide evidence for a more natural reformulation of type IIB theory in terms of F-theory.	interpretation 0.46	0.83	0.77	asserted 0.94
s3	Compactification of M-theory on a manifold $K$ which admits elliptic fibration is equivalent to compactification of F-theory on $K\times S^1$.	result 0.88	0.50	0.16	asserted 1.00
s4	A large class of $N=1$ theories in 6 dimensions are obtained by compactification of F-theory on Calabi-Yau threefolds.	result 0.79	0.51	0.11	asserted 1.00
s5	A class of phenomenologically promising compactifications of F-theory is on $Spin(7)$ holonomy manifolds down to 4 dimensions.	result 0.47	0.55	0.08	asserted 0.78
s6	This may provide a concrete realization of Witten's proposal for solving the cosmological constant problem in four dimensions.	interpretation 1.00	0.16	0.07	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.88.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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6D SCFTs and Phases of 5D Theories
hep-th 2017-03 accept novelty 7.0 of 10

Circle reduction of a 6D (1,0) SCFT yields between one and four 5D SCFTs, and the 5D quiver obtained from its tensor branch is a separate phase of the same theory, connected by flop transitions.
Terminal Singularities, Milnor Numbers, and Matter in F-theory
hep-th 2016-12 conditional novelty 7.0 of 10

Uncharged localised hypermultiplets in six-dimensional F-theory are counted by the Milnor numbers of Q-factorial terminal singularities, and the resulting spectra pass anomaly cancellation.
S-folds and 4d N=3 superconformal field theories
hep-th 2016-02 accept novelty 7.0 of 10

Only five variants of N=3-preserving S-folds exist, labeled by (k,ell) = (3,1), (3,3), (4,1), (4,4), (6,1), as shown by matching F-theory discrete torsion against M-theory orbifold charges.
F-theory and 2d (0,2) Theories
hep-th 2016-01 conditional novelty 7.0 of 10

Compactifying F-theory on singular elliptic Calabi-Yau five-folds yields 2d N=(0,2) gauge theories whose matter content, holomorphic couplings, and anomalies are read off from the geometry of the elliptic fibration.
N=3 four dimensional field theories
hep-th 2015-12 conditional novelty 7.0 of 10

The paper constructs the first examples of four-dimensional N=3 superconformal field theories outside supergravity, as F-theory orbifolds of N=4 super Yang-Mills by combined R-symmetry and SL(2,Z) actions.
A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua
hep-th 2015-10 conditional novelty 7.0 of 10

A Monte Carlo walk over toric threefold bases connected to P3 estimates about 10^48 such bases, peaked near h^{1,1} = 82, with roughly 76% carrying a non-Higgsable SU(3)xSU(2) pair.
Tensor Hierarchy and Generalized Cartan Calculus in SL(3)$\times$SL(2) Exceptional Field Theory
hep-th 2015-01 conditional novelty 7.0 of 10

The authors build the SL(3)xSL(2) exceptional field theory, introducing a nilpotent Cartan-like differential that organizes a five-level tensor hierarchy and yields an action claimed to encode D=11 or type IIB supergravity.
Non-Higgsable clusters for 4D F-theory models
hep-th 2014-12 conditional novelty 7.0 of 10

Only nine simple gauge algebras and five two-factor products, including SU(3) x SU(2), can arise as non-Higgsable clusters in 4D F-theory, while larger connected configurations can branch, loop, and form long chains.
Complete Intersection Fibers in F-Theory
hep-th 2014-11 accept novelty 7.0 of 10

A general algorithm puts complete-intersection elliptic fibers into Weierstrass form, enabling a classification of 3134 toric Mordell-Weil groups and new F-theory models with distinct 10-plet charges and Z4 symmetry.
Elliptic Genus of E-strings
hep-th 2014-11 accept novelty 7.0 of 10

A new 2d (0,4) O(n) gauge theory reproduces the E-string elliptic genus and provides the all-genus refined topological string partition function on the canonical bundle over half-K3.
Landscaping with fluxes and the E8 Yukawa Point in F-theory
hep-th 2014-04 conditional novelty 7.0 of 10

An integral monodromy basis for Calabi-Yau fourfold periods is constructed and cross-checked, then used to argue that integer fluxes can drive F-theory moduli to orbifold and gauge-enhancement loci.
Sections, multisections, and U(1) fields in F-theory
hep-th 2014-04 accept novelty 7.0 of 10

Genus-one fibrations without a global section sit inside the Weierstrass moduli space, and nearly every six-dimensional U(1) gauge symmetry arises from Higgsing an SU(2) with adjoint matter.
Box Graphs and Singular Fibers
hep-th 2014-02 accept novelty 7.0 of 10

Decorated box graphs, sign-colored representation graphs obeying flow rules, enumerate the crepant resolutions and higher-codimension fiber types of elliptically fibered Calabi-Yau fourfolds, revealing new non-Kodaira...
F-theory on Genus-One Fibrations
hep-th 2014-01 conditional novelty 7.0 of 10

Dropping the section requirement on the elliptic fibration gives a valid F-theory limit, with corrected hypermultiplet counting and new monodromy breakings such as E6 to G2.
F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections
hep-th 2013-03 accept novelty 7.0 of 10

The elliptic curve with two rational points is realized as the Calabi-Yau onefold in the del Pezzo surface dP2, yielding a complete, anomaly-consistent U(1) x U(1) matter spectrum for F-theory.
All Complete Intersection Calabi-Yau Four-Folds
hep-th 2013-03 conditional novelty 7.0 of 10

The authors enumerate complete intersection Calabi-Yau four-folds in products of projective spaces, producing 921,497 configuration matrices with Euler characteristics from 0 to 2610, nearly all of them elliptically fibered.
U(1) symmetries in F-theory GUTs with multiple sections
hep-th 2012-11 accept novelty 7.0 of 10

Factorised Tate models give a systematic global construction of U(1) symmetries in F-theory SU(5) GUTs, and the SU(5) x U(1)_PQ model contains charge-10 singlets incompatible with a single E8 decomposition.
Superconformal Partition Functions and Non-perturbative Topological Strings
hep-th 2012-10 conditional novelty 7.0 of 10

The non-perturbative refined topological string is defined as a triple product of perturbative amplitudes with modular-inverted couplings, and this reproduces the known single-M5-brane index.
M-theory and Type IIA Flux Compactifications
hep-th 2012-08 conditional novelty 7.0 of 10

New higher-derivative M-theory couplings involving fluxes and curvatures generate M5-brane charge, evading the supergravity no-go theorem and permitting Minkowski flux compactifications of M-theory and type IIA.
Toric bases for 6D F-theory models
hep-th 2012-04 conditional novelty 7.0 of 10

A complete computer enumeration finds 61,539 toric bases for 6D F-theory, with tensor multiplet number from 0 to 193, and confirms that the toric monomial count reproduces the gravitational anomaly condition on every base.
Gauge Fluxes in F-theory and Type IIB Orientifolds
hep-th 2012-02 accept novelty 7.0 of 10

The universal spectral cover flux of F-theory SU(5) models is identified with the Type IIB gauge flux of the massive diagonal U(1), and the extra massless U(1) flux is matched to a specific linear combination of brane fluxes.
Classifying bases for 6D F-theory models
hep-th 2012-01 accept novelty 7.0 of 10

The paper gives the complete list of curve configurations on an F-theory base that unavoidably carry gauge symmetry, fixing the minimal gauge and matter content of any six-dimensional F-theory vacuum.
Yukawas, G-flux, and Spectral Covers from Resolved Calabi-Yau's
hep-th 2011-08 accept novelty 7.0 of 10

An explicit resolution of elliptically fibered Calabi-Yau fourfolds with A4 singularities yields exact Chern classes, proves a conjectured Euler-characteristic shift, and derives the local Higgs-bundle spectral cover ...
Tate's algorithm and F-theory
hep-th 2011-06 accept novelty 7.0 of 10

The Tate forms used to build F-theory models fail for several singularity types and can be globally obstructed; the authors give a corrected local ansatz and a concrete obstruction example.
On Flux Quantization in F-Theory
hep-th 2010-11 conditional novelty 7.0 of 10

Smooth F-theory fourfolds always have integrally quantized four-form flux; half-integral quantization arises only from 7-brane singularities on non-spin surfaces, matching the Freed-Witten anomaly.
The N=1 effective action of F-theory compactifications
hep-th 2010-08 conditional novelty 7.0 of 10

A Legendre-transform lift from three-dimensional M-theory on a Calabi-Yau fourfold yields the leading-order four-dimensional N=1 Kahler potential and holomorphic gauge coupling functions of F-theory.
Global aspects of the space of 6D N = 1 supergravities
hep-th 2010-08 accept novelty 7.0 of 10

For 6D N=1 supergravities with fewer than nine tensor multiplets, only finitely many gauge groups and matter contents are anomaly-free, and each such theory carries an integral lattice that must embed in the F-theory ...
Codimension-3 Singularities and Yukawa Couplings in F-theory
hep-th 2009-01 conditional novelty 7.0 of 10

Local field-theory models of codimension-3 singularities in F-theory show the up-type Yukawa matrix is rank-1 at leading order for generic complex structure, and identify spectral surfaces as the natural home of chira...
GUTs and Exceptional Branes in F-theory - II: Experimental Predictions
hep-th 2008-06 conditional novelty 7.0 of 10

Internal U(1) hypercharge flux on a del Pezzo seven-brane breaks SU(5) to the Standard Model and, combined with curvature-induced wavefunction repulsion, yields an exponentially small mu term and Dirac neutrino masses...
GUTs and Exceptional Branes in F-theory - I
hep-th 2008-02 accept novelty 7.0 of 10

F-theory on a local elliptic fourfold is described by a partially twisted eight-dimensional Yang-Mills theory on the seven-brane surface, with chiral matter on curve defects whose intersections generate Yukawa couplings.
A Geometry for Non-Geometric String Backgrounds
hep-th 2004-06 accept novelty 7.0 of 10

T-fold string backgrounds, glued by T-duality transition functions, can be reformulated as conventional geometry on a doubled T^{2n} fibration in which physical spacetime is a locally chosen null T^n slice.

See all 189 Pith citations

How later papers used it

472 later papers cite it

Evidence: substantial. 466 of the citing papers share no author with it, and 19 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 715 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

2 later papers by other authors are listed below: 1 built on it and 1 cited it as support. 1 more paper by the same authors is listed: 1 cited it as support.

Builds on it 1 of 3 papers with this label

Used the idea of this paper and generalized it to the eight-derivative level.

On effective field theory of F-theory beyond leading order (2018) · 0.99 · paraphrased: the licence does not allow quoting

Cites it as support 2 of 7 papers with this label · 1 by the same authors

Said its finding that the manifold has no integrable complex structure but only an almost complex structure is consistent with this paper's prediction.

Supersymmetric Configurations, Geometric Transitions and New Non-Kahler Manifolds (2010) · 0.97 · paraphrased: the licence does not allow quoting

By the same authors

Said that this F-theory setup avoids the problem of net charge in a compact space because the 7-branes source non-abelian flux that can cancel with themselves, as in this paper.

The String Landscape, the Swampland, and the Missing Corner (2017) (same authors) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

458 arXiv papers cite this in 693 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 · 258 sentences without a confident label

Contests it 1 paper

More recently, it has been claimed that F-theory [ref] admits a 12d action [ref] , taking the form [equation] Upon closer examination, we find [ref] is just 11d supergravity integrated over a spectator dimension[ We provide support for this claim in Appendix [ref]. ].

A 12d Origin of the Type IIB String Theory: A Review · A 10+2d Theory? · contest 0.82

Builds on it 2 papers

We use eq. higherlast to show that the on-source curvature vanishes for F-theory axio-dilaton compactifications of 10D Type IIB supergravity with arbitrary codimension-two sources, generalizing a known result when the sources are supersymmetric [ref] .

On Brane Back-Reaction and de Sitter Solutions in Higher-Dimensional Supergravity · Example: the axio-dilaton and 10D Type IIB supergravity · 0.98
This correspondence is an adiabatic extension of an $8$-dimensional duality [ref] (i.e. heterotic theory compactified on $T^2$ related to F-theory on a $K3$ surface), fibered over a shared base manifold $B_n-1$ to obtain lower-dimensional dualities.

Multiple Fibrations in Calabi-Yau Geometry and String Dualities · Multiple Fibrations and Heterotic/F-theory Duality · 0.85

Uses its method or tool 6 papers

In order to control the contributions of the 7-branes to the curvature and to the resulting potential energy in five dimensions, we will use F theory [ref] .

Dual Purpose Landscaping Tools: Small Extra Dimensions in AdS/CFT · $AdS_5×Small$ Examples · 0.97
As in [ref] , our approach to understand the non-geometry of the heterotic string will be to use the map to F-theory [ref] .

Ubiquity of non-geometry in heterotic compactifications · Introduction · 0.94
We use F-theory [ref] to study these questions, as this approach gives a global and nonperturbative picture of the largest currently understood set of string vacua.

Applications of Flux Compactifications in F-Theory · Introduction · 0.84

Uses its data or values 1 paper

For instance, the F-theory [ref] geometry with $O(10^272,000)$ flux vacua [ref] and the exact ensemble [ref] of $4/3 ×2.96 ×10^755$ F-theory geometries exhibit $33$ and and $762 ±11$ gauged dark sectors, respectively.

On Axion Reheating in the String Landscape · Introduction · 0.81

Compares with it 3 papers

This number can easily be compared to the moduli of $K3$ considered as an elliptic fibration over $ℂℙ^1$ [ref] , defined by the Weierstrass equation y^2=x^3+f_8(a,b) x+g_12(a,b) where $[a:b]$ are the homogeneous coordinates of $ℂℙ^1$ and $f_8$ and $g_12$ are homogeneous polynomials of degree 8 and 12 respectively.

D7-Brane Motion from M-Theory Cycles and Obstructions in the Weak Coupling Limit · General remarks on $K3$ · 0.81
This functional form is also the same as the F-theory case [ref] , in which one takes $e^φ(z)∝η(ρ)^-2z^-N/12$ so that the transverse space has conical deficit angle $πN/ 12$.

Exotic Branes in String Theory · Near-ring analysis · 0.88
It is similar to the situation in F-theory [ref] , where the area of the $T^2$ is fixed and the modulus of the torus is not a dynamical parameter.

Semi-Flatland · Reduction to seven dimensions · 0.95

Cites it as support 5 papers

Later, through the study of the 6d (2,0) gravity theory [ref] , and the 7-brane backreacted vacua [ref] , more evidence for an effective 12d interpretation of the type IIB string theory have emerged.

A 12d Origin of the Type IIB String Theory: A Review · Introduction · 0.99
A convincing argument that they correspond to gauge transformations of the target space theory, uses the duality between Type IIB string theory with M-theory on $T^2$ [ref] (or the duality with F-theory [ref] ). $Ω$ and $(-1)^F_L$ then correspond to a parity transformation on $T^2$ [ref] , and once again, it is an spacetime diffeomorphism, and the symmetries are gauged in the target space.

The AdS/$\mathsf{C}$-$\mathsf{P}$-${\mathsf T}$ Correspondence · Holographic $𝖯$-symmetry · 0.97
This picture is also necessitated by heterotic duality [ref] : at a generic point in the moduli space of an $E_8$ gauge bundle the theory is completely Higgsed and the $240$ W-bosons of $E_8$ are massive; in the dual F-theory picture the heterotic bundle moduli map to complex structure moduli which deform the $E_8$ singularity and give rise to the necessary two-cycles.

Matter From Geometry Without Resolution · Introduction · 0.96

Cites it as background 333 papers

Orientifold compactifications of Type IIB string theory on a Calabi-Yau threefold in the presence of D3 and D7 branes can be geometrically described by F-theory compactified on a Calabi-Yau fourfold [ref] .

Chern class identities from tadpole matching in type IIB and F-theory · Introduction · 0.93
Static codimension two configurations with varying axion-dilaton are the starting point for F-theory [ref] .

Expanding F-Theory · F-theory Revisited · 0.89
In this work, we initiate the study of the D(-1)-instanton superpotential in type IIB compactification by focusing on the Sen-limit [ref] in F-theory [ref] .

D(-1)-Instanton Superpotential In String Theory · Introduction · 0.82

Discussion

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