CFT's From Calabi-Yau Four-folds

C. Vafa; E. Witten; S. Gukov

REVIEW

CFT's From Calabi-Yau Four-folds

S. Gukov, C. Vafa, E. Witten

High Energy Physics - Theory · Submitted 9 Jun 1999 · 42 references

Pith's verdict: The flux superpotential of Eq. (2.38) is derived twice and holds up; the A_m/Kazama-Suzuki identification is a real counting check, with one presentational gap in the section 3 soliton reduction that is smaller than the stress-test note suggests.

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CitationC. Vafa, E. Witten, S. Gukov. "CFT's From Calabi-Yau Four-folds." arXiv:hep-th/9906070 (1999). https://pith.science/paper/TOX6BSAJ

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Editor's note

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Two things to take from this. First, Eq. (2.38), W = (1/2π)∫Ω∧G, is the flux superpotential most of the later moduli-stabilization literature runs on, and it is derived here twice — once from the Becker-Becker supersymmetry conditions, once from a domain-wall tension argument. The second derivation is the one I trust more, because it does not route through the ten-dimensional equations of motion. Second, the A_{m-1} four-fold singularity is matched to the level-one Kazama-Suzuki coset, with k=1 giving the A_m N=2 minimal model. That is a real result: a wrapped-brane soliton count equals the soliton count of an exactly solvable coset, and the two counts are computed independently, so the agreement is a check rather than a restatement. The number of vacua equals the dimension of the k-th antisymmetric tensor of SU(m); the solitons correspond to the roots. Neither count uses the coset, which is what makes the match worth something. Where it is soft. Section 3 reduces a supersymmetric four-cycle to a one-dimensional problem on the z_1 plane, assuming the cycle is Lagrangian with respect to the fiber Kähler form. The paper says outright that this 'is not necessarily true' with mixed terms, and patches it with a t→∞ rescaling. The stress-test note is right that the sentence about BPS states being t-independent addresses the calibration phases, not the Lagrangian condition. I would call this a presentational gap rather than a hole: what the argument actually needs is deformation-invariance of the BPS index, which is standard, and the paper gestures at it without stating it cleanly. The explicit cycles at finite t may not be the ones constructed; the count is what is matched. Worth a footnote, not a rewrite. Two smaller things. The D and E extension is explicitly conjectural — the paper says so, and the Hermitian-symmetric-space pattern makes it plausible but not proved. And the no-scale conclusion in the revised Appendix II leans partly on [40], listed as 'to appear'; the paper's own derivation in Appendix II carries most of the weight, so this is minor. Who should read it: anyone working on flux compactification or on the LG/coset correspondence. The superpotential section is required reading; the singularity classification is a useful reference. Peer review: yes, send it out. The central formula is right and the A_m identification is a genuine check. The revision history — retracting the AdS claim after Polchinski's comments — is a good sign, not a bad one.

Referee report

Significance

This is an influential paper connecting singularity geometry to exactly solvable conformal field theories. The vacuum-counting and flux-quantization analysis in §2.1–2.2 is clean and self-contained, and the domain-wall tension argument for the superpotential in §2.4 is a genuine independent check on the supersymmetry derivation. The identification of wrapped-brane solitons with Landau-Ginzburg solitons in §3–4 is a real check rather than a restatement, and the paper is honest about its limitations: it flags the fiber-Lagrangian assumption in §3, labels the D and E identifications as conjectural in §4.1, and openly revises the G_{4,0} claim from AdS to no-scale in Appendix II following Polchinski's comments. The result, if correct, gives a large class of exactly solvable CFTs from string geometry and a testing ground for higher-dimensional analogues.

Major objections (1)

[§3, paragraph beginning 'Strictly speaking we have constructed the supersymmetric cycle...'] The paper acknowledges that the cycle may not be Lagrangian with respect to the full Kähler form when mixed terms f_i dz_1 ∧ dz_i^* are present, and patches this by rescaling a_i → t a_i and taking t → ∞. The justification given is that 'the BPS states will be the same for all t, because the condition of the BPS charges getting aligned does not change as we change t (the BPS charges only receive an overall rescaling).' This sentence addresses the calibration condition (the phases of Z_i = ∫_{S_i} Ω), but not the Lagrangian condition, which depends on the Kähler form that is also being rescaled. The t-independence of the BPS spectrum is therefore asserted rather than proved. Since the identification with Kazama-Suzuki models in §4 rests on this reduction, the paper should either supply an index argument for the t-independence of the BPS spectrum or state explicitly that the identification is established only in the t → ∞ limit.

Minor objections (11)

[§2.4] The phrase 'we will somewhen loosely call W̃ a superpotential' contains a typo: 'somewhen' should be 'sometimes'.
[Acknowledgements] The phrase 'J. Polchinski hor helpful' contains a typo: 'hor' should be 'for'.
[Reference [3]] The title contains a typo: 'Catastrphes' should be 'Catastrophes'.
[Reference [34]] The title contains a typo: 'Landau-Ginzburge' should be 'Landau-Ginzburg'.
[Acknowledgements] The grant number 'RFBR No 98-02-1 6575' has an extraneous space and should be corrected.
[Reference [40]] Reference [40] is listed as 'to appear'. The no-scale supergravity interpretation in Appendix II relies on it. The paper should either provide the reference or note that the derivation in Appendix II is self-contained.
[Abstract] The abstract says 'we identify some of these theories with certain Kazama-Suzuki coset models', but §4.1 says 'it is natural to conjecture' for the D and E cases. The abstract should distinguish the established A_{m-1} identification from the conjectural D and E cases.
[Abstract] The abstract says 'in many instances in two or three dimensions one obtains nontrivial superconformal field theories', but §2.6 says 'in a few cases in three dimensions'. The abstract should be more precise about the three-dimensional case.
[§5.2] The claim that 'there is only one option for how the SU(2)_R that acts on the cohomology of a Kähler manifold can act on the L^2 harmonic forms on W: they transform with spin 1' is asserted without derivation. A brief justification would help the reader.
[§2.6] The claim that 'the classical singularity survives quantum corrections' is argued via the absence of instantons because L has only two-cycles. This is plausible but could be stated more carefully, e.g., by explicitly noting that there are no compact one-, three-, five-, or six-cycles on X that could support a wrapped instanton.
[§2.6 and §4.1] The paper uses the Cecotti-Vafa classification [23] to argue that the soliton spectrum determines the CFT. This classification is established for massive deformations of N=2 minimal models. For the D and E cases, which are Kazama-Suzuki models but not necessarily minimal models, the paper says 'it is natural to conjecture'. The paper should state more explicitly which parts of the identification rest on the classification and which rest on the LG-KS correspondence [32].

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The paper

The paper claims

Wrapped branes at a Calabi-Yau four-fold singularity reproduce exactly solvable two-dimensional conformal field theories.

The reading

The paper asks what happens to string theory when the internal Calabi-Yau four-fold develops an isolated singularity, and argues that the answer is often a nontrivial interacting quantum field theory rather than a free or empty one. The authors classify the possible vacua by the flux of the M-theory three-form potential, build the domain walls that interpolate between them out of wrapped branes, and read off an effective superpotential from the tension of those walls. Near the singularity the wrapped-brane solitons turn out to obey exactly the same first-order equation as the solitons of a two-dimensional Landau-Ginzburg model, $dW = P_m^{(n-2)/2}dz_1$. For the $A$-$D$-$E$ singularities this lets the authors name the resulting two-dimensional theories: they are the level-one Kazama-Suzuki coset models, including the $N=2$ unitary minimal models, which are exactly solvable. If the identification holds, a purely geometric computation in string theory reproduces the spectrum of a large known family of rational conformal field theories.

What carries the argument

The load-bearing device is the effective superpotential $W = frac{1}{2pi}int_X Omega wedge G$, together with its Kähler analogue $tilde W = int_X K wedge K wedge G$. The first turns the conditions $G^{0,4} = G^{1,3} = 0$ for unbroken supersymmetry into the statement that $W$ and $dW$ vanish, and it also gives the tension of a domain wall across which $G$ jumps by $2pi[S]$ as the change in $W$. The second reproduces primitiveness, $K wedge G = 0$. On the geometry side, the wrapped-brane volume integral reduces to $int_I |P_m(z_1)|^{(n-2)/2}dz_1$, which is minimized exactly when the image of the cycle in the $W$-plane is a straight line between critical points, the same rule that governs solitons in a Landau-Ginzburg theory with superpotential $W$. Matching the two descriptions of the same soliton is what licenses the identification with the coset model.

Extended reading notes

Core claim

The central claim is that M-theory, Type IIA, or F-theory on a Calabi-Yau four-fold with an isolated $A$-$D$-$E$ singularity, with the flux at infinity fixed to its minimal value so that no membranes remain, flows in the infrared to a nontrivial superconformal field theory whose soliton spectrum is that of a level-one Kazama-Suzuki coset model. The argument runs through three steps. First, the flux of the three-form potential is classified by a coset $H^4(X;Z)/i(H^4_{cpct}(X;Z)) = Z_m$ for an $A_{m-1}$ singularity, so a vacuum is a point in the weight lattice of $SU(m)$ modulo the root lattice, and the number of vacua is the dimension of the $k$-fold antisymmetric tensor representation. Second, the domain walls between these vacua are branes wrapped on vanishing four-cycles, and their tension equals the change in a superpotential $W = (1/2pi)int_X Omega wedge G$. Third, minimizing the wrapped-brane volume reproduces the Landau-Ginzburg soliton condition, and the soliton counting rule of Cecotti and Vafa then fixes the conformal field theory. The authors extend the same logic to the $D$ and $E$ singularities, whose level-one cosets are built on Hermitian symmetric spaces, and note in a revised appendix that a $(4,0)$ piece of the flux breaks supersymmetry and gives a classically scale-invariant no-scale vacuum rather than an AdS one.

Load-bearing premise

The derivation assumes that a supersymmetric four-cycle can be built by taking a minimal cycle in the elliptic fiber over a line segment in the base, which the authors themselves say is not necessarily true once the Kähler form has mixed terms, and which they justify only by a rescaling limit in which those mixed terms are dominated by the fiber terms.

Editorial extensions

If this is right

Type IIA on a deformed $A_{m-1}$ four-fold singularity with flux at infinity labelled by $k$ is the massive deformation of the level-one coset $SU(m)/(SU(m-k) times SU(k) times U(1))$, with $k=1$ giving the $A_m$ $N=2$ minimal model.
The $D_n$, $E_6$, and $E_7$ singularities give the other level-one Kazama-Suzuki models built on Hermitian symmetric spaces, with the number of vacua equal to the dimension of the corresponding representation ($2n$ for the vector, $2^{n-1}$ for the spinor, $27$ for $E_6$, $56$ for $E_7$); $E_8$ has trivial center and yields no such model.
Because the coset models are exactly solvable, the construction supplies a geometric derivation of their spectra and suggests that their correlation functions can be extracted from string theory on the singularity.
In three dimensions a $Phi^3$ superpotential is expected to flow to a nontrivial fixed point, so a few of the M-theory compactifications are candidates for three-dimensional superconformal theories; in four dimensions the same superpotentials are expected to flow to free theories.
The singularity analysis extends beyond hypersurfaces: for the hyper-Kähler cone $T^*CP^2$ the flux quantization law forces a nonzero $G$, and there is exactly one model, with $Phi = 3/8$, that has two vacua.

Reading between the lines

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

If the wrapped-cycle solitons and the Landau-Ginzburg solitons agree for all deformations and not only in the rescaling limit used to justify them, the correspondence could be turned around: the known modular data of the Kazama-Suzuki models would predict the BPS spectrum of four-branes on vanishing cycles, which is a computation one could attempt independently.
The same dictionary should apply to non-supersymmetric or less supersymmetric singularities, since the soliton condition $dW = P^{(n-2)/2}dz_1$ is a statement about calibrated cycles and does not use the full holonomy of the four-fold.
The count of vacua as the dimension of an antisymmetric tensor representation suggests a direct check against the chiral ring of the coset model, which is known to be the cohomology ring of the Hermitian symmetric space.
The no-scale conclusion for $(4,0)$ flux, corrected in the revised appendix, implies that any four-fold compactification with nonzero $int Omega wedge G$ is classically flat but not supersymmetric, so the superpotential is a diagnostic of supersymmetry breaking rather than of vacuum energy.

Original abstract

We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a Calabi-Yau four-fold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated singularity, one often generates massless chiral superfields and a superpotential, and in many instances in two or three dimensions one obtains nontrivial superconformal field theories. In the case of two dimensions, we identify some of these theories with certain Kazama-Suzuki coset models, such as the N=2 minimal models.

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We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a Calabi-Yau four-fold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated singularity, one often generates massless chiral superfields and a superpotential, and in many instances in two or three dimensions one obtains nontrivial superconformal field theories. In the case of two dimensions, we identify some of these theories with certain Kazama-Suzuki coset models, such as the N=2 minimal models.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a Calabi-Yau four-fold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes.	contribution 0.85	0.81	0.06	asserted 0.89
s1	We analyze the vacuum and soliton structure of these models, and show that near an isolated singularity, one often generates massless chiral superfields and a superpotential, and in many instances in two or three dimensions one obtains nontrivial superconformal field theories.	result 0.81	0.94	0.76	asserted 0.94
s2	In the case of two dimensions, we identify some of these theories with certain Kazama-Suzuki coset models, such as the N=2 minimal models.	result 0.90	0.89	0.14	asserted 0.87

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

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Small-volume regimes of flop curves in 4d N=1 F-theory require non-perturbative states—blow-up moduli and D3-string excitations—to complete the locally enhanced N=2 spectrum.
Cosmic superstrings in large volume compactifications: PTAs, LISA and time-varying tension
astro-ph.CO 2025-04 conditional novelty 7.0 of 10

Cosmic superstrings with a time-varying tension in large-volume string compactifications generically predict a LISA-band spectral drop and a high-frequency f^4 boost, encoding the mass and decay of the volume modulus.
Dimensional Reduction and K\"ahler Metric for Metric Moduli in Imaginary Self-Dual Flux
hep-th 2025-01 conditional novelty 7.0 of 10

A 10D supergravity reduction derives the 4D moduli space metric for Kähler moduli and, for the first time, for complex structure flat directions in warped type IIB flux compactifications.
Chiral global embedding of Fibre Inflation with $\overline{\rm D3}$ uplift
hep-th 2024-12 conditional novelty 7.0 of 10

Explicit construction of a chiral global embedding of Fibre Inflation with anti-D3 uplift on an h^{1,1}=4 K3-fibered Calabi-Yau, using magnetised D7-branes, a Whitney brane, and O3-planes at a conifold tip, with viabl...
Asymptotically Free AdS3/CFT2
hep-th 2021-08 conditional novelty 7.0 of 10

String theory on AdS3 x squashed S3 with AdS radius below the string length is proposed to be dual to a symmetric product of sigma models on R_phi x S3 deformed by a phi-dependent Z2 twist operator.
F-theory flux vacua at large complex structure
hep-th 2021-05 unverdicted novelty 7.0 of 10

At large complex structure in F-theory, the F-term potential simplifies to V = Z^{AB} ρ_A ρ_B, yielding two families of flux vacua with all complex structure moduli fixed, one with bounded saxion vevs and one with unb...
$G_4$ Flux, Algebraic Cycles and Complex Structure Moduli Stabilization
hep-th 2020-09 conditional novelty 7.0 of 10

An explicit algebraic-cycle G4 flux on the Fermat sextic fourfold stabilizes all 426 complex structure moduli but overshoots the M2-tadpole bound, while a Greene-Plesser symmetric flux leaves 285 flat directions.
Coulomb and Higgs Branches from Canonical Singularities: Part 0
hep-th 2020-07 conditional novelty 7.0 of 10

The magnetic quiver of a 5d SCFT engineered by a canonical threefold singularity is the electric quiver of the 4d SCFT from the same singularity, with one abelian flavor symmetry gauged.
Higher-Form Symmetries in 5d
hep-th 2020-05 accept novelty 7.0 of 10

The 1-form symmetry of a 5d theory built from M-theory on a non-compact Calabi-Yau threefold is the quotient of relative by absolute second homology, computable as the Smith normal form of the intersection matrix of c...
IIB flux non-commutativity and the global structure of field theories
hep-th 2019-08 accept novelty 7.0 of 10

The paper traces the global-structure ambiguity of 6d (2,0) theories to non-commuting RR flux boundary conditions in IIB on C^2/Γ, reproducing the known classification of 4d N=4 theories and adding new results on dual...
Uplifting Runaways
hep-th 2018-09 conditional novelty 7.0 of 10

Antibranes in a warped deformed conifold drive the tip-sphere deformation to zero, and the orientifold tadpole budget then forbids the large warping needed for a small positive cosmological constant in perturbative Type IIB.
The Kreuzer-Skarke Axiverse
hep-th 2018-08 conditional novelty 7.0 of 10

Calabi-Yau threefolds with large Picard number have very narrow Kähler cones, so requiring every curve volume to exceed the string scale forces the manifold and its cycles to grow as a high power of the Picard number,...
Geometric classification of 4d $\mathcal{N}=2$ SCFTs
hep-th 2018-01 conditional novelty 7.0 of 10

A universal dimension formula read off from monodromy around normal rays in the Coulomb branch reproduces known Lagrangian and hypersurface results and yields a totient-function count of allowed Coulomb branch dimensions.
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.
ADE Little String Theory on a Riemann Surface (and Triality)
hep-th 2015-06 conditional novelty 7.0 of 10

The partition function of the ADE (2,0) little string on a three-punctured sphere equals the q-deformed ADE Toda three-point conformal block, with explicit quivers and vertex operators given for all ADE groups.
Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2
hep-th 2015-04 accept novelty 7.0 of 10

Toroidal compactification of 6d (1,0) SCFTs, solved via mirror symmetry, yields many inequivalent 4d N=2 SCFTs, including class S theories whose curves emerge from the mirror geometry.
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.
F-term Axion Monodromy Inflation
hep-th 2014-04 conditional novelty 7.0 of 10

F-term axion monodromy, built from massive Wilson lines and fluxed p-form axions, offers a supersymmetric framework for large-field inflation with a controlled eta problem.
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...
Fivebranes and 4-manifolds
hep-th 2013-06 conditional novelty 7.0 of 10

A proposed dictionary maps 4-manifolds to 2d N=(0,2) theories, under which Vafa-Witten partition functions become elliptic genera and the gluing measure becomes the index of a (0,2) vector multiplet.
Infinitely many N=2 SCFT with ADE flavor symmetry
hep-th 2012-10 conditional novelty 7.0 of 10

For each ADE Lie group G, an infinite tower of 4D N=2 SCFTs D(G,s) is argued to exist, with flavor symmetry G and beta-function contribution s/[2(s+1)] adjoint hypermultiplets.
Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes
hep-th 2012-03 accept novelty 7.0 of 10

For M-theory on a Calabi-Yau fourfold, conifold transitions survive background G-flux precisely when the canonical bundle of the singular curve splits into two line bundles, and a new class of mixed horizontal/vertica...
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.
D3-brane Potentials from Fluxes in AdS/CFT
hep-th 2010-01 conditional novelty 7.0 of 10

The D3-brane potential in a warped conifold receives its leading infrared term from the square of imaginary anti-self-dual flux, with the full spectrum of scaling dimensions classified and matched to the dual conforma...
SUSY Breaking in Local String/F-Theory Models
hep-th 2009-06 conditional novelty 7.0 of 10

Gravity-mediated soft terms in LARGE-volume local string models are suppressed by Ms/MP relative to the gravitino mass, so TeV soft masses imply a gravitino above 10^8 GeV and a volume of order 10^6-10^7.
Towards an Explicit Model of D-brane Inflation
hep-th 2007-06 conditional novelty 7.0 of 10

For the Kuperstein D7 embedding, the full D3-brane inflaton potential including moduli-stabilization corrections can be fine-tuned to support slow-roll inflation near an inflection point, while ACR embeddings admit no...
D6-branes and torsion
hep-th 2006-03 accept novelty 7.0 of 10

Torsional (co)homology classes of twisted tori explain why RR tadpoles shift, why D6-brane moduli are lifted, how the D-brane discretum appears, and how the Freed-Witten anomaly arises in type IIA flux vacua.
Generalized N=1 Orientifold Compactifications and the Hitchin functionals
hep-th 2006-02 conditional novelty 7.0 of 10

The paper derives N=1 Kähler potentials and superpotentials for type II orientifolds on SU(3)-structure manifolds and proposes generalized half-flat manifolds as mirrors of Calabi-Yau compactifications with NS-NS flux.
The QCD Axion and Moduli Stabilisation
hep-th 2006-02 conditional novelty 7.0 of 10

A clean supergravity no-go theorem forbids supersymmetric moduli stabilization with a flat QCD axion, and a large-volume construction gives f_a approximately the geometric mean of the Planck and SUSY-breaking scales.
More Dual Fluxes and Moduli Fixing
hep-th 2006-02 conditional novelty 7.0 of 10

Adding a new set of S-dual non-geometric fluxes to type IIB orientifolds yields N=1 Minkowski vacua in which the dilaton, complex structure, and Kahler moduli are all fixed.
Nongeometric Flux Compactifications
hep-th 2005-08 accept novelty 7.0 of 10

A T-duality invariant superpotential for type II torus compactifications is constructed that includes nongeometric Q and R fluxes alongside H-flux and geometric flux, making the IIA and IIB vacuum distributions identical.
Dynamics of warped compactifications and the shape of the warped landscape
hep-th 2005-07 conditional novelty 7.0 of 10

A ten-dimensional formula for the potential of warped flux compactifications is derived, yielding a slope-dominance criterion for de Sitter vacua and a warping-corrected definition of holomorphic Kahler moduli.
Fluxes, moduli fixing and MSSM-like vacua in a simple IIA orientifold
hep-th 2005-06 conditional novelty 7.0 of 10

Adding RR, NS and metric fluxes to a T^6/(Omega(-1)^F_L I_3) Type IIA orientifold stabilizes all closed-string moduli in supersymmetric AdS vacua, and one such vacuum hosts a three-generation MSSM-like intersecting D6...
Soft Supersymmetry Breaking in KKLT Flux Compactification
hep-th 2005-03 conditional novelty 7.0 of 10

In KKLT flux compactification with all moduli stabilized, anomaly mediation is always comparable in size to modulus mediation, so the soft supersymmetry-breaking terms are a fixed mixture of the two rather than either...
Distributions of nonsupersymmetric flux vacua
hep-th 2004-11 accept novelty 7.0 of 10

Non-supersymmetric flux vacua correspond to eigenvectors of the superpotential Hessian, and in the generic random-matrix ensemble the count of metastable pure F-breaking vacua with near-zero cosmological constant scal...
Building a Better Racetrack
hep-th 2004-04 conditional novelty 7.0 of 10

Explicit Calabi-Yau orientifold compactifications are exhibited in which D3-instanton and gaugino-condensation effects stabilize every Kähler modulus, together with a no-go argument that rules out the one-modulus case.
Distributions of flux vacua
hep-th 2004-04 accept novelty 7.0 of 10

Flux vacua in IIb Calabi-Yau compactification follow a computable density built from special geometry, strongly enhanced near conifold points, with a cosmological constant distribution that is uniform near zero.
New Supersymmetric String Compactifications
hep-th 2002-11 accept novelty 7.0 of 10

T-dualizing type IIB flux vacua on T^6/Z2 yields a new class of supersymmetric compactifications on twisted tori, with a superpotential that includes flux and geometric-twist contributions.
Mirror Symmetry in Generalized Calabi-Yau Compactifications
hep-th 2002-11 conditional novelty 7.0 of 10

Type IIB on a Calabi-Yau threefold with electric NS three-form flux is mirror to type IIA on a half-flat SU(3)-structure manifold, and the two four-dimensional effective actions agree.
RR Flux on Calabi-Yau and Partial Supersymmetry Breaking
hep-th 1999-12 accept novelty 7.0 of 10

NS-NS and RR three-form fluxes on a non-compact Calabi-Yau threefold act as N=2 Fayet-Iliopoulos terms, give mass to part of the spectrum, and break N=2 supersymmetry to N=1.
The heterotic superpotential and the alpha' expansion
hep-th 2026-10 conditional novelty 6.0 of 10

Varying the heterotic superpotential with the recursive Hull connection reproduces Bergshoeff-de Roo conditions through alpha'^3 and gives conjectural all-order F-term equations.
Data analysis of millions of Type IIB flux vacua
hep-th 2026-10 accept novelty 6.0 of 10

A dataset of 1,296,633 reliable flux vacua (127,241 supersymmetric) is produced and used to train classifiers and regressors that predict vacuum properties from flux invariants.
Tetra-quadric CY Threefold and Assisted Fibre Inflation
hep-th 2026-08 conditional novelty 6.0 of 10

Three Kähler moduli on a tetra-quadric CY orientifold collectively drive assisted fibre inflation with individual shifts near 2.2 M_p instead of the effective 5.7 M_p.
Spectator Axions in String Inflation and Primordial Black Holes
hep-th 2026-07 conditional novelty 6.0 of 10

In Fibre Inflation, spectator axions with f≲0.1 M_Pl leave the PBH power spectrum unchanged, while for f≳0.1 M_Pl and exponentially small non-perturbative prefactors they can raise P_R from 1.9e-4 to 3.5e-3 and enable...

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402 later papers cite it

Evidence: substantial. 392 of the citing papers share no author with it, and 23 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 575 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

7 later papers by other authors are listed below: 2 used its method or tool, 1 compared its own results with it and 4 cited it as support.

Uses its method or tool 2 of 10 papers with this label

By utilizing the special geometry of the underlying manifold, we write down the flux-induced superpotential, i.e., Gukov-Vafa-Witten (GVW) type superpotential [ref] , as a function of untwisted and twisted moduli fields.

Stabilization of a twisted modulus on a mirror of rigid Calabi-Yau manifold (2024) · Introduction · 0.81
Used the flux-induced superpotential of this paper to transform the scalar potential into the standard N=1 supergravity form.

Metastable Flux Configurations and de Sitter Spaces (2007) · 0.81 · paraphrased: the licence does not allow quoting

Compares with it 1 of 3 papers with this label

Checked its gauge/gravity setup by matching this paper's GVW superpotential to the gauge theory effective superpotential.

Gauge/gravity duality and the interplay of various fractional branes (2008) · 0.95 · paraphrased: the licence does not allow quoting

Cites it as support 4 of 6 papers with this label

Since such deformations are non-normalizable due to [ref] , the fluctuations around the vev become non-dynamical and thus do not give rise to a full-fledged branch of the moduli space of the 3d theory, but rather to a set of couplings.

M-theory geometric engineering for rank-0 3d $\mathcal{N}=2$ theories (2024) · Zero modes and the deformed phase · 0.98
Found that this group is the same as the co-root lattice given in this paper.

ADE Little String Theory on a Riemann Surface (and Triality) (2015) · 0.97 · paraphrased: the licence does not allow quoting
Used this paper's result that exactly these scalars obtain an F-term in a non-supersymmetric flux background to argue they render the U(1) gauginos massive.

U(1) Mediation of Flux Supersymmetry Breaking (2008) · 0.95 · paraphrased: the licence does not allow quoting
Used this paper's result that solutions of the complex structure F-terms are left invariant in going from weak to strong warping to justify its procedure.

Thraxions: Ultralight Throat Axions (2018) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

405 arXiv papers cite this in 582 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 · 356 sentences without a confident label

Contests it 1 paper

This means that the field theories we construct cannot[We note that it was suggested previously, incorrectly, that these singularities lead to AdS$_4$ holographic duals [ref] .] flow to dual conformal fixed points in the IR.

AdS_4/CFT_3 duals from M2-branes at hypersurface singularities and their deformations · Introduction · contest 0.87

Builds on it 5 papers

This can be considered as the generalization of the computation first performed in [ref] , where the dimensional reduction of the kinetic terms of the NS-NS and R-R type IIB three-form on a (non-toroidal) CY three-fold gives the no-scale scalar potential described in supergravity language by the tree-level Kähler potential for the complex structure/axio-dilaton moduli and the Gukov-Vafa-Witten (GVW) superpotential [ref] .

Relating Double Field Theory to the Scalar Potential of N=2 Gauged Supergravity · Introduction · 0.95
Building on work on flux superpotentials [ref] and their description in generalised geometry [ref] , we conjectured in [ref] that the superpotential is given by the singlet part of the intrinsic torsion of the $ψ$ structure and explicitly showed this was true for the examples of $2$ in M-theory and generic $𝒩=1$ backgrounds of type II theories.

Heterotic backgrounds via generalised geometry: moment maps and moduli · The superpotential · 0.90
This is a generalization of the Gukov-Vafa-Witten (GVW) superpotential [ref] , and can be extracted, for example, from the general expression in [ref] , see also [ref] .

The unwarped, resolved, deformed conifold: fivebranes and the baryonic branch of the Klebanov-Strassler theory · The superpotential · 0.89

Uses its method or tool 10 papers

The Kähler potential for the above set of fields is obtained from standard dimensional reduction [ref] as [equation] The superpotential is obtained from the generalized Gukov-Vafa-Witten formula [ref] [equation] After inserting the various forms from Eq. [ref] and [ref] and using Eq. [ref] as well as the properties of the half-flat mirror basis given in Section [ref] this leads to [equation] The first term arises from the non-vanishing torsion of the internal space and the second term is due to the non-vanishing H-flux induced by the gauge bundle.

Moduli Stabilising in Heterotic Nearly K\"ahler Compactifications · Kähler potential and superpotential · 0.92
With $Π=(X^0,X^1,F_0,F_1)^T$ and [equation] the GVW superpotential of flux compactifications [ref] becomes [equation] For the bounded type IV active representative [equation] the axiodilaton equation $D_SW=0$ at the benchmark saxion gives [equation] The numerical residual satisfies $|D_SW|<10^-12$.

Penumbral Inflation from Calabi-Yau Boundaries · Flux polynomial · 0.88
By utilizing the special geometry of the underlying manifold, we write down the flux-induced superpotential, i.e., Gukov-Vafa-Witten (GVW) type superpotential [ref] , as a function of untwisted and twisted moduli fields.

Stabilization of a twisted modulus on a mirror of rigid Calabi-Yau manifold · Introduction · 0.81

Compares with it 3 papers

Nonetheless, the number of localized modes that are found via our techniques is always equal or greater than the number of normalizable deformations à la [ref] .

M-theory geometric engineering for rank-0 3d $\mathcal{N}=2$ theories · Zero modes and the deformed phase · 0.93
In this subsection, we make an important consistency check of our gauge/gravity set up by matching the GVW superpotential [ref] to the gauge theory effective superpotential considered in the previous subsection.

Gauge/gravity duality and the interplay of various fractional branes · The Gukov-Vafa-Witten superpotential · 0.95
Notice that the effective potential ([ref]) does not exactly reproduce the Gukov-Vafa-Witten superpotential [ref] _GVW=∫_MΩ_CY∧(F_3+i e^-Φ H) . in the subcase of warped IIB CY compactifications.[On the other hand, $_GVW$ can be obtained from ([ref]) by truncating it in the naive way.

On moduli and effective theory of N=1 warped flux compactifications · $$-moduli and the superpotential · 0.98

Cites it as support 3 papers

Since such deformations are non-normalizable due to [ref] , the fluctuations around the vev become non-dynamical and thus do not give rise to a full-fledged branch of the moduli space of the 3d theory, but rather to a set of couplings.

M-theory geometric engineering for rank-0 3d $\mathcal{N}=2$ theories · Zero modes and the deformed phase · 0.98
Indeed, R3 has been advocated by Gukov, Vafa and Witten as a necessary condition for a sound SCFT [ref] .

Geometric classification of 4d $\mathcal{N}=2$ SCFTs · Behavior along the discriminant · 0.98
This group is indeed the same as the co-root lattice, $Λ^∨=H_cmpt^2(X, ℤ)$ [ref] .

ADE Little String Theory on a Riemann Surface (and Triality) · Monopoles on $ℝ×T^2$ · 0.97

Cites it as background 173 papers

Such fluxes give a potential to the complex structure moduli at tree level, which can be expressed in the resulting three-dimensional $𝒩=2$ theory in terms of the Gukov-Vafa-Witten (GVW) superpotential [ref] [equation] The minima of the induced scalar potential are solutions of the F-term equations $D_IW=0$, $I=1,...,h^1,3(X)$.

$G_4$ Flux, Algebraic Cycles and Complex Structure Moduli Stabilization · Introduction · 0.93
In this section we discuss compactifications of F-theory to four space-time dimensions with a focus on the induced flux superpotential [ref] inherited from the M-theory description of F-theory [ref] .

Computing Brane and Flux Superpotentials in F-theory Compactifications · Flux superpotentials in F-theory · 0.87
In flux compactifications non-renormalisation is not so simple, as the superpotential does depend explicitly on the dilaton through the Gukov, Vafa, Witten (GVW) superpotential [ref] .

Anti-D3 branes and moduli in non-linear supergravity · Introduction · 0.86

Discussion

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