Pith. sign in

REVIEW 4 major objections 4 minor 121 references

Towards the Non-Perturbative Completion of 4d N=1 Effective Theories of Gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 4d N=1 string compactifications need non-perturbative completion: local supersymmetry at small volume demands states — blow-up moduli, D3-string excitations, a 7-brane vector — absent perturbatively.

desk verdict A serious, partly explicit proposal that 4d N=1 F-theory needs non-perturbative light states at small volume, but the central spectrum is fixed by hand and the paper admits it. read the letter →

arxiv 2510.23698 v2 pith:I52KUNXE submitted 2025-10-27 hep-th

Gonzalo F. Casas , Max Wiesner This is my paper · ORCID
classification hep-th MSC 81T3083E3014J32 PACS 11.25.Mj11.25.-w
keywords F-theorynon-perturbativecompletion4dN=1supersymmetryfloptransitionsenhancedD3-branestringsCalabi-Yaufourfoldsheteroticduality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that four-dimensional $N=1$ effective theories of gravity obtained from string compactifications are incomplete at the perturbative level: in small-volume regimes, consistency requires additional light states of non-perturbative origin. To make this concrete, the authors exploit subsectors that locally exhibit enhanced supersymmetry, where the enhancement dictates exactly which degrees of freedom are missing. The central example is a shrinkable flop curve in an F-theory compactification: the perturbative Type IIB orientifold contains only a massless $N=1$ chiral multiplet plus a massive $N=1$ vector multiplet, while the enhanced symmetry requires a massless $N=2$ hypermultiplet and a massive $N=2$ vector multiplet. The paper identifies the completing states — a blow-up modulus, D3-brane string excitations, and a massive 7-brane vector — and extends the picture to complex-structure moduli, flux vacua, a unifying heterotic dual, and transitions without supersymmetry enhancement. If correct, the results imply that any complete $4d$ $N=1$ theory of gravity must include non-perturbative states invisible to the perturbative spectrum, and that locally enhanced supersymmetry is the tool that makes them visible.

What carries the argument

The load-bearing machinery is the birational factorization of a flop: the curve $C_0$ is blown up into an exceptional divisor $E \simeq P^1\times P^1$ whose two $P^1$ factors, $C'$ and $C''$, yield a four-phase structure of the blown-up Kähler moduli space; Phase IV (both curves small) is where the completion is visible. Three concrete mechanisms carry the argument: (i) the blow-up supplies the complex modulus $T_E$ and two axions $b_1^E, b_2^E$, which with $t_0$ complete the massless $N=2$ hypermultiplet; (ii) a D3-brane string wrapped on $C''$ has worldsheet vacuum energy $E_0=-1/2$ and tension $|t'' + \beta e^{-2\pi t'} + i/2|$, and its $n=1$ excitation becomes massless exactly at the flop locus once $\beta=1/2$, providing the charged chiral $\chi_+$; (

What would settle it

Compute the exact $[p,q]$-string instanton corrections to the tension of the D3-brane wrapped on $C''$ (or the low-lying worldsheet spectrum of the non-critical string on $O(-1)\oplus O(0)$): if the $n=1$ excitation is not massless at $t'=t''=0$, or if the correction coefficient differs from $\beta=1/2$, the identification of $\chi_+$ fails. A concrete target: determine $E_0$ and the spectrum of the worldsheet theory from first principles and check the mass formula $M_n^2 = |\alpha(n/\sqrt{2} \cdot T + E_0)|^2$ used in Eq. (3.35).

Watch

Extended reading notes

Core claim

Take a shrinkable flop curve $C_0$ in the base of an elliptically fibered Calabi-Yau fourfold that avoids the $O7$-plane ($C_0\cdot \bar{K}_{B_3}=0$, normal bundle $O(-1)\oplus O(-1)$). Perturbative Type IIB orientifolds see only a massless $N=1$ chiral multiplet and a massive $N=1$ vector multiplet at small volume, but local enhanced supersymmetry demands a massless $N=2$ hypermultiplet and a massive $N=2$ vector multiplet. The paper identifies the missing states as non-perturbative: blowing up $C_0$ into an exceptional divisor $E$ adds the modulus $T_E$ completing the hypermultiplet; the $n=1$ excitation of a D3-brane string on $C''$ gives the charged chiral; the massive vector becomes a Stückelberg-massive 7-brane $U(1)$. Central r

Load-bearing premise

The paper assumes that the non-perturbative correction to the D3-brane string tension is a single exponential $\beta e^{-2\pi t'}$ with $\beta$ fixed to $1/2$ by imposing the $N=2$ completion (rather than computed), and that the low-lying excitation masses follow the critical-string formula — if either fails, the identified charged state $\chi_+$ would not complete the massive $N=2$ vector multiplet.

Editorial extensions

If this is right

  • 4d N=1 effective theories of gravity obtained from string compactifications are not complete at the perturbative level: small-volume regimes necessarily contain additional light states of non-perturbative origin, enlarging the moduli space beyond the perturbative description.
  • The light spectrum of a flop curve with C0·\bar K_{B3}=0 is fixed by local supersymmetry to be a massless N=2 hypermultiplet plus a massive N=2 vector multiplet, with concrete microscopic identifications (blow-up modulus, D3-string excitation, 7-brane vector).
  • Embedding the local sector in a compact fourfold changes the D3-brane tadpole by -31 units and removes 124 complex structure deformations; spacetime-filling D3/M2-branes make the transition possible at any point in complex structure moduli space.
  • Gravity-induced N=2→N=1 breaking splits the locus where three states become massless into components separated by an exponentially small amount, breaking the N=2 mass degeneracy by O(e^{-T_{D,mov}}) and removing the restored U(1) point.
  • In the complex structure sector, supersymmetric W=0 flux vacua can be realized at loci of enhanced supersymmetry through tensionless domain walls, and a heterotic dual unifies the flop and flux transitions as NS5-brane nucleations (on a base curve vs. on the elliptic fiber).

Reading between the lines

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

  • Editorial extension: the same logic suggests that any small-volume limit of a 4d N=1 theory — not just flop curves — should host non-perturbative light states; the enhanced-supersymmetry sectors are simply the cases where the completion can be computed, so the paper is evidence for a general completeness principle.
  • Editorial extension: the β=1/2 fixing is a testable prediction in disguise: a first-principles computation of the [p,q]-string instanton sum on C'' either confirms β=1/2 (supporting the completion) or produces a different coefficient, which would force an alternative identification of the charged state.
  • Editorial extension: the 7-brane/closed-string identification of the massive vector suggests that similar 'chameleon' gauge fields — closed-string U(1)s that become open-string U(1)s across a transition — may be generic in 4d N=1 transitions and could be searched for in explicit compact models.
  • Editorial extension: the exponential mass hierarchy between N=2 multiplet partners (mass differences ~ e^{-T_{D,mov}}) could in principle be probed in concrete compactifications, giving a sharp quantitative signature of gravity-induced supersymmetry breaking in protected local sectors.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper argues that 4d N=1 F-theory/Type IIB orientifold compactifications with locally enhanced supersymmetry require non-perturbative completion in the small-volume regime. It focuses on a shrinkable flop curve C0 in the base B3 with normal bundle O(-1)⊕O(-1) and C0·\bar K_{B3}=0. The perturbative orientifold spectrum there contains only a massless N=1 chiral multiplet and a massive N=1 vector multiplet, whereas local N=2 supersymmetry requires a massless N=2 hypermultiplet and a massive N=2 vector multiplet. The paper proposes that the missing states arise from the blow-up of C0 to an exceptional divisor E≃F0, adding T_E and axions b1^E,b2^E, together with a massive 7-brane vector Υ and a charged chiral χ+ coming from the n=1 excitation of a D3-brane string on C''. It computes explicit topological changes, including δχ=-744, δn_D3=-31, and δh^{3,1}=-124, and discusses global supersymmetry breaking, the complex-structure sector, and a heterotic dual NS5-brane picture. The key identification of χ+ and χ0, however, relies on uncomputed non-perturbative mass formulas that are fixed by requiring the N=2 completion.

Significance. If correct, the paper would establish a general principle: 4d N=1 corners of string theory require non-perturbative light states, and local supersymmetry enhancement is a reliable guide to identifying them. The paper contains several explicit and checkable computations, notably the Chern-class change under blow-up (§3.4.1) and the Freed–Witten induced b''2=1/2 shift (3.32). It also makes falsifiable predictions, e.g., the mass scaling of χ+ in (3.38) and of χ0 in (3.42), and the requirement of extra three-cycles after blow-up. The heterotic dual perspective provides a unifying qualitative picture. However, the central spectrum completion is not yet a derivation: the values β=1/2 and δh^{2,1}=1, and the form of the χ0 mass, are imposed rather than computed. The current status is therefore a well-motivated scenario with a clear route to checking or falsifying the claims.

major comments (4)
  1. [§3.3.2, Eqs. (3.36)–(3.38)] The identification of χ+ is load-bearing but imposed, not derived. The classical D3-string formula (3.35) places the n=1 massless point at t''=1/2; the shift to t'=t''=0 is achieved by the conjectured correction TS''=|t''+β e^{-2πt'}+i/2|, and Eq. (3.37) fixes β=1/2 precisely by demanding the N=2 completion. The text states 'we do not compute these corrections from first principle here', and footnote 13 assumes the critical-string mass formula for a non-critical E-like string. If β differs, or the mass formula has an O(1) shift or different n-dependence, χ+ is massive at t0=0 and the claimed massive N=2 vector multiplet is not completed. This requires either an independent computation/justification of β or an explicit reframing as a conjecture with stated assumptions.
  2. [§3.2.2, Eq. (3.53) and arguments 1–3] The input δh^{2,1}=1 is not derived. The explicit Chern-class computation (3.46)–(3.50) fixes only δχ=-744, i.e., δh^{3,1}+δh^{1,1}-δh^{2,1}=-124. With δh^{1,1}=1, the physical arguments in items 1–3 set δh^{2,1}=1 and hence δh^{3,1}=-124. The two extra three-cycles Γ1,Γ2 then provide the axions b1^E,b2^E forming χ0. If the actual Hodge number change differs, χ0 and hence the massive N=2 vector completion are absent. The three arguments are consistency/heuristic statements, not a computation. The paper should either provide a concrete F-theory model with a computed δh^{2,1} or clearly identify this as an additional assumption.
  3. [§3.3.3, Eq. (3.42)] The χ0 mass formula is likewise imposed. The text states 'we did not derive (3.42) from first principles'; the form is selected so that m^2_χ0 vanishes at t'+t''→0, as needed for the uncharged chiral to complete the massive vector multiplet. This is a boundary condition, not independent evidence. A different instanton/prefactor structure or additional contributions would alter the mass hierarchy and the completion in Phase IV. The role of this formula should therefore be presented as a postulated form, not as a result, unless supported by an independent computation.
  4. [§3.4.1–§3.4.2] The global embedding argument rests on the identification of a 248-dimensional field-theory sector built from 31 M2-branes and 124 complex-structure degenerations. The arithmetic 8×31=248 and the codimension statement in (3.57) are suggestive, but the claim that these two different ingredients combine into one decoupled sector, and that this makes the transition possible at generic complex structure, is not demonstrated. This is used to argue that the blow-up is a consistent deformation of compact fourfolds. The authors should either develop this sector more concretely or mark the global consistency claim as conjectural.
minor comments (4)
  1. [Eq. (3.49)] The displayed integral appears to have a typo: the integrand should be c1(B3)^3 (or the subscript/superscript is lost), not c1(B3) alone.
  2. [Eq. (3.35)] The notation for α is dimensionally implicit; please state explicitly that α is an order-one dimensionless constant, since it appears inside the mass formula alongside dimensionful tensions.
  3. [§3.2.3] Minor wording: 'Phase II and III' should be 'Phases II and III'.
  4. [Fig. 3] The caption refers to an 'orange line' representing the χ+ massless locus, but the figure is schematic and the line is not visible in the preprint text; please ensure the figure and caption are consistent.

Circularity Check

3 steps flagged · score 6.0 of 10

Phase IV masslessness is imposed, not derived: Eq. (3.37) fixes β=1/2 by requiring χ+ to be massless at t0=0, and Eq. (3.42) is likewise chosen so that χ0 vanishes at t',t''→0; the non-perturbative completion is therefore constructed from the N=2 target rather than independently predicted.

  1. self definitional [Sec. 3.3.2, Eqs. (3.36)–(3.38) and footnote 13]
    "While we do not compute these corrections from first principle here, we notice that the requirement of having the d.o.f. making up a light N=2 vector multiplet in phase IV implies β≠0. More precisely, we have to impose that the field χ+ becomes massless at the locus {t0=0}⊂ M_B3 ... Imposing that the string excitation with n=1 becomes massless at this point fixes β=1/2."

    The desired conclusion—that χ+ is the light charged chiral completing the massive N=2 vector multiplet at t0=0—is used as the condition that fixes the unknown coefficient β. The subsequent 'prediction' m^2_χ+ ≃ α²π²(t')²/4 near the origin is then an algebraic consequence of this tuning, not an independent check. The same target condition is also what selects β≠0 at all. Footnote 13 adds a further unverified input: the low-n mass formula of the non-critical string is assumed to be the critical-string formula, so if the true correction or mass formula differed, the phase-IV completion would fail.

  2. self definitional [Sec. 3.3.3, Eq. (3.42)]
    "the mass of χ0 thus has the form m^2_χ0/M10^2 ≃ α e^{-2πTE}(1−e^{−2π(t′+t′′)})^2 ... Let us stress that, again, we did not derive (3.42) from first principles. Instead, we were guided by the physics that arises at the origin of the Higgs branch of a 4d N=2 field theory. Imposing the existence of a massless, uncharged N=1 chiral multiplet then led to the expression (3.42)."

    The mass formula is chosen so that m_χ0 → 0 exactly at t′,t′′ → 0, i.e., it is constructed to satisfy the N=2 completion that the paper claims to demonstrate. The subsequent identification of χ0 as the required neutral chiral multiplet therefore does not test the completion; it is the completion written as an ansatz. The paper explicitly admits this by saying the expression was 'imposed' from the N=2 Higgs-branch physics rather than derived.

1 more flagged steps
  1. self definitional [Sec. 3.2.2, item 1 (δh^{2,1}=1)]
    "As we explain in more detail below, these four additional scalar fields are the degrees of freedom required to form 4d N=2 supermultiplets. Therefore, due to the enhanced supersymmetry, we expect the blow-up to introduce an additional h2,1."

    The Hodge-number shift δh^{2,1}=1 is inferred from the requirement that the blow-up supply the four scalars needed for N=2 multiplets; the later statement that the blow-up adds the degrees of freedom required for N=2 is therefore a restatement of the input rather than an independent geometric derivation. In the local, non-compact model no independent topological computation fixes δh^{2,1}; it is set by the supersymmetry-enhancement target.

full rationale

The geometric scaffolding of the paper—birational factorization of the flop, the Chern-class computation δχ=−744, the D3/M2-tadpole bookkeeping, and the heterotic reinterpretation—is largely self-contained and not circular. However, the central Phase IV claim is constructed rather than independently derived. The coefficient β in the non-perturbative tension correction (3.36) is fixed in (3.37) by imposing that the n=1 D3-string excitation χ+ be massless at t0=0, which is precisely the N=2 completion the paper advertises; (3.42) for χ0 is likewise imposed so that it vanishes at t′,t′′→0. The paper explicitly states that these corrections are not computed from first principles and that the mass formula assumes a critical-string form (footnote 13). Thus the advertised result—that non-perturbative effects supply the missing light states completing the N=2 multiplets—rests on parameters and ansätze chosen to realize that very completion. This is a partial, real circularity in the quantitative spectrum claim, even though the geometric identification of the candidate states (blow-up modulus TE, axions b_i^E, D3-string excitations, 7-brane vector) gives the proposal independent conceptual content. No load-bearing self-citation chain was found: the self-citations present (e.g., [40], [48]) are contextual and do not by themselves force the central conclusion.

Assumptions & free parameters 4 free parameters · 9 assumptions · 3 invented entities

The central derivation pulls in several inputs that the reader did not pay for upstream. The most important are β and the χ0 mass formula, both explicitly chosen to produce the N=2 spectrum, and δh^{2,1}=1, chosen so the blow-up supplies the needed axions. The remaining assumptions are standard string-duality/geometry facts or qualitative arguments. The genuine first-principles content lies in the topological computations of §3.4.1 and the mass-formula application (3.35).

free parameters (4)
  • β (non-perturbative correction to D3-string tension on C') = 1/2
    Fixed in Eq. (3.37) so the n=1 excitation of the D3-string wrapped on C'' is massless at t'=t''=0 (the locus t0=0). Authors: 'While we do not compute these corrections from first principle here ... imposing that the string excitation with n=1 becomes massless at this point fixes β=1/2.'
  • α (order-one constants in mass formulas) = unspecified
    Appears in the string mass formula (3.35), the mass (3.38), and the χ0 mass formula (3.42); never fixed or derived.
  • γ (O(1) coefficient in massless loci under global SUSY breaking) = O(1)
    In Eqs. (3.65) and (3.67), the loci where χ± is massless are shifted by t0=±γ e^{-T_D,mov} with 0<γ∼O(1); no derivation given.
  • δh^{2,1}=1 (change in Hodge number upon blow-up) = 1
    Adopted in §3.2.2 via three arguments (N=2 completion, no local D3-brane charge, 6d parallel); not computed from the geometry of the blow-up, but required for the transition to work.
assumptions (9)
  • domain assumption Small-volume limit of a curve with normal bundle O(-1)⊕O(-1) is described by the origin of an N=2 Higgs branch (conifold transition physics)
    Invoked in §3.1.1 citing [12,13,71]; underlies the entire N=2 completion argument in Section 3.
  • domain assumption A curve C0 in the base of an F-theory fourfold with C0·\bar K_{B3}=0 realizes a local subsector with enhanced supersymmetry (local SU(3) holonomy in the fourfold)
    Used throughout §3; for the orientifold picture, C0 does not intersect the O7-plane, so local N=2 enhancement is assumed exact enough to dictate the spectrum even in the quantum regime.
  • ad hoc to paper Mass formula for low-lying excitations of the non-critical D3-brane string resembles the critical string formula
    Footnote 13 explicitly assumes this for Eq. (3.35); if false, the identification of χ+ as a massless n=1 state changes.
  • ad hoc to paper Non-perturbative corrections to the D3-string tension have the single-instanton form βe^{-2πt'} with β=1/2, and the χ0 mass has the form (3.42)
    Eqs. (3.36)-(3.37) and (3.42); the paper states these are not derived from first principles but chosen to reproduce the N=2 spectrum.
  • ad hoc to paper The Pfaffian A_E of the Euclidean D3-instanton on E depends nontrivially on the axions b1^E,b2^E and vanishes at b1=b2=0, generating their Planck-scale masses along M_B3
    §3.2.2; asserted via [81,82] without computing the five-brane partition function.
  • standard math Euler characteristic formula χ(X4)=12∫(c1c2+30c1^3) for smooth Weierstrass models and the tadpole condition χ/24=n_D3+½∫G4^2
    Used in §3.4.1 and §4, from [73-75]; standard F-theory results accepted as background.
  • domain assumption Existence of a heterotic dual for K3-fibered elliptically fibered CY fourfolds, with spectral-bundle data η1, η2
    Section 5 relies on F-theory/heterotic duality [58,59,105,106]; the paper explicitly says the dual picture is qualitative.
  • domain assumption Loci in complex structure moduli space where periods reduce to K3 periods (with W=∂W=0 flux vacua) exist and are realized by enhanced SUSY
    Section 4 builds on [57]; the paper adds a tensionless-domain-wall argument but assumes such loci exist.
  • ad hoc to paper For point blow-ups (§6), the Pfaffian vanishes and its derivative vanishes when an M2-brane sits on the exceptional divisor
    Eqs. (6.4)-(6.5) are assumed so that W_np=0 at the transition point; not derived.
invented entities (3)
  • Light states χ0 and χ± completing the massive N=2 vector multiplet in Phase IV
    purpose: Provide the missing degrees of freedom demanded by local enhanced SUSY in the small-volume regime of the flop curve: χ+ charged chiral from D3-string on C'' (or C'), χ0 neutral chiral from B2/C2 axions, Υ massive vector from 7-brane gauge field on E.
    They are composites of known string objects (D3-branes on curves, 2-form axions) and a 7-brane gauge field, but their masses and the β=1/2 condition are imposed, not derived; no observable handle outside the construction.
  • Extra three-cycles Γ1, Γ2 (δh^{2,1}=1) producing axions b1^E, b2^E after blow-up
    purpose: Supply the second chiral multiplet needed to form the massless N=2 hypermultiplet; cancel local D3-brane charge.
    Required by the N=2 completion argument in §3.2.2; not demonstrated in a concrete compact CY fourfold.
  • Local field theory sector M_FT of 31 M2-branes + 124 complex-structure degenerations
    purpose: Account for the change in Euler characteristic/tadpole in the compact transition and make the transition possible for any complex structure.
    Proposed in §3.4.1 on dimensional grounds (248=8×31); interpretive and not tested.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Towards the Non-Perturbative Completion of 4d N=1 Effective Theories of Gravity." pith.science (2026). https://pith.science/paper/I52KUNXE

@misc{pith2026251023698,
  author       = {Pith},
  title        = {Pith review of: Towards the Non-Perturbative Completion of 4d N=1 Effective Theories of Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I52KUNXE}},
  note         = {Machine review of arXiv:2510.23698}
}
abstract

We show that four-dimensional $\mathcal N=1$ effective theories of gravity obtained from string compactifications require a non-perturbative completion, as additional light states of non-perturbative origin must be incorporated in the small volume regime to obtain a consistent low-energy description. This completion becomes concrete in subsectors that locally exhibit enhanced supersymmetry, where the enhancement predicts the existence of additional light degrees of freedom absent in the perturbative description. Motivated by analogous setups in six-dimensional $\mathcal N=(1,0)$ theories, we focus on the K\"ahler moduli space of F-theory compactifications on Calabi--Yau fourfolds to four dimensions, where shrinkable curves not intersected by 7-planes realize such supersymmetry-enhanced subsectors. Guided by the enhanced supersymmetry, we use F-theory to identify the degrees of freedom missing in the small volume regime of the perturbative Type IIB description. A consistent embedding of these local subsectors into a four-dimensional $\mathcal N=1$ theory of gravity requires an appropriate inclusion of complex structure moduli and spacetime-filling D3-branes. We also discuss supersymmetry enhancement in the complex structure sector and study how a heterotic dual description gives a unifying picture of the different F-theory sectors with enhanced supersymmetry. Finally, we comment on cases without local supersymmetry enhancement.

Figures

Figures reproduced from arXiv: 2510.23698 by the authors.

Figure 1
Figure 1. The conifold singularity of the 4d N = 2 theory separates the Higgs from the Coulomb branch. The Higgs branch is parameterized by the volume of C˜ 0, while the Coulomb branch is parameterized by the volume of A. In the Higgs branch, a massive photon arises by reducing C4 over the chain Σ3. In the Coulomb branch, the same amount of bosonic degrees of freedom arises from a charged hypermultiplet corresponding to D3-br… view at source ↗
Figure 2
Figure 2. Schematic representation of the birational factorization of a flop transition. The left diagram corresponds to the 4-fold X4 containing the flop curve C0, whereas the right one corresponds to X˜ 4 after a flop transition. The diagram in the middle shows the 4-fold Xˆ 4 for which the curve C0 has been blown up into an extra exceptional divisor E. due to the enhanced supersymmetry, the moduli space in the vicinity of … view at source ↗
Figure 3
Figure 3. A sketch of the phase structure of MXb4 . The Phase I corresponds to the classical geometric description of F-theory on Xb4. Phases II and III correspond to a regime in which the volumes of the cycles C ′ and C ′′ differ hierarchically. The deep interiror of these phases can respectively be identified with the moduli space of the theory compactified on MX4 and MXˇ4 . Phase IV corresponds to a small-volume regime for… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: A sketch of the fibration structure of the Calabi Yau 4-fold Xb4 considered in this chapter T 2 → P 1 F → B2 = F1. The fiber P 1 f degenerates over C ′ in the union of two rational curves intersecting over a point corresponding to a blow-up of the base curve of F1. in …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

121 extracted references · 112 linked inside Pith

  1. [1]

    Vafa,The String landscape and the swampland,hep-th/0509212

    C. Vafa,The String landscape and the swampland,hep-th/0509212

    arXiv

  2. [2]

    Palti,The Swampland: Introduction and Review,Fortsch

    E. Palti,The Swampland: Introduction and Review,Fortsch. Phys.67(2019), no. 6 1900037, [arXiv:1903.06239]

    arXiv 2019

  3. [3]

    Gra˜ na and A

    M. Gra˜ na and A. Herr´ aez,The Swampland Conjectures: A Bridge from Quantum Gravity to Particle Physics,Universe7(2021), no. 8 273, [arXiv:2107.00087]

    arXiv 2021

  4. [4]

    van Beest, J

    M. van Beest, J. Calder´ on-Infante, D. Mirfendereski, and I. Valenzuela,Lectures on the Swampland Program in String Compactifications,arXiv:2102.01111

    arXiv

  5. [5]

    N. B. Agmon, A. Bedroya, M. J. Kang, and C. Vafa,Lectures on the string landscape and the Swampland,arXiv:2212.06187

    arXiv

  6. [6]

    P. S. Green and T. Hubsch,Connecting Moduli Spaces of Calabi-yau Threefolds, Commun. Math. Phys.119(1988) 431–441

    1988

  7. [7]

    Candelas, P

    P. Candelas, P. S. Green, and T. Hubsch,Finite Distances Between Distinct Calabi-yau Vacua: (Other Worlds Are Just Around the Corner),Phys. Rev. Lett.62(1989) 1956. 53

    1989

  8. [8]

    P. S. Green and T. Hubsch,Possible Phase Transitions among Calabi-Yau Compactifications,Phys. Rev. Lett.61(1988) 1163

    1988

Show all 121 references
  1. [9]

    Candelas, P

    P. Candelas, P. S. Green, and T. Hubsch,Rolling Among Calabi-Yau Vacua, Nucl. Phys. B330(1990) 49

    1990

  2. [10]

    P. S. Aspinwall, B. R. Greene, and D. R. Morrison,Calabi-Yau moduli space, mirror manifolds and space-time topology change in string theory,Nucl. Phys. B416(1994) 414–480, [hep-th/9309097]

    1994 arXiv

  3. [11]

    P. S. Aspinwall, B. R. Greene, and D. R. Morrison,Multiple mirror manifolds and topology change in string theory,Phys. Lett. B303(1993) 249–259, [hep-th/9301043]

    1993 arXiv

  4. [12]

    Strominger,Massless black holes and conifolds in string theory,Nucl

    A. Strominger,Massless black holes and conifolds in string theory,Nucl. Phys. B451(1995) 96–108, [hep-th/9504090]

    1995 arXiv

  5. [13]

    B. R. Greene, D. R. Morrison, and A. Strominger,Black hole condensation and the unification of string vacua,Nucl. Phys. B451(1995) 109–120, [hep-th/9504145]

    1995 arXiv

  6. [14]

    Ooguri and C

    H. Ooguri and C. Vafa,Summing up D instantons,Phys. Rev. Lett.77(1996) 3296–3298, [hep-th/9608079]

    1996 arXiv

  7. [15]

    Saueressig and S

    F. Saueressig and S. Vandoren,Conifold singularities, resumming instantons and non-perturbative mirror symmetry,JHEP07(2007) 018, [arXiv:0704.2229]

    2007 arXiv

  8. [16]

    Marchesano and M

    F. Marchesano and M. Wiesner,Instantons and infinite distances,JHEP08 (2019) 088, [arXiv:1904.04848]

    2019 arXiv

  9. [17]

    Baume, F

    F. Baume, F. Marchesano, and M. Wiesner,Instanton Corrections and Emergent Strings,JHEP04(2020) 174, [arXiv:1912.02218]

    2020 arXiv

  10. [18]

    ´Alvarez-Garc ´ ıa, D

    R. ´Alvarez-Garc ´ ıa, D. Kl¨ awer, and T. Weigand,Membrane limits in quantum gravity,Phys. Rev. D105(2022), no. 6 066024, [arXiv:2112.09136]

    2022 arXiv

  11. [19]

    Robles-Llana, M

    D. Robles-Llana, M. Rocek, F. Saueressig, U. Theis, and S. Vandoren, Nonperturbative corrections to 4D string theory effective actions from SL(2,Z) duality and supersymmetry,Phys. Rev. Lett.98(2007) 211602, [hep-th/0612027]

    2007 arXiv

  12. [20]

    Robles-Llana, F

    D. Robles-Llana, F. Saueressig, U. Theis, and S. Vandoren,Membrane instantons from mirror symmetry,Commun. Num. Theor. Phys.1(2007) 681–711, [arXiv:0707.0838]

    2007 arXiv

  13. [21]

    Alexandrov, B

    S. Alexandrov, B. Pioline, F. Saueressig, and S. Vandoren,D-instantons and twistors,JHEP03(2009) 044, [arXiv:0812.4219]

    2009 arXiv

  14. [22]

    Alexandrov,D-instantons and twistors: Some exact results,J

    S. Alexandrov,D-instantons and twistors: Some exact results,J. Phys. A42 (2009) 335402, [arXiv:0902.2761]. 54

    2009 arXiv

  15. [23]

    Alexandrov,Twistor Approach to String Compactifications: a Review,Phys

    S. Alexandrov,Twistor Approach to String Compactifications: a Review,Phys. Rept.522(2013) 1–57, [arXiv:1111.2892]

    2013 arXiv

  16. [24]

    Ooguri and C

    H. Ooguri and C. Vafa,On the Geometry of the String Landscape and the Swampland,Nucl. Phys.B766(2007) 21–33, [hep-th/0605264]

    2007 arXiv

  17. [25]

    S.-J. Lee, W. Lerche, and T. Weigand,Emergent strings from infinite distance limits,JHEP02(2022) 190, [arXiv:1910.01135]

    2022 arXiv

  18. [26]

    McOrist and I

    J. McOrist and I. V. Melnikov,Half-Twisted Correlators from the Coulomb Branch,JHEP04(2008) 071, [arXiv:0712.3272]

    2008 arXiv

  19. [27]

    Kreuzer, J

    M. Kreuzer, J. McOrist, I. V. Melnikov, and M. R. Plesser,(0,2) Deformations of Linear Sigma Models,JHEP07(2011) 044, [arXiv:1001.2104]

    2011 arXiv

  20. [28]

    Becker, M

    K. Becker, M. Becker, M. Haack, and J. Louis,Supersymmetry breaking and alpha-prime corrections to flux induced potentials,JHEP06(2002) 060, [hep-th/0204254]

    2002 arXiv

  21. [29]

    von Gersdorff and A

    G. von Gersdorff and A. Hebecker,Kahler corrections for the volume modulus of flux compactifications,Phys. Lett. B624(2005) 270–274, [hep-th/0507131]

    2005 arXiv

  22. [30]

    M. Berg, M. Haack, and B. Kors,String loop corrections to Kahler potentials in orientifolds,JHEP11(2005) 030, [hep-th/0508043]

    2005 arXiv

  23. [31]

    M. Berg, M. Haack, and E. Pajer,Jumping Through Loops: On Soft Terms from Large Volume Compactifications,JHEP09(2007) 031, [arXiv:0704.0737]

    2007 arXiv

  24. [32]

    M. Berg, M. Haack, and J. U. Kang,One-Loop Kahler Metric of D-Branes at Angles,JHEP11(2012) 091, [arXiv:1112.5156]

    2012 arXiv

  25. [33]

    M. Berg, M. Haack, J. U. Kang, and S. Sj¨ ors,Towards the one-loop K¨ ahler metric of Calabi-Yau orientifolds,JHEP12(2014) 077, [arXiv:1407.0027]

    2014 arXiv

  26. [34]

    Kim,On string one-loop correction to the Einstein-Hilbert term and its implications on the K¨ ahler potential,JHEP07(2023) 044, [arXiv:2302.12117]

    M. Kim,On string one-loop correction to the Einstein-Hilbert term and its implications on the K¨ ahler potential,JHEP07(2023) 044, [arXiv:2302.12117]

    2023 arXiv

  27. [35]

    T. W. Grimm, R. Savelli, and M. Weissenbacher,Onα ′ corrections in N=1 F-theory compactifications,Phys. Lett. B725(2013) 431–436, [arXiv:1303.3317]

    2013 arXiv

  28. [36]

    T. W. Grimm, J. Keitel, R. Savelli, and M. Weissenbacher,From M-theory higher curvature terms toα ′ corrections in F-theory,Nucl. Phys. B903(2016) 325–359, [arXiv:1312.1376]

    2016 arXiv

  29. [37]

    Weissenbacher,F-theory vacua andα ′-corrections,JHEP04(2020) 032, [arXiv:1901.04758]

    M. Weissenbacher,F-theory vacua andα ′-corrections,JHEP04(2020) 032, [arXiv:1901.04758]

    2020 arXiv

  30. [38]

    Klaewer, S.-J

    D. Klaewer, S.-J. Lee, T. Weigand, and M. Wiesner,Quantum corrections in 4d N= 1 infinite distance limits and the weak gravity conjecture,JHEP03(2021) 252, [arXiv:2011.00024]. 55

    2021 arXiv

  31. [39]

    Cicoli, F

    M. Cicoli, F. Quevedo, R. Savelli, A. Schachner, and R. Valandro,Systematics of theα’ expansion in F-theory,JHEP08(2021) 099, [arXiv:2106.04592]

    2021 arXiv

  32. [40]

    Wiesner,Light strings and strong coupling in F-theory,JHEP04(2023) 088, [arXiv:2210.14238]

    M. Wiesner,Light strings and strong coupling in F-theory,JHEP04(2023) 088, [arXiv:2210.14238]

    2023 arXiv

  33. [41]

    Cvetiˇ c and M

    M. Cvetiˇ c and M. Wiesner,Nonperturbative resolution of strong coupling singularities in 4D N=1 heterotic M-theory,Phys. Rev. D110(2024), no. 10 106008, [arXiv:2408.12458]

    2024 arXiv

  34. [42]

    Intriligator, H

    K. Intriligator, H. Jockers, P. Mayr, D. R. Morrison, and M. R. Plesser, Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes,Adv. Theor. Math. Phys.17(2013), no. 3 601–699, [arXiv:1203.6662]

    2013 arXiv

  35. [43]

    Gendler, M

    N. Gendler, M. Kim, L. McAllister, J. Moritz, and M. Stillman,Superpotentials from singular divisors,JHEP11(2022) 142, [arXiv:2204.06566]

    2022 arXiv

  36. [44]

    L. B. Anderson, C. R. Brodie, and J. Gray,Branes and bundles through conifold transitions and dualities in heterotic string theory,Phys. Rev. D108(2023), no. 10 106018, [arXiv:2211.05804]

    2023 arXiv

  37. [45]

    A. Font, A. Herr´ aez, and L. E. Ib´ a˜ nez,The Swampland Distance Conjecture and Towers of Tensionless Branes,JHEP08(2019) 044, [arXiv:1904.05379]

    2019 arXiv

  38. [46]

    Lanza, F

    S. Lanza, F. Marchesano, L. Martucci, and I. Valenzuela,Swampland Conjectures for Strings and Membranes,JHEP02(2021) 006, [arXiv:2006.15154]

    2021 arXiv

  39. [47]

    Lanza, F

    S. Lanza, F. Marchesano, L. Martucci, and I. Valenzuela,The EFT stringy viewpoint on large distances,JHEP09(2021) 197, [arXiv:2104.05726]

    2021 arXiv

  40. [48]

    C. F. Cota, A. Mininno, T. Weigand, and M. Wiesner,The asymptotic Weak Gravity Conjecture for open strings,JHEP11(2022) 058, [arXiv:2208.00009]

    2022 arXiv

  41. [49]

    Marchesano and M

    F. Marchesano and M. Wiesner,4d strings at strong coupling,JHEP08(2022) 004, [arXiv:2202.10466]

    2022 arXiv

  42. [50]

    Martucci, N

    L. Martucci, N. Risso, and T. Weigand,Quantum gravity bounds onN= 1 effective theories in four dimensions,JHEP03(2023) 197, [arXiv:2210.10797]

    2023 arXiv

  43. [51]

    G. F. Casas, L. E. Ib´ a˜ nez, and F. Marchesano,Yukawa couplings at infinite distance and swampland towers in chiral theories,JHEP09(2024) 170, [arXiv:2403.09775]

    2024 arXiv

  44. [52]

    Grieco, I

    A. Grieco, I. Ruiz, and I. Valenzuela,EFT strings and dualities in 4dN= 1, arXiv:2504.16984

    arXiv

  45. [53]

    Weigand,Lectures on F-theory compactifications and model building,Class

    T. Weigand,Lectures on F-theory compactifications and model building,Class. Quant. Grav.27(2010) 214004, [arXiv:1009.3497]

    2010 arXiv

  46. [54]

    Weigand,F-theory,PoSTASI2017(2018) 016, [arXiv:1806.01854]

    T. Weigand,F-theory,PoSTASI2017(2018) 016, [arXiv:1806.01854]. 56

    2018 arXiv

  47. [55]

    D. R. Morrison and C. Vafa,Compactifications of F theory on Calabi-Yau threefolds. 1,Nucl. Phys. B473(1996) 74–92, [hep-th/9602114]

    1996 arXiv

  48. [56]

    Witten,Phase transitions in M theory and F theory,Nucl

    E. Witten,Phase transitions in M theory and F theory,Nucl. Phys. B471 (1996) 195–216, [hep-th/9603150]

    1996 arXiv

  49. [57]

    T. W. Grimm and D. van de Heisteeg,Exact flux vacua, symmetries, and the structure of the landscape,JHEP01(2025) 005, [arXiv:2404.12422]

    2025 arXiv

  50. [58]

    D. R. Morrison,The Geometry underlying mirror symmetry, alg-geom/9608006

    arXiv

  51. [59]

    D. R. Morrison and C. Vafa,Compactifications of F theory on Calabi-Yau threefolds. 2.,Nucl. Phys. B476(1996) 437–469, [hep-th/9603161]

    1996 arXiv

  52. [60]

    Sagnotti,A Note on the Green-Schwarz mechanism in open string theories, Phys

    A. Sagnotti,A Note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B294(1992) 196–203, [hep-th/9210127]

    1992 arXiv

  53. [61]

    M. B. Green and J. H. Schwarz,Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory,Phys. Lett. B149(1984) 117–122

    1984

  54. [62]

    Aldazabal, A

    G. Aldazabal, A. Font, L. E. Ibanez, and F. Quevedo,Heterotic / heterotic duality in D = 6, D = 4,Phys. Lett. B380(1996) 33–41, [hep-th/9602097]

    1996 arXiv

  55. [63]

    Horava and E

    P. Horava and E. Witten,Heterotic and Type I string dynamics from eleven dimensions,Nucl. Phys. B460(1996) 506–524, [hep-th/9510209]

    1996 arXiv

  56. [64]

    Horava and E

    P. Horava and E. Witten,Eleven-dimensional supergravity on a manifold with boundary,Nucl. Phys. B475(1996) 94–114, [hep-th/9603142]

    1996 arXiv

  57. [65]

    Seiberg and E

    N. Seiberg and E. Witten,Comments on string dynamics in six-dimensions, Nucl. Phys. B471(1996) 121–134, [hep-th/9603003]

    1996 arXiv

  58. [66]

    Witten,Small instantons in string theory,Nucl

    E. Witten,Small instantons in string theory,Nucl. Phys. B460(1996) 541–559, [hep-th/9511030]

    1996 arXiv

  59. [67]

    M. F. Atiyah,On analytic surfaces with double points,Proceedings of the Cambridge Philosophical Society54(1958) 36–41

    1958

  60. [68]

    Hayashi, C

    H. Hayashi, C. Lawrie, and S. Schafer-Nameki,Phases, Flops and F-theory: SU(5) Gauge Theories,JHEP10(2013) 046, [arXiv:1304.1678]

    2013 arXiv

  61. [69]

    Sen,Orientifold limit of F theory vacua,Phys

    A. Sen,Orientifold limit of F theory vacua,Phys. Rev. D55(1997) R7345–R7349, [hep-th/9702165]

    1997 arXiv

  62. [70]

    Demirtas, M

    M. Demirtas, M. Kim, L. McAllister, J. Moritz, and A. Rios-Tascon,Small cosmological constants in string theory,JHEP12(2021) 136, [arXiv:2107.09064]

    2021 arXiv

  63. [71]

    B. R. Greene, D. R. Morrison, and C. Vafa,A Geometric realization of confinement,Nucl. Phys. B481(1996) 513–538, [hep-th/9608039]. 57

    1996 arXiv

  64. [72]

    Reid,Minimal model of canonical 3-folds,Adv

    M. Reid,Minimal model of canonical 3-folds,Adv. Stud. Pure Math.1(1983) 131–180

    1983

  65. [73]

    Dasgupta and S

    K. Dasgupta and S. Mukhi,A Note on low dimensional string compactifications,Phys. Lett. B398(1997) 285–290, [hep-th/9612188]

    1997 arXiv

  66. [74]

    Sethi, C

    S. Sethi, C. Vafa, and E. Witten,Constraints on low dimensional string compactifications,Nucl. Phys. B480(1996) 213–224, [hep-th/9606122]

    1996 arXiv

  67. [75]

    Dasgupta, G

    K. Dasgupta, G. Rajesh, and S. Sethi,M theory, orientifolds and G - flux, JHEP08(1999) 023, [hep-th/9908088]

    1999 arXiv

  68. [76]

    Klemm, B

    A. Klemm, B. Lian, S. S. Roan, and S.-T. Yau,Calabi-Yau fourfolds for M theory and F theory compactifications,Nucl. Phys. B518(1998) 515–574, [hep-th/9701023]

    1998 arXiv

  69. [77]

    Denef,Les Houches Lectures on Constructing String Vacua,Les Houches87 (2008) 483–610, [arXiv:0803.1194]

    F. Denef,Les Houches Lectures on Constructing String Vacua,Les Houches87 (2008) 483–610, [arXiv:0803.1194]

    2008 arXiv

  70. [78]

    T. W. Grimm,The N=1 effective action of F-theory compactifications,Nucl. Phys. B845(2011) 48–92, [arXiv:1008.4133]

    2011 arXiv

  71. [79]

    Greiner and T

    S. Greiner and T. W. Grimm,On Mirror Symmetry for Calabi-Yau Fourfolds with Three-Form Cohomology,JHEP09(2016) 073, [arXiv:1512.04859]

    2016 arXiv

  72. [80]

    Greiner and T

    S. Greiner and T. W. Grimm,Three-form periods on Calabi-Yau fourfolds: Toric hypersurfaces and F-theory applications,JHEP05(2017) 151, [arXiv:1702.03217]

    2017 arXiv

  73. [81]

    Witten,Five-brane effective action inM-theory.,J

    E. Witten,Five-brane effective action inM-theory.,J. Geom. Phys.22(1997) 103–133, [hep-th/9610234]

    1997 arXiv

  74. [82]

    O. J. Ganor,A Note on zeros of superpotentials in F theory,Nucl. Phys. B499 (1997) 55–66, [hep-th/9612077]

    1997 arXiv

  75. [83]

    T. W. Grimm,Non-Perturbative Corrections and Modularity in N=1 Type IIB Compactifications,JHEP10(2007) 004, [arXiv:0705.3253]

    2007 arXiv

  76. [84]

    S. Chen, D. van de Heisteeg, and C. Vafa,Symmetries and M-theory-like vacua in four dimensions,JHEP07(2025) 258, [arXiv:2503.16599]

    2025 arXiv

  77. [85]

    Atiyah, J

    M. Atiyah, J. M. Maldacena, and C. Vafa,An M theory flop as a large N duality,J. Math. Phys.42(2001) 3209–3220, [hep-th/0011256]

    2001 arXiv

  78. [86]

    Monnee, T

    J. Monnee, T. Weigand, and M. Wiesner,K-Points and Type IIB/Heterotic Duality with NS5-Branes,arXiv:2510.02435

  79. [87]

    D. S. Freed and E. Witten,Anomalies in string theory with D-branes,Asian J. Math.3(1999) 819, [hep-th/9907189]

    1999 arXiv

  80. [88]

    Del Zotto and G

    M. Del Zotto and G. Lockhart,On Exceptional Instanton Strings,JHEP09 (2017) 081, [arXiv:1609.00310]. 58

    2017 arXiv

  81. [89]

    J. Kim, K. Lee, and J. Park,On elliptic genera of 6d string theories,JHEP10 (2018) 100, [arXiv:1801.01631]

    2018 arXiv

  82. [90]

    S.-J. Lee, W. Lerche, and T. Weigand,Tensionless Strings and the Weak Gravity Conjecture,JHEP10(2018) 164, [arXiv:1808.05958]

    2018 arXiv

  83. [91]

    Candelas, A

    P. Candelas, A. Font, S. H. Katz, and D. R. Morrison,Mirror symmetry for two parameter models. 2.,Nucl. Phys. B429(1994) 626–674, [hep-th/9403187]

    1994 arXiv

  84. [92]

    D. R. Morrison and M. R. Plesser,Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties,Nucl. Phys. B440(1995) 279–354, [hep-th/9412236]

    1995 arXiv

  85. [93]

    C. F. Cota, A. Klemm, and T. Schimannek,Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities,JHEP11(2019) 170, [arXiv:1910.01988]

    2019 arXiv

  86. [94]

    Grana,Flux compactifications in string theory: A Comprehensive review, Phys

    M. Grana,Flux compactifications in string theory: A Comprehensive review, Phys. Rept.423(2006) 91–158, [hep-th/0509003]

    2006 arXiv

  87. [95]

    Denef, M

    F. Denef, M. R. Douglas, and S. Kachru,Physics of String Flux Compactifications,Ann. Rev. Nucl. Part. Sci.57(2007) 119–144, [hep-th/0701050]

    2007 arXiv

  88. [96]

    Aldazabal, D

    G. Aldazabal, D. Badagnani, L. E. Ibanez, and A. M. Uranga,Tadpole versus anomaly cancellation in D = 4, D = 6 compact IIB orientifolds,JHEP06 (1999) 031, [hep-th/9904071]

    1999 arXiv

  89. [97]

    Aldazabal, A

    G. Aldazabal, A. Font, L. E. Ibanez, and G. Violero,D = 4, N=1, type IIB orientifolds,Nucl. Phys. B536(1998) 29–68, [hep-th/9804026]

    1998 arXiv

  90. [98]

    Gukov, C

    S. Gukov, C. Vafa, and E. Witten,CFT’s from Calabi-Yau four folds,Nucl. Phys. B584(2000) 69–108, [hep-th/9906070]. [Erratum: Nucl.Phys.B 608, 477–478 (2001)]

    2000 arXiv

  91. [99]

    Haack and J

    M. Haack and J. Louis,M theory compactified on Calabi-Yau fourfolds with background flux,Phys. Lett. B507(2001) 296–304, [hep-th/0103068]

    2001 arXiv

  92. [100]

    Witten,On flux quantization in M theory and the effective action,J

    E. Witten,On flux quantization in M theory and the effective action,J. Geom. Phys.22(1997) 1–13, [hep-th/9609122]

    1997 arXiv

  93. [101]

    Becker and M

    K. Becker and M. Becker,M theory on eight manifolds,Nucl. Phys. B477 (1996) 155–167, [hep-th/9605053]

    1996 arXiv

  94. [102]

    Hulek and H

    K. Hulek and H. Verrill,On modularity of rigid and nonrigid calabi-yau varieties associated to the root latticea 4, 2005

    2005

  95. [103]

    Jockers, S

    H. Jockers, S. Kotlewski, and P. Kuusela,Modular Calabi-Yau fourfolds and connections to M-theory fluxes,JHEP12(2024) 052, [arXiv:2312.07611]. 59

    2024 arXiv

  96. [104]

    A. P. Braun and R. Valandro,G 4 flux, algebraic cycles and complex structure moduli stabilization,JHEP01(2021) 207, [arXiv:2009.11873]

    2021 arXiv

  97. [105]

    S.-J. Lee, W. Lerche, and T. Weigand,Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions,JHEP08(2019) 104, [arXiv:1901.08065]

    2019 arXiv

  98. [106]

    Friedman, J

    R. Friedman, J. Morgan, and E. Witten,Vector bundles and F theory, Commun. Math. Phys.187(1997) 679–743, [hep-th/9701162]

    1997 arXiv

  99. [107]

    Curio and R

    G. Curio and R. Y. Donagi,Moduli in N=1 heterotic / F theory duality,Nucl. Phys. B518(1998) 603–631, [hep-th/9801057]

    1998 arXiv

  100. [108]

    Haghighat, G

    B. Haghighat, G. Lockhart, and C. Vafa,Fusing E-strings to heterotic strings: E+E→H,Phys. Rev. D90(2014), no. 12 126012, [arXiv:1406.0850]

    2014 arXiv

  101. [109]

    S.-J. Lee, W. Lerche, G. Lockhart, and T. Weigand,Quasi-Jacobi forms, elliptic genera and strings in four dimensions,JHEP01(2021) 162, [arXiv:2005.10837]

    2021 arXiv

  102. [110]

    Klemm, P

    A. Klemm, P. Mayr, and C. Vafa,BPS states of exceptional noncritical strings, Nucl. Phys. B Proc. Suppl.58(1997) 177, [hep-th/9607139]

    1997 arXiv

  103. [111]

    Lerche, P

    W. Lerche, P. Mayr, and N. P. Warner,Noncritical strings, Del Pezzo singularities and Seiberg-Witten curves,Nucl. Phys. B499(1997) 125–148, [hep-th/9612085]

    1997 arXiv

  104. [112]

    O. J. Ganor and A. Hanany,Small E(8) instantons and tensionless noncritical strings,Nucl. Phys. B474(1996) 122–140, [hep-th/9602120]

    1996 arXiv

  105. [113]

    Witten,Nonperturbative superpotentials in string theory,Nucl

    E. Witten,Nonperturbative superpotentials in string theory,Nucl. Phys. B474 (1996) 343–360, [hep-th/9604030]

    1996 arXiv

  106. [114]

    Kallosh, A.-K

    R. Kallosh, A.-K. Kashani-Poor, and A. Tomasiello,Counting fermionic zero modes on M5 with fluxes,JHEP06(2005) 069, [hep-th/0503138]

    2005 arXiv

  107. [115]

    Billo, L

    M. Billo, L. Ferro, M. Frau, F. Fucito, A. Lerda, and J. F. Morales,Flux interactions on D-branes and instantons,JHEP10(2008) 112, [arXiv:0807.1666]

    2008 arXiv

  108. [116]

    T. W. Grimm, M. Kerstan, E. Palti, and T. Weigand,On Fluxed Instantons and Moduli Stabilisation in IIB Orientifolds and F-theory,Phys. Rev. D84 (2011) 066001, [arXiv:1105.3193]

    2011 arXiv

  109. [117]

    Bianchi, A

    M. Bianchi, A. Collinucci, and L. Martucci,Magnetized E3-brane instantons in F-theory,JHEP12(2011) 045, [arXiv:1107.3732]

    2011 arXiv

  110. [118]

    Bianchi, A

    M. Bianchi, A. Collinucci, and L. Martucci,Freezing E3-brane instantons with fluxes,Fortsch. Phys.60(2012) 914–920, [arXiv:1202.5045]. 60

    2012 arXiv

  111. [119]

    Bianchi, G

    M. Bianchi, G. Inverso, and L. Martucci,Brane instantons and fluxes in F-theory,JHEP07(2013) 037, [arXiv:1212.0024]

    2013 arXiv

  112. [120]

    I. Bena, J. Blab¨ ack, M. Gra˜ na, and S. L¨ ust,The tadpole problem,JHEP11 (2021) 223, [arXiv:2010.10519]

    2021 arXiv

  113. [121]

    Denef, M

    F. Denef, M. R. Douglas, B. Florea, A. Grassi, and S. Kachru,Fixing all moduli in a simple f-theory compactification,Adv. Theor. Math. Phys.9(2005), no. 6 861–929, [hep-th/0503124]. 61

    2005 arXiv

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.