Kahler Potentials of Chiral Matter Fields for Calabi-Yau String Compactifications

Daniel Cremades; Fernando Quevedo; Joseph P. Conlon

REVIEW

Kahler Potentials of Chiral Matter Fields for Calabi-Yau String Compactifications

Joseph P. Conlon, Daniel Cremades, Fernando Quevedo

High Energy Physics - Theory · Submitted 26 Sep 2006 · 26 references

Pith's verdict: A genuinely useful scaling method with the right τ_s exponent and a sign typo in the headline volume factor.

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CitationDaniel Cremades, Fernando Quevedo, Joseph P. Conlon. "Kahler Potentials of Chiral Matter Fields for Calabi-Yau String Compactifications." arXiv:hep-th/0609180 (2006). https://pith.science/paper/SOUDHWKK

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The thing to know up front: the method is the contribution, and it's a good one. Conlon, Cremades and Quevedo extract the Kähler moduli dependence of the matter metric by inverting the supergravity Yukawa formula. Since T is forbidden from the perturbative superpotential by the PQ shift symmetry, the scaling of the physical Yukawa is directly the scaling of K̃. That observation turns an intractable Calabi-Yau computation into a wavefunction-overlap counting exercise. The toroidal check against Font–Ibáñez and Lüst–Mayr–Richter–Stieberger comes out right, which is the kind of external validation that makes the method credible. The λ ∈ [0,1] bound from V₁₂₃ ⊂ V_ij is a nice touch, and the μ-term argument in section 3.2 is a free bonus that I didn't expect to hold up as well as it does. The authors are also honest about what they can't do: no complex structure dependence, leading power only, classical weak coupling. Take those as given, not as findings of mine. Now the soft spots. The stress-test is right about the sign. Eq. (3.6) with p=1 gives K̃ ~ τ_b^{-1} ~ V^{-2/3}, and volume independence of Ŷ requires exactly that; but eqs. (3.9), (3.24) and (6.1) print V^{+2/3}. Section 4's single-modulus cross-check, K = -3log(T+T̄) + C*C/(T+T̄)^{2/3}, also gives the negative sign. So the derivation and the τ_s^{1/3} are right and the volume exponent in the headline equation is a typo — but it's a typo in the equation people will quote, and anyone plugging (6.1) into a soft-term calculation without rederiving will get the wrong volume scaling. Eq. (3.7)'s exponent is also off by a factor of two in the first term. Worth saying plainly. Second, the load-bearing physical premise — that rescaling the cycle only rescales wavefunction normalisation, never shape — is argued, not proven. The argument (shape is fixed by complex structure, which hasn't changed) is reasonable and I believe it in the dilute-flux limit, but it deserves one sentence acknowledging it could fail once τ_s is not large. Third, the 'minimal model' with one small cycle and four stacks distinguished by flux on homologically trivial 2-cycles is sketched rather than constructed, and the λ values in 3.1.3 for the more complicated geometries are order-of-magnitude estimates. Who should read it: anyone computing soft terms in LVS, or anyone who needs matter metrics on a CY they can't do CFT on. It's a method paper and it has been used. Peer review: yes, send it out. The typos need fixing but the physics is sound and the technique is worth a referee's time.

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Major objections (1)

[§3.1.1–3.1.2, §4, §6; Eqs. (3.6), (3.7), (3.9), (3.23), (3.24), (6.1)] The volume exponent in the paper's central result appears with the wrong sign, and the paper's own equations contradict each other. From Eq. (3.6), K̃_α = τ_b^{-p_α}k_α, together with p_α = 1 (Eq. 3.8), one has K̃ ~ τ_b^{-1} ~ V^{-2/3}. The locality argument of §3.1.1 requires the same: with K̂ = -2 ln V, e^{K̂/2} = V^{-1}, so volume independence of Ŷ in Eq. (2.3) forces (K̃^3)^{1/2} ~ V^{-1}, i.e. K̃ ~ V^{-2/3}. However, Eqs. (3.9), (3.23), (3.24) and the headline Eq. (6.1) all print K̃ ~ τ_s^{1/3}V^{+2/3}. Taken at face value, Eq. (3.9) gives Ŷ ~ V^{-2}τ_s^{-3λ/2}, which is not volume-independent and so contradicts §3.1.1. The correct result is K̃_αβ̄ = τ_s^{1/3}V^{-2/3}k_αβ̄(φ). The same slip appears in Eq. (3.7), where the printed exponent -3+(p_α+p_β+p_γ)/2 should be -3/2+(p_α+p_β+p_γ)/2; with the printed value and p=1 one gets Ŷ ~ V^{-1}, contradicting the sentence immediately following Eq. (3.7) that 'this implies p_α+p_β+p_γ = 3'. The consistency check at the end of §4 also only works with the negative sign: substituting τ_s → τ_b, λ = 1/3 and V ~ τ_b^{3/2} into (3.23) gives τ_b^{-2/3}, matching (4.2), whereas the printed V^{+2/3} gives τ_b^{+4/3}. Since Eq. (6.1) is the paper's headline result and is quoted as input to the companion soft-term paper [26], these signs should be corrected throughout before publication.

Minor objections (9)

[Eq. (3.40)] The second line reads k_H1(τ_i) but should read k_H2(τ_i), by analogy with Eq. (3.39).
[Eq. (6.1)] The left-hand side is printed as K̂_αβ̄; it should be K̃_αβ̄, the matter metric, consistent with Eq. (3.24).
[Eq. (5.2)] The prefactor contains 'A2k', which is ambiguous; presumably this is A_k^2, the square of the area of the k-th torus. Please make the notation explicit.
[Eq. (3.25)] The relation α = 1/(2τ) should be checked against the standard D7-brane normalisation 1/g^2 = Re T, which gives α = g^2/(4π) = 1/(4πτ); the factor of 2π depends on conventions and should be stated.
[Eqs. (3.42) and (3.43)] These two equations are identical to each other, and (3.42) simply restates (3.37); one of them can be dropped.
[§3.2] The argument that the superpotential µ must vanish assumes µ is independent of V. This follows from the shift symmetry of T, but the paper should say so explicitly, since the whole argument rests on it.
[§3.1.2] The sentence 'loop corrections are suppressed by ∼ 4πτ ∼ 100 and are at the percent level' is garbled; presumably the suppression factor is 1/(4πτ) ∼ 1%.
[§6] The statement that bifundamental fields behave like D3-brane fields and D7 Wilson lines would be more useful with the explicit scalings for those cases quoted alongside.
[§3.1.3] The bound λ ∈ [0,1] is derived from the heuristic relation (3.35) and the assertion V_123 ⊂ V_ij; the assumptions entering this bound should be stated more precisely, since the bound is presented as a general result.

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The paper

The paper claims

Physical Yukawa couplings fix how chiral matter scales with the Kähler moduli in large-volume type IIB flux compactifications, giving the matter metric needed for soft supersymmetry breaking.

The reading

The Kähler potential for visible matter is the least controlled ingredient of a string-derived $N=1$ supergravity, and until now it had been computed only for flat toroidal backgrounds. This paper argues that one does not need a full Calabi-Yau computation to get the leading moduli dependence: because Kähler moduli are forbidden from appearing in the superpotential by a shift symmetry, the physical Yukawa couplings depend on them only through the matter metric, and those Yukawa couplings can be obtained from the classical overlap of normalised wavefunctions. Combining locality (matter sits on a small cycle inside a large bulk, so the Yukawa strength cannot know about the bulk volume) with the scaling of normalised wavefunctions under a rescaling of the cycle volume gives the modular weights directly. For bifundamental matter on magnetised D7 branes in the large-volume models the result is $K̃_{αβ̄} = τ_s^{1/3} V^{2/3} k_{αβ̄}(φ)$, the leading term in a weak-coupling expansion in the small cycle volume. If right, this supplies exactly the missing input for canonically normalising matter fields and for computing gravity-mediated soft terms in these compactifications.

What carries the argument

The load-bearing device is the identification of the physical Yukawa coupling with a triple overlap of canonically normalised wavefunctions, combined with the shift symmetry $Im(T_i) → Im(T_i) + ε_i$ that keeps Kähler moduli out of the superpotential. Together these turn an unknown non-holomorphic quantity, the matter metric $K̃_{αβ̄}$, into a scaling problem: measure how the overlap integral responds to a uniform rescaling of the compact cycle, then read off the modular weight from the supergravity formula relating $Ŷ$ to $K̃$. The paper's one-dimensional Gaussian toy model makes the logic visible — rescaling the line rescales the overlap by $√λ$ without any need to evaluate the integral — and the dimensional reduction of the DBI action on a magnetised 4-cycle supplies the realistic version of the same statement.

Extended reading notes

Core claim

The central claim is a determination of the Kähler moduli dependence of the matter metric for chiral bifundamental fields on wrapped D7 branes in large-volume IIB flux compactifications. The argument runs through the supergravity relation $Ŷ_{αβγ} = e^{K/2} Y_{αβγ}/(K̃_α K̃_β K̃_γ)^{1/2}$. Since the Kähler moduli $T_i$ cannot appear in the tree-level superpotential, they cannot appear in $Y_{αβγ}$, so any scaling of the physical Yukawa coupling with $τ_i = Re(T_i)$ is inherited entirely from the matter metric. The authors then compute that scaling by treating the Yukawa as a triple overlap of normalised wavefunctions: because all Standard Model matter is localised on the small cycle, the overlap is insensitive to the overall volume, which forces the volume exponent to be $p_α = 1$ universally; and because a rescaling $τ_s → βτ_s$ only rescales the normalisation of the wavefunctions, $ψ → ψ/√β$, the physical Yukawa scales as $τ_s^{-1/2}$, giving $λ = 1/3$. The result is $K̃_{αβ̄} = τ_s^{1/3} V^{2/3} k_{αβ̄}(φ)$, equation (6.1) of the paper. The authors extend the same bookkeeping to more elaborate local geometries (three D7 stacks meeting pairwise, overlaps on 2-cycles) obtaining $λ = 1/2$ and $λ = 1/6$, argue for a general bound $λ ∈ [0,1]$, treat the single-Kähler-modulus KKLT-like case (giving $K = -3 log(T+T̄) + C^*C/(T+T̄)^{2/3}$), check the toroidal case against known results, and use the same scaling to argue that the superpotential $μ$ term must vanish in these models.

Load-bearing premise

The argument assumes that rescaling the small cycle's volume only rescales the normalisation of the matter wavefunctions and never changes their shape, so that the physical Yukawa coupling scales as a clean power of the cycle volume.

Editorial extensions

If this is right

The canonically normalised matter fields in large-volume IIB models acquire a definite moduli dependence, $K̃ = τ_s^{1/3} V^{2/3} k(φ)$, which fixes the mass basis and the normalisation entering every observable quantity.
Gravity-mediated soft supersymmetry breaking terms can now be computed with the bifundamental matter metric appropriate to the MSSM rather than the adjoint or Wilson-line metrics previously used as stand-ins.
The superpotential $μ$ term is argued to vanish in the large-volume limit, since a non-zero $μ$ would give a physical mass growing above the string scale as $V → ∞$; a physical $μ$ can still arise through the Giudice-Masiero operator.
The scaling expansion in $τ_s$ is identified with the gauge coupling expansion on the D7 worldvolume, so the leading power is a controlled approximation with percent-level corrections at realistic coupling.
The same techniques, applied to toroidal compactifications, reproduce the known matter metrics such as $K̃_{12} ∼ 1/√{t_3}$, providing a consistency check on the method.

Reading between the lines

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

If the volume exponent $p_α = 1$ is universal as argued, then flavour-changing neutral currents from off-diagonal matter metrics are suppressed by the same overall volume factor for every flavour, so the flavour problem in these models reduces to the unknown complex structure dependence $k_{αβ̄}(φ)$ rather than to Kähler moduli.
The same scaling argument could be inverted: a future explicit computation of a physical Yukawa coupling's cycle-volume scaling in a genuine Calabi-Yau would directly test the wavefunction-normalisation premise, and any deviation from $τ_s^{-1/2}$ would signal that the Kähler modulus is changing wavefunction shapes after all.
Because the method only reaches moduli with a perturbative shift symmetry, the complex structure dependence of the matter metric remains entirely open; a complementary argument using $U$-moduli would need a setting where those moduli are absent from $Y_{αβγ}$, which the paper does not supply.
The $λ ∈ [0,1]$ bound is a useful consistency check on any proposed local brane geometry: a model whose implied modular weight falls outside this window would be inconsistent with the classical large-volume limit.

Original abstract

The Kahler potential is the least understood part of effective N=1 supersymmetric theories derived from string compactifications. Even at tree-level, the Kahler potential for the physical matter fields, as a function of the moduli fields, is unknown for generic Calabi-Yau compactifications and has only been computed for simple toroidal orientifolds. In this paper we describe how the modular dependence of matter metrics may be extracted in a perturbative expansion in the Kahler moduli. Scaling arguments, locality and knowledge of the structure of the physical Yukawa couplings are sufficient to find the relevant Kahler potential. Using these techniques we compute the `modular weights' for bifundamental matter on wrapped D7 branes for large-volume IIB Calabi-Yau flux compactifications. We also apply our techniques to the case of toroidal compactifications, obtaining results consistent with those present in the literature. Our techniques do not provide the complex structure moduli dependence of the Kahler potential, but are sufficient to extract relevant information about the canonically normalised matter fields and the soft supersymmetry breaking terms in gravity mediated scenarios.

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The abstract, sentence by sentence

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

The Kahler potential is the least understood part of effective N=1 supersymmetric theories derived from string compactifications. Even at tree-level, the Kahler potential for the physical matter fields, as a function of the moduli fields, is unknown for generic Calabi-Yau compactifications and has only been computed for simple toroidal orientifolds. In this paper we describe how the modular dependence of matter metrics may be extracted in a perturbative expansion in the Kahler moduli. Scaling arguments, locality and knowledge of the structure of the physical Yukawa couplings are sufficient to find the relevant Kahler potential. Using these techniques we compute the `modular weights' for bifundamental matter on wrapped D7 branes for large-volume IIB Calabi-Yau flux compactifications. We also apply our techniques to the case of toroidal compactifications, obtaining results consistent with those present in the literature. Our techniques do not provide the complex structure moduli dependence of the Kahler potential, but are sufficient to extract relevant information about the canonically normalised matter fields and the soft supersymmetry breaking terms in gravity mediated scenarios.

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	The Kahler potential is the least understood part of effective N=1 supersymmetric theories derived from string compactifications.	background 0.92	0.05	0.03	asserted 1.00
s1	Even at tree-level, the Kahler potential for the physical matter fields, as a function of the moduli fields, is unknown for generic Calabi-Yau compactifications and has only been computed for simple toroidal orientifolds.	gap 0.98	0.04	0.04	asserted 1.00
s2	In this paper we describe how the modular dependence of matter metrics may be extracted in a perturbative expansion in the Kahler moduli.	contribution 0.60	0.82	0.15	hedged possibility 0.95
s3	Scaling arguments, locality and knowledge of the structure of the physical Yukawa couplings are sufficient to find the relevant Kahler potential.	method 0.94	0.48	0.41	asserted 0.99
s4	Using these techniques we compute the `modular weights' for bifundamental matter on wrapped D7 branes for large-volume IIB Calabi-Yau flux compactifications.	result 0.84	0.96	0.41	asserted 1.00
s5	We also apply our techniques to the case of toroidal compactifications, obtaining results consistent with those present in the literature.	result 0.73	0.94	0.07	asserted 0.76
s6	Our techniques do not provide the complex structure moduli dependence of the Kahler potential, but are sufficient to extract relevant information about the canonically normalised matter fields and the soft supersymmetry breaking terms in gravity mediated scenarios.	limitation 1.00	0.82	0.22	asserted 0.97

What kind of paper the classifier reports this is: method or tool 0.58 · evidence from proof or derivation 0.92.

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sources read

arXiv listing abstract v2

not examined

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

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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

Figures

**Figure 1.** A one-modulus toy model, illustrating three Gaussian matter wavefunctions. The Yukawa coupling is determined by the integrated overlap of the normalised wavefunctions. The Yukawa coupling is exponentially sensitive to the values of the ξi . In the correspondence with IIB compactifications, ξi/a corresponds to the complex structure moduli, determining the shape of the wavefunctions, whereas the size of the line corresponds to the K¨ahler moduli. We now consider rescaling the size of the line (the ‘K¨ahler modulus’), without altering the shape of the wavefunctions (the ‘complex…

**Figure 2.** The same toy model wavefunctions after rescaling the size of the real line. The loci and breadths of the wavefunctions have also been rescaled and so the relative shapes are unaltered. However the physical Yukawa coupling scales parametrically with the size of the real line and its absolute magnitude has changed. seek to move beyond toy examples and apply the above strategy to IIB Calabi-Yau flux compactifications. 3. The Large-Volume Model We now apply the above ideas to realistic examples. We will use two geometries, first that of the large-volume models of [14, 15] and then that…

**Figure 3.** The physical picture: Standard Model matter is supported on a small blow-up cycle located within the bulk of a very large Calabi-Yau. The volume of the Calabi-Yau sets the gravitino mass and is responsible for the weak/Planck hierarchy. 3.1.1 Volume Dependence In the large-volume limit we can factorise K˜ α as in (2.5), K˜ α = τ −pα b kα(τi , φ). (3.6) τb ∼ V2/3 is the size of the large 4-cycle and we use φ to denote both dilaton and complex structure moduli. While we have included an index pα, from the picture of figure 3 we expect universality as locality implies all matter f

**Figure 4.** In the minimal geometry, there is only one small 4-cycle. The different brane stacks of the Standard Model are distinguished by having different magnetic fluxes on the internal 2-cycles of the 4-cycle. In the minimal model above, these 2-cycles are not inherited from the Calabi-Yau and only exist as cycles in the geometry of the 4-cycle. Four distinct brane stacks are required to realise the Standard Model, and we schematically show how these stacks are distinguished by different choices of magnetic flux. In the large-volume scenario illustrated in figures 3 and 4, the Standard…

Figures reproduced from arXiv:hep-th/0609180 by the authors.

References

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In two LARGE-volume string inflation models the inflaton couples more strongly to hidden-sector fields than to Standard Model fields, so viable reheating requires the hidden sector to be a pure N=1 SYM theory with a m...
F-theory, GUTs, and the Weak Scale
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In local F-theory GUTs the requirement that the Higgs mu term land near the weak scale forces gauge mediation at F about 10^17 GeV squared, which simultaneously yields an axion decay constant near 10^12 GeV and tan be...
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The leading string-loop correction to the Kähler potential cancels from the scalar potential at leading order whenever the correction is homogeneous of degree -2 in two-cycle volumes, an 'extended no-scale structure' ...
Astrophysical and Cosmological Implications of Large Volume String Compactifications
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How later papers used it

47 later papers cite it

Evidence: substantial. 38 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 28 of the 82 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

1 later paper by other authors is listed below: 1 used its method or tool. 2 more papers by the same authors are listed: 1 used its method or tool and 1 used its data or values.

Uses its method or tool 2 of 6 papers with this label · 1 by the same authors

Followed this paper's work, in which the leading modular dependence of the two functions was estimated through simple scaling arguments, to specify the functions of the moduli fields.

Natural Quintessence in String Theory (2012) · 0.94 · paraphrased: the licence does not allow quoting

By the same authors

Followed an argument similar to this paper's for the matter fields.

Towards Realistic String Vacua From Branes At Singularities (2008) (same authors) · 0.91 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 2 papers with this label · 1 by the same authors

By the same authors

Used this paper's result for the Kähler metric of the chiral matter fields.

Astrophysical and Cosmological Implications of Large Volume String Compactifications (2007) (same authors) · 0.89 · paraphrased: the licence does not allow quoting

How the literature uses it

44 arXiv papers cite this in 79 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 · 64 sentences without a confident label

Builds on it 1 paper

The volume scaling of the Kähler potential for the matter fields $ϕ_j$ can be inferred to be (following the same philosophy of [ref] and assuming a diagonal structure): $K≃∑_j τ_1^1/3^-2/3|ϕ_j|^2$.

Testing String Vacua in the Lab: From a Hidden CMB to Dark Forces in Flux Compactifications · Anisotropic case · 0.95

Uses its method or tool 6 papers

Following the rescaling arguments in [ref] , we set $p=1$ and note that the leading order terms cancel in all soft terms, leaving only the volume suppressed contributions.

Moduli Redefinitions and Moduli Stabilisation · Redefining $T_a$ in the Kähler potential · 0.95
The idea [ref] is to recall that the physical Yukawa couplings $Y_αβγ$ among three fields $Φ_α$ after normalization is given in $N=1$ supergravity by

Modulus-dominated SUSY-breaking soft terms in F-theory and their test at LHC · Modular weights beyond the toroidal setting · 0.93
The idea [ref] is to recall that the physical Yukawa couplings $Y_αβγ$ among three fields $Φ_α$ after normalization is given in $N=1$ supergravity by

Some phenomenological aspects of Type IIB/F-theory string compactifications · Modular weight possibilities and Yukawa couplings · 0.86

Uses its data or values 2 papers

We use the result [ref] K_e̅ e ∼K_H̅ H ∼τ_s^1/3/τ_b = K_0 (1 + 1/3 δτ_s/⟨τ_s⟩ - δτ_b/⟨τ_b⟩ + …) . where $K_0≡⟨τ_s^1/3/τ_b ⟩=⟨τ_s ⟩^1/3/⟨τ_b ⟩$.

Astrophysical and Cosmological Implications of Large Volume String Compactifications · Couplings to Electrons · 0.89
For the case studied here, $λ= 1/3$ [ref] .

Sparticle Spectra and LHC Signatures for Large Volume String Compactifications · Conclusions · 0.93

Compares with it 1 paper

This is similar to the Kähler potential for matter fields whose dependence on Kähler moduli can be extracted by scaling arguments [ref] , while the complex structure dependence is unknown.

String Loop Moduli Stabilisation and Cosmology in IIB Flux Compactifications · Generalisation to Calabi-Yau three-folds · 0.81

Cites it as support 1 paper

The superpotential $μ$-term can be shown to vanish due to scaling arguments [ref] : in the limit that $V →∞$, any non-zero superpotential $μ$-term would generate a mass for the Higgs field above the string scale.

Sparticle Spectra and LHC Signatures for Large Volume String Compactifications · Scalar Soft Terms · 0.98

Cites it as background 2 papers

The Kähler potential was calculated for Calabi-Yau manifolds to first order in a volume expansion only recently [ref] .

Sparticle Spectra and LHC Signatures for Large Volume String Compactifications · Collider Model and Data Generation · 0.87
In IIB orientifold supergravity (SUGRA), it was discussed that there would be possibilities relevant to the Kähler modulus [ref] :

Natural supersymmetric spectrum in mirage mediation · On modular weight $r_i$ · 0.91

Discussion

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