G_2-manifolds and associative submanifolds via semi-Fano 3-folds

Alessio Corti; Johannes Nordstr\"om; Mark Haskins; Tommaso Pacini

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G_2-manifolds and associative submanifolds via semi-Fano 3-folds

Alessio Corti, Mark Haskins, Johannes Nordström, Tommaso Pacini

Differential Geometry · Submitted 18 Jul 2012 · 75 references

Pith's verdict: A solid, load-bearing paper: it fixes Kovalev's matching argument, extends the construction to semi-Fano blocks, and gets diffeomorphism types out for the first time.

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CitationAlessio Corti, Johannes Nordstr\"om, Mark Haskins, Tommaso Pacini. "G_2-manifolds and associative submanifolds via semi-Fano 3-folds." arXiv:1207.4470 (2012). https://pith.science/paper/AU2D7YWI

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Bottom line: this is the paper that puts the twisted connected sum construction on a working footing, and it earns its claims. The two things to know are that it corrects Kovalev's false Lemma 6.47 and rebuilds the K3 matching argument properly, and that it opens the construction to semi-Fano blocks, which is what makes the rigid associative 3-folds possible. The integral cohomology and p1 computations, and the resulting diffeomorphism-type determinations for 2-connected examples, are genuinely new — nobody had done that for compact G2 manifolds before, including Joyce's examples. The tables in §7 are real work, not decoration: 82 almost-diffeomorphism types from rank-one Fano pairs alone, with the divisibility of p1 pinned down case by case. The persistence result (Prop. 5.15) is clean and the rigidity criterion (Lemma 5.11) is the right observation. I checked the Mayer-Vietoris bookkeeping in Theorem 4.9 and the totally-real/Baire step in Prop. 6.18; both hold up. The stress-test note is right that there is no internal objection here. Where I would push back is on proportion, not correctness. Section 8's G2-transitions are programmatic — the authors say as much, and the metric realisation is explicitly future work, so treat that section as a research agenda rather than a result. The H3 torsion-free hypothesis on the semi-Fano Y is assumed and not proved in general; the authors flag it and note they know of no counterexample, which is honest but leaves a gap in the stated generality of (ii). And the matching is generic existence, not classification: the paper is upfront that it does not understand all matchings of a given pair, which is why the moduli-component question in (vii) stays speculation. None of these touch the central claims. The analytic existence and the semi-Fano deformation theory sit in the companion papers, so the contribution here is the gluing, the topology, and the associatives — that is the right division of labour, not a weakness. Audience: G2 geometers, Calabi-Yau/algebraic geometers who care about weak Fano and lattice polarised K3s, and anyone working on calibrated submanifolds or enumerative invariants for G2. Send it to a serious referee; expect a long report on §6 and §7, and expect it to come back positive.

Referee report

Significance

The paper is a substantial and careful contribution. Its most concrete advances are the integral (not merely rational) computation of the cohomology and first Pontrjagin class of twisted connected sums, the resulting determination of the (almost-)diffeomorphism type of many compact G2-manifolds via the Wilkens classification, and the construction of compact rigid associative 3-folds from rigid holomorphic curves. The explicit tables are a genuine asset: they make the arithmetic of the matching problem checkable and give the reader concrete 7-manifolds rather than existence statements. The orthogonal-gluing formalism is a clean organising device that will be reused. The paper is also candid about what it does not do: the analytic existence theory is delegated to [37], the semi-Fano deformation theory to [18], and the metric realisation of G2-transitions is explicitly left as a future prospect. These are appropriate delegations to published work, not gaps in the present argument.

Minor objections (8)

[§7, Example No 1] The text states that for the non-primitive perpendicular matching of two quartic blocks, 'although Theorem 4.9 shows that M is simply-connected and has H2(M) = 0 it is no longer 2-connected.' This appears to be an error: with π1(M) = 0 and H2(M;Z) = 0, the Hurewicz theorem gives π2(M) ≅ H2(M) = 0, so M is still 2-connected. What fails is the torsion-freeness of H4(M), and hence the hypothesis of Theorem 4.24; the sentence should be corrected accordingly.
[§4, Remark 4.14 and the remark following Lemma 4.29] Lemma 4.13 establishes that the two summands Tor(L/N- + T+) and Tor(L/N+ + T-) of Tor H4(M) are isotropic for the torsion linking form, and the subsequent paragraph identifies the pairing between them. The further assertion that 'the isomorphism class of the torsion-linking form is determined by the isomorphism class of H4(M)' does not follow from isotropy alone, since a decomposition of a finite abelian group into a pair of isotropics with perfect pairing is not canonical in general. Since this remark is used to justify that the pair (H4(M), p1(M)) suffices for the almost-diffeomorphism classification of 2-connected twisted connected sums with torsion, a sentence of justification (or a restriction to the cases actually used) would help.
[§7, discussion of No 5–6] The sentence 'No 5a and 5g are both 2-connected with b3(M) = 95, so are diffeomorphic by Theorem 4.24' cites only the almost-diffeomorphism classification. Diffeomorphism additionally requires the inertia-group computation of Corollary 4.34 (here div p1 = 4, so I(M) = Θ7); the citation should include Corollary 4.34.
[Notation, §3–§6] The letter T is used both for the neck-length parameter in Theorem 3.12 and Proposition 5.15 and for the transcendental lattice N^⊥ in §4 and §6, and K is used both for ker ρ in (4.1) and for the complex structure K± on the asymptotic K3. The clashes are resolvable from context but slow the reader down; a notational remark or a different symbol for one of each pair would help.
[Abstract, item (v), versus §8] The abstract states 'We describe geometric transitions between G2-metrics on different 7-manifolds', whereas §8 describes G2-transitions as 'a convenient organisational principle' and places their realisation at the level of metric geometry among future prospects. The abstract phrasing could be softened to match the body, so that readers do not take (v) as a construction of families of G2-metrics.
[Introduction and §5] The priority claims that the constructed associatives are 'the first compact associative 3-folds in compact G2-manifolds that are proven to be rigid' and that these are 'the first compact G2-manifolds for which the diffeomorphism type is known' are strong. They are plausible given the literature the authors cite, but a date-stamped qualification (or a footnote pointing to the relevant prior literature on rigidity of associatives) would make them easier to assess.
[§6, Proposition 6.2 and Definition 6.3] The period condition is written as '⟨k∓, ±k0⟩', which is terse: it would help to spell out explicitly that the period of S+ is ⟨k-, k0⟩ and that of S- is ⟨k+, -k0⟩, and to record which sign convention is forced by the requirement r*ωK- = -ωK+.
[Table 3 caption] The caption writes 'b3(M) = b4(M) = b + 23' without saying that b is the row label of the first column; a short clarification would prevent the reader from looking for a symbol b in the body.

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The paper

The paper claims

A pair of asymptotically cylindrical Calabi–Yau 3-folds built from semi-Fano 3-folds can be glued into a compact 7-manifold with holonomy exactly $G_2$, and the paper shows how to find such pairs and read off the topology of the result.

The reading

The paper is about manufacturing compact Riemannian 7-manifolds whose holonomy group is the exceptional group $G_2$, by a procedure called the twisted connected sum: take two non-compact Calabi–Yau 3-folds that look like a cylinder $S^1$ times a K3 surface at infinity, cut off their ends, and glue them together with a twist that exchanges the two circle directions. The hard step is not the gluing but the matching: the two K3 surfaces at infinity must be related by a hyper-Kähler rotation, and finding such pairs is a lattice-arithmetic problem about K3 periods. The authors' move is to enlarge the supply of usable halves from Fano 3-folds to semi-Fano 3-folds, a much larger class where the anticanonical morphism contracts curves but never divisors, and then to show that a purely lattice-theoretic condition (a primitive embedding of the sum of the two polarising lattices into the K3 lattice) suffices to produce matching data. Because these halves are algebraically explicit, the resulting 7-manifolds come with computable integral cohomology, torsion, and first Pontrjagin class, which for many examples is enough to pin down the diffeomorphism type outright. A second payoff is that rigid holomorphic curves in the halves survive the gluing as rigid associative 3-folds, giving some of the first compact associative submanifolds in compact $G_2$-manifolds proved to be rigid.

What carries the argument

The load-bearing device is the reformulation of the hyper-Kähler rotation condition as a statement about periods: a triple $(k_+, k_-, k_0)$ of positive classes in $L_mathbb{R}$, with $k_pm$ orthogonal to $N_pm$ and $k_0$ orthogonal to both, whose span gives the period point of each marked K3 surface and whose $k_pm$ restrict from Kähler classes on the building blocks. Existence of such a triple is proved by a transversality argument: the relevant real submanifold of the Griffiths period domain is totally real of maximal dimension precisely when $N_+$ and $N_-$ meet orthogonally, so it must meet the dense open set of periods realised by the family. Around this sit two supporting tools: the ACyl Calabi–Yau existence theorem reducing analysis to projective geometry, and the Wilkens classification of 2-connected 7-manifolds by $(H^4, p_1)$.

Extended reading notes

Core claim

The central claim is that asymptotically cylindrical Calabi–Yau 3-folds of semi-Fano type are legitimate building blocks for the twisted connected sum construction, and that the matching problem for them can be solved systematically by ``orthogonal gluing''. Concretely, given two deformation families of semi-Fano 3-folds with polarising lattices $N_+$ and $N_-$, one forms the orthogonal push-out $W = N_+ + N_-$ and looks for a primitive embedding $W hookrightarrow L$ into the K3 lattice; combined with the deformation theory of $N$-polarised semi-Fano 3-folds, this yields periods and Kähler classes giving a hyper-Kähler rotation, hence a compact $G_2$-manifold. The authors then compute the integral cohomology of the glued 7-manifold from the building blocks, including the torsion of $H^3$ and $H^4$ via $L/(N_+ + N_-)$ and its relatives, and compute $p_1(M) = -2Y(c_2(Z_+), c_2(Z_-))$ by an explicit gluing of classifying maps. For the 2-connected examples with torsion-free $H^4$ this reduces the diffeomorphism classification to the pair $(b_3, operatorname{div} p_1)$, and the paper reports 82 distinct almost-diffeomorphism types from rank-one Fano blocks alone, with the diffeomorphism type determined in most cases.

Load-bearing premise

The matching step leans on a genericity statement carried over from the authors' earlier paper: that essentially every period point and every class in the relevant Kähler cone is actually realised by a genuine semi-Fano 3-fold in the family, and the required lattice embedding is then verified case by case rather than proved in general.

Editorial extensions

If this is right

Every pair of semi-Fano 3-folds whose polarising lattices have ranks summing to at most 11 and admit a primitive orthogonal push-out into the K3 lattice yields a compact $G_2$-manifold, so the known supply of examples grows from hundreds to hundreds of thousands of candidate pairs.
For 2-connected outputs with torsion-free $H^4$, the pair $(b_3(M), operatorname{div} p_1(M))$ determines the almost-diffeomorphism type, and since $operatorname{div} p_1 in {4,8,12,16,24,48}$ the diffeomorphism type is fixed except when the divisibility is 16 or 48.
A rigid holomorphic curve $C$ in a building block persists, for sufficiently long neck, to a rigid associative $S^1 imes C$ in the glued $G_2$-manifold, so counting associative 3-folds becomes a question about counting rigid curves in the semi-Fano halves.
The same smooth 7-manifold frequently arises from several different pairs of building blocks, and different matchings of the same pair can give topologically distinct 7-manifolds, which makes the moduli space of $G_2$-metrics on a fixed manifold a genuine question rather than a formality.
Flopping a semi-Fano 3-fold, or passing from a Fano to a semi-Fano by a conifold transition, produces a corresponding ``$G_2$-transition'' between the resulting 7-manifolds, organising the geography of examples.

Reading between the lines

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

The orthogonal-gluing criterion is arithmetic and cheap to test, so the bottleneck has shifted: the binding constraint is now knowing the Kähler cone $operatorname{Amp}_Z$ of each building block, which the paper must verify by hand in the non-perpendicular cases. A systematic computation of these cones for toric semi-Fanos would convert the case-by-case work into a genuine mass-production theorem.
Because the number of rigid associatives is a topological invariant of the matching data but not of the underlying smooth 7-manifold, one could try to use counts of rigid associatives as a crude probe of which connected component of the $G_2$-moduli space a metric lies in; the paper stops short of this, but its examples with the same 7-manifold and different associative counts are exactly the test
The torsion computation via $L/(N_+ + N_-)$ suggests a converse design principle: choose overlattices of $N_+ \perp N_-$ with prescribed discriminant groups to engineer $H^3(M)$ torsion deliberately, rather than discovering it after the fact.
If the deformation-theoretic genericity input were to fail for some semi-Fano family, the failure would be visible as a period domain not swept out by that family, which is in principle checkable by computing the image of the period map for that family.

Original abstract

We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of the paper are: (i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. (iii) We construct many new topological types of compact G_2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in arXiv:1206.2277. (iv) We obtain much more precise topological information about twisted connected sum G_2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G_2-manifolds. (v) We describe "geometric transitions" between G_2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. (vi) We construct many G_2-manifolds that contain rigid compact associative 3-folds. (vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. This leads to speculation that the moduli space of G_2-metrics on a given 7-manifold may consist of many different connected components.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of the paper are: (i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. (iii) We construct many new topological types of compact G_2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in arXiv:1206.2277. (iv) We obtain much more precise topological information about twisted connected sum G_2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G_2-manifolds. (v) We describe "geometric transitions" between G_2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. (vi) We construct many G_2-manifolds that contain rigid compact associative 3-folds. (vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. This leads to speculation that the moduli space of G_2-metrics on a given 7-manifold may consist of many different connected components.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction.	contribution 0.93	0.93	0.25	asserted 1.00
s1	Some of the main contributions of the paper are: (i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction.	contribution 0.99	0.94	0.12	asserted 0.99
s2	(ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction.	contribution 0.97	0.92	0.15	asserted 1.00
s3	(iii) We construct many new topological types of compact G_2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in arXiv:1206.2277.	contribution 0.91	0.96	0.20	asserted 1.00
s4	(iv) We obtain much more precise topological information about twisted connected sum G_2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G_2-manifolds.	contribution 0.85	0.97	0.15	asserted 1.00
s5	(v) We describe "geometric transitions" between G_2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds.	contribution 0.95	0.95	0.08	asserted 1.00
s6	(vi) We construct many G_2-manifolds that contain rigid compact associative 3-folds.	contribution 0.92	0.97	0.10	asserted 1.00
s7	(vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein.	contribution 0.63	0.95	0.21	asserted 1.00
s8	This leads to speculation that the moduli space of G_2-metrics on a given 7-manifold may consist of many different connected components.	interpretation 0.98	0.18	0.17	speculative 0.89

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from proof or derivation 1.00.

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arXiv listing abstract v3

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Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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How the literature uses it

47 arXiv papers cite this in 272 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 · 185 sentences without a confident label

Contests it 2 papers

Let $M$ be $2$-connected, let $q_M^∘$ be the linking family of $M$ as defined in Definition [ref] and let $-q^∙_M$ be the linking family of $M$ as defined in [ref] (we have introduced the sign to correct the mistake in [ref] : see Remark [ref].) Comparing these definitions, we see that for each $σ∈(π)$

The classification of 2-connected 7-manifolds · The almost diffeomorphism classification · contest 0.87
For this reason, in both this paper and in [ref] we chose not to rely on arguments from [ref] , and to give proofs or alternative references for the main results we need.

Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds · Introduction · contest 0.83

Builds on it 3 papers

Hence Theorem [ref] is a development [ref] in dimension $7$ where $2$-connected $7$-manifolds are classified up to diffeomorphism by the smooth equivalence class of the characteristic form of a $3$-connected coboundary.

The classification of 2-connected 7-manifolds · The generalised Eells–Kuiper invariant · 0.95
In this section we study an explicit example of [ref] , performing a number of new computations necessary to uncover interesting physical aspects of this M-theory vacuum, as well as studying topology changing transitions to other $G_2$ manifolds and M-theory vacua.

The Landscape of M-theory Compactifications on Seven-Manifolds with $G_2$ Holonomy · A Rich Example · 0.84
It is also possible to do a similar two-stage (or multiple-stage) resolution process for examples of $$-manifolds coming from the `twisted connected sum' constructions of [ref] .

A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi-Hanson spaces · Resolving orbifolds T⁷/Γ in several stages · 0.90

Uses its method or tool 8 papers

In this case we use the “twisted connected sum” construction of closed s [ref] ; it was shown in [ref] that this construction yields large numbers of closed s that are 2-connected, with torsion-free cohomology, and how to compute the invariants required to apply homeomorphism classification results of Wilkens.

Exotic G_2-manifolds · Introduction · 0.95
[ref] , which is a particular recipe to fulfill the matching condition for suitable pairs of building blocks $(Z_,S_)$.

Effective action from M-theory on twisted connected sum $G_2$-manifolds · Construction of $G_2$-manifolds · 0.94
Using the topological progress of [ref] , we also compute the form of membrane instanton corrections to the superpotential (for the first time in a compact model) and $G_2$ topology changing transitions.

The Landscape of M-theory Compactifications on Seven-Manifolds with $G_2$ Holonomy · A Rich Example · 0.83

Uses its data or values 2 papers

These give rise to [ref] at least $50$ million matching pairs, arising only from ACyl Calabi-Yau threefolds from semi-Fano building blocks of rank at most two or from toric semi-Fano threefolds; given the limited nature of this search, many more can probably be obtained by considering higher rank building blocks.

The Landscape of M-theory Compactifications on Seven-Manifolds with $G_2$ Holonomy · The Landscape of M-theory on $G_2$ Manifolds · 0.87
In [ref] it was shown that $b_3$ is often of $O(30)$, but it was also argued convincingly that this ensemble is only scratching the surface.

TASI Lectures on Remnants from the String Landscape · Axions · 0.80

Compares with it 3 papers

We begin by explaining the relationship between the two definitions of linking family and showing that Theorem [ref] is equivalent to [ref] .

The classification of 2-connected 7-manifolds · The almost diffeomorphism classification · 0.99
Using this identification we rewrite the Dirac operator $: Γ(NL_θ) →Γ(NL_θ)$ as follows [equation] [equation] (The formula in ([ref]) is identical with the formula in [ref] and differs from the one in [ref] by the sign (-1), noting that $d^* α_V = - * d * α_V$.)

McLean's second variation formula revisited · The normal bundle of an associative submanifold and its associated Dirac operator · 0.96
For comparison, twisted connected sums involving only rank 1 Fanos realise 82 different manifolds, and 46 different values of $b_3(M)$ [ref] .

Exotic G_2-manifolds · Perpendicular matching · 0.96

Cites it as support 1 paper

Now [ref] states that all linking families defined on $(π)$ arise as the quadratic linking families of $2$-connected $M$ and that any isomorphism of linking families defined on $(π)$ is realised by an almost diffeomorphism.

The classification of 2-connected 7-manifolds · The almost diffeomorphism classification · 0.97

Cites it as background 34 papers

In this section we focus on the construction of $G_2$-manifolds as put forward by Kovalev [ref] and further developed by Corti et al. [ref] .

Effective action from M-theory on twisted connected sum $G_2$-manifolds · Kovalev's construction of $G_2$-manifolds · 0.84
To fix notation, let us begin with a quick informal review of the construction of TCS $G_2$-holonomy manifolds [ref] .

Mirror Symmetry for $G_2$-Manifolds: Twisted Connected Sums and Dual Tops · Mirror symmetry for TCS: heuristics · 0.97
We begin by recalling quadratic linking families as defined in [ref] and explaining their relationship to connected sum splittings as above.

The classification of 2-connected 7-manifolds · The almost diffeomorphism classification · 0.87

What it cites · 15 sentences without a confident label

Contests 1 paper

We explain below in more detail how together with [ref] and [ref] this paper addresses problems with both steps (i) and (ii) in Kovalev's original paper [ref] —and therefore puts the twisted connected sum construction on a firm foundation—and also extends substantially the settings in which solutions to (i) and (ii) can be constructed.

Asymptotically cylindrical Calabi-Yau manifolds · Twisted connected sums and rotations · contest 0.86

Compares with 1 paper

Our definition of `positive' agrees with that of Joyce [ref] , while Hitchin [ref] uses `positive' where we use `-form'.

The geometry of three-forms in six and seven dimensions · s on a vector space · 0.96

Cites as support 1 paper

Whenever the $V_±$ in the theorem have holonomy $3$, [ref] implies that their fundamental groups are finite and generated by the $^1$ factors in the cylindrical ends.

Asymptotically cylindrical Calabi-Yau manifolds · The gluing procedure · 0.98

Cites as background 4 papers

See [ref] for further discussion of this and other related issues.

Asymptotically cylindrical Calabi-Yau manifolds · Asymptotically cylindrical Calabi–Yau $3$-folds · 0.94
For some recent physical predictions from M-theory on -spaces see [ref] ; see also [ref] for some other aspects of M-theory on -spaces.

Mixed Wino-Axion Dark Matter in String/M Theory and the 130 GeV Gamma-line "Signal" · Connections to M-theory · 0.91
For some recent physical predictions from M-theory on -spaces see [ref] ; see also [ref] for some other aspects of M-theory on -spaces.

Bounds on Scalar Masses in Theories of Moduli Stabilization · Connections to M-theory · 0.91

Discussion

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