The Universe as a Domain Wall

Andre Lukas; Burt A. Ovrut; Daniel Waldram; K.S. Stelle

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The Universe as a Domain Wall

Andre Lukas, Burt A. Ovrut, K.S. Stelle, Daniel Waldram

High Energy Physics - Theory · Submitted 27 Mar 1998 · 52 references

Pith's verdict: A genuinely influential paper: the 5D gauged supergravity of heterotic M-theory and its exact BPS domain wall, with the truncation-consistency check deferred to an in-preparation reference.

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arXiv IDhep-th/9803235 (version 2)

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CitationAndre Lukas, Burt A. Ovrut, Daniel Waldram, K.S. Stelle. "The Universe as a Domain Wall." arXiv:hep-th/9803235 (1998). https://pith.science/paper/WSP7F37O

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Editor's note

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This is the paper that made the domain-wall picture of heterotic M-theory concrete, and it holds up. The core result is that the five-dimensional effective theory of the strongly coupled E8 x E8 string, restricted to the universal multiplets, is a gauged N=1 supergravity with four-dimensional boundaries, and that it admits an exact BPS three-brane domain wall (eq. 4.2) whose linearization reproduces Witten's deformed Calabi-Yau background. The key technical move is retaining the non-zero mode of the four-form, G_ABCD proportional to the Kahler form times epsilon(x11) (eq. 3.3), which gauges a U(1) of the universal hypermultiplet and generates the bulk potential. That is the piece people had been missing, and it is what makes the domain wall possible at all. The derivation is explicit, the action (3.6) is matched carefully against known five-dimensional supergravity, and the Killing spinor check (eq. 4.4) is done rather than asserted. The constant alpha is fixed by Calabi-Yau data through eq. 2.11, not tuned, so there is no hidden fitting here. The brane-language reading of Witten's background as wrapped five-branes is a nice piece of physical bookkeeping and it earns its place. The soft spots are real but bounded. First, the consistency of the truncation to universal modes plus the single non-zero mode is stated after eq. (3.8) and then deferred to ref. [40], which is listed as in preparation. That is the load-bearing assumption, and a reader cannot check it from this paper alone. Second, the fermionic completion and the precise gauging of the SU(2) R-symmetry are reported as the outcome of an investigation rather than derived in the text. Third, 'exact' means exact in the truncated five-dimensional theory; the authors say plainly in Section 4 that exactness in the full eleven-dimensional theory is not established, and they cite Strominger on loop corrections. That is honest scoping, not overreach, but it should be read as such. The footnote on the Yang-Mills coupling normalization is also candid about a live disagreement in the literature. None of this changes the central claim, which is correctly stated as a result about the truncated effective theory. Who should read it: anyone working on brane-world cosmology, five-dimensional supergravity, or the heterotic M-theory effective action. It is the reference point for that whole line. It deserves a serious referee, and the one thing I would press on is making the truncation argument self-contained rather than pointing at a forthcoming paper.

Referee report

Significance

If the construction holds, this is a foundational result: it identifies gauged five-dimensional supergravity with four-dimensional boundaries, rather than four-dimensional N=1 supergravity, as the correct intermediate-energy arena for the strongly coupled heterotic string, and it supplies the background on which any subsequent reduction to four dimensions must be performed. The paper earns credit on several specific counts. The reduction ansatz (Eqs. 3.1–3.4) is written out explicitly, so the reader can in principle repeat the computation. The resulting action is not merely asserted to be supersymmetric; it is matched term by term against the known form of five-dimensional supergravity, with the hypermultiplet metric derived from the Kähler potential (Eq. 3.11) and the Killing vector identified. The parameter α is fixed by Calabi–Yau data in Eq. (2.11), so the domain-wall slope in Eq. (4.2) is a parameter-free prediction rather than a fitted quantity. The linearization check against Witten's order-κ^{2/3} background is a genuine falsifiable comparison, and the brane-language reinterpretation in terms of wrapped five-branes is a substantive conceptual contribution. The result has aB

Major objections (3)

[§3, paragraph following Eq. (3.8)] The consistency of the truncation to universal modes plus the single non-zero mode (Eq. 3.3) is load-bearing for everything that follows, and the manuscript supports it with one sentence: 'We have checked the consistency of the truncation which leads to the above action by an explicit reduction of the 11-dimensional equations of motion to five dimensions.' The details are deferred to ref. [40], which is listed as 'in preparation' and therefore cannot be consulted. Because the central claim is that Eq. (3.6) is the effective theory, a reader needs to know which components of the eleven-dimensional equations of motion were checked and, in particular, whether setting the massive Calabi–Yau harmonics to zero induces linear terms that would source them. A short sketch — the components checked, the order in κ at which the check was performed, and the fate of the dropped harmonics — would make the claim self-contained without requiring the full computation. I want to be clear that this is a request for displayed evidence, not an assertion that the truncation fails; the paper does scope its claims to the truncated theory.
[§3, paragraph following Eq. (3.10)] The identification of the action (3.6) as a gauged version of N=1 five-dimensional supergravity is the paper's headline claim, but the supporting argument is compressed to 'investigation of the other terms in the action, including the fermions, shows that the resulting five-dimensional theory is precisely a gauged form of supergravity.' The fermionic action is never displayed, and the variations (3.12) are quoted rather than derived, with the text stating only that they follow 'either by a reduction of the 11-dimensional gravitino variation (2.7) or by generalizing the known five-dimensional transformations.' Since the gauging of a U(1) subgroup of the SU(2) automorphism group, alongside the U(1) isometry of M_Q, is a structural statement about the theory rather than a property of the bosonic action alone, the manuscript would be considerably stronger if it either displayed the fermionic completion or demonstrated that the bosonic action together with (3.12) closes on the five-dimensional N=1 algebra. As written, the reader must take the gauged-supergravity identification largely on trust.
[§4, Eqs. (4.2)–(4.3) and the comparison with Eqs. (2.9)–(2.10)] The correspondence with Witten's deformed Calabi–Yau background is one of the two headline results, but the text states only that 'it is straightforward to show that the linearized version of (4.2) ... coincides with Witten's solution (2.9), (2.10) upon appropriate matching of the integration constants.' The matching is never exhibited. Given that the eleven-dimensional solution is known only to order κ^{2/3}, the reader would benefit from seeing the expansion of Eq. (4.2) to first order in α written out, together with the explicit values of a0, b0 and c0 in terms of R0, V0 and the orbifold radius that reproduce Eqs. (2.10a)–(2.10c). This is a request for a displayed check rather than a doubt about the result; the check is short and would convert an assertion into a verifiable statement.

Minor objections (8)

[Eq. (3.8)] Please check the normalization of the five-dimensional Newton constant. With the reduction ansatz (3.1), the eleven-dimensional Einstein–Hilbert term reduces to −(1/2κ²)∫√−g₅ √Ω R₅, which gives κ₅² = κ²/v rather than the printed κ₅² = κ²v. The printed form is dimensionally inconsistent (κ²v has dimension length^15, whereas κ₅² must have dimension length^3), and it is also inconsistent with the accompanying expression for α_GUT in the same equation, which is dimensionless only if κ₅² = κ²/v. If the authors use a different convention for v, this should be stated explicitly.
[Eqs. (3.6c) and (4.3)] The two boundary cosmological terms enter with opposite signs, and the junction conditions that fix the slope of H in Eq. (4.2) depend on the relative sign. It would help to show the Israel-type matching at y = 0 and y = πρ explicitly, so that the reader can verify that the signs in Eq. (3.6c) are precisely those required by Eq. (4.3).
[§3, paragraph following Eq. (3.10)] The statement that the U(1) charge 'is quantized since, suitably normalized, tr R∧R is an element of H^{2,2}(X,Z)' carries physical weight, since it is the five-brane charge quantization argument. The phrase 'suitably normalized' does the work here; please state the normalization explicitly, or cite where it is fixed.
[Footnote 1 and Eq. (2.11)] The footnote notes that the alternative normalization of the Yang–Mills coupling found in ref. [10] multiplies the coefficients in Eq. (2.3) and Eq. (2.4) by 2^{−1/3}. Since α in Eq. (2.11) is built from these coefficients, this rescaling propagates directly into the slope of the harmonic function in Eq. (4.2). It would be useful to state this rescaling explicitly at the point where the domain-wall solution is presented, so that the numerical prediction is unambiguous.
[§2, Eqs. (2.10a)–(2.10e)] The linearized corrections are quoted from ref. [20] after dropping all non-zero-eigenvalue harmonics. A sentence explaining why those harmonics decouple at this order — rather than merely asserting that they 'will be of no relevance to the low energy theory' — would help the reader judge the reliability of the truncation.
[Notation, §2 and §3] The symbol ε is used both for the step function ε(x11) in Eqs. (2.10d) and (3.3) and, with indices, for the Levi-Civita symbol ε^{αβγδδ} in Eqs. (3.6a) and (3.9). A distinct symbol for the step function would remove a recurring source of confusion.
[Reference [36]] The year is printed as '(1084)'; this should read '(1984)'.
[§3, scope of the boundary sector] The paper sets boundary gauge matter to zero and defers the general case to ref. [40]. It would be worth stating explicitly, in one sentence, whether the domain-wall solution (4.2) is expected to survive when boundary gauge fields are excited, or whether that is part of what is deferred.

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The paper

The paper claims

The five-dimensional effective theory of the strongly coupled heterotic string is a gauged supergravity whose vacuum is a pair of parallel three-branes, and four-dimensional spacetime is the worldvolume of one of them.

The reading

The paper asks what the strongly coupled heterotic string looks like in the regime where the orbifold dimension is larger than the Calabi-Yau space, so that the universe appears five-dimensional before it appears four-dimensional. Its answer is that the low-energy theory of the universal fields is not plain five-dimensional supergravity but a gauged version of it, with four-dimensional boundary actions living on the orbifold planes and a potential in the bulk. The gauge field is the graviphoton, and what it gauges is a $U(1)$ inside the isometry group of the universal hypermultiplet's coset $SU(2,1)/SU(2)/U(1)$; the charge is fixed by a constant $a$ built from $a$ times the integral of $a$ over the Calabi-Yau space. The payoff is an exact three-brane domain wall: two parallel branes sitting on the orbifold planes, whose linearization reproduces Witten's deformed Calabi-Yau background, and which the authors propose as the correct background for reducing to four dimensions, with four-dimensional spacetime identified with the brane worldvolume. The authors acknowledge that they do not know whether the solution survives higher-derivative corrections in the full theory.

What carries the argument

The load-bearing device is the retention of a non-zero mode of the four-form field strength in the internal Calabi-Yau directions, equation (3.3). Unlike an ordinary Kaluza-Klein zero mode, it cannot be turned off: its coefficient is fixed by Calabi-Yau data through the boundary Bianchi-identity sources, essentially by the integral of the Kahler form wedged with the curvature-squared term. That fixed coefficient is what gauges the universal hypermultiplet, what produces the bulk potential that a domain wall requires, and what makes the two boundary source terms match the bulk solution. The authors verify that after dualizing the three-form to a scalar $a$, the hypermultiplet scalars $(V,a,a,ar{a})$ have a Kahler potential $K_Q = -a(S+ar S - 2Car C)$ with $S = V + aar a + ia$ and $C = a$, whose metric is quaternionic with coset $SU(2,1)/SU(2)/U(1)$, and that the shift of $a$ is a Killing vector of that metric.

Extended reading notes

Core claim

The central claim is that the effective five-dimensional theory of the strongly coupled heterotic string, restricted to the gravity multiplet and the universal hypermultiplet, is a gauged $N=1$ supergravity with four-dimensional boundaries, and that this theory has an exact BPS three-brane domain-wall solution. The gauging is forced by a piece of the eleven-dimensional four-form field strength that has no five-dimensional zero mode, a so-called non-zero mode, taken proportional to the Kahler form of the Calabi-Yau space and to the step function across the orbifold, $G_{ABCD} = (a/6),a_{ABCD}{}^{EF}a_{EF},a(x^{11})$. Keeping this term does two things at once: it turns ordinary derivatives of the hypermultiplet scalars into gauge-covariant ones with respect to a $U(1)$ Killing vector, and it generates a bulk potential of size $a^2/V^2$. The boundary Yang-Mills and curvature terms supply matching cosmological terms of opposite sign on the two planes, and the two together fix the solution to be a pair of parallel three-branes at $x^{11}=0$ and $x^{11}=a$, with metric functions built from a harmonic function $H = (a/3)a|y| + c_0$. The solution is exact in the five-dimensional theory, whereas Witten's deformed Calabi-Yau background was constructed only to first order in $a$; expanding the new solution to that order reproduces it.

Load-bearing premise

The load-bearing premise is that keeping only the universal supermultiplets and a single non-zero mode proportional to the Kahler form, while dropping all massive Calabi-Yau harmonics, is a consistent truncation whose exact solutions really capture the eleven-dimensional physics; the authors state they have checked this by an explicit reduction of the equations of motion, with the details deferred to a forthcoming paper.

Editorial extensions

If this is right

If the reduction is right, the correct background for deriving four-dimensional $N=1$ supergravity from the strongly coupled heterotic string is not a Calabi-Yau times Minkowski space but a five-dimensional domain wall, and the four-dimensional theory should be obtained by reducing on that wall rather than on the original product space.
The five-dimensional theory carries a potential, so flat five-dimensional space is not a solution without decompactifying the Calabi-Yau; any cosmological or phenomenological analysis of the intermediate-energy regime has to start from the domain-wall background instead.
The gauged $U(1)$ charge of the axion-like scalar is fixed by a topological quantity of the Calabi-Yau space and is therefore quantized, matching the quantization of five-brane charge in the brane picture of Witten's background.
In brane language, Witten's deformed Calabi-Yau background is a collection of five-branes wrapped on two-cycles and confined to the orbifold planes, which become three-branes spanning the fixed planes when reduced to five dimensions.
Because the domain-wall solution is exact in the five-dimensional theory rather than first order in $a$, it provides a candidate all-orders completion of the deformed Calabi-Yau background, subject to corrections from higher-derivative terms.

Reading between the lines

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

If four-dimensional spacetime is really a three-brane worldvolume, then the warping of the fifth dimension should leave an imprint on four-dimensional couplings; the $H^{1/2}$ and $H^3$ factors in the solution suggest that gauge and gravitational couplings on the two planes are naturally different, which is the kind of effect that would show up as a small mismatch between unification-scale predict
The mechanism here is general: any compactification where a form field strength has a fixed internal flux will gauge a symmetry of the moduli space and generate a potential, so the same construction should apply to other string compactifications with quantized internal flux, and the resulting domain walls should be the generic vacuum rather than a special case.
A reader could test the picture by asking whether the second-order terms in the eleven-dimensional expansion of Witten's background, which the original construction did not compute, agree with the series expansion of $H^{1/2}$ and $H^3$; a disagreement would mean the exact solution is an artifact of the truncation rather than a genuine completion.

Original abstract

It is shown that the effective five-dimensional theory of the strongly coupled heterotic string is a gauged version of N=1 five-dimensional supergravity with four-dimensional boundaries. For the universal supermultiplets, this theory is explicitly constructed by a generalized dimensional reduction procedure on a Calabi-Yau manifold. A crucial ingredient in the reduction is the retention of a "non-zero mode" of the four-form field strength, leading to the gauging of the universal hypermultiplet by the graviphoton. We show that this theory has an exact three-brane domain wall solution which reduces to Witten's "deformed" Calabi-Yau background upon linearization. This solution consists of two parallel three-branes with sources provided by the four-dimensional boundary theories and constitutes the appropriate background for a reduction to four dimensions. Four-dimensional space-time is then identified with the three-brane worldvolume.

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It is shown that the effective five-dimensional theory of the strongly coupled heterotic string is a gauged version of N=1 five-dimensional supergravity with four-dimensional boundaries. For the universal supermultiplets, this theory is explicitly constructed by a generalized dimensional reduction procedure on a Calabi-Yau manifold. A crucial ingredient in the reduction is the retention of a "non-zero mode" of the four-form field strength, leading to the gauging of the universal hypermultiplet by the graviphoton. We show that this theory has an exact three-brane domain wall solution which reduces to Witten's "deformed" Calabi-Yau background upon linearization. This solution consists of two parallel three-branes with sources provided by the four-dimensional boundary theories and constitutes the appropriate background for a reduction to four dimensions. Four-dimensional space-time is then identified with the three-brane worldvolume.

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s0	It is shown that the effective five-dimensional theory of the strongly coupled heterotic string is a gauged version of N=1 five-dimensional supergravity with four-dimensional boundaries.	result 0.84	0.86	0.50	asserted 1.00
s1	For the universal supermultiplets, this theory is explicitly constructed by a generalized dimensional reduction procedure on a Calabi-Yau manifold.	method 0.94	0.77	0.06	asserted 1.00
s2	A crucial ingredient in the reduction is the retention of a "non-zero mode" of the four-form field strength, leading to the gauging of the universal hypermultiplet by the graviphoton.	method 0.95	0.72	0.08	asserted 1.00
s3	We show that this theory has an exact three-brane domain wall solution which reduces to Witten's "deformed" Calabi-Yau background upon linearization.	result 0.95	0.94	0.41	asserted 1.00
s4	This solution consists of two parallel three-branes with sources provided by the four-dimensional boundary theories and constitutes the appropriate background for a reduction to four dimensions.	result 0.63	0.78	0.20	asserted 1.00
s5	Four-dimensional space-time is then identified with the three-brane worldvolume.	interpretation 0.86	0.71	0.18	asserted 1.00

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

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Out Of This World Supersymmetry Breaking
hep-th 1998-10 accept novelty 8.0 of 10

The super-Weyl anomaly unavoidably transmits supersymmetry breaking to gauginos and scalars, and in a brane-sequestered geometry this anomaly mediation dominates, fixing soft masses in terms of renormalization group b...
Supersymmetric Galileons
hep-th 2011-02 conditional novelty 7.0 of 10

Supersymmetrizing the ghost condensate to repair its fermion instability yields the conformal Galileon, and the resulting N=1 theory has a stable vacuum that violates the null energy condition.
Curvature Singularities: the Good, the Bad, and the Naked
hep-th 2000-02 accept novelty 7.0 of 10

Singular five-dimensional gravity solutions are physically admissible only if the scalar potential stays bounded above along the solution, a criterion that matches known AdS/CFT examples and rules out several proposed...
Modeling the fifth dimension with scalars and gravity
hep-th 1999-09 conditional novelty 7.0 of 10

A first-order superpotential method generates exact five-dimensional scalar-gravity solutions, stabilizes brane spacing, and shows the graviton is massless with no transverse-traceless tachyons.
On Kaluza-Klein States from Large Extra Dimensions
hep-ph 1998-11 accept novelty 7.0 of 10

Each Kaluza-Klein level of the (4+n)-dimensional graviton holds one massive spin-2, (n-1) spin-1 and n(n-1)/2 spin-0 states; the spin-1 states decouple and the scalars couple only through their trace, the dilaton.

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Grand Unification at Intermediate Mass Scales through Extra Dimensions
hep-ph 1998-06 conditional novelty 7.0 of 10

Extra spacetime dimensions make gauge couplings run as power laws, so they unify at intermediate or TeV scales instead of the usual 10^16 GeV.
The dark dimension, proton decay, and the length of the M-theory interval
hep-th 2025-10 conditional novelty 6.0 of 10

Proton decay limits force the M-theory interval in heterotic E8×E8 compactifications to be R ≲ 2.7×10^-28 m, ruling it out as a micron-sized dark dimension.
Dynamical Cobordism and Swampland Distance Conjectures
hep-th 2021-07 conditional novelty 6.0 of 10

Running solutions with dynamical tadpoles end at finite-distance cobordism walls where scalars reach infinite moduli-space distance, yielding new scaling relations between wall distance, moduli distance, and the Swamp...
Cosmological Perturbations Through a Non-Singular Ghost-Condensate/Galileon Bounce
hep-th 2014-04 accept novelty 6.0 of 10

Super-horizon curvature perturbations pass through a non-singular ghost-condensate/Galileon bounce essentially unchanged, and are provably momentarily frozen at one specific time near the bounce for all wavelengths.
A Cosmological Super-Bounce
hep-th 2013-10 conditional novelty 6.0 of 10

A non-singular cosmic bounce preceded by an ekpyrotic contraction phase can be embedded in N=1 supergravity without ghost instabilities, at the cost of several hand-tuned functions and one added stabilizing term.
The Atiyah Class and Complex Structure Stabilization in Heterotic Calabi-Yau Compactifications
hep-th 2011-07 conditional novelty 6.0 of 10

Three computations of how a holomorphic vector bundle constrains the complex structure moduli of a Calabi-Yau threefold are argued to agree, and explicit bundles that freeze 80 of 82 moduli are exhibited.
Stabilizing All Geometric Moduli in Heterotic Calabi-Yau Vacua
hep-th 2011-01 conditional novelty 6.0 of 10

All geometric moduli of a smooth heterotic Calabi-Yau compactification can be stabilized without NS flux by combining bundle holomorphy, anomalous U(1) D-terms, and non-perturbative effects, with an explicit AdS example.
Supersymmetric P(X,phi) and the Ghost Condensate
hep-th 2010-12 accept novelty 6.0 of 10

A superfield recipe builds N=1 supersymmetric extensions of P(X,phi) scalar theories, and its ghost-condensate application shows the extra supersymmetric fields pick up wrong-sign gradient terms.
New Ekpyrotic Cosmology
hep-th 2007-02 conditional novelty 6.0 of 10

Merging a two-field ekpyrotic contraction phase with a ghost-condensate bounce produces a non-singular early universe that yields a scale-invariant, potentially red density spectrum without inflation.
Generating Ekpyrotic Curvature Perturbations Before the Big Bang
hep-th 2007-02 conditional novelty 6.0 of 10

Entropic perturbations generated during an ekpyrotic contraction can be converted into scale-invariant curvature perturbations at a moduli-space bounce, yielding the observed amplitude and a spectral tilt in the range...
New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls
hep-th 2001-03 accept novelty 6.0 of 10

A new dual formulation of Type IIA supergravity, coupled to a supersymmetric O8-plane action, gives a BPS D8-O8 domain-wall solution with mass quantized in string units and a maximum wall separation set by the string ...
Exploring Positive Monad Bundles And A New Heterotic Standard Model
hep-th 2009-11 conditional novelty 5.0 of 10

An exhaustive scan rules out all positive two-term monad bundles on complete intersection Calabi-Yau manifolds as sources of the exact Standard Model spectrum, and a semi-positive monad on the bi-cubic is shown to yield it.
Matter localization and resonant deconfinement in a two-sheeted spacetime
hep-th 2006-03 reject novelty 4.0 of 10

In a two-sheeted spacetime model, environmental interactions, chiefly gravity, localize fermions on one sheet, and resonant transfer to the other sheet becomes possible at a matching frequency.
Cosmological Tests of Gravity
astro-ph.CO 2010-04 accept novelty 3.0 of 10

Modified gravity models can be organized by three screening mechanisms, and current astrophysical tests remain consistent with general relativity while upcoming surveys will sharpen the constraints.
Ekpyrotic and Cyclic Cosmology
astro-ph 2008-06 accept novelty 3.0 of 10

A contracting ekpyrotic phase before a brane collision can solve the standard cosmological puzzles and generate near-scale-invariant, strongly non-gaussian perturbations, with a blue gravitational-wave spectrum that d...
Large and infinite extra dimensions
hep-ph 2001-04 accept novelty 3.0 of 10

A pedagogical survey of brane-world models in which extra dimensions are large or infinite, covering how matter is trapped on a brane and what observable signatures large, warped or infinite extra dimensions would produce.
Quintom Cosmology: Theoretical implications and observations
hep-th 2009-09 accept novelty 2.0 of 10

Quintom dark energy, in which the equation of state crosses w = -1, is reviewed as an observationally motivated but theoretically challenging extension of standard cosmology.
Brane-world gravity
gr-qc 2003-12 accept novelty 2.0 of 10

A comprehensive 2003 survey of Randall-Sundrum brane-world gravity, covering the covariant projection formalism, brane cosmology, the black hole problem, and cosmological perturbation theory, with an explicit list of ...

How the literature uses it

69 arXiv papers cite this in 109 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 · 50 sentences without a confident label

Builds on it 1 paper

The analysis in this paper has a direct, but non-trivial, extension to strongly coupled heterotic $M$-theory [ref] .

Supersymmetric Hidden Sectors for Heterotic Standard Models · Introduction · 0.85

Uses its method or tool 1 paper

We will be taking the approach developed in Ref. [ref] , which was originally used to study 5D warped brane models with a scalar field [ref] .

Brane cosmological solutions in six-dimensional warped flux compactifications · Introduction · 0.97

Compares with it 4 papers

where $α̂_GUT$ is a parameter given by[As discussed in [ref] , the expression for $$ presented here is two times larger than the result given in [ref] .]

Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models · Split Wilson Lines · 0.87
In the limit $β,γ→0$, this model is reduced to the 5D effective theory of heterotic M-theory derived in Ref. [ref] on-shell.

Off-shell dimensional reduction of 5D orbifold supergravity · 5D gauged supergravity with universal hypermultiplet · 0.97
Note that this solution is in Gaussian Normal (GN) gauge, ($g_yy = 1, g_yμ=0$), a different gauge from the one originally written down in [ref] .

On black holes in heterotic braneworlds · Heterotic M-theory · 0.97

Cites it as support 1 paper

The perturbative superpotential $𝒲$ can be shown to be independent of the dilaton or the Kähler moduli [ref] .

Heterotic $M$-Theory Hidden Sectors with an Anomalous $U(1)$ Gauge Symmetry · Hidden Sector Lagrangian · 0.98

Cites it as background 44 papers

In this section, we describe the braneworld setup of Lukas, Ovrut, Stelle and Waldram (LOSW) [ref] , and give the background solution of heterotic M-theory.

On black holes in heterotic braneworlds · Overview of the heterotic braneworld and perturbation theory · 0.89
This theory was further compactified to five-dimensional “heterotic” M-theory on a Calabi–Yau threefold in [ref] and then, by integrating out the $S^1/ℤ_2$ modes, to a four-dimensional effective $E_8 ×E_8$ heterotic string theory [ref] .

Explicit Soft Supersymmetry Breaking in the Heterotic M-Theory $B-L$ MSSM · Introduction · 0.95
In this paper, we calculate and discuss the potential energy for the complex structure moduli, the Kähler moduli and the dilaton in heterotic M-theory vacua [ref] whose hidden sector contains a line bundle with an anomalous $U(1)$ structure group appropriately embedded into $E_8$ [ref] .

Moduli Axions, Stabilizing Moduli and the Large Field Swampland Conjecture in Heterotic M-Theory · Introduction · 0.90

Discussion

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