Supersymmetric Domain Wall and RG Flow from 4-Dimensional Gauged N=8 Supergravity

Changhyun Ahn; Kyungsung Woo

REVIEW

Supersymmetric Domain Wall and RG Flow from 4-Dimensional Gauged N=8 Supergravity

Changhyun Ahn, Kyungsung Woo

High Energy Physics - Theory · Submitted 14 Nov 2000 · 32 references

Pith's verdict: The A1-eigenvalue-as-superpotential construction holds up, but Eq. (26) is misprinted in exactly the coefficient the paper advertises, and the SO(3)xSO(3) appendix defers its data to an earlier arXiv version.

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arXiv IDhep-th/0011121 (version 2)

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CitationChanghyun Ahn, Kyungsung Woo. "Supersymmetric Domain Wall and RG Flow from 4-Dimensional Gauged N=8 Supergravity." arXiv:hep-th/0011121 (2000). https://pith.science/paper/7RAG4RDJ

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Editor's note

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The thing to know up front: the central construction survives scrutiny, but one of the three equations carrying it is misprinted, and the misprint sits in the equation the paper advertises as its punchline. That is a fixable transcription error, not a broken result, but it needs fixing before anyone quotes it. What is actually new here is narrower than the abstract suggests. The scalar potentials are reproductions, and the authors say so plainly: the SU(3) potential matches Warner, the SO(5) potentials match Romans. The new content is the identification of the A1 eigenvalue z3 as a superpotential, the two algebraic identities (25), the first-order BPS equations (31) on the four-parameter SU(3) slice, and the explicit SO(3)xSO(3) potential in Section 2.3. The BPS equations are derived twice, once by completing the energy functional into squares and once from the vanishing of the spin-1/2 and spin-3/2 variations, and the two routes agree. That double derivation, together with the external match to published potentials, is the strongest evidence in the paper and it is real evidence. On the soft spot: the stress-test note is right. Combining (23) with (25) gives an alpha coefficient of (2/3)/(p^2 q^2), not (2/3)p^2 q^2 as printed in (26), and the same for phi. But the BPS equations (31) carry 1/(3p^2 q^2) and 1/(3r^2 t^2), which is exactly what the corrected (26) requires, so the qualitative claim survives and the printed equation does not. While they are at it, the lambda equation in (29) has 3/4 where the kinetic normalization 3/8 gives 4/3. Two smaller points. The identities (25) are asserted rather than derived, but z3 is explicit in (24), so this is a finite differentiation any reader can do; the reader's report overweights that concern. The SO(3)xSO(3) appendix, by contrast, defers its 28-beins and A2 components to the original arXiv version, which genuinely blocks independent checking of the Section 2.3 potential, and I would ask for it. Who this is for: people in the holographic RG flow and gauged supergravity corner who want the explicit superpotentials and BPS equations for these sectors. It is not a broad-audience paper. Recommendation: send it out. The physics is sound and the errors are transcription-level.

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Major objections (4)

[Eq. (26)] Eq. (26) prints the α and φ gradient coefficients as (2/3)p²q² and (1/2)r²t². Combining Eq. (23) with Eq. (25) gives (2/3)/(p²q²) and (1/2)/(r²t²) instead: writing z3 = W e^{-2iβ}, Eq. (25) gives ∂_λβ = ∂_αW/(4√2 pq W), so 4W²(∂_λβ)² = (∂_αW)²/(8p²q²), and the α coefficient in V is (16/3)/(8p²q²) = (2/3)/(p²q²). The same computation for φ gives (1/2)/(r²t²). The BPS equations (31) and the field-space metric from (27) confirm the inverted form. Since Eq. (26) is the equation advertised as showing W is the 'true superpotential', this needs correction.
[Eq. (25)] Eq. (25) is introduced with 'We have found that...' and no derivation. These identities are load-bearing: without them the cross terms in the energy functional do not combine into a total derivative and the BPS bound (30) does not follow. Since z3 is given explicitly in Eq. (24), the identities are checkable, but the paper should provide at least a sketch of the derivation or a check at a representative point.
[§3.2 and §3.3] In §3.2 (SO(5)+) and §3.3 (SO(3)×SO(3)), the first-order equations ∂_rλ = ±(8√2/7)g∂_λW and ∂_rλ1 = ±4√2g∂_λ1W do not have fixed points at the non-supersymmetric critical points, because ∂W does not vanish there. The paper acknowledges this ('at nonsupersymmetric SO(7)+ fixed point, it does not [vanish]'), but still describes the solutions as flows interpolating between the two vacua. A solution of a first-order equation with nonzero velocity at a point passes through it rather than stopping. The analytic solution in §3.3, λ1(r) = log[(1+e^{√2g(c-r)})/(1-e^{√2g(c-r)})], indeed passes through the SO(3)×SO(3) point and continues; it does not interpolate. The claim of a 'non-BPS domain-wall solution interpolating between supersymmetric SO(8) vacuum and nonsupersymmetric SO(3)×SO(3) one' is therefore not supported by the solution as written.
[Abstract] The abstract states that 'the deformation of seven sphere S^7 gives rise to non-trivial RG flow in three-dimensional boundary conformal field theory from UV fixed point to IR fixed point.' For the SU(3) sector, the paper constructs the BPS equations but does not integrate them; no explicit interpolating flow is presented. The only explicit solution is the SO(3)×SO(3) one discussed above. The abstract should be softened to reflect that the paper constructs the flow equations, not the flows themselves.

Minor objections (4)

[Eq. (31), φ equation] The first expression ∓(1/(2rt))ig(...) evaluates to ±(√2/(4r²t²))g∂_φW, but the printed result is ±(√2/(3r²t²))g∂_φW. This is a factor 4/3 discrepancy.
[Passim] There are several typographical errors: 'ingradients' (p. 5), 'scalalr' (p. 6), 'repect' (p. 11), 'ealier' (p. 16), 'wih' (p. 18), 'conseqence' (p. 19), and 'Breitlohner' (p. 25).
[Eqs. (12) and (22)] The symbols p, q, r, t are used in Eq. (12) for hyperbolic functions of λ, λ' and again in Eq. (22) for hyperbolic functions of λ_i. The paper notes this, but it is confusing; a different notation would help.
[§2.3] The paper states that the SO(3)×SO(3) scalar potential was 'never written' explicitly. It would be good to cite the closest previous work (Warner [20]) and state precisely what is new in the explicit expression.

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The paper

The paper claims

A single eigenvalue of the theory's A-tensor is shown to act as the superpotential that governs domain walls and renormalization group flow between fixed points in four-dimensional gauged N=8 supergravity.

The reading

The paper asks whether the known critical points of four-dimensional gauged $N=8$ supergravity — the theory obtained by gauging the SO(8) subgroup of $E_7$ — can all be organized around a single object, a superpotential, in the way that simpler truncations of the theory already were. Working in three invariant sectors (SU(3), SO(5) and SO(3)$_6$×SO(3)), the authors build the 28-bein fields explicitly from the scalar and pseudo-scalar expectation values, extract the $A_1$ and $A_2$ tensors, and read off the scalar potential. In the SU(3) sector they identify one eigenvalue, $z_3$, of the $A_1$ tensor whose magnitude reproduces the potential as a sum of squares of its own field derivatives minus a term proportional to its square — exactly the shape needed for a superpotential. Why a reader should care: if that identification holds, the second-order equations of motion collapse into first-order gradient flow, the energy functional becomes a sum of complete squares with a total-derivative remainder, and the resulting domain-wall solutions are the holographic image of renormalization group trajectories running from an ultraviolet fixed point to an infrared one in the dual three-dimensional conformal field theory. The authors check the first-order equations twice, once by squaring the energy density and once by demanding that the spin-$1/2$ and spin-$3/2$ supersymmetry variations vanish in the bosonic background, and the two routes agree.

What carries the argument

The load-bearing device is a pair of algebraic identities relating the magnitude of the complex superpotential to the phase of its complex conjugate: $∂_α log|z_3| = 2√2 pq ∂_λ Arg z_3^*$ and $∂_φ log|z_3| = 2√2 rt ∂_{λ'} Arg z_3^*$. These identities are what allow the eight cross terms in the squared energy density to combine into a total derivative $4√2 g e^{3A}∂_r|z_3|$, leaving a sum of complete squares plus a boundary term and yielding the BPS bound $E ≥ -2√2 g [e^{3A(∞)}W(∞) - e^{3A(-∞)}W(-∞)]$. Supporting machinery includes the explicit 28-bein matrices $u^{IJ}{}_{KL}$, $v_{IJKL}$ built by exponentiating the self-dual four-form expectation values, and the observation that the $A_2$-tensor components $y_3$ and $y_7$ are proportional to $e^{iα}∂_{λ}z_3^*$ and $e^{iφ}∂_{λ'}z_3^*$ respectively — which is why the spin-$1/2$ variation reproduces the same first-order equations.

Extended reading notes

Core claim

The central claim is that in the SU(3)-invariant sector of gauged $N=8$ supergravity, the eigenvalue $z_3$ of the $A_1$ tensor supplies a genuine superpotential $W=|z_3|$ for the four-parameter subspace spanned by $λ,λ',α,φ$, so that the scalar potential takes the form $V=g^2[16/3(∂_λ W)^2 + 2/(3p^2q^2)(∂_α W)^2 + 4(∂_{λ'}W)^2 + 1/(2r^2t^2)(∂_φ W)^2 - 6W^2]$. From this the authors derive first-order BPS equations, $∂_rλ = ±8√2 g/3 ∂_λ W$ and its companions, together with $∂_rA = ∓√2 g W$ for the scale factor, and show that every solution of these first-order equations also solves the second-order Einstein and scalar equations. The same construction is carried through the SO(5) sectors, where only non-supersymmetric, unstable critical points appear and are already contained in the SU(3) analysis, and through the SO(3)×SO(3) sector, where the full scalar potential as a function of seven parameters is written out explicitly for the first time and a one-dimensional flow along $λ_1$ is solved analytically.

Load-bearing premise

The two algebraic identities linking the derivative of the superpotential's magnitude with respect to $α$ and $φ$ to the derivative of its phase with respect to $λ$ and $λ'$ are asserted rather than derived, and without them the cross terms in the energy functional do not collapse into a total derivative and the BPS bound does not follow.

Editorial extensions

If this is right

Each supersymmetric critical point of $W$ gives a stable AdS$_4$ vacuum with $V = -6g^2|z_3|^2$, and the three known supersymmetric points in the SU(3) sector (SO(8), G$_2$, SU(3)×U(1)) sit at $∂_λ W = ∂_{λ'}W = ∂_α W = ∂_φ W = 0$.
The monotonicity relation $∂_r^2 A = -2g^2[8/3(∂_λ W)^2 + 2(∂_{λ'}W)^2 + 1/(3p^2q^2)(∂_α W)^2 + 1/(3r^2t^2)(∂_φ W)^2] ≤ 0$ gives a c-function along the flow, so the holographic RG trajectory is monotone by construction.
Near a critical point the mass matrix takes the form $M^2_{ij} = 2g^2W^2 U_{ik}(U_{kj} - 3δ_{kj})$, so the eigenvalues of $U$ directly fix the conformal dimensions $Δ$ of the dual operators and hence which perturbations are relevant at the ultraviolet end.
In the SO(3)×SO(3) sector the analytic solution $λ_1(r) = log[(1+e^{√2 g(c-r)})/(1-e^{√2 g(c-r)})]$, $A(r) = (1/4)(3√2 gr + log[2 sinh √2 g(c-r)])$ is a non-BPS wall interpolating between the supersymmetric SO(8) vacuum and the non-supersymmetric SO(3)×SO(3) vacuum at $Λ = -14g^2$.

Reading between the lines

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

The identities relating $log|z_3|$ to $Arg z_3^*$ look like a Cauchy-Riemann-type structure on the scalar manifold; if they hold beyond the SU(3) slice, they would suggest the four-parameter sector inherits a complex-geometric structure from $E_7$/SU(8) rather than from supersymmetry alone, which the authors themselves hint at when noting that the SU(3)×U(1) point does not satisfy them.
Because the superpotential is a single eigenvalue of a matrix built from the 56-bein, the same recipe could in principle be applied to any invariant sector once the 28-beins are known — the SO(3)×SO(3) case with seven parameters is the natural next test, and the paper only solves its one-dimensional slice.
The non-BPS SO(3)×SO(3) flow is claimed stable by a positive-energy argument rather than by a Breitenlohner-Freedman check; computing the scalar masses at $Λ = -14g^2$ would settle whether the infrared end is genuinely a healthy fixed point or only a saddle of the potential.
If the $A_1$ eigenvalue is the right organizing object, the number of supersymmetries at a vacuum should equal the degeneracy of eigenvalues of $A_1$ whose magnitude equals $√{-Λ/6g^2}$; this gives a cheap diagnostic that can be run on any newly found critical point without solving the fluctuation equations.

Original abstract

By studying various, known extrema of 1) SU(3) sectors, 2) SO(5) sectors and 3) SO(3)xSO(3) sectors of gauged N =8 supergravity in four-dimensions, one finds that the deformation of seven sphere \S^7 gives rise to non-trivial renormalization group(RG) flow in three-dimensional boundary conformal field theory from UV fixed point to IR fixed point. For SU(3) sectors, this leads to four-parameter subspace of the supergravity scalar-gravity action and we identify one of the eigenvalues of A_1 tensor of the theory with a superpotential of scalar potential that governs RG flows on this subspace. We analyze some of the structure of the superpotential and discuss first-order BPS domain-wall solutions, using some algebraic relations between superpotential and derivatives of it with respect to fields, that determine a (super)symmetric kink solution in four-dimensional N =8 supergravity, which generalizes all the previous considerations. The BPS domain-wall solutions are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N=8 supergravity. For SO(5) sectors, there exist only nontrivial nonsupersymmetric critical points that are unstable and included in SU(3) sectors. For SO(3)xSO(3) sectors, we construct the scalar potential(never been written) explicitly and study explicit construction of first-order domain-wall solutions.

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

By studying various, known extrema of 1) SU(3) sectors, 2) SO(5) sectors and 3) SO(3)xSO(3) sectors of gauged N =8 supergravity in four-dimensions, one finds that the deformation of seven sphere \S^7 gives rise to non-trivial renormalization group(RG) flow in three-dimensional boundary conformal field theory from UV fixed point to IR fixed point. For SU(3) sectors, this leads to four-parameter subspace of the supergravity scalar-gravity action and we identify one of the eigenvalues of A_1 tensor of the theory with a superpotential of scalar potential that governs RG flows on this subspace. We analyze some of the structure of the superpotential and discuss first-order BPS domain-wall solutions, using some algebraic relations between superpotential and derivatives of it with respect to fields, that determine a (super)symmetric kink solution in four-dimensional N =8 supergravity, which generalizes all the previous considerations. The BPS domain-wall solutions are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N=8 supergravity. For SO(5) sectors, there exist only nontrivial nonsupersymmetric critical points that are unstable and included in SU(3) sectors. For SO(3)xSO(3) sectors, we construct the scalar potential(never been written) explicitly and study explicit construction of first-order domain-wall solutions.

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	By studying various, known extrema of 1) SU(3) sectors, 2) SO(5) sectors and 3) SO(3)xSO(3) sectors of gauged N =8 supergravity in four-dimensions, one finds that the deformation of seven sphere \S^7 gives rise to non-trivial renormalization group(RG) flow in three-dimensional boundary conformal field theory from UV fixed point to IR fixed point.	result 0.87	0.81	0.79	asserted 1.00
s1	For SU(3) sectors, this leads to four-parameter subspace of the supergravity scalar-gravity action and we identify one of the eigenvalues of A_1 tensor of the theory with a superpotential of scalar potential that governs RG flows on this subspace.	result 0.49	0.88	0.08	asserted 1.00
s2	We analyze some of the structure of the superpotential and discuss first-order BPS domain-wall solutions, using some algebraic relations between superpotential and derivatives of it with respect to fields, that determine a (super)symmetric kink solution in four-dimensional N =8 supergravity, which generalizes all the previous considerations.	contribution 0.88	0.93	0.14	asserted 1.00
s3	The BPS domain-wall solutions are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N=8 supergravity.	background 0.51	0.27	0.05	asserted 1.00
s4	For SO(5) sectors, there exist only nontrivial nonsupersymmetric critical points that are unstable and included in SU(3) sectors.	result 1.00	0.67	0.05	asserted 1.00
s5	For SO(3)xSO(3) sectors, we construct the scalar potential(never been written) explicitly and study explicit construction of first-order domain-wall solutions.	contribution 0.98	0.95	0.08	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

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not examined

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levels run

P abstract

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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.** The plots of V (on the left) and W (on the right), with λ 1 on the horizontal axis. Scalar potential V , at λ 1 = 0, is the maximally supersymmetric and locally maximum while superpotential W at that point is locally minimum. The cosmological constant is −6. We took g 2 as 1 for simplicity. At λ 1 = 2 sinh−1 2 = 2.89, V has locally minimum and is nonsupersymmetric and the cosmological constant is −14. See also [31]. 24

Figures reproduced from arXiv:hep-th/0011121 by the authors.

References

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How the literature uses it

38 arXiv papers cite this in 99 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 · 54 sentences without a confident label

Builds on it 3 papers

In this sense, we have found the 11-dimensional solution for the 4-dimensional supersymmetric flow presented in [ref] along the constraints and the corresponding $SU(3)$-invariant RG flow to the IR point of 3-dimensional dual gauge theory is the curve ($m_1 ≠m_2$) connecting the $SO(8)$ point to the $SU(3) ×U(1)_R$ point in Figure 1 of [ref] .

Holographic N=1 Supersymmetric Membrane Flows · An $N=1$ $SU(3)$-invariant supersymmetric flow · 0.80
For nonzero values for $θ$ and $ψ$, the previous six $4 ×4$ block diagonal matrices [ref] where $θ=0=ψ$ are generalized to be the present three $8 ×8$ block diagonal matrices.

Are There Any New Vacua of Gauged N=8 Supergravity in Four Dimensions? · The $SU(3)$-invariant sector of gauged $N=8$ supergravity with four scalar space · 0.97
One would expect that the flow to the $SU(3) ×U(1)$ invariant fixed point [ref] could be easily promoted to a smooth flow with chemical potentials in global $AdS$ in a manner very analogous to the result here [ref] .

Supersymmetric Charged Clouds in AdS_5 · Conclusions · 0.80

Uses its method or tool 4 papers

We impose the nontrivial $AdS_4$ radial coordinate dependence of vacuum expectation values subject to the 4-dimensional RG flow equations [ref] .

Towards An N=1 SU(3)-Invariant Supersymmetric Membrane Flow In Eleven-Dimensional Supergravity · Introduction · 0.87
Then the supersymmetric flow equations with ([ref]) and ([ref]) are described as [ref] d /d r = 8√(2)/3 g ∂_ W , d '/d r = 2 √(2) g _' W, d α/d r = √(2)/3p^2 q^2 g _ W, d ϕ/d r = √(2)/3r^2 t^2 g _ϕ W , d A/d r = - √(2) g W.

Comments on Holographic Gravity Dual of N=6 Superconformal Chern-Simons Gauge Theory · The holographic RG flows for $SU(3)$-invariant sector in four dimensions · 0.85
Since the remaining four kinetic terms plus the scalar potential in ([ref]) depend on only $, ', α$ and $ϕ$ and do not depend on $θ$ and $ψ$, one can recombine those four kinetic terms with the corresponding scalar potential parts in terms of the squares of the four derivatives of superpotential as in [ref] .

Are There Any New Vacua of Gauged N=8 Supergravity in Four Dimensions? · The $SU(3)$-invariant sector of gauged $N=8$ supergravity with four scalar space · 0.90

Compares with it 4 papers

As a check of consistency, the expressions in refs [ref] for the pure electric SO(8) gauging are exactly recovered[By redefining the fields as $p=cosh(λ)$ , $q=sinh(λ)$ , $r=cosh(λ')$ and $t=sinh(λ')$, the mass term $𝒜^11$ in ([ref]) is written as 𝒜^11=e^i ω ( p^3 (r^4+t^4 e^4 i ϕ)+6 p q^2 r^2 t^2 e^2 i (α+ϕ) ) + e^-i ω ( e^3 i α q^3 (r^4+t^4 e^4 i ϕ) + 6 q p^2 r^2 t^2 e^i (α+2 ϕ) ) , ]

On new maximal supergravity and its BPS domain-walls · Gravitino-gravitino terms · 0.97
The superpotential $W$ appearing in ([ref]) is the same as the one in [ref] except the factor $√(2)$ in front of $'$(or $ρ$) for the hyperbolic functions of ([ref]).

The Gauge Dual of Gauged N=8 Supergravity Theory · The holographic supersymmetric RG flows in four dimensions · 0.87
The $ẑ_1$ does not contain the field $θ$ or $ψ$ and coincides with the one by putting the constraints $θ=0=ψ$ [ref] .

Are There Any New Vacua of Gauged N=8 Supergravity in Four Dimensions? · The $SU(3)$-invariant sector of gauged $N=8$ supergravity with four scalar space · 0.82

Cites it as background 23 papers

The holographic supersymmetric RG flow equations connecting $N=8$ $SO(8)$-invariant fixed point to $N=2$ $SU(3) ×U(1)$-invariant fixed point have been found in [ref] (See also [ref] for earlier work).

The Gauge Dual of Gauged N=8 Supergravity Theory · Introduction · 0.83
The $SU(3)$-invariant sector of the scalar manifold of gauged $N=8$ supergravity [ref] in four dimensions has been studied in [ref] .

Comments on Holographic Gravity Dual of N=6 Superconformal Chern-Simons Gauge Theory · The holographic RG flows for $SU(3)$-invariant sector in four dimensions · 0.90
The holographic supersymmetric RG flow equation connecting $N=8$ $SO(8)$ point to $N=2$ $SU(3) ×U(1)$ point has been studied in [ref] where the $U(1)$ symmetry here can be identified with $U(1)_R$ symmetry of 3-dimensional theory coming from the $N=2$ supersymmetry while those from $N=8$ $SO(8)$ point to $N=1$ $G_2$ point has been studied in [ref] .

New N=2 Supersymmetric Membrane Flow In Eleven-Dimensional Supergravity · Introduction · 0.89

Discussion

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