REVIEW
Supersymmetry Constraints on Type IIB Supergravity
High Energy Physics - Theory · Submitted 11 Aug 1998 · 24 references
Pith's verdict: A real derivation of a previously conjectured result, with a coefficient chain worth checking but not worth dismissing.
T0 · AI review
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Referee report
The paper
The paper claims
Supersymmetry alone fixes the modular form multiplying the $R^4$ correction in type IIB supergravity, turning a conjecture about Laplace eigenfunctions on $SL(2, Z)$ into a derivation.
The reading
What carries the argument
The engine is the Noether procedure applied at order $(alpha')^3$, combined with the supercovariant completion $hat G$ of the three-form field strength and the modular covariant derivatives $D_w = i(tau_2 partial_tau - i w/2)$ and $bar D_w$, which raise and lower $U(1)$ charge and map a $(w,-w)$ form to $(w+1,-w-1)$. Two Laplacians on such forms, $nabla^{2(-)}_w = 4 D_{w-1} bar D_{-w}$ and $nabla^{2(+)}_w = 4 bar D_{-w-1} D_w$, differ by $2w$; this bookkeeping is what converts the pair of first-order equations into eigenvalue statements and lets the weight-zero eigenvalue $3/4$ propagate to weight twelve. The other load-bearing ingredient is the classical $lambda^2 lambda^{*2}$ interaction, whose normalization is not tabulated in the literature and is derived here from the mixing of three classical terms under the lowest-order dilatino and gravitino variations.
Extended reading notes
Core claim
The central claim is that the two terms in $S^{(3)}$ with maximal fermion number, $f^{(12,-12)}lambda^{16}$ and $f^{(11,-11)}hat G lambda^{14}$, are linked by supersymmetry variations that mix with nothing else at that order. Requiring the $lambda^{16}psi^*_mu epsilon$ variation to cancel gives $D_{11} f^{(11,-11)} = -frac{4}{3cdot 144} f^{(12,-12)}$, where $D_w$ is the modular covariant derivative. A second variation, the $lambda^{16}lambda^*epsilon^*$ channel, receives a contribution from a classical $lambda^2lambda^{*2}$ contact term whose coefficient $c = -3/128$ the authors fix in an appendix, and a further relation comes from requiring the supersymmetry algebra to close on the dilatino up to its equation of motion, $32 D_{11} g = f^{(12,-12)}$. Eliminating the transformation function $g$ yields a second first-order equation, and the pair reduces to $nabla^{2(-)}_{12} f^{(12,-12)} = (-132 + 3/4) f^{(12,-12)}$ and $nabla^{2(+)}_{11} f^{(11,-11)} = (-132 + 3/4) f^{(11,-11)}$. The first reproduces exactly the equation conjectured earlier for the sixteen-dilatino coefficient, and hence, through the linearized supersymmetry relation between the terms, for the coefficient of $R^4$ as well.
Load-bearing premise
The derivation assumes, by inspection rather than by systematic proof, that no term with sixteen dilatinos and one gravitino can be produced when the order-$(alpha')^3$ corrections to the supersymmetry transformations act on the classical action; if such a term exists, the first-order relation changes and the eigenvalue moves.
Editorial extensions
If this is right
- The $R^4$ coefficient is fixed to $D^{12} f^{(0,0)}$ with $f^{(0,0)}$ the weight-zero Eisenstein series of eigenvalue $3/4$, so the entire tower of order-$(alpha')^3$ interactions is determined by one function whose weak-coupling behaviour is a tree-level term, a one-loop term, and an infinite series of D-instanton corrections.
- Because the maximal-fermion terms are the ones that mix least, the same strategy should constrain the order-$(alpha')^5$ terms, where the paper sketches how new structures such as $lambda^{16}hat G^4$ and $lambda^{16}hat G^3hat G^*$ enter and how the corresponding functions would be fixed.
- The nine-dimensional compactification inherits the method: the $R^4$ coefficient of the form $V^{-1/2} f^{(0,0)}(Omega,bar Omega) + 2pi^2/3V$ can in principle be derived directly rather than through indirect duality arguments.
- More generally, the technique gives a route to the moduli dependence of protected interactions in lower-dimensional compactifications, including cases where the duality group becomes exceptional or infinite-dimensional.
Reading between the lines
- The eigenvalue $-132 + 3/4$ is not an independent input: it is what twelve covariant derivatives do to the weight-zero eigenvalue $3/4$, so the whole result is one number dressed by the weight bookkeeping. That makes it a sharp target — an independent string computation that produced any other shift would be a clean failure.
- Because the derivation leans only on the maximal-fermion terms, it should be insensitive to the detailed bosonic action. That suggests the same bookkeeping could reach the $D^4R^4$ and higher-derivative couplings whose coefficients are currently known only at low genus.
- The step I would want checked first is the order-$(alpha')^5$ extension the paper sketches but does not carry out: there $delta^{(4)}S^{(3)}$ mixes with $delta^{(0)}S^{(5)}$ and $delta^{(7)}S^{(0)}$, and whether the generalized Eisenstein series with eigenvalue $1/4 + g^2$ survives that mixing is a genuine test rather than a restatement.
- A reader outside this literature might notice that the argument never uses the self-duality of the five-form in an essential way; the authors note that compactifying on a circle would sidestep the missing covariant action entirely, which suggests the constraints are a property of the equations of motion rather than of any particular Lagrangian.
Original abstract
Supersymmetry is used to derive conditions on higher derivative terms in the effective action of type IIB supergravity. Using these conditions, we are able to prove earlier conjectures that certain modular invariant interactions of order alpha' **3 relative to the Einstein-Hilbert term are proportional to eigenfunctions of the Laplace operator on the fundamental domain of SL(2,Z). We also discuss how these arguments generalize to terms of higher order in alpha', as well as to compactifications of supergravity.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Supersymmetry is used to derive conditions on higher derivative terms in the effective action of type IIB supergravity. Using these conditions, we are able to prove earlier conjectures that certain modular invariant interactions of order alpha' **3 relative to the Einstein-Hilbert term are proportional to eigenfunctions of the Laplace operator on the fundamental domain of SL(2,Z). We also discuss how these arguments generalize to terms of higher order in alpha', as well as to compactifications of supergravity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Supersymmetry is used to derive conditions on higher derivative terms in the effective action of type IIB supergravity. | method 0.81 | 0.81 | 0.11 | asserted 1.00 |
| s1 | Using these conditions, we are able to prove earlier conjectures that certain modular invariant interactions of order alpha' **3 relative to the Einstein-Hilbert term are proportional to eigenfunctions of the Laplace operator on the fundamental domain of SL(2,Z). | result 0.98 | 0.92 | 0.80 | asserted 1.00 |
| s2 | We also discuss how these arguments generalize to terms of higher order in alpha', as well as to compactifications of supergravity. | contribution 0.55 | 0.79 | 0.06 | asserted 0.66 |
What was examined, and what was not
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References
How later papers used it
97 later papers cite it
1 later paper by other authors is listed below: 1 built on it.
Builds on it 1 paper
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Generalized this paper's work by implementing the Noether procedure to the required order in the derivative expansion, also taking into account the corrected supersymmetry transformations, to construct a class of such terms in the effective action.
Supersymmetry constraints on the R^4 multiplet in type IIB on T^2 (2011) · 0.98 · paraphrased: the licence does not allow quoting
How the literature uses it
91 arXiv papers cite this in 155 sentences
Papers citing it · 115 sentences without a confident label
Builds on it 1 paper
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In order to construct a class of such terms in the effective action, we shall implement the Noether procedure to the required order in the derivative expansion, also taking into account the corrected supersymmetry transformations, generalizing the work of [ref] .
Supersymmetry constraints on the R^4 multiplet in type IIB on T^2 · Introduction · 0.98
Uses its method or tool 4 papers
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The reason we want to do the calculation in the type IIB theory is because we want to impose the constraints implied by S–duality which allow us to obtain explicit expressions for these amplitudes at high genera, along the lines of [ref] .
Constraining gravitational interactions in the M theory effective action · The leading terms in the M theory effective action · 0.90
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From noether one can judiciously choose maximally fermionic terms in the effective action [ref] and construct corrected supersymmetry transformations such that noether is satisfied.
The structure of the R^8 term in type IIB string theory · Relations among the moduli dependent couplings of the various non–BPS interactions · 0.86
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We will now explain the arguments for obtaining the eigenvalue equation [ref] for the coefficient function $ℰ^10_00$ in ten dimensions using constraints from supersymmetry following [ref] .
Eisenstein series and automorphic representations · Supersymmetry constraints* · 0.90
Uses its data or values 1 paper
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Using the ten dimensional values $λ^(10)_(0,0)=3/4$, $λ^(10)_(1,0)=15/4$ and $λ^(10)_(0,1)=12$ determined in [ref] , we deduce that the coefficients of the terms discussed in [ref] satisfy the following set of Laplace eigenvalue equations with source terms, [equation]
String theory dualities and supergravity divergences · Constraints on the coefficients · 0.99
Compares with it 2 papers
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This term corresponds to an effective $ℓ_s^-2 R^4$ coupling in $AdS_5$, which, in the flat space limit, matches the $ℓ_s^-2 R^4$ contribution to the Type IIB graviton S-matrix as computed at finite string coupling $g_s$ in [ref] .
Modular Invariance in Superstring Theory From ${\cal N} = 4$ Super-Yang-Mills · Conclusion · 0.94
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The resulting coefficients were found to match the predictions of space-time supersymmetry [ref] and S-duality [ref] in Type IIB string theory [ref] and [ref] .
Identities among higher genus modular graph tensors · Introduction · 0.84
Cites it as background 29 papers
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D-instanton contribution to the four graviton amplitude in type IIB string theory was predicted by Green and Gutperle by requiring the amplitude to be S-duality invariant [ref] .
Muti-instanton Amplitudes in Type IIB String Theory · Introduction · 0.83
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Many D-instanton induced terms were predicted using S-duality invariances of various compactified string theories [ref] , but except for the early attempts [ref] , the direct systematic computation of these amplitudes from first principles has not been carried out.
Normalization of Type IIB D-instanton Amplitudes · Introduction · 0.97
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For type IIB string theory in 10 dimensions, an exact answer for the eight-derivative, $R^4$ coupling consistent with duality invariance under $G_10=SL(2,)$ was proposed in the seminal work [ref] , and verified in a number of subsequent works [ref] .
$R^4$ couplings and automorphic unipotent representations · Introduction · 0.89
Discussion
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