REVIEW
Effects of D-instantons
High Energy Physics - Theory · Submitted 20 Jan 1997 · 39 references
Pith's verdict: The one-instanton R^4 calculation is solid; the SL(2,Z) Eisenstein completion is an honestly labelled conjecture whose instanton coefficients the paper never actually computes.
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Referee report
The paper
The paper claims
A single D-instanton may fix the exact coefficient of the $R^4$ interaction in type IIB string theory, and the way it does so suggests the perturbative series stops at one loop.
The reading
What carries the argument
The load-bearing mechanism is the integration over the sixteen fermionic zero modes of the D-instanton background. In supergravity these modes are generated by applying the sixteen broken supersymmetries to the classical solution, producing the chain of one-point functions for $λ$, $ψ_μ$, $G_{μνρ}$ and the metric; in string theory the same modes appear as fermionic open-string vertex operators attached to a disk with Dirichlet boundary conditions, and the disk amplitudes reproduce the field-theory tadpoles after LSZ reduction. Because a physical amplitude needs sixteen powers of the Grassmann modulus $ε_0$, only processes with enough external states survive, and the resulting contact terms are the instanton corrections. The second piece of machinery is the generalized Eisenstein series $f_s(τ,τ̄)=∑_{(p,n)≠(0,0)}(τ_2/|p+nτ|^2)^s$ at $s=3/2$, which is manifestly $SL(2,ℤ)$-invariant, satisfies the Laplace eigenvalue equation $τ_2^2(∂_{τ_1}^2+∂_{τ_2}^2)E^*=s(s-1)E^*$, and after Poisson resummation produces exactly the tree-level $ζ(3)$ term, the one-loop term, and the sum over instanton charges.
Extended reading notes
Core claim
The paper's central claim is that the D-instanton's contribution to the $R^4$ interaction is not an isolated curiosity but the first term of an $SL(2,ℤ)$-invariant whole. Two ingredients carry it. First, the authors compute the one-instanton background directly, in supergravity and independently on the string world-sheet, by identifying the sixteen fermionic zero modes generated by the broken supersymmetries and integrating over them together with the ten bosonic collective coordinates; this yields an explicit $e^{2πiτ_0}$ correction with the same $R^4$ tensor structure as the tree-level and one-loop terms. Second, T-duality in nine euclidean dimensions relates multiply-wrapped D-particle world-lines to D-instantons of charge $p=mn$, giving a spectrum of multiply-charged instantons with actions $2πi|mn|τ$. Combining these with the known tree-level coefficient $aζ(3)τ_2^{3/2}$ and one-loop coefficient $bτ_2^{-1/2}$ singles out the generalized Eisenstein series $f(τ,τ̄)=∑τ_2^{3/2}/|p+nτ|^3$, whose expansion reproduces both perturbative terms and then a sum of single multiply-charged instanton contributions with no further perturbative terms. The authors are explicit that they have 'by no means proven' this identification, only that it satisfies several stringent constraints.
Load-bearing premise
The argument needs the perturbative $R^4$ terms to vanish beyond one loop, a non-renormalization statement the authors justify only heuristically and explicitly say has not appeared in the literature and deserves further study.
Editorial extensions
If this is right
- The $R^4$ coefficient in type IIB is conjectured to be exactly $f(τ,τ̄)=∑_{(p,n)≠(0,0)}τ_2^{3/2}/|p+nτ|^3$, so a single function of the complex coupling encodes tree level, one loop, and every instanton sector at once.
- Perturbative corrections to $R^4$ vanish beyond one loop in the zero-instanton sector; the genus expansion of this particular coupling truncates.
- The multi-instanton contributions to $R^4$ come only from single instantons of multiple charge $p=mn$, not from separated unit instantons, which instead give higher-derivative interactions because of extra fermionic zero modes.
- The instanton terms come with an infinite series of perturbative corrections around each charge, beginning with a $τ_2$-independent term that an explicit D-instanton calculation should be able to reproduce as a check.
- Fixed-angle four-graviton scattering on a Dirichlet disk falls off as a power of the invariants rather than exponentially, so the D-instanton produces point-like behaviour in a theory whose fundamental scattering is normally soft.
Reading between the lines
- If the truncation is real, the $R^4$ coefficient becomes a protected observable: any candidate nonperturbative definition of type IIB has to reproduce the Eisenstein series, which is a much sharper target than matching the two known perturbative coefficients.
- The same function appears as the spectral sum of the Laplacian on a two-torus with modular parameter $τ$, so the $R^4$ coupling is secretly a heat-kernel-like count of Kaluza–Klein momenta and windings; this is the kind of structure that would let eleven-dimensional or M-theoretic formulations make direct contact with ten-dimensional instanton sums.
- The point-like fixed-angle behaviour sits uneasily with the expectation that D-particle physics is governed by the eleven-dimensional Planck scale while the D-instanton moduli space knows only the ten-dimensional one; comparing the two scales in a controlled amplitude looks like the cleanest way to see whether the power law survives the inclusion of multiply-charged instantons.
- Because the fermionic zero-mode count is what forces the truncation, the same argument should apply to other protected couplings with the same number of external states, which suggests a small family of $SL(2,ℤ)$-invariant functions indexed by the eigenvalue $s$ rather than a one-off accident.
Original abstract
Scattering of fundamental states of type IIB supergravity and superstring theory is discussed at low orders in perturbation theory in the background of a D-instanton. The integration over fermionic zero modes in both the low energy supergravity and in the string theory leads to explicit nonperturbative terms in the effective action. These include a single instanton correction to the known tree-level and one-loop $R^4$ interactions. The `spectrum' of multiply-charged D-instantons is deduced by T-duality in nine dimensions from multiply-wound world-lines of marginally-bound D-particles. This, and other clues, lead to a conjectured SL(2,Z) completion of the $R^4$ terms which suggests that they are not renormalized by perturbative corrections in the zero-instanton sector beyond one loop. The string theory unit-charged D-instanton gives rise to point-like effects in fixed-angle scattering, raising unresolved issues concerning distance scales in superstring theory.
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
Scattering of fundamental states of type IIB supergravity and superstring theory is discussed at low orders in perturbation theory in the background of a D-instanton. The integration over fermionic zero modes in both the low energy supergravity and in the string theory leads to explicit nonperturbative terms in the effective action. These include a single instanton correction to the known tree-level and one-loop $R^4$ interactions. The `spectrum' of multiply-charged D-instantons is deduced by T-duality in nine dimensions from multiply-wound world-lines of marginally-bound D-particles. This, and other clues, lead to a conjectured SL(2,Z) completion of the $R^4$ terms which suggests that they are not renormalized by perturbative corrections in the zero-instanton sector beyond one loop. The string theory unit-charged D-instanton gives rise to point-like effects in fixed-angle scattering, raising unresolved issues concerning distance scales in superstring theory.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Scattering of fundamental states of type IIB supergravity and superstring theory is discussed at low orders in perturbation theory in the background of a D-instanton. | contribution 0.86 | 0.67 | 0.07 | asserted 0.94 |
| s1 | The integration over fermionic zero modes in both the low energy supergravity and in the string theory leads to explicit nonperturbative terms in the effective action. | result 0.52 | 0.84 | 0.14 | asserted 1.00 |
| s2 | These include a single instanton correction to the known tree-level and one-loop $R^4$ interactions. | result 0.99 | 0.78 | 0.18 | asserted 1.00 |
| s3 | The `spectrum' of multiply-charged D-instantons is deduced by T-duality in nine dimensions from multiply-wound world-lines of marginally-bound D-particles. | method 0.55 | 0.81 | 0.06 | asserted 1.00 |
| s4 | This, and other clues, lead to a conjectured SL(2,Z) completion of the $R^4$ terms which suggests that they are not renormalized by perturbative corrections in the zero-instanton sector beyond one loop. | interpretation 0.89 | 0.53 | 0.46 | hedged possibility 0.68 |
| s5 | The string theory unit-charged D-instanton gives rise to point-like effects in fixed-angle scattering, raising unresolved issues concerning distance scales in superstring theory. | result 0.44 | 0.77 | 0.11 | asserted 0.99 |
What was examined, and what was not
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- P abstract
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Figures
Figures reproduced from arXiv:hep-th/9701093 by the authors.
References
How the literature uses it
181 arXiv papers cite this in 341 sentences
Papers citing it · 236 sentences without a confident label
Builds on it 3 papers
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The goal of this paper is to apply the framework of D-instanton perturbation theory to compute D-instanton contributions to the S-matrix of type IIB superstring theory in ten-dimensional Minkowskian spacetime, beyond the leading order results of [ref] .
D-instanton Effects in Type IIB String Theory · Introduction · 0.97
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The instanton measure $μ_3/2(ℓ_1, ℓ_2)$ generalizes the familiar D$(-1)$ instanton measure $μ_3/2(N)$ of [ref] which is also known to capture the effects of pure charge $N$ $A$-type D2-brane instantons [ref] (corresponding to $ℓ_2=0$).
Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1) · Eisenstein Series and the Exact Universal Hypermultiplet Geometry · 0.88
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We have generalized a single D instanton calculation [ref] into that for multiple D instantons by introducing matrix Majorana-Weyl spinor fields.
Vertex operators of Type IIB matrix model via calculation of disk amplitudes · Conclusion · 1.00
Uses its method or tool 4 papers
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In order to obtain these solutions we will simply sum over images of the $α'$ corrections under the duality group following [ref] .
Non-Perturbative Corrections and Modularity in N=1 Type IIB Compactifications · Introduction · 0.98
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As explained in [ref] , the invariance under the discrete subgroup $SL(2,)$ can be restored by summing over images, using similar techniques as the ones used for $R^4$ couplings in toroidal compactifications [ref] .
D-instantons and twistors · S-duality and A-type D-instanton corrections · 0.86
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Writing the polarization tensor as $ζ^μ_rν_r = ζ^(μ_r ζ̃^ν_r)$, and evaluating the fermionic integrals in a special frame as described in [ref] , in terms of its $SO(8)$ components, one obtains the following result for $⟨h⟩_4$ [In both, string theory and supergravity, the four-graviton scattering result is given as an integral of the product of four factors of “$⟨h⟩_4$" defined as the tadpole associated with the disk with four fermion zero modes coupled to the graviton and the self dual fourth-rank antisymmetric tensor: [equation] $ϵ_0$ corresponding to the broken supersymmetry - the only covariant combination of four $ϵ_0$'s, two physical momenta and the physical polarization tensor.] : [equation] where: $γ^ij_ab = 1/2γ^[i_aȧγ^j]_ȧb, γ^i$ being the generators of $Cl_8$ Clifford algebra and $R_ijmn ≡k_ik_mζ_(jζ̃_n)$ is the linearized curvature.
On ${\cal M}$-Theory Dual of Large-$N$ Thermal QCD-Like Theories up to ${\cal O}(R^4)$ and $G$-Structure Classification of Underlying Non-Supersymmetric Geometries · $O(l_p^6)$ Corrections to the Background of CITEMARK60Z in the MQGP Limit · 0.85
Uses its data or values 5 papers
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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
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Its dependence with respect to the moduli space parametrised by ${𝒱_2, τ}$ has been already computed in [ref] , and it is captured by the following non-trivial function (see also [ref] )
On the Species Scale, Modular Invariance and the Gravitational EFT expansion · M-theory on $T^2$ · 0.84
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More concretely, the correction determined by [ref] is S^(3)_ℛ^4= α'^3g_s^3/2/32 πG ∫d^10 x ∫d^16 θ √(-g)f^(0,0)(τ,τ̅) [(θΓ^mnpθ) (θΓ^qrsθ)ℛ_mnpqrs]^4 + c.c. where $f^(0,0)(τ,τ̅)$ is a modular form [ref] .
Quantum corrections to eta/s · Introduction · 0.83
Compares with it 6 papers
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We show that, whereas S-duality in the large-radius limit requires the inclusion of tree-level and non-perturbative effects [ref] , the leading small-radius couplings we find exhibit S-duality invariance independently.
The emergence of inherently 9-dimensional one-loop effective action from T-duality · Introduction · 0.88
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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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It is remarkable that the new solution is more proper to be known as an equivalent for the famous D-instanton solution of AdS$_5$/CFT$_4$ duality studied in [ref] and [ref] .
New Instantons in AdS_4/CFT_3 from D4-Branes Wrapping Some of CP^3 · Introduction · 0.98
Cites it as support 2 papers
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However, this symmetry descends from the usual S-duality action in ten-dimensional string theory, and there is overwhelming evidence that the modular subgroup $SL(2,)$ remains a symmetry at the quantum level (see [ref] and citations thereof).
S-duality in Twistor Space · Quantum corrections · 0.97
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Furthermore, in a number of examples from string theory it was actually possible to show that the observable is given by an Eisenstein series itself [ref] .[That Eisenstein series are mostly not square-integrable is no problem in these cases since the object computed (part of a scattering amplitude) is not a wavefunction and not required to be normalisable.]
Eisenstein series and automorphic representations · A physicist's possible answer · 1.00
Cites it as background 71 papers
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In Section 2, we review the basic setup and results of [ref] inclusive of a summary of the results of [ref] on the conjectured $SL(2,ℤ)$ completion of the $D=11$ supergravity action by studying four-graviton scattering in a D-instanton background and a non-renormalization beyond one loop at $O(R^4)$ in the zero-instanton sector.
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If the Euclidean world-volume of a brane is wrapped around an appropriate compact manifold, we obtain a so-called D-instanton [ref] that induces a non-perturbative gravitational interaction involving the axion and dilaton fields (see review [ref] and the next section).
D-instantons and multiparticle production in N=4 SYM · Introduction · 0.96
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The one-instanton contribution to closed string scattering was studied in type IIB string theory in [ref] , and in $c=1$ string theory in [ref] .
Multi-Instanton Calculus in c = 1 String Theory · Multiple ZZ-instantons from the worldsheet perspective · 0.83
Discussion
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