In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen vanishes. Our proof uses the reformulation of ansatz in terms of even cosets, theta functions, and specifically the theory of the $Γ_{00}$ linear system on Jacobians introduced by van Geemen and van der Geer.
At the two-loop level, where the amplitudes were computed by D'Hoker and Phong, we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera.
We show that the higher genus 4-point superstring amplitude is strongly constrained by the geometry of moduli space of Riemann surfaces. A detailed analysis leads to a natural proposal which satisfies several conditions. The result is based on the recently derived Siegel induced metric on the moduli space of Riemann surfaces and on combinatorial products of determinants of holomorphic abelian differentials.
For a calculation of divergent fermion string amplitudes a regularization procedure invariant under the supermodular group is constructed. By this procedure superstring amplitudes of an arbitrary genus are calculated using both partition functions and superfield vacuum correlators computed early. A finiteness of superstring amplitudes and related topics are discussed.
Combining certain identities for modular forms due to Igusa with Schottky-Jung relations, we study the cosmological constant for the recently proposed ansatz for the chiral superstring measure in genus 5. The vanishing of this cosmological constant turns out to be equivalent to the long-conjectured vanishing of a certain explicit modular form of genus 5 on the moduli of curves M_5, and we disprove this conjecture, thus showing that the cosmological constant for the proposed ansatz does not vanish identically. We exhibit an easy modification of the genus 5 ansatz satisfying factorization constr…
In this paper we continue the program pioneered by D'Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansätze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in…
The vanishing of the cosmological constant and the non-renormali-zation theorem are verified at two loops by explicit computation using the hyperelliptic language and the newly obtained chiral measure of D'Hoker and Phong. A set of identities is found which is used in the verification of the non-renormalization theorem and leads to a great simplification of the calculation of the four-particle amplitude at two loops.
We show that a recently conjectured expression for the superstring three-point amplitude, in the framework of the Cacciatori, Dalla Piazza, van Geemen - Grushevsky ansatz for the chiral measure, fails to vanish at three-loop, in contrast with expectations from non-renormalization theorems. Based on analogous two-loop computations, we discuss the possibility of a non-trivial correction to the amplitude and propose a natural candidate for such a contribution. Thanks to a new remarkable identity, it is reasonable to expect that the corrected three-point amplitude vanishes at three-loop, recoverin…
Superstring amplitudes of an arbitrary genus are calculated through super-Schottky parameters by a summation over the fermion strings. For a calculation of divergent multi-loop fermion string amplitudes a supermodular invariant regularization procedure is used. A cancellation of divergences in the superstring amplitudes is established. Grassmann variables are integrated, the superstring amplitudes are obtained to be explicitly finite and modular invariant.
A ten-dimensional super-Poincare covariant formalism for the superstring was recently developed which involves a BRST operator constructed from superspace matter variables and a pure spinor ghost variable. A super-Poincare covariant prescription was defined for computing tree amplitudes and was shown to coincide with the standard RNS prescription. In this paper, picture-changing operators are used to define functional integration over the pure spinor ghosts and to construct a suitable $b$ ghost. A super-Poincare covariant prescription is then given for the computation of N-point multiloop ampl…
We extend spacetime duality to superspace, including fermions in the low-energy limits of superstrings. The tangent space is a curved, extended superspace. The geometry is based on an enlarged coordinate space where the vanishing of the d'Alembertian is as fundamental as the vanishing of the curl of a gradient.
We use a previously derived integral representation for the four graviton amplitude at two loops in Superstring theory, whose leading term for vanishing momenta gives the two-loop contribution to the R^4 term in the Effective Action. We find by an explicit computation that this contribution is zero, in agreement with a general argument implying the vanishing of the R^4 term beyond one loop.
At the tree level, the maximally helicity violating amplitudes of N gauge bosons in open superstring theory and of N gravitons in supergravity are known to have simple representations in terms of tree graphs. For superstrings, the graphs encode integral representations of certain generalized Gaussian hypergeometric functions of kinematic invariants while for supergravity, they represent specific kinematic expressions constructed from spinor-helicity variables. We establish a superstring/supergravity correspondence for this class of amplitudes, by constructing a mapping between the positions of…
We extend the Berkovits-Maldacena prescription for MHV amplitudes of the open superstring to the closed superstring, showing that in the α'=0 limit it reduces to the result of supergravity found recently by Hodges. We also verify that this prescription calculates the correct superstring tree level MHV amplitude for 4 gravitons including α' corrections.
The super-Poincare covariant formalism for the superstring is used to compute massless four-point two-loop amplitudes in ten-dimensional superspace. The computations are much simpler than in the RNS formalism and include both external bosons and fermions.
The hybrid formalism for the superstring is used to compute one-loop amplitudes with an arbitrary number of external d=4 supergravity states. These one-loop N-point amplitudes are expressed as Koba-Nielsen-like formulas with manifest d=4 supersymmetry.
Based on the recent developments of explicit computations at 2 loops in superstring theory in the covariant RNS formalism, we propose an explicit formula for the arbitrary loop 4-particle amplitude in superstring theory. We prove that this formula passes two very difficult tests: modular invariance and factorization. If proved, this shows that superstring theory is not only finite order by order in perturbation theory but is also exceptionally simple.
An open superstring field theory action has been proposed which does not suffer from contact term divergences. In this paper, we compute the on-shell four-point tree amplitude from this action using the Giddings map. After including contributions from the quartic term in the action, the resulting amplitude agrees with the first-quantized prescription.
We compute one-loop threshold corrections to R^4 terms in N= 5,6 supergravity vacua of Type II superstrings. We then discuss non-perturbative corrections generated by asymmetric D-brane instantons. Finally we derive generating functions for MHV amplitudes at tree level in N = 5, 6 supergravities.
It is shown that the four-particle amplitude of superstring theory at two loops obtained in [1,2] is equivalent to the previously obtained results in [3,4,5]. Here the ${\bf Z}_2$ symmetry in hyperelliptic Riemann surface plays an important role in the proof.
The pure spinor formalism for the superstring has recently been used to compute massless four-point two-loop amplitudes in a manifestly super-Poincare covariant manner. In this paper, we show that when all four external states are Neveu-Schwarz, the two-loop amplitude coincides with the RNS result.
The discovery of pure spinor formalism makes the computation of superstring scattering amplitudes possible. In this paper, we will illustrate how computer algebra system Cadabra is used in computing the supersymmetric amplitude in pure spinor formalism and provide the source code that computes the tree-level massless 5-gluon amplitude.
The Type II Superstring amplitude to 1-loop order is given by an integral of $\vartheta$-functions over the moduli space of tori, which diverges for real momenta. We construct the analytic continuation which renders this amplitude well defined and finite, and we find the expected poles and cuts in the complex momentum plane.
In these proceedings, the multiloop amplitude prescription using the super-Poincare invariant pure spinor formalism for the superstring is reviewed. Unlike the RNS prescription, there is no sum over spin structures and surface terms coming from the boundary of moduli space can be ignored. Massless N-point multiloop amplitudes vanish for N<4, which implies (with two mild assumptions) the perturbative finiteness of superstring theory. And R^4 terms receive no multiloop corrections in agreement with the Type IIB S-duality conjecture of Green and Gutperle.
We study in detail the factorization of the newly obtained two-loop four-particle amplitude in superstring theory. In particular some missing factors from the scalar correlators are obtained correctly, in comparing with a previous study of the factorization in two-loop superstring theory. Some details for the calculation of the factorization of the kinematic factor are also presented.
Multi-loop interaction amplitudes in the theory of the closed, oriented superstrings are obtained by the integration of local amplitudes which are represented by a sum of the spinning string local amplitudes. The last local amplitudes are given explicitly through super-Schottky group parameters and interaction vertex coordinates on the $(1|1)$ complex supermanifold. The integration is ambiguous under those replacements of the integration variables which admix Grassmann variables to the boson ones. So the calculation is guided by a preservation of local symmetries of the superstring. The obtain…
The unitarity equations for the boson interaction amplitudes in the superstring theory are used to calculate the interaction amplitudes including the Ramond states, which are 10-fermion and Ramond bosons. The n-loop, 4-point amplitude with two massless Neveu-Schwarz bosons and two massless Ramond states is obtained explicitly. It is shown that, in addition, the unitarity equations require some integral relations for local functions determining the amplitude. For the tree amplitude the validness of the above integral relations is verified.
The spin-4/3 fractional superstring is characterized by a chiral algebra involving a spin-4/3 current on the world-sheet in addition to the energy-momentum tensor. These currents generate physical state conditions on the fractional superstring Fock space. Scattering amplitudes of these physical states are described which satisfy both spurious state decoupling and cyclic symmetry (duality). Examples of such amplitudes are calculated using an explicit $c=5$ realization of the spin-4/3 current algebra. This representation has three flat coordinate boson fields and a global SO(2,1) Lorentz symmetr…
We revisit the tree-level closed superstring amplitude and identify its alpha'-expansion as series with single-valued multiple zeta values as coefficients. The latter represent a subclass of multiple zeta values originating from single-valued multiple polylogarithms at unity. Moreover, the alpha'-expansion of the closed superstring amplitude can be cast into the same algebraic form as the open superstring amplitude: the closed superstring amplitude essentially is the single-valued version of the open superstring amplitude. This fact points into a deeper connection between gauge and gravity amp…
The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states.
We study the annulus amplitudes in the (2,4) minimal superstring theory using the continuum worldsheet approach. Our results reproduce the semiclassical behavior of the wavefunctions of FZZT-branes recently studied in hep-th/0412315 using the dual matrix model. We also study the multi-point functions of neutral FZZT-branes and find the agreement between their semiclassical limit and the worldsheet annulus calculation.
Feynman amplitudes of light-cone gauge string field theory for Type II superstrings are shown to be equivalent to those of the covariant first quantized formulation. In order to regularize the contact term divergences, we consider the theory in a linear dilaton background $Φ_{\mathrm{dilaton}}=-iQX^{1}$. We show that the scattering amplitudes are correctly reproduced in the limit $Q\to0$, even with Ramond sector external lines.
We study the tree level scattering or emission of n closed superstrings from a decaying non-BPS brane in Type II superstring theory. We attempt to calculate generic n-point superstring disk amplitudes in the rolling tachyon background. We show that these can be written as infinite power series of Toeplitz determinants, related to expectation values of a periodic function in Circular Unitary Ensembles. Further analytical progress is possible in the special case of bulk-boundary disk amplitudes. These are interpreted as probability amplitudes for emission of a closed string with initial conditio…
In this work we report on recent progress in the calculation of open superstring scattering amplitudes, at tree level, with more than four external massless states. We also report on the corresponding terms in the low energy effective lagrangian.
Multiloop superstring amplitudes are calculated in the explicit form by the solution of Ward identities. A naive generalization of Belavin-Knizhnik theorem to the superstring is found to be incorrect since the period matrix turns out to be depended on the spinor structure over the terms proportional to odd moduli. These terms appear because fermions mix bosons under the two-dim. supersymmetry transformations. The closed, oriented superstring turns out to be finite, if it possesses the ten-dimensional supersymmetry, as well as the two-dimentional one. This problem needs a further study.
Using pure spinors, the superstring was recently quantized in a manifestly ten-dimensional super-Poincaré covariant manner and a covariant prescription was given for tree-level scattering amplitudes. In this paper, we prove that this prescription is cyclically symmetric and, for the scattering of an arbitrary number of massless bosons and up to four massless fermions, it agrees with the standard Ramond-Neveu-Schwarz prescription.
We derive the complete five-gluon scattering amplitude at tree level, within the context of Open Superstring theory. We find the general expression in terms of kinematic factors, and also find its complete expansion up to ${\cal O}({α'}^3)$ terms. We use our scattering amplitude to test three non-equivalent ${\cal O}({α'}^3)$ effective lagrangians that have recently been matter of some controversy.
We evaluate all next-to-maximal helicity violating (NMHV) six-gluon amplitudes in type I open superstring theory in four dimensions, at the disk level, to all orders in alpha'. Although the computation utilizes supersymmetric Ward identities, the result holds for all compactifications, even for those that break supersymmetry and is completely model-independent. Together with the maximally helicity violating (MHV) amplitudes presented in the previous work, our results provide the complete six-gluon disk amplitude.
In two recent papers, a new method was developed for calculating ten-dimensional superstring amplitudes with an arbitrary number of loops and external massless particles, and for expressing them in manifestly Lorentz-invariant form. By explicitly checking for divergences when the Riemann surface degenerates, these amplitudes are proven to be finite. By choosing light-cone moduli for the surface and comparing with the light-cone Green-Schwarz formalism, these amplitudes are proven to be unitary.
We discuss supersymmetric Ward identities relating various scattering amplitudes in type I open superstring theory. We show that at the disk level, the form of such relations remains exactly the same, to all orders in alpha', as in the low-energy effective field theory describing the alpha'-> 0 limit. This result holds in D=4 for all compactifications, even for those that break supersymmetry. We apply SUSY relations to the computations of N-gluon MHV superstring amplitudes, simplifying the existing results for N<7 and deriving a compact expression for N=7.
We calculate the chiral string amplitude in pure spinor formalism and take four point amplitude as an example. The method could be easily generalized to $N$ point amplitude by complicated calculations. By doing the usual calculations of string theory first and using a special singular gauge limit, we produce the amplitude with the integral over Dirac $δ$-functions. The Bosonic part of the amplitude matches the CHY amplitude and the Fermionic part gives us the supersymmetric generalization of CHY amplitude. Finally, we also check the dependence on boundary condition for heterotic chiral string …
In type II superstring theory, the vacuum amplitude at a given loop order $g$ can receive contributions from the boundary of the compactified, genus $g$ supermoduli space of curves $\overline{\mathfrak M}_g$. These contributions capture the long distance or infrared behaviour of the amplitude. The boundary parametrises degenerations of genus $g$ super Riemann surfaces. A holomorphic projection of the supermoduli space onto its reduced space would then provide a way to integrate the holomorphic, superstring measure and thereby give the superstring vacuum amplitude at $g$-loop order. However, su…
Using the pure spinor formalism, a particular superstring scattering amplitude involving one closed string and $N$ open string vertex operators in $\mathrm{AdS}{}_5 \times \mathrm{S}^5$ is studied. It is shown that the tree-level amplitude containing one supergravity state and $N$ super-Yang-Mills states located on D3-branes near the AdS${}_5$ boundary can be expressed as a $d=4$ ${\cal N}=4$ harmonic superspace integral in terms of the supergravity and super-Yang-Mills superfields.
We study the odd spin structure contributions to the multiloop amplitudes of light-cone gauge superstring field theory. We show that they coincide with the amplitudes in the conformal gauge with two of the vertex operators chosen to be in the pictures different from the standard choice, namely (-1,-1) picture in the type II case and -1 picture in the heterotic case. We also show that the contact term divergences can be regularized in the same way as in the amplitudes for the even structures and we get the amplitudes which coincide with those obtained from the first-quantized approach.
We use the non-minimal pure spinor formalism to compute in a super-Poincare covariant manner the four-point massless one and two-loop open superstring amplitudes, and the gauge anomaly of the six-point one-loop amplitude. All of these amplitudes are expressed as integrals of ten-dimensional superfields in a "pure spinor superspace" which involves five $θ$ coordinates covariantly contracted with three pure spinors. The bosonic contribution to these amplitudes agrees with the standard results, and we demonstrate identities which show how the $t_8$ and $ε_{10}$ tensors naturally emerge from integ…
Very recently, Grushevsky continued D'Hoker and Phong's program of finding the chiral superstring measure from first principles by constructing modular forms satisfying certain factorization constraints. He has proposed an ansatz in genus 4 and conjectured a possible formula for the superstring measure in any genus, subject to the condition that certain modular forms admit holomorphic roots. In this note we want to give some evidence that Grushevsky's approach seems to be very fruitful.
We compute explicitly the four-particle amplitude in superstring theories by using the hyperelliptic language and the newly obtained chiral measure of D'Hoker and Phong. Although the algebra of the intermediate steps is a little bit involved, we obtain a quite simple expression for the four-particle amplitude. As expected, the integrand is independent of all the insertion points. As an application of the obtained result, we show that the perturbative correction to the $R^4$ term in type II superstring theories is vanishing point-wise in (even) moduli space at two loops.
We compute the massless five-point amplitude of open superstrings using the non-minimal pure spinor formalism and obtain a simple kinematic factor in pure spinor superspace, which can be viewed as the natural extension of the kinematic factor of the massless four-point amplitude. It encodes bosonic and fermionic external states in supersymmetric form and reduces to existing bosonic amplitudes when expanded in components, therefore proving their equivalence. We also show how to compute the kinematic structures involving fermionic states.
On-shell superspace techniques are used to quantify R-symmetry violation in type IIB superstring theory amplitudes in a flat background in ten dimensions. This shows the existence of a particularly simple class of non-vanishing amplitudes in this theory which violate R-symmetry maximally. General properties of the class and some of its extensions are established which at string tree level are shown to determine the first three non-trivial effective field theory contributions to all multiplicity. This leads to a natural conjecture for the exact analytic part of the first two of these.
We consider how to normalize the scattering amplitudes of 4D heterotic superstrings in a Minkowski background. We fix the normalization of the vacuum amplitude (the string partition function) at each genus, and of every vertex operator describing a physical external string state in a way consistent with unitarity of the $S$-matrix. We also provide an explicit expression for the map relating the vertex operator of an incoming physical state to the vertex operator describing the same physical state, but outgoing. This map is related to hermitean conjugation and to the hermiticity properties of t…
In this paper the pure spinor formalism is used to obtain a compact expression for the superstring N-point disk amplitude. The color ordered string amplitude is given by a sum over (N-3)! super Yang-Mills subamplitudes multiplied by multiple Gaussian hypergeometric functions. In order to obtain this result, the cohomology structure of the pure spinor superspace is exploited to generalize the Berends-Giele method of computing super Yang-Mills amplitudes. The method was briefly presented in [1], and this paper elaborates on the details and contains higher-rank examples of building blocks and ass…
Using the non-minimal version of the pure spinor formalism, manifestly super-Poincare covariant superstring scattering amplitudes can be computed as in topological string theory without the need of picture-changing operators. The only subtlety comes from regularizing the functional integral over the pure spinor ghosts. In this paper, it is shown how to regularize this functional integral in a BRST-invariant manner, allowing the computation of arbitrary multiloop amplitudes. The regularization method simplifies for scattering amplitudes which contribute to ten-dimensional F-terms, i.e. terms in…
Widely spread cruel misconceptions and mistakes in the calculation of multi-loop superstring amplitudes are exposed. Correct calculations are given. It is shown that the cardinal mistake in the gauge fixing procedure presents ab ovo in the Verlinde papers. The mistake was reproduced in following proposals including the recent papers. The modular symmetry of the multi-loop superstring amplitudes is clarified, an incorrectness of previous conjectures being shown. It is shown that the Berezin-type integral versus boson and fermion moduli is doubt under non-split transformations mixing fermion int…
We derive fully covariant expressions for all two-point scattering amplitudes of two massless closed strings from a Dirichlet $p$-brane. This construction relies on the observation that there is a simple relation between these D-brane amplitudes in type II superstring theory and four-point scattering amplitudes for type I open superstrings. From the two-point amplitudes, we derive the long range background fields for the D-branes, and verify that as expected they correspond to those of extremally charged $p$-brane solutions of the low energy effective action.
We propose a procedure to determine the moduli-space integrands of loop-level superstring amplitudes for massless external states in terms of the field theory limit. We focus on the type II superstring. The procedure is to: (i) take a supergravity loop integrand written in a BCJ double-copy representation, (ii) use the loop-level scattering equations to translate that integrand into the ambitwistor string moduli-space integrand, localised on the nodal Riemann sphere, and (iii) uplift that formula to one on the higher-genus surface valid for the superstring, guided by modular invariance. We sho…
We reexamine genus one super-Green functions with general boundary conditions twisted by $(α, β)$ for $(σ, τ)$ directions in the eigenmode expansion and derive expressions as infinite series of hypergeometric functions. Using these, we compute one-loop superstring amplitudes with non-maximal supersymmetry, taking an example of massless vector emissions of open string type I ${\cal Z}_2$ orbifold.
