Pure Spinor Research Articles

Complete N-point superstring disk amplitude I. Pure spinor computation

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 Mafra et al. (2011) [1], and this paper elaborates on the details and contains higher-rank examples of building blocks and associated cohomology objects. But the main achievement of this work is to identify these field-theory structures in the pure spinor computation of the superstring amplitude. In particular, the associated set of basis worldsheet integrals is constructively obtained here and thoroughly investigated together with the structure and properties of the amplitude in Mafra et al. (2011) [2], arXiv:1106.2646 [hep-th].

Nilpotency of the b ghost in the non-minimal pure spinor formalism

The b ghost in the non minimal pure spinor formalism is not a fundamental field. It is based on a complicated chain of operators and proving its nilpotency is nontrivial. Chandia proved this property in arXiv:1008.1778, but with an assumption on the non-minimal variables that is not valid in general. In this work, the b ghost is demonstrated to be nilpotent without this assumption.

Complete N-point superstring disk amplitude II. Amplitude and hypergeometric function structure

Using the pure spinor formalism in part I (Mafra et al., preprint [1]) we compute the complete tree-level amplitude of N massless open strings and find a striking simple and compact form in terms of minimal building blocks: the full N-point amplitude is expressed by a sum over (N−3)! Yang–Mills partial subamplitudes each multiplying a multiple Gaussian hypergeometric function. While the former capture the space–time kinematics of the amplitude the latter encode the string effects. This result disguises a lot of structure linking aspects of gauge amplitudes as color and kinematics with properties of generalized Euler integrals. In this part II the structure of the multiple hypergeometric functions is analyzed in detail: their relations to monodromy equations, their minimal basis structure, and methods to determine their poles and transcendentality properties are proposed. Finally, a Gröbner basis analysis provides independent sets of rational functions in the Euler integrals.

Harmonic superspace from the AdS 5 × S 5 pure spinor formalism

On-shell supergravity vertex operators in an $AdS_5\times S^5$ background are described in the pure spinor formalism by the zero mode cohomology of a BRST operator. After expanding the pure spinor BRST operator in terms of the $AdS_5$ radius variable, this cohomology is computed using N=4 harmonic superspace variables and explicit superfield expressions are obtained for the behavior of supergravity vertex operators near the boundary of $AdS_5$.

Loop amplitudes in maximal supergravity with manifest supersymmetry

We present a description for amplitude diagrams in maximal supergravities obtained by dimensional reduction from D=11, derived from a field theory point of view using the pure spinor formalism. The advantages of this approach are the manifest supersymmetry present in the formalism, and the limited number of interaction terms in the action. Furthermore, we investigate the conditions set by this description in order for amplitudes in maximal supergravity to be finite in the ultraviolet limit. Typically, there is an upper limit to the dimension, set by the loop order, which for an arbitrary number of loops is no larger than two. In four dimensions, the non-renormalisation power of the formalism fails for the 7-loop contribution to the 4-point amplitude, all of which is in clear agreement with previous work.

Semiclassical equivalence of Green–Schwarz and pure–spinor/hybrid formulations of superstrings in AdS5 × S5 and AdS2 × S2 × T6

We demonstrate the equivalence between the worldsheet one-loop partition functions computed near classical string solutions in the Green–Schwarz and in the pure–spinor formulations of superstrings in AdS5 × S5. While their bosonic sectors are the same in the conformal gauge, their fermionic sectors superficially appear to be very different (first versus second-derivative kinetic terms, presence versus absence of fermionic gauge symmetry). Still, we show that the quadratic fluctuation spectrum of 16 fermionic modes of the pure–spinor formulation is the same as in the Green–Schwarz superstring and the contribution of the extra ‘massless’ fermionic modes cancels against that of the pure–spinor ghosts. We also provide evidence for a similar semiclassical equivalence between the Green–Schwarz and the hybrid formulations of superstrings in AdS2 × S2 × T6 by studying several particular examples of string solutions.

PURE SPINOR FORMALISM FOR ${\rm Osp}({\mathcal N}\vert 4)$ BACKGROUNDS

We start from the Maurer–Cartan (MC) equations of the [Formula: see text] superalgebras satisfied by the left-invariant superforms realized on supercoset manifolds of the corresponding supergroups and we derive some new pure spinor constraints. They are obtained by "ghostifying" the MC forms and extending the differential d to a BRST differential. From the superalgebras [Formula: see text] we single out different subalgebras [Formula: see text] associated with the different cosets [Formula: see text]: each choice of ℍ leads to a different weakening of the pure spinor constraints. In each case, the number of parameter is counted and we show that in the cases of Osp (6|4)/ U (3)× SO (1, 3), Osp (4|4)/ SO (3) × SO (1, 3) and finally Osp (4|4)/ U (2) × SO (1, 3) the bosonic and fermionic degrees of freedom match in order to provide a c = 0 superconformal field theory. We construct both the Green–Schwarz and the pure spinor sigma model for the case Osp (6|4)/ U(3) × SO (1, 3) corresponding to AdS 4 ×ℙ3. The pure spinor sigma model can be consistently quantized.

Pure spinor superstring in $ \mathrm{Ad}{{\mathrm{S}}_4}\times \mathbb{C}{{\mathrm{P}}^3} $ with unconstrained ghosts

We construct the action for the pure spinor superstring in the coset description of AdS_4 x CP^3 superspace, using the variables which solve the pure spinor condition. As a test of the consistency of the approach, we use the background field method to verify the absence of central charge at the second order in the expansion and to show the one-loop finiteness of the effective action.

Pure spinors, twistors, and emergent supersymmetry

Starting with a classical action whose matter variables are a d=10 spacetime vector $x^m$ and a pure spinor $\lambda^\alpha$, the pure spinor formalism for the superstring is obtained by gauge-fixing the twistor-like constraint $\partial x^m (\gamma_m \lambda)_\alpha =0$. The fermionic variables $\theta^\alpha$ are Faddeev-Popov ghosts coming from this gauge-fixing and replace the usual (b,c) ghosts coming from gauge-fixing the Virasoro constraint. After twisting the ghost-number such that $\theta^\alpha$ has ghost-number zero and $\lambda^\alpha$ has ghost-number one, the BRST cohomology describes the usual spacetime supersymmetric states of the superstring.

FERMIONIC T-DUALITY: A SNAPSHOT REVIEW

Through a self-dual mapping of the geometry AdS5 ×S5, fermionic T-duality provides a beautiful geometric interpretation of hidden symmetries for scattering amplitudes in [Formula: see text] super-Yang–Mills. Starting with Green–Schwarz sigma-models, we consolidate developments in this area into this small review. In particular, we discuss the translation of fermionic T-duality into the supergravity fields via pure spinor formalism and show that a general class of fermionic transformations can be identified directly in the supergravity. In addition to discussing fermionic T-duality for the geometry AdS4 × ℂ P 3, dual to [Formula: see text] ABJM theory, we review work on other self-dual geometries. Finally, we present a short round-up of studies with a formal interest in fermionic T-duality.

$ \mathcal{N} $ = 2 vacua in generalized geometry

Abstract We find the conditions on compactifications of type IIA to four-dimensional Minkowski space to preserve $ \mathcal{N} $ = 2 supersymmetry in the language of Exceptional Generalized Geometry (EGG) and Generalized Complex Geometry (GCG). In EGG, off-shell $ \mathcal{N} $ =2 supersymmetry requires the existence of a pair of SU(2)R singlet and triplet algebraic structures on the exceptional generalized tangent space that encode all the scalars (NS-NS and R-R) in vector and hypermultiplets respectively. We show that on shell $ \mathcal{N} $ = 2 requires, except for a single component, these structures to be closed under a derivative twisted by the NS-NS and R-R fluxes. We also derive the corresponding GCG-type equations for the two pairs of pure spinors that build up these structures.

Cornering the unphysical vertex

In the classical pure spinor worldsheet theory of AdS5xS5 there are some vertex operators which do not correspond to any physical excitations. We study their flat space limit. We find that the BRST operator of the worldsheet theory in flat space-time can be nontrivially deformed without deforming the worldsheet action. Some of these deformations describe the linear dilaton background. But the deformation corresponding to the nonphysical vertex differs from the linear dilaton in not being worldsheet parity even. The nonphysically deformed worldsheet theory has nonzero beta-function at one loop. This means that the classical Type IIB SUGRA backgrounds are not completely characterized by requiring the BRST symmetry of the classical worldsheet theory; it is also necessary to require the vanishing of the one-loop beta-function.

On semiclassical analysis of pure spinor superstring in an AdS 5 × S 5 background

Abstract Relation between semiclassical analyses of Green-Schwarz and pure spinor formalisms in an AdS 5 × S 5 background is clarified. It is shown that the two formalisms have identical semiclassical partition functions for a simple family of classical solutions. It is also shown that, when the classical string is furthermore rigid, this in turn implies that the two formalisms predict the same one-loop corrections to spacetime energies.

Conformal field theories in six-dimensional twistor space

This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space–time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the six-dimensional case in which twistor space is the 6-quadric Q in CP7 with a view to applications to the self-dual (0,2)-theory. We show how spinor-helicity momentum eigenstates have canonically defined distributional representatives on twistor space (a story that we extend to arbitrary dimension). These yield an elementary proof of the surjectivity of the Penrose transform. We give a direct construction of the twistor transform between the two different representations of massless fields on twistor space (H2 and H3) in which the H3s arise as obstructions to extending the H2s off Q into CP7.We also develop the theory of Sparling’s ‘Ξ-transform’, the analogous totally real split signature story based now on real integral geometry where cohomology no longer plays a role. We extend Sparling’s Ξ-transform to all helicities and homogeneities on twistor space and show that it maps kernels and cokernels of conformally invariant powers of the ultrahyperbolic wave operator on twistor space to conformally invariant massless fields on space–time. This is proved by developing the six-dimensional analogue of the half-Fourier transform between functions on twistor space and momentum space. We give a treatment of the elementary conformally invariant Φ3 amplitude on twistor space and finish with a discussion of conformal field theories in twistor space.

Fermionic coset, critical level $ \mathcal{W}_4^{{(2)}} $ -algebra and higher spins

The fermionic coset is a limit of the pure spinor formulation of the AdS5xS5 sigma model as well as a limit of a nonlinear topological A-model, introduced by Berkovits. We study the latter, especially its symmetries, and map them to higher spin algebras. We show the following. The linear A-model possesses affine $\AKMSA{pgl}{4}{4}_0$ symmetry at critical level and its $\AKMSA{psl}{4}{4}_0$ current-current perturbation is the nonlinear model. We find that the perturbation preserves $\mathcal{W}^{(2)}_4$-algebra symmetry at critical level. There is a topological algebra associated to $\AKMSA{pgl}{4}{4}_0$ with the properties that the perturbation is BRST-exact. Further, the BRST-cohomology contains world-sheet supersymmetric symplectic fermions and the non-trivial generators of the $\mathcal{W}^{(2)}_4$-algebra. The Zhu functor maps the linear model to a higher spin theory. We analyze its $\SLSA{psl}{4}{4}$ action and find finite dimensional short multiplets.

Generalized structures of ten-dimensional supersymmetric solutions

Four-dimensional supersymmetric type II string theory vacua can be described elegantly in terms of pure spinors on the generalized tangent bundle T+T*. In this paper, we apply the same techniques to any ten-dimensional supersymmetric solution (not necessarily involving a factor with an AdS4 or Minkowski4 metric) in type II theories. We find a system of differential equations in terms of a form describing a "generalized ISpin(7) structure". This system is equivalent to unbroken supersymmetry, in both IIA and IIB. One of the equations reproduces in one fell swoop all the pure spinors equations for four-dimensional vacua.

Notes on the overall coefficient of the two‐loop superstring amplitude using pure spinor

AbstractUsing the results recently obtained for computing integrals over (non‐minimal) pure spinor superspace, we compute the coefficient of the massless two‐loop four‐point amplitude from first principles. This proceeding is a very short summary on the paper [arXiv:1003.0678 [hep‐th]], which was done with the collaboration of Carlos R. Mafra.

The geometry of pure spinor space

We investigate the complex geometry of D=10 pure spinor space. The K\"ahler structure and the corresponding metric giving rise to the desired Calabi-Yau property are determined, and an explicit covariant expression for the Laplacian is given. The metric is not that of a cone obtained by embedding pure spinor space in a flat space of unconstrained spinors. Some directions for future studies, concerning regularisation and generalisation to eleven dimensions, are briefly discussed.

On projections to the pure spinor space

A family of covariant non-linear projections from the space of SO(10) Weyl spinors onto the space of pure SO(10) Weyl spinors is presented. The Jacobian matrices of these projections are related to a linear projector which was previously discussed in pure spinor string literature and which maps the antighost to its gauge invariant part. Only one representative of the family leads to a Hermitian Jacobian matrix and can itself be derived from a scalar potential. Comments on the SO(1,9) case are given as well as on the non-covariant version of the projection map. The insight is applied to the ghost action of pure spinor string theory, where the constraints on the fields can be removed using the projection, while introducing new gauge symmetries. This opens the possibility of choosing different gauges which might help to clarify the origin of the pure spinor ghosts. Also the measure of the pure spinor space is discussed from the projection point of view. The appendix contains the discussion of a toy model which served as a guideline for the pure spinor case.

The Konishi multiplet at strong coupling

We introduce a method to compute from first principles the anomalous dimension of short operators in N=4 super Yang-Mills theory at strong coupling, where they are described in terms of superstring vertex operators in an anti-de Sitter background. We focus on the Konishi multiplet, dual to the first massive level of the superstring, and compute the one-loop correction to its anomalous dimension at strong coupling, using the pure spinor formalism for the superstring.