Pure Spinor Research Articles

On String Integrability: A Journey through the Two‐Dimensional Hidden Symmetries in the AdS/CFT Dualities

One of the main topics in the modern String Theory are the AdS/CFT dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, that is, the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality. We review the fundamental concepts and properties of integrability in two‐dimensional σ‐models and in the AdS/CFT context. The first part is focused on the AdS5/CFT4 duality, especially the classical and quantum integrability of the type IIB superstring on AdS5 × S5 which is discussed in both pure spinor and Green‐Schwarz formulations. The second part is dedicated to the AdS4/CFT3 duality with particular attention to the type IIA superstring on AdS4 × ℂP3 and its integrability. This review is based on the author′s PhD thesis discussed at Uppsala University the 21st September 2009.

Covariant representation theory of the Poincaré algebra and some of its extensions

There has been substantial calculational progress in the last few years for gauge theory amplitudes which involve massless four dimensional particles. One of the central ingredients in this has been the ability to keep precise track of the Poincaré algebra quantum numbers of the particles involved. Technically, this is most easily done using the well-known four dimensional spinor helicity method. In this article a natural generalization to all dimensions higher than four is obtained based on a covariant version of the representation theory of the Poincaré algebra. Covariant expressions for all possible polarization states, both bosonic and fermionic, are constructed. For the fermionic states the analysis leads directly to pure spinors. The natural extension to the representation theory of the on-shell supersymmetry algebra results in an elementary derivation of the supersymmetry Ward identities for scattering amplitudes with massless or massive legs in any integer dimension from four onwards. As a proof-of-concept application a higher dimensional analog of the vanishing helicity-equal amplitudes in four dimensions is presented in (super) Yang-Mills theory, Einstein (super-)gravity and superstring theory in a flat background.

Decoupling of unphysical states in the minimal pure spinor formalism I

This is the first of a series of two papers where decoupling of unphysical states in the minimal pure spinor formalism is investigated. The multi-loop amplitude prescription for the minimal pure spinor superstring formulated in hep-th/0406055 involves the insertion of picture changing operators in the path integral. These operators are BRST closed in a distributional sense and depend on a number of constant tensors. One can trace the origin of these insertions to gauge fixing, so the amplitudes are formally independent of the constant tensors. We show however by explicit tree-level and one-loop computations that the picture changing operators are not BRST closed inside correlators and the amplitudes do depend on these constant tensors. This is due to the fact that the gauge fixing condition implicit in the existing minimal amplitude prescription is singular and this can lead to Lorentz violation and non-decoupling of BRST exact states. As discussed in hep-th/0406055, a manifestly Lorentz invariant prescription can be obtained by integrating over the constant tensors and in the sequel to this paper, it is shown that when one includes these integrations unphysical states do decouple to all orders despite the fact that the PCO's are not BRST closed inside correlators.

Pure spinor superfields, with application toD= 3 conformal models

Pure spinor superfields, with application to<i>D</i>= 3 conformal models

Simplifying the tree-level superstring massless five-point amplitude

We use the pure spinor formalism to obtain the supersymmetric massless five-point amplitude at tree-level in a streamlined fashion. We also prove the equivalence of an OPE identity in string theory with a subset of the Bern-Carrasco-Johansson five-point kinematic relations, and demonstrate how the remaining BCJ identities follow from the different integration regions over the open string world-sheet, therefore providing a first principles derivation of the (supersymmetric) BCJ identities.

Towards a manifestly supersymmetric action for 11-dimensional supergravity

We investigate the possibility of writing a manifestly supersymmetric action for 11-dimensional supergravity. The construction involves an explicit relation between the fields in the super-vielbein and the super-3-form, and uses non-minimal pure spinors. A simple cubic interaction term for a single scalar superfield is found.

NEW BRST CHARGES IN RNS SUPERSTRING THEORY AND DEFORMED PURE SPINORS

We show that new BRST charges in RNS superstring theory with nonstandard ghost numbers, constructed in our recent work, can be mapped to deformed pure spinor (PS) superstring theories, with the nilpotent pure spinor BRST charge QPS = ∮λαdα still retaining its form but with singular operator products between commuting spinor variables λα. Despite the OPE singularities, the pure spinor condition λγmλ = 0 is still fulfilled in a weak sense, explained in the paper. The operator product singularities correspond to introducing interactions between the pure spinors. We conjecture that the leading singularity orders of the OPE between two interacting pure spinors is related to the ghost number of the corresponding BRST operator in RNS formalism. Namely, it is conjectured that the BRST operators of minimal superconformal ghost pictures n &gt; 0 can be mapped to nilpotent BRST operators in the deformed pure spinor formalism with the OPE of two commuting spinors having a leading singularity order λ(z)λ(w) ~ O(z-w)-2(n2+6n+1). The conjecture is checked explicitly for the first nontrivial case n = 1.

One-loop superstring amplitude from integrals on pure spinors space

In the Type II superstring the 4-point function for massless NS-NS bosons at one-loop is well known [1][14]. The overall constant factor in this amplitude is very important because it needs to satisfy the unitarity and S-duality conditions [14]. This coefficient has not been computed in the pure spinor formalism due to the difficulty to solve the integrals on the pure spinors space. In this paper we compute it by using the non-minimal pure spinor formalism and we will show that the answer is in perfect agreement with the one given in [14].

Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory

The constraints imposed by maximal supersymmetry on multi-loop contributions to the scattering of four open superstrings in the U(N) theory are examined by use of the pure spinor formalism. The double-trace term k2 t8(tr F2)2 (where k represents an external momentum and F the Yang-Mills field strength) only receives contributions from L ⩽ 2 (where L is the loop number) while the single-trace term k2 t8(tr F4) receives contributions from all L. These statements are verified up to L = 5, but arguments based on supersymmetry suggest they extend to all L. This explains why the single-trace contributions to low energy maximally supersymmetric Yang-Mills field theory are more divergent in the ultraviolet than the double-trace contributions. We also comment further on the constraints on closed string amplitudes and their implications for ultraviolet divergences in = 8 supergravity.

Domain wall flow equations and [formula omitted] structure compactifications

We study supersymmetric domain wall solutions in four dimensions arising from the compactification of type II supergravity on an SU ( 3 ) × SU ( 3 ) structure manifold. Using a pure spinor approach, we show that the supersymmetry variations can be reinterpreted as a generalisation of the Hitchin flow equations and describe the embedding of an SU ( 3 ) × SU ( 3 ) structure manifold into a G 2 × G 2 structure manifold. We find a precise agreement between the four- and ten-dimensional supergravity results. The flow equations derived here should have applications in constructing the gravity duals of Chern–Simons-matter conformal field theories.

Regularizing cubic open Neveu-Schwarz string field theory

After introducing non-minimal variables, the midpoint insertion of Y\bar Y in cubic open Neveu-Schwarz string field theory can be replaced with an operator N_\rho depending on a constant parameter \rho. As in cubic open superstring field theory using the pure spinor formalism, the operator N_\rho is invertible and is equal to 1 up to a BRST-trivial quantity. So unlike the linearized equation of motion Y\bar Y QV=0 which requires truncation of the Hilbert space in order to imply QV=0, the linearized equation N_\rho QV=0 directly implies QV=0.

Simplifying and extending theAdS5×S5pure spinor formalism

Although the AdS5 × S5 worldsheet action is not quadratic, some features of the pure spinor formalism are simpler in an AdS5 × S5 background than in a flat background. The BRST operator acts geometrically, the left and right-moving pure spinor ghosts can be treated as complex conjugates, the zero mode measure factor is trivial, and the b ghost does not require non-minimal fields. Furthermore, a topological version of the AdS5 × S5 action with the same worldsheet variables and BRST operator can be constructed by gauge-fixing a G/G principal chiral model where G = PSU(2,2|4). This topological model is argued to describe the zero radius limit that is dual to free = 4 super-Yang-Mills and can also be interpreted as an ``unbroken phase'' of superstring theory.

Decoupling of unphysical states in the minimal pure spinor formalism II

This is the first of a series of two papers where decoupling of unphysical states in the minimal pure spinor formalism is investigated. The multi-loop amplitude prescription for the minimal pure spinor superstring formulated in hep-th/0406055 involves the insertion of picture changing operators in the path integral. These operators are BRST closed in a distributional sense and depend on a number of constant tensors. One can trace the origin of these insertions to gauge fixing, so the amplitudes are formally independent of the constant tensors. We show however by explicit tree-level and one-loop computations that the picture changing operators are not BRST closed inside correlators and the amplitudes do depend on these constant tensors. This is due to the fact that the gauge fixing condition implicit in the existing minimal amplitude prescription is singular and this can lead to Lorentz violation and non-decoupling of BRST exact states. As discussed in hep-th/0406055, a manifestly Lorentz invariant prescription can be obtained by integrating over the constant tensors and in the sequel to this paper, it is shown that when one includes these integrations unphysical states do decouple to all orders despite the fact that the PCO's are not BRST closed inside correlators.

E7(7)formulation ofN= 2 backgrounds

In this paper we reformulate N = 2 supergravity backgrounds arising in type II string theory in terms of quantities transforming under the U-duality group E7(7). In particular we combine the Ramond-Ramond scalar degrees of freedom together with the O(6,6) pure spinors which govern the Neveu-Schwarz sector by considering an extended version of generalised geometry. We give E7(7)-invariant expressions for the Kähler and hyperkähler potentials describing the moduli space of vector and hypermultiplets, demonstrating that both correspond to standard E7(7) coset spaces. We also find E7(7) expressions for the Killing prepotentials defining the scalar potential, and discuss the equations governing N = 1 vacua in this formalism.

Pure spinor vertex operators in Siegel gauge and loop amplitude regularization

Since the b ghost in the pure spinor formalism is a composite operator depending on non-minimal variables, it is not trivial to impose the Siegel gauge condition b0V = 0 on BRST-invariant vertex operators. Using the antifield vertex operator V* of ghost-number +2, we show that Siegel gauge unintegrated vertex operators can be constructed as b0V* and Siegel gauge integrated vertex operators as ∫ dz b−1b0V*. These Siegel gauge vertex operators depend on the non-minimal variables, so scattering amplitudes involving these operators need to be regularized using the prescription developed previously with Nekrasov. As an example of this regularization prescription, we compute the four-point one-loop amplitude with four Siegel gauge integrated vertex operators. This is the first one-loop computation in the pure spinor formalism that does not require unintegrated vertex operators.

Universal BPS structure of stationary supergravity solutions

We study asymptotically flat stationary solutions of four-dimensional supergravity theories via the associated G/H ∗ pseudo-Riemannian non-linear sigma models in three spatial dimensions. The Noether charge C associated to G is shown to satisfy a characteristic equation that determines it as a function of the four-dimensional conserved charges. The matrix C is nilpotent for non-rotating extremal solutions. The nilpotency degree of C is directly related to the BPS degree of the corresponding solution when they are BPS. Equivalently, the charges can be described in terms of a Weyl spinor |C i of Spin ∗ (2N), and then the characteristic equation becomes equivalent to a generalisation of the Cartan pure spinor constraint on |C i . The invariance of a given solution with respect to supersymmetry is determined by an algebraic ‘Dirac equation’ on the Weyl spinor |C i . We explicitly solve this equation for all pure supergravity theories and we characterise the stratified structure of the moduli space of asymptotically Taub–NUT black holes with respect to their BPS degree. The analysis is valid for any asymptotically flat stationary solutions for which the singularities are protected by horizons. The H ∗ -orbits of extremal solutions are identified as Lagrangian submanifolds of nilpotent orbits of G, and so the moduli space of extremal spherically symmetric black holes is identified as a Lagrangian subvariety of the variety of nilpotent elements of g. We also generalise the notion of active duality transformations to an ‘almost action’ of the three-dimensional duality group G on asymptotically flat stationary solutions.

(NON)TRIVIALITY OF PURE SPINORS AND EXACT PURE SPINOR–RNS MAP

All the BRST-invariant operators in pure spinor formalism in d = 10 can be represented as BRST commutators, such as [Formula: see text] where λ+ is the U(5) component of the pure spinor transforming as [Formula: see text]. Therefore, in order to secure nontriviality of BRST cohomology in pure spinor string theory, one has to introduce "small Hilbert space" and "small operator algebra" for pure spinors, analogous to those existing in RNS formalism. As any invariant vertex operator in RNS string theory can also represented as a commutator V = {Q brst , LV} where L = -4c∂ξξe-2ϕ, we show that mapping [Formula: see text] to L leads to identification of the pure spinor variable λα in terms of RNS variables without any additional nonminimal fields. We construct the RNS operator satisfying all the properties of λα and show that the pure spinor BRST operator ∮λαdα is mapped (up to similarity transformation) to the BRST operator of RNS theory under such a construction.

Higher-loop amplitudes in the non-minimal pure spinor formalism

We analyze the properties of the non-minimal pure spinor formalism. We show that Siegel gauge on massless vertex operators implies the primary field constraint and the level-matching condition in closed string theory by reconstructing the integrated vertex operator representation from the unintegrated ones. The pure spinor integration in the non-minimal formalism needs a regularisation. To this end we introduce a new regulator for the pure spinor integration and an extension of the regulator to allow for the saturation of the fermionic d-zero modes to all orders in perturbation. We conclude with a preliminary analysis of the properties of the four-graviton amplitude to all genus order.

A note on T-dualities in the pure spinor heterotic string

In this note we study the preservation of the classical pure spinor BRST constraints under super T-duality transformations. We also determine the invariance of the one-loop conformal invariance and of the local gauge and Lorentz anomalies under the super T-dualities.

T-duality, generalized geometry and non-geometric backgrounds

We discuss the action of O(d,d), and in particular T-duality, in the context of generalized geometry, focusing on the description of so-called non-geometric backgrounds. We derive local expressions for the pure spinors descibing the generalized geometry dual to an SU(3) structure background, and show that the equations for N=1 vacua are invariant under T-duality. We also propose a local generalized geometrical definition of the charges f, H, Q and R appearing in effective four-dimensional theories, using the Courant bracket. We then address certain global aspects, in particular whether the local non-geometric charges can be gauged away in, for instance, backgrounds admitting a torus action, as well as the structure of generalized parallelizable backgrounds.