Pure Spinor Research Articles

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.

Superstrings onAdS4×CP3from supergravity

We derive from a general formulation of pure spinor string theory on type IIA backgrounds the specific form of the action for the AdS_4 x P^3 background. We provide a complete geometrical characterization of the structure of the superfields involved in the action.

The one-loop open superstring massless five-point amplitude with the non-minimal pure spinor formalism

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.

Superstring perturbation theory

The state of superstring perturbation theory is reviewed with an emphasis on the state of the pure spinor superstring perturbation theory. We begin with a brief summary of the state of perturbation theory in the Ramond–Neveu–Schwarz and in the Green–Schwarz formulations of the superstring. Then we proceed to a quick review of the minimal and non-minimal pure spinor formulations of the superstring and discuss the multi-loop amplitude prescriptions in each of them. We end with a summary and open questions.

Type I supergravity effective action from pure spinor formalism

Using the pure spinor formalism, we compute the tree-level correlation functions for three strings, one closed and two open, in N=1 D=10 superspace. Expanding the superfields in components, the respective terms of the effective action for the type I supergravity are obtained. All terms found agree with the effective action known in the literature. This result gives one more consistency test for the pure spinor formalism.

The classical exchange algebra ofAdS5×S5string theory

The classical exchange algebra satisfied by the monodromy matrix of AdS5× S 5 string theory in the Green-Schwarz formulation is determined by using a first-order Hamiltonian formulation and by adding to the Bena-PolchinskiRoiban Lax connection terms proportional to constraints. This enables in particular to show that the conserved charges of this theory are in involution. This result is obtained for a general world-sheet metric. The same exchange algebra is obtained within the pure spinor description of AdS5 × S 5 string theory. These results are compared to the one obtained

Poisson structures and generalized Kähler submanifolds

Let X be a compact Kähler manifolds with a non-trivial holomorphic Poisson structure β . Then there exist deformations { ( J β t , ψ t ) } of non-trivial generalized Kähler structures with one pure spinor on X . We prove that every Poisson submanifold of X is a generalized Kähler submanifold with respect to ( J β t , ψ t ) and provide non-trivial examples of generalized Kähler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bihermitian structures constructed from Poisson structures.

Vertex Algebroids over Veronese Rings

We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.

Exploring pure spinor string theory onAdS4× ℂℙ3

In this paper we formulate the pure spinor superstring theory on AdS_4 x CP^3. By recasting the pure spinor action as a topological A-model on the fermionic supercoset Osp(6|4)/SO(6)xSp(4) plus a BRST exact term, we prove the exactness of the sigma-model. We then give a gauged linear sigma-model which reduces to the superstring in the limit of large volume and we study its branch geometry in different phases. Moreover, we discuss possible D-brane boundary conditions and the principal chiral model for the fermionic supercoset.

The pure spinor formulation of superstrings**These lectures have been given at the RTN Winter School on Strings, Supergravity and Gauge Theories, CERN (2008).

In this paper we outline the construction of pure spinor superstrings. We consider both the open and closed pure spinor superstrings in critical and noncritical dimensions and on flat and curved target spaces with RR flux. We exhibit the integrability properties of pure spinor superstrings on curved backgrounds with RR fluxes.

Superfield actions forN= 8 andN= 6 conformal theories in three dimensions

The manifestly supersymmetric pure spinor formulations of the Bagger-Lambert-Gustavsson models with N = 8 supersymmetry and the Aharony-Berg\-man-Jafferis-Maldacena models with N = 6 supersymmetry are given. The structures of the pure spinors are investigated in both cases, and non-degenerate measures are formed using non-minimal sets of variables, allowing for the formulation of an action principle.

N= 8 superfield formulation of the Bagger-Lambert-Gustavsson model

We reformulate the Bagger-Lambert-Gustavsson model using an N = 8 superspace, thus making the full supersymmetry manifest. The formulation is based on appropriate ``pure spinor wave functions'' for the Chern-Simons and matter multiplets. The Lagrangian has an extremely simple structure, essentially containing a Chern-Simons-like term for the gauge field wave function and a minimally coupled matter wave function. No higher order interactions than cubic are present. The consistency of the setup relies on an interplay between the algebraic structures of the 3-algebra and of the pure spinors.

Perturbative super-Yang-Mills from the topologicalAdS5×S5sigma model

A topological sigma model based on the pure spinor formalism was recently proposed for the small radius limit of the AdS_5xS^5 superstring. Physical states in this model can be constructed by connecting holes on the worldsheet with Wilson lines of the worldsheet gauge field. The contribution of these states to the topological amplitude is claimed to reproduce the usual Feynman diagram expansion of gauge-invariant super-Yang-Mills correlation functions.

Note about redefinition of the BRST operator for pure spinor string in a general background

We discuss an analysis presented by Berkovits in Ref 1 in the context of classical mechanics and perform its extension to the case of pure spinor string in a general background.

Aspects of quantum integrability for pure spinor superstring inAdS5×S5

We consider the monodromy matrix for the pure spinor IIB superstring on AdS5 × S5 at leading order at strong coupling, in particular its variation under an infinitesimal and continuous deformation of the contour. Such variation is equivalent to the insertion of a local operator. Demanding the BRST-closure for such an operator rules out its existence, implying that the monodromy matrix remains contour-independent at the first order in perturbation theory. Furthermore we explicitly compute the field strength corresponding to the flat connections up to leading order and directly check that it is free from logarithmic divergences. The absence of anomaly in the coordinate transformation of the monodromy matrix and the UV-finiteness of the curvature tensor finally imply the integrability of the pure spinor superstring at the first order.

Yang-Mills Chern-Simons corrections from the pure spinor superstring

Nilpotency of the pure spinor BRST operator in a curved background implies superspace equations of motion for the background. By computing one-loop corrections to nilpotency for the heterotic sigma model, the Yang-Mills Chern-Simons corrections to the background are derived.

New supersymmetric flux vacua with intermediate SU(2) structure

We find new supersymmetric four-dimensional Minkowski flux vacua of type II string theory on nilmanifolds and solvmanifolds. We extend the results of M. Grana, R. Minasian, M. Petrini, and A. Tomasiello to the case of intermediate SU(2) structures (the two internal supersymmetry parameters are neither parallel nor orthogonal). As pointed out recently by P. Koerber and D. Tsimpis, intermediate SU(2) structures are possible when one considers mixed orientifold projection conditions. To find our vacua, we rewrite these projection conditions in a more tractable way by introducing new variables: the projection basis. In these variables, the SUSY conditions become also much simpler to solve, and we find three new vacua. In addition, we find that these variables correspond to the SU(2) structure appearing with the dielectric pure spinors, objects introduced and discussed by R. Minasian, M. Petrini, A. Zaffaroni, and N. Halmagyi, A. Tomasiello, in the AdS/CFT context. Besides, our solutions provide some intuition on what a dynamical SU(3) x SU(3) structure solution could look like.

Pure spinor partition function and the massive superstring spectrum

We explicitly compute up to the fifth mass-level the partition function of ten-dimensional pure spinor worldsheet variables including the spin dependence. After adding the contribution from the (x^{\mu}, \theta^{\alpha}, p_{\alpha}) matter variables, we reproduce the massive superstring spectrum. Even though pure spinor variables are bosonic, the pure spinor partition function contains fermionic states which first appear at the second mass-level. These fermionic states come from functions which are not globally defined in pure spinor space, and are related to the b ghost in the pure spinor formalism. This result clarifies the proper definition of the Hilbert space for pure spinor variables.

Hilbert space of curved βγ systems on quadric cones

We clarify the structure of the Hilbert space of curved \beta\gamma systems defined by a quadratic constraint. The constraint is studied using intrinsic and BRST methods, and their partition functions are shown to agree. The quantum BRST cohomology is non-empty only at ghost numbers 0 and 1, and there is a one-to-one mapping between these two sectors. In the intrinsic description, the ghost number 1 operators correspond to the ones that are not globally defined on the constrained surface. Extension of the results to the pure spinor superstring is discussed in a separate work.

Pure Spinor Geometry: Its Possible Role in Quantum Physics

Recently pure spinor geometry has allowed, after decades of desperate efforts, some definite steps forward for the solution of one of the main unsolved problems of theoretical physics: that of quantum gravity; thus confirming the role of pure spinors in quantum physics. In this paper we try to show that it might be a fundamental role, since pure spinors not only are at the origin of strings, now often adopted in quantum field theory, where they substitute the Euclidean concept of point-event, but also they provide quite simple explanations to several aspects of elementary particle phenomenology, often cumbersome and obscure. Furthermore they naturally define compact manifolds in momentum spaces where a purely geometrical way may be found for the representation and easy solution of some dynamical quantum problems, as first indicated, already a long time ago, by V. Fock.