Pure Spinor Research Articles

Supersymmetric backgrounds from generalized Calabi‐Yau manifolds

AbstractWe show that the supersymmetry transformations for type II string theories on six‐manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kähler form eiJ and the holomorphic form O. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd‐rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: eiJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly O is closed in IIB. This means that supersymmetric SU(3)‐structure manifolds are always complex in IIB while they are twisted symplectic in IIA. Modulo a different action of the B‐field, these are all generalized Calabi‐Yau manifolds, as defined by Hitchin.

The spinorial geometry of supersymmetric IIB backgrounds

We investigate the Killing spinor equations of IIB supergravity for one Killing spinor. We show that there are three types of orbits of Spin(9, 1) in the space of Weyl spinors which give rise to Killing spinors with stability subgroups and G2. We solve the Killing spinor equations for the and invariant spinors, give the fluxes in terms of the geometry and determine the conditions on the spacetime geometry imposed by supersymmetry. In both cases, the spacetime admits a null, self-parallel, Killing vector field. We also apply our formalism to examine a class of backgrounds which admit one and two pure spinors as Killing spinors and investigate the geometry of the spacetimes.

Origin of pure spinor superstring

The pure spinor formalism for the superstring, initiated by N. Berkovits, is derived at the fully quantum level starting from a fundamental reparametrization invariant and super-Poincare invariant worldsheet action. It is a simple extension of the Green-Schwarz action with doubled spinor degrees of freedom with a compensating local supersymmetry on top of the conventional kappa-symmetry. Equivalence to the Green-Schwarz formalism is manifest from the outset. The use of free fields in the pure spinor formalism is justified from the first principle. The basic idea works also for the superparticle in 11 dimensions.

Quantum Consistency of the Superstring in AdS5 × S5 Background

Using arguments based on BRST cohomology, the pure spinor formalism for the superstring in an AdS5 ? S5 background is proven to be BRST invariant and conformally invariant at the quantum level to all orders in perturbation theory. Cohomology arguments are also used to prove the existence of an infinite set of non-local BRST-invariant charges at the quantum level.

BRST Cohomology and Nonlocal Conserved Charges

A relation is found between nonlocal conserved charges in string theory and certain ghost-number two states in the BRST cohomology. This provides a simple proof that the nonlocal conserved charges for the superstring in an AdS_5 x S^5 background are BRST-invariant in the pure spinor formalism and are kappa-symmetric in the Green-Schwarz formalism.

Covariant multiloop superstring amplitudes

In this article, the multiloop amplitude prescription using the super-Poincaré 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. Also, R 4 terms receive no multiloop contributions in agreement with the Type IIB S-duality conjecture of Green and Gutperle. To cite this article: N. Berkovits, C. R. Physique 6 (2005).

Relating the Green-Schwarz and Pure Spinor Formalisms for the Superstring

Although it is not known how to covariantly quantize the Green-Schwarz (GS) superstring, there exists a semi-light-cone gauge choice in which the GS superstring can be quantized in a conformally invariant manner. In this paper, we prove that BRST quantization of the GS superstring in semi-light-cone gauge is equivalent to BRST quantization using the pure spinor formalism for the superstring.

Topological M-theory from pure spinor formalism

We construct multiloop superparticle amplitudes in 11d using the pure spinor formalism. We explain how this construction emerges in the superparticle limit of the multiloop pure spinor superstring amplitudes prescription. We then argue that this construction points to some evidence for the existence of a topological M theory based on a relation between the ghost number of the full-fledged supersymmetric critical models and the dimension of the spacetime for topological models. In particular, we show that the extensions at higher orders of the previous results for the tree and one-loop level expansion for the superparticle in 11 dimensions is related to a topological model in 7 dimensions.

Higher-Dimensional Twistor Transforms using Pure Spinors

Hughston has shown that projective pure spinors can be used to construct massless solutions in higher dimensions, generalizing the four-dimensional twistor transform of Penrose. In any even (euclidean) dimension d = 2n , projective pure spinors parameterize the coset space SO(2n)/U(n), which is the space of all complex structures on 2n . For d = 4 and d = 6, these spaces are 1 and 3 and the appropriate twistor transforms can easily be constructed. In this paper, we show how to construct the twistor transform for 6$> d>6 when the pure spinor satisfies nonlinear constraints, and present explicit formulas for solutions of the massless field equations.

On the b-antighost in the pure spinor quantization of superstrings

Recently Berkovits has constructed a picture raised, compound field bB which is used to compute higher loop amplitudes in the pure spinor approach of superstrings. On the other hand, in the twisted and gauge-fixed, superembedding approach with n=2 world-sheet (w.s.) supersymmetry that reproduces the pure spinor formulation, a field b appears quite naturally as the current of one of the two twisted charges of the w.s. supersymmetry, the other being the BRST charge. In this Letter we study the relation between b and bB. We shall show that bZ, where Z is a picture raising operator, and bB belong to the same BRST cohomological class. This result is of importance since it implies that the cumbersome singularity which is present in b, is in fact harmless if b is combined with Z.

Mirror Symmetric SU(3)-Structure Manifolds with NS Fluxes

When string theory is compactified on a six-dimensional manifold with a nontrivial NS flux turned on, mirror symmetry exchanges the flux with a purely geometrical composite NS form associated with lack of integrability of the complex structure on the mirror side. Considering a general class of T3-fibered geometries admitting SU(3) structure, we find an exchange of pure spinors (e iJ and Ω) in dual geometries under fiberwise T–duality, and study the transformations of the NS flux and the components of intrinsic torsion. A complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with NS flux.

Supersymmetric backgrounds from generalized Calabi–Yau manifolds

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kähler form e i J and the holomorphic form Ω. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: e i J is closed under the action of the twisted exterior derivative in IIA theory, and similarly Ω is closed in IIB. This means that supersymmetric SU(3)-structure manifolds are always complex in IIB while they are twisted symplectic in IIA. Modulo a different action of the B-field, these are all generalized Calabi–Yau manifolds, as defined by Hitchin. To cite this article: M. Graña et al., C. R. Physique 5 (2004).

Covariant one-loop amplitudes in [formula omitted

We generalize to the eleven-dimensional superparticle Berkovits' prescription for loop computations in the pure spinor approach to covariant quantization of the superstring. Using these ten- and eleven-dimensional results, we compute covariantly the following one-loop amplitudes: C ∧ X 8 in M-theory; B ∧ X 8 in type II string theory and F 4 in type I. We also verify the consistency of the formalism in eleven dimensions by recovering the correct classical action from tree-level amplitudes. As the superparticle is only a first approximation to the supermembrane, we comment on the possibility of extending this construction to the latter. Finally, we elaborate on the relationship between the present BRST language and the spinorial cohomology approach to corrections of the effective action.

Multiloop Amplitudes and Vanishing Theorems using the Pure Spinor Formalism for the Superstring

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 amplitudes. One can easily prove that massless N-point multiloop amplitudes vanish for N<4, confirming the perturbative finiteness of superstring theory. One can also prove the Type IIB S-duality conjecture that $R^4$ terms in the effective action receive no perturbative contributions above one loop.

Supersymmetric backgrounds from generalized Calabi-Yau manifolds

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form eiJ and the holomorphic form Ω. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: eiJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly Ω is closed in IIB. Modulo a different action of the B–field, this means that supersymmetric SU(3)-structure manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An equivalent, and somewhat more conventional, description is given as a set of relations between the components of intrinsic torsions modified by the NS flux and the Clifford products of RR fluxes with pure spinors, allowing for a classification of type II supersymmetric vacua on six-manifolds. We find in particular that supersymmetric six-manifolds are always complex for IIB backgrounds while they are twisted symplectic for IIA.

Vertex Operators for Closed Superstrings

We construct an iterative procedure to compute the vertex operators of the closed superstring in the covariant formalism given a solution of IIA/IIB supergravity. The manifest supersymmetry allows us to construct vertex operators for any generic background in presence of Ramond-Ramond (RR) fields. We extend the procedure to all massive states of open and closed superstrings and we identify two new nilpotent charges which are used to impose the gauge fixing on the physical states. We solve iteratively the equations of the vertex for linear x-dependent RR field strengths. This vertex plays a role in studying non-constant C-deformations of superspace. Finally, we construct an action for the free massless sector of closed strings, and we propose a form for the kinetic term for closed string field theory in the pure spinor formalism.

On the Covariant Quantization of Type II Superstrings

In a series of papers Grassi, Policastro, Porrati and van Nieuwenhuizen have introduced a new method to covariantly quantize the GS-superstring by constructing a resolution of the pure spinor constraint of Berkovits' approach. Their latest version is based on a gauged WZNW model and a definition of physical states in terms of relative cohomology groups. We first put the off-shell formulation of the type II version of their ideas into a chirally split form and directly construct the free action of the gauged WZNW model, thus circumventing some complications of the super group manifold approach to type II. Then we discuss the BRST charges that define the relative cohomology and the N=2 superconformal algebra. A surprising result is that nilpotency of the BRST charge requires the introduction of another quartet of ghosts.

The Automorphic Membrane

We present a 1-loop toroidal membrane winding sum reproducing the conjectured $M$-theory, four-graviton, eight derivative, $R^4$ amplitude. The $U$-duality and toroidal membrane world-volume modular groups appear as a Howe dual pair in a larger, exceptional, group. A detailed analysis is carried out for $M$-theory compactified on a 3-torus, where the target-space $Sl(3,\Zint)\times Sl(2,\Zint)$ $U$-duality and $Sl(3,\Zint)$ world-volume modular groups are embedded in $E_{6(6)}(\Zint)$. Unlike previous semi-classical expansions, $U$-duality is built in manifestly and realized at the quantum level thanks to Fourier invariance of cubic characters. In addition to winding modes, a pair of new discrete, flux-like, quantum numbers are necessary to ensure invariance under the larger group. The action for these modes is of Born-Infeld type, interpolating between standard Polyakov and Nambu-Goto membrane actions. After integration over the membrane moduli, we recover the known $R^4$ amplitude, including membrane instantons. Divergences are disposed of by trading the non-compact volume integration for a compact integral over the two variables conjugate to the fluxes -- a constant term computation in mathematical parlance. As byproducts, we suggest that, in line with membrane/fivebrane duality, the $E_6$ theta series also describes five-branes wrapped on $T^6$ in a manifestly U-duality invariant way. In addition we uncover a new action of $E_6$ on ten dimensional pure spinors, which may have implications for ten dimensional super Yang--Mills theory. An extensive review of $Sl(3)$ automorphic forms is included in an Appendix.

Harmonic superspaces from superstrings

We derive harmonic superspaces for N=2,3,4 SYM theory in four dimensions from superstring theory. The pure spinors in ten dimensions are dimensionally reduced and yield the harmonic coordinates. Two anticommuting BRST charges implement Grassmann analyticity and harmonic analyticity. The string field theory action produces the action and field equations for N=3 SYM theory in harmonic superspace.

Self-Dual Super-Yang-Mills as a String Theory in (x, ) Space

Different string theories in twistor space have recently been proposed for describing ${\cal N}=4$ super-Yang-Mills. In this paper, my Strings 2003 talk is reviewed in which a string theory in $(x,\theta)$ space was constructed for self-dual ${\cal N}=4$ super-Yang-Mills. It is hoped that these results will be useful for understanding the twistor-string proposals and their possible relation with the pure spinor formalism of the $d=10$ superstring.