Pure Spinor Research Articles

Taming the 11D pure spinor b-ghost

We provide an alternative compact expression for the 11D pure spinor b-ghost by introducing a new set of negative ghost number operators made out of non-minimal pure spinor variables. Using the algebraic properties satisfied by these operators, it will be straightforwardly shown that left{Q,bright}=frac{P^2}{2} , as well as {b, b} = QΩ. As an application of this novel formulation, the ghost number two vertex operator will easily be obtained in a completely covariant manner from a standard descent relation, the ghost number three vertex operator will be shown to satisfy the generalized Siegel gauge condition, and the 11D supergravity two-particle superfield will be constructed in a quite simple way.

D=5 holomorphic Chern-Simons and the pure spinor superstring

The physical states of D=5 holomorphic Chern-Simons theory correspond to on-shell D=10 open superstring states in the cohomology of q+, where q+ is one of the 16 spacetime supersymmetry generators. Scattering amplitudes of these states can be computed either using the usual Ramond-Neveu-Schwarz (RNS) superstring prescription with N=1 worldsheet supersymmetry, or using a topological ĉ=5 string theory with twisted N=2 worldsheet supersymmetry.It will be argued that the relation between D=5 holomophic Chern-Simons and the RNS superstring is identical to the relation between the pure spinor superstring and the recently constructed B-RNS-GSS superstring which has both N=1 worldsheet supersymmetry and D=10 spacetime supersymmetry. Physical states of the pure spinor superstring correspond to on-shell B-RNS-GSS states which are in the cohomology of λαqα, where λα is a D=10 pure spinor. And scattering amplitudes of these states can be computed either using the full B-RNS-GSS superstring prescription with N=1 worldsheet supersymmetry, or using the pure spinor superstring amplitude prescription with twisted N=2 worldsheet supersymmetry. This should be useful for proving equivalence of the RNS and pure spinor amplitude prescriptions and for clarifying the relation of their multiloop subtleties.

Constraints in the BV formalism: Six-dimensional supersymmetry and its twists

We formulate the abelian six-dimensional N=(2,0) theory perturbatively, in a generalization of the Batalin–Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimal twists at the perturbative level. This calculation hinges on the existence of an L∞ action of the supersymmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appears naturally in the pure spinor superfield formalism, but understanding it requires developing a presymplectic generalization of the BV formalism, inspired by Dirac's theory of constraints. The holomorphic twist consists of symplectic-valued holomorphic bosons from the N=(1,0) hypermultiplet, together with a degenerate holomorphic theory of holomorphic one-forms from the N=(1,0) tensor multiplet, which can be seen to describe the infinitesimal intermediate Jacobian variety. We check that our formulation and our results match with known ones under various dimensional reductions, as well as comparing the holomorphic twist to Kodaira–Spencer theory. Matching our formalism to five-dimensional Yang–Mills theory after reduction leads to some issues related to electric–magnetic duality; we offer some speculation on a nonperturbative resolution.

Maximally Twisted Eleven-Dimensional Supergravity

We perform the maximal twist of eleven-dimensional supergravity. This twist is partially topological and exists on manifolds of G_2 times SU(2) holonomy. Our derivation starts with an explicit description of the Batalin–Vilkovisky complex associated to the three-form multiplet in the pure spinor superfield formalism. We then determine the L_infty module structure of the supersymmetry algebra on the component fields. We twist the theory by modifying the differential of the Batalin–Vilkovisky complex to incorporate the action of a scalar supercharge. We find that the resulting free twisted theory is given by the tensor product of the de Rham and Dolbeault complexes of the respective G_2 and SU(2) holonomy manifolds as conjectured by Costello.

Synchronization of Gursey System

Gursey Model, the only possible four-dimensional pure spinor model, proposed as a possible basis for a unitary description of elementary particles. The model exhibits chaotic behaviors depending on the system parameter values. In this study, we investigate the synchronization of chaotic dynamic in the Gursey wave equation that has particle-like solutions derived classical field equations. Numerical results for synchronization of the Gursey system are performed to indicate the accuracy of the used method.

Perspectives on the pure spinor superfield formalism

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety. We illustrate our discussion with many examples in various dimensions.

Thirring İnstantonlarının Lyapunov Üstelleri

Recently, nonlinear differential equations corresponding to pure spinor instanton solutions were obtained by using Heisenberg anzat in the 2D Thirring Model, which has been used as a toy model in Quantum field theory. In addition, the evolution of spinor type instanton solutions in phase space was investigated according to the change in the constant parameter β. Spinor instanton dynamics is a special case in which nonlinear terms play an important role. Chaos describes certain nonlinear dynamical systems that depend very precisely on initial conditions. Lyapunov exponents are an important method for measuring stability and deterministic chaos in dynamical systems. Lyapunov exponents characterize and quantify the dynamics of small perturbations of a state or orbit in state space. In this study, Lyapunov spectrum of spinor type instanton solutions was investigated by examining the largest local and global Lyapunov exponents. As a result of the Lyapunov Spectrum, it was determined that the spinor type instanton solutions exhibit chaotic behavior at parameter value β = 2. Periodic and quasi periodic behaviors were detected when the parameter values were β < 2. In cases of β > 2, weak chaotic behaviors were observed. This study demonstrates that Thirring Instantons, which are spinor type instanton solutions, exhibit chaotic properties.

Lax pairs for new [formula omitted]-symmetric coset σ-models and their Yang-Baxter deformations

Two-dimensional σ-models with ZN-symmetric homogeneous target spaces have been shown to be classically integrable when introducing WZ-terms in a particular way. This article continues the search for new models of this type now allowing some kinetic terms to be absent – analogously to the Green-Schwarz superstring σ-model on Z4-symmetric homogeneous spaces. A list of such integrable ZN-symmetric (super)coset σ-models for N≤6 and their Lax pairs is presented. For arbitrary N a big class of integrable models is constructed that includes both the known pure spinor and Green-Schwarz superstring on Z4-symmetric cosets.Integrable Yang-Baxter deformations of this class of ZN-symmetric (super)coset σ-models can be constructed in same way as in the known Z2- or Z4-cases. Deformations based on solutions of the modified classical Yang-Baxter equation, the so-called η-deformation, require deformation of the constants defining the Lagrangian and the corresponding Lax pair. Homogeneous Yang-Baxter deformations (i.e. those based on solutions to the classical Yang-Baxter equation) leave the equations of motion and consequently the Lax pair invariant and are expected to be classically equivalent to the undeformed model.As an example, the relationship between Z3-symmetric homogeneous spaces and nearly (para-)Kähler geometries is revisited. Confirming existing literature it is shown that the integrable choice of WZ-term in the Z3-symmetric coset σ-model associated to a nearly Kähler background gives an imaginary contribution to the action.

On the stability of string theory vacua

Vacuum compactifications may suffer from instabilities under small perturbations or tunnel effects; both are difficult to analyze. In this paper we consider the issue from a higher-dimensional perspective. We first look at how stability works for supersymmetric vacua, where it is widely expected to hold. We first show that the nucleation of brane bubbles in type II AdS compactifications is forbidden in the probe approximation by a simple argument involving pure spinors and calibrations. We then adapt familiar positive-energy theorems directly to M-theory and type II supergravity, rather than to their effective lower-dimensional reductions, also showing how to consistently include localized sources. We finally initiate an analysis of how these arguments might be extended to non-supersymmetric vacua. In M-theory, at the lower-derivative level, we find that the most natural modifications fail to stabilize the skew-whiffed and Englert vacua.

KK-like relations of α′ corrections to disk amplitudes

Inspired by the definition of color-dressed amplitudes in string theory, we define analogous color-dressed permutations replacing the color-ordered string amplitudes by their corresponding permutations. Decomposing the color traces into symmetrized traces and structure constants, the color-dressed permutations define BRST-invariant permutations, which we show are elements of the inverse Solomon descent algebra and we find a closed formula for them. We then present evidence that these permutations encode KK-like relations among the different α′ corrections to the disk amplitudes refined by their MZV content. In particular, the number of linearly independent amplitudes at a given α′ order and MZV content is given by (sums of) Stirling cycle numbers. In addition, we show how the superfield expansion of BRST invariants of the pure spinor formalism corresponding to α′2ζ2 corrections is encoded in the descent algebra.

Superspace BRST/BV Operators of Superfield Gauge Theories

We consider the superspace BRST and BV description of 4D,N=1 super-Maxwell theory and its non-abelian generalization Super Yang–Mills. By fermionizing the superspace gauge transformation of the gauge superfields, we define the nilpotent superspace BRST symmetry transformation (𝓈). After introducing an appropriate set of anti-superfields and defining the superspace antibracket, we use it to construct the BV-BRST nilpotent differential operator (s) in terms of superspace covariant derivatives. The anti-superfield independent terms of s provide a superspace generalization of the Koszul–Tate resolution (δ). In the linearized limit, the set of superspace differential operators that appear in s satisfy a nonlinear algebra which can be used to construct a BRST charge Q, without requiring pure spinor variables. Q acts on the Hilbert space of superfield states, and its cohomology generates the expected superspace equations of motion.

Type IIB superstring vertex operator from the -8 picture

A new procedure was recently proposed for constructing massless Type IIB vertex operators in the pure spinor formalism. Instead of expressing these closed string vertex operators as left-right products of open string vertex operators, they were instead constructed from the complex N=2 d=10 superfield whose lowest real and imaginary components are the dilaton and Ramond-Ramond axion. These Type IIB vertex operators take a simple form in the -8 picture and are related to the usual vertex operators in the zero picture by acting with picture-raising operators. In this paper, we compute explicitly this picture-raising procedure and confirm this proposal in a flat background. Work is in progress on confirming this proposal in an AdS5×S5 background.

Green-Schwarz and pure spinor formulations of chiral strings

Bosonic and RNS chiral strings have been defined from a singular gauge fixing of the respective Polyakov and spinning string actions, enforcing, among other things, the finite nature of their physical spectra. Except for the heterotic case, the tensionless limits of such chiral models have been shown to describe the same field theories predicted by their ambitwistor analogues. In this paper, we study the Green-Schwarz formulation for Type II and heterotic superstrings in a singular gauge. After performing a light-cone gauge analysis, their physical spectra are shown to match those of RNS chiral strings, and their respective tensionless limits are found to describe the same field theories predicted by RNS ambitwistor strings. Their pure spinor counterparts are then introduced by making use of the Oda-Tonin method. In doing so, symmetries hidden in the pure spinor ambitwistor string action become manifest, proposals motivating the sectorized pure spinor BRST charges find simple grounds, and integrated vertex operators emerge naturally.

10D super-Yang-Mills scattering amplitudes from its pure spinor action

Using the pure spinor master action for 10D super-Yang-Mills in the gauge b0V = QΞ, tree-level scattering amplitudes are calculated through the perturbiner method, and shown to match those obtained from pure spinor CFT techniques. We find kinematic numerators made of nested b-ghost operators, and show that the Siegel gauge condition b0V = 0 gives rise to color-kinematics duality satisfying numerators whose Jacobi identity follows from the Jacobi identity of a kinematic algebra.

Pin(d, d) covariance of pure spinor equations for supersymmetric vacua and non-Abelian T-duality

In a supersymmetric compactification of Type II supergravity, preservation of mathcal{N} = 1 supersymmetry in four dimensions requires that the structure group of the generalized tangent bundle TM ⨁ T∗M of the six dimensional internal manifold M is reduced from SO(6) to SU(3) × SU(3). This topological condition on the internal manifold implies existence of two globally defined compatible pure spinors Φ1 and Φ2 of non-vanishing norm. Furthermore, these pure spinors should satisfy certain first order differential equations. In this paper, we show that non-Abelian T-duality (NATD) is a solution generating transformation for these pure spinor equations. We first show that the pure spinor equations are covariant under Pin(d, d) transformations. Then, we use the fact NATD is generated by a coordinate dependent Pin(d, d) transformation. The key point is that the flux produced by this transformation is the same as the geometric flux associated with the isometry group, with respect to which one implements NATD. We demonstrate our method by studying NATD of certain solutions of Type IIB supergravity with SU(2) isometry and SU(3) structure.

Two-loop superstring five-point amplitudes. Part III. Construction via the RNS formulation: even spin structures

The contribution from even spin structures to the genus-two amplitude for five massless external NS states in Type II and Heterotic superstrings is evaluated from first principles in the RNS formulation. Using chiral splitting with the help of loop momenta this problem reduces to the evaluation of the corresponding chiral amplitude, which is carried out using the same techniques that were used for the genus-two amplitude with four external NS states. The results agree with the parity-even NS components of a construction using chiral splitting and pure spinors given in earlier companion papers [29] and [33].

Manifest spacetime supersymmetry and the superstring

The algebra of spacetime supersymmetry generators in the RNS formalism for the superstring closes only up to a picture-changing operation. After adding non-minimal variables and working in the “large” Hilbert space, the algebra closes without picture-changing and spacetime supersymmetry can be made manifest. The resulting non-minimal version of the RNS formalism is related by a field redefinition to the pure spinor formalism.

SL(5) Supersymmetry

AbstractWe consider supersymmetry in five dimensions, where the fermionic parameters are a 2‐form under . Supermultiplets are investigated using the pure spinor superfield formalism, and are found to be closely related to infinite‐dimensional extensions of the supersymmetry algebra: the Borcherds superalgebra , the tensor hierarchy algebra and the exceptional superalgebra E(5, 10). A theorem relating and E(5, 10) to all levels is given.

Normal form of nilpotent vector field near the tip of the pure spinor cone

Pure spinor formalism implies that supergravity equations in space-time are equivalent to the requirement that the worldsheet sigma-model satisfies certain properties. Here we point out that one of these properties has a particularly transparent geometrical interpretation. Namely, there exists an odd nilpotent vector field on some singular supermanifold, naturally associated to space-time. Is it true that all supergravity fields are encoded in this vector field, as coefficients in its normal form, and the nilpotence is equivalent to the target space equations of motion? We show that this is approximately correct. The normal form is parametrized by some tensor fields, which satisfy hyperbolic equations. These equations are slightly weaker than the full supergravity equations.

AdS5 \xd7 S5 supergravity vertex operators

In any type II superstring background, the supergravity vertex operators in the pure spinor formalism are described by a gauge superfield. In this paper, we obtain for the first time an explicit expression for this superfield in an AdS5 × S5 background. Previously, the vertex operators were only known close to the boundary of AdS5 or in the minus eight picture. Our strategy for the computation was to apply eight picture raising operators in the minus eight picture vertices. In the process, a huge number of terms are generated and we have developed numerical techniques to perform intermediary simplifications. Alternatively, the same numerical techniques can be used to compute the vertices directly in the zero picture by constructing a basis of invariants and fitting for the coefficients. One motivation for constructing the vertex operators is the computation of AdS5 × S5 string amplitudes.