Pure Spinor Research Articles

Maxwell–Dirac system in cosmology

Within the scope of a Bianchi type-I (BI) cosmological model, we study the interacting system of spinor and electromagnetic fields and its role in the evolution of the Universe. In some earlier studies, it was found that in the case of a pure spinor field, the presence of nontrivial nondiagonal components of energy–momentum tensor (EMT) leads to some severe restrictions both on the space–time geometry and/or spinor field itself, whereas in the case of electromagnetic field with induced nonlinearity, such components impose severe restrictions on metric functions and the components of the vector potential. It is shown that in the case of interacting spinor and electromagnetic fields, the restrictions are not as severe as in the other cases and in this case, a nonlinear and massive spinor field with different components of vector potential can survive in a general Bianchi type-I space–time.

Beta deformed sigma model and strong deformation coupling limit

We study the beta deformation of the superstring in AdS5 × S5 at all orders in the deformation parameter, employing the pure spinor formalism. This is necessary in order to study the regime of strong deformation parameter, which in the field side is related to fishnet theories. We compare the pure spinor sigma model approach to the previously known supergravity description. We find a complete agreement. Moreover, the BRST structure of the worldsheet model provides a natural explanation of the peculiar features of the worldsheet model in the fishnet limit. In particular, we study the degeneracy of the sigma model Lagrangian. We show that the BRST structure is responsible for a particularly “tame” degeneration of the fishnet sigma-model.

Marginally deformed AdS5/CFT4 backgrounds in Type IIB

Multi-parameter families of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 0 Type IIA and Type IIB AdS5 solutions are presented, promoting to N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 in some special cases. The G-Structure description of each N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 solution is given, requiring an Abelian T-Duality of the G-Structure conditions and Pure Spinors. Investigations at the boundaries are performed for a two-parameter family of Type IIA and a three-parameter family of Type IIB solutions, finding the presence of orbifold singularities in some backgrounds. All parameters drop out of the Holographic Central Charge calculation, pointing to marginal deformations in the dual CFT description.

D-terms in generalised complex geometry

Generalised Complex Geometry provides a natural interpretation of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetry conditions for warped solutions of type II supergravity as differential equations on polyforms on the internal manifold. Written in this language the supersymmetry conditions correspond to calibration conditions for probe D-branes: D-string, domain-wall or space-filling branes, depending on the directions they span in the non-compact four-dimensional space. The BPS condition corresponding to the calibration of space-filling D-branes has been reformulated by Tomasiello, eliminating the explicit dependence on the metric. We generalise this derivation to the case of non-supersymmetric backgrounds violating the domain-wall and D-string calibration conditions. We use this reformulation to derive constraints that the ten-dimensional solutions with BPS space-filling sources must respect in order to dimensionally reduce to solutions of four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supergravity with non-vanishing F-terms and potentially non-vanishing D-terms. We give the equations of motion for the class of type II vacua satisfying these constraints in the language of pure spinors. We investigate how restrictive these constraints are for the class of type IIB SU(3) backgrounds with BPS space-filling O5-planes.

4D Chern-Simons and the pure spinor AdS5×S5 superstring

Four-dimensional Chern-Simons (4DCS) theory is useful for understanding integrable sigma-models and constructing new ones. In this paper, we show how to derive the complete pure spinor AdS5×S5 superstring sigma-model from 4DCS theory with defects. The matter sector of this sigma model was previously derived by Costello and Yamazaki, and we propose here that the pure spinor ghosts come from gauge-fixing meromorphic transformations of 4DCS which lead to the usual pure spinor Lax connection including the ghost contribution. Published by the American Physical Society 2024

New non-supersymmetric flux vacua from generalised calibrations

We construct a new class of non-supersymmetric ten-dimensional type II flux vacua, by studying first order differential equations which are deformations of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 1 supersymmetry conditions. We do so within the context of Generalised Complex Geometry, where there is a natural interpretation of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 1 supersymmetry conditions in terms of calibration conditions for probe D-branes, called D-string, domain-wall or space-filling branes, depending on them wrapping two, three or four non-compact dimensions. We focus on the class of non-supersymmetric vacua violating the D-string calibration condition, and write down their general equations of motion in the language of pure spinors. We solve them for a subclass of vacua, where the deformation of the calibration condition is dictated by the foliated geometry of the internal space. We also construct backgrounds violating both the domain-wall and D-string calibration conditions, generalising the one-parameter DWSB class of backgrounds introduced in Lüst et al. We present several explicit solutions with SU(2) and SU(3) structures, and we investigate briefly their associated low energy effective theories.

Double space T dualization and coordinate dependent RR field

In this article, we examine T dualization in the double space formalism of type II superstring theory in pure spinor formulation. All background fields are constant except the Ramond-Ramond field, which depends infinitesimally on bosonic coordinates xμ. In double space, T dual transformations are represented as permutations of the starting xμ and dual coordinates yμ. Combining these two sets of coordinates into the double coordinate ZM=(xμ,yμ), while demanding that the T dual double coordinate has the same T dual transformation law as the initial ones, we obtain how background fields transform under T duality. Comparing these results with ones obtained using the Buscher T dualization procedure, we conclude that these two approaches are equivalent for the considered choice of the background fields. Published by the American Physical Society 2024

Canonical Supermultiplets and Their Koszul Duals

The pure spinor superfield formalism reveals that, in any dimension and with any amount of supersymmetry, one particular supermultiplet is distinguished from all others. This “canonical supermultiplet” is equipped with an additional structure that is not apparent in any component-field formalism: a (homotopy) commutative algebra structure on the space of fields. The structure is physically relevant in several ways; it is responsible for the interactions in ten-dimensional super Yang–Mills theory, as well as crucial to any first-quantised interpretation. We study the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty $$\\end{document} algebra structure that is Koszul dual to this commutative algebra, both in general and in numerous examples, and prove that it is equivalent to the subalgebra of the Koszul dual to functions on the space of generalised pure spinors in internal degree greater than or equal to three. In many examples, the latter is the positive part of a Borcherds–Kac–Moody superalgebra. Using this result, we can interpret the canonical multiplet as the homotopy fiber of the map from generalised pure spinor space to its derived replacement. This generalises and extends work of Movshev–Schwarz and Gálvez–Gorbounov–Shaikh–Tonks in the same spirit. We also comment on some issues with physical interpretations of the canonical multiplet, which are illustrated by an example related to the complex Cayley plane, and on possible extensions of our construction, which appear relevant in an example with symmetry type G2×A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2 \ imes A_1$$\\end{document}.

Ransformations of Currents in Sigma-Models with Target Space Supersymmetry

We develop a framework for systematic study of symmetry transformations of sigma-model currents in a special situation, when symmetries have a well-defined projection onto the target space. We then apply this formalism to pure spinor sigma-models, and describe the resulting geometric structures in the target space (which in our approach includes the pure spinor ghosts). We perform a detailed study of the transformation properties of currents, using the formalism of equivariant cohomology. We clarify the descent procedure for the ''universal'' deformation corresponding to changing the overall scale of the worldsheet action. We also study the contact terms in the OPE of BRST currents, and derive some relations between currents and vertex operators which perhaps have not been previously acknowledged. We also clarify the geometrical meaning of the ''minimalistic'' BV action for pure spinors in AdS.

Constructing massive superstring vertex operators from massless vertex operators using the pure spinor formalism

The vertex operator for the first massive states of the open superstring is constructed in terms of d=10 super Yang-Mills superfields using the OPE's of massless vertex operators in the pure spinor formalism.

Twisting pure spinor superfields, with applications to supergravity

Twisting pure spinor superfields, with applications to supergravity

Tree-level color-kinematics duality from pure spinor actions

Tree-level color-kinematics duality from pure spinor actions

Scattering three closed strings off a Dp-brane in pure spinor formalism

We compute the disk amplitude of three closed strings in the pure spinor formalism. Among others, this amplitude probes tree-level gravitational interactions in the presence of Dp-branes. After disentangling holomorphic and anti-holomorphic closed string coordinates on the disk by means of introducing monodromy phases we find a compact expression for the disk amplitude of three closed strings in terms of open superstring six-point amplitudes. Furthermore, we provide the low-energy expansion (in the inverse string tension) of our amplitude and discuss some relevant Dp-brane couplings associated to it. Finally, we write down an expression for the general structure of the disk amplitude of any number nc of closed strings in terms of pure open string amplitudes involving 2nc open strings.

Superspace expansion of the 11D linearized superfields in the pure spinor formalism, and the covariant vertex operator

11D pure spinors have been shown to successfully describe 11D supergravity in a manifestly super-Poincaré covariant manner. The feasibility of its actual usage for scattering amplitude computations requires an efficient manipulation of the superfields defining linearized 11D supergravity. In this paper, we directly address this problem by finding the superspace expansions of these superfields, at all orders in θ, from recursive relations their equations of motion obey in Harnad-Shnider-like gauges. After introducing the 11D analogue of the 10D ABC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{ABC} $$\\end{document} superparticle, we construct, for the first time, a fully covariant vertex operator for 11D supergravity by making use of the linearized 11D superfields. Notably, we show that this vertex reproduces the Green-Gutperle-Kwon 11D operators in light-cone gauge.

Supersymmetric non-Abelian DBI equations from an open pure spinor superstring

Supersymmetric non-Abelian DBI equations from an open pure spinor superstring

Aberration of gravitational waveforms by peculiar velocity

ABSTRACT One key prediction of General Relativity is that gravitational waves are emitted with two independent polarizations. Any observation of extra polarization mode, spin-1 or spin-0, is consequently considered a smoking gun for deviations from General Relativity. In this paper, we show that the velocity of merging binaries with respect to the observer gives rise to spin-1 polarization in the observer frame even in the context of General Relativity. These are pure projection effects, proportional to the plus and cross polarizations in the source frame, hence they do not correspond to new degrees of freedom. We demonstrate that the spin-1 modes can always be rewritten as pure spin-2 modes coming from an aberrated direction. Since gravitational waves are not isotropically emitted around binary systems, this aberration modifies the apparent orientation of the binary system with respect to the observer: the system appears slightly rotated due to the source velocity. Fortunately, this bias does not propagate to other parameters of the system (and therefore does not spoil tests of General Relativity), since the impact of the velocity can be fully reabsorbed into new orientation angles.

Remarks on the Integral Form of D=11 Supergravity

We make some considerations and remarks on D = 11 supergravity and its integral form. We start from the geometrical formulation of supergravity and by means of the integral form technique we provide a superspace action that reproduces (at the quadratic level) the recent formulation of supergravity in pure spinor framework. We also make some remarks on Chevalley-Eilenberg cocycles and their Hodge duals.

One loop mass renormalization of massive states using pure spinor formalism

As a check of the first massive integrated vertex operator in the pure spinor formalism constructed in arXiv:1802.04486, we compute the one loop 2-point function of the stable non BPS massive states in SO(32) heterotic string theory. This allows us to compute the one loop renormalized mass of these states using the pure spinor formalism. Our results are in agreement with the corresponding results obtained by Sen using the RNS formalism. This provides an instance of the equivalence between the RNS and the pure spinor formalism for the massive states at loop level.

The Derived Pure Spinor Formalism as an Equivalence of Categories

We construct a derived generalization of the pure spinor superfield formalism and prove that it exhibits an equivalence of dg-categories between multiplets for a supertranslation algebra and equivariant modules over its Chevalley-Eilenberg cochains. This equivalence is closely linked to Koszul duality for the supertranslation algebra. After introducing and describing the category of supermultiplets, we define the derived pure spinor construction explicitly as a dg-functor. We then show that the functor that takes the derived supertranslation invariants of any supermultiplet is a quasi-inverse to the pure spinor construction, using an explicit calculation. Finally, we illustrate our findings with examples and use insights from the derived formalism to answer some questions regarding the ordinary (underived) pure spinor superfield formalism.

A Minimal b Ghost

AbstractThe b ghost, or b operator, used for fixing Siegel gauge in the pure spinor superfield formalism, is a composite operator of negative ghost number, satisfying , where q is the pure spinor differential (BRST operator). It is traditionally constructed using non‐minimal variables. However, since all cohomology has minimal representatives, it seems likely that there should be versions of physically meaningful operators, also with negative ghost number, using only minimal variables. The purpose of this letter is to demonstrate that this statement holds by providing a concrete construction in super‐Yang–Mills theory, and to argue that it is a general feature in the pure spinor superfield formalism.