We derive the Free Differential Algebra for type IIA supergravity in 10 dimensions in the string frame. We provide all fermionic terms for all curvatures. We derive the Green-Schwarz sigma model for type IIA superstring based on the FDA construction and we check its invariance under kappa-symmetry. Finally, we derive the pure spinor sigma model and we check the BRST invariance. The present derivation has the advantage that the resulting sigma model is constructed in terms of the superfields appearing in the FDA and therefore one can directly relate a supergravity background with the corresponding sigma model. The complete explicit form of the BRST transformations is given and some new pure spinor constraints are obtained. Finally, the explicit form of the action is given.
We construct the type IIA Green-Schwarz superstring action on a ten-dimensional IIA plane wave background with 24 supersymmetries keeping the full 32 fermionic coordinates. Starting from the symmetry superalgebra for the maximally supersymmetric eleven dimensional plane wave background, we obtain the eleven dimensional superfields. The Kaluza-Klein reduction leads to the ten dimensional superfields for the IIA plane wave background, from which the type IIA superstring action is constructed. We show that the superstring action reduces correctly to the previously known light-cone gauge fixed act…
The compactification of type IIA superstring on n-dimensional Calabi-Yau manifolds is discussed. In particular, a conjecture is given for type IIA extremal p-dimensional black brane attractors corresponding to Ads_{p+2}\times S^{8-2n-p} horizon geometries.
The twistor--like formulation of the type IIA superstring $σ$--model in D=10 is obtained by performing a dimensional reduction of the recently proposed twistor--like action of the supermembrane in D=11. The superstring action is invariant under local, worldsheet $(n,n)$ supersymmetry where $3\leq n \leq 8$ and is classical equivalent to the standard Green--Schwarz action (at least for $n=8)$
We show that the M-theory/IIA and IIA/IIB superstring dualities together with the diffeomorphism invariance of the underlying theories require the presence of certain p-brane bound states in IIA and IIB superstring theories preserving 1/2 of the spacetime supersymmetry. We then confirm the existence of IIA and IIB supergravity solutions having the appropriate p-brane bound states interpretation.
This paper considers the Free Differential Algebra and rheonomic parametrization of type IIA Supergravity, extended to include the BRS differential and the ghosts. We consider not only the ghosts lambda's of supersymmetry but also the ghosts corresponding to gauge and Lorentz transformations. In this way we can derive not only the BRS transformations of fields and ghosts but also the standard pure spinor constraints on lambda's. Moreover the formalism allows to derive the action for the pure spinor formulation of type IIA superstrings in a general background, recovering the action first obtain…
We study the low-lying excitations of Type IIA superstring theory in a plane wave background with 24 supersymmetries. In the light-cone gauge, the superstring action has ${\cal N}=(4,4)$ supersymmetry and is exactly solvable, since it is quadratic in superstring coordinates. We obtain explicitly the spectrum of the Type IIA supergravity fluctuation modes in the plane wave background and give its correspondence with the spectrum of string states from the zero-mode sector of the light-cone superstring Hamiltonian.
In light of the present LHC Run II at $\sqrt{s}=13$ $TeV$, string y standard-like model is studied. Concretely, a singlet $S $ scalar-extended SM given in terms four stacks of intersecting D6-branes in a type IIA superstring compactification producing a large gauge symmetry is examined. The involved scales are dealt with. According to the dark matter relic density, the mass of the scalar dark matter beyond the SM $m_{S}\lesssim 10^{3}GeV$ and the corresponding Higgs portal couplings $λ_{SH}\lesssim 10^{-8}$ are approached.
We calculate the entropy of 4-charge extremal black holes in Type IIA supersting theory by using Sen's entropy function method. Using the low energy effective actions in both $10D$ and 4D, we find precise agreements with the Bekenstein-Hawking entropy of the black hole. We also calculate the higher order corrections to the entropy and find that they depend on the exact form of the higher order corrections to the effective action.
In the previous paper, the authors pointed out correspondence of a supersymmetric double-well matrix model with two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background from the viewpoint of symmetries and spectrum. In this paper we further investigate the correspondence from dynamical aspects by comparing scattering amplitudes in the matrix model and those in the type IIA theory. In the latter, cocycle factors are introduced to vertex operators in order to reproduce correct transformation laws and target-space statistics. By a perturbative treatment of the Ramond-R…
Motivated by the M-theory compactification physics, we study a type IIA superstring quiver model with an open string axion field associated with the U(1)_{PQ} Peccei-Quinn global symmetry using D6-brane physics. This realization extends the symmetry of the SM-Like and contributes to the mediation of the electroweak symmetry breaking. The associated axion energy scales are approached in terms of D2 and D6-branes in type IIA compactifications on Calabi-Yau threeforlds. Using the fermion mass scales as well as string theory results, the allowed axion window is discussed in terms of intersecting D…
Witten proposed that the low energy physics of strongly coupled D=10 type-IIA superstring may be described by D=11 supergravity. To explore the stringy aspects of the underlying theory we examine the stringy massive states. We propose a systematic formula for identifying non-perturbative states in D=10 type-IIA superstring theory, such that, together with the elementary excited string states, they form D=11 supersymmetric multiplets multiplets in SO(10) representations. This provides hints for the construction of a weakly coupled D=11 theory that is dual to the strongly coupled D=10 type IIA s…
We study lower-dimensional superstrings in the double-spinor formalism introduced by Aisaka and Kazama. These superstrings can be consistently quantized and are equivalent to the lower-dimensional pure-spinor superstrings proposed by Grassi and Wyllard. The unexpected physical spectrum of the pure-spinor superstrings may thus be regarded as a manifestation of noncriticality. We also discuss how to couple these covariant superstrings to the compactified degrees of freedom described by the N=2 superconformal field theory.
A brane picture in Type IIA superstring for the Yang Monopole is reconsidered. It makes use of D2 and D4-branes wrapped on cycles in the K3 surface. When the model was first presented some problems concerning the charges of the monopoles arised. In this paper, they are shown to be cured by the model itself. Surprisingly, the incompatibility between the multi-charge configuration and the spherical symmetry of the Yang Monopole is seen in the brane description as the emergence of the enhancon shell and the fuzzy geometry. This consistency is deep and surprising, and is the point that triggered t…
In this lectures 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.
It is shown that the pure spinor formulation of the heterotic superstring in a generic gravitational and super Yang-Mills background has vanishing one-loop beta functions.
In the previous paper, the authors pointed out correspondence between a supersymmetric double-well matrix model and two-dimensional type IIA superstring theory on a Ramond-Ramond background from the viewpoint of symmetry and spectrum. This was confirmed by agreement between planar correlation functions in the matrix model and tree-level amplitudes in the superstring theory. In order to investigate the correspondence further, in this paper we compute correlation functions to all order of genus expansion in the double scaling limit of the matrix model. One-point functions of operators protected …
A covariant action for closed D=11 superstring with local $κ$-symmetry and global supersymmetry transformations obeying the algebra $\{Q_α,Q_β\}=CΓ^{μν}P_μn_ν$ is suggested. Physical sector variables of the model and their dynamics exactly coincide with those of the D=10 type IIA Green-Schwarz superstring. It is shown that action of the D=10 type IIA Green-Schwarz superstring can be considered as a partially gauge fixed version of the D=11 superstring action.
We solve the Killing spinor equations of standard and massive IIA supergravities for a Killing spinor whose isotropy subgroup in Spin(9, 1) is SU(4) and identify the geometry of the spacetime. We demonstrate that the Killing spinor equations impose some mild constraints on the geometry of the spacetime which include the existence of a time-like Killing vector field which leaves the fields and the Killing spinor invariant.
We study to what extent it is possible to generalise Berkovits' pure-spinor construction in d=10 to lower dimensions. Using a suitable definition of a ``pure'' spinor in d=4,6, we propose models analogous to the d=10 pure-spinor superstring in these dimensions. Similar models in d=2,3 are also briefly discussed.
In this paper we study the pure spinor formulation of the superstring in $AdS_5\times S^5$ around point particle solutions of the classical equations of motion. As a particular example we quantize the pure spinor string in the BMN background.
We lift the bosonic AdS_4 x CP^3 solution of type IIA supergravity preserving 24 supersymmetries to a D=10 superspace which has 32 Grassmann-odd directions. The type IIA superspace is obtained from D=11 via dimensional reduction of the coset superspace OSp(8|4)/SO(7) x SO(1,3) by realizing the latter as a Hopf fibration over the former. This construction generalizes to superspace the Hopf fibration of S^7 as a U(1) bundle over CP^3, and is suitable for writing the explicit form of Green-Schwarz-type actions encoding the dynamics of the type IIA string and branes in the AdS_4 x CP^3 superbackgr…
It is proven that the classical pure spinor superstring in an AdS_5 x S^5 background has a flat current depending on a continuous parameter. This generalizes the recent result of Bena, et al. for the classical Green-Schwarz superstring.
IIA supergravity backgrounds preserving one supersymmetry locally admit four types of Killing spinors distinguished by the orbits of $Spin(9,1)$ on the space of spinors. We solve the Killing spinor equations of IIA supergravity with and without cosmological constant for Killing spinors representing two of these orbits, with isotropy groups $Spin(7)$ and $Spin(7)\ltimes\mathbb{R}^8$. In both cases, we identify the geometry of spacetime and express the fluxes in terms of the geometry. We find that the geometric constraints of backgrounds with a $Spin(7)\ltimes\mathbb{R}^8$ invariant Killing spin…
Starting with a classical action where a pure spinor $λ^α$ is only a fundamental and dynamical variable, the pure spinor formalism for superparticle and superstring is derived by following the BRST formalism. In this formalism, not only the string variable $x^m$ but also the space-time spinor $θ^α$ are emerged as the Faddeev-Popov (FP) ghosts of a topological symmetry and its reducible symmetry. This study suggests that the fundamental theory behind the pure spinor formalism of the superstring might be a topological field theory.
The tree-level operator product expansion coefficients of the matter currents are calculated in the pure spinor formalism for type IIB superstring in the AdS(5)*S(5) background.
We show that the Lax pair of the pure spinor superstring in AdS5xS5 admits a generalization where the generators of the superconformal algebra are replaced by the generators of some infinite-dimensional Lie superalgebra.
A simplified pure spinor superstring $b$ ghost in a curved heterotic background was constructed recently. The $b$ ghost is a composite operator and it is not holomorphic. However, it satisfies $\bar\partial b = [ Q , Ω]$, where $Q$ is the BRST charge. In this paper, we find a possible $Ω$.
Using the RNS-like fermionic vector variables introduced in arXiv:1305.0693, the pure spinor $b$ ghost in a curved heterotic superstring background is easily constructed. This construction simplifies and completes the $b$ ghost construction in a curved background of arXiv:1311.7012.
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.
We continue the study of the d=2,4,6 pure-spinor superstring models introduced in [1]. By explicitly solving the pure-spinor constraint we show that these theories have vanishing central charge and work out the (covariant) current algebra for the Lorentz currents. We argue that these super-Poincare covariant models may be thought of as compactifications of the superstring on CY_{4,3,2}, and take some steps toward making this precise by constructing a map to the RNS superstring variables. We also discuss the relation to the so called hybrid superstrings, which describe the same type of compacti…
We present an integrable deformation of the $AdS_{5}\times S^{5}$ pure spinor action based on homological perturbation theory. Its equations of motion, Lax connection and BRST symmetry are discussed. The resulting model describes a pure spinor superstring in a generalized supergravity background.
In a previous paper, the BRST cohomology in the pure spinor formalism of the superstring was shown to coincide with the light-cone Green-Schwarz spectrum by using an SO(8) parameterization of the pure spinor. In this paper, the SO(9,1) Lorentz generators are explicitly constructed using this SO(8) parameterization, proving the Lorentz invariance of the pure spinor BRST cohomology.
We analyze the decomposition of recently constructed unfaithful spinor representations of K(E10) under its SO(9) x SO(9), and SO(9) x SO(2) subgroups, respectively, where K(E10) is the `maximal compact' subgroup of the hyperbolic Kac--Moody group E(10). We show that under these decompositions, respectively, one and the same K(E10) spinor gives rise to both the fermionic fields of IIA supergravity, and to the (chiral) fermionic fields of IIB supergravity. This result is thus the fermionic analogue of the decomposition of E(10) under its SO(9,9) and SL(9) x SL(2) subgroups, respectively, which y…
In these lecture notes presented at the 2015 Villa de Leyva Summer School, we give an introduction to superstring theory. We begin by studying the particle and superparticle in order to get a better understanding on the superstring side. Afterwards, we review the pure spinor formalism and end by computing the scattering amplitude for three gravitons at tree-level.
In this thesis are presented two aplications of the sigma model for the superstring in the pure spinor formulation. The first aplication concerns the computation of the one-loop conformal invariance for the type II superstring, resulting in equations of motion written in superspace for the background fields coupled to the superstring. The second application is related to the BRST invariance of the heterotic superstring at the quantum level, which allows to find stringy corrections to the ten-dimensional super Yang-Mills/supergravity constraints.
Notes taken by Oscar A. Bedoya of lectures of Nathan Berkovits in June 2009 at the Galileo Galilei Institute School "New Perspectives in String Theory". Topics include the pure spinor superparticle and superstring, tree and loop amplitudes, Ramond-Ramond backgrounds, and open problems.
We present a geometric formulation of type-IIA and -IIB superstring theories in which the Wess-Zumino term is second order in the supersymmetric currents. The currents are constructed using supergroup manifolds corresponding to superalgebras: the IIA superalgebra derived from M-algebra and the IIB superalgebra obtained by a T-duality transformation of the IIA superalgebra. We find that a slight modification of the IIB superalgebra is needed to describe D-string theories, in which the U(1) gauge field on the worldsheet is explicitly constructed in terms of D-string charges. A unification of the…
We study the compactification of the pure spinor superstring down to four dimensions. We find that the compactified string is described by a conformal invariant system for both the four dimensional and for the compact six dimensional variables. The four dimensional sector is found to be invariant under a non-critical N=2 superconformal transformations.
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 $\hat c$=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 the …
The discovery of pure spinor formalism makes the computation of superstring scattering amplitudes possible. In this paper, we will illustrate how computer algebra system Cadabra is used in computing the supersymmetric amplitude in pure spinor formalism and provide the source code that computes the tree-level massless 5-gluon amplitude.
The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states.
We examine the surface term for the BRST transformation of the open pure spinor superstring in an AdS_5 x S^5 background. We find that the boundary condition to eliminate the surface term leads to a classification of possible configurations of 1/2 supersymmetric D-branes.
In the framework of the pure spinor approach of superstring theories, we describe the Y-formalism and use it to compute the picture raised b-field. At the end we discuss briefly the new, non-minimal formalism of Berkovits and the related non-minimal b-field.
The SO(32) heterotic superstring on a Calabi-Yau manifold can spontaneously break supersymmetry at one-loop order even when it is unbroken at tree-level. It is known that calculating the supersymmetry-breaking effects in this model gives a relatively accessible test case of the subtleties of superstring perturbation theory in the RNS formalism. In the present paper, we calculate the relevant amplitudes in the pure spinor approach to superstring perturbation theory, and show that the regulator used in computing loop amplitudes in the pure spinor formalism leads to subtleties somewhat analogous …
We examine the possibility of constructing analytical solutions describing marginal deformations in the open superstring field theory that is based on the non-minimal pure-spinor formalism. It is found out that some methods used for constructing solutions of bosonic and RNS string field theories do not seem to generalize to the pure-spinor case, while other methods do lead to reliable analytical solutions.
It has recently been shown that the ten-dimensional superstring can be quantized using the BRST operator $Q=\ointλ^αd_α$ where $λ^α$ is a pure spinor satisfying $λγ^m λ=0$ and $d_α$ is the fermionic supersymmetric derivative. In this paper, the pure spinor version of superstring theory is formulated in a curved supergravity background and it is shown that nilpotency and holomorphicity of the pure spinor BRST operator imply the on-shell superspace constraints of the supergravity background. This is shown to lowest order in $α'$ for the heterotic and Type II superstrings, thus providing a compac…
We worked out the Batalin-Fradkin-Tyutin (BFT) conversion program of second class constraints to first class constraints in the GS superstring using light cone coordinates. By applying this systematic procedure we were able to obtain a gauge system that is equivalent to the recent model proposed by Berkovits and Marchioro to relate the GS superstring to the pure spinor formalism.
The pure spinor formalism for the superstring has recently been used to compute massless four-point two-loop amplitudes in a manifestly super-Poincare covariant manner. In this paper, we show that when all four external states are Neveu-Schwarz, the two-loop amplitude coincides with the RNS result.
We construct the action for the pure spinor superstring in the coset description of AdS_4 x CP^3 superspace, using the variables which solve the pure spinor condition. As a test of the consistency of the approach, we use the background field method to verify the absence of central charge at the second order in the expansion and to show the one-loop finiteness of the effective action.
We compute the one-loop beta function for the Type II superstring using the pure spinor formalism in a generic supergravity background. It is known that the classical pure spinor BRST symmetry puts the background fields on-shell. In this paper we show that the one-loop beta functions vanish as a consequence of the classical BRST symmetry of the action.
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^μ, θ^α, p_α) 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 cla…
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.
The OPE between the composite $b$ ghost and the unintegrated vertex operator for massless states of the pure spinor superstring is computed and shown to reproduce the structure of the bosonic string result. The vanishing of the double pole singularity is Lorentz gauge and the single pole is shown to be equal to the corresponding integrated vertex operator.
The double-spinor formalism, proposed by Aisaka and Kazama, provides a basis for the pure-spinor formalism, which allows manifestly super-Poincare covariant quantization of superstrings. We extend it to the case of backgrounds realized by coset superspaces. A general method constructing reparametrization invariant action is given using two concrete examples, the flat space-time and AdS(5)xS(5). We find that it is natural to double not only the spinor coordinates but the whole superspace for describing double-spinor superstrings on such backgrounds. The reparametrization invariant action has lo…
We perform a detailed study of the type IIA superstring in AdS(4) x CP(3). After introducing suitable bosonic light-cone and fermionic kappa worldsheet gauges we derive the pure boson and fermion SU(2|2) x U(1) covariant light-cone Hamiltonian up to quartic order in fields. As a first application of our derivation we calculate energy shifts for string configurations in a closed fermionic subsector and successfully match these with a set of light-cone Bethe equations. We then turn to investigate the mismatch between the degrees of freedom of scattering states and oscillatory string modes. Since…
Mason and Skinner recently constructed a chiral infinite tension limit of the Ramond-Neveu-Schwarz superstring which was shown to compute the Cachazo-He-Yuan formulae for tree-level d=10 Yang-Mills amplitudes and the NS-NS sector of tree-level d=10 supergravity amplitudes. In this letter, their chiral infinite tension limit is generalized to the pure spinor superstring which computes a d=10 superspace version of the Cachazo-He-Yuan formulae for tree-level d=10 super-Yang-Mills and supergravity amplitudes.
